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3129
|
dbpedia
|
3
| 87
|
https://www.mycause.com.au/page/85683/international-volunteer-lifeguard
|
en
|
International Volunteer Lifeguard
|
[
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] |
[] |
[] |
[
""
] | null |
[] | null |
This January I will be travelling to Chile to volunteer as a Lifeguard for the International Surf Lifesaving Association. On this trip I will be volunteeri
|
en
|
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|
mycause
|
https://www.mycause.com.au/page/85683/international-volunteer-lifeguard
| |||||
3129
|
dbpedia
|
1
| 51
|
https://lifeguards.com.au/nsw/als-lifeguards-take-training-international/
|
en
|
ALS Lifeguards Take Training International
|
[
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] |
[] |
[] |
[
""
] | null |
[
"cororke"
] |
2023-06-29T00:02:43+00:00
|
en
|
NSW
|
https://lifeguards.com.au/nsw/als-lifeguards-take-training-international/
|
How would you like training Thai Lifeguards on a tropical beach? How about teaching community CPR at a beachfront bar or singing Karaoke with a chorus of Vietnamese lifeguards?
Seven lucky ALS lifeguards from NSW participated in tours to Vietnam and Phuket Thailand earlier this year to do exactly that.
This project began in 2020 with a successful DFAT grant application from the ALS to provide training and lifesaving development to various lifeguard and lifesaving entities in ASEAN (Association of Southeast Asian Nations) countries. Called the ASEAN Friendship Grant; the application was designed to assist these groups to further develop their lifesaving endeavours to Australian standards.
ALS lifeguard David Field had worked with these group over the past 15 years and the groups willingly provided letters of support for the grant application. COVID-19 delayed the implementation but finally, with the deadline approaching, two tours went ahead in early 2023.
Vietnam: March 2023
Lifeguard group included Lachlan Field, Heli Murray, Mariah Jones, David Field and Susan Eke. Goals Achieved included: re-certifying 52 public lifeguards to Australian standards; qualifying newly employed public lifeguards to Australian standards; supporting public lifeguard managers in their roles; reconnecting and strengthening ties between public lifeguard bodies in Vietnam and Australia; conducting community CPR and children’s water safety activities in cooperation with local lifeguard groups.
The activities took place in collaboration with Danang City Lifeguards, Nha Trang Bay Management Lifeguards and SwimVietnam. A full speed tour with lots of travel and training in different venues. The language barrier was overcome by demonstration and excellent translations from the Surf Life Saving Vietnam representative, Thin Vu.
Although there are some women employed as trained water safety staff in hotels in Vietnam there has only been one woman employed as a lifeguard on a public beach.
For those who attended the highlight was the kids water safety on Danang Beach with around 85 kids and the lifeguard team running things.
Phuket: April 2023
This tour was run in collaboration with Phuket Lifeguard Services aka Phuket Surf Lifesaving Thailand, Thailand Surfing Federation and Australian Consul General to Phuket. The goals were similar to the Vietnam project. On tour were Scott McCartney, Chloe Jones and David Field.
Two teams of lifeguards were trained, a beginning group and an advanced group. During the three days training at Naiharn Beach there were numerous interruptions as Chloe and Scott carried out rescues on people swimming outside the flags. Scott went to the aid of two Russian swimmers, where the man said he was “OK” and his partner said “I am Not!”.
He brought the woman to shore and the man remained in the rip and was later seen escaping by climbing up the barnacle-covered rocks. The next day Chloe helped out the overstretched lifeguards by rescuing an eight-year-old Thai girl.
When Chloe brought her to the beach via a board she was scooped up by the enthusiastic lifeguard trainees in a text book demonstration of the two person pick up and carry – and returned to her mother.
The tour ended with a kid’s water safety activity on Patong Beach and meeting with senior officials from local and Thai national government and followed by dinner out with the Consul General. The goal here is to once again have the 11 Andaman Beaches of Phuket operating under a consistent approach to Beach Management based on Australian standards.
ALS’ Chris O’Rorke, who managed the project, said that the remainder of the funding will be allocated to providing online support for lifesaving entities throughout the region. Future grant applications will be sought and will be open to qualified ALS staff in order to continue international training and drowning prevention programs.
|
||||||
3129
|
dbpedia
|
1
| 5
|
https://www.ilsf.org/about/organogram/regional-branches/americas/
|
en
|
ILS Americas Region – International Life Saving Federation
|
[
"https://www.ilsf.org/wp-content/uploads/2019/01/tapestry_logo_0.gif"
] |
[] |
[] |
[
""
] | null |
[] | null |
en
|
https://www.ilsf.org/about/organogram/regional-branches/americas/
| ||||||||
3129
|
dbpedia
|
1
| 10
|
https://www.surflifesaving.org.nz/
|
en
|
Surf Lifesaving NZ
|
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] |
[] |
[] |
[
""
] | null |
[] | null |
/images/favicon.ico
|
https://www.surflifesaving.org.nz/
|
SLSNZ is a charity and the national association representing 74 Surf Life Saving Clubs across Aotearoa, 4,500+ volunteer Surf Lifeguards who patrol at over 80 locations through summer.
Become a Surf Lifeguard
We are a swimming nation. New Zealand boasts 15,000km of coastline and 90% of us live within 40 minutes of a beach but our glorious surf can be a deadly playground. Find out how you can join us.
Become a Surf Lifeguard
Help us save lives.
Help us reduce the number of beach drownings in New Zealand with a donation to Surf Life Saving New Zealand.
Donate today
|
|||||||
3129
|
dbpedia
|
2
| 12
|
https://www.ilsf.org/organisation/uk-slsgb/
|
en
|
Surf Life Saving Great Britain – International Life Saving Federation
|
[
"https://www.ilsf.org/wp-content/uploads/2019/01/tapestry_logo_0.gif"
] |
[] |
[] |
[
""
] | null |
[] | null |
en
|
https://www.ilsf.org/organisation/uk-slsgb/
|
The Surf Life Saving Association of Great Britain was founded in 1955 and was a founder member of WLS. Surf Life Saving is an exciting and healthy recreational activity that embodies voluntary public service.
It helps people to reach high standards of physical fitness, swimming ability and life saving skills. It is a water sport that involves the use of boards, skis and boats. It involves competition for members of all age groups as individuals and as members of teams.
The Surf Life Saving Association of Great Britain has over 4.000 members. They patrol Britain’s beaches voluntarily, in their own time and at their own expense. Members carry out training and qualify Proficiency awards so that they are able to:
Advise people how to avoid getting into danger in the water.
React sensibly to emergencies, effect necessary rescues and administer the required first aid.
Take part in Surf Life Saving Competitions and Championships for which relevant proficiency awards are generally a pre-requisite.
Mission Statement
To provide a safe and enjoyable environment on our beaches.
Objectives
|
|||||||
3129
|
dbpedia
|
2
| 86
|
https://olympics.com/en/paris-2024/sports/surfing
|
en
|
Paris 2024 Olympics
|
https://img.olympics.com/images/image/private/t_social_share_thumb/f_auto/primary/npvokfsuku44obof2abt
|
https://img.olympics.com/images/image/private/t_social_share_thumb/f_auto/primary/npvokfsuku44obof2abt
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[] |
[] |
[
""
] | null |
[] | null |
The world's top Surfing athletes will gather at the Paris 2024 Olympics (Jul 26-Aug 11, 2024). Get the latest Olympic Surfing updates here.
|
en
|
/_pr/topic-assets/favicon/paris2024/favicon.ico
|
https://olympics.com/en/paris-2024/sports/surfing
| ||||
3129
|
dbpedia
|
3
| 72
|
https://www.cityofrehoboth.com/news/rbp-again-host-action-packed-lifeguard-competition-year-towers-beach-location
|
en
|
RBP again to host action-packed lifeguard competition, this year at Towers Beach location
|
https://www.cityofrehoboth.com/themes/custom/cityofrehoboth/favicon.ico
|
https://www.cityofrehoboth.com/themes/custom/cityofrehoboth/favicon.ico
|
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] |
[] |
[] |
[
"rehoboth beach town ordinance",
"beach regulations",
"boardwalk rules",
"board meetings",
"committee minutes",
""
] | null |
[] |
2024-07-01T09:34:21-04:00
|
The Rehoboth Beach Patrol will once again host lifeguard-athletes from throughout the Mid-Atlantic region as they compete in sand and surf events this July for the right to compete in the United States Lifesaving Association’s national competition.
|
en
|
/themes/custom/cityofrehoboth/favicon.ico
|
City of Rehoboth
|
https://www.cityofrehoboth.com/news/rbp-again-host-action-packed-lifeguard-competition-year-towers-beach-location
|
The Rehoboth Beach Patrol will once again host lifeguard-athletes from throughout the Mid-Atlantic region as they compete in sand and surf events this July for the right to compete in the United States Lifesaving Association’s national competition.
The 2024 United States Lifesaving Association’s Mid-Atlantic Regional Championships will begin at 9:30 am Wednesday, July 10. The beach venue for this year’s competition will move from Rehoboth Beach to Towers Beach in Delaware Seashore State Park due to construction of Rehoboth’s beach patrol headquarters at Baltimore Avenue.
Image
This will be the 27th consecutive year that the Rehoboth Beach Patrol has been selected to host this event, which draws 250-350 lifeguard-competitors from New York to Virginia.
“We are excited to have the opportunity to host the 43rd annual lifesaving championships just as we have for more than 25 years,” says Rehoboth Beach Patrol Captain Jeff Giles. “We’re looking forward to an expanded competition with additional events. The day is filled with action-packed competition that can’t be seen in any other forum.”
This event showcases the skills and knowledge necessary to be an ocean lifeguard. The event schedule includes competitions traditionally held in Rehoboth such as a 90-meter sprint, 2K beach run, 4 x 90 soft-sand beach relay, 400-meter surf swim, swimmer rescue race, landline rescue race, paddleboard rescue race, run-swim-run, surf dash, and beach flags. In addition, surf skis and board events, which traditionally have been held in New Jersey, will be held in Delaware for the first time this year.
“The level of local and regional talent competing will put on a show not to be missed,” says fifth-year Rehoboth Beach Patrol guard Sophia Gulotti. “We performed well last year, and everyone is excited to see who has what it takes to win a championship and qualify for the nationals in South Padre Island, Texas.”
The United States Lifesaving Association is America’s nonprofit professional association of beach lifeguards and open water rescuers. USLA works to reduce the incidence of death and injury in the aquatic environment through public education, national lifesaving standards, training programs, promotion of high levels of lifeguard readiness, and other means.
Established in 1921, the Rehoboth Beach Patrol is considered one of the nation's leading lifesaving agencies with 103 years of excellence in safety and lifesaving competition as well as a 700-member alumni association prepared to guide the RBP to continued excellence.
|
||
3129
|
dbpedia
|
3
| 7
|
https://cslsa.org/Exchange-Programs.html
|
en
|
California Surf Lifesaving Association
|
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"California Surf Lifesaving Association"
] | null |
[] | null |
California Surf Lifesaving Association
|
/images/website/1291/graphics/favicon.png
| null |
Exchange Programs
Home
Exchange Programs
Professional lifeguards throughout the world have a rich tradition of exchanging lifesaving techniques, equipment, training methods and ideas. The CSLSA is proud to be a leader in this area by encouraging and promoting lifeguard exchanges within the CSLSA region and with New Zealand.
Inter-Agency CSLSA lifeguard exchange programs
Any professional lifeguard interested in visiting and observing the operation of a CSLSA Southern California or Hawaii Lifeguard Agency should be self-motivated and able to initiate, organize and schedule an exchange. He or she should also consider the following guidelines:
Possess a minimum of three summers of open water or surf lifeguard experience.
Exchange with and represent an organization with USLA Certification.
Obtain a letter of recommendation for an exchange from your lifeguard Chief, Captain or Lieutenant.
Develop a contact person such as the training officer from the lifeguard agency you wish to visit.
Identify an area of interest or focus such as front line or tower response, Rescue vessels or Junior Lifeguard program.
Use your scheduled days off or get approval for paid training during the exchange.
CSLSA and Auckland, New Zealand exchange program
This exchange program started in 1969 and has a rich tradition of exchanging professional and volunteer lifesaving techniques and equipment. Each year two visiting lifeguards from New Zealand visit Southern California beaches in July for six weeks. Similarly, two lifeguards from California visit the coastline of New Zealand during January for six weeks. The application is contained within the Annual Notice (below).
2023/2024 California/New Zealand Lifeguard Exchange:
TO: All current members of the California Surf Life Saving Association (CSLSA).th
WHAT: The CSLSA has again been invited by the Auckland, New Zealand SLSA representatives to participate in a lifeguard exchange program. This is the 55n year this exchange has taken place between Auckland/Northern Region and California. Delegates will tour various beaches observing and participating in lifesaving practices and procedures. This program is designed to enrich lifesaving knowledge and perpetuate international lifesaving camaraderie.
WHEN: Late December, 2024 through Mid-February, 2025 (approximate dates; actual dates decided by you and partner)
RESPONSIBILITY: The two CSLSA members will be responsible for their own food and transportation costs to and from Auckland, New Zealand. Surf Lifesaving Northern Region will provide accommodations throughout the exchange. The CSLSA will sponsor a portion of the airfare to New Zealand and provide a stipend upon delivering a report to the Board of Directors (upon satisfactorily completing the tour).
HOW TO APPLY:
1. Applicant must have a minimum of three summers of open water lifeguard experience and three years membership in the United States Lifesaving Association (USLA);
2. Applicant must presently be a member of the CSLSA in good standing;
3. Applicant must provide resume and cover page including name, mailing address, contact phone number, e-mail address, and agency.
4. Must have two (2) letters of recommendation. One from each of the following: A. Your Chief Lifeguard, Captain, or Lieutenant B. Your CSLSA Chapter President or other Executive Officer.
5. Must complete an oral interview with final approval given by the CSLSA Executive Board
6. Applicants from USLA certified agencies are highly desirable.
E-MAIL TO: E-mail your completed application with all documents to: exchanges@cslsa.org
DEADLINE: All materials MUST be e-mailed and time-stamped by Wednesday, September 12th, 2024 at 11:59pm. LATE APPLICATIONS WILL NOT BE ACCEPTED.
PLEASE CLICK TO DOWNLOAD AND READ THE OFFICIAL INFORMATION
2024 Exchange Delegates
Click here to Download the Powerpoint in PDF Form!
Vianne Kelly, Los Angeles City
Jackson Lawrence, City of Seal Beach
|
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3129
|
dbpedia
|
2
| 90
|
https://lifeguards.com.au/nsw/als-lifeguards-take-training-international/
|
en
|
ALS Lifeguards Take Training International
|
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[] |
[
""
] | null |
[
"cororke"
] |
2023-06-29T00:02:43+00:00
|
en
|
NSW
|
https://lifeguards.com.au/nsw/als-lifeguards-take-training-international/
|
How would you like training Thai Lifeguards on a tropical beach? How about teaching community CPR at a beachfront bar or singing Karaoke with a chorus of Vietnamese lifeguards?
Seven lucky ALS lifeguards from NSW participated in tours to Vietnam and Phuket Thailand earlier this year to do exactly that.
This project began in 2020 with a successful DFAT grant application from the ALS to provide training and lifesaving development to various lifeguard and lifesaving entities in ASEAN (Association of Southeast Asian Nations) countries. Called the ASEAN Friendship Grant; the application was designed to assist these groups to further develop their lifesaving endeavours to Australian standards.
ALS lifeguard David Field had worked with these group over the past 15 years and the groups willingly provided letters of support for the grant application. COVID-19 delayed the implementation but finally, with the deadline approaching, two tours went ahead in early 2023.
Vietnam: March 2023
Lifeguard group included Lachlan Field, Heli Murray, Mariah Jones, David Field and Susan Eke. Goals Achieved included: re-certifying 52 public lifeguards to Australian standards; qualifying newly employed public lifeguards to Australian standards; supporting public lifeguard managers in their roles; reconnecting and strengthening ties between public lifeguard bodies in Vietnam and Australia; conducting community CPR and children’s water safety activities in cooperation with local lifeguard groups.
The activities took place in collaboration with Danang City Lifeguards, Nha Trang Bay Management Lifeguards and SwimVietnam. A full speed tour with lots of travel and training in different venues. The language barrier was overcome by demonstration and excellent translations from the Surf Life Saving Vietnam representative, Thin Vu.
Although there are some women employed as trained water safety staff in hotels in Vietnam there has only been one woman employed as a lifeguard on a public beach.
For those who attended the highlight was the kids water safety on Danang Beach with around 85 kids and the lifeguard team running things.
Phuket: April 2023
This tour was run in collaboration with Phuket Lifeguard Services aka Phuket Surf Lifesaving Thailand, Thailand Surfing Federation and Australian Consul General to Phuket. The goals were similar to the Vietnam project. On tour were Scott McCartney, Chloe Jones and David Field.
Two teams of lifeguards were trained, a beginning group and an advanced group. During the three days training at Naiharn Beach there were numerous interruptions as Chloe and Scott carried out rescues on people swimming outside the flags. Scott went to the aid of two Russian swimmers, where the man said he was “OK” and his partner said “I am Not!”.
He brought the woman to shore and the man remained in the rip and was later seen escaping by climbing up the barnacle-covered rocks. The next day Chloe helped out the overstretched lifeguards by rescuing an eight-year-old Thai girl.
When Chloe brought her to the beach via a board she was scooped up by the enthusiastic lifeguard trainees in a text book demonstration of the two person pick up and carry – and returned to her mother.
The tour ended with a kid’s water safety activity on Patong Beach and meeting with senior officials from local and Thai national government and followed by dinner out with the Consul General. The goal here is to once again have the 11 Andaman Beaches of Phuket operating under a consistent approach to Beach Management based on Australian standards.
ALS’ Chris O’Rorke, who managed the project, said that the remainder of the funding will be allocated to providing online support for lifesaving entities throughout the region. Future grant applications will be sought and will be open to qualified ALS staff in order to continue international training and drowning prevention programs.
|
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3129
|
dbpedia
|
1
| 21
|
https://isasurf.org/learning/isa-ils-water-safety-course/
|
en
|
ILS Water Safety Course — International Surfing Association
|
https://isasurf.org/wp-content/uploads/2021/04/favicon_2020.ico
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https://isasurf.org/wp-content/uploads/2021/04/favicon_2020.ico
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2021-04-24T12:34:15+00:00
|
en
|
https://isasurf.org/wp-content/uploads/2021/04/favicon_2020.ico
|
International Surfing Association
|
https://isasurf.org/learning/isa-ils-water-safety-course/
|
Learn How to Organize an ISA/ILS International Surf and SUP Instructor Aquatic Rescue and Safety Course here.
View List of Approved Presenters here.
The International Surfing Association (ISA) and the International Lifesaving Federation (ILS) have entered into a key strategic partnership together to establish a universally recognized, global standard water safety course and certification for surfing and stand-up paddle (SUP) coaches and instructors. In its first year (2017) this course was wildly successful being run 27 times in 17 countries for 290 surf and SUP instructors. Since the partnership began in 2017 there have been 341 ISA/ILS Water Safety Courses run for 4,282 instructors in over 50 countries.
The ISA/ILS International Surf and SUP Instructor Aquatic Rescue and Safety Course is specific for the surf instructing environment and available for all ISA and ILS Members to access. The course can be presented by ILS Member Organizations and those with formal endorsed trainer/assessor qualification.
The aims of this course are as follows:
Establish a global minimum standard for Surf/SUP instructors water safety certification as the current water safety requirements vary from country to country. This will translate into a transportable safety accreditation that will benefit traveling coaches.
Complement the ISA Coaching and Instructing Program Courses that currently require all ISA accredited instructors to hold a valid, recognized water safety accreditation
Promote water safety world-wide.
Enhance the value of the ISA Coaching Certification and provide more opportunities to our program participants.
ISA Course Presenter Tim Jones.
Course Details:
The course is run over two days. It contains theory, practical components, and group activities. Assessment is competency based and successful completion is based upon the following:
Full attendance of the training course and all practical assessments
Basic Surf Fitness Test
Competency in each of the Practical Assessments
Completion of course participant workbook
Basic Surf Fitness Test: Run-swim-run to demonstrate surf skill competency, based on small to moderate surf conditions and to be completed within a 5-minute timeframe: Minimum of 100m run, negotiation of the inner surf zone with a swim of at least 100m through the surf and impact zone, 100m run. Course presenters may extend the Basic Surf Fitness Test depending on conditions and adjust the time-frame accordingly.
Practical Assessments: Unconscious board rescue, basic first aid techniques in various scenarios, and provision of basic emergency care, including CPR.
Please find an overview of the ISA/ILS International Surf and SUP Instructor Aquatic Rescue and Safety Course HERE
All participants will receive an ISA/ILS Water Safety Accreditation that is valid for two years. This accreditation fulfills the minimum water safety requirement for the ISA Coaching and Instructing Program and includes international ILS-endorsement and recognition
All course material will be delivered electronically to the Course Organizer or to the Official Course Presenters.
Renewal: The ISA/ILS Water Safety Accreditation must be refreshed every two years
Please Note: ***Requirements of individual member nations internal requirements in relation to CPR and fitness recertification must also be considered.
Who Can Present:
All presenters of the ISA/ILS International Surf and SUP Instructor Aquatic Rescue and Safety Course must be approved by the ISA and the presentation of this course is available to the following:
Individuals with formal ILS endorsed trainer/assessor lifesaving qualifications or those that demonstrate underpinning knowledge and ability to present the course. These individuals must submit a CV + qualifications to the ISA/ILS for review
ILS Member Organizations and Lifesaving Trainers with formal lifesaving qualifications and underpinning knowledge of the course competencies. To find the ILS Member in your nation view the ILS Membership Directory
All interested course presenters can apply to the ISA Office by sending a CV, lifesaving qualifications, and lifesaving experience to [email protected]
Please note:
***Course Presenters must have completed the course or demonstrate clear underpinning knowledge of the competencies
***The course will be offered in all countries of the world except those with existing surf/SUP instructor specific water safety accreditation courses (Great Britain, Ireland, Brazil, and South Africa) In these countries the following organizations must be consulted prior to the running of this course: Surf Lifesaving Great Britain (Great Britain), Irish Water Safety (Ireland), the Sociedade Brasileira Salvamento Aquatico (Brazil), and Lifesaving South Africa (South Africa).
***Preference of presenting this course given to existing ISA Course Presenters with formal recognized lifesaving qualifications.
Course Development History:
The ISA requires all ISA Certified Coaches and Instructors to maintain a valid lifesaving or water safety accreditation. In upholding this requirement the ISA noticed a lack of consistency in water safety accreditations for Surf and SUP coaches and that many had to use courses that were more time consuming than necessary, difficult to access, and providing non-applicable content. Additionally there was a lack of available courses in some countries where ISA Coaching courses were taking place. The ISA worked with ISA Course Presenter and ILS accredited trainer Jamo Borthwick, to develop the base of the content and the ILS, with their extensive water safety experience, have reviewed, contributed, and joined ISA in making this course available.
The ISA was pleased to reach an agreement with ILS at the end of 2016 and roll-out the ISA/ILS International Surf and SUP Instructor Aquatic Rescue and Safety Course in 2017. This water safety accreditation course is fully endorsed by the International Lifesaving Federation (ILS) and all participants will receive an internationally endorsed water safety accreditation. In 2017 this course was run 27 times in 17 countries for 290 Surf and SUP instructors.
Summary:
The new ISA/ILS International Surf and SUP Instructor Aquatic Rescue and Safety Course will be complementary to your ISA Coaching and Instructing Program Courses and allow your instructors the opportunity to obtain an internationally endorsed water safety accreditation in conjunction with their coaching and instructing accreditation.
As Course Organizers, ISA members have the following options for water safety course delivery:
|
|||
3129
|
dbpedia
|
3
| 38
|
https://www.israellifesaving.org/about
|
en
|
Israel Life Saving Federation
|
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[] |
[] |
[
""
] | null |
[] | null |
en
|
Israel Life Saving
|
https://www.israellifesaving.org/about
|
Israel Life Saving Federation (ILSF) is a nonprofit, community-based volunteer organization based on the world's leading water-safety model - Surf Life Saving Australia (SLSA). Our goal is to prevent drownings across Israel through education and practical courses catered to all age groups from as young as 7 years old.
The surf-rescue model seeks to teach volunteers and passersby to respond and offer first aid in the case of ocean drowning incidents. Courses include water awareness, ocean lifesaving and first aid, combined with fitness and competitions. Participants learn and practice skills in the sand and water with equipment including specialized floats and surfboards, and compete in running, swimming and surfing.
While new in Israel, surf lifesaving is a broad, well-established social movement that has saved hundreds of thousands of lives around the world. Established in Australia in 1907, the movement has 180,000 volunteers in Australia alone, and is also popular in English-speaking countries including New Zealand, Ireland, South Africa and the United Kingdom. Members serve as volunteer lifeguards on their local beaches and participate in competitive surf sports.
ISLF, Israel's local surf lifesaving federation, was founded in 2019 by Paul Hakim, a Silver Medallion surf lifesaver and beach patrol manager from Sydney, his brother Danny Hakim, and Lisa Segelov, all originally from Australia. In 2021, volunteers saved or assisted more than 70 people in the water, both while volunteering at official swimming events and as passersby on Israel’s beaches.
The organization's operations during 2021 included leading beachside courses for children, teens and adults, and volunteering on the water safety team for major sporting events including Israel's first Ironman competition in Tiberias and the Olympic marathon swimming qualifier in Eilat. ISLF is also active in lobbying the Knesset for effective water safety education.
In July 2022, ISLF hosted the first-ever surf-lifesaving event at the Maccabiah games. Other activities in 2022 have included courses for all ages on beaches around the country, including in Tel Aviv, Herzliya, Ashkelon, Dor and Jisr al-Zarqa.
ISLF President Paul Hakim notes that many beach drownings occur in broad daylight, after the lifeguards have left for the day. The beaches are often still full of people, and if only one person were to know how to respond, lives could be saved.
“Coming from Australia, surf lifesaving is a normal thing, and coming to Israel it’s so lacking. This is a huge vacuum in Israel that we have the potential to fill,” explains Hakim, referring to the large number of drownings at Israel’s beaches every summer. “Israel is known as the start-up nation when it comes to technology, and Australia is known as the best nation for water safety and prevention. We’re bringing Australian expertise to Israel.”
Israel Life Saving Federation (ILSF) is a member of the European Life Saving Federation and is an official associate member of the International Life Saving Federation. It is a registered nonprofit in Israel.
|
||||||
3129
|
dbpedia
|
3
| 80
|
https://www.guidestar.org/profile/33-0511304
|
en
|
California Surf Lifesaving Association
|
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[
""
] | null |
[] | null |
TO PROMOTE BEACH SAFETY AWARENESS AND PROFESSIONAL LIFEGUARD STANDARDS THROUGH PUBLIC EDUCATION, TRAINING PROGRAMS, EXCHANGE PROGRAMS, JUNIOR LIFEGUARD ...
|
en
|
https://cdn.candid.org/favicon.ico
|
https://www.guidestar.org/profile/33-0511304
|
Build relationships with key people who manage and lead nonprofit organizations with GuideStar Pro. Try a low commitment monthly plan today.
Analyze a variety of pre-calculated financial metrics
Access beautifully interactive analysis and comparison tools
Compare nonprofit financials to similar organizations
Want to see how you can enhance your nonprofit research and unlock more insights? Learn More about GuideStar Pro.
Build relationships with key people who manage and lead nonprofit organizations with GuideStar Pro. Try a low commitment monthly plan today.
Analyze a variety of pre-calculated financial metrics
Access beautifully interactive analysis and comparison tools
Compare nonprofit financials to similar organizations
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Report successfully added to your cart!
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Financial data for the most recent year (2023)*
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|
|||||
3129
|
dbpedia
|
1
| 60
|
https://www.redcross.org/
|
en
|
American Red Cross
|
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[
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"disaster relief",
"CPR certification",
"donate blood"
] | null |
[] | null |
Every 8 minutes the American Red Cross responds to an emergency. Support the Red Cross. Join us today by making a donation.
|
en
|
/etc/designs/redcross/shared/favicon.ico
|
https://www.redcross.org
|
This all-inclusive, simple-to-use, all-hazard app, guides you to prepare for climate-affected hazards and lets you customize 40 different severe weather alerts to help keep you and your loved ones safe.
Available in Spanish
Compatible with Apple Watch and Android wearable devices
|
|||||
3129
|
dbpedia
|
2
| 62
|
https://www.shorebeach.com/index.php/2-uncategorised/29-what-is-shore-beach-service
|
en
|
What is Shore Beach Service?
|
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[
"Hilton Head Island",
"SC",
"South Carolina",
"beaches",
"beach",
"lifeguard stand",
"lifeguards",
"beach chairs rentals",
"beach safety"
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[
"Sasha Sweeney"
] | null |
en
| null |
If you have visited Hilton Head Island’s beaches, you are probably familiar with Shore Beach Service. We are on the beach in the lifeguard highchairs, under the blue & gray umbrellas, or driving the red beach patrol vehicles. If you are new to the island, let us help you have a great vacation!
Hilton Head Island is blessed with some of the safest waters on the east coast, as gentle sloping beaches and offshore sandbars generally create calm surf conditions. However, always be cautious in the ocean and please follow our beach safety tips and beach rules to ensure a safe and enjoyable day.
Serving Hilton Head since 1974, Shore Beach Service is the island’s premier beach equipment rental service. We provide the rental of umbrellas and chairs anywhere throughout the beach.
We are the only company on Hilton Head Island that can set up the equipment for you anywhere on the beach daily, saving you the hassle so you can have a more relaxing and enjoyable vacation.
See our Rentals page for rates and other info.
WE ARE OPEN FOR 2024!
Shore Beach Service provides the beach patrol for Hilton Head Island’s 13.5 miles of beach. The beach patrol includes about 80 lifeguards from across the country and around the world.
As a member of the South Atlantic region of the United States Lifesaving Association (USLA) since 1989 and a Certified Agency of the USLA for Open Water Lifeguarding since 1995, our lifeguards are held to the USLA’s high standards of training. We are always looking for physically fit individuals with outstanding customer service skills to join our team.
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https://fire.lacounty.gov/junior-lifeguard-program/
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en
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Junior Lifeguard
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#Page Content
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/wp-content/uploads/2019/08/cropped-LACoFD-Logo-Color-32x32.png
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County of Los Angeles Fire Department
|
https://fire.lacounty.gov/junior-lifeguard-program/
|
Deputy Fire Chief Robert Harris started his career at the age of 19 after he was appointed reserve firefighter with the City of Montebello Fire Department.
Chief Harris began his service with the County of Los Angeles Fire Department on May 8, 1992. During his tenure with the Department, he has promoted through the ranks from Firefighter, Fire Fighter Paramedic, Fire Inspector, Fire Fighter Specialist, Fire Captain, Battalion Chief, Assistant Fire Chief, Acting Deputy Fire Chief, and Deputy Fire Chief, Central Regional Operations Bureau, effective June 1, 2024.
Over the years, Chief Harris has attended Dillard University (New Orleans), the University of Southern California, Long Beach Community College, and Compton Community College. Chief Harris is a graduate of Columbia Southern University with a Bachelor of Science degree in Fire Administration.
Throughout his 33 years of service with the County of Los Angeles Fire Department, Chief Harris has also served as a member of the Department’s international Urban Search and Rescue Team. He has traveled and provided rescue efforts around the globe. For more than four years, he served as the program manager of our elite rescue team and was the Department’s primary point-of-contact with our state, federal, and international partners.
In his spare time, he enjoys spending time with family and riding motorcycles. Chief Harris has a true passion for mentoring others who are seeking a career in the fire service. He believes in the motto, “each one, reach one”.
Fire Chief Anthony C. Marrone has been a member of the County of Los Angeles Fire Department for 38 years and a chief officer for the past 26 years. Prior to his appointment by the County of Los Angeles Board of Supervisors as the tenth Fire Chief and Forester and Fire Warden, Chief Marrone served as the Interim Fire Chief.
Chief Marrone leads one of the largest metropolitan emergency services agencies in the United States, providing traditional fire and life safety services to more than 4.1 million residents and commercial business customers in 60 cities served by the Department, in addition to 120 unincorporated areas of Los Angeles County within its 2,311-square-mile service delivery area. The Department operates out of 176 fire stations, with 5,000 emergency responders and business professionals operating with an annual budget of just over $1.6 billion. In addition, the Department provides lifeguard, air and wildland, hazardous materials, homeland security, health hazardous materials, forestry, and urban search and rescue services throughout the County. The Department’s urban search and rescue team, known internationally as USA-2, is one of only two highly specialized teams available for international response through a cooperative agreement with the Bureau for Humanitarian Assistance.
Chief Marrone’s well-established career combines broad experience in both emergency and business operations with an extensive list of accomplishments and assignments, including leading and managing Business Operations, the Leadership and Professional Standards Bureau, Special Services Bureau, Emergency Medical Services Bureau, East Regional Operations Bureau, and Central Regional Operations Bureau, in addition to special projects. He has also directly managed routine and complex wildland fires and other significant all-risk incidents.
During his career with the Department, Chief Marrone has served on the Los Angeles County Emergency Preparedness Commission, the FIRESCOPE Board of Directors, Legal Exposure Reduction Committee, County Emergency Operations Center Team Lead, and as an Incident Commander on one of the Department’s three Incident Management Teams.
Chief Marrone looks forward to collaborating with the Board of Supervisors, labor unions, stakeholder organizations, members of the Department, and the residents and communities we serve, to further stabilize the Department’s budget, increase our diversity, equity, and inclusion efforts, and expand emergency services and community risk reduction programs.
Deputy Fire Chief William L. Mayfield Jr. is a 35-year veteran of the County of Los Angeles Fire Department (LACoFD). He began his fire service career in 1988. He has held the ranks of firefighter, firefighter specialist, fire captain, battalion chief, assistant fire chief, and deputy fire chief.
Chief Mayfield currently manages the Training and Emergency Medical Services Bureau. Emergency medical services is responsible for paramedic training, certification, equipment, quality improvement, and legal aspects for all basic and advanced emergency medical services provided by the LACoFD. Training Services is responsible for training all new firefighters and conducting ongoing in-service training sessions for all members. Training Services develops training materials, organizes classes and training programs for recruits and refresher courses for other Department personnel.
Chief Mayfield’s previous assignment was to command and lead the Central Regional Operations Bureau. In that role, he provided leadership for four divisions, seven battalions, 55 fire stations within 22 cities, and over 1,600 firefighting and lifeguard personnel. The Lifeguard Division provides water rescue and medical services to 11 cities and consists of specialized and trained professionals who protect Catalina Island and 72 miles of sandy beaches and open water.
As an assistant fire chief, Chief Mayfield’s assignment was managing Division 4 of the East Regional Operations Bureau, consisting of three battalions, 12 cities and 25 fire stations. Chief Mayfield has been a chief officer for over
18 years. He has worked multiple operational, administrative, and special assignments throughout his career, including Fire Prevention, Command and Control, and several field commands.
As an incident commander, Chief Mayfield also assists in managing the LACoFD’s Incident Management Team 1.
Deputy Fire Chief Mike Inman has worked as a fire service professional for over 40 years. Chief Inman started his career in 1983, as a reserve firefighter with the Monterey Park Fire Department. In 1986, he was hired as a firefighter with the County of Los Angeles Fire Department (LACoFD) and has promoted through the ranks as a firefighter paramedic, firefighter specialist, fire captain, battalion chief, and assistant fire chief.
In November 2023, Chief Inman was assigned to the East Regional Operations Bureau and was officially promoted to deputy fire chief on March 1, 2024. In this role, he led and managed four divisions with 1,400 personnel, 10 battalions, and 76 fire stations serving 34 cities.
Currently, Chief Inman is assigned to the Special Services Bureau where he leads a team of nearly 300 professional staff with 113 dedicated dispatch personnel dispatching more than 449,000 calls for services each year, 94 professional trades personnel that provide facility maintenance and oversee construction of new and replacement structures for over 260 facilities, and over 55 mechanics responsible for repairing, outfitting, and maintaining a fleet of more than 1,900 Department vehicles and emergency apparatus. Chief Inman also oversees the LACoFD’s Equipment Development Committee.
Well versed in emergency management, Chief Inman became a pioneer in the urban search and rescue program that included several national deployments. For over 23 years, he has held various incident command system positions as part of the United States Forest Service Type 2 Incident Management Team (IMT) and served as the operations section chief and operations branch director with the Cal Fire Type 1 IMT. Since 1998, Chief Inman has served in various capacities on the LACoFD Incident Management Teams.
Chief Inman is a certified California State Fire Marshal Chief Officer, and a graduate of the Executive Leadership Development Program and Public Safety Leadership Program at the University of Southern California, Sol Price School of Public Policy. He teaches FEMA and California Incident Command Certification System (CICCS) courses and is a qualified incident commander, operations section chief, safety officer, and division supervisor.
.
Deputy Fire Chief Dennis Breshears started his fire service career at the age of 23 after he was appointed to the Monrovia Fire Department on February 24, 1994. Chief Breshears then accepted a position with the Orange County Fire Authority in 1999 and began his service with the County of Los Angeles Fire Department (LACoFD) in December 2000.
Chief Breshears steadily promoted through the ranks from firefighter, firefighter paramedic, firefighter specialist, fire captain, battalion chief, assistant fire chief, and deputy fire chief on December 16, 2021.
Over the years, Chief Breshears attended Lutheran High School (La Verne, California), Glendale College, Cal Poly San Luis Obispo, Kaplan University, and California State University, Long Beach. He holds a Bachelor’s degree in Fire and Emergency Management and a Master’s degree in Emergency Services Administration.
For more than three years, Chief Breshears served as the Chief of the Professional Performance Section. As a captain, he also completed a special assignment as the LACoFD’s incident command system coordinator. Chief Breshears is a state-certified instructor and Type II Operations Section Chief and Deputy Incident Commander for the LACoFD Incident Management
Team 1.
In June 2021, Chief Breshears was selected to serve on the FIRESCOPE Task Force. He is currently assigned as the deputy fire chief over the North Regional Operations Bureau.
In his spare time, he enjoys spending time with his family and restoring classic cars. While he is very proud of his profession, he considers being a father to three amazing daughters as his greatest accomplishment in life.
As the Chief Deputy of Business Operations, Theresa Barrera oversees the Fire Department’s $1.6 billion budget and provides executive oversight of the Administrative Services, Special Services, and Prevention Services Bureaus, the Planning and Grants and Executive Support Divisions, and the Compliance Office.
Chief Barrera joined the Fire Department in 2004 and served as the Assistant Chief and Chief of the Financial Management Division. In 2022, Chief Barrera was appointed as Deputy Fire Chief of the Administrative Bureau and worked closely with internal and external stakeholders to improve standard business practices, ensure administrative and fiscal compliance, and foster a workforce that is representative of the communities we serve. In 2023, Chief Barrera was appointed as the Chief Deputy of Business Operations.
Prior to joining the Fire Department, Chief Barrera held various fiscal positions at the Los Angeles County Department of Public Social Services and Auditor-Controller. Chief Barrera received a Bachelor of Science degree in Accounting from the University of Southern California.
Chief Deputy Jon F. O’Brien has worked as a fire service professional for over 33 years. Chief O’Brien started his career as a volunteer firefighter with the City of Sierra Madre. After graduating from high school, he completed paramedic training at the Los Angeles County Paramedic Training Institute and was hired by the City of Monrovia as a full-time firefighter/paramedic until he joined the County of Los Angeles in 1999.
Chief O’Brien has served in several operational and administrative assignments, promoting through the ranks to his current position as Chief Deputy of Emergency Operations. Along the way, he has worked as a flight medic in the Department’s Air Operations Section, a fire crew supervisor in the Camps Section, a recruit training captain, and a field battalion chief.
In April 2014, Chief O’Brien was assigned to the Emergency Medical Services (EMS) Bureau and was responsible for the education and training of the Department’s 3,200 emergency medical technicians and 1,250 paramedics. The following year in November 2015, Chief O’Brien was promoted to Assistant Fire Chief and assigned to Division VI in the Central Regional Operations Bureau. In July 2017, he returned to the EMS Bureau as Acting Deputy Fire Chief and was officially promoted to Deputy Fire Chief in December 2017.
In April 2020, Chief O’Brien was assigned as Deputy Fire Chief of the North Regional Operations Bureau where he oversaw the cities of Palmdale, Lancaster, Santa Clarita, and La Cañada Flintridge, as well as the Air & Wildland Division and the Technical Operations Section. He was also the incident commander of the Department’s Incident Management Team 1.
In October 2022, Chief O’Brien assumed the role of Acting Chief Deputy of Emergency Operations. As Chief Deputy, Chief O’Brien serves as second in command and is responsible for overseeing the Fire Department’s three Operations bureaus (North, Central, and East), as well as the Air and Wildland Division and the Lifeguard Division. Chief O’Brien is also responsible for the Training and Emergency Medical Services Bureau and the Homeland Security Section.
On August 23, 2023, was officially assigned as Chief Deputy of Emergency Operations.
Chief O’Brien received his Associate of Arts degree in fire science at Mount San Antonio Community College and his Bachelor of Science degree in public policy and management at the University of Southern California. He currently represents the Department on the FIRESCOPE Operations Team and the Los Angeles County Measure B Advisory Committee.
Born to immigrant parents who moved to the United States from Greece, Deputy Fire Chief Eleni Pappas was raised in Jersey City, New Jersey. After graduating from Saint Dominic’s Academy High School, she was accepted into the University of Southern California where she competed as a varsity rower, helping her team win the prestigious San Diego Crew Cup. She graduated with a bachelor’s degree during the civil unrest in May 1992.
From her college apartment, she watched Los Angeles burn and was impressed with the fire engines and tiller trucks racing across the city to extinguish the fires. She decided then and there to become a firefighter.
By 1996, Chief Pappas earned her paramedic license from Daniel Freeman Paramedic School in Inglewood. She then worked as an EMT for Goodhew Ambulance where she ran 9-1-1 calls with the Los Angeles County Fire Department (LACoFD).
A few years later, she was hired by Ventura County Fire Department and served as a firefighter for one year. She then became a firefighter with the City of Los Angeles and served there for four years. In 2001, after the attack on the World Trade Center, she was accepted into the Department’s Recruit Academy and graduated from the 109th Recruit Class later that year.
Since joining the LACoFD, she has promoted through every rank and is currently the highest-ranking woman in the Department’s history and the first-ever woman to obtain the ranks of Assistant Fire Chief and now Deputy Fire Chief.
She has worked in all three regional operations bureaus and in all 22 operations battalions. Since her promotion as a chief officer in September 2012, Chief Pappas has spent three years as the co-chairperson of the Equipment Development Committee and also managed the Department-wide implementation of the electronic patient care reporting (ePCR) system. As an Assistant Fire Chief, she was assigned to Division VI in the Central Regional Operations Bureau and managed the Fire Explorers youth mentoring program. Currently, she is assigned to the Special Services Bureau where she manages the Command and Control, Construction and Maintenance, and Fleet Services Divisions.
In August 2021, Chief Pappas successfully completed her master’s degree in Emergency Management from Cal State Long Beach.
At home, Chief Pappas enjoys gardening and spending time with her family, their dogs, parakeets, and bearded dragon. Raising her daughter is her greatest accomplishment. She cherishes spending quality time with her beautiful 12-year-old daughter who is the center of her life.
Deputy Fire Chief Vince A. Peña has been with the Los Angeles County Fire Department since 1981. Chief Peña has held the positions of firefighter, firefighter paramedic, firefighter specialist, fire camp foreman, fire captain, battalion chief, assistant fire chief, deputy fire chief, and acting chief deputy.
As a chief officer, battalion chief assignments have included Battalion 5 in Malibu, Battalion 16 in Covina, Battalion 20 in Inglewood, and Battalion 2 in San Dimas. He also served as the camp section battalion chief for the paid camps and heavy equipment unit. As an assistant fire chief, he was assigned to Division 2 in the east San Gabriel Valley and the Air & Wildland Division.
In Chief Peña’s assignment as the deputy fire chief of the North Regional Operations Bureau, he oversaw the cities of Palmdale, Lancaster, Santa Clarita, and La Cañada Flintridge, as well as the Technical Operations Section and the Air & Wildland Division. He has also served as operations section chief for the Department on many large wildland incidents and was the incident commander of the Department’s Incident Management Team 1.
Since October 2022, and following his assignment as acting chief deputy, Chief Peña has served as the deputy fire chief over the East Regional Operations Bureau.
Chief Peña attended East Los Angeles College, the University of La Verne, and the Executive Leadership Development Program for the County of Los Angeles. He also instructs incident command courses for the Fire Department and throughout the country.
Deputy Fire Chief Thomas C. Ewald has served in the professional fire services for 35 years. Chief Ewald started his career as a firefighter with the City of Cedar Rapids, Iowa Fire Department in 1986. While working in the Midwest, he attended community college and completed paramedic training. Chief Ewald joined the Los Angeles County Fire Department in 1992 where he rose through the ranks serving as firefighter, paramedic, captain, battalion chief, and assistant chief.
Chief Ewald has served as a firefighter paramedic at Universal Studios and West Hollywood; a fire inspector in East Los Angeles; an apparatus engineer in Carson and Pomona; an engine company captain in Southgate and South Los Angeles; a staff captain for the Central Regional Operations Bureau Deputy; a field battalion chief in El Monte, Commerce, Palos Verdes; the chief of Technical Operations, overseeing local, national and international Urban Search and Rescue Operations; assistant chief in Division I, covering the South Bay and Catalina Island, and as the assistant chief, overseeing the Air and Wildland Division.
During his career, Chief Ewald has been called upon to respond to manmade and natural disasters across the county and worldwide with notable incidents, including Hurricane Katrina (New Orleans), Hurricane Dean (Belize), Cyclone Nargis (Camp H.S. Smith Hawaii), 2011 Japan Earthquake and Tsunami (Ofunato) , 2011 New Zealand Earthquake (Christchurch) and 2015 Typhoon Maysak (Micronesia).
In December 2017, Chief Ewald was promoted to the rank of Deputy Fire Chief where he oversaw the Department’s Special Services Bureau consisting of three divisions: Fleet Services, Command and Control, and Construction & Maintenance.
On April 1, 2021, Chief Ewald’s tour of duty ended at Special Services and he assumed command of the Central Regional Operations Bureau. By October 2022, Chief Ewald was then assigned to oversee the North Regional Operations Bureau.
Chief Ewald holds a Bachelor of Science degree in Fire Prevention Administration from Cogswell Polytechnical College and a Master of Science degree in Leadership from the University of Southern California’s Sol Price School of Public Policy. In 2018, Chief Ewald attended the Senior Executives in State and Local Government Program at Harvard University’s Kennedy School of Government. Chief Ewald is a qualified Type 2 Incident Commander and Type 2 Operation Section Chief and provides leadership to the Department’s Incident Management Team Two.
Chief Ewald resides in Southern California with his wife and four children.
Anderson Mackey is an Acting Deputy Fire Chief for the Los Angeles County Fire Department, currently overseeing Training and the Emergency Medical Services Bureau.
Chief Mackey was born and raised in the City of Los Angeles and is a 33-year veteran of the Department. After graduating from the fire academy, he was assigned to Fire Station 103 in Pico Rivera. He was later assigned to Fire Station 105 in Compton as his second probationary station. In November 1989, Fire Fighter Mackey transferred to Fire Station 8 in West Hollywood. In January 1991, he volunteered to attend the Paramedic Training Institute. After successful completion of the six-month program, he was re-assigned to Fire Station 8 as a Fire Fighter Paramedic. In February 1992, he transferred to Fire Station 7 where he remained for the next six years. In October 1998, Mackey transferred to Fire Station 161 in Hawthorne and, 11 months later, was promoted to the rank of Fire Fighter Specialist. He was then re-assigned to Fire Station 58 in Ladera Heights. In February 2000, Mackey promoted to the rank of Fire Captain and was assigned to Fire Station 83 in Rancho Palos Verdes. By November 2000, he transferred to Fire Station 173 in Inglewood. In November 2006, Chief Mackey volunteered to head the Recruitment Unit where he managed over 50 recruiters who volunteered to give career presentations at high schools, colleges/universities, career fairs, and community events. In November 2010, Chief Mackey was promoted to the rank of Battalion Chief and was assigned to Battalion 10 in El Monte, and then Battalion 8 in Whittier. In 2011, he was transferred to Battalion 20 in Inglewood. Two years later, Chief Mackey was re-assigned to the Employee Services Section where he worked directly for the Fire Chief. In February 2018, he promoted to the rank of Assistant Fire Chief.
Chief Mackey received his diploma of completion at Dillard University, New Orleans for the Executive Development Institute. He resides in Pasadena with his beautiful wife Carmen and two lovely daughters, Denver and Blu. In his spare time, he enjoys golf, swimming, skiing, and spending time with his family and friends.
Commonly known as the Los Angeles County Fire Department, the Consolidated Fire Protection District of Los Angeles County (CFPD) is a dependent special district. As a dependent special district, the Los Angeles County Board of Supervisors acts as the CFPD’s board of directors. Fire protection districts are governed by the Fire Protection District Law of 1987 (Health & Safety Code, Section 13800 et al). The CFPD has the additional responsibilities for the Forester & Fire Warden (F&FW). In 1992, the duties of the F&FW were assigned to the CFPD and those responsibilities are found in the Los Angeles County Code 2.20.
The CFPD has a civilian oversight committee that annually reviews expenditures of the CFPD’s special tax to ensure it is expended in the manner approved by voters in 1997. Authority for the oversight committee is found in the establishing resolution for the special tax. The committee has seven members, one each appointed by each member of the Board of Supervisors, one appointed by the City Selection committee, and the director of the Los Angeles County Economy and Efficiency Committee.
Chief Deputy Dawnna B. Lawrence is the first female Chief Deputy of the Los Angeles County Fire Department.
In her role as Chief Deputy of Business Operations, Chief Lawrence oversees the Fire Department’s $1.3 billion budget and more than 800 employees in the Administrative, Prevention, and Special Services Bureaus.
Chief Lawrence initially came to the Fire Department in October 2012 as the Deputy Chief of the Administrative Services Bureau, where she served as the financial advisor to Fire Chief Daryl L. Osby in the midst of fiscal challenges, stemming from the 2008 Recession. In June 2015, Chief Lawrence was appointed to Chief Deputy of Business Operations and continues to work closely with internal and external stakeholders to ensure the financial future of the Fire Department is stable and sustainable.
Chief Lawrence is dedicated to creating an inclusive environment for all Fire Department team members through comprehensive action and sustainable policies and practices, in addition to fostering a workforce that is truly representative of the communities we serve.
Prior to joining the Fire Department, Chief Lawrence devoted 20 years climbing the ranks in administrative services at the Los Angeles County Department of Public Works (DPW), culminating in her appointment to Chief Financial Officer (CFO) in 2006. As CFO, Chief Lawrence was responsible for managing the DPW’s $2 billion operating budget. Her efforts and achievement earned Chief Lawrence a CFO of the Year nomination in the September 2012 issue of the Los Angeles Business Journal.
Earlier in her career, Chief Lawrence spent a collective five years at the Department of Health Services and the Department of the Auditor-Controller. Chief Lawrence received her Bachelor of Arts degree in Business Administration from California State University, Fullerton. She is also a member of the Government Finance Officers Association.
Chief Lawrence and her two sons are long-time residents of South Pasadena.
John R. Todd is a Registered Professional Forester in the State of California and he was employed as a forester by the Los Angeles County Fire Department from 1988 to 2012. In April 2012, John was promoted to the rank of deputy fire chief of the Prevention Services Bureau (PSB). The PSB is comprised of the Fire Prevention Division, the Forestry Division and the Health Hazardous Materials Division. Members of the Bureau serve the citizens of Los Angeles County by completing inspections and educating the community about the benefits of proper safety practices, completing building, sprinkler, and fire alarm plan checks, protecting natural resources, providing conservation education programs and advice to interested groups, using technology to assess weather, fuel moisture, and fire danger, and protecting public health and the environment from accidental releases and improper handling, storage, transportation, and disposal of hazardous materials and wastes.
John received a Bachelor of Science in Natural Resources Management from Cal Poly, San Luis Obispo in 1988. He has also completed many advanced courses in leadership, the Incident Command System, fire behavior, protection of resources, and urban search and rescue.
The dry, sunny climate and variable terrain of Southern California combine to create an environment where wildfires are a part of the natural ecosystem and an almost year-round occurrence. This ecosystem fosters a diverse fire-adapted community of plants and animals. Although human caused wildfires far outnumber naturally occurring wildfires within Los Angeles County, both have the potential to create situations where structures in the Wildland Urban Interface can be at risk. All vegetation will burn, even though irrigation has created a deceptively lush landscape of ornamental plants.
Following the loss of lives and structures during the 1993 wildfire season, the Los Angeles County Board of Supervisors created the Wildfire Safety Panel to offer recommendations that would help reduce the threat to life and property in areas prone to wildfires. One of the recommendations was to follow the findings of the Wildland Urban lnterface Task Force and another was to enforce the provisions of the Bates Bill. Jurisdictional Fire Departments were required to establish a set of guidelines and landscape criteria for all new construction in Fire Hazard Severity Zones. As a result, Fuel Modification Plans became a requirement within Los Angeles County beginning in 1996.
In the areas served by the County of Los Angeles Fire Department, all new construction, remodeling fifty percent or greater, construction of certain outbuildings and accessory structures over 120 square feet, parcel splits and subdivision/developments within areas designated as Fire Hazard Severity Zones will require a Fuel Modification Plan approval before the applicable land division, Conditional Use Permit, or Building Permit will be approved. The County of Los Angeles Fire Department Forestry Division’s Fuel Modification Unit is responsible for processing, reviewing, and approving these plans.
Cal Fire is responsible for the mapping and revisions to all Fire Hazard Severity Zones across the state. These zone designations establish minimum standards for building construction and exterior landscape features in an effort to mitigate the increasing losses from our cycle of wildfire vents. Cal Fire designates the Severity Zones for all State Responsibility Areas (SRAs). In Local Responsibility Areas (LRAs), the jurisdictional county or city determines the Severity Zones with approval from the state that are then adopted by local ordinance or city councils.
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https://www.ilsf.org/
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International Life Saving Federation
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[
""
] | null |
[] | null |
en
|
https://www.ilsf.org/
|
This Bulletin advises that, for clarity, the LWC 2024 Event Management Committee has approved the marking of a 2m line back line for the Line Throw event. The new Technical…
Read More
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https://alisonanddon.com/2024/02/20/nippers-at-the-beach-surf-lifesaving-in-australia/
|
en
|
Nippers At The Beach – surf lifesaving in Australia
|
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2024-02-20T00:00:00
|
February 2023 — As I look down the beach I can hardly believe what I’m seeing! It looks like it could be a surf lifesaving carnival and I've never been to one. Or some kind of carnival anyway, and with my usual curiosity I want to see what's going on.I immediately head towards it, even…
|
en
|
https://secure.gravatar.com/blavatar/d9eb6f460d0fa9d140a1f2af775d879ab8fc48ca597b9a4f4bdac37c42b586dd?s=32
|
Adventures in Wonderland
|
https://alisonanddon.com/2024/02/20/nippers-at-the-beach-surf-lifesaving-in-australia/
|
February 2023 — As I look down the beach I can hardly believe what I’m seeing! It looks like it could be a surf lifesaving carnival and I’ve never been to one. Or some kind of carnival anyway, and with my usual curiosity I want to see what’s going on.
I immediately head towards it, even if I can only have a quick look and take a few photos. We’re trying to find the off-leash area for the dog and have discovered the carnival in the process. I so badly want to investigate, at least a little!
A childhood memory: a picture of surf lifesavers, five or six of them, pushing a big wooden boat with oars out into the surf, their heads covered in cotton caps tied under the chin. Another picture: a surf lifesaving carnival, 1950’s, probably in Sydney, a parade of men, again with the caps, marching with the leader proudly holding a big flag aloft displaying the club colours. It was probably on Bondi Beach, even then a famous, almost mythical place.
Things have changed a lot since then. The wooden boats have been replaced by inflatables with outboard motors, and with jet skis; helicopters roam the skies above popular beaches;
eight-foot surfboards have been replaced by much shorter and lighter rescue boards with hand grips; there is now a national governing body, and Surf Life Saving Australia has grown to be one of the biggest volunteer organisations in the world. There are almost 190,000 members, and over the years more than 685,000 people have been saved from drowning. There are now 314 clubs around the country, and it’s all funded by government grants, fundraising, corporate sponsorships, and community donations. One thing hasn’t changed – they all still wear the little caps tied under the chin.
SLSA exists to save lives, create great Australians and build better communities. Through its coastal safety, lifesaving, education, sport and recreation programs and services, SLSA generates significant social and economic benefits for the Australian community each year.
A recent audit demonstrated that for every dollar invested $20 is returned to the economy; the yearly worth to the community is $6.5 billion; and over sixteen million volunteer hours are logged each year. Australia’s intrinsic and passionate beach culture results in 300 million beach goers visiting the coast every year, but Australian beaches are among the most dangerous and unpredictable in the world. SLSA, a unique not-for-profit community organisation, has grown from the need to manage the hazards associated with this inherent beach culture.
Apart from local carnivals and competitions, that are the training ground for skilled lifesavers, there is The Aussies – the annual Australian Surf Life Saving Championships where members from Australia’s 314 surf clubs compete in more than 480 beach and ocean events. It is the largest event of its kind in the world, and compares in size to the Commonwealth Games, with as many as 5 to 7 thousand competitors in youth, open, and masters categories. Events cover all the skills required to save a life, from water skills to beach skills to first aid and team events – including sprinting, board rescue, wading, surf swimming, ironman/woman and a surf boat competition. It’s huge! Surf lifesaving has become a national sport.
But what we come across is not The Aussies. What we come across is the nippers.
Nippers includes the Junior Development Program which is designed to ensure children have fun at the beach, while participating in lessons that will provide them with a pathway to become a qualified surf lifesaver and a junior competitor. As they progress through the various age groups, nippers will undergo lessons in wading, running, ocean swimming, board paddling, and lifesaving skills as well as learning the basics of resuscitation and first aid.
This event is the Far South Coast Branch Junior Carnival 2023 on Broulee Beach. It’s a riot of kids in electric pink and acid green,
which I initially think are club colours. But no, they are simply for visibility. Then I notice every child has his/her age category marked on their leg or upper arm – Under11, U12, etc. The oldest are U14. Later a bit of sleuthing tells me that the participating clubs are Moruya, Pambula, Tathra, Broulee, Bermagui, Batemans Bay, and Narooma – all names I’m familiar with, all towns on the NSW far south coast. Each club has a tent and mountains of equipment.
There must be hundreds of people here: officials, race supervisors, parents, families, and of course the NSW branch of SLSA in its iconic red and yellow colours.
I watch the kids prepare for, and then run relays.
Meanwhile out on the water there are paddling races. I think they keep their legs bent to stop their feet slowing them down by dragging in the water. That’s my guess anyway. They’re paddling on something that seems to be the offspring of a marriage between a boogie board and a surf board.
Here you can see a rescue board with the handgrips along both sides for a drowning person to hold on to.
This next race went across the sand rather than along it. And weirdly the kids began lying on their bellies facing away from the direction they were to run in. They jump up and turn and run like mad things for a flag baton across the sand. There’s one less baton than the number of kids entered.
Of course there are swimming races. I’m in awe of the way the kids hurl themselves into the surf, dive through the waves and swim into deeper and deeper water. These are not short swims. Out to the buoys and back is several hundred metres. These kids below are all only nine years old; you can see the U10 on their legs. You know when they hurl themselves into the surf without hesitation that they’ve been doing it since they could walk, and probably started learning to swim before they could walk.
And naturally there’s a lot of hanging out with your mates, playing in the sand.
And those caps? They’re for visibility, a clear form of in-water identification. At carnivals the cap colours indicate the club. On beaches that are patrolled by SLSA they are worn by lifesavers while on active duty. The red and yellow quartered patrol cap has been an iconic symbol of the surf lifesaving movement for almost a century and makes lifesavers instantly recognizable.
The difference between lifesavers and lifeguards is lifesavers are volunteers who are part of surf lifesaving clubs. Lifeguards are paid professionals who work for the Australian Lifeguard Service, local council, or an alternate service provider. Where possible, lifeguards are recruited through the volunteer base provided by SLSA. Apart from first aid and resuscitation certificates, lifeguards must pass the following fitness tests: A minimum 400m swim, 800m run, 400m rescue board paddle and 800m run in the surf, all in under 25 minutes, plus an 800 metre pool swim in under 14 minutes, and rescue scenarios utilising a Rescue Board and Rescue Tube. They are clearly serious athletes.
Of course I knew there was surf lifesaving volunteers on Aussie beaches, and I knew that they were always recognizable by their bright red and yellow colours, but I had no idea the organisation had gotten so big. And so necessary. Last year sadly 125 people drowned at Australian beaches, but nearly 10,000 were rescued.
|
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dbpedia
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| 0
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https://www.islasurf.org/
|
en
|
International Surf Lifesaving Association
|
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"https://googleads.g.doubleclick.net/pagead/viewthroughconversion/962542949/?guid=ON&script=0"
] |
[] |
[] |
[
""
] | null |
[
"ISLA Surf"
] |
2017-01-26T05:03:47-08:00
|
The International Surf Lifesaving Association exists to advance professional lifesaving development in areas of need around the globe.
|
en
|
International Surf Lifesaving Association
|
https://www.islasurf.org/
|
I AM AN INDIVIDUAL
Are you a lifeguard looking to take your lifesaving skills to the frontlines of drowning prevention? Then you have come to right place. Learn more about the ISLA Experience to start your journey as an ISLA Volunteer.
I AM AN AGENCY REPRESENTATIVE
Are you an agency, government organization, or group looking for resources to build your lifesaving organization? ISLA Provides lifesaving resources to places in need. Learn more at our Request ISLA Services page.
Featured Products
ISLA Surf Shop
We provide the lowest price on products that are tested in every terrain and condition. Proceeds fund ISLA projects around the globe.
Become an ISLA member to immediately access special pricing on premium apparel, tech gear, swag and more!
|
|||||
3129
|
dbpedia
|
1
| 83
|
https://www.thepost.co.nz/nz-news/350366130/welly-winter-swim-no-biggie-emergency-responders
|
en
|
The Post
|
[] |
[] |
[] |
[
""
] | null |
[] | null |
en
|
assets/icon/Favicon-Post-192x192.png
| null | |||||||
3129
|
dbpedia
|
3
| 22
|
https://www.surffestival.org/shield
|
en
|
Wieland Shield
|
https://static.parastorage.com/client/pfavico.ico
|
https://static.parastorage.com/client/pfavico.ico
|
[
"https://static.wixstatic.com/media/05acd2_67611617929745b3aefbcfb294ff72f5.jpg/v1/fill/w_977,h_221,al_c,lg_1,q_80,enc_auto/05acd2_67611617929745b3aefbcfb294ff72f5.jpg",
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] |
[] |
[] |
[
"6-Man",
"Crum",
"Intracrew",
"Lifeguard",
"Paddle",
"Pier-to-Pier",
"Saikley",
"Swim",
"Taplin"
] | null |
[] | null |
en
|
https://static.parastorage.com/client/pfavico.ico
|
surffestival
|
https://www.surffestival.org/shield
|
The Wieland Shield represents a long-standing relationship between the Los Angeles County Lifeguards and the Victorian State Surf Life Saving Association of Australia. The exchange of ocean lifesaving knowledge, skills, and techniques has continued unbroken for the past five decades.
Upon the invitation of the Surf Life Saving Association of Australia (SLSA), the LA Lifeguards participated in the first International Surf Life Saving Carnival which was held in conjunction with the 1956 Olympic Games held in Melbourne, Victoria, Australia.
Los Angeles County Lifeguards hosted the first Australian SLSA team to tour the USA in 1965 and presented the team before the LA County Board of Supervisors.
In 1967 LA Lifeguards participated in an educational and competition tour of Australia and New Zealand. The tour included a return to Melbourne from Feb. 21 to 28, and competitions were staged at Portsea and Ocean Grove. The Australian responsible for hosting the Americans was George Nutbeam. The manager of the USA team was LA County Lifeguard and Surf Lifesaving Association of America President Bob Burnside.
|
|||
3129
|
dbpedia
|
3
| 59
|
https://www.latimes.com/socal/daily-pilot/news/tn-hbi-1220-lifeguards-charity-20121218-story.html
|
en
|
Lifeguarding around the world
|
https://ca-times.brightspotcdn.com/dims4/default/1ff2a7a/2147483647/strip/true/crop/600x315+0+45/resize/1200x630!/quality/75/?url=https%3A%2F%2Fwww.trbimg.com%2Fimg-50d0c10c%2Fturbine%2Ftn-hbi-lifeguard-charity-1.jpg-20121218
|
https://ca-times.brightspotcdn.com/dims4/default/1ff2a7a/2147483647/strip/true/crop/600x315+0+45/resize/1200x630!/quality/75/?url=https%3A%2F%2Fwww.trbimg.com%2Fimg-50d0c10c%2Fturbine%2Ftn-hbi-lifeguard-charity-1.jpg-20121218
|
[
"https://ca-times.brightspotcdn.com/a6/d6/eea0f1094fb281dbea09e0aa79cd/art-caltimes-trademark-3x.png"
] |
[] |
[] |
[
"Huntington Beach",
"Huntington Beach Lifeguards",
"Orange County",
"International Surf",
"Raquel Lizarraga"
] | null |
[
"Britney Barnes"
] |
2012-12-18T18:00:33+00:00
|
Four junior lifeguard instructors found charity designed to teach drowning prevention in other countries as well as offer services locally.
|
en
|
/apple-touch-icon.png
|
Daily Pilot
|
https://www.latimes.com/socal/daily-pilot/news/tn-hbi-1220-lifeguards-charity-20121218-story.html
|
HUNTINGTON BEACH — In some countries, lifeguards aren’t a given, and if they are, there isn’t always uniform life-saving procedures in place or the equipment and infrastructure needed to save lives.
“I think we take it for granted that we have lifeguards here all the time,” said Raquel Lizarraga, 25, who volunteers with the Huntington Beach-based International Surf Lifesaving Assn.
It was the disparity in lifeguard services that brought together four Huntington Beach Lifeguards to volunteer their time and expertise to found the International Surf Lifesaving Assn., or ISLA, which works to prevent deaths from drowning by sharing and exchanging life-saving procedures with established lifeguard agencies to areas without any lifeguard around the world. The organization also offers its services locally, helping to patrol the beaches and give classes in Laguna Beach.
Since 2008, lifeguards and other emergency personnel have volunteered their time to give assistance in Chile, Dominican Republic, Ecuador, Macedonia, Mexico, New Zealand and Nicaragua. The organization is getting requests for help in Macedonia, Philippines, South Africa and Mozambique, said lifeguard Henry Reyes, 32.
“I think the coolest part of saving lives through ISLA is...we’re actually able to expand our rescues exponentially,” Reyes said.
ISLA receives its funding through donations. Most of the volunteers travel to the various countries on their own dime, according to Reyes.
A group of ISLA volunteers are spending their New Year’s meeting with officials in Peru and assisting Chile’s private lifesaving agency to patrol the busy beaches and exchange information.
“We help show them basic things that we seem to take for granted here,” said Huntington Beach lifeguard Tyler Erwin, 25.
The organization was founded by Reyes and his then fellow Huntington Beach Junior Lifeguard instructors Peter Eich, Scott Hunthausen and Olin Patterson in 2008. It started after Hunthausen, now living in Texas, returned from studying abroad in Nicaragua where his host brother drowned.
“[That] really kick started it off and moved all of us into action,” Reyes said.
Since they started working for ISLA, the volunteers have discovered that lifeguard conditions vary greatly.
Chile and Brazil have developed lifeguard infrastructure, but Nicaragua doesn’t really have anything, Reyes said. Some places rely on volunteer lifeguards who only work on the weekends while other places have private contractors, he said.
“It’s a completely different animal down there,” Erwin said.
ISLA conducts lifeguard certification courses, bringing down equipment and teaching how to identify swimmers in need of rescue, about the different rip tides and currents and how to use equipment, like buoys.
“It can be frustrating when you don’t have the support you think you need, but it can be rewarding when you are helping them,” Erwin said.
In Chile, ISLA will be working with Argentina lifeguards to partner with Chile’s private lifesaving agency Servicios Especiales Acuaticos Ltda, or SEAL, which has more than 200 lifeguards, Reyes said.
The group will help patrol the coast of Pichilemu, exchange lifesaving techniques as well as meet with lifeguard officials in Chile, Peru, Argentina, and Uruguay to discuss drowning prevention strategies and exchange programs.
“I just find it really interesting how the different places adapt with the resources they have there,” Erwin said.
Britney.barnes@latimes.com
Twitter: @britneyjbarnes
Want To Help?
|
||
3129
|
dbpedia
|
2
| 15
|
https://discover.pbcgov.org/parks/pages/ocean-rescue.aspx
|
en
|
Parks & Recreation Ocean Rescue
|
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] |
[] |
[] |
[
""
] | null |
[] | null |
en
|
/SiteImages/Icon/seal.ico
|
https://discover.pbcgov.org:443/parks/pages/ocean-rescue.aspx
|
Return to Beaches
Palm Beach County Ocean Rescue Lifeguards
Palm Beach County Ocean Lifeguards are a team of well trained, highly skilled professionals. They specialize in preventative actions, ocean rescues, and emergency medical assistance to beachgoers. Palm Beach County employs 66 year-round lifeguards and 40 seasonal lifeguards for 14 parks. These men and women are entrusted with protecting and servicing an estimated 5.2 million beachgoers every year, averaging 200 rescues, 630 bathing assists, 16,000 first aids, 113,000 preventative actions, reuniting 30 lost children with their parents and 9,000 participants in their public education events.
But who exactly are these men and women who make up Palm Beach County’s Ocean Rescue?
The 106 guards range in age from the early 20s to the early 60s. A surprising number hold college degrees. Most are married and have children. Former teachers, accountants, firefighters, writers, stock brokers, air traffic controllers, real estate agents, business people, and lawyers have all worn the Palm Beach County lifeguard uniform. Most have worked for Palm Beach County for over 10 years; 17 over 20; and a dozen over 30.
Palm Beach County Ocean Rescue is an agency certified by the United States Lifesaving Association (USLA). While all lifeguards hold certifications in lifeguarding, CPR/AED and Emergency Medical Response- first aid, most surpass this, a majority are Emergency Medical Technicians (EMT), several are certified paramedics, in SCUBA, and some have a boat captain license.
Regardless of age and work experience, Palm Beach County Ocean Lifeguards are all exceedingly skilled and trained ocean athletes. All lifeguards train daily to hone their swimming, running, rowing, paddling, and rescue skills to ensure not only the public’s safety but also their own. Formal training drills, led by Training Officers, are conducted at least twice per month to keep skills sharp. To maintain their jobs, they must biannually pass a timed swimming and running test, which is a requirement of each USLA agency.
Aiding the effectiveness of the lifesaving operation is the use of a wide range of equipment. Most inlet parks have Rigid Hull Inflatable Rescue Boats or a Rescue Water Craft, and All Terrain Vehicles are in operation at most beaches. Each lifeguard tower is fully stocked with first aid supplies, a back board, oxygen, resuscitation equipment, an Automated External Defibrillator, and rescue tubes and paddle boards.
The skills and rigorous training of these dedicated professionals has yielded some notable awards:
1996 – Beach Patrol of the Year by the Florida Beach Patrol Chief’s Association
2003 – First place in the Bill Shearer International Basic Life Support Competition at CLINCON
2004 – First place in the USLA National Surf Lifesaving Championships
2016 – Beach Patrol of the Year by the Florida Beach Patrol Chief's Association
2018 – First place in the Southeast Regional USLA Surf Lifesaving Championships
Seven Palm Beach County Ocean Lifeguards have been named Lifeguard of the Year in the state of Florida
2020 – Beach Patrol of the Year by the Florida Beach Patrol Chief’s Association
2021 – First place in the Southeast Regional USLA Surf Lifesaving Championships
2022 – First place in the Southeast Regional USLA Surf Lifesaving Championships
Palm Beach County Ocean Lifeguards are a diverse group of highly trained and dedicated individuals who offer an outstanding life saving service to all visitors of County beaches.
Lifeguard Tryouts
Ocean Rescue is conducting a seasonal Ocean Lifeguard test. The testing process is in two stages.
1. A Run/Swim/Run continuously for approximately 20 – 25 minutes
|
||||||
3129
|
dbpedia
|
1
| 17
|
https://lifesaving.com.au/
|
en
|
Surf Life Saving Queensland
|
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] |
[] |
[] |
[
""
] | null |
[] |
2024-07-19T00:00:00
|
Want to be involved in surf lifesaving? We have a role to suit any age and any background. Learn More There’s nothing better than Queensland
beaches but would…
|
en
|
Surf Life Saving Queensland
|
https://lifesaving.com.au/
|
Our valued partners help us deliver on our vision of zero preventable deaths in Queensland public waters.
A partnership with Surf Life Saving Queensland offers these businesses a marketing edge coupled with a strong sense of community commitment that delivers positive results.
|
|||||
3129
|
dbpedia
|
2
| 42
|
https://today.usc.edu/lifeguard-travels-the-world-while-earning-usc-master-of-public-health-online/
|
en
|
Lifeguard travels the world while earning USC Master of Public Health online
|
[
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] |
[] |
[] |
[
""
] | null |
[
"Larissa Puro"
] |
2018-03-16T15:00:04+00:00
|
University of Southern California News
|
en
|
/wp-content/themes/usc-communications-2023/favicon/apple-touch-icon.png
|
USC Today
|
https://today.usc.edu/lifeguard-travels-the-world-while-earning-usc-master-of-public-health-online/
|
The moment San Clemente surfer Jonathan Robinson plunged into the Pacific to pass a lifeguarding test at age 17, his passion became his profession.
Seven years of protecting the Orange County coastline groomed Robinson to be where he is today. The USC Master of Public Health student dreams of an international career in emergency medicine, saving lives wherever he can.
The aspiring physician and MPH student is pursuing his degree online through the Keck School of Medicine of USC in global health leadership while traveling the world for the International Surf Lifesaving Association (ISLA). He serves as an emergency medical technician at a childrens hospital and volunteers as a youth program director and CPR instructor.
Drowning as a public health issue
Robinson is committed to saving swimmers’ lives through ISLA. Drowning is the third leading cause of unintentional injury death worldwide, according to the World Health Organization. While all economies and regions face burden and death from drowning, low- and middle-income countries account for more than 90 percent of unintentional drowning deaths.
Dedicated to open-water lifesaving and preventing drownings, the association helps people secure aquatic safety in their own coastal communities. The organization supports lifeguard-training programs and exchanges, equipment donations, purchasing connections and technology to sustain a global network of lifeguards that share information, techniques, stories and culture.
Robinsons ISLA trip to Nicaragua was a collaboration with the Red Cross. He and other team members slept in a warehouse alongside Nicaraguan lifeguards and members of the police force, fire department and military.
Our shared learning was tested each day on the beaches with thousands of people in our waters, he said. I was surrounded by a network of global first-responders and lifeguards, united in their mission to combat injury or death from drowning.
After that trip, he was hooked.
Planning other lifeguarding projects
Now, as a watch commander, Robinson spends less time scanning the horizon these days in order to work with association administrators to help plan projects, facilitate inter-agency coordination, identify goals for response teams and organize trip logistics.
Google Translate has become my best friend in receiving and sending emails to Turkey, Thailand, Chile, China and Nicaragua, Robinson said. But software and language skills only go so far in handling the multifaceted challenges of international work.
The skills I have learned from the MPH program in performing cultural assessments and consulting with local experts has helped fill in the gaps, he said. At ISLA, we never assume we are the sole experts in host countries; we are not there to teach or to lead; we are there to partner, collaborate and to empower.
This month, Robinson will travel to China and return for a third time to Nicaragua to assist in the training of local lifeguards and support for beaches. In the last year, he has led international teams to Turkey and Thailand, where they assisted training for lifeguards in drowning prevention, open-water rescue and medical skills.
Flexible, applicable education
Between work hours, local volunteer work and global trips, Robinson is careful to stay on top of his coursework.
The beauty of being an online student is self-managing my time for studies, he said.
That flexibility is a cornerstone of USCs online MPH degree.
Many of our online MPH students, like Jonathan, are working professionals, proving you dont have to put your life on hold to get a prestigious graduate degree, said the programs director Shubha Kumar, assistant professor of clinical preventive medicine. It is challenging to juggle school on top of a career and public service, but we work with each student individually to ensure they stay on track with their goals.
And after college?
USC was Robinsons top choice for pursing a graduate degree.
I sought a graduate program that could strengthen my professional skills and build connections within the public health sector, he said. His mentor, Mellissa Withers, USC associate professor of clinical preventive medicine, has connected him to fellow Trojans working in his field, and she advises him on career goals.
Having a faculty adviser like her is why I chose USCs MPH program and why being a member of the Trojan Family is different than any other, he said.
After completing his degree, Robinson plans to specialize in emergency medicine while leading research on protocols and policy reform. He advises all students looking to study and work in public health to prepare to be transformed by the people with whom they work.
The communities and programs you partner with will show you the resiliency of the human spirit, he said. Public health is as much about promoting healthier behaviors and outcomes as it is about being advocates for individuals who just want to be heard.
|
||||
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|
dbpedia
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1
| 2
|
https://www.ilsf.org/
|
en
|
International Life Saving Federation
|
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] |
[] |
[] |
[
""
] | null |
[] | null |
en
|
https://www.ilsf.org/
|
This Bulletin advises that, for clarity, the LWC 2024 Event Management Committee has approved the marking of a 2m line back line for the Line Throw event. The new Technical…
Read More
|
|||||||
3129
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dbpedia
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2
| 9
|
https://cslsa.org/History.html
|
en
|
California Surf Lifesaving Association
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California Surf Lifesaving Association
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History
Home
About CSLSA
History
When Australia was chosen to host the 1956 summer Olympics, lifesavers there decided to hold an invitational lifesaving competition to be known as the Australian Olympic International Surf Championships at Torquay Beach, outside Melbourne, Victoria. [Australia Medallion] The Honorable Judge Adrian Curlewis of Australia appointed Arthur Parkens, an Australian lifesaving instructor, to solicit participation from the United States. California lifeguards and a contingent from the Territory of Hawaii decided to participate. Both teams were required trained and awarded, "The Australian Surflifesavers Medallion," so as to meet the international competition standards required for the event.
The California lifeguards organized themselves under the banner of the Surf Life Saving Association of America (SLSA), although they were solely from the Los Angeles County and Los Angeles City lifeguard agencies. This was the first American lifeguard association of its kind, even if its name was a bit grand considering its narrow scope.
Team members from the SLSA included Team Captain Rusty Williams of Los Angeles County (LACO), Team Coach Kirby Temple (LACO), Team Manager Herb Barthels, Sr. of Los Angeles City (LACity), Tad Devine (Santa Monica City), Bob Burnside (LACO), Mike Bright (LACO), Greg Noll (LACO), Dave Ballinger (LACO), Chick McIlroy (LACO), Paul McIlroy (LACO), Sheridan Byerly (LACO), and Roger Jensen (LACO). The Hawaiian lifeguard team included Dr. Don Gustuson, Team Manager Harry Shaeffer, Team Coach Tom Shaeffer, Tom Moore, Tom Zahn, Dan Durego, Tim Guard, L. Honka, Peter Balding, and Shaky Felez.
Picture on left: Zahn, Noll and Bright.
The event was held on November 26, 1956 and drew an immense crowd of 115,000 spectators. The legendary "Duke" Paoa Kahanamoku of Hawaii served as the honorary event Chairman. In addition to the American and Australian teams, lifeguard teams from South Africa, Great Britain, Ceylon, and New Zealand participated.
As for the Americans, the Hawaiian Territory placed first in the Beach Relay. Tad Devine of California placed second in the swim. Bob Burnside of California placed third in the belt race.
Perhaps more importantly than the competition itself, lifelong relationships were built around this historic event and both countries were to benefit tremendously. The American rescue tube and rescue buoy were first introduced to Australia on this tour, later to become staples of Australian lifesaving gear as they were in the US. Tom Zahn, Tom Moore, and Bob Burnside brought Malibu Balsa Surfboards with them, the first total Australian exposure to the Malibu Surfboard. When they departed Australia, the boards were left behind, which revolutionized surfing in Australia.
After the event, LA County Lifeguard Chief Bud Stevenson decided to use SLSA in his efforts to upgrade professional lifeguarding. Chief Stevenson appointed Bob Burnside as President of the nascent organization and Lt. Don Hill as Secretary. Despite the broadly embracing name of Surf Life Saving Association of America, the early focus was to remain on Los Angeles County issues.
Bob Burnside called for representatives from as many Southern California lifeguard agencies as possible to attend a concept meeting at Santa Monica Lifeguard Headquarters in the winter of 1963. In attendance were Vince Moorhouse (Huntington Beach), Max Bowman (Huntington Beach), Don Rohrer (LA City), Dick Heineman (LA City), Tim Dorsey (Seal Beach), host Jim Richards (Santa Monica), and a representative from Long Beach.
The group agreed that they should establish a truly national organization, based on the structure of the Australian association, to be called the Surf Life Saving Association of America.
The early organization established Southern and Northern Chairmen of the State of California, and a temporary Executive Board was formed to establish a constitution, bylaws, and method of equal representation for the association. This put in place all the necessary criteria for affiliation with the organization by local chapters, allowing each chapter to participate equally in the first election of officers, which took place in 1965. In the meantime, temporary chairmen took charge.
In 1964, Huntington Beach's newly dedicated lifeguard headquarters was adopted as the center for SLSA activities. In that same year, Howard Lee of LA County designed the national logo, which is still in use today. His design was influenced by a similar design that Tad Devine of the 1956 Australia team had created for the team uniform. Both are strikingly similar to the logo of the United States Life-Saving Service, an arm of the United States government, which had rescued shipwrecked sailors during the 1800s and 1900s, before being merged with the Revenue Cutter Service to form the US Coast Guard.
The National Surf Life Saving Association of America is Born
In 1965, the SLSA title was dropped in favor of the National Surf Life Saving Association (NSLSA) and the first election officers of was held, at this time for a one-year term of office. They were President Bob Burnside, Vice-President Dick Hazard (San Clemente), Treasurer Max Bowman, Secretary Don Rohrer, and Sergeant at Arms Tim Dorsey. The goals and objectives were identified and weekly meetings were agreed upon, rotating among different lifeguard headquarters for over a year. In another 1965 development, Australia was invited to send their national competition team to compete in the US.
A year earlier, in 1964, ABC television's Wide World of Sports had filmed a lifeguard competition at Huntington Beach. During the competition, lifeguards Mike Henry and Pete Orth of Carpenteria, California, lost control of their dory on a 10 foot wave and crashed into the Huntington Beach pier. It became one of the Great Moments of 1964, replayed repeatedly for television audiences throughout America.
Building on this memory, in 1965 Wide World of Sports invited the NSLSA and the touring Australian team, to fly to the East Coast and compete in a first ever East Coast/West Coast lifeguard competition. This televised, international event was held at Montauk Point on Long Island, New York. At this event, the concept of a truly national affiliation under the umbrella of NSLSA took seed. Also in 1965, Santa Cruz became the first lifesaving association outside Southern California to join.
The year 1966 saw a new election of officers, with Bob Burnside remaining as president, Phil Stubbs of San Clemente as vice-president, Jack Buck as Secretary, and Don Rohrer as treasurer, with Tim Dorsey remaining as Sergeant at Arms. Also in 1966, the California Chief Lifeguard Association, which had first formed in the late 1930s, reconvened and appointed Vince Moorhouse as chairman. They conferred $431.80 from their association bank account to the NSLSA treasury, along with their blessings and pledge to support the organization.
On August 25, 1967, it was decided to change the term of office to two years. Mike Henry of California State (north) was elected president, Phil Stubbs of San Clemente remained vice-president, Bob Burnside moved to secretary, Dick Heinemann of LA City as treasurer, and Tim Dorsey continuing on as Sergeant at Arms.
In 1967, NSLSA sent a competition team to Ft. Lauderdale, Florida to compete in the first recognized East versus West lifeguard championships, continuing the national affiliation concept among all the agencies involved. Teams from New York to Miami and the West Coast team battled it out in a rousing competition.
The East Coast/West Coast competition helped further an effort to make NSLSA a truly national organization. Lt. Jim Holland of the Miami Beach Patrol was appointed to act as East Coast liaison for NSLSA. He was responsible for bringing into the first Florida chapters into NSLSA: Miami Beach and Boca Raton. Secretary Bob Burnside flew to Florida to tour Florida beaches with Lt. Holland in an effort to further increase Eastern affiliations.
It was also during 1967 that the Australians invited the NSLSA affiliated lifeguards back for a competition tour that included several unusual feats. This included a stunning win by the 16 year old Huntington Beach lifeguard Spike Beck in the Australian National Championship Junior Belt Race. At the New South Wales championships Australian veteran "Spas" Hearst, Bob Burnside, Paul Mathies (LACO), Jim Richards (Santa Monica), and Ruby Kroon teamed up for a binational win in the surfboat race.
Pictured left to right: Ray Bray, Spike Beck, Jerry McGraw, Joe Metzger
In 1969, the change in officers found Phil Stubbs elected president, Bob Shea of San Diego vice president, Logan Lockabey of California State secretary, Dick Heinemann treasurer, and Tim Dorsey still watching the door as Sergeant at Arms. That year, the first international educational exchange was undertaken with a visit to Auckland, New Zealand by Max Bowman (Huntington Beach), Phil Stubbs (San Clemente), and Logan Lockabey (Newport Beach). It was also in that year that NSLSA received membership in the Council for National Cooperation in Aquatics (CNSA).
In 1969 that the Dade County (Florida) Board of Supervisors requested that NSLSA representatives journey to Miami and review lifeguard procedures there in the wake of a rash of ocean drownings. Bob Burnside and Phil Stubbs handled this task, with Paul Cocke (LA County) and Bill Richardson (Huntington Beach) assisting. The outcome included recommendations that resulted in installation of a communication system, new vehicles and equipment, new qualification requirements, increased funding, and the hiring of Lt. Holland as Chief of the Dade County Lifeguard Division. It was the first demonstration of the potentially power of NSLSA to improve lifesaving standards nationwide. In 1970 Hempstead Beach, New York joined NSLSA. It was the first member chapter from the upper East Coast. The NSLSA newsletter changed its name that year to Ocean Lifeguard Magazine. The editor was Tim Dorsey.
World Life Saving was created in 1971 in Australia, to include the national lifesaving federations of Australia, Great Britain, New Zealand, South Africa, and the US. Chief Vince Moorehouse of Huntington Beach was appointed the NSLSA International Liaison to WLS, then President from 1976-80. Max Bowman served from 1988-1993. On 24 February 1993, WLS merged with FIS to form the International Life Saving Federation (ILS). By that time, WLS represented more than 20 full member national lifesaving organizations.
In 1971, President Phil Stubbs was reelected, Eric Lucas of Laguna Beach was elected vice president, Logan Lockabey remained as secretary, Bill Ward became secretary, and Tim Dorsey stayed on for another term as Sergeant at Arms. In that year, recommended beach standards and certification were first completed for all lifeguard classifications in an effort to improve standardization and professionalism. Hempstead Beach withdrew their membership that year, after only one year in the organization.
Two other organizations requested NSLSA professional help in 1971, Big Surf surf park in Tempe, Arizona and the State of New York. In the case of the latter, a team of 11 members were sent to help in training and appraisal of New York lifeguard practices, but when they arrived they learned that they had been summoned amidst a job action and were being enlisted to ensure lifeguard protection in the case of a strike. This event turned out to create some seriously bad feelings between New York lifeguards and the NSLSA, which were not to subside for many years.
In 1972, the Internal Revenue Service granted NSLSA tax-exempt status as a not for profit, educational organization. World Life Saving held its Board of Directors meeting in Huntington Beach that year, the first international lifesaving meeting in America. The NSLSA newsletter, under the title "Certification," also mentions that NSLSA was preparing the groundwork for a training certificate to be issued to newly trained lifeguards [and] studying a proposal from the YMCA of America to certify lifeguards for that organization."
In 1973, the sixth Executive Board was elected to include Vince Moorhouse (Huntington Beach) as president, Eric Lucas (Long Beach) as vice president, Buddy Belshe of Newport Beach as secretary, Bill Ward (Long Beach) as treasurer, and Tim Dorsey sergeant at arms. That year also saw the development of agreements on standardization of beach warning flags and the first international training officers exam.
South Africa was the next destination for an education and competition tour. An 11 man team was sent in 1974 that included Max Bowman, John Patty (Long Beach), Tim Dorsey, Mark Bodenbender (Huntington Beach), BI Gerald, (Huntington Beach), Buddy Belshe (Newport Beach), Bill Owen (San Diego), Sheridan Byerly (San Clemente), Richard Marks (LA City), Paul Mathies (LACO), and Topper Harock (Newport Beach).
In 1974, NSLSA conducted a site review and beach lifeguard service survey for the City of Santa Cruz. This process demonstrated potential of the NSLSA to help influence management of beach lifesaving organizations. It was a process that was used many times again in future years.
In 1975, Eric Lucas (Long Beach) stepped up to become president, with Buddy Belshe (Newport Beach) as vice-president, Dick Miller of Long Beach as secretary, and Max Bowman (Huntington Beach) as treasurer. Also in 1975, Vince Moorehouse (Huntington Beach) was elected president of World Life Saving.
The United States Lifesaving Association is Created
Over the years, NSLSA had been very successful in organizing national and international exchanges of information, competitions, and public education efforts to help reduce drowning. Progress made initially to embrace East Coast agencies however, had languished and the organization remained largely an association of California lifeguards and a few chapters from Florida.
Some felt that the organization should remain a surf lifesaving organization, barring participation from lifeguards at lakes, rivers, and similar venues. Others felt that all lifeguards at natural, open water locales should be eligible for membership. One of these was Sheridan Byerly (now of San Clemente), who had been a member of the 1956 Australian team.
The 1977 elections were a turning point. It was anticipated by many that Buddy Belshe would be elected president that year, but in an unexpected upset, Sheridan Byerly was elected instead. Considering that he had never been elected to an NSLSA post before that time, his election was a surprising event. As it turns out, it was to portend further change. In that same year, Dick Miller was re-elected vice-president, Max Bowman treasurer, and Larry Gibson of Newport Beach secretary. Partway through the two-year term of office, Gibson resigned his post and Byron Wear of San Diego was appointed to replace him. Wear would later become the first USLA Executive Director until 1984. Many years later he was to be elected to the San Diego City Council.
A priority for Byerly was the push to make NSLSA a truly national organization. A debate occurred over opening the guidelines for membership to allow personnel from lake, river and similar venues. It was hotly contested as many felt that the association should incorporate only ocean agencies and that bringing in other areas that would not necessarily be year-round operations, would shift the power to part time and non-ocean agencies, thus creating a philosophical difference of priorities. Nonetheless, Byerly persevered. Changes to the bylaws were drafted and plans were laid to create regions throughout the United States with their own presidents and executive boards.
In February 1979, Byerly and Wear took leave of their jobs and began recruiting work in Florida, encouraging further participation. At that time, Florida membership centered on Boca Raton and Dade County. They met with lifeguards from many agencies, including Joe Wooden and Tom Renick of Volusia County.
In May 1979, the NSLSA Board of Directors met in Santa Cruz, knowing that the debate over broadening the membership scope of the organization was coming to a head. President Byerly chaired a meeting thick with heated and passionate discussion about the course of the organization's future. Ultimately, the NSLSA Board of Directors voted to change the name of the organization to the United States Lifesaving Association (USLA) and adopt the various bylaw changes that had been drafted. It was a truly historic event, which set the stage for a broader and more embracing organization. It was agreed that members could include, any member of an ocean, bay, lake, river, or open water lifesaving or rescue service, including chiefs, directors, and their equivalent.
Since that time, the United States Lifesaving Association, an idea launched in California, has thrived, having a major and very positive influence nationally over drowning prevention and lifeguard training standards. The first truly national competition was conducted in 1980 in San Diego, California under the USLA banner, a tradition that has continued annually ever since. In the spring of 2014, USLA celebrated their 50th Anniversary at a Board of Directors meeting and Educational Conference hosted by the CSLSA’s Huntington Beach chapter. Click HERE to view the program agenda.
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https://andyjoyce.info/tag/surf-steps/
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Surf Steps: Surf, SUP and Safety.
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2024-04-25T11:59:49+00:00
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Posts about Surf Steps written by kiwiandyjoyce
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en
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Surf Steps: Surf, SUP and Safety.
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https://andyjoyce.info/tag/surf-steps/
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Part 2: Time to shine!
“You may be the only person left who believes in you, but it’s enough. It takes just one star to pierce a universe of darkness. Never give up.” — Richelle E. Goodrich, Author
For the next 17 years, we poured our hearts into our surf school “Surf Steps,” One of the most fulfilling yet challenging endeavours was running our “Junior Surf Club” for over a decade. It was a labour of love that demanded endless liaison, organisation, and nurturing—both of the young athletes and their sometimes equally passionate parents! Despite the challenges, it was incredible witnessing these individuals evolve into top-level surfers, coaches, lifeguards, and physiotherapists. filled me with immense pride. Even today, I am full of pride in what was achieved and cherish the bonds forged during those formative years, realising the profound impact our coaching and mentorship had on their lives and also importantly, mine.
During this time in 2007 and after two decades, I finally pursued my long-held dream of attending university. This opened doors to new opportunities and perspectives in Sport. Stepping back from the management of the day-to-day operations of the school in 2011 allowed me to explore some of these new avenues, recognising many areas for surfing development. So I endeavoured to address these areas’ lack of resources by also becoming a trainer and mentor of surf and SUP coaches, as well as a water safety instructor.
In 2013, I embarked on my first international sports development consultancy project with the Danish National Surf Federation, representing the International Surfing Association (ISA). This has continued into many more projects since and has allowed me to impart invaluable skills to many, but also provided an opportunity to make tangible drowning prevention improvements and create work opportunities in many countries. Contributing to the improvements in the sport, the safety and well-being of individuals worldwide has been immensely gratifying.
The pivotal year of 2020 heralded a new chapter in my career journey with the offer of the Regional Development Advisor position created by the ISA as it became a recognised Olympic sport. The incredibly varied nature of this role nowadays ensures that no two days are alike, there are so many uncharted territories to try to provide development solutions to! We also closed our Surf School this year. COVID-19 effects, and the continuing lack of support and restrictions in operation from our school’s landlords, the local government, finally led us to this decision. This allowed me to focus more time on my new career direction and shed a workload we no longer had the passion for and wanted to carry. In 2023 I was offered the ISA International Development Officer role which is the role I fulfil at present.
From establishing the first surf school in our local area to spearheading groundbreaking projects on a global scale, I feel my path has been now defined as a pursuit of excellence and a commitment to pushing boundaries. It’s still a daily challenge to decide how far to push these or when not to!
As I reflect on my journey, I am reminded of how hard it was at times to persevere in the pursuit of my desire to do something beneficial for me and others. Despite grappling with many moments of self-doubt and anxiety, I have emerged stronger and more resilient.
To all those grappling with uncertainty and apprehension about pursuing their dreams, I urge you to take that leap of faith. Embrace your inner brightness, for it has the power to illuminate even the darkest of paths. With perseverance and unwavering belief in yourself, you have the potential to carve a path that not only fulfils your aspirations but also leaves a lasting impact on the world.
What inspired YOU to Coach or to change your career path? Let me know in the comments😊
Part 1: Beginnings
Reflecting on my past, I realised I stumbled into coaching almost by chance. I understand now that it was my desires, life events, and the environments I found myself in all played a role in steering me towards this path.
I’m going back to my early years in this journey, when I began surfing at the age of twelve in Christchurch, NZ. As I approached the end of high school in the late 80s, our economy was in recession and we didn’t have a lot of money, so it was too challenging for me to pursue the University pathway. Struggling to secure good employment, I sought a route that would allow me to earn a tertiary qualification while being gainfully employed. This quest led me to explore opportunities within the Air Force, prompting me to undertake their entrance exams.
Performing well in the exams, I was encouraged by the recruiter to pursue aircrew selection. However, I “choked” during the impromptu public speaking assessment, hindered by insecurities a lot of 17-year-old have and paralysed with performance anxiety. Although I was urged to reapply for the next intake, financial pressures led me to seek alternative paths.
During the selection process, I connected with a fellow candidate already serving in the military, who suggested considering choosing an engineering trade if I couldn’t afford to wait. Following his advice, I enrolled as an aircraft engineering cadet, embarking on a career that allowed me to travel extensively, indulging in my passion for surfing and honing my knack for fixing things.
As I entered my thirties and embraced fatherhood, a newfound sense of responsibility and desire to do different and make a difference began to take root. The experience of nurturing my young daughter had quickly shifted my focus from self to others, prompting me to contemplate what changes I could make in my life to do different.
The idea of becoming a surf coach emerged.
Eager to share my love for the ocean and my wealth of experiences with others, I underwent training as a surf coach, balancing this newfound passion with my existing career in engineering. This eventually led to a part-time coaching role for a chain of surf shops across the south of England.
Becoming a coach marked a pivotal moment in my vocational journey, where my skills and passions took precedence over mere financial considerations. The success of a “girls only” surf event that I organised for the shop chain propelled me into a full-time coaching and sports equipment trainer role. However, as fate would have it, the company underwent a takeover, resulting in redundancy. As a part of my severance package, I negotiated then retainment of the lease agreement with the local council for the surf school, along with all the equipment and my company car. I still have two original beginner boards from 2002 that are still in use to this day.
Stay tuned for Part 2, where I’ll delve into what happened next!
Local surfers from the social group “Sisters of Stoke (SOS)” hit the beach near Bournemouth pier on St. Patrick’s Day Sunday last week, not only to catch the cool spring waves but to learn bystander rescue techniques from local surf, SUP and safety training organisation Surf Steps.
The term ‘bystander’ describes any member of the public, be they family, friend, or stranger, attempting to rescue someone in distress. However, it can be extremely hazardous for the bystanders if they don’t understand and assess the risks or have knowledge of rescue principles/techniques.
This is the first time this training programme has run in the UK. SALT (Surfers Awareness in Lifesaving Techniques) was designed by a team of Huntington Beach lifeguards, in California, US, to provide aquatic emergency response guidelines and techniques to surfing members of the public.
The program teaches The 3R’s:
Recognition of potential hazards and victims.
Reaction to people in need, with consideration of one’s own safety first.
Response from lifeguards, emergency services and surfers.
The Sisters of Stoke (SOS) said much of what they learned on Sunday was new to them. The SALT program changed the way they think about people who end up in unsafe situations in the water. SALT teaches that a crisis can happen to anyone at any time and that surfers should have an obligation to look after one another, especially when there is no lifeguard patrol.
“It’s rewarding to be in a program that’s not just about surfing — it’s about helping out others,” “It is going to have a ripple effect. … With all the people here today, you know they are going to save lives in the future.” Mar, SOS
The group took turns in training being the person in trouble and then, rescuer, then assistance. Andy, Surf Steps’ trainer, showed the group how to identify if the person was in trouble, how to secure them, and then how to start the person’s safe return to shore.
“We realised that many bystander rescues were conducted by surfers who are already out there surfing. If you surf long enough, it is not a matter of if you will rescue someone — it’s when.” Andy, Surf Steps
SALT instructors tell participants that many rescues will also need a response from lifeguards or other emergency medical services. A big issue is that many untrained people trying to help in these situations without knowledge and coordination can make it more difficult.
In the US, The SALT program is run by Huntington Beach Marine Safety Division and is free to any local surf groups. Those interested can email marinesafetyeducation@surfcity-hb.org to set up a class.
Interested in similar programs for Surf or SUP groups in your local area? Contact Andy, Director of Surf Steps by email: andy@surfsteps.co.uk
Big thanks to Sistas of Surf for being part of our top 1st-time trial of SALT!
Last and no means least, Ian and Terry at Havana Beach Hotel for being the stand-up members of the local community by providing their awesome venue for the classroom and changing facilities! Legends:-)
Walton on Thames, Elmbridge, Surrey August 21-23 2020
8 SUP coaches from Ditton’s Paddle Boarding took part in the first Coach Rescue course since the UK lockdown in March.
The course was hosted at a superb private location right on the banks of the river Thames arranged by Brett Scillitoe, Admiral of Dittons, with Surfing England and SLSGB trainer/assessor Andy Joyce.
Subjects covered included practical skills, safety, communication, risk analysis and assessment, SUP rescue, and First Aid.
The days started with theory and dry land practical sessions, followed in the afternoon by a lot of enjoyable water-based practical sessions in the surprisingly mild temperatures of the summertime Thames.
“It was great to be able to get back to doing what I love, helping people become safer coaches in and on the water and enable them to perform quality rescues in emergency situations. The students finally got the long-waited opportunity to complete the internationally recognised course and showed this in the way they applied themselves during the courses.” Andy Joyce
” Andy was a great instructor and all the students commented on how well he guided them through all parts of the course, with care and consideration. Nothing was ever a bother to go through again. They were all very keen to get going and practice their new skills.” Brett Scillitoe
Surf Steps provide this and other safety courses that are suitable for all watersports users, not just lifesavers and coaches.
If you are interested in any information regarding these, feel free to contact Trainer: Andy (andy@surfsteps.co.uk) or Ditton’s Paddle Boarding https://dpbclub.co/contact-us/
Last week, after a long time in lockdown, the Surfing England Adaptive (Para) Surf crew got together on Monday the 20th July in Porthtowan for a long-needed team catch up and surf!
Blessed with great weather and easy conditions, a lot of the core team made it for the day with some new surfers joining as well. We also had the great surprise of Bruno Hansen, Danish prone surfing world champion turning up to say hi!
It provided a great opportunity to look at equipment, technical aspects of surfing, and renewing and developing the relationships between the group as a whole.
The day was arranged by Andy Joyce, Surfing England’s Para Surf Team Manager and Melissa Reid, Britain’s first woman’s World Champion surfer (VI) at her home break of Porthtowan. We were grateful for the amazing support by the following companies and people:
-The Wave Project for organisational assistance and the use of their beach wheelchair.
-Spike Kane and The Wave for the use of the prone board for Tash.
–Snugg and Swell for the use of their boards and wetsuits.
–Caroline Kearsley Photography (ckearsleyphotography@gmail.com) for the amazing shots of the group featured here. For more of her work, click here
The team are looking at having another training session in September, stay tuned for more of the action!
If you are interested in finding out more and/or supporting the Para Surf team in any way, please do not hesitate to contact Andy ( +44 7941508531 or andy@surfsteps.co.uk)
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https://floridadep.gov/rcp/fcmp/content/beach-warning-flag-program
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en
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Beach Warning Flag Program
|
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The safety and enjoyment of Floridas public beaches are affected by changes in tide and surf conditions. To minimize the risks of drowning or serious injury, the Florida Coastal Management Program worked with the Florida Beach Patrol Chiefs Association, the United States Lifesaving Association (USLA) and the International Life Saving Federation to develop a uniform warning
|
en
|
https://floridadep.gov/sites/default/files/favicon_6.ico
|
https://floridadep.gov/rcp/fcmp/content/beach-warning-flag-program
|
The safety and enjoyment of Florida’s public beaches are affected by changes in tide and surf conditions. To minimize the risks of drowning or serious injury, the Florida Coastal Management Program worked with the Florida Beach Patrol Chiefs Association, the United States Lifesaving Association (USLA) and the International Life Saving Federation to develop a uniform warning flag program for use by Florida’s beachfront communities.
Why does Florida need a uniform warning flag system?
Many residents and visitors travel to different parts of the state to enjoy Florida’s beautiful public beaches, and many beach communities post warning flags. Differences in flag colors, sizes and symbols from place-to-place can confuse beach goers, thereby decreasing the effectiveness of efforts to improve public safety. The Florida Legislature decided that a uniform flag system would provide the best measure of safety and, in 2005, amended section 380.276, F.S., to require that all public beaches displaying warning flags using only the flags developed for the state’s warning program.
How does it work?
Florida’s beach warning flag program uses flags in four colors accompanied by interpretive signs along the beach to explain the meaning of each color. To the extent funds are available, warning flags and interpretive signs are provided free of charge to local governments that provide public beach access. The communities that receive the free warning flags and interpretive signs are responsible for installing, properly using, and maintaining the flags and signs.
Are flags used to warn of the presence of rip currents?
The beach flags provide general warnings about overall surf conditions and do not specifically advise the public of the presence of rip currents. However, increased awareness of natural conditions that pose a significant risk at the beach, such as rip currents, is a critical element to improve public safety. Therefore, in addition to this warning system, the FCMP also distributes rip current educational signs to local governments and public parks in the state of Florida if funds are available.
Since 2004, FCMP has distributed these comprehensive national signs that were developed through the combined efforts of the National Oceanic and Atmospheric Administration’s National Weather Service (NWS) and SeaGrant, and the USLA. To further your understanding on the dynamics and dangers of rip currents, FCMP encourages you to consult the professional advice provided by the NWS and the USLA.
What are the dimensions of the flags and signs?
Each flag measure 29.25”H. x 39”W. The signs are 30” x 36”.
Specifications for Beach Warning Flags and Signs
Beach Warning Flag Program Legislative History
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https://www.slsgb.org.uk/
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en
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Life Saving Education, Safety & Sport
|
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[
"Doodlebug Admin"
] |
2014-08-06T14:26:24+00:00
|
Surf Life Saving GB. About Us · Support Us · Become A Club Member · Education · SLSGB Accredited Training Centre Courses.
|
en
|
Surf Life Saving GB
|
https://www.slsgb.org.uk/
|
Welcome to Surf Life Saving Great Britain
We are a Search and Rescue charity of over 10,000 members.
Our purpose is saving lives and preventing drowning. Our volunteers have been providing lifeguard training and patrols for 70 years. Starting at age 7 members develop the skills, confidence, fitness and water safety awareness to become Lifesavers at 16. Our Surf lifeguards have the highest beach safety standards.
We train and operate volunteer inland and inshore search and rescue teams, are full members of the UK Search and Rescue Operators Group and are leaders in water safety risk management.
Our wholly owned subsidiary, SLSGB Training Ltd, provides training resources for Police Authorities and other organisations who work close to water. We offer bespoke safety advisory and consultancy services and are International leaders in the development of Flood Response and Rescue resilience programmes and key person training.
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https://emergencyservices.honolulu.gov/ocean-safety-lifeguard-services/the-history-of-ocean-safety/
|
en
|
Honolulu Emergency Services Department
|
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2022-04-20T19:32:01+00:00
|
A Timeline of Our History 1901 The Moana Hotel opens in Waikīkī in March, ushering in an era of tourism, and with it emerges a crew of extremely talented and ge
|
en
|
Honolulu Emergency Services Department
|
https://emergencyservices.honolulu.gov/ocean-safety-lifeguard-services/the-history-of-ocean-safety/
|
The Moana Hotel opens in Waikīkī in March, ushering in an era of tourism, and with it emerges a crew of extremely talented and generally jovial group of men called, 'The Waikīkī Beachboys.' The Waikīkī Beachboys appear on the scene with a revival of traditional Hawaiian watersports, surfboard riding, and outrigger canoe riding. They are described as "both a cause and byproduct of the booming Hawaiʻi tourist trade; they earn a living primarily by lifeguarding and giving surf lessons and canoe rides, and spend their free time surfing, swimming and playing music."
Territory of Hawaiʻi Act 201 officially establishes a "Life-Saving Patrol" for Waikīkī Beach, and has come to be celebrated as the birth of the Ocean Safety Division on Oʻahu. Act 201 called for assignment to Waikīkī Beach of two members not less than 18 and not older than 40 years of age, who were "selected for their strength and ability as swimmers and boatmen, and shall be expert in the use of the methods of resuscitation, both with and without apparatus, apparently drowned persons." These first city lifeguards were paid $75 a month. The City & County of Honolulu officially celebrates this event as the birth of its lifeguarding service.
Ralph "Buster" Wallwork joins Ocean Safety as a new Lifeguard and would end up rising through the ranks to Captain, and then to Chief of Operations in 1982 until his retirement in 1994
The Honolulu Fire Department begins making shoreline patrols around the island of Oʻahu by pulling a Lifeguard from the Natatorium into the fire truck.
Mākaha Beach Park caretaker and legendary waterman Richard "Buffalo" Keaulana is appointed as the first lifeguard at Mākaha.
The first lifeguard tower on the North Shore is erected at Sunset Beach, and then another at Waimea Bay.
Eddie Aikau becomes the first city lifeguard to work the North Shore. He is tasked with covering all of the beaches and surf breaks between Haleʻiwa and Sunset, using an old city hearse converted to a city lifeguard vehicle complete with rescue board on the roof. He bases at Waimea Bay every day and drives into town to check in once a month at the Natatorium.
While surfing Waimea Bay in the Eddie Aikau Big Wave Invitational lifeguard Brian Keaulana wiped out for a two-set hold down. While tumbling underwater he began thinking, “God here I am, in the same position of the people I see when they drown." When he came up, the next wave was even bigger. There next to him in the impact zone was a friend Squiddy Sanchez on a stand up Jet Ski saying, "Brian, you alright?" and pulled him from the impact zone. This sparked a vision in Brian who immediately worked with fellow Lifeguards Terry Ahue and Mel Puʻu to retrofit personal watercraft for surf rescue.
Personal Watercraft (PWC) are incorporated in the lifeguard program as a rescue tool for the North and Leeward Coast Districts for high surf rescue.
Keawaʻula Beach at Kaʻena Point State Park is located at the westernmost point on the island of Oʻahu. The beach is exposed to high surf; a strong shore break; and a strong, often severe, current. The remote, pristine site attracts many surfers, sunbathers, swimmers, and waders. The combination of dangerous physical features and heavy use by patrons increases the risk for water-related injury and death. From 1985 to 1991, two drownings and 40 near-drownings occurred at Keawaʻula Beach. Although the State of Hawaiʻi does not provide lifeguards, it elected to contract with the City and County of Honolulu to place lifeguards at Keawaʻula Beach beginning in January 1992.
Moi Hole - An amazing surf rescue happened in 1993 when 26-year-old tourist Hugh Alexander was knocked off a 20-foot rock ledge near Yokohama Beach on the Waiʻanae coast and was trapped in a sea cave known as a “Moi Hole” for more than two hours as it was pounded by 10 to 15 foot waves. Legendary waterman and lifeguard Brian Keaulana, dodging the giant waves, attempted to drive a Jet Ski with lifeguard Craig Davidson on a rescue sled up to the cave to rescue the man. But the vehicle ran onto a rock which pierced the hull, disabling it. Davidson swam out through the waves and was picked up by a rescue helicopter. Keaulana swam to shore, raced to his house in Mākaha to get another Jet Ski. This time, with new partner Earl Bungo, they gunned the Jet Ski back into the boiling surf and managed to pluck Alexander from the mouth of the cave. But then a big wave knocked both Alexander and Bungo from the sled and against the rocks. Keaulana, putting his own life on the line again, charged back into the watery mêlée, putting the Jet Ski between the men and rocks. Once the two were back on board, Keaulana shot the Jet Ski seaward, just as a new set of large waves came rolling in. Keaulana got an award from the U.S. Lifesaving Association in San Diego for the rescue.
The Ocean Safety and Lifeguard Services Division, in a City wide re-organization, is moved from the Department of Parks and Recreation to the newly created Emergency Services Department, along with the Emergency Medical Services Division.
Jan. 15, 1998 – The Queen of Mākaha Rell Sunn passed away after a long battle with cancer. One of the most popular and famous lifeguards in Ocean Safety history, she was a fixture at Tower 47 in Mākaha from 1977 through the late 1980s, and has inspired countless women to follow her footsteps in the ali‘i red shorts and gold shirt. She was the City’s first woman to lifeguard the Leeward Coast.
Ocean Safety announces that it will sponsor and manage its own Junior Lifeguard Program for the first time in more than a decade. In the intervening years (2013-2022), city lifeguards formed a non-profit in order to run the wildly successful program themselves.
Ocean Safety conducts one of the largest recruit class in its history (30 new Guards) in order to continue meeting an increasing demand signal for service.
Lt. Chelsea Kahalepauole-Bizik is promoted in December, becoming the service's first official female supervisor.
Lifeguard Luke Shepardson takes two hours of vacation time to surf two heats of the prestigious Eddie Aikau Invitational while assigned to Tower 29 at Waimea Bay, and wins in a storybook finish to a great day of surfing on Jan. 22, 2023.
Mayor Rick Blangiardi announces in an April "State of the City Address," that his administration will study whether or not to make Ocean Safety a separate county department.
The second class of Ocean Safety EMTs -- the first since 2005 -- graduates 24 from the state's Emergency Medical Technician school at Kapiʻolani Community College.
Beginning on the South Shore in July, and wrapping up on the North Shore by Labor Day, all towers move to a 10-hour day from 8 a.m. to 6:30 p.m. -- marking arguably the biggest operational change in the service in decades.
A new tower is installed in August at Kahe Point, marking the first time the county added a tower since the late 1990s.
Ocean Safety stages its first ever Fall Junior Lifeguard Program during the Department of Education's Oct. 9-13 break, conducting very popular 1-day clinics at five different beach parks, reaching more than 600 keiki.
An "Ocean Safety Department Task Force" is poised to deliver a recommendation in January to Mayor Blangiardi on whether or not the service should be a separate standalone public safety agency -- like Police, Fire, and EMS -- in the City & County of Honolulu.
A special Mahalo to retired Ocean Safety Chief Ralph "Buster" Wallwork, and former Chief John Silberstein, for their work on this brief history.
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|
en
|
Odisha: Foreign marine safety experts provide training to Puri lifeguards
|
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[
"Debabrata Mohapatra"
] |
2023-10-08T23:08:00+05:30
|
Jonathan Robinson and Nick Schriver (both from California), Richard Bedford from Australia and Emilio Cavazos from Mexico reached Odisha last week fol
|
en
|
The Times of India
|
https://timesofindia.indiatimes.com/city/bhubaneswar/odisha-foreign-marine-safety-experts-provide-training-to-puri-lifeguards/articleshow/104265848.cms
|
Debabrata Mohapatra is an Assistant Editor at The Times of India, Bhubaneswar. He had been writing for TOI from Puri since 2006 before joining the Bhubaneswar bureau in August 2010. He covers crime, law & order and Congress.
|
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https://www.mlive.com/news/grand-rapids/2013/10/making_waves_lifeguard_trainin.html
|
en
|
Making waves: Lifeguard training in Lake Michigan draws students from across U.S.
|
https://www.mlive.com/pf/resources/images/mlive/favicon.ico?d=1381
|
https://www.mlive.com/pf/resources/images/mlive/favicon.ico?d=1381
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[
"Andrew Krietz | akrietz@mlive.com",
"Andrew Krietz",
"akrietz@mlive.com"
] |
2013-10-08T14:15:05+00:00
|
Adam DeBoer practically grew up in the water, spending most of his summer weekends in Lake Michigan and eventually becoming a swimmer during college.
|
en
|
/pf/resources/images/mlive/favicon.ico?d=1381
|
mlive
|
https://www.mlive.com/news/grand-rapids/2013/10/making_waves_lifeguard_trainin.html
|
Lifeguards are instructed in water rescue technique in Lake Michigan at Camp Geneva. ( Mark Copier | MLive.com )
HOLLAND, MI — Adam DeBoer practically grew up in the water, spending most of his summer weekends in Lake Michigan.
The former college swimmer now is a police officer who loves plying his sport in the big lake. He understands it's a powerful force whose conditions can turn dangerous without warning.
Still, his recent jump into 50-some degree water still brought a shock, all in the name of safety.
“The water was cold. That was the worst part," laughed DeBoer, one of 17 people who braved Lake Michigan's chill last weekend to earn surf lifeguard certification for open waters — a designation that includes the big lake.
The three-day event drew participants from across the country. But it was the high number of drownings in the Great Lakes region — and the desire to return lifeguards to beaches here — that was the impetus for holding the training in West Michigan.
Dave Benjamin, executive director of the Great Lakes Surf Rescue Project, and event organizers have made it their mission to institute a region-wide standard of lifeguarding on the Great Lakes.
The organization, with assistance from Bob Pratt, its director of education, reached out to the International Surf Lifesaving Association to set up the three-day certification course at Holland’s Camp Geneva.
Related: Surf lifeguard certification training begins Friday to help protect swimmers in rough waters
Typically, pool lifeguard certification isn’t enough to prepare someone for working lakeside, Benjamin said. A pool doesn’t have rough waters, fluctuating temperatures and a changing lakebed we’re accustomed to, thanks to our natural lakes, he said.
When the surf challenge kicked off last Friday, everyone understood it wasn’t going to be a typical day at the beach.
"We send half of the group (into the lake) and say, basically, 'Drown!' — and the other group, 'OK, go get them,'" said William Koon, vice president of operations for ISLA.
“(We’re going) to give people the tools they need to prevent a situation … empowering them to go beyond our training.”
Changing tides
At least 60 people have drowned in the Great Lakes so far this year, and at least 322 people have lost their lives in water-related incidents in the region since 2010, according to the Great Lakes Surf Rescue Project.
“Those statistics blew my mind,” said Koon, who grew up and lifeguards in the area of the iconic Huntington Beach, Calif. “If there’s more than a couple drownings a year (in that area of California), people freak out.
“It’s a big deal.”
Across West Michigan, Koon said many people seem to react with apathy to news of another Great Lakes drowning.
Benjamin agrees and wants to change that attitude. Events like last weekend's are part of that plan.
Bootcamp training
Day one of training started promptly at 8 a.m. with a 500-meter swim to two buoys and back.
“It taxed me; it really pushed me to my limits,” Bruce Macartney, a teacher at Forest Hills Eastern High School, said of the three-day event. “It was very humbling, but I feel very grateful that I made it out alive.”
Training was comprised of at least five physical activities in the lake, coupled with hours of classroom lecture. Certification was earned only after passing an instructor-led mock drowning with a final exam.
One of the first lessons involved the very basics of the water: Waves, rip currents and the conditions that create them.
The course’s students — even its six instructors — traveled on their own dime to attend. Some hailed from Connecticut, Florida, Minnesota and North Carolina.
However, not all are lifeguards.
Macartney teaches a class called "Gone Boarding" and has his students design and build anything from surfboards to snowboards and more.
Although students are tested in the water before going out to test their boards, the program for him is meant to provide an extra layer of security, Macartney said.
DeBoer grew up in Holland but now works for the South Haven Police Department. He said despite having a popular summer beach, the department’s staff runs the gamut in swimming ability.
“You have some guys who say they won’t go in the water because they don’t like it. They’re not comfortable, they can’t swim,” DeBoer said.
The city has throw rings on its piers and uses a flag system to warn beachgoers about any potential dangers, DeBoer said.
Lifeguards haven’t been on the city’s beaches in at a decade, largely thanks to budget cuts. It's a story that has played out across the region for years, Benjamin said.
Related: Should public beaches along the Great Lakes in Michigan have lifeguards?
Guarding the lakes
The course is a step forward for organizers with Great Lakes Surf Rescue Project’s goal to conduct additional, specialized training for those wanting to lifeguard in the Great Lakes.
Ideally, Benjamin and his team of volunteers want to train the next generation of lifeguards and put them back on the beaches where they’re needed, he said.
The weekend of military-like bootcamp training resulted in all of its participants earning certificates, Benjamin said. ISLA organizers say a handful of students typically don’t finish the training.
“I’m ecstatic. It's amazing that we could have this high of a completion level with such a wide range of people attending the class, even guys in their 50s,” Benjamin said.
During the freezing winter of 2010, Benjamin had a near-drowning experience while surfing along the Northern Indiana shoreline. The waves knocked him to the lakebed and pushed him closer to some jagged, icy rocks.
Benjamin panicked, but forced himself to stop and think. He knew what to do — stay calm and relax before floating to shore in his wetsuit.
More people need the judgement to be safe out in the open waters, and lifeguards will be at the front lines, Benjamin said.
“Doing all of this is pretty simple: It’s to help prevent accidents and educate ourselves to help us serve more people when they’re in need,” Benjamin said.
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https://en.wikipedia.org/wiki/International_Surf_Lifesaving_Association
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en
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International Surf Lifesaving Association
|
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https://en.wikipedia.org/static/favicon/wikipedia.ico
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2013-10-16T01:25:51+00:00
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/static/apple-touch/wikipedia.png
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https://en.wikipedia.org/wiki/International_Surf_Lifesaving_Association
|
The International Surf Lifesaving Association (ISLA) is a nonprofit organization that advances professional lifesaving development to areas in need around the globe. ISLA uses latest technology to identify areas in need. ISLA advances development through lifeguard training programs and aquatic rescue operation consults, lifeguard exchanges, equipment donations, and by integrating with drowning prevention organizations to share information, techniques, stories, and culture.[1]
Mission statement
[edit]
The International Surf Lifesaving Association (ISLA) exists to advance professional lifesaving development to areas in need around the globe.[2]
History
[edit]
Four Huntington Beach Junior Lifeguard Instructors; Peter Eich, Scott Hunthausen, Olin Patterson, and Henry Reyes started ISLA in 2008. ISLA was formed after Scott returned home from a study abroad semester in Nicaragua where he experienced the drowning of his host family's son friend, and witnessed the alarming drowning rates where over 60 people drowned during the 4-day Semana Santa holiday (Easter).[3]
Organization structure
[edit]
ISLA has a two two-tier system of leadership, there are two separate boards, an Executive Board for day-to-day business and a Board of Directors for supervising the President. The ISLA President is hired by the Board of Directors and presides over the Executive Board and the ISLA Chairman is elected by the Board of Directors and presides over the Board of Directors, currently the roles of President and Chairman is held by one individual. The Executive Board is currently made up of the ISLA President, VP of Sales, VP of Operations, and the VP of Marketing. The ISLA Vice Presidents oversee International Ambassadors, management staff (also known as Marine Safety Officers MSO's), and volunteers.[4]
Humanitarian projects
[edit]
2008: Transcontinental Bike Ride[5]
2009: Nicaragua
2010: Nicaragua, New Zealand
2011: Ecuador, Nicaragua,[6] Dominican Republic[7]
2012: Nicaragua,[8] Dominican Republic(Spring),[9] Mexico(Summer), Dominican Republic(Fall), Mexico(Fall), New Zealand, Peru,[10] Chile[11]
2013: Dominican Republic, Nicaragua, Greece, Macedonia, United States,[12] Mexico(Fall), England, Germany
Global Drowning Tracker©
[edit]
GDT Version 1.0
[edit]
In 2013 at the National Drowning Prevention Symposium in Fort Lauderdale, Florida ISLA launched the world's first Global Drowning Tracker© (www.drowningtracker.com). Although the software is just a working prototype, The Global Drowning Tracker© allows people around the world to input statistics on drownings, and promotes awareness by utilizing social media. The end result is a tool that enables researchers, lifesavers and drowning prevention experts a real-time snapshot of where resources need to be allocated to prevent drownings.[13] Global Drowning Tracker© V.1.0 was presented to a Congressional Committee in Washington D.C. as part of International Water Safety Day on May 15, 2013.[14]
GDT Version 2.0
[edit]
Currently the International Surf Lifesaving Association is seeking sponsors in the international drowning prevention community to co-develop the next version of the Global Drowning Tracker© software.[15] GDT Version 2.0 proposed features include:
Bulk data import/export
Report multiple victims in one incident
Enhanced administrative interface and controls
Dynamic IP language translation for top 20 languages
Real-time interactive map
Improved SMS Text Messaging incident reporting
ICD code and Utstein integration for scientific reporting
Dynamically adjusting screen resolution for a full range of devices
Certification
[edit]
Overview
[edit]
The ISLA Lifeguard Certification Program was developed to raise international lifeguarding standards and training programs to a minimum professional standard. There are two types of ISLA Open Water Lifeguard Certifications; Basic, and Advanced.[16]
Basic Open Water Lifeguard
[edit]
Course Length: 3 days / 30 hours.
Course description
[edit]
The ISLA Basic Open Water Lifeguard Course is designed for people with little or no background in open water lifeguarding. It provides an introductory exposure to both the theoretical and practical components of open water lifeguard subjects such as lifeguard operations, aquatic injury prevention, and basic open water rescue. This course does not include a certification in CPR or First Aid.[17]
Course objectives
[edit]
At the end of this course, the trainee should have the ability to:
Prevent an accident based on knowledge of physical and social conditions.
Recognize dangerous aquatic conditions and hazardous areas.
Identify a victim in distress.
Discern appropriate responses to a variety of different situations taking into account personal ability and scene safety.
Effectively execute a basic rescue in open water.
Course prerequisites
[edit]
Any student wishing to participate in the ISLA Basic Open Water Lifeguard course must meet the following requirements:
16 Years of age or older (with minors consent form for those under 18)
Ability to swim 500 meters (550 yards) without stopping
Ability to tread water for five minutes
Ability to dive to a depth of three meters (Ten Feet)
Advanced Open Water Lifeguard
[edit]
Course Length: 10 days/ 100 hours
The advanced course explores all the subjects in the Basic Open Water Lifeguard Course more in depth, along with the inclusion of a First Aid and CPR certification.
References
[edit]
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https://www.thepress.co.nz/nz-news/350367884/schools-scrambling-relief-teachers-shortage-slams-canterbury
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The Press
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http://www.worldpaddleawards.com/organisation/international-lifesaving-federation-ils
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International Lifesaving Federation (ILS)
|
http://www.worldpaddleawards.com/files/misc/ils.jpg
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The International Lifesaving Federation (ILS) is the world authority for drowning prevention and lifesaving sport.
Operating its Headquarters out of Leuven in Belgium, the Secretary General of the ILS is Dr. Harald Vervaecke.
“The ILS leads, supports and collaborates with national and international organisations engaged in drowning prevention to ensure best practices in water safety, water rescue, lifesaving, lifeguarding and lifesaving sport.”
The ILS decentralises its affairs under the management of four regional entities in Africa, the Americas, Asia-Pacific and Europe.
With regards to lifesaving sport, the ILS is the governing body responsible for event sanctioning - every two years it organises the Lifesaving World Championships called the Rescue Series, with further continental championships.
Lifesaving sport is also part of the 2017 World Games and World Masters Games. Additionally it is recognised by the International Military Sports Council (CISM) with events featuring on the Military World Championships.
At the International Olympic Committee (IOC) Session in Atlanta in 1996, the ILS was recognised as a full-voting Association of the IOC Recognised International Sports Federations (ARISF).
International lifesaving activities date back to 1898 when the first World Congress was held in Marseille, France. As a result the Fédération Internationale de Sauvetage Aquatique (FIS) and World Life Saving (WLS) were set up, eventually merging in 1993 and in turn to become the International Lifesaving Federation as it is known today.
Website: www.ilsf.org
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https://www.guidestar.org/profile/26-3679133
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International Surf Lifesaving Association
|
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Find and check a charity using Candid's GuideStar. Look up 501(c)(3) status, search 990s, create nonprofit organizations lists, and verify nonprofit information.
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en
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https://cdn.candid.org/favicon.ico
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https://www.guidestar.org/profile/26-3679133
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Want more data?
A GuideStar Pro report containing the following information is available for this organization:
Download it now for $125.
Need the ability to download nonprofit data and more advanced search options?
Consider a Pro Search subscription.
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https://www.surflifesaving.org.nz/about-us/what-we-do
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en
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What We Do
|
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https://www.surflifesaving.org.nz/about-us/what-we-do
|
SAVING LIVES SINCE 1910
Surf Life Saving New Zealand (SLSNZ) is the leading beach and coastal safety, drowning prevention and rescue authority in Aotearoa. We are truly unique, delivering proactive lifeguarding and essential emergency rescue services, a range of public education beach safety programmes, member education, training and development, as well as a highly respected sport.
We do all this as a charity and rely on the generosity of the public, commercial partners, foundations and trusts for donations and financial contributions in order to lead and support our incredible front-line volunteer lifeguarding services. SLSNZ is the national association representing 74 surf lifesaving clubs with 18,000+ members, including more than 4,500 volunteer Surf Lifeguards. Our lifeguards patrol over 80 locations each summer and provide emergency call-out rescue services throughout Aotearoa, saving hundreds of lives each year and ensuring thousands return home safe after a day at the beach.
Even though our volunteer Surf Lifeguards have kept thousands of people safe on our beaches and saved thousands of lives, New Zealand’s beach and coastal fatal drowning rates have increased over the last 5 years compared to the previous 5 years. We are committed to changing this. Our vision is "no-one drowns on our beaches". It's what drives us and why we are "In It For Life".
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https://dofehillary.org.nz/partner/surf-lifesaving/
|
en
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Surf Lifesaving – The Duke of Edinburgh's International Award
|
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2022-10-02T08:49:27+00:00
|
Surf Life Saving New Zealand is the leading beach and coastal safety, drowning prevention and rescue authority in Aotearoa. They deliver proactive lifeguarding and essential emergency rescue services, a range of public education beach safety programmes, member education, training and development, as well as a highly respected sport. They do all this as a charity and rely on the generosity of the public, commercial partners, foundations and trusts for donations and financial contributions in order to lead and support our incredible front-line volunteer lifeguarding services. SLSNZ is the national association representing 74 surf lifesaving clubs with 18,000+ members, including more than 4,500 volunteer Surf Lifeguards. SLSNZ lifeguards patrol over 80 locations each summer and provide emergency call-out rescue services throughout Aotearoa, saving hundreds of lives each year and ensuring thousands return home safe after a day at the beach.
|
en
|
The Duke of Edinburgh's International Award
|
https://dofehillary.org.nz/partner/surf-lifesaving/
|
If you are a Surf Lifesaver, your mahi may count towards your Award. Surf Lifesaving have active Award Units in Bethells Beach and Mairangi Bay, so if you live in those area you may be able to connect with a Surf Lifesaving Award Leader. You can only claim Recognition of Prior Learning (RPA) if you are doing your Award through a Surf Lifesaving Award Unit.
If you are a Surf Lifesaver at another location, you can still use your lifesaving activities towards your Award. You will need to provide your Award Leader with information on how your activities meet criteria for your Award sections and find an adult at Surf Lifesaving to be your Assesor.
Note that the requirements of the Award must be met, so you may need additional activity time to complete your Award. This could be in the activity you are already doing and/or another activity.
Below is a list of many of the activities that fit with each of the Award sections. You can use other suitable activities, however these are the ones that are an integral part of Surf Life Saving.
Service: Your time on patrol as a lifeguard counts towards this, along with any of the other volunteer activity that you do around your club, school, sport or community. Coach or manage a sports team, fundraise for a charity, volunteer at the SPCA, become a leader at a youth club.
Physical: All of your surf sport training and competition activities contribute to this, as well as any other sports you may do such as soccer or netball, athletics, skiing, kayaking, kickboxing, horse riding, running, dancing
Skills: All the surf lifeguard awards and qualifications you do can count towards this, along with skills you develop in other areas eg: play a musical instrument, learn a craft such as jewellery making, referee or umpire for a sport, learn sign language, drama and theatre skills
Adventurous Journey or Exploration: This is something that you will plan for, it involves being part of a team and completing an expedition for a set number of days. Journeys do not have to be done on foot, they can also be done on horseback, in a boat, on a bicycle – in fact anything without an engine!
|
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https://www.lwc2024.com/info/ambassadors-and-committee/
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en
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AMBASSADORS AND COMMITTEE – LWC 2024
|
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https://www.lwc2024.com/info/ambassadors-and-committee/
|
On behalf of the Local Organising Committee and Surf Life Saving Queensland, it is my pleasure to invite all ILS Member Nations to participate in the Lifesaving World Championships 2024 on Australia’s Gold Coast. This is the third Lifesaving World Championships I have had the honour of chairing and with the support of everyone involved, it is my goal to deliver the most inclusive, inspiring and best ever attended Lifesaving World Championships for your enjoyment.
The support offered by the Queensland Government, the City of Gold Coast and our important sponsors, will help make the Lifesaving World Championships 2024 a very memorable experience for our international visitors and all Australians who participate. The beauty and vast range of experiences available on the Gold Coast will make the visit very memorable for competitors, officials, their families and friends from around the World.
Ocean and beach events will be conducted on the pristine beach and parklands at Kurrawa, Broadbeach while all pool competition will be held in the city’s first-class Aquatic Centre at Southport. All meetings, functions and assemblies will be conducted in the Broadbeach precinct close to the beach and surrounded by a range of excellent accommodation and hospitality venues.
If you want to see the World’s best surf lifesavers in action including lifesavers from at least 50 nations and more than 7,000 competitors and officials, we invite you to come along. The Lifesaving World Championships have not been held in Queensland since 1988 and Surf Life Saving Queensland is doing everything to plan and deliver a world class event.
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http://www.coab.us/29/Ocean-RescueLifeguards
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The Atlantic Beach Official Website!
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http://www.coab.us/images/favicon.ico
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Atlantic Beach Ocean Rescue
The Atlantic Beach Ocean Rescue organization operates under an Advanced Certification from the United States Lifesaving Association. Members are certified in emergency medical services, and they are capable of providing advanced levels of first aid in emergencies.
The staffing of lifeguards on the beach is seasonal, but the core leadership of the organization is comprised of long term, dedicated individuals. Selection and training are high priorities, especially during the spring months. During the summer, lifeguards staff and patrol the beach on a daily basis, providing guidance and direction relative to hazardous conditions, marine animals, and first aid.
The Atlantic Beach Ocean Rescue organization participates in training and professional competitions with other lifeguard organizations throughout the region, the state, the country, and the world.
The hours for the Atlantic Beach Lifeguards are Monday - Friday from 10 a.m. - 5 p.m., Saturday - Sunday from 10 a.m. to 5:30 p.m.
Lifeguards full time beach sitting starts on Memorial weekend and ends Labor Day weekend.
2024 Lifeguard Event Calendar
2024 Ocean Lifeguard General Information and Application
2024 Atlantic Beach Ocean Rescue Lifeguards - New Hire Introduction
Jr. Lifeguard Information:
PROGRAM SESSION DATES
Session 1- June 17 through June 21 (Age: 9-11) From 9:00am-1:00pm
Session 2- June 24 through June 28 (Age: 12-15) From 9:00am-1:00pm
2024 Atlantic Beach Jr Life Guard Brochure
2024 Application-Release Form
Click the below link to access the public link for lightning detection for the city
Thorguard (Lightning Warning Detection)
Atlantic Beach Lifeguard Station
One Ahern Street
Atlantic Beach, FL 32233
Phone: 904-247-5883
Emergencies: 911
Email Atlantic Beach Lifeguard Station
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https://patch.com/california/los-angeles/kerlan-jobe-partners-california-surf-lifesaving-association
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Kerlan-Jobe Partners with California Surf Lifesaving Association
|
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""
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[
"Trace Longo"
] |
2022-06-21T18:22:33+00:00
|
Kerlan-Jobe Partners with California Surf Lifesaving Association - Los Angeles, CA - Cedars-Sinai Kerlan-Jobe Institute is Official Healthcare Sponsor of the 2022 West Coast Lifeguard and Junior Lifeguard Championships
|
en
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https://cdn.patchcdn.com/assets/layout/icons/logo/favicon.ico
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Los Angeles, CA Patch
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https://patch.com/california/los-angeles/kerlan-jobe-partners-california-surf-lifesaving-association
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Cedars-Sinai Kerlan-Jobe Institute is Official Healthcare Sponsor of the 2022 West Coast Lifeguard and Junior Lifeguard Championships
LOS ANGELES – Cedars-Sinai Kerlan-Jobe Institute, a world leader in the diagnosis, treatment and prevention of sports-related injuries and illnesses, today announced a new partnership with the California Surf Lifesaving Association (CSLSA) as the Official Healthcare Sponsor of the 2022 West Coast Lifeguard and Junior Lifeguard Championships on July 22nd and 23rd.
The California Surf Lifesaving Association is a non-profit (501-C3 tax exempt) organization chartered to promote beach safety awareness and professional open-water lifesaving standards. The 30 CSLSA Chapters/Agencies share the mission, goals and objectives of the CSLSA. The CSLSA (West Coast Region) is one of nine regions of the United States Lifesaving Association, which is affiliated with the International Life Saving Federation.
“We are excited to partner with the California Surf Lifesaving Association because of its commitment to increasing awareness about open-water beach safety and our shared focus on injury prevention,” said Daniel Kharrazi, M.D., sports medicine specialist and orthopedic surgeon, at Cedars-Sinai Kerlan-Jobe Institute. “As the proud and preferred provider of Southern California’s courageous first responders, our extensive experience in the fields of sports medicine, outpatient surgery, and rehabilitation will help keep lifeguards healthy and, on the beach, so they can continue to do what they do best – protect and save lives.”
As open-water lifesavers, CSLSA promotes beach safety awareness and professional lifeguard standards through public education, training programs, exchange programs, junior lifeguard programs, competition and other means. The ultimate goal is to prevent and reduce aquatic injuries, accidents and death at open-water beaches in the United States and throughout the world.
Bill Humphreys, President of the California Surf Lifesaving Association, stated: “As first responders in a challenging and often dangerous environment, lifeguards must remain in top physical condition as they serve the public. Injuries to lifeguards can be minimized but cannot be prevented entirely. When injuries to lifeguards do occur, top-notch care provided by Cedars-Sinai Kerlan-Jobe Institute is needed in order to restore the health and functions of the lifeguard. Because of its outstanding reputation and history of serving first responders, the California Surf Lifesaving Association is very pleased to enter into a partnership with Cedars-Sinai Kerlan-Jobe Institute as the Official Healthcare Sponsor. This is a partnership which will directly benefit and serve both lifeguards and the public for many years to come.”
For more information about the Cedars-Sinai Kerlan-Jobe Institute, please visit www.kerlanjobe.org. For more information about the California Surf Lifesaving Association, please visit www.cslsa.org.
Cedars-Sinai Kerlan-Jobe Institute
Cedars-Sinai Kerlan-Jobe Institute is a world leader in the diagnosis, treatment and prevention of sports-related injuries and illnesses. As the sports division of the Number 3 nationally ranked Cedars-Sinai Department of Orthopaedics, institute physicians provide comprehensive care for a broad range of adult and pediatric orthopaedic conditions. Patients benefit from world-renowned experts in multi-specialty orthopaedics and the physicians who take care of professional athletes are the same ones who treat you. For more information, or make an appointment, visit www.kerlanjobe.org.
California Surf Lifesaving Association
The California Surf Lifesaving Association is a non-profit (501-C3 tax exempt) organization chartered to promote beach safety awareness and professional open-water lifesaving standards. The 30 CSLSA Chapters/Agencies share the mission, goals and objectives of the CSLSA. The CSLSA (west coast region) is one of nine regions of the United States Lifesaving Association (USLA) which in turn is affiliated with the International Life Saving Federation (ILS). As open-water lifesavers, our MISSION is to promote Beach Safety awareness and Professional Lifeguard standards through public education, training programs, exchange programs, junior lifeguard programs, competition and other means. The ultimate goal is to prevent and reduce aquatic injuries, accidents and death at open-water beaches in the United States and throughout the world. Our mission is to promote beach safety awareness and professional open-water lifesaving standards. Our members include lifeguards, junior lifeguards, and open water rescue professionals from California (Santa Cruz south to the Mexican border) and Arizona. We accomplish our mission through public education, junior lifeguard programs, training programs, exchange programs, competition and other means. The CSLSA works to prevent and reduce aquatic injuries, accidents and fatalities at open-water beaches.
|
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Instagram
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3
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https://www.academia.edu/10495837/On_Proofs_and_Types_in_Second_Order_Logic
|
en
|
On Proofs and Types in Second Order Logic
|
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[] |
[] |
[
""
] | null |
[
"Paolo Pistone",
"univ-amu.academia.edu"
] |
2015-02-03T00:00:00
|
In my dissertation I address some questions concerning the proof theory of second order logic and its constructive counterpart, System F (Girard 1971). These investigations follow two distinct (though historically related) viewpoints in proof theory,
|
https://www.academia.edu/10495837/On_Proofs_and_Types_in_Second_Order_Logic
|
In my dissertation I address some questions concerning the proof theory of second order logic and its constructive counterpart, System F (Girard 1971). These investigations follow two distinct (though historically related) viewpoints in proof theory, which are compared throughout the text: on the one side, the proof theoretic semantics tradition inaugurated by Dummett and Prawitz (Prawitz 1971, Dummett 1991), focusing on the analysis of the inferential content of proofs; on the other side, the interactionist tradition arising from Kleene's realizability (Kleene 1945) and the Tait/Girard reducibility technique (Tait 1967, Girard 1971), which interprets proofs as untyped programs and focuses, rather, on the behavioral content of proofs, i.e. the way in which they interact through the cut-elimination algorithm. A distinction is made between the issues of justifying and understanding ("explaining why" and "explaining how", as in Girard 2000) impredicative reasoning, i.e. between non elementary results like the Hauptsatz and the combinatorial analysis of proofs, seen as programs, i.e. recursive objects. As for justifi cation, an epistemological analysis of the circularity involved in the second order Hauptsatz is developed; it is shown that the usual normalization arguments for second order logic do not run into the vicious circularity claimed by Poincaré and Russell, but involve a diff erent, epistemic, form of circularity. Still, this weaker circularity makes justifi cation, in a sense, pointless; in particular, some examples of inconsistent higher order theories admitting epistemically circular normalization arguments are discussed. As for the explanation issue, a constructive and combinatorial (i.e. independent from normalization) analysis of higher order order quantifi cation is developed along two directions, with some related technical results. The fi rst direction arises from the parametric and dinatural interpretations of polymorphism (Reynolds 1983, Girard-Scott-Scedrov 1992), which provide a clear mathematical meaning to Carnap's defense of impredicative quanti fication (Carnap 1983). In particular, the violation of the parametric condition leads to paradoxes which are often ignored in the philosophical literature (with the exception of Longo-Fruchart 1997). The analysis of the combinatorial content of these interpretations leads to a 1-completeness theorem (every normal closed -term in the universal closure of a simple type is typable in simple type theory), which connects the interactionist and the inferential conceptions of proof. The second direction follows the analysis of the typing conditions of the -terms associated with intuitionistic second order proofs. To the \vicious circles" in the proofs there correspond recursive (i.e. circular) speci fications for the types of the -terms. The geometrical structure of these vicious circles is investigated (following Lechenadec 1989, Malecki 1990, Giannini - Ronchi Della Rocca 1991), leading to a combinatorial characterization of typability in some inconsistent extension of System F: since, as Girard's paradox shows, a typable term need not be normalizing, one is indeed naturally led to consider not normalizing theories. Such investigations go in the direction both of a mathematical understanding of the structure generated by the vicious circles of impredicative theories and of the development of a proof-theoretic analysis of potentially incorrect or uncertain proofs.
|
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1
| 19
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https://www.cambridge.org/core/journals/philosophy-of-science/article/abs/connecting-the-revolutionary-with-the-conventional-rethinking-the-differences-between-the-works-of-brouwer-heyting-and-weyl/9376F606893AEDB9AD887548C59D6099
|
en
|
Connecting the Revolutionary with the Conventional: Rethinking the Differences between the Works of Brouwer, Heyting, and Weyl
|
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[
""
] | null |
[
"Kati Kish Bar-On"
] | null |
Connecting the Revolutionary with the Conventional: Rethinking the Differences between the Works of Brouwer, Heyting, and Weyl - Volume 90 Issue 3
|
en
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/core/cambridge-core/public/images/favicon.ico
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Cambridge Core
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https://www.cambridge.org/core/journals/philosophy-of-science/article/abs/connecting-the-revolutionary-with-the-conventional-rethinking-the-differences-between-the-works-of-brouwer-heyting-and-weyl/9376F606893AEDB9AD887548C59D6099
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7589
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3
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https://archive.org/stream/StudiesInLogicAndTheFoundationsOfMathematics121AnneS.TroelstraDirkVanDalenConstr/%2528Studies%2520in%2520Logic%2520and%2520the%2520Foundations%2520of%2520Mathematics%2520121%2529%2520Anne%2520S.%2520Troelstra%252C%2520Dirk%2520van%2520Dalen-Constructivism%2520in%2520mathematics_%2520An%2520introduction.%2520Volume%25201-N_djvu.txt
|
en
|
( Studies In Logic And The Foundations Of Mathematics 121) Anne S. Troelstra, Dirk Van Dalen Constructivism In Mathematics An Introduction. Volume 1 N : Free Download, Borrow, and Streaming : Internet
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https://archive.org/services/img/StudiesInLogicAndTheFoundationsOfMathematics121AnneS.TroelstraDirkVanDalenConstr
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https://archive.org/services/img/StudiesInLogicAndTheFoundationsOfMathematics121AnneS.TroelstraDirkVanDalenConstr
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(Studies in Logic and the Foundations of Mathematics 121) Anne S. Troelstra, Dirk van Dalen-Constructivism in mathematics_ An introduction. Volume 1-N
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en
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Internet Archive
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https://archive.org/details/StudiesInLogicAndTheFoundationsOfMathematics121AnneS.TroelstraDirkVanDalenConstr
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Search the history of over 866 billion web pages on the Internet.
Search the Wayback Machine
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https://link.springer.com/article/10.1007/s10670-021-00397-7
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On the Costs of Classical Logic
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2021-06-03T00:00:00
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This article compares classical (or KF-like) and nonclassical (or PKF-like) axiomatisations of the fixed-point semantics developed by Kripke (J Philos 72(1
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en
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SpringerLink
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https://link.springer.com/article/10.1007/s10670-021-00397-7
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The use of nonclassical logics is extensive in the development of formal theories of truth. Probably the most popular semantic theory of truth formulated in nonclassical logic is Kripke’s fixed-point semantics Kripke (1975). This semantic conception of truth lends itself to being axiomatised in classical logic as well as in noncalssical logic. Classical axiomatisations are variants of the theory known as Kripke-Feferman (KF); nonclassical axiomatisations are variants of the nonclassical theory known as Partial-Kripke-Feferman (PKF).Footnote 1
KF and PKF (and variants thereof) have been often compared in the literature, and they have both been defended and criticised on different grounds. Defenders of KF (e.g. Halbach 2014; Halbach and Nicolai 2018) point out that KF is mathematically much stronger than PKF—KF proves more arithmetical statements than PKF—and proof-theoretic strength is often taken to be a virtue of a theory of truth. Thanks to recent work by Halbach and Nicolai, we have now a better grasp on the source of this asymmetry in the proof-theoretic strengths of KF and PKF: Halbach and Nicolai (2018) have crucially shown that, once the schema of induction is restricted to the arithmetical vocabulary, KF and PKF have the same truth-free consequences and they prove the same sentences to be true. This indicates clearly that the deductive weakness of PKF does not arise from a lack of truth-theoretic resources that are on the contrary available to KF. It is the use of nonclassical logic that, paralysing inductive reasoning on sentences containing the truth predicate \(\mathrm {Tr}\), has profound consequences on the arithmetical statements derivable in the system.Footnote 2
Defenders of PKF (e.g. Field 2008; Horsten 2009), on the other hand, argue that PKF is more adequate from a conceptual point of view, notwithstanding its mathematical weakness.Footnote 3 Hartry Field, for example, argues that PKF-like theories are far superior than KF-like theories. One of the main advantages of PKF is that it exploits what (Field 2008, p. 69) calls the “raison d’être of fixed points”, i.e., the complete equivalence between \(\varphi \) and \(\mathrm {Tr} (\varphi )\), for any sentence, in any context. It is known, in fact, that the truth predicate of PKF is fully transparent, which means that it is always possible to infer \(\mathrm {Tr} (\varphi )\) from \(\varphi \), and vice versa. By contrast, a problem often associated with KF is its asymmetry, that is the fact that there are sentences \(\varphi \) such that \(\varphi \) is derivable, but \(\mathrm {Tr} (\varphi )\) is not.Footnote 4
In this paper, we will also compare KF and PKF. However, the line of investigation will be different. We will not rely on the strength on KF to dismiss the use of nonclassical logic, nor shall we rely on the asymmetry of KF to argue that PKF is conceptually more adequate. We will not rely on the transparency of \(\mathrm {Tr}\) in PKF to discard the use of classical logic, nor shall we emphasise that in PKF “nothing like sustained ordinary reasoning can be carried on” (Feferman 1984, p. 95). The reason is that we are not going to compare KF and PKF qua theories of truth simpliciter. And we are not going to ask whether one theory is more adequate than the other, in absolute terms. What we would like to do, instead, is to compare KF and PKF qua axiomatisations of the fixed-point semantics. We would like to ask whether one theory is more adequate than the other as axiomatisation of the Kripkean conception of truth.
The interest in this analysis has been triggered by the already mentioned paper by Halbach and Nicolai—titled “On the costs of nonclassical logic”—where they also compare KF and PKF qua axiomatisations of the fixed-point semantics. Halbach and Nicolai (2018) warn that the costs of abandoning classical logic might go beyond the initial expectations of the defender of PKF. The crucial claims defended by Halbach and Nicolai are that
the use of nonclassical logic of PKF does not affect the truth-theoretic content [and] the [Kripkean] conception of truth [...] is captured equally well by the nonclassical theory PKF and by the classical theory KF. (Halbach and Nicolai 2018, p. 228; p. 241)
Clearly, the proof-theoretic weakness of PKF, in itself, is not enough to justify the rejection of nonclassical logic in axiomatisations of the fixed-point semantics. It could for example be the case that the Kripkean conception of truth can be adequately axiomatised only within a nonclassical system, in which case we would need to use nonclassical logic, despite its costs. But if – as claimed by Halbach and Nicolai (2018)—KF captures the Kripkean conception of truth as adequately as PKF, then this would nourish philosophical doubts vis-à-vis the use of nonclassical logic: Why would we cripple our ordinary reasoning if we can axiomatise the target conception of truth using classical logic? If, however, one can show that PKF is more adequate than KF (let us reiterate: not as a truth theory simpliciter, but as axiomatisation of the fixed-point semantics), then one would have to reconsider whether employing nonclassical logics is or is not worth. That is why we would like to ask whether KF and PKF can be taken to capture the Kripkean conception of truth equally well and/or whether they can be said to have the same truth-theoretic content.
The answer emerging from some results presented in this article will be “it depends”. On the one hand, by strengthening the central results by Halbach and Nicolai (2018) and Nicolai (2018), it will be shown that there are compelling reasons for believing that some variants of KF and PKF are, truth-theoretically, on a par. On the other hand, it will also be shown that there are other variants of KF and PKF that cannot be said to capture the Kripkean conception of truth equally well, nor can they be taken to have the same truth-theoretic content. By inspecting these theories, it turns out that there is one specific aspect of the Kripkean conception of truth that can be captured only by the PKF-variants, but not by the corresponding KF-variants. The aspect we are referring to is that in fixed-point models, ‘being untrue’ and ‘being false’ (i.e., having a true negation) are coextensive properties. As it will become clear below, this is not an accidental property of the Kripkean conception of truth, but rather one of its constitutive features. It can be shown that while every variant of PKF does embody the coextensiveness of untruth and falsity, the same cannot be said for KF: within some of its variants, being untrue and being false are different properties. This discrepancy between proof theory and semantics, it will be argued, indicates that not only nonclassical logic, but also classical logic has its own costs, and that the lack of a transparent truth-predicate may well not be its highest.
1.1 Structure and Key Points of the Paper
What follows can be divided into two parts and a technical appendix. The first part comprises Sect. 2 and Sect. 3. In Sect. 2 we explain what is meant by ‘Kripkean conception of truth’, why Halbach and Nicolai (2018) maintain that this conception is embodied equally well by KF and PKF, and why they claim that KF and PKF have the same truth-theoretic content. This amounts to introducing two criteria: the first establishes when an axiomatic system can be said to capture a semantic construction; the second is a method for comparing the truth-theoretic content of two axiomatic systems. In §3, relying on two counterexamples (Example 6 and Example 8), we first point out that the criteria employed by Halbach and Nicolai (2018) are too coarse-grained, and we then suggest a way to refine them. We conclude the first part of the article by implementing the new, finer-grained criteria: We show that the central theorems of Halbach and Nicolai (2018) and Nicolai (2018) can be made stronger, thereby providing additional support to the claim that some variants of KF and PKF can be seen as being, truth-theoretically, equivalent. The second part, §4 and §5, is the reason behind the title of this article: It will be shown that two variants of KF cannot be said to capture the Kripkean conception of truth as adequately as their PKF-counterparts, which amounts to the price we have to pay for axiomatising the Kripkean semantics using classical logic.
1.2 Preliminaries and Notation
Formalism is kept to a minimum in the main body of the paper, and all technical details are contained in an appendix. However, the philosophical points suggested here rely on some original formal results, without which our claims about KF and PKF would lose both support and interest. Hence, some formalism will be used below, for which we now fix a modicum of notation and preliminaries, assuming the reader being familiar with Kripke (1975). As background theory of syntax we use Peano arithmetic, PA. \(\mathcal {L}_{\textsf {PA} }\) is the language of arithmetic, and \(\mathcal {L} _\mathrm {Tr} \) is \(\mathcal {L}_{\textsf {PA} } \cup \{\mathrm {Tr} \}\). We abuse notation and write \(\mathrm {Tr} (\varphi )\) instead of \(\mathrm {Tr} \ulcorner {\varphi }\urcorner \), where \(\ulcorner {\varphi }\urcorner \) is the numeral of the Gödel number of \(\varphi \). Given a theory \(\mathrm {Th}\), we let ‘\(\mathrm {Th} \vdash \varphi \)’ symbolise a definition of the \(\mathrm {Th}\)-derivability of \(\varphi \).
A four-valued model for \(\mathcal {L} _\mathrm {Tr} \) is a structure \(\langle \mathcal {N}, (E, A)\rangle \) such that \(\mathcal {N}\) is a model for \(\mathcal {L}_{\textsf {PA} }\) and (E, A) is a pair of subsets of \(|{\mathcal {N}}|\) (the support of \(\mathcal {N}\)) interpreting \(\mathrm {Tr}\). We also refer to structures \(\langle \mathcal {N}, (E, A)\rangle \) as FDE-structures. The standard model of PA is denoted by \(\mathbb {N}\). The relation expresses that \(\varphi \) is FDE-satisfied in the structure \(\langle \mathbb {N}, (E, A)\rangle \). Moreover, we say that an FDE-structure \(\langle \mathbb {N}, (E, A)\rangle \) is
a \(\textsf {K} \mathsf {3}\)-structure, if \(E \cap A = \varnothing \),
a LP-structure, if \(E \cup A = \omega \),
a \(\textsf {KS} \mathsf {3}\)-structure, if it is either \(\textsf {K} \mathsf {3}\) or LP,
a CL-structure, if \(A = \omega - E\).
\(\textsf {Sent} \) is the set of (codes of) sentences, and \(\textsf {NSent} :=\omega -\textsf {Sent} \). The Kripke Jump is a function \(\Phi :\wp (\omega )^2\longrightarrow \wp (\omega )^2\) on pairs of subsets of natural numbers defined as
A fixed-point of \(\Phi \) is a pair (E, A) such that \(\Phi (E, A)=(E, A)\). A fixed-point model for \(\mathcal {L} _\mathrm {Tr} \) is an FDE-structure \(\langle \mathbb {N}, (E, A)\rangle \) where \((E, A)=\Phi (E, A)\). A fixed-point model is
consistent, if it is a \(\textsf {K} \mathsf {3}\)-structure,
complete, if it is a LP-structure,
symmetric, if it is a \(\textsf {KS} \mathsf {3}\)-structure.
By KF and PKF we mean the truth-systems that admit both truth-value gluts (sentences which are both true and false) and truth-value gaps (sentences which are neither true nor false). We work with KF and PKF formulated over two-sided sequent calculi. \(\textsf {KF} _\textsf {S} \) and \(\textsf {PKF} _\textsf {S} \) are so called symmetric variants, i.e., they are theories excluding the simultaneous occurrence of gaps and gluts. \(\textsf {KF} _\textsf {S} \) can be obtained from KF by adding the axiom
and \(\textsf {PKF} _\textsf {S} \) can be obtained from PKF by adding the axiom
Both GoG and GG stand for “gaps or gluts”.
We also study variations on the induction schema.Footnote 5 In particular, given \(\mathrm {Th}\in \{\textsf {KF} , \textsf {KF} _\textsf {S} , \textsf {PKF} , \textsf {PKF} _\textsf {S} \}\), we let
\(\mathrm {Th}^- \) be the theory obtained from \(\mathrm {Th}\) by restricting the induction schema to arithmetical-formulae;
\(\mathrm {Th}^\mathsf{int }\) be the theory obtained from \(\mathrm {Th}^- \) by replacing the restricted version of induction with the schema of internal induction;
\(\mathrm {Th}^+\) be the theory obtained by extending \(\mathrm {Th}\) with the schema of transfinite induction up to any ordinal below \(\varepsilon _0\).Footnote 6
As mentioned in the introduction, we want to verify whether KF and PKF can be said to capture the Kripkean conception of truth equally well. Following Halbach and Nicolai (2018), by ‘Kripkean conception of truth’ we mean the nonclassical understanding of it, that is, we understand KF as a classical system that describes a compositional, self-applicable, transparent, nonclassical notion of truth (compare (Halbach and Nicolai 2018, Sect. 2)). Technically, this means that we will be looking at classes of fixed-point models in their nonclassical version, with truth-value gaps and/or gluts.Footnote 7\(^,\)Footnote 8
2.1 \(\varvec{\mathbb {N}}\)-categoricity
The claim that an axiomatic system captures a certain semantic construction is in need of clarification, especially if the logic of the formal system does not coincide with the logic of the semantic construction. A criterion commonly employed in literature (and in particular by Halbach and Nicolai (2018)), establishing when an axiomatic theory of truth can be said to capture a semantic theory, is the criterion known as ‘\(\mathbb {N}\)-categoricity’. The idea behind this criterion, introduced by Fischer et al. (2015), is that an axiomatic system S captures a semantic theory \(\mathfrak {S}\) if the standard models of S are exactly the standard models in \(\mathfrak {S}\), where a ‘standard model’ is a model expanding the standard model \(\mathbb {N}\) with an interpretation for \(\mathrm {Tr}\).
In order to make this idea rigorous and provide a formal definition, we restrict our attention to a specific kind of semantic theories of truth. Since we will be dealing only with classical and fixed-point models, we define a semantic theory of truth to be a class \(\mathfrak {S}\) of models \(\langle \mathcal {N}, (E, A)\rangle \) for \(\mathcal {L} _\mathrm {Tr} \), where \(\mathcal {N}\) interprets \(\mathcal {L}_{\textsf {PA} }\), and the pair (E, A) is an interpretation for \(\mathrm {Tr}\), where E (A) is called the (anti-)extension of \(\mathrm {Tr}\). An important observation about classical and fixed-point models is that A is definable via E. More precisely:
Remark 1
In classical models, the anti-extension A is definable as the complement of the extension E.
Given a fixed-point \((E, A)=\Phi (E, A)\) of the Kripke Jump \(\Phi \), the anti-extension A is definable via E as
$$\begin{aligned} A:=\{\varphi \in \textsf {Sent} \mid \lnot \varphi \in E\}\cup \textsf {NSent}. \end{aligned}$$
For notational convenience, then, we often drop A and we let a semantic truth theory be a class \(\mathfrak {S}\) of structures \((\mathcal {N}, E)\) where \(\mathcal {N}\) is an interpretation of \(\mathcal {L}_{\textsf {PA} }\) and E an interpretation of \(\mathrm {Tr}\), implicitly assuming that A can be defined via E. Also, we often write that \(E=\Phi (E)\) is a fixed-point of \(\Phi \).
Having clarified the type of semantic theory we will be dealing with, we can now introduce the \(\mathbb {N}\)-categoricity criterion.Footnote 9
Criterion 2
(\(\mathbb {N}\)-categoricity) Let \(\mathrm {Th}\) be an axiomatic \(\mathcal {L} _\mathrm {Tr} \)-theory formulated in the logic L; let be the class of standard models of \(\mathrm {Th}\); let \(\mathfrak {S}\) be a semantic theory, and let \(\mathfrak {S} ^\mathbb {N}:=\{(\mathbb {N}, E)\mid (\mathbb {N}, E)\in \mathfrak {S} \}\) be the class of standard models in \(\mathfrak {S} \). \(\mathrm {Th}\) is an adequate axiomatisation of \(\mathfrak {S} \) if and only if,
$$\begin{aligned} (\mathbb {N}, E)\in \mathfrak {M} ^\mathbb {N}\ \text{ if } \text{ and } \text{ only } \text{ if } \ (\mathbb {N}, E)\in \mathfrak {S} ^\mathbb {N}. \end{aligned}$$
Criterion 2 imposes that the class of the extensions of \(\mathrm {Tr}\) in standard models of \(\mathrm {Th}\) be ‘sound’ and ‘complete’ relative to the class of the extensions of \(\mathrm {Tr}\) in standard models in \(\mathfrak {S}\). Sound in the sense that, whenever E is the extension of \(\mathrm {Tr}\) in a \(\mathbb {N}\)-model of \(\mathrm {Th}\), then E is the extension of \(\mathrm {Tr}\) in a \(\mathbb {N}\)-model in \(\mathfrak {S}\); complete in the sense that, whenever E is the extension of \(\mathrm {Tr}\) in a \(\mathbb {N}\)-model in \(\mathfrak {S}\), then E is the extension of \(\mathrm {Tr}\) in a \(\mathbb {N}\)-model of \(\mathrm {Th}\). More intuitively, this criterion requires that \(\varphi \) is true in a \(\mathbb {N}\)-model of \(\mathrm {Th}\) precisely when \(\varphi \) is true in a \(\mathbb {N}\)-model in \(\mathfrak {S}\).
It is worth observing that the Criterion 2 does not imply that \(\mathfrak {M} ^\mathbb {N} =\mathfrak {S} ^\mathbb {N} \). In fact, the logic L in which the theory \(\mathrm {Th}\) is formulated need not coincide with the logic of the semantic theory \(\mathfrak {S}\). As a consequence, the way in which A is definable in the structures of \(\mathfrak {M} ^\mathbb {N} \) need not be equivalent to the way in which A is definable in the structures of \(\mathfrak {S}^\mathbb {N} \). For example, if we let \(\mathfrak {S}^\mathbb {N} \) be the class of fixed-point models for \(\mathcal {L} _\mathrm {Tr} \), and if we let L be classical logic, then \(\mathfrak {S}^\mathbb {N} \) would contain FDE-structures \(\langle \mathbb {N}, (E, A)\rangle \) such that \(E=\Phi (E)\) and \(A:=\{\varphi \in \textsf {Sent} \mid \lnot \varphi \in E\}\cup \textsf {NSent} \), whereas \(\mathfrak {M} ^\mathbb {N} \) would contain classical structures \(\langle \mathbb {N}, (E, A)\rangle \) where \(E=\Phi (E)\) and \(A=\omega -E\).
It is well known that both KF and PKF are—in the sense just described—adequate axiomatisations of the fixed-point semantics. That is,
Theorem 3
(Feferman 1991; Halbach and Horsten 2006) Let \(\mathcal {K}:=\{(\mathbb {N}, E)\mid E=\Phi (E)\}\) be the class of fixed-point models for \(\mathcal {L} _\mathrm {Tr} \); let \(\mathcal {M} ^\textsf {KF} :=\{(\mathbb {N}, E)\mid (\mathbb {N}, E)\models \textsf {KF} \}\) be the class of standard models of KF; let be the class of standard models of PKF. Then
Relative to the \(\mathbb {N}\)-categoricity criterion, then, KF and PKF can be said to capture the Kripkean conception of truth equally well.Footnote 10 The qualification before the comma is necessary. As emphasized by Halbach and Nicolai (2018), in fact, the \(\mathbb {N}\)-categoricity criterion can at best be seen as a necessary condition an axiomatic theory has to satisfy in order to capture a semantic construction. A number of issues are connected with Criterion 2,Footnote 11 and it is not difficult to see that even if two axiomatic theories \(\mathrm {Th}\) and \(\mathrm {Th'}\) are \(\mathbb {N}\)-categorical axiomatisations of a semantic theory \(\mathcal {M}\), this does not imply that \(\mathcal {M}\) is captured equally well by \(\mathrm {Th}\) and \(\mathrm {Th'}\), nor can we infer that \(\mathrm {Th}\) and \(\mathrm {Th'}\) embody the same conception of truth. An example showing this point vividly will be provided below (see Example 8). Before that, however, let us explain how Halbach and Nicolai compare KF and PKF qua formal systems, and why they claim that these theories have the same truth-theoretic content.
2.2 Inner Theory
After having established that there is a precise sense in which KF and PKF can be taken to embody the same conception of truth, Halbach and Nicolai (2018) move on to compare the two theories qua formal systems. A comparison of the conceptual aspects of axiomatic theories of truth—what could be called their ‘truth-theoretic content’—is a non-trivial task, and sophisticated methods of analysis have been developed. Unfortunately, several of these tools, like relative interpretability or the finer-grained relative truth definability introduced by Fujimoto (2010), cannot be used for comparing KF and PKF: These methods are not suitable for comparing theories formulated in different logics.
In the spirit of Reinhardt (1985, 1986), and following up on the line of investigation initiated by Halbach and Horsten (2006), Halbach and Nicolai compare KF and PKF via what is usually called their ‘inner theory’, i.e., they compare the sets of sentences that are provably true in each theory.Footnote 12 Recall that ‘\(\mathrm {Th} \vdash \varphi \)’ symbolises a definition of the \(\mathrm {Th}\)-derivability of \(\varphi \). Then
Definition 4
(Inner Theory) Let \(\mathrm {Th}\) be a truth theory formulated in \(\mathcal {L} _\mathrm {Tr} \). The inner theory of \(\mathrm {Th}\), \(\mathrm {I} \mathrm {Th} \), is defined thus:
$$\begin{aligned} \mathrm {I} \mathrm {Th}:= \{ \varphi \in \textsf {Sent} \mid \mathrm {Th} \vdash \mathrm {Tr} (\varphi ) \}. \end{aligned}$$
Concentrating on the set of provably true sentences allows a neat comparison between KF-like and PKF-like theories. The reason is that
$$\begin{aligned} {\textsf {PKF}} \vdash \varphi \text{ if, } \text{ and } \text{ only } \text{ if, } \textsf {PKF} \vdash \mathrm {Tr} (\varphi ). \end{aligned}$$
which means that PKF-like theories are identical to their internal theories. An important consequence of this observation is that, whereas it is not useful to ask whether KF is contained in PKF,Footnote 13 a key question is whether the inner theory of KF is contained in PKF and/or vice versa.Footnote 14
Halbach and Nicolai (2018) and Nicolai (2018), building on previous results by Halbach and Horsten (2006), have analysed the relationship between the inner theories of several variants of KF and corresponding variants of PKF. They have shown a thorough equivalence between them: Given a KF-like theory, there is a corresponding PKF-like theory such that the set of provably true sentences of the former coincide with the set of theorems of the latter. In other words, they proved the following remarkableFootnote 15
Theorem 5
(Halbach and Horsten 2006; Halbach and Nicolai 2018; Nicolai 2018) Let \((\textsf {PKF} _\star , \textsf {KF} _\star )\) range over the following theory-pairs
Then
$$\begin{aligned} \textsf {PKF} _\star = \textsf {I} \textsf {KF} _\star . \end{aligned}$$
Let us emphasise that the claim that KF and PKF embody the same conception of truth is a consequence of their being \(\mathbb {N}\)-categorical axiomatisations of the fixed-point semantics, not a consequence of their having the same inner theory.Footnote 16 The analysis of the inner theories of KF and PKF is a second step: After having established that a classical and a nonclassical system are both \(\mathbb {N}\)-categorical axiomatisations of a certain semantics, one can compare these theories qua formal systems, asking whether they prove the same sentences to be true, or whether one theory is truth-theoretically stronger than the other.
This section begins by refining the criteria just introduced. Examples 6 and 8 below show that they are too coarse-grained. As it will become evident, both examples indicate that the set of provably not-true sentences plays a crucial role:Footnote 17 Some of the aspects relating a formal system and a semantic theory on the one hand, and some of the aspects relating and/or differentiating two formal systems on the other, materialise only if we take into account what the theories prove to be not-true. In what follows, then, we include the concept of the anti-extension of \(\mathrm {Tr}\) in a refinement of the above criteria and we discuss the implications of this inclusion. As mentioned in the introduction, this finer-grained analysis (i) will strengthen the central results presented in the previous section, i.e., Theorems 3 and 5, and (ii) will show that there is a key aspect of the Kripkean semantics that some variants of KF cannot capture as adequately as their PKF-counterparts.
Before we begin, a caveat is in order. A tacit assumption underlying the present study is that it is possible (in principle, at least) for a classical system to embody a transparent, nonclassical conception of truth. The results presented in this section contain insofar a defence of classical logic, as it will be shown that a truth-theoretical equivalence between classical and nonclassical systems can be established under criteria which are stricter than those commonly employed. Of course, the nonclassical logician can always reject this assumption and argue that the very same idea of axiomatising the Kripkean conception of truth using classical logic is a non-starter, as KF is asymmetric. In fact, more is true: (Halbach and Horsten 2006, p. 688) have observed that “KF is essentially asymmetric”, in the sense that KF cannot be consistently closed under the rules of Necessitation and Co-Necessitation
In other words, we cannot even consistently postulate that outer theory (what is provable) and inner theory (what is provably true) of KF are identical. But being symmetric, it may be argued, is a sine qua non for being considered an adequate axiomatisation of the fixed-point semantics, hence no KF variant could be said to embody the Kripkean conception of truth.Footnote 18 As already mentioned in the Introduction, however, in this article we would like to shift the focus of contention, examining the relationship between KF and PKF from a different perspective: We will neither rely on the asymmetry of KF to argue against classical logic, nor on the weakness of PKF to argue against nonclassical logic.
3.1 Refining Inner Theory
One may wonder whether analysing the relationship between the inner theory of a classical system with the inner theory of a nonclassical alternative is sufficient to understand how they are related to each other from a conceptual point of view. For, if one is interested in comparing the truth-theoretic aspects of two theories, concentrating only on what is provably true in each system might not be enough. There could in fact be other aspects relating and/or differentiating the theories which may emerge only if we implement a finer-grained method of comparison.
The set of provably true sentences, no doubt, is an important component in the analysis of the truth-theoretic content of a truth theory. Observe, however, that for any theory \(\mathrm {Th}\) we can consider the following sets:Footnote 19
Let \(\mathrm {Th} ^E, \mathrm {Th} ^A\), and \(\mathrm {Th} ^F\) be the extension, anti-extension, and falsity-set, respectively, induced by \(\mathrm {Th}\). Analysing the relationship between these sets, we can read off from \(\mathrm {Th}\) considerably more than \(\mathrm {I} \mathrm {Th} \) (here \(\mathrm {Th} ^E\)). For instance, one could ask whether \(\mathrm {Th}\) thinks that being untrue and being false are coextensive properties, i.e., one could ask whether \(\mathrm {Th} ^A = \mathrm {Th} ^F\)? Or one could ask whether \(\mathrm {Th}\) thinks that every sentence is either true or untrue, i.e., whether \({\textsf {Sent}} \subseteq \mathrm {Th} ^E \cup \mathrm {Th} ^A\)? Or whether there are sentences that are both true and untrue, i.e., whether \(\mathrm {Th} ^E \cap \mathrm {Th} ^A \ne \varnothing ?\) And so forth.
Arguably, statements about being not-true and statements about being false are ipso facto statements about truth. It thus seems that the inner theory of \(\mathrm {Th}\) is only one component of its overall view on truth. \(\mathrm {Th} ^A\), and its relationship with \(\mathrm {Th} ^E\), is not less relevant. To see this point vividly, consider the following
Example 6
Let \(\mathrm {Th}\) and \(\mathrm {Th'}\) be two \(\mathcal {L} _\mathrm {Tr} \)-theories, and suppose that \(\mathrm {Th} ^E = \mathrm {Th'} ^E\), i.e., suppose
$$\begin{aligned} \mathrm {Th} \vdash \mathrm {Tr} (\varphi ) \ \text{ iff } \ \mathrm {Th'} \vdash \mathrm {Tr} (\varphi ). \end{aligned}$$
Suppose further that \(\mathrm {Th} ^A \ne \mathrm {Th'} ^A\). Then we could for instance have \(\mathrm {Th} ^A = \mathrm {Th} ^F\), i.e.
but possibly \(\mathrm {Th'} ^A \not \subset \mathrm {Th'} ^F\), i.e.
This means that ‘being not-true’ and ‘being false’ are coextensive properties according to \(\mathrm {Th}\), but not according to \(\mathrm {Th'}\). Therefore, if we were to compare \(\mathrm {Th}\) and \(\mathrm {Th'}\) only with respect to the sentences they prove to be true, we’d wrongly be lead to the conclusion that \(\mathrm {Th}\) and \(\mathrm {Th'}\) had the same truth-theoretic content, even though there is a straightforward sense in which \(\mathrm {Th}\) ’s and \(\mathrm {Th'}\) ’s views on truth are different and, to some extent, incompatible. \(\square \)
Our idea behind the refinement of the criterion for comparing KF and PKF qua formal systems is now rather simple: We will ask not only how the sets
$$\begin{aligned} \{\varphi \in {\textsf {Sent}} \mid {\textsf {KF}} \vdash \mathrm {Tr} (\varphi )\} \ \text{ and } \ \{\varphi \in \textsf {Sent} \mid \textsf {PKF} \vdash \varphi \} \end{aligned}$$
are related to each other. We will in addition ask how the sets
$$\begin{aligned} \{\varphi \in {\textsf {Sent}} \mid {\textsf {KF}} \vdash \mathrm {Tr} (\varphi )\} \ \text{ and } \ \{\varphi \in \textsf {Sent} \mid \textsf {PKF} \vdash \varphi \} \end{aligned}$$
are related to each other.
Since the use of ‘inner theory’ is already quite extensive in the literature, we use ‘truth-theoretic content’ instead, as this notion has been used by Halbach and Nicolai (2018) interchangeably with inner theory.Footnote 20
Definition 7
(Truth-theoretic content) Let \(\mathrm {Th}\) be a \(\mathcal {L} _\mathrm {Tr} \)-theory. The truth-theoretic content of \(\mathrm {Th}\), \(\mathrm {Th} ^\text {ttc} \), is defined as follows.
Using the notation introduced above, we have
$$\begin{aligned} \mathrm {Th} ^{\mathrm{ttc}} := \langle {\mathrm {Th} ^E, \mathrm {Th} ^A}\rangle . \end{aligned}$$
According to Definition 7, a sentence \(\varphi \) is part of the truth-theoretic content of \(\mathrm {Th}\) whenever \(\mathrm {Th}\) has something to say about the truth status of \(\varphi \), i.e., whenever \(\varphi \) is provably true, or provably untrue.
3.2 Refining \(\mathbb {N}\)-categoricity
A counterexample very similar to Example 6 can be constructed to show that \(\mathbb {N}\)-categoricity criterion, too, needs to be refined.
Example 8
Let \(\mathfrak {S}\) be a semantic theory such that the anti-extension A can be defined via E, say in a Kripkean fashion as
$$\begin{aligned} A:=\{\varphi \mid \lnot \varphi \in E\}, \end{aligned}$$
so that ‘being not-true’ and ‘being false’ are coextensive properties.
Now let \(\mathrm {Th}\) and \(\mathrm {Th'}\) be two theories formulated in the logics L and \(L'\), respectively. Suppose further that both \(\mathrm {Th}\) and \(\mathrm {Th'}\) are \(\mathbb {N}\)-categorical axiomatisations of \(\mathcal {M}\), i.e., suppose
This is compatible with two further assumptions, i.e.
\(\mathrm {Th}\) and \(\mathrm {Th'}\) can hardly be taken to embody the same conception of truth, let alone to capture \(\mathfrak {S}\) equally well: In \(\mathrm {Th}\) ‘being not-true’ and ‘being false’ are coextensive properties, in \(\mathrm {Th'}\) they are not. Indeed, one could argue that \(\mathrm {Th'}\) does not embody the conception of truth conveyed by \(\mathfrak {S}\), as they don’t agree on whether untruth and falsity are coextensive. It seems thus uncontroversial that Criterion 2 is too coarse-grained to establish whether two theories can be said to capture a certain semantics equally well. \(\square \)
Our idea behind the refinement of the \(\mathbb {N}\)-categoricity criterion is now as simple as the idea behind the refinement of the notion of inner theory. We will not only require that the class of the extensions of \(\mathrm {Tr}\) in \(\mathbb {N}\)-models of \(\mathrm {Th}\) be ‘sound’ and ‘complete’ relative to the class of the extensions of \(\mathrm {Tr}\) in \(\mathbb {N}\)-models in \(\mathfrak {S}\). We will additionally require that the class of the anti-extension of \(\mathrm {Tr}\) in \(\mathbb {N}\)-models of \(\mathrm {Th}\) be ‘sound’ and ‘complete’ relative to the class of the anti-extensions of \(\mathrm {Tr}\) in \(\mathbb {N}\)-models in \(\mathfrak {S}\). To put it intuitively: Whereas Criterion 2 only requires that \(\varphi \) is true in a standard model of \(\mathrm {Th}\) iff \(\varphi \) is true in a standard model in \(\mathfrak {S}\), the finer-grained criterion introduced below additionally requires that \(\varphi \) is not-true in a standard model of \(\mathrm {Th}\) iff \(\varphi \) is not-true in a standard model in \(\mathfrak {S}\).
To give this idea formal expression, let us introduce some notation. Recall first that the semantic theories we are concerned with are classes of structures \(\langle \mathcal {N}, (E, A)\rangle \), such that the anti-extension A can be defined via the extension E. And recall that for simplicity’s sake we have so far denoted these semantic theories as classes of structures \((\mathcal {N}, E)\), without explicit mention of the definable anti-extension A. It goes without saying that these semantic theories can equivalently be presented from the opposite perspective. That is, the semantic theories we are dealing with can be defined as classes of structures \(\langle \mathcal {N}, (E, A)\rangle \), such that E is definable via A. For example, in classical logic E can be defined as \(A^c\), i.e., as the complement of A, and in fixed-point models we can define E as \(E := \{ \varphi \in {\textsf {Sent}} \mid \lnot \varphi \in A \}\). So we can equivalently define semantic theories as classes of structures \((\mathcal {N}, A)\), without explicit mention of the definable extension E. For the purposes of this article, it is indeed sometimes convenient to conceive of models for \(\mathcal {L} _\mathrm {Tr} \) as structures \((\mathcal {N}, A)\) instead of \((\mathcal {N}, E)\). Hoping that our notation will not cause any confusion, we occasionally write
instead of
It is important to emphasise that there is no difference in the definition of the satisfaction relation . That is to say, and are just two notational variants expressing the same relation between structures \(\langle \mathbb {N}, (E, A)\rangle \) and sentences \(\varphi \). For example, in the case of a classical model \((\mathbb {N}, E)\) where E and A are disjoint and complementary, we do not define \((\mathbb {N}, A)\models \mathrm {Tr} (\varphi ) \text { iff } \varphi \in A\), but we understand it simply as a notational variant of the standard definition of the classical satisfaction relation, i.e., we have
$$\begin{aligned} (\mathbb {N}, A)\models \mathrm {Tr} (\varphi ) \ \text{ iff } \ \varphi \in A^c = E. \end{aligned}$$
The reason for introducing this notational variant, as it will become clear in the following definition, is that it allows to express neatly the intuitive idea exposed above, according to which not only the class of the extensions of \(\mathrm {Tr}\), but also the class of its anti-extensions has to be sound and complete relative to \(\mathbb {N}\)-models of the semantic theory.
Criterion 9
(\(\mathbb {N}\)-categoricity\(^+\)) Let \(\mathrm {Th}\) be an axiomatic \(\mathcal {L} _\mathrm {Tr} \)-theory formulated in the logic L. Let X denote either E or A. Let be the class of standard models of \(\mathrm {Th}\), and let \(\mathfrak {S} ^\mathbb {N}:=\{(\mathbb {N}, X)\mid (\mathbb {N}, X)\in \mathfrak {S} \}\) be the class of standard models of a semantic theory \(\mathfrak {S} \). \(\mathrm {Th}\) is an adequate axiomatisation of \(\mathfrak {S} \) if and only if,
Intuitively, Criterion 9 imposes two requirements. First, it requires that a sentence \(\varphi \) is true in a \(\mathbb {N}\)-model of \(\mathrm {Th}\) precisely when this sentence is true in some \(\mathbb {N}\)-model in \(\mathfrak {S}\) (just as Criterion 2). Second, it requires that a sentence \(\varphi \) is untrue in a \(\mathbb {N}\)-model of \(\mathrm {Th}\) precisely when this sentence is untrue in some \(\mathbb {N}\)-model in \(\mathfrak {S}\).Footnote 21
3.3 Implementing the New Criteria
We conclude the first part of this article with two observations showing that the connection between the variants of KF and PKF studied by Halbach and Nicolai (2018) and Nicolai (2018) is fairly strong. To begin with, although it is not difficult to see that PKF (\(\textsf {PKF} _\textsf {S} \)) is an \(\mathbb {N}\)-categorical\(^+\) axiomatisation of the class of four-valued (symmetric) fixed-points of \(\Phi \), the same can be observed for the KF-variants as well, which is somewhat surprising.
Theorem 10
(i) KF and PKF are \(\mathbb {N}\)-categorical\(^+\) axiomatisations of the class of four-valued fixed-points of \(\Phi \). (ii) \(\textsf {KF} _\textsf {S} \) and \(\textsf {PKF} _\textsf {S} \) are \(\mathbb {N}\)-categorical\(^+\) axiomatisations of the class of symmetric fixed-points.
Second, having established that these variants of KF and PKF can be said to capture the Kripkean conception of truth equally well, we can compare them qua formal systems. It turns out that these theories not only agree on what is true, but they agree on what it untrue, too.
Proposition 11
Let \((\textsf {PKF} _\star , \textsf {KF} _\star )\) range over the following theory-pairs
Then
$$\begin{aligned} \textsf {PKF} _\star ^\text {ttc} = \textsf {KF} _\star ^\text {ttc} . \end{aligned}$$
Detailed proofs of the last two observations are provided in the Appendix. Here we just sketch the proof idea.
Proof of idea of Theorem 10
Theorem 10 implies two different claims. The first, already known, is that E is the extension of \(\mathrm {Tr}\) in a \(\mathbb {N}\)-model of KF iff E is the extension of a fixed-point of \(\Phi \). The second claim is that A is the anti-extension of \(\mathrm {Tr}\) in \(\mathbb {N}\)-model of KF iff A is the anti-extension of a fixed-point of \(\Phi \). The second claim, in other words, means that
The left-to-right direction of this second claim can be shown by induction on the complexity of formulae: Given a model \((\mathbb {N}, A)\models \textsf {KF} \), one shows that, for \(E=\{\varphi \in {\textsf {Sent}} \mid \lnot \varphi \in A\}\),
which amounts to showing that \(\Phi (E,A)=(E,A)\). As for the right-to-left direction, one checks that, given a fixed point (E, A) of \(\Phi \), the structure \((\mathbb {N}, A)\) classically satisfies the axioms of KF. \(\square \)
Proof of idea of Proposition 11
The key to the proof is the fact that for any theory \(\mathrm {Th} \in \{\textsf {KF} _\star , \textsf {PKF} _\star \}\), we have
$$\begin{aligned} \mathrm {Th} \vdash \lnot \mathrm {Tr} (\varphi ) \ \text{ iff } \ \mathrm {Th} \vdash \mathrm {Tr} (\lnot \varphi ), \end{aligned}$$
which in other words means \({\textsf {KF}} _\star ^A = {\textsf {KF}} _\star ^F\) and \({\textsf {KF}} _\star ^A={\textsf {PKF}} _\star ^F\). The equivalence stated in the Proposition now easily follows. We already know that \(\textsf {KF} _\star \) and \(\textsf {PKF} _\star \) agree on what is true, i.e., \({\textsf {KF}} _\star ^E={\textsf {PKF}} _\star ^E\). Since being false is understood as having a true negation, it follows that \({\textsf {KF}} _\star ^F={\textsf {PKF}} _\star ^F\). But since \({\textsf {KF}} _\star ^F= {\textsf {KF}} _\star ^A\) and \({\textsf {PKF}} _\star ^F = {\textsf {PKF}} _\star ^A\), we get \({\textsf {KF}} _\star ^A={\textsf {PKF}} _\star ^A\), and hence the desired conclusion \(\textsf {PKF} _\star ^\text{ ttc } =\textsf {KF} _\star ^\text{ ttc }\). \(\square \)
Let us conclude the first part of this article with a brief digression, mentioning a possible further refinement of the criteria just introduced. Criterion 9 requires that a sentence \(\varphi \) is true (untrue) in a standard model of \(\mathrm {Th}\) precisely when this sentence is true (untrue) in a standard model in \(\mathfrak {S}\). Definition 7 singles out the set of sentences which are provably (un)true in \(\mathrm {Th}\). In both cases, the focus is on truth-theoretic sentences. A referee, though, correctly pointed out that there is no reason why we should consider only truth-theoretic sentences and not truth-theoretic inferences as well. Indeed, a natural extension of both criteria, that would make them more broadly applicable, could be based on the inclusion of truth-theoretic inferences. Of course, there might be disagreement on what should count as a truth-theoretic inference, but the following seems to be a plausible option: Whereas a logical inference is a reasoning from a (possibly empty) set of premises \(\varphi , \psi , \xi \dots \) to a conclusion \(\sigma \), a truth-theoretic inference could be understood as a reasoning from a (possibly empty) set of premises \(\mathrm {Tr} (\varphi ), \mathrm {Tr} (\psi ), \mathrm {Tr} (\xi ) \dots \) to a conclusion \(\mathrm {Tr} (\sigma )\).Footnote 22
One way of extending Criterion 9, then, could consist in requiring that the internal logic (understood as provable truth-theoretic inferences) of a theory \(\mathrm {Th}\) be closed under the logic of the intended semantics. For example, suppose the intended semantics \(\mathfrak {S}\) is formulated over the logic L. Assume further that the inference from \(\varphi , \psi , \xi \dots \) to \(\sigma \) is an L-valid inference. Then one could argue that \(\mathrm {Th}\) captures \(\mathfrak {S}\) only if (in addition to being \(\mathbb {N}\)-categorical\(^+\)) \(\mathrm {Th}\) has enough resources to infer \(\mathrm {Tr} (\sigma )\) from the premises \(\mathrm {Tr} (\varphi ), \mathrm {Tr} (\psi ), \mathrm {Tr} (\xi ) \dots \).Footnote 23 Such an extension would overcome some inadequacies of Criteria 2 and 9.Footnote 24 Similarly, a more in depth comparison of two axiomatic systems (qua formal systems) could consist in investigating not only their respective truth-theoretic content (understood as provable truth-theoretic facts), but also their truth-theoretic logic (understood as their internal logic).Footnote 25 This would improve the analysis of the systems, as there are theories that prove the same truth-theoretic facts, but do not perform the same truth-theoretic reasoning.Footnote 26
The results presented in the previous section feed the philosophical doubts raised by Halbach and Nicolai vis-à-vis the use of nonclassical logic in this context, as we have seen that there is a stronger sense in which classical and nonclassical systems can be taken to be truth-theoretically equivalent.Footnote 27 So why should we axiomatise the fixed-point semantics using nonclassical logic, if we can obtain an equally good axiomatisation using classical logic? Why should we give up on sustained ordinary reasoning, and severely impede our inductive reasoning, if we can obtain the same truth-theoretic content working within a classical and powerful system?
It is indeed difficult to find a convincing answer to these questions. However, so far we have only considered some of the KF- and PKF-variants. Indeed, in the concluding remarks, Halbach and Nicolai point out that
[of] course, our results here are only a case study. The base theory can be varied, the assumptions on truth, and the way classical logic is restricted can be varied. In other base theories different schemata may be employed. We don’t intend to embark on the enterprise of browsing through all possible combinations. We think the burden of proof is with those who advocate a restriction of classical logic. (Halbach and Nicolai 2018, p. 251)
We now take the burden of proof, and we analyse two other variants of KF and PKF, namely variants restricting the class of models to consistent fixed-points, and variants restricting the class of models to complete fixed-points. It will turn out that for these variants different philosophical considerations are in order.Footnote 28
Let us label theories with consistency principles \(\textsf {KF} _\textsf {cs} \) and \(\textsf {PKF} _\textsf {cs} \), while theories with completeness principles \(\textsf {KF} _\textsf {cp} \) and \(\textsf {PKF} _\textsf {cp} \). \(\textsf {KF} _\textsf {cs} \) can be obtained from KF plus the axiom
(Cons) forces the truth predicate to be consistent: no sentence \(\varphi \) can be both true and false. Symmetrically, \(\textsf {KF} _\textsf {cp} \) can be obtained from KF plus the axiom
(Comp) forces the truth predicate to be complete: every sentence \(\varphi \) is either true or false.
As for the PKF-counterparts of these KF-variants, they have been introduced and investigated in Castaldo and Stern (2020).Footnote 29\(\textsf {PKF} _\textsf {cs} \) can be obtained from PKF plus an axiom schema imposing consistency, e.g.
Symmetrically, we can obtain the system \(\textsf {PKF} _\textsf {cp} \) from PKF plus an axiom schema imposing completeness, e.g.
Having defined (so to speak) the theories, let us see how they are related to each other. It is known that they are all \(\mathbb {N}\)-categorical axiomatisations of the intended class of fixed-points. Hence, relative to the \(\mathbb {N}\)-categoricity criterion, \(\textsf {KF} _\textsf {cs} \) and \(\textsf {PKF} _\textsf {cs} \) embody the consistent declination of the Kripkean conception of truth equally well. And similarly, \(\textsf {KF} _\textsf {cp} \) and \(\textsf {PKF} _\textsf {cp} \) capture the class of complete fixed-points equally well. Moreover, it has been shown that these pairs of theories prove the same sentence to be true:
Proposition 12
Let \((\textsf {PKF} _\star , \textsf {KF} _\star )\) range over the pairs
$$\begin{aligned}&\!\!\!(\textsf {PKF} _\textsf {cs} ^-, \textsf {KF} _\textsf {cs} ^-), (\textsf {PKF} _\textsf {cs} , \textsf {KF} ^\mathsf {int} _\textsf {cs} ), (\textsf {PKF} _\textsf {cs} ^+, \textsf {KF} _\textsf {cs} ), (\textsf {PKF} _\textsf {cp} ^-, \textsf {KF} _\textsf {cp} ^-), (\textsf {PKF} _\textsf {cp} , \textsf {KF} ^\mathsf {int} _\textsf {cp} ),\\&\!\!\!(\textsf {PKF} _\textsf {cp} ^+, \textsf {KF} _\textsf {cp} ). \end{aligned}$$
Then
$$\begin{aligned} {\textsf {KF}} _\star ^E = {\textsf {PKF}} _\star ^E. \end{aligned}$$
Proof
This is a corollary of Proposition 4.15 of Castaldo and Stern (2020).Footnote 30\(\square \)
If we were now to employ the criteria presented in §2, we would have to say that, e.g., \(\textsf {KF} _\textsf {cs} \) and \(\textsf {PKF} _\textsf {cs} \) embody the consistent declination of the Kripkean conception of truth equally well, and we would have to say that they have the same truth-theoretic content.
However, even though it is easily observed that \(\textsf {PKF} _\textsf {cs} \) and \(\textsf {PKF} _\textsf {cp} \) are \(\mathbb {N}\)-categorical\(^+\) axiomatisations of the class of consistent, respectively complete, fixed-points, the same cannot be said for the consistent and complete versions of KF:
Observation 13
(i) \(\textsf {KF} _\textsf {cs} \) is not an \(\mathbb {N}\)-categorical\(^+\) axiomatisation of the class of consistent fixed-point models. (ii) \(\textsf {KF} _\textsf {cp} \) is not an \(\mathbb {N}\)-categorical\(^+\) axiomatisation of the class of complete fixed-point models.
Proof
(i) Let \(\tau \) be a Truth-teller. There are consistent fixed-points \((E_{\textsf {cs}}, A_\textsf {cs})\) of \(\Phi \), e.g. the least, such that \(\tau \wedge \lnot \tau \notin A_\textsf {cs}\), hence \((\mathbb {N}, A_\textsf {cs})\not \models \lnot \mathrm {Tr} (\tau \wedge \lnot \tau )\). However, \(\textsf {KF} _\textsf {cs} \vdash \lnot \mathrm {Tr} (\tau \wedge \lnot \tau )\), hence \((\mathbb {N}, A_\textsf {cs})\not \models \textsf {KF} _\textsf {cs} \). (ii) For \(\textsf {KF} _\textsf {cp} \), dual considerations hold. \(\square \)
Moreover, it is easy to see that neither \(\textsf {KF} _\textsf {cs} \)-and-\(\textsf {PKF} _\textsf {cs} \) nor \(\textsf {KF} _\textsf {cp} \)-and-\(\textsf {PKF} _\textsf {cp} \) agree on what is not-true.
Observation 14
For \((\textsf {PKF} _\star , \textsf {KF} _\star )\) as in Proposition 12,
$$\begin{aligned} \textsf {PKF} _\star ^\text {ttc} \ne \textsf {KF} _\star ^\text {ttc} . \end{aligned}$$
Proof
There are many sentences that \(\textsf {KF} _\textsf {cs} \) deem not-true and that are independent from \(\textsf {PKF} _\textsf {cs} \). Dually, there are many sentences that \(\textsf {PKF} _\textsf {cp} \) deem not-true that are not so in \(\textsf {KF} _\textsf {cp} \). Again, the sentence \(\tau \wedge \lnot \tau \), for \(\tau \) a Truth-teller, is an example. \(\square \)
The asymmetry of \(\textsf {KF} _\textsf {cs} \) and \(\textsf {KF} _\textsf {cp} \) is stronger than the asymmetry of KF and \(\textsf {KF} _\textsf {S} \). One can say that KF and \(\textsf {KF} _\textsf {S} \) are weakly asymmetric, as they do not prove some of their theorems to be true. \(\textsf {KF} _\textsf {cs} \) and \(\textsf {KF} _\textsf {cp} \), however, are strongly asymmetric: the former, in fact, proves some of its theorems to be not-true, while the latter proves some of its theorems to be false. The example chosen more frequently involves a liar sentence \(\lambda \), and it is often pointed out that \(\textsf {KF} _\textsf {cs} \vdash \lambda \wedge \lnot \mathrm {Tr} (\lambda )\).Footnote 31 This strong asymmetry has been the target of a forceful criticism by several authors, such as Leon Horsten or Hartry Field. (Horsten 2011, p. 127), for example, emphasises that \(\textsf {KF} _\textsf {cs} \) “proves sentences that by its own lights are untrue” (emphasis in the original), and he takes this as a “sure mark of philosophical unsoundness.” (Field 2008, p. 132) additionally observes that \(\textsf {KF} _\textsf {cs} \) not only proves \(\varphi \wedge \lnot \mathrm {Tr} (\varphi )\) for certain degenerate sentences like Liars, but \(\textsf {KF} _\textsf {cs} \) proves instances of its own axioms to be untrue.Footnote 32 It seems undeniable that declaring some of its own theorems untrue, or false, is an odd feature of a truth system. And it seems—to use (Field 2008, p.132)’s words—“highly peculiar” to postulate an axiom schema, if one is then going to declare some of its instances untrue. In other words, being strongly asymmetric can be considered to be a problem per se.Footnote 33
That being said, however, this is not the claim defended in this paper. It will not be argued that declaring some of its own theorems and axioms untrue or false implies philosophical unsoundness. Let us reiterate on the fact that we are not comparing KF and PKF qua theories of truth simpliciter, in their own light. We are comparing them qua axiomatisations of the fixed-point semantics. In the next and last section we would thus like to focus on an additional and different problem of \(\textsf {KF} _\textsf {cs} \) and \(\textsf {KF} _\textsf {cp} \), which is in a sense a consequence of their strong asymmetry, and which is philosophically relevant for the present line of investigation on KF and PKF.
Have we shown that the class of consistent, respectively complete, fixed-points is captured more adequately by \(\textsf {PKF} _\textsf {cs} \) and \(\textsf {PKF} _\textsf {cp} \) rather than by \(\textsf {KF} _\textsf {cs} \) and \(\textsf {KF} _\textsf {cp} \)? What we have shown, so far, is that \(\textsf {KF} _\textsf {cs} \) and \(\textsf {KF} _\textsf {cp} \) are not \(\mathbb {N}\)-categorical\(^+\) axiomatisations of the consistent, respectively complete, fixed-point models. But this, it may be argued, is not enough to claim that, say, \(\textsf {KF} _\textsf {cs} \) does not embody the Kripkean conception of true in its consistent declination.Footnote 34 Establishing that the class of anti-extensions of \(\mathrm {Tr}\) in standard models of \(\textsf {KF} _\textsf {cs} \) is neither sound nor complete relative to the class of anti-extensions of \(\mathrm {Tr}\) in consistent fixed-point models is—one could argue—not necessarily a problem. In fact—the argument might continue—if a sentence \(\varphi \) is never in the extension of \(\mathrm {Tr}\) in consistent fixed-points, then there is a precise sense in which \(\varphi \) is not true. For example, classical contradictions, including those involving ungrounded sentences like Truth-tellers \(\tau \), are never in the extension of \(\mathrm {Tr}\), hence they might be taken to be not true. And so the fact that \(\textsf {KF} _\textsf {cs} \vdash \lnot \mathrm {Tr} (\tau \wedge \lnot \tau )\) but \(\textsf {PKF} _\textsf {cs} \not \vdash \lnot \mathrm {Tr} (\tau \wedge \lnot \tau )\) only shows that \(\textsf {KF} _\textsf {cs} \) is, but \(\textsf {PKF} _\textsf {cs} \) is not, able to capture this aspect of the consistent fixed-points semantics.
Let us suppose, for the sake of argument, that this line of reasoning justifying the provability of sentences such as \(\lnot \mathrm {Tr} (\tau \wedge \lnot \tau )\), on the ground that \(\tau \wedge \lnot \tau \) is never the extension on \(\mathrm {Tr}\), is indeed sound.Footnote 35 But even if one concedes that \(\textsf {KF} _\textsf {cs} \) captures the aspect of the consistent fixed-points semantics according to which, say, contradictions are not true, we do not believe that the same cannot be said for \(\textsf {PKF} _\textsf {cs} \). Indeed, we believe that \(\textsf {PKF} _\textsf {cs} \) does considerably better in this respect. It is in fact not difficult to see that we can distinguish between (in particular) two ways of being not true. The first is the one we have just introduced: There are sentences, like \(\tau \wedge \lnot \tau \), that are never true but not always untrue, i.e., sentences that are always outside the extension of \(\mathrm {Tr}\), however they are not always in its anti-extension. But there are also sentences, like \(0 + 0 \ne 0\), that are never true and always untrue, i.e., sentences that are always in the anti-extension of \(\mathrm {Tr}\). Now, \(\textsf {KF} _\textsf {cs} \) does not see the difference between these two ways of being something other than true, in the sense that \(\lnot \mathrm {Tr} (\tau \wedge \lnot \tau )\) and \(\lnot \mathrm {Tr} (0 + 0 \ne 0)\) are both theorems of \(\textsf {KF} _\textsf {cs} \):
By contrast, \(\textsf {PKF} _\textsf {cs} \) nicely captures this difference, as while \(\lnot \mathrm {Tr} (0 + 0 \ne 0)\) is a theorem of \(\textsf {PKF} _\textsf {cs} \), \(\mathrm {Tr} (\tau \wedge \lnot \tau )\) is only an antitheorem:
In other words, \(\textsf {PKF} _\textsf {cs} \) knows the difference between \(\tau \wedge \lnot \tau \) and \(0 + 0 \ne 0\): the former is never true, the latter is always untrue.Footnote 36
We would like to provide a final and crucial argument supporting the claim that \(\textsf {KF} _\textsf {cs} \) and \(\textsf {PKF} _\textsf {cs} \) cannot be taken to embody the Kripkean conception of truth equally well. Recall that in fixed-point models the anti-extension can be defined via the extension as follows:Footnote 37
The fact that A can be defined as the set of sentences whose negations are element of E, it should be observed, is not just a technical detail. As noted by (Soames 1999, p.181), “a guiding intuition behind Kripke’s formal construction is the idea that the status of the claim that a sentence S is true or that S is not true is dependent on the prior status of S or \(\lnot \)S.” This means that the status of \(\lnot \mathrm {Tr} (\varphi )\) depends on the status of \(\lnot \varphi \), upon which in turn depends the status of \(\mathrm {Tr} (\lnot \varphi )\). In other words, one of the distinctive features of the partial conception of truth presented by Kripke (1975) is that being not-true and being false are coextensive properties.
Crucially, Proposition 11 shows that this philosophical insight can be elegantly captured within a classical proof system. In fact, by inspecting the proof of Proposition 11, we realise that there exists a remarkable symmetry between semantics and proof theory: Extension and anti-extension induced by KF and \(\textsf {KF} _\textsf {S} \), and by PKF and \(\textsf {PKF} _\textsf {S} \), are related to each other in the exact same way as extension and anti-extension are related to each other in fixed-point models, that is
This is good news for the classical logician: the pleasing symmetry between semantics and proof theory can be preserved within classical systems. Yet, at the same time the costs of classical logic emerge precisely in the realisation that this symmetry is lost when we try to axiomatise consistent (or complete) fixed-point models within a classical system. It is here that we can see the impact of classical logic on what could be called ‘truth-theoretic reasoning’. In \(\textsf {KF} _\textsf {cs} \), the reasoning that leads us from \(\lnot \mathrm {Tr} (\varphi )\) to \(\mathrm {Tr} (\lnot \varphi )\) is no longer valid: If we know that a sentence is not-true, we cannot conclude that it is false. There is a precise sense, then, in which one can say that \(\textsf {PKF} _\textsf {cs} \) does, and \(\textsf {KF} _\textsf {cs} \) does not (at least, not as adequately as \(\textsf {PKF} _\textsf {cs} \)) capture the fixed-point semantics: Being not-true and being false are coextensive properties only in \(\textsf {PKF} _\textsf {cs} \), but not in \(\textsf {KF} _\textsf {cs} \), i.e.,
In sum, not only nonclassical logic, but also classical logic has its own costs: We cannot axiomatise the class of consistent (or complete) fixed-points in classical logic as adequately as we can do within a nonclassical system. Having established that both frameworks have their own virtues and their own costs, a natural further step in this investigation would consist in assessing which option is best. The nonclassical logician should reflect on whether the cost of losing proof-theoretic strength surpasses the virtue of having a transparent truth predicate. The classical logician should reflect on whether the possibility of using ordinary reasoning is enough to justify the loss of some insights of the intended semantics. These questions will not be discussed here.
Let us conclude by providing a response to Halbach and Nicolai’s warning, according to which “the incision [of nonclassical logic] doesn’t hit our truth-theoretic reasoning; it hits mathematical reasoning at its heart” (Halbach and Nicolai 2018, p. 229). Whereas the mathematical costs of nonclassical logic can be quantified as the transfinite distance between \(\omega ^\omega \) and \(\varepsilon _0\), the truth-theoretic costs of classical logic can be identified with the impossibility of reasoning about truth in the intended way. In response, or better: in addition, to Halbach and Nicolai’s warning, it may then be held that the incision of classical logic doesn’t hit our mathematical reasoning; it hits our truth-theoretic reasoning at its heart.
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Intuitionistic mathematics
Intuitionistic mathematics
Idea
Intuitionistic mathematics (often abbreviated INT) is the earliest full-blown variety of constructive mathematics, done according to the mathematical principles developed by L.E.J. Brouwer through his philosophy of intuitionism.
Beware that this terminology is not consistent across mathematics. Not infrequently the word “intuitionistic” is used to refer simply to constructive mathematics in general, or to constructive logic, or to impredicative set theory done in constructive logic. This page is about Brouwer’s intuitionism, which is a specific variety of constructive mathematics that (unlike neutral constructive mathematics) uses axioms that contradict classical mathematics. For more on this see at Terminology below.
The first main philosophical idea is that mathematical truth (a true statement) can only be attained in one’s mind, by carefully arranging one’s concepts and constructions in such a way that there remains absolutely no doubt that every aspect of the statement is verified, unambiguously, without reliance on any ‘outside’ assumption, for instance about the platonic nature of reality. (The principle of excluded middle is such an assumption, and Brouwer gave mathematical counterexamples to its validity, eventually leading to the foundational crisis in mathematics (Grundlagenstreit) around 1930, and Brouwer’s conflict with Hilbert).
The second main philosophical idea is that the mind works over time. One does not have everything ready and done from the start. Infinity is a (perceived) idealized property of time, but one cannot have completed any infinite process or construction on any given day. Infinity is potential, not actual.
The main mathematical idea then becomes that we can only build mathematical structures and truths starting from the natural numbers 0,1,...0, 1, ... (where the dots indicate potential, not actual infinity). Natural numbers are definite and precise, but most real numbers, such as π\pi, have a very different nature. We can only form something that we call ‘π\pi’ through a never-finished process of approximation.
Nonetheless, a substantial part of mathematics can be built up (‘constructed’) in this way. What mainly differentiates intuitionistic mathematics from constructive mathematics are two added axioms.
Brouwer ‘deduced’ two axiomatic insights, notably ‘continuous choice’ and a transfinite-induction based knowledge principle called `the Bar Theorem'. 'Continuous choice' conflicts with classical mathematics, the Bar Theorem is classically true (it boils down to stating that any open cover of Baire space is inductive).
Kleene & Vesley (in Foundations of Intuitionistic Mathematics, 1965) offered a clean axiomatic approach which is nowadays called FIM. Kleene proved that FIM is equiconsistent with classical mathematics. Kleene also proved that theorems of FIM are recursively realizable, which shows the computational content of FIM.
Terminology
Terminological ambiguity is often present in constructive mathematics and its varieties. Intuitionistic mathematics (INT) includes axioms that contradict classical logic; but people in non-foundational disciplines often use “intuitionistic” to mean roughly the same as “constructive mathematics” (say: mathematics without the principle of excluded middle, usually with computational/algorithmic content and some restriction on impredicativity, but nothing added that contradicts classical mathematics).
There are a variety of ways to use the term ‘intuitionistic’. We list them here, roughly from the most specific to the most general, and contrast (where appropriate) with the term ‘constructive’:
Intuitionism is an early-20th-century philosophy of mathematics developed by Brouwer, according to which mathematics is a free creation of a mind, and valid results are about what that mind creates (rather than about an external reality, as in platonism, or about nothing, as in formalism). From this controversial starting point, Brouwer drew even more controversial conclusions about both mathematics and logic (which he saw as derived from mathematics, rather than conversely as in logicism?). Intuitionism is one particular philosophy of constructivism.
Intuitionistic mathematics is the mathematics along the lines of the mathematics that Brouwer came up with. However, it's not necessary to accept Brouwer's philosophy to practise intuitionistic mathematics; conversely, one may accept Brouwer's philosophical starting place but not his conclusions about the resulting mathematics. Intuitionistic mathematics is one particular variety of constructive mathematics.
One example of intuitionistic mathematics (which nicely shows that intuitionism is not a matter of “belief” but of subject) is type II computable mathematics (see for instance Bauer 05, section 4.3.1).
Intuitionistic set theory? is a set theory, generally proffered as a foundation of mathematics, intended to capture intuitionistic mathematics. As the terminology is usually used (for example in the name of IZF, intuitionistic Zermelo-Frankel set theory), ‘intuitionistic’ means that excluded middle fails but power sets are included (making it impredicative). In contrast, ‘constructive’ set theory (such as CZF, constructive Zermelo-Frankel set theory) has function sets but not power sets (making it weakly predicative). The former is technically convenient, but the latter is better motivated. That said, Brouwer's mathematics was even more predicative, making both of these set theories stronger than he would accept.
Intuitionistic type theory is generally proffered as a foundation of mathematics that is (in most of its forms) both constructive and predicative. For purposes of comparing type theory to set theory, it might be nice if ‘intuitionistic’ and ‘constructive’ were distinguished for type theories as they are for set theories, but they aren't. (Then again, there was never much sense in making that distinction for set theories using that terminology.)
There is variant of the NuPrl type theory with choice sequences: PDF.
Intuitionistic logic is the logic that intuitionistic mathematics, set theory, and type theory use, which lacks the principle of excluded middle; other forms of constructive mathematics also use intuitionistic logic, which is therefore also known as constructive logic.
Mathematical principles
Brouwer did not believe in a rigourous formalization of mathematics, for various reasons (amongst which the mathematical incompleteness of formal systems, as later proved by Gödel's incompleteness theorem). He saw mathematical logic and formal systems as a correct part of mathematics, but held that this part could capture the essence nor the scope of mathematics in a meaningful way. Nonetheless he admitted that certain often-used arguments and mental constructions could be formalized `as an abbreviation'.
Brouwer’s student Arend Heyting (who later succeeded Brouwer as professor in Amsterdam) formalized what is now known as intuitionistic logic. He also is largely responsible for the Brouwer-Heyting-Kolmogorov interpretation (BHK) of intuitionistic logic, BHK can be seen as a precursor to realizability.
In hindsight, we may say that intuitionistic mathematics is done in a pretopos identified as Set.
We have the axiom of infinity and countable choice, as in classical mathematics.
We have the classically false principles of continuity? and continuous choice?.
We have the fan theorem and bar theorem, which are classically true but fail in Russian constructivism.
There's also some stuff about choice sequences that is highly philosophical, involving mathematical principles such as the Kripke's schema.
Although it's not enough to derive all of the above, much of the essence of intuitionistic mathematics, or at least intuitionistic analysis, can be summarized in the theorem that every (set-theoretic) function from the unit interval to the real line is uniformly continuous.
Intuitionistic mathematics is often regarded as a specialization of Bishop's constructive mathematics obtained by adding the above principles, but this is somewhat questionable if it refers to what Bishop actually did (and in particular the fact that he worked with Bishop sets); see Bishop's constructive mathematics for discussion.
Feel
In intuitionistic mathematics, already set theory behaves a lot like topology, a point stressed by Frank Waaldijk (web). He uses the Kleene-Vesley? system. Fourman’s continuous truth makes this remark precise using topos theory.
Although intuitionistic mathematics does not accept all function sets (much less power sets), it seems to allow for both inductive and coinductive structures; see a Café comment. The reluctance to add function spaces is similar to the problem of function spaces in topology; see nice category of spaces.
Related concepts
intuitionistic logic
BHK interpretation
constructive mathematics
intuitionistic type theory
realizability
choice sequence
References
For more see also the references at constructive mathematics.
The roots of Brouwer’s intuitionism are apparently in his PhD thesis
L.E.J. Brouwer, Over de Grondslagen der Wiskunde, PhD thesis, Universiteit van Amsterdam, 1907.
A reference written by Brouwer and still in print in English is
L.E.J. Brouwer, Brouwer’s Cambridge Lectures on Intuitionism, edited by D. van Dalen
Early exposition and endorsement:
Hermann Weyl, Über die neue Grundlagenkrise der Mathematik, Zürich 1920 (gdz, purl:PPN266833020_0010)
Hermann Weyl, Die heutige Erkenntnislage in der Mathematik, Weltkreis-Verlag, Erlangen 1926. (GoogleBooks)
Original articles on the formal logic of intuitionism (intuitionistic logic):
Arend Heyting, Die formalen Regeln der intuitionistischen Logik. I, II, III. Sitzungsberichte der Preußischen Akademie der Wissenschaften, Physikalisch-Mathematische Klasse (1930) 42-56, 57-71, 158-169
abridged reprint in:
Karel Berka, Lothar Kreiser (eds.), Logik-Texte, De Gruyter (1986) 188-192 [doi:10.1515/9783112645826]
Arend Heyting, Die intuitionistische Grundlegung der Mathematik, Erkenntnis 2 (1931) 106-115 [jsotr:20011630, pdf]
Andrey Kolmogorov, Zur Deutung der intuitionistischen Logik, Math. Z. 35 (1932) 58-65 [doi:10.1007/BF01186549]
Hans Freudenthal, Zur intuitionistischen Deutung logischer Formeln, Comp. Math. 4 (1937) 112-116 [numdam:CM_1937__4__112_0]
Arend Heyting, Bemerkungen zu dem Aufsatz von Herrn Freudenthal “Zur intuitionistischen Deutung logischer Formeln”, Comp. Math. 4 (1937) 117-118 [doi:CM_1937__4__117_0]
L. E. J. Brouwer, Points and Spaces, Canadian Journal of Mathematics 6 (1954) 1-17 [doi:10.4153/CJM-1954-001-9]
Arend Heyting, Intuitionism: An introduction, Studies in Logic and the Foundations of Mathematics, North-Holland (1956, 1971) [ISBN:978-0720422399]
Georg Kreisel, Section 2 of: Mathematical Logic, in T. Saaty et al. (ed.), Lectures on Modern Mathematics III, Wiley New York (1965) 95-195
Formalization in terms of realizability (the Kleene-Vesley topos), and hence proof that intuitionistic mathematics is consistent:
Stephen Kleene, R. E. Vesley, The foundations of intuitionistic mathematics, North-Holland (1965)
General review:
Stanford Encyclopedia of Philosophy, Intuitionism in the Philosophy of Mathematics
See also texts on constructive mathematics, such as:
Anne Sjerp Troelstra, Dirk van Dalen, Constructivism in Mathematics – An introduction, Vol 1, Studies in Logic and the Foundations of Mathematics 121, North Holland (1988) [ISBN:9780444702661]
Reviews with further developments:
Frank Waaldijk, On the foundations of constructive mathematics – especially in relation to the theory of continuous functions (web)
(with a focus on constructive analysis).
Discussion of an approach to general topology in intuitionistic mathematics:
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https://www.scribd.com/document/120125693/History-Of-Maths-1900-To-The-Present
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History of Maths 1900 To The Present
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History Of Maths 1900 To The Present - Free ebook download as PDF File (.pdf), Text File (.txt) or read book online for free. The Big Book of Mathematics, Principles, Theories, and Things. Part IV
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In Memoriam
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valid until 2019, see new site: https://www.aips.be
Paul Busch (1955-2018)
Paul Busch, born February 2, 1955, passed away June 9, 2018, after a short illness. He studied physics, mathematics and philosophy at the University of Cologne, where he received his doctorate in theoretical physics in 1982. His research interests focused on the foundations of quantum phyiscs and quantum information, in particular, the quantum theory of measurement, quantum theory and relativity, the theory of positive operator valued (POV) measures and the notion of unsharp quantum reality. He held post doctoral research positions at the University of Cologne, where he worked with Peter Mittelstaedt, and the Max Planck Intitute for Biophysical Chemistry in Göttingen. He was appointed University Docent in Theoretical Physics in Cologne in 1988, and was awarded the title of Professor there in 1994. He held appointments as visiting professor and research scholar at a number of institutions including Florida Atlantic University (1986), the University of Heidelberg (in Philosophy, 1994), Harvard University (hosted by Roy Glauber, Lyman Laboratory of Physics, 1994/5), and the Perimeter Institute of Theoretical Physics (2005-2007). He also held an adjunct professorship at the Department of Physics and Astronomy, University of Turku. He was professor of mathematical physics at the University of Hull and was Head of Department there from 2002 before joining the University of York in 2005. He was one of the editors of the “Foundations of Physics” journal and co-editor of the Springer monograph series "Fundamental Theories of Physics". He has published over 90 papers and is co-author of two monographs, The Quantum Theory of Measurement (Springer 1991/96) and Operational Quantum Physics (1995/97) (see http://paulbusch.wixsite.com/research-page and his CV there). He was elected Member of the Académie Internationale de Philosophie des Sciences in 2008, and Member of the Foundational Questions Institute (FQXi) in 2015. He has been awarded a Royal Society Leverhulme Trust Senior Research Fellowship (2017/18), and from 2016-2018, he was President of the International Quantum Structures Association.
(Brigitte Falkenburg)
Bernard d'Espagnat (1921-2015)
Bernard d’Espagnat, né le 22 août 1921 à Fourmagnac et mort le 1er août 2015 à Paris, est un physicien français.
Il est membre de l'Académie des sciences morales et politiques à partir de 1996, professeur émérite de l'Université Paris Orsay, désormais Université Paris Sud, membre de l'Académie internationale de philosophie des sciences (Bruxelles) à partir de 1975 et de l'université interdisciplinaire de Paris.
À partir de la fin des années 1960, il se distingue par ses travaux sur les enjeux philosophiques de la mécanique quantique et, en particulier, par sa conception du « réel voilé », qui constitue une approche originale du réalisme en physique.
Il est le fils du peintre Georges d’Espagnat (1870-1950), auquel il a consacré un ouvrage.
Bernard d'Espagnat a obtenu son doctorat à l'École polytechnique, qu'il a intégrée en 1942, et à l'Institut Henri-Poincaré (promotion Louis de Broglie). Il débute sa carrière en tant que chercheur au Centre national de la recherche scientifique(CNRS), de 1947 à 1957. Durant cette période, il travaille également avec le physicien Enrico Fermi à Chicago (1951-1952) et sur un projet de recherche mené par Niels Bohr à l'Institut de Copenhague (1953-1954). Il poursuit sa carrière scientifique en rejoignant le Centre d'études et de recherches nucléaires (CERN), à Genève, et comme physicien théorique à l'institution qui succède au CERN, l'Organisation européenne pour la recherche nucléaire (1954-1959).
De 1959 à sa retraite, en 1987, il enseigne à la Faculté des sciences de la Sorbonne. Il dirige le Laboratoire de physique théorique et des particules élémentaires de l'Université Paris-Sud 11, à Orsay (1980-1987). Par ailleurs, il est professeur invité à l’Université du Texas à Austin, en 1977, et à l'Université de Californie à Santa Barbara, en 1984. En 2009, il se voit décerner le prix Templeton.
Il a en particulier contribué à éclaircir les enjeux théoriques des expériences d’Alain Aspect sur le paradoxe EPR.
Ouvrages:
•Conceptions de la physique contemporaine ; les interprétations de la mécanique quantique et de la mesure (1965)
•Conceptual Foundations of Quantum Mechanics (1971)
•À la recherche du réel - Le regard d’un physicien, Gauthier-Villars, 1979 réédition Pocket, 1991
•Un Atome de sagesse, propos d’un physicien sur le réel voilé (1982)
•Nonseparability and the Tentative Descriptions of Reality (1984)
•Une incertaine réalité - Le monde quantique, la connaissance et la durée, Gauthier-Villars, 1985 réédition Fayard, 1993
•Penser la science ou les enjeux du savoir (1990)
•Georges d’Espagnat (1990)
•Regards sur la matière des quanta et des choses (en collaboration avec Étienne Klein) (1993)
•Le réel voilé - Analyse des concepts quantiques, Fayard, 1994
•Physique et réalité, un débat avec Bernard d’Espagnat (1997)
•Ondine et les feux du savoir. Carnets d'une petite sirène (1998)
•Traité de physique et de philosophie (2002)
•Candide et le physicien (avec Claude Saliceti, 2008)
Michael Detlefsen ( 1948 - 2019 )
Michael Detlefsen (20 October 1948 – 21 October 2019) was a McMahon-Hank Professor of Philosophy at the University of Notre Dame. His areas of special interest were logic, history of mathematics, philosophy of mathematics and epistemology.
Detlefsen began his academic career as an Assistant, and then Associate Professor, at the University of Minnesota, Duluth in 1975. He remained there until 1983 but would also hold a position as a visiting scholar at the University of Split, Croatia, from 1981 to 1982. He began teaching at Notre Dame as a Visiting Associate Professor in 1983 and became an Associate Professor there in 1984. He was promoted to full professor in 1989 and installed as a McMahon-Hank Professor of Philosophy in 2008. He held visiting professorships at the University of Split (1981–1982), the University of Konstanz (1987–1988, 1994), and at the Paris Diderot University (2007). He held a senior chaire d'excellence with the Agence Nationale de la Recherche (ANR) in France from 2007 through 2011. He was a past president of the Philosophy of Mathematics Association (PMA).
Scholarly work
Detlefsen wrote a number of works on the foundational ideas of the German mathematician David Hilbert, and other major nineteenth and twentieth century foundational thinkers, held research fellowships from various foundations including the International Research and Exchange Commission.
Stanislas Docks (1901-1985)
Le Père Stanislas Dockx o.p., fondateur des deux Académies AIPS et AISR.
Stanislas Dockx est né à Anvers (Belgique) le 31 mars 1901 dans une ancienne famille de commerçants anversois. Il fut d'abord orienté vers des études de commerce mais il fut vite bien plus intéressé par les problèmes philosophiques, sociologiques et religieux. Le 7 novembre 1920, il entra dans l'Ordre de St Dominique. Il fit son noviciat au couvent de La Sarte, près de Huy. De 1921 à 1924, il reçut sa formation philosophique au couvent de Gand. Au-delà des cours reçus, il tenait à lire directement les auteurs dans le texte. Il s'intéressa particulièrement à la métaphysique, à la logique, à la cosmologie, à la psychologie rationnelle et expérimentale. Vinrent ensuite les années d'études théologiques au couvent de Louvain. Il demanda alors l'autorisation de poursuivre des études en vue de l'obtention d'un doctorat en physique-mathématique mais ce projet fut retardé car il lui fut demandé d'assurer des cours (Ecriture Sainte, cosmologie, sciences naturelles) au Studium dominicain de La Sarte (de 1929 à 1936). Il eut cependant l'occasion, pendant ce temps, de suivre des cours de physique et de mathématique aux universités de Liège et de Louvain. Il élabora à ce moment une "Théorie fondamentale du système périodique des éléments" (éditée en 1959). En 1936, il fut nommé directeur du collège international des étudiants de l'université (dominicaine) de Fribourg (en Suisse). Mais il suivit lui-même des cours complémentaires de théologie et obtint, en 1938, le diplôme de docteur en théologie avec une thèse sur la théologie thomiste de la grâce ("Fils de Dieu par grâce", éditée en 1948). Il fut alors appelé au Studium dominicain de Varsovie où il assura (en latin) les cours de métaphysique et de théologie fondamentale. De retour en Belgique en 1939, il assura les cours de logique, de philosophie des sciences, de métaphysique et de cosmologie au Studium (philosophique) de Gand et ensuite des cours de théologie au Studium de Louvain, jusqu'en 1956. C'est pendant ces années qu'il fonda l'Institut International des Sciences Théoriques et tout d'abord la "classe des sciences profanes" (qui devint l'Académie Internationale de Philosophie des Sciences) dont le premier symposium eut lieu au Palais des Académies de Bruxelles en 1947. Un deuxième symposium eut lieu à Paris en 1949. Mais le suivant n'eut lieu qu'en 1961, étant donné ses obligations d'enseignement (et, de 1956 à 58, sa présence à Rome comme Pénitencier Apostolique à la basilique Sainte-Marie Majeure). A l'occasion du Concile Vatican II (1963-65), il fut sollicité comme expert ("peritus") par le patriarche grec-catholique Maximos IV. Le rapport qu'il rédigea dans le cadre de cette fonction ("Des pouvoirs dans l'Eglise",) fut distribué à tous les Pères conciliaires. La thèse, suivant laquelle le pouvoir des évêques tient à leur ordination comme évêque et non d'une délégation à partir du pape, fut largement discutée et se retrouva dans les textes conciliaires. Il fut aussi la cheville ouvrière de la levée des excommunications (de 1054) entre l'Eglise orthodoxe et l'Eglise catholique (le 7 décembre 1965) et des visites de Paul VI au Patriarche Athénagoras de Constantinople (25 juillet 1967) et de celui-ci à Rome (le 26 octobre 1967). C'est dans ce milieu des experts conciliaires qu'il trouva le vivier des théologiens (dont Mgr Ratzinger ...) désireux de continuer une recherche ouverte en théologie. Sollicités par le Père Dockx (qui avait également sollicité des théologiens des autres confessions chrétiennes), ils acceptèrent de devenir les membres fondateurs de l' Académie Internationale des Sciences Religieuses (la "classe des sciences sacrées" pressentie dès 1944). Le premier symposium de fondation eut lieu, dans la foulée, dans l'ancien couvent dominicain de Constance en mai 1964 sur le thème de la collégialité épiscopale. Agé de 63 ans, le Père Dockx se consacra dès lors entièrement aux Académies, jusqu'à ses 80 ans. Ses qualités humaines d'amabilité, de délicatesse, de discrétion, non moins que sa largeur d'esprit, son souci de la liberté de recherche et de parole et d'une recherche pointue chacun dans ses compétences donnèrent l'empreinte à son oeuvre. Après quelques accrocs de santé, il mourut le 7 novembre 1985 à Bruxelles, âgé de 84 ans.
Jean Gayon (1949-2018)
Professeur à l’Université Paris 1 Panthéon-Sorbonne, ancien directeur de l’IHPST, Jean Gayon s’est éteint le 28 avril 2018, à quelques semaines de son 69ème anniversaire.
Chercheur de double formation en philosophie et en biologie, Jean Gayon a consacré la majeure partie de ses recherches à l’histoire et à la philosophie de la biologie contemporaine, ses objets de prédilection étant les fondements conceptuels de la théorie de l’évolution, l’histoire de la génétique et de la biométrie. Il a aussi développé des réflexions éclairantes sur les aspects sociaux, politiques et éthiques des sciences de la vie et de la santé à l’époque contemporaine (en particulier sur l’eugénisme, le problème des races humaines et la biodiversité). Et il a aussi travaillé sur la philosophie générale des sciences et sur les rapports entre celle-ci et l’histoire des sciences : un sujet qui l’a occupé dans les derniers mois de sa vie, dans lesquels il n’a jamais cessé de travailler et d’animer la recherche collective au sein de l’IHPST et ailleurs.
Enseignant d’abord dans le secondaire, il a ensuite commencé sa carrière académique à l’Université de Bourgogne, où il a été nommé maître de conférences (1985-1990), puis professeur (1990-1997). Il a rejoint Paris en tant que professeur à l’Université Paris 7 (1997-2001), puis à l’Université Paris 1 Panthéon-Sorbonne (2001-2016), avant d’y être professeur émérite (depuis septembre 2016).
De 2010 à 2016 il a dirigé l’Institut d’Histoire et de Philosophie des Sciences et des Techniques.
Reçu à l’agrégation de philosophie en 1972 et devenu professeur de lycée à Maurepas (Yvelines), il retourne à l’université pendant neuf ans pour suivre une formation universitaire en biologie, du DEUG au DEA d’évolution de Claudine Petit, obtenu à l’université Paris 7 en 1983. Elle lui donne un pouvoir d’analyse exceptionnel dans les questions d’histoire de la biologie auxquelles il consacre les trente années suivantes.
Il soutient en 1989, sous la direction de François Dagognet, une thèse d’Etat sur la « théorie de la sélection naturelle : Darwin et après-Darwin ». Extrêmement documenté, nourri d’archives, ce travail explique comment ‘l’hypothèse de sélection naturelle’ avancée par Darwin se retrouve corroborée plusieurs décennies plus tard par la biologie évolutive moderne, selon un cheminement complexe qui passe par les biométriciens, Pearson et Galton, les premiers mendéliens, des études empiriques, puis la génétique des populations.
Le texte se clôt avec une analyse philosophique de la théorie dite neutraliste de Motoo Kimura, expliquant comment ce neutralisme prolonge plutôt qu’il ne récuse le sélectionnisme darwinien. Il deviendra un livre publié en français, Darwin et l’après Darwin (Kimé, 1992) puis en anglais sous le titre Darwinism’s struggle for survival en 1998 chez Cambridge University Press (1). Lu et commenté aussi bien par des philosophes et historiens des sciences que par des biologistes, ce texte est maintenant un classique de la littérature de philosophie / histoire de la biologie.
Suite à un long séjour aux Etats-Unis à cette époque, Jean Gayon a noué des contacts amicaux et professionnels avec la communauté des historiens et philosophes de la biologie, dont la structuration institutionnelle était alors en cours. Il sera l’un des membres fondateurs de la société de philosophie et d’histoire de biologie (ISHPSSB) dans les années 1990. Ces contacts américains, comme Richard Burian (Virginia Tech), Robert Brandon (Duke) ou Phillip Sloan (Notre Dame), seront étroitement associés à ses travaux, par exemple dans le domaine de l’histoire de la génétique.
Formé dans ce qu’il est convenu d’appeler la tradition de l’épistémologie historique française, proche de George Canguilhem et de Jacques Roger, Jean Gayon a initié un rapprochement entre celle-ci et la philosophie des sciences anglo-saxonne d’inspiration plus analytique, qui fut décisif pour ceux qu’il a formés, à Paris 7 puis Paris 1, et surtout pour l’orientation de l’IHPST, à partir de 2002.
L’œuvre philosophique de Jean Gayon, qui inclut outre le livre sur le darwinisme, plusieurs dizaines d’ouvrages dirigés et des centaines d’articles en anglais, français et parfois espagnol dans des revues internationales de philosophie ou d’histoire des sciences, couvre un grand nombre de questions qui peuvent se répartir selon les axes suivants:
L’épistémologie de la biologie évolutive, en particulier la question du concept de fonction, le rôle des concepts économiques en biologie évolutive, mais aussi l’histoire du transformisme - et ses grandes figures depuis Lamarck.
L’eugénisme - avec un intérêt massif pour Fisher, auquel Darwin et l'après-Darwin consacre des analyses qui font référence, et plus récemment l’amélioration humaine, human enhancement, qui communique avec les anciens projets eugénistes d’avant-guerre en leur ajoutant une dimension techniciste et médicale.
Les races, qui furent en leur temps un objet de préoccupation majeur pour les biologistes évolutionnaires fondateurs de la théorie classique (Théorie Synthétique) de l’évolution, souvent engagés dans une entreprise de réfutation scientifique des racismes.
La notion d’hérédité - aussi bien du point de vue de l’histoire de la génétique, de celui de son sens philosophique et son histoire au long cours.
L’épistémologie française, avec les figures régulièrement questionnées de Canguilhem et surtout Bachelard, et plus généralement l’interrogation sur les relations entre histoire des sciences et philosophie des sciences.
Professeur, Jean Gayon a formé comme directeur de thèses plusieurs générations d’étudiants et plusieurs ont obtenu le prix de thèse de Paris 1 ou le prix de la chancellerie de Paris. Les plus anciens sont aujourd’hui pour beaucoup d’entre eux chercheurs ou enseignants-chercheurs dans l’université française, spécialistes de philosophie et d’histoire de la biologie et de la médecine, mais parfois aussi de philosophie générale des sciences. Il a enseigné inlassablement en Licence et en Master, et introduit aux questions d’histoire de la biologie ou de philosophie de la génétique et de l’évolution des apprentis philosophes qui se tournèrent vers d’autres spécialités. Il n’est pas interdit de penser que le paysage français de philosophie académique de la biologie est en grande partie tributaire de Jean Gayon.
A l’Université Paris 1 Panthéon-Sorbonne, il a dirigé, de 2001 à 2005, le DEA d’Histoire et de Philosophie des Sciences, ancêtre du Master LOPHISC, l’Ecole Doctorale de philosophie de 2002 à 2010, le collège des Ecoles Doctorales de l’université Paris 1 de 2008 à 2010.
Responsable de l’équipe de philosophie de la biologie et de la médecine à l’IHPST depuis son arrivée, il a induit le tournant vers la philosophie des sciences qui caractérise cette équipe, en particulier par la constitution de réseaux de collaborations avec des philosophes étrangers (USA, Mexique, Canada, Royaume Uni…) comme avec des institutions de biologie (Muséum National d’Histoire Naturelle, Collège de France etc.)
Sous son impulsion se sont déroulés depuis 2004 plusieurs programmes de recherche qui jouèrent un rôle structurant pour cette équipe et qui ont abrité séminaires, colloques internationaux ou groupes de lecture : une ACI sur les fonctions, une ANR sur les rapports entre économie et évolution, un projet de ‘politique scientifique’ de Paris 1 sur l’histoire de la Théorie Synthétique de l’Evolution…
Comme directeur de l’IHPST, il a œuvré avec équilibre et enthousiasme à renforcer le rôle de cette institution sur la scène scientifique française et internationale, sans jamais privilégier son domaine de recherche, mais s’efforçant au contraire de faire prospérer toutes les équipes, en un contexte d’intense réciprocité. Puis, devenu professeur émérite, il avait récemment pris la direction de l’équipe ‘Histoire et philosophie des sciences’ pour y mener un travail de fond sur la manière d’articuler de nouveau ces deux disciplines, après l’inflexion vers la ‘philosophie des sciences’ des quinze dernières années.
Membre de la Leopoldina, membre titulaire de l’Académie internationale de philosophie des sciences, deux fois membre senior de l’IUF, membre de l’Institut international de philosophie, il avait participé au comité stratégique du CNRS dans les années 2000. Il a été aussi membre actif du Comité national d’histoire et philosophie des sciences (Académie des sciences), affilié à l’ICSU, et membre du Conseil d'Administration du CNHPS.
L’IHPST lui a consacré en 2017 des journées d’études pendant lesquelles des collègues philosophes et biologistes, français, mexicains et nord-américains, ont contribué à situer sa démarche. Ces communications font l’objet d’un ouvrage qui sort ce mois-ci (2).
Avant de nous quitter, il accepta aussi le projet d’un long livre d’interview de 525 pages mené par Victor Petit (3). Il y situe sa carrière par rapport aux traditions d’histoire des sciences en France et à la philosophie en général. Sa parution prochaine marquera une date dans l’histoire des recherches sur la connaissance.
Ce décès prématuré est une très grande perte pour toute la communauté philosophique française et internationale, et une très grande tristesse pour ses amis et ses proches. Nous ne perdons pas seulement un collègue. Nous perdons un maître, un guide, un exemple et un ami : un homme apparemment réservé, mais capable de grands élans d’amitié et d’humanité. Sans lui, nous nous sentons tous plus pauvres, autant comme scientifiques, que comme personnes. Nous ne l’oublierons pas, et tâcherons de continuer son œuvre et son enseignement.
Le directeur de l'IHPST, au nom de toute l'équipe.
(1) Gayon Jean. 1998. Darwinism's struggle for survival: heredity and the hypothesis of natural selection, Cambridge University Press, 516 p.
(2) Philosophie, histoire, biologie. Mélanges offerts à Jean Gayon, sous la direction de Francesca Merlin & Philippe Huneman. Paris, Editions Materiologiques.
(3) La connaissance de la vie aujourd’hui, Paris: ISTE, à paraître. Ce livre inclut comme le précédent une liste exhaustive des publications de Jean Gayon.
Gilles G. Granger (1920-2016)
Gilles-Gaston Granger (January 28, 1920- August 24, 2016) was a French philosopher who contributed to philosophy of social sciences, philosophy of logic and mathematics, and history of philosophy, writing on philosophers such as Aristotle, Condorcet, Bolzano, and Wittgenstein. Interested in the variety of methods underlying scientific reasoning, he created in 1964, developed and directed until 1986 the Center for Comparative Epistemology at the University of Aix-en Provence. Elected in 1986 at College de France with a chair of the same name, he has left a substantial bibliography including nineteen books and one hundred and fifty articles. His work has been translated in nine languages.
Gaston Granger was born and grew up in Paris. His father was a carpenter. His mother and younger sister both died from tuberculosis when he was still a toddler. Given his excellent results in primary school, he was encouraged to become a school teacher, a cursus which, at the time, did not require a baccalaureate. One of his father's clients, however, himself a Professor, understood that the young man had exceptional intellectual talents. Thanks to his help, young Granger was admitted to Lycée Henri IV, and became in 1940 a student in the prestigious Ecole Normale Supérieure. He passionately attended Jean Cavaillès' seminar until the latter was arrested as a resistant in August 1942. Cavaillès' work on the Philosophy of Mathematics may have motivated him and his classmate Jules Vuillemin to study mathematics. Along with other ENS students who opposed force labour in Germany, however, he had to leave ENS to join a Resistance group in Creuse in 1943. After the war, he kept his resistant code name of Gilles.
He was nominated in 1947 Professor of Philosophy at the University of São Paulo, where he taught and published articles in Portuguese. Nominated in the University of Rennes (France) in 1955, he defended a State Doctoral thesis comprising two books about Economic Methodology and the Social Mathematics of Condorcet.
On line with these works, his next major opus, Formal Thought and the Sciences of Man (1960), deals with the mathematical models that contribute to formalization in the human sciences. It is based on the transcendental claim that mathematics have an a priori role in producing scientific knowledge. Scientific objectification is seen as a symbolic process through which what Grangerwill later call "formal contents" are generated. Scrutinized from the viewpoint of science in the making, the computational properties of the languages of science across fields of inquiry (linguistics, economics, psychology) are emphasized as reflecting specific rational practices of interest to the philosopher of science.
After two years spent in Congo as a Director of the Ecole Normale Supérieure of Central Africa, he was nominated Professor at the University of Aix-en Provence in 1964. Within a few years, he managed to set up an active research group. As a "Center for Comparative Epistemology", it specialized in the study of the modes of knowledge production across scientific fields, with a strong emphasis on the philosophy of mathematics and logic. With its research seminar and his library, this CNRS-funded unit would soon attract students and researchers from around the world, in particular from Canada, where Granger has been a regular guest speaker and invited Professor. A major event organized in 1969 by the Center was the Conference entitled "Wittgenstein and the problem of the philosophy of science". That same year, Granger published a book on Wittgenstein, followed, much later, by a translation of Wittgenstein's Tractatus Logico-Philosophicus.
Granger's Essay on the Philosophy of Style, published in 1968, highlights stylistic variations in the formal analysis of magnitude, geometry, vectors, linguistic, and action theory. These variations are seen as the proper object of philosophy, whose goal is to interpret individual ways of construing the relation between form and content. Philosophical knowledge is now seen as intimately linked to the exploration of stylistic comparisons between scientific modelizations. Pour une Connaissance Philosophique, published in 1988, develops further the concept of philosophical knowledge as an interpretive endeavour, focussing on the stylistic variations and their specific types of formal contents.
From 1986 to 1990, Granger held the chair of Comparative Epistemology at College de France. He authored no less that eight more books after his retirement. Gilles Granger was Doctor Honoris Causa of the Universities of São Paulo, (Brazil) and Sherbrook (Canada). He has exerted a deep influence on his students and colleagues. Two books have been devoted to his work.
Publications
•Lógica e filosofia das ciências, São Paulo: Editora Melhoramentos, 1955.
• La Raison, Paris, Presses Universitaires de France, 1955, 1984.
• Méthodologie économique, Paris: Presses Universitaires de France, 1955.
• La mathématique sociale du Marquis de Condorcet, Paris, Presses Universitaires de France, 1956,
Paris: Odile Jacob, 1989.
• Pensée formelle et sciences de l'homme, Paris, Aubier-Montaigne, 1960, augmented edition: 1967, Paris:
Archives Karéline, 2010.
• Wittgenstein, Paris: Seghers, 1969.
• Essai d'une philosophie du style, Paris: Armand Colin, 1969, Paris: Odile Jacob, 1987.
• La Théorie aristotélicienne de la science, Paris: Aubier, 1976.
• Langages et épistémologie, Paris, Klincksieck, 1979.
• Leçon inaugurale, Paris: Collège de France, 1987.
• Pour la connaissance philosophique, Paris: Odile Jacob, 1988.
• Invitation à la lecture de Wittgenstein, Aix-en-Provence: Alinéa, 1990.
• La Vérification, Paris: Odile Jacob, 1992.
• La Science et les sciences, Paris: Presses Universitaires de France, coll. "Que sais-je ?", 1993.
• Formes, opérations, objets, Paris: Vrin, 1994.
• Le probable, le possible et le virtuel, Paris: Odile Jacob, 1995.
• L'irrationnel, Paris: Odile Jacob, 1998.
• La pensée de l'espace, Paris: Odile Jacob, 1999.
• Sciences et réalité, Paris: Odile Jacob, 2001.
• Philosophie langage science, Paris: EDP Sciences, 2003.
Translations
• Wittgenstein, L. Carnets 1914-1916. Paris: Gallimard, 1971.
• Wittgenstein, L. Tractatus Logico-Philosophicus. Paris: Gallimard, 1972.
Collective Books on Gilles-Gaston Granger's Philosophy:
• La Connaissance Philosophique. Essais sur la philosophie de Gilles Gaston Granger. Joëlle Proust &
Elisabeth Schwartz (eds.), Paris: Presses Universitaires de France, 1995.
• La pensée de Gilles-Gaston Granger, (Antonia Soulez, ed.), Paris: Hermann, 2010.
(by Joëlle Proust).
Adolf Grünbaum (1923-2018)
From Wikipedia, the free encyclopedia
Adolf Grünbaum (May 15, 1923 – November 15, 2018) was a German-American philosopher of science and a critic of psychoanalysis, as well as Karl Popper's philosophy of science. He was the first Andrew Mellon Professor of Philosophy at the University of Pittsburgh from 1960 until his death, and also served as Co-Chairman of its Center for Philosophy of Science (from 1978), Research Professor of Psychiatry (from 1979), and Primary Research Professor in the Department of History and Philosophy of Science (from 2006). His works include Philosophical Problems of Space and Time (1963), The Foundations of Psychoanalysis (1984), and Validation in the Clinical Theory of Psychoanalysis (1993).
Life and career
Adolf Grünbaum's family left Nazi Germany in 1938 and emigrated to the United States. He received a B.A. with twofold High Distinction in Philosophy and in Mathematics from Wesleyan University, Middletown, Connecticut in 1943.
During the second world war, Grünbaum was trained at Camp Ritchie, Maryland, and thus was one of the Ritchie Boys. He was stationed in Berlin and interrogated highly-placed Nazis, returning to the United States in 1946.
Grünbaum obtained both his M.S. in physics (1948) and his Ph.D in philosophy (1951) from Yale University. He was a chaired professor of Philosophy at Lehigh University, Bethlehem, Pennsylvania (1956–1960), after rising through the ranks there, starting in 1950, becoming a full professor in 1955.
In the fall of 1960, Grünbaum left Lehigh University to join the faculty of the University of Pittsburgh, where he became the first Andrew Mellon Professor of Philosophy. In that year, he also became the founding Director of that University's Center for Philosophy of Science, serving as Director until 1978. He and the colleagues he recruited then built world-class Philosophy and History and Philosophy of Science Departments at the university. Several of these colleagues had come from Yale University's Philosophy Department, starting in 1962. During this recruitment period the University of Pittsburgh appointed Nicholas Rescher, Wilfrid Sellars, Richard Gale, Nuel Belnap, Alan Ross Anderson, and Gerald Massey, among others.
In 2003, Grünbaum resigned from the Department of Philosophy at the University of Pittsburgh, while retaining his lifetime tenured Mellon Chair and all of his other affiliations at that university.
Grünbaum served as President of both the American Philosophical Association (Eastern Division) and the Philosophy of Science Association (two terms). He was the director of the Center for Philosophy of Science from 1960 to 1978. He was the president of the Division of Logic, Methodology and Philosophy of Science of the International Union of History and Philosophy of Science (IUHPS) in 2004–2005 and then automatically became president of the IUHPS from 2006 to 2007. He is also a Fellow of the American Academy of Arts and Sciences.
He received the Senior U.S. Scientist Prize from the Alexander von Humboldt Foundation (Germany, 1985), the Fregene Prize for science from the Italian Parliament (1998) and the Wilbur Lucius Cross Medal for outstanding achievement from Yale University (1990). Also, in May 1995, he received an honorary doctorate from the University of Konstanz in Germany and, in 2013, an honorary doctorate of philosophy from the University of Cologne in Germany. In 2013, he received the Großes Bundesverdienstkreuz from the Federal Republic of Germany.
Grünbaum was Jewish. He died in November 2018 at the age of 95.
Philosophical work
Grünbaum was the author of nearly 400 articles and book chapters as well as books on space-time and the critique of psychoanalysis. He is often viewed as part of the American brand of logical empiricism, associated especially with Hans Reichenbach.
Grünbaum did not embrace the prevailing — especially among physical scientists — Popperian philosophy of science, leading to some notoriety in the 1960s after he was ridiculed in print by the iconic physicist Richard Feynman. A much-quoted exchange followed Grünbaum's neo-Leibnizian suggestion that the flow of time might be an illusion only in conscious entities, in which Feynman asked whether dogs, then cockroaches, were sufficiently conscious entities. Reportedly as a mark of further disdain, Feynman refused to let his name be printed, becoming instead the easily recognizable "Mr. X".
Some 40 years later, writer Jim Holt would characterize Grünbaum as, in the 1950s, "the foremost thinker about the subtleties of space and time," and as, by the 2000s, "arguably the greatest living philosopher of science." Holt portrays a rationalist Grünbaum who rejects any hint of mysteriousness in the cosmos (a "great rejector").
Selected publications
· Modern Science and Zeno's Paradoxes (first edition, 1967; second edition, 1968)
· Geometry and Chronometry in Philosophical Perspective (1968)
· Philosophical Problems of Space and Time (first edition, 1963; second edition, 1973)
· The Foundations of Psychoanalysis (1984)
· Validation in the Clinical Theory of Psychoanalysis: A Study in the Philosophy of Psychoanalysis (1993)
· Collected Works, Volume 1 (ed. by Thomas Kupka): Scientific Rationality, the Human Condition, and 20th Century Cosmologies, Oxford University Press 2013. Volume 2: The Philosophy of Space & Time (ed. by Thomas Kupka), is forthcoming 2019; Volume 3: Lectures on Psychoanalysis (ed. by Thomas Kupka & Leanne Longwill), is forthcoming 2019 as well (both also with OUP).
Peter Kemp (1937-2018)
Peter Kemp, born January 24, 1937, passed away August 4, 2018. Originally educated in theology at the University of Aarhus, Denmark, but very early on he became interested in philosophy, in particular French philosophy, which he studied while attending various universities in Germany and France as a research scholar. Later he befriended Paul Ricoeur and his first major book Théorie de L’Engagement (1973) was influenced by his interaction with Ricoeur. A major feature of Kemp’s character was his commitments to humanistic principles and his active engagement in the public debate about many important cultural and societal issues. He believed that each and every human being was irreplaceable and this made him very critical towards modern technology. He saw the naïve use of technology as a threat to our understanding of human beings and to the way we treat them. For many years Peter Kemp was the director of the Center for Ethics and Law at the University of Copenhagen. In the years before retirement he spent his time as a professor of pedagogical philosophy at the University of Aarhus. Peter Kemp wrote approximately 40 books, some of which introduced French philosophy to a broader Danish audience. Another part of his oeuvre was written in French. Between 2003 and 2008 Kemp was elected as the President of the “Federation Internationale des Sociétés de Philosophie” (FISP). In 2016 he was appointed lifelong member of the “Académie des Sciences Morales et Politiques” (ASMP). A theme in Kemp’s philosophy was that we are all global citizens and with his death we missed a true humanist.
(Jan Faye)
Georg Kreisel (1923-2015)
Georg Kreisel (September 15, 1923 in Graz – March 1, 2015 in Salzburg) was an Austrian-born mathematical logician who studied and worked in Great Britain and America. Kreisel came from a Jewish background; his family sent him to England before the Anschluss, where he studied mathematics at Trinity College, Cambridge and then, during World War II, worked on military subjects. After the war he returned to Cambridge and received his doctorate. He taught at the University of Reading until 1954 and then worked at the Institute for Advanced Study from 1955 to 1957. Subsequently he taught at Stanford University and the University of Paris. Kreisel was appointed a professor at Stanford University in 1962 and remained on the faculty there until he retired in 1985.
Kreisel worked in various areas of logic, and especially in proof theory, where he is known for his so-called "unwinding" program, whose aim was to extract constructive content from superficially non-constructive proofs.
Kreisel was elected to the Royal Society in 1966; Kreisel remained a close friend of Francis Crick whom he had met in the Royal Navy during WWII.
While a student at Cambridge, Kreisel was the student most respected by Ludwig Wittgenstein. Ray Monk writes, "In 1944--when Kreisel was still only twenty-one--Wittgenstein shocked Rush Rhees by declaring Kreisel to be the most able philosopher he had ever met who was also a mathematician."
Kreisel was also a close friend of the Anglo-Irish philosopher and novelist Iris Murdoch. They met at Cambridge in 1947 during Murdoch's year of study there. Peter Conradi reports that Murdoch transcribed Kreisel's letters into her journals over the next fifty years. According to Conradi, "For half a century she nonetheless records variously Kreisel's brilliance, wit and sheer 'dotty' solipsistic strangeness, his amoralism, cruelty, ambiguous vanity and obscenity." Murdoch dedicated her 1971 novel An Accidental Man to Kreisel and he became a (partial) model for several characters in other novels, including Marcus Vallar in The Message to the Planet and Guy Openshaw in Nuns and Soldiers.
After retirement Kreisel lived in Salzburg, Austria. He wrote several biographies of mathematicians including Kurt Gödel, Bertrand Russell and Luitzen Egbertus Jan Brouwer.
Jean Ladrière (1921-2007)
Jean Ladrière naquit à Nivelles (Belgique) en 1921. Au cours de ses études de philosophie à l'Université de Louvain, il s'intéressa particulièrement à la logique formelle (cours du prof. Dopp) et à la philosophie sociale (cours du chanoine Jacques Leclercq).
Son intérêt pour la logique formelle l'amena à faire des études complémentaires en mathématiques pures. Il consacra sa thèse de doctorat en philosophie (présentée en 1949) à l'étude des "implications du théorème de Gödel pour la théorie de la démonstration". Il rédigea encore un mémoire de licence en mathématiques sur les fonctions récursives (1951) et une thèse d'agrégation (1957) sur "Les limitations internes des formalismes" où, à partir des problèmes posés par le théorème de Gödel et des théorèmes apparentés, il montrait que le langage mathématique ne pouvait pas, ultimement, se passer du langage naturel. C'était donc le thème, très kantien, des limites de la rationalité qui était repris mais dans une perspective nouvelle, comme une exigence interne d'ouverture (et donc non interprété comme une limite) à d'autres champs de sens.
Dès 1958, il fut chargé d'enseigner la philosophie des sciences et des mathématiques à l'Institut Supérieur de Philosophie de Louvain et, par un regroupement bien réfléchi de cours, fit de la philosophie des sciences une section à part entière. A l'occasion de son enseignement, très suivi, il ne cessa de développer les problèmes d'épistémologie des sciences et fut un des premiers à introduire les épistémologues anglo-saxons et du Cercle de Vienne (dont, particulièrement, Wittgenstein) dans le monde philosophique francophone et d'en dégager les enjeux (grâce à sa maîtrise des langues, sa très large érudition et son magistral esprit de synthèse). De là, il s'intéressa aussi aux problèmes plus généraux de philosophie du langage, ce qui lui permettait de rejoindre son intuition première d'ouverture souhaitable ou même nécessaire entre les différents "champs de sens" ou langages, dont le langage théologique. Ce dernier et, plus largement, la confrontation entre la raison et la foi tint une place importante dans la réflexion de l'auteur. Il rédigea de très nombreux articles sur tous ces sujets (partiellement rassemblés dans trois volumes sous le titre: Articulation du sens, 1970,1984, 2004) et un livre: La science, le monde et la foi (Casterman, Tournai, 1972).
Son intérêt pour les sujets de société ne fut pas en reste. Au niveau académique, il reprit le cours de philosophie sociale du chanoine Leclercq pendant de nombreuses années. Il fut co-fondateur d'un Centre de recherche et d'information socio-politique (le CRISP, qui fait toujours référence comme Centre d'analyse indépendant et pertinent sur l'actualité socio-politique belge). En 1973, il publia: Vie sociale et destinée (Duculot, Gembloux), reprenant des thèmes-clés de ses cours. Dans les années '70, l'Unesco lui demanda de rédiger un rapport sur le défi de la science et de la technique par rapport aux cultures. Ce rapport fut publié sous le titre: Les enjeux de la rationalité (Aubier-Montaigne, Paris, 1977). Dans les années '80, il soutint la création et participa activement à l'activité d'un Centre d'études bioéthiques attaché à la faculté de médecine de l'Université de Louvain. En 1999 paraissait encore un volume: L'éthique dans l'univers de la rationalité (Fides, Montréal, 1999), rassemblant diverses contributions et formant une véritable introduction philosophique à l'éthique appliquée.
Il fit toute sa carrière à l'Université de Louvain (où il dirigea une centaine de thèses de doctorat !). Tous ses étudiants se souviendront de sa manière magique de mettre en valeur leurs (parfois maigres) connaissances. Il fut professeur invité dans de très nombreuses universités étrangères (en France, Portugal, Pologne, Congo, Chili, Brésil, Japon, Etats-Unis). Il participa à de multiples colloques et congrès. Toutes les personnes qui l'ont rencontré ont pu apprécier son extrême modestie et sa délicatesse infinie. Il reçut le titre de docteur honoris causa d'une dizaine d'universités. Il fut membre (et exerça des présidences) de nombreuses associations internationales de philosophie (dont l'AIPS et l'AISR).
Peter Mittelstaedt (1929-2014)
Peter Mittelstaedt (* 24. November 1929 in Leipzig; † 21. November 2014) war ein deutscher Physiker und Wissenschaftstheoretiker.
Leben:
Nach dem Studium der Physik an den Universitäten Jena, Bonn und Göttingen, mit Promotionsabschluss in Göttingen in Theoretischer Physik im Jahre 1956 bei Werner Heisenberg, hatte Mittelstaedt Forschungsaufenthalte beim CERN in Genf sowie am MIT in Cambridge, USA und am damaligen Max-Planck-Institut für Physik und Astrophysik in München, wo er sich auch im Jahre 1961 habilitierte. Seit 1965 war Mittelstaedt Professor für Theoretische Physik an der Universität zu Köln, seine Hauptarbeitsgebiete waren Philosophie der Naturwissenschaft, Wissenschaftstheorie und Logik sowie Grundlagen der Relativitätstheorie und Quantentheorie. 1968 und 1969 war er Dekan der Mathematisch-Naturwissenschaftlichen Fakultät, 1970 und 1971 Rektor der Universität, 1971 bis 1973 Prorektor, später Prorektor für Forschung in Köln. Mittelstaedt wurde 1995 emeritiert.
Publikationen:
Bücher
Philosophische Probleme der modernen Physik, BI Wissenschaftsverlag 1963 (7. Aufl. 1989)
Klassische Mechanik, BI 1970 (2. Aufl. 1995)
Die Sprache der Physik: Aufsätze und Vorträge, BI 1972
Der Zeitbegriff in der Physik – physikalische und philosophische Untersuchungen zum Zeitbegriff in der klassischen und relativistischen Physik, BI Wissenschaftsverlag 1976 (3. Aufl. 1989)
Quantum Logic, Dordrecht, Reidel 1978
Sprache und Realität in der modernen Physik, BI 1986, mit Paul Busch, Pekka J. Lahti The Quantum Theory of Measurement, Springer, Lecturenotes in Physics, 1991, 2. Auflage 1996
The Interpretation of Quantum Mechanics and the Measurement Process, Cambridge University Press 1998 (paperback 2004) mit Paul A. Weingartner Laws of Nature, Springer 2005
Rational Reconstructions of Modern Physics, Springer, Dordrecht 2011, ISBN 978-94-007-0076-5.
Herausgeber (Auszug)
Symposium on the Foundations of Modern Physics mit Pekka J. Lahti, 1985, 1987, 1990
Symposium on the Foundations of Modern Physics mit Paul Busch und Pekka J. Lahti, 1993
Wissenschaftliche Artikel siehe Homepage
Jesús Mosterín (1941-2017)
Jesús Mosterín (Bilbao, 24 sept. 1941-Barcelona, 4 oct. 2017)
On the 4th of October, 2017, we heard the sad news that AIPS member Jesús Mosterín passed away. He was a highly relevant intellectual in the Spanish milieu, and well known abroad for his contributions to the philosophy of cosmology and biology. It seems not too far-fetched to say that he was a Spanish Russell: a logician by training, founder of the Barcelona group in logic, who unfolded his talent and opened up his interests to a much broader range of issues, becoming a public figure. His views were always marked by a rational, objectivist approach to the questions at stake, which earned him a reputation of being ‘a rationalist.’ But he was, more than that, a thinker of life, of the value of life.
Jesús Mosterín died of cancer caused by exposition to asbestos, an illness about which he spoke openly and with great lucidity two years ago (‘Una cita con la parca’, El País, March 2015). Lucidity is indeed something that comes naturally to mind when speaking of Jesús – an enlightened attitude, an openness of mind without dogmas, but with great rational demands.
He began studying mathematical logic in Germany, Münster, before coming back to Spain and settling in Barcelona. After 1996, he would abandon the Univ. of Barcelona and become a member of the Consejo Superior de Investigaciones Científicas (CSIC). In his early phase, as professor at Barcelona and founder of its logic group, he published texts on elementary logic, on set theory, on second-order logic. But during the 1970s some experiences would change his life and orientation. He started working for the editorial house Salvat and collaborating with renowned naturalist Felix Rodriguez de la Fuente in the publication of an encyclopaedia of animal life, Fauna. This would lead to a life-long engagement with related topics, including his opposition to bullfighting, reflections on the question of animal rights, and the presidency of Proyecto Gran Simio (Great Ape). He wrote books such as ¡Vivan los animales!, A favor de los toros, El triunfo de la compasión, El reino de los animales and El derecho de los animales, as well as many papers on these issues, thus becoming a well known public figure in these debates in Spain and Latin America.
As he wrote, “It is in our hands to take on the role of lucid guardians of the biosphere, or else abdicate our responsibility and become drunken witnesses of the disaster that we ourselves are causing.” En nuestras manos está asumir nuestro papel de guardianes lúcidos de la biosfera, o abdicar de nuestra responsabilidad y asistir como testigos borrachos al desastre que nosotros mismos estamos provocando.
The expansion of Mosterín’s range of interests had become clear in 1978 with the publication of Racionalidad y acción humana (several editions up to 2008). In this and other works, such as Filosofía de la cultura, his approach was marked by a highly interdisciplinary perspective, combining ideas from science with philosophical reflections. In his approach to culture, e.g., he liked to emphasize the different forms of cultural life in animals, concluding that we are not the only cultural animal. This line of work culminated with the publication in 2006 of La naturaleza humana, a book in which Mosterín fights to establish the great role of biological traits in human life and behaviour, against philosophical (or other) attempts to insist on the indeterminacy of the human.
Special mention deserves his Spanish edition of Kurt Gödel, Obras completas (Madrid: Alianza Editorial, 1981, 2006), uniting in a single volume all published works of Gödel, which came out before the English edition of collected works prepared by Feferman. And more recently the edition of Rudolf Carnap’s Untersuchungen zur allgemeinen Axiomatik, prepared with Thomas Bonk (Darmstadt: Wissenschaftliche Buchgesellschaft, 2000).
A particularly relevant contribution to the literature in philosophy of physics is the paper written jointly with J. Earman, A critical look at inflationary cosmology. Philos. Sci. 66 (1999), no. 1, 1–49.
The Spanish-speaking literature in philosophy of science is indebted to Mosterín for a very relevant reference book, the lengthy Diccionario de Lógica y Filosofía de la Ciencia, written in collaboration with Roberto Torretti (Madrid: Alianza Editorial, 2002; second edn. 2010). But there are many other contributions that could be mentioned here, among which I shall mention the book Conceptos y teorías en la ciencia (3rd edn, 2000). Los lógicos (2000, 2007), which offers highly readable presentations of the life and work of key figures in the history of modern logic – Frege, Russell, Cantor, Gödel, von Neumann, Turing. And the collection of papers Ciencia viva: Reflexiones sobre la aventura intelectual de nuestro tiempo (2001, 2006).
All of these works have gone through two or more editions, which is a clear indication of the public following of Mosterín’s well-informed, clear and insightful discussions of intellectual topics.
(by José Ferreirós)
Ilya Prigogine (1917-2003)
Ilya Prigogine (25 janvier 1917 à Moscou - 28 mai 2003) est un physicien et un chimiste belge d'origine russe. Il a reçu le prix Nobel de chimie en 1977, après avoir reçu la Médaille Rumford en 1976.
Il est connu surtout pour sa présentation sur les structures dissipatives et l'auto-organisation des systèmes, qui ont changé les approches par rapport aux théories classiques basées sur l'entropie. Ce en quoi il révèle une théorie parallèle à la théorie du chaos. Dans La Nouvelle Alliance. La métamorphose de la science, Prigogine développe la thèse suivante : la science classique considérait les phénomènes comme déterminés et réversibles, ce qui est en contradiction avec l'expérience courante. L'irréversibilité des phénomènes temporels caractéristique de la thermodynamique (non linéaire) réconcilie la physique avec le sens commun, tout en faisant date dans l'histoire de la thermodynamique.
Biographie
Il étudia la chimie à l'Université libre de Bruxelles en Belgique.
Ilya Prigogine explique ainsi son parcours : jeune émigré de Moscou d'origine juive, exilé en Allemagne puis en Belgique à Bruxelles pour fuir le nazisme , il voulut comprendre comment on arrivait à devoir fuir son propre pays. Il aborda la politique mais fut contraint d'étudier le droit. Voulant comprendre le comportement d'un accusé, il étudia la psychologie. Pour comprendre clairement la psychologie et la science du comportement, il buta sur le fonctionnement du cerveau humain. Ainsi, il étudia la biologie, la chimie et enfin la biochimie. En poussant plus loin pour comprendre les interactions chimiques, il étudia la physique des particules. De la physique, il passa à l'astrophysique et à la cosmologie. Il aborda alors les questions fondamentales : la matière, le vide, le temps et son sens unique (la flèche du temps). Pour comprendre la flèche du temps il dut étudier les structures dissipatives.
En 1977, il est lauréat du prix Nobel de chimie « pour ses contributions à la thermodynamique hors équilibre, particulièrement la théorie des structures dissipatives ».
Il cofonda le centre qui porte son nom à l'Université du Texas à Austin.
Il laissa également son nom à la Haute École Libre de Bruxelles Ilya Prigogine (HELB IP), associée à l'Université libre de Bruxelles (ULB). Il était membre de l'Académie roumaine.
Distinctions et récompenses
Il a reçu le prix Francqui en 1955 et le titre de docteur honoris causa de l'Université Jagellon de Cracovie en 1981. Il est lauréat du prix Nobel de chimie en 1977.
Publications
Introduction à la thermodynamique des processus irréversibles, Dunod, 1968, (ISBN2-87647-169-8)
Structure, stabilité et fluctuations - avec P. Glansdorff, Masson, 1971, (ISBN2-2252-9690-1)
Physique, temps et devenir - Masson, 1980, (ISBN2-2256-6792-6)
La Nouvelle alliance - avec Isabelle Stengers, Gallimard, 1986, (ISBN2-0703-2324-2)
Entre le temps et l'éternité - avec Isabelle Stengers, Fayard, 1988, (ISBN2-2130-2172-4)
A la rencontre du complexe - avec Grégoire Nicolis, Presses universitaires de France, 1992, (ISBN2-1304-3606-4)
Les Lois du chaos (Le leggi del caos) - Flammarion, 1993, transcription de deux conférences données à l'université de Milan en 1992, (ISBN2-0821-0220-3)
Thermodynamique, des moteurs thermiques aux structures dissipatives - avec Dilip Kondepudi, Odile Jacob, 1996,(ISBN2-7381-0646-3)
La Fin des certitudes, Odile Jacob, 1996, (ISBN2-7381-0330-8)
L'Homme devant l'incertain - Odile Jacob, 2001, (ISBN2-7381-0831-8)
Le Monde s'est-il créé tout seul ?, avec Henri Atlan, Joël De Rosnay, Albert Jacquard, Jean-Marie Pelt et Trinh Xuan Thuan, Albin Michel, 2008, (ISBN2-2261-7855-4)
Erhard Scheibe (1927-2010)
Erhard Scheibe was born in Berlin on September 24, 1927, a few days after the physicist Niels Bohr gave his famous “Como” lecture. Indeed, philosophical problems concerning the relations between quantum mechanics and classical physics stood at the centre of Scheibe’s work throughout his life. In 1946, he began to study mathematics and physics in Göttingen, where he belonged to the circle of young scholars gathered around Werner Heisenberg and Carl-Friedrich von Weizsäcker. In 1955, he there received his doctorate with a thesis in mathematics. Then he was working with Heisenberg at the Max Planck Institute for Physics in Göttingen. In 1957, he proceeded to Hamburg as von Weizsäcker’s assistant. He obtained his Habilitation in 1963 in Hamburg with a philosophical study of quantum mechanics, Die kontingenten Aussagen in der Physik [The Contingent Propositions of Physics] (Athenäum, 1964). From 1964 to 1982 was a full Professor of Philosophy at Göttingen, and from 1982 to his retirement in 1992, at Heidelberg. In 1973, he published The Logical Analysis of Quantum Mechanics (Pergamon Press, 1973). His lectures and essays covered themes from the concept of cause to scientific explanation, from Leibniz to Kant, from the laws of physics to the laws of nature, from the structure of physical theories to the progress of physics, not to mention the many technical articles on quantum mechanics and special or general relativity. Scheibe’s reputation grew. He became, not just one of the most important philosophers of the exact sciences, but the one who shaped the philosophy of physics in the German speaking world. In 1977, he became a full member of the Academy of the Sciences of Göttingen. In 1979-80, he was a fellow at the Center for Interdisciplinary Research (ZIF) at Bielefeld and at the Center for Philosophy of Science in Pittsburgh. In 1982, he became a corresponding member of the Academy of Science and Letters of Mainz, and of the Académie Internationale de la Philosophie des Sciences. In 1987, he spent the winter term as an invited professor at the University of California in Irvine. In the academic year 1987/88, he was a fellow at the Wissenschaftskolleg [Institute for Advanced Studies] in Berlin. In 1988, he held the Presidency of the Leibniz Committee of the Academy in Göttingen. From 1989 to 2002, he was a member of the Editorial Board of the journal Philosophia Naturalis. After his retirement in 1992, he published a monumental investigation of the unity of physics in the face of incommensurable theories, Die Reduktion physikalischer Theorien. Ein Beitrag zur Einheit der Physik [The Reduction of Physical Theories. A Contribution to the Unity of Physics] (Springer, Vol. I: 1997, Vol. II: 1999). A collection in English of Scheibe’s most important papers and lectures followed, making a great deal more of his work accessible to the English speaking philosophical world. Between Rationalism and Empiricism: Selected Papers in the Philosophy of Physics (Springer, 2001). In 2003, Erhard Scheibe was made an honorary member of the Gesellschaft für Analytische Philosophie (GAP). His final book, Die Philosophie der Physiker [The Philosophy of the Physicists] (Beck, 2006, 2007), was wrested from the ravages of an increasingly debilitating illness, and Scheibe passed away on January 7, 2010, in Hamburg. (See The Philosopher of the Physicists. The Legacy of Erhard Scheibe. In: General Journal for Philosophy of Science 42 (2011), 1-15.)
(Brigitte Falkenburg)
Dudley Shapere (1928 - 2016)
Dudley Shapere was an internationally prominent philosopher of science. He studied at Harvard University as an undergraduate and graduate, receiving a doctorate in philosophy in 1957. Subsequently, he taught at Ohio State University, the University of Chicago, the University of Illinois, and The University of Maryland at College Park. In 1984 he was appointed the Z. Smith Reynolds Professor of History and Philosophy of Science at Wake Forest, from where he retired and became a Professor Emeritus in 2001.
Shapere was a titular member of our Academy, as well as a fellow of the American Association for the Advancement of Science, the American Philosophy of Science Association, the American Psychological Association, the History of Science Society, and the American Philosophical Association. During his career, he held visiting appointments at numerous universities and research centers, notably Rockefeller University, Harvard University, and The Institute for Advanced Study, Princeton. He served at the U.S. National Science Foundation as Program Director in History and Philosophy of Science, 1966-1975. Shapere lectured worldwide regularly; his topics of highest demand were conceptual change, the concept of observation in science and philosophy, and the philosophical impact of evolutionary ideas in science. The author of numerous articles in scholarly journals, his books include Reason and the Search for Knowledge, Philosophical Problems of Natural Science, and Galileo: A Philosophical Study.
Dudley Shapere is survived by his wife of 42 years, Hannah Hardgrave; their daughters Elizabeth and Christine Anne; his son Alfred and daughter Catherine by his previous marriage to Alfreda Bingham; and five grandchildren
(by Alberto Cordero)
Patrick Suppes (1922-2014)
Patrick Colonel Suppes (/ˈsʊpɪs/; March 17, 1922 – November 17, 2014) was an American philosopher who made significant contributions to philosophy of science, the theory of measurement, the foundations of quantum mechanics, decision theory, psychology and educational technology. He was the Lucie Stern Professor of Philosophy Emeritus at Stanford University and until January 2010 was the Director of the Education Program for Gifted Youth also at Stanford.
Suppes was born on March 17, 1922, in Tulsa, Oklahoma. He grew up as an only child, later with a half brother George who was born in 1943 after Patrick had entered the army. His grandfather, C.E. Suppes, had moved to Oklahoma from Ohio. Suppes' father and grandfather were independent oil men. His mother died when he was a young boy. He was raised by his stepmother, who married his father before he was six years old. His parents did not have much formal education.
Suppes began college at the University of Oklahoma in 1939, but transferred to the University of Chicago in his second year, citing boredom with intellectual life in Oklahoma as his primary motivation. In his third year, at the insistence of his family, Suppes attended the University of Tulsa, majoring in physics, before entering the Army Reserves in 1942. In 1943 he returned to the University of Chicago and graduated with a B.S. in meteorology, and was stationed shortly thereafter at the Solomon Islands to serve during World War II.
Suppes was discharged from the Army Air Force in 1946. In January 1947 he entered Columbia University as a graduate student in philosophy as a student of Ernest Nagel and received a PhD in 1950. In 1952 he went to Stanford University, and from 1959 to 1992 he was the director of the Institute for Mathematical Studies in the Social Sciences (IMSSS). (He was later to become the Lucie Stern Professor of Philosophy, Emeritus, at Stanford.
In the 1960s Suppes and Richard C. Atkinson (the future president of the University of California) conducted experiments in using computers to teach math and reading to schoolchildren in the Palo Alto area. Stanford's Education Program for Gifted Youth and Computer Curriculum Corporation (CCC, now named Pearson Education Technologies) are indirect descendants of those early experiments. At Stanford, Suppes was instrumental in encouraging the development of high-technology companies that were springing up in the field of educational software up into the 1990s, (such as Bien Logic).
One computer used in Suppes and Atkinson's Computer-assisted Instruction (CAI) experiments was the specialized IBM 1500 Instructional System. Seeded by a research grant in 1964 from the U.S. Department of Education to the Institute for Mathematical Studies in the Social Sciences at Stanford University, the IBM 1500 CAI system was initially prototyped at the Brentwood Elementary School (Ravenswood City School District) in East Palo Alto, California by Suppes. The students first used the system in 1966.
During the 1950s and 1960s Suppes collaborated with Donald Davidson on decision theory, at Stanford. Their initial work followed lines of thinking which had been anticipated in 1926 by Frank P. Ramsey, and involved experimental testing of their theories, culminating in the 1957 monograph Decision Making: An Experimental Approach. Such commentators as Kirk Ludwig trace the origins of Davidson's theory of radical interpretation to his formative work with Suppes.
In 1965 he was elected as a member of the National Academy of Sciences for his work on mathematical psychology.
On November 13, 1990, President George H. W. Bush awarded Suppes with the prestigious President's National Medal of Science for work in Behavioral and Social Science.
In 1994 he was inducted as a Fellow of the Association for Computing Machinery. He is the laureate of the 2003 Lakatos Award for his contributions to the philosophy of science.
He is a member of the Norwegian Academy of Science and Letters.
Bibliography:
Suppes, Patrick; Arrow, Kenneth J.; Karlin, Samuel (1960). Mathematical models in the social sciences, 1959: Proceedings of the first Stanford symposium. Stanford, California: Stanford University Press. ISBN 9780804700214.
Including: Suppes, Patrick (1960), Stimulus-sampling theory for a continuum of response, pp. 348–363.
Suppes, Patrick (1972) (1960). Axiomatic Set Theory. Dover. Spanish translation by H. A. Castillo, Teoria Axiomatica de Conjuntos.
Suppes, Patrick (1984).Probabilistic Metaphysics, Blackwell Pub; Reprint edition (October 1986)
Humphreys, P. (1994). Patrick Suppes: Scientific Philosopher, Synthese Library (Springer-Verlag).
Vol. 1: Probability and Probabilistic Causality.
Vol. 2: Philosophy of Physics, Theory Structure and Measurement, and Action Theory.
Suppes, Patrick (1999) (1957). Introduction to Logic. Dover. Spanish translation by G. A. Carrasco, Introduccion a la logica simbolica. Chinese translation by Fu-Tseng Liu.
Suppes, Patrick (2002). Representation and Invariance of Scientific Structures. CSLI (distributed by the University of Chicago Press).
Suppes, Patrick; Hill, Shirley (2002) (1964). A First Course in Mathematical Logic. Dover. Spanish translation.
Suppes, Patrick; Luce, R. Duncn; Krantz, David; Tversky, Amos (2007) (1972). Foundations of Measurement, Vols. 1–3. Dover
Anne Sjerp Troelstra (1939-2019)
Anne Sjerp Troelstra (Maartensdijk,1939– Blaricum, 2019)was full professor of pure mathematics and foundations of mathematics at the University of Amsterdam. He described his research interests as ‘history and philosophy of constructivism; metamathematics of systems based on intuitionistic logic; proof theory'. Parts of his work have been taken up in computer science as well.
Troelstra was an academic grandson of L.E.J. Brouwer, and began his contributions to the latter’s foundational program with the dissertation Intuitionistic General Topology, supervised by Arend
Heyting, and defended in 1966. A one-year leave in 1966-1967 at Stanford, where his host was Georg Kreisel, was perhaps his second most formative experience, for both the topics and the convictions behind his main line of work, the formal metamathematics of intuitionistic mathematics. This influence was first seen in Troelstra's widely-read Principles of Intuitionism of 1969, which
originated in a lecture series he had given at the Summer Conference on Proof Theory and Intuitionism, Buffalo, NY, in 1968.
In 1970 Troelstra succeeded Heyting as full professor, and he remained at the University of Amsterdam until his retirement in 2000. Over the years, his work naturally became somewhat more diverse, and in particular he increasingly wrote also on the history and philosophy of intuitionism (and of constructive mathematics in general).
Another scientific and leisure interest of his was botany, which extended to accounts of journeys made to investigate natural history (botany, zoology, mineralogy, geology). This led to three books in Dutch, as well as the voluminous Bibliography of Natural History Travel Narratives.
Besides the Académie Internationale de Philosophie des Sciences, he was an elected member of the Royal Dutch Academy of Sciences (1976) and corresponding member of the Bavarian Academy of Sciences (1996).
In 1996 the Technische Universität München awarded him the F.L. Bauer prize for internationally outstanding contributions to theoretical computer science, citing his contributions to making intuitionistic logic useful in extracting algorithms and programs from mathematical proofs.
Troelstra's scientific archive is kept at the Noord-Hollands Archief in Haarlem.
His was a stern, ironical, and very scientific mind.
Principal publications:
- Principles of Intuitionism (Lecture Notes in Mathematics 95,Heidelberg: Springer, 1969)
- (with Georg Kreisel) Formal systems for some branches of intuitionistic analysis (Annals of Mathematical Logic, 1 (1970),
229-387)
- (with Jeff Zucker, Craig Smorynski, and William Howard) Metamathematical Investigations of Intuitionistic Arithmetic and
Analysis (Lecture Notes in Mathematics 344, Heidelberg: Springer, 1973)
- Analyzing choice sequences, Journal of Philosophical Logic 12 (1983), 197-260.
- (with Jane Kister and Dirk van Dalen) Omega-Bibliography of mathematical Logic. Vol. VI: Proof Theory and Constructive Mathematics (Heidelberg: Springer, 1987)
- (with Dirk van Dalen) Constructivism in Mathematics (Amsterdam:Elsevier, 1988, 2 volumes)
- (with Helmut Schwichtenberg) Basic Proof Theory (Cambridge Uiversity Press, 1996)
- Bibliography of Natural History Travel Narratives (Leiden: Brill, 2016)
Daniel Vanderveken (1949-2019)
Daniel Vanderveken était un Philosophe Québécois.
Docteur en Philosophie de l'Université catholique de Louvain, il était Professeur à l'Université du Québec à Trois-Rivières et membre titulaire de l'Institut International de Philosophie. Spécialiste de la philosophie analytique et de la sémiotique, membre de l'Académie internationale de philosophie des sciences, directeur du Groupe de recherche sur la communication et le discours, il avait fondé avec le philosophe américain John Searle la logique des actes de discours. Il avit reçu le Prix d'excellence en recherche dans la catégorie « arts, sciences humaines et sociales ainsi que sciences de la gestion ».
Quelques-uns de ses ouvrages:
Actions, Rationalité & Décision - Actions, Rationality & Decision, avec Denis Fisette (codir.), Londres, Volume 6, College Publications, collection Cahiers de Logique et d'Épistémologie, 2008.
Logic, Thought and Action dans la collection Logic (dir.), Epistemology and the Unity of Science de Springer, 2005.
Les actes de discours : Essai de philosophie du langage et de l'esprit sur la signification des énonciations, Liège et Bruxelles, Pierre Mardaga, 1988.
Foundations of Illocutionary Logic, avec John Searle, Cambridge University Press, 1985.
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Sadly, we have to inform you that Anne Troelstra, after a short illness, suddenly died on March 7th, aged 79. A world-renowned eminent researcher, a supportive colleague, a teacher who ...
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Sadly, we have to inform you that Anne Troelstra, after a short illness, suddenly died on March 7th, aged 79. A world-renowned eminent researcher, a supportive colleague, a teacher who trained many students to become important scholars, is no longer with us. Beginning in 1957 as a mathematics student, he remained at the University of Amsterdam all his life, except for a number of visiting professorships. He rose quickly to take his place as the successor of Brouwer and Heyting. As a full professor from 1970, he was the universally recognized authority on intuitionism and constructivism in general, leaving behind a number of books that will remain landmarks for many years to come. Retiring in 2000, he made a further name for himself as an author on natural history travel narratives. Anne was regularly seen at the ILLC until this year. His impressive personality, always intensely occupied with his present interests, will be greatly missed.
Funeral announcement card
Anne Sjerp Troelstra (August 10, 1939–March 7, 2019)
Anne Troelstra was born on August 10th, 1939, in Maartensdijk. In 1957, he enrolled as a student of mathematics at the University of Amsterdam – and eventually his interests converged on intuitionism with Arend Heyting as his advisor. Students he was close to include Olga Bakker and E.W. Beth’s students Dick de Jongh and Hans Kamp. With Dick de Jongh, he even wrote a pioneering paper on intuitionistic propositional logic, published in 1966, that contained the first definition of the central notion of a p-morphism, as well as the simplest form of the duality between Heyting algebras and relational frames. After obtaining his master’s degree in 1964, Anne at once became an assistant professor, according to a custom of the time. It took him just two years from there to finish his dissertation, supervised by Heyting. Besides intuitionism, a main interest of Heyting was geometry and perhaps not accidentally Anne’s PhD thesis was a study of intuitionistic topology. This subject made him aware of the role of continuity in intuitionistic mathematics, a concept that was to play an important part in his research in the years to come, in many different forms.
Anne then obtained a scholarship to Stanford to visit Georg Kreisel, and spent the academic year 1966-7 there, with Olga whom he had married the year before. Anne sharpened and modified Kreisel’s ideas on choice sequences, and together they created formalizations of analysis resulting in a large article where the typically intuitionistic concept of a lawless sequence of numbers that successfully evades description by any fixed law, reached its final form. In August 1968, Anne played a central role in the famous Buffalo Conference on Intuitionism and Proof Theory, a meeting of all important logicians of the day with an interest in constructivism. He gave a series of ten lectures there, which turned into his first book, published in 1969, that contained the core of his seminal ideas on intuitionistic formal systems and their meta-mathematical investigation. Back home in 1968, he became a lector (associate professor) in 1968, and a full professor in 1970. Further early recognition was to follow. In 1976, he became a member of the Dutch Royal Academy of Sciences.
The meta-mathematics of intuitionistic systems was a chaotic jumble of results when Anne entered it. Here he showed his greatest strength: creating order in a vast and diverse area. In 1973, the order was there in his book Metamathe-matical Investigation of Intuitionistic Arithmetic and Analysis. Especially striking are the clarification of the properties of different models, various types of realizability, and functional interpretations. The last chapters on special topics were written by Jeff Zucker, Craig Smorynski, and Bill Howard, but the lion’s share had been written by Anne, editor and architect of the whole. This book, known in the community as ‘Springer 344’ still functions as a landmark for serious researchers in the subject. Over time, this work developed into the much larger Constructivism in Mathematics, in two volumes co-authored with Dirk van Dalen, published in 1988, the standard text on constructivism right until today.
Of course, Anne also published in depth on special topics. A central notion in the study of intuitionistic formal systems is realizability, introduced by Stephen Kleene in the 1940’s. Anne’s thorough studies of the subject resulted in a long article in the proceedings of the Second Scandinavian Logic Symposium of 1971. The interest remained with him for life. In 1998, a chapter on realizability by his hand came out in the Handbook of Proof Theory. Characteristically, Anne’s text had been finished a few years before, faithfully meeting his deadline, but delays by other authors kept him updating, somewhat grumblingly, with all new results in the area. What he published had to be the complete state of the art.
Other topics pursued in depth by Anne, through the 70s, 80s and 90s are the history of intuitionism and the philosophical basis of the theory of choice sequences. In his important 1977 book Choice Sequences, he proved lawless sequences to be essentially just a figure of speech by an elimination theorem, showing how statements about lawless sequences can be expressed in a theory containing only lawlike sequences. But he did stress that the notion of lawless sequence still serves a purpose as a clear notion derived by informal rigor.
Moving beyond intuitionism proper over the years, Anne broadened his scope to proof theory in general and wrote two more books which again created new order in diverse fields. In 1992, a textbook Lectures on Linear Logic came out proposing improved formats for a then still only partially understood new paradigm. He contributed the majority of the chapters in a book with Helmut Schwichtenberg called Basic Proof Theory, 1996, that is still a standard resource.
An important part of Anne’s life were his PhD students, of whom he supervised 17, and with many of whom he maintained a close relationship. His first PhD student Daniel Leivant finished a thesis in 1975 on the meta-mathematics of intuitionistic arithmetic, and later made his career in computer science. Initially a scarce commodity, in the 1980s, the number of PhD students increased, and Anne’s students wrote on a broad variety of topics, such as intuitionistic meta-mathematics, combinatory algebra, category theory, Martin-Löf type theory, bounded arithmetic, linear logic, and provability logic. Many of these topics reflected the introduction by Anne, often in a close collaboration with Dirk van Dalen in Utrecht, of new topics on the Dutch scene. These students then carried the torch further by themselves. For instance, Ieke Moerdijk became an international leader at the interface in topos theory and logic and category theory generally, while Jaap van Oosten became a worldwide authority on realizability. In the Netherlands alone, four of Anne’s students have become full professors, in mathematics, computer science, AI and philosophy. But Anne was an dedicated teacher at all academic levels, whose precision, clarity and scholarship influenced generations of students in Amsterdam.
Anne retired in 2000, but not to rest. All his life, he had a deep interest in natural history and a wide knowledge of the plants of the Netherlands and abroad. His annual linocuts of plants discovered on his travels in Europe were famous. To those on a walk while listening to him, what looked like an ordinary city street to the untrained eye would turn into a rich landscape of flora, history, and natural wonders. This very year 2019, an article by him will appear on new species of blackberries, his special interest. Anne also made a further name for himself as an author on natural history travel narratives, chronicling the exotic characters and adventures of the past denied to the average academic of today. His major Bibliography of Natural History Narratives was published in 2016.
Anne will be missed in the first place because he will no longer be there to tell us what he thinks about a question that you may have about constructivism. You always knew that you would get a completely honest answer from somebody who knew all the issues and had already thought much further than you. But it is just as much the personal qualities that will be missed. Anne was a very special, and to some, occasionally intimidating person: penetrating, honest, critical, ironic, sharp at times, but always open to arguments and unfailingly supportive of his students and colleagues. He will be deeply missed by all.
Our thoughts go out to his wife Olga and to his daughters Willemien and Ine.
Johan van Benthem
Dick de Jongh
Prof. Troelstra's archive is available at the the Noord-Hollands Archief. The index of the archive is filed as X-2003-01 in ILLC's Technical Notes Series. A list of publications and his CV are available here and here.
When Anne and Olga came to California in 1966 a lifelong friendship began, eventually bringing them and their teenage children to visit Yiannis and me and our teenage children in Greece. We admired Willemien and Ine's beautiful travel journals, learned from Olga the restorative power of tea with bread and good Dutch cheese, and were amazed by the family's botanical erudition. Anne taught me to recognize Phlomis fruticosa, Malva sylvestris and other common Greek wildflowers. In later years he and Olga visited us for botanical adventures and good fellowship, and we visited them in Muiderberg, where Anne guided us on forest walks and longer excursions of which vivid memories remain. We exchanged books and handcraft, cards and photos, looking forward to our next meeting.
At conferences, and in papers, books and correspondence, I have learned more about intuitionistic logic, arithmetic and analysis from Anne than from anyone else. He was a firm but kind critic, encouraging even modest advances by others, always giving fair credit. He took seriously his position as Brouwer's and Heyting's successor at the University of Amsterdam and once confessed to feeling exhausted by the responsibility to understand every new development concerning intuitionism. At retirement he donated much of his own mathematical library to the University of Athens, where he was a visiting lecturer for the graduate program in logic. His generous spirit has enriched innumerable lives. We miss Anne's presence but his essence lives on in books and papers and letters and in all our memories.
Joan Rand Moschovakis
Anne Troelstra
Around 1977, I was a master's student under Dirk van Dalen and Henk Barendregt in Utrecht. We students attended the `baby seminar' under supervision of Jan Willem Klop. That year's baby seminar was concerned with a book that was simply called 344. We students found the book somewhat hard to study, since it gave many details but little motivation. Nevertheless, we learned a lot from the book. The book was written by Anne Troelstra.
From 344, I learned, among many other things, about Heyting Arithmetic and Kleene Realizability. In my career, I returned again and again to Anne's presentation and I used the things I learned in my work. For example, Anne characterized the theory of Kleene realizability as Heyting Arithmetic plus an extended version of Church's Thesis. My paper on the Completeness Principle can be viewed as an answer to the question: what happens if we do the same thing Anne did when Kleene Realizability is replaced by the provability translation?
I met Anne for the first time in 1977/78 when he visited Dirk van Dalen at the Mathematical Institute in Utrecht (now the Freudenthal Building). Even if Anne looked, at first sight, somewhat stern, he turned out to be friendly and accessible. He was always prepared to answer questions and to discuss problems. At a certain point, I started calling Anne `Anne'. Dirk van Dalen noted this and asked `Did professor Troelstra specifically invite you to call him `Anne'?' No, but it felt appropriate. After that I avoided, for some time, any form of address for Anne and then returned to `Anne'. There was never any sign that Anne disapproved of `Anne'.
Anne did not care much for worldly fame. For, him the main motivation was understanding a thing in all possible detail. This was a defining property of his personality. We see it both in his work in logic and in his work on botanical travel stories.
I did not see Anne very often in the last years, but, now and then, we attended the same meeting. It was always good seeing him and talking to him. It is hard to understand that now this is not possible anymore.
Albert Visser
In my formative years around 1975 in the logic school of Dirk van Dalen and Henk Barendregt, at the Mathematical Institute, Boedapestlaan 6 in Utrecht, I was happily sharing a room with Albert Visser, adjacent to the rooms of Dirk and Henk and at the
same fourth floor corridor as the room of our friend Jeff Zucker. We had a well-attended Intercity Colloquium, bi-weekly taking place alternatingly at the institutes in Amsterdam and Utrecht. Anne Troelstra, Dick de Jongh, Roel de Vrijer were regular participants,
together with visitors in those years such as Walter van Stigt, David Isles and Craig Smorynski, and many more short or long term visiting friends and colleagues from abroad. As a junior Ph.D-student under supervision of Dirk and Henk my (too) difficult assignment was to assist a group of newcoming students, including Albert, in digesting and mastering various chapters of Anne’s famous Springer LNM 344, shortly known as ‘344’.
The encounter with 344 did not influence me as deeply as would have been desirable, but in my subsequent development to a theoretical computer scientist I did greatly profit from the treatment in 344 of important recursion theory theorems such as the ones of Myhill-Shepherdson and Kreisel-Lacombe-Schoenfield. They were instrumental in a later paper (1982) by Jan Bergstra and me about parametrized data types, continuing Jan’s well-known series of papers together with John Tucker from Swansea, analyzing the theory of abstract data types.
I used to drive our Utrecht group, Dirk, Henk, Jeff, Albert in my small car to the Amsterdam sessions of the Intercity seminar, and dually, drive the Amsterdam participants, Anne, Dick, Roel after the Utrecht sessions back to Utrecht Central Station. I remember Anne’s stimulating comments, when I had made good progress with my PhD-thesis, about term rewriting systems and lambda calculus. I remember Anne as a true scholar and a gentleman. He was an example, a role model for junior logicians and computer scientists. At the event of his emeritate, I thanked him for his continual inspiration. I remember Anne’s facial expression, somewhat amused and ironic.
Later, the past fifteen years, I encountered Anne many times in the meetings of the Section Mathematics of the Academy. His life and work will be for many of us, in logic and computer science, an ever-lasting inspiration and enrichment.
Jan Willem Klop
I first saw Anne in action when he taught "Analysis II", the major stumbling block for beginning mathematics students. It was whispered in our group that he was very clever, having become an Associate Professor at a very early age, but it would be saying too much that the field of Analysis came sparklingly alive. What did come alive was my image of Anne, he looked very much then like he looked all through his life: serious, sharp, erudite, and with his technical subjects at his fingertips. Later on, I took his all courses on intuitionism, even though I chose the path of model theory eventually, having tasted the sinful delights of getting mathematical results without constructivist proofs.
Anne, characteristically, never held my choice against me, nor did he object to my broader activism in philosophy, linguistics, computation, and even further areas over time. Anne was clearly a mathematical logician from the heartland, but he saw pursuing wider frontiers of logic as good for the whole field, rather than as a threat to established rank and dogma.
I owe a lot to Anne. He helped me at a crucial stage of my dissertation project in 1976, he had confidence in me when I was appointed as his collega proximus in 1986 (an honor that I am still vividly aware of after all these years), and in the formative time of the ILLC, he was quietly but persistently supportive, even though governance and organizational activism were not among his favorites. For many years, our offices were side by side, and our contacts happened daily. That does not mean Anne never criticized his next-door neighbor, sometimes with the aid of a list of points on his whiteboard, but always for good reasons: and in his turn, he was always open to arguments, and able to change his mind.
Familiarity may breed contempt, as is said, but it can also breed respect. Over the years, I came to appreciate Anne's qualities as a researcher and as a person more and more. I also admired his starting a new life after retirement, rather than following the inertia that people call 'still going strong in one's field'. With Anne gone, my world in Amsterdam looks reduced: it has lost a dimension.
Johan van Benthem
Arriving in September 1961 in Amsterdam for a master study with E.W. Beth I soon found myself in contact with one of Heyting's students: Anne Troelstra. Since our subjects were close we participated in a number of the same classes. He was meticulous, neat, always ahead of deadlines, everything I was not. After a while, I had to recognize he was very clever as well. We shared a love of intuitionistic logic, and looked at the basics together. Soon we found the 14 equivalence classes of formulas with only p, q and implication. More seriously, we delved into my subject, the theory of the models later called Kripke models, and established some important results. He stayed here for a PhD with Heyting, I left for one in the U.S. When I returned to take up a position at the UvA, he had already a solid reputation. It was always a safe feeling that he was there. When I had gone back again to the U.S. for two years of teaching, and on wanting to return found it was less easy than I had expected to find a position again in the Netherlands he was able to ease me back into the UvA. We both worked in intuitionism but from different angles. When we discussed issues in that area he often surprised me, suddenly showing insights I didn't suspect. This remained so even when his main interest had shifted from logic to natural history. Life will be different now that he is no longer there.
Dick de Jongh
I first met Anne Troelstra at the Mathematisches Forschungsinstitut Oberwolfach. It was at the time when he just had finished his marvellous lecture notes volume entitled "Metamathematical Investigations of Intuitionistic Arithmetic and Analysis". This impressive piece of work quickly became the standard source of knowledge for at least a generation of mathematics students with an interest in the logical foundations of constructive mathematics. In fact, at the time it was the most-read volume of the whole series of Springer Lecture Notes in Mathematics at the Mathematical Institute Library of LMU Munich. Later it was extended to the almost encyclopedic two-volume book "Constructivism in Mathematics", which he wrote together with Dirk van Dalen, and later again I had the great pleasure to be his coauthor in the book "Basic Proof Theory", with appeared in two editions around the turn of the century. For many years we also cooperated in organizing the regular workshop on Mathematical Logic at the Mathematisches Forschungsinstitut Oberwolfach.
In all these years he strongly impressed me by the clarity and originality of his mathematical work, and also his ability to relate it to the vast literature of his field. He was an absolutely honest person, who always insisted to give other researchers the deserved credit for their work. Apart from mathematics he had many other interests, which he pursued with similar quality and endurance as his mathematical work.
I will very much miss him as a dear colleague and friend.
Helmut Schwichtenberg
One of the great pleasures of the academic life is that occasionally you meet (and become friends with) people who look different, talk strangely and generally have nothing in common with you except math---in this case; and so it was that Joan and I met and became friends with Anne and Olga back in the sixties, when we were all very young. Joan had a lot to talk about with Anne, of course, but I, too, always enjoyed the scientific exchanges with him: he was a tolerant intuitionist, and I had started in constructive mathematics, reading Heyting's little book in '57 or '58, probably before Anne. (I lost my constructive faith in Grad School but like many lapsed Catholics, I never lost my respect for the faithful or lingering guilt for my dropping out.)
There have been many trips over the years---some with children---by the Troelstras to Greece and by us to the Netherlands, most recently three years ago. Many happy memories, mostly of talk about plants.
A very vivid one (some years back) is of a walk in Parnitha, the tallest of the four mountains that surround Athens. It was chilly (Spring or Fall most likely) and Anne was in his element, latin names of species pouring out of him; until he turned silent and said softly that he had seen more species that morning than exist in Holland---the single, greatest praise of the Greek countryside I ever heard.
We will miss him.
Yiannis Moschovakis
Professor Troelstra has been one of my main scientific mentors and supporters when I was young and needed support the most. He not only invited me to my first Oberwolfach Meeting in 1990 but also to my first talk abroad (the Intercity Logic Seminar between Amsterdam and Utrecht) again in 1990.
He was co-referee for both my master and my PhD theses and no scientific work had a greater influence on me than his Springer LNM 344 "Metamathematical Investigation of Intuitionistic Arithmetic and Analysis" from 1973 which is only book I had to buy twice since the first copy disintegrated due to its intensive use. There was a time when I remembered the page numbers on which certain facts were presented in this book.
He examplified for me the ideal scholar. I will always remember him in the highest regards and will be forever grateful to him.
Ulrich Kohlenbach
In the last quarter of the 20th century, Anne Troelstra, with my assistance, organized an Amsterdam-Münster logic contact between our two institutions: Every second year, a few Amsterdam logicians under Anne's leadership would visit the logic institute at Münster, some of them as well as some Münster logicians would present their recent research, and every other year, it would be the other way around. For many young logicians from Münster -possibly also from Amsterdam- this was their first encounter with the international logic scene, and it certainly influenced and improved the work on intuitionism and constructive mathematics at the Münster institute considerably.
This contact was also a basis for a deep friendship between the Troelstra and the Diller family, resulting in a number of visits to their respective homes at Muiderberg and Münster. We, the Dillers at Münster, are deeply moved by the sudden and unexpected death of Anne, and our feelings of sympathy are particularly with Olga.
Justus Diller
It must have been in the second year of my studying maths, in the spring of 1983, when I first met Professor Troelstra when I attended a logic seminar. We read Dana Scott's Lectures on a Mathematical Theory of Computation, and I was immediately tasked with presenting Chapter 3.
Half a year later, I was student-assistent under him. I helped with bibliographical research for the Omega-bibliography, of which Troelstra and Diller compiled the part on Constructivism.
Troelstra could be rather rigid when it came to social conventions. Dutch, like many other continental European languages, has a polite/formal form and a familiar form (different ways to say "you"). In those days, it was still customary to address a student in the polite form, whereas for a colleague one would of course use the familiar form. When I did my bibliographical chores, I was a colleague and was talked to in the familiar form. When, after a discussion about this work, I asked him whether I could ask a question about his course in lambda calculus, which I was following, he switched at once to the polite form.
I accompanied him on trips to Muenster in Germany. Once a year, a delegation of teachers, postdocs, PhD students and undergraduates went for a short weekend-seminar to the department of Diller and Pohlers (and once a year, the Muensteraner reciprocated with a visit to Amsterdam). Once when we passed a sculpture of a nymph-like person, her naked torso rising out of a rough substance, he commented: "the artist probably thought: if I reveal more of her body, it can only lead to disappointment". He had definitely a sense of humour.
Maybe once a year, he invited the whole logic group to his place for dinner. At the end of the evening he would put a large crate, full with books, on the floor, and everyone was invited to take what (s)he liked.
In the fall of 1986 I had been working on my master thesis, but my assistentship had ended and I had several jobs, so the thesis work had been put on hold. Troelstra was not in the habit of telling you what he thought of your work, and naturally I assumed I would do my master's some time and then find a position outside academia. But I was in for a surprise, when early february 1987 he was on the phone. "You should finish your thesis and get your master diploma, for you can start by March 1 as PhD student." It was not a question. At the ceremony for my master diploma (my parents were present) he called me a "rough diamond", which was sort of a compliment, I suppose. The emphasis was on "rough".
As a PhD student I of course got to know Troelstra better as a mathematician. I did my first steps in topos theory, following work by Martin Hyland. Although he had not worked in this field, he always held it in some kind of timorous esteem, although he stressed the importance of applications. Abstraction had its limits. He had lots of criticisms, mainly on my style of writing ("too succinct", "too terse"); later I realized that he taught me a lot during this period. Reading my manuscripts was sometimes a strain for him, and he complained that supervising me would shorten his life by a year. I came to appreciate how he could "feel" a result, even if he was not familiar with the formal details.
Again, he was not going to give you any compliments so after 3 years of work on my thesis I was just dead certain that it had been a total failure. I would leave academia without a thesis. But again there was a surprise: "you should do some research in the library for an overview of the field. And gather your results, for it is about time to start writing up".
One aspect of doing a PhD with Troelstra was that, starting right after the ceremony, one should no longer use the polite form with him and call him "Anne". This took some getting used to, but I managed eventually.
In the northern Dutch province of Friesland (which is not where Anne was born or grew up, but is, I presume, where his ancestors came from) the name "Anne" is a common boys' name. A funny story was told at the memorial conference which Ieke Moerdijk, Harold Schellinx and I organized to celebrate Anne's 60th birthday: when he first met Dana Scott, he told him that he had thought Dana was a girl. He probably didn't reflect on how his own name might sound to an English speaker.
I am grateful to have been able to study under a scholar of Anne's calibre. His staunch honesty will remain a beacon for me the rest of my life.
Jaap van Oosten
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Homotopy Type Theory Univalent Foundations of MathematicsT HE U NIVALENT F OUNDATIONS P ROGRAM I NSTITUTE FOR A DVANCE...
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https://epdf.tips/homotopy-type-theory-univalent-foundations-of-mathematics.html
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T HE U NIVALENT F OUNDATIONS P ROGRAM I NSTITUTE FOR A DVANCED S TUDY
Homotopy Type Theory Univalent Foundations of Mathematics The Univalent Foundations Program Institute for Advanced Study
“Homotopy Type Theory: Univalent Foundations of Mathematics” c 2013 The Univalent Foundations Program
Book version: first-edition-611-ga1a258c MSC 2010 classification: 03-02, 55-02, 03B15 This work is licensed under the Creative Commons Attribution-ShareAlike 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by-sa/3.0/. This book is freely available at http://homotopytypetheory.org/book/.
Acknowledgment Apart from the generous support from the Institute for Advanced Study, some contributors to the book were partially or fully supported by the following agencies and grants: • Association of Members of the Institute for Advanced Study: a grant to the Institute for Advanced Study • Agencija za raziskovalno dejavnost Republike Slovenije: P1–0294, N1–0011. • Air Force Office of Scientific Research: FA9550-11-1-0143, and FA9550-12-1-0370. This material is based in part upon work supported by the AFOSR under the above awards. Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the author(s) and do not necessarily reflect the views of the AFOSR.
• Engineering and Physical Sciences Research Council: EP/G034109/1, EP/G03298X/1. • European Union’s 7th Framework Programme under grant agreement nr. 243847 (ForMath). • National Science Foundation: DMS-1001191, DMS-1100938, CCF-1116703, and DMS-1128155. This material is based in part upon work supported by the National Science Foundation under the above awards. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.
• The Simonyi Fund: a grant to the Institute for Advanced Study
Preface IAS Special Year on Univalent Foundations A Special Year on Univalent Foundations of Mathematics was held in 2012-13 at the Institute for Advanced Study, School of Mathematics, organized by Steve Awodey, Thierry Coquand, and Vladimir Voevodsky. The following people were the official participants. Peter Aczel
Eric Finster
Alvaro Pelayo
Benedikt Ahrens
Daniel Grayson
Andrew Polonsky
Thorsten Altenkirch
Hugo Herbelin
Michael Shulman
Steve Awodey
Andr´e Joyal
Matthieu Sozeau
Bruno Barras
Dan Licata
Bas Spitters
Andrej Bauer
Peter Lumsdaine
Benno van den Berg
Yves Bertot
Assia Mahboubi
Vladimir Voevodsky
Marc Bezem
¨ Per Martin-Lof
Michael Warren
Thierry Coquand
Sergey Melikhov
Noam Zeilberger
There were also the following students, whose participation was no less valuable. Carlo Angiuli
Guillaume Brunerie
Egbert Rijke
Anthony Bordg
Chris Kapulkin
Kristina Sojakova
In addition, there were the following short- and long-term visitors, including student visitors, whose contributions to the Special Year were also essential. Jeremy Avigad Cyril Cohen Robert Constable Pierre-Louis Curien Peter Dybjer Mart´ın Escardo´ Kuen-Bang Hou Nicola Gambino
Richard Garner Georges Gonthier Thomas Hales Robert Harper Martin Hofmann Pieter Hofstra Joachim Kock Nicolai Kraus
Nuo Li Zhaohui Luo Michael Nahas Erik Palmgren Emily Riehl Dana Scott Philip Scott Sergei Soloviev
iv
About this book We did not set out to write a book. The present work has its origins in our collective attempts to develop a new style of “informal type theory” that can be read and understood by a human being, as a complement to a formal proof that can be checked by a machine. Univalent foundations is closely tied to the idea of a foundation of mathematics that can be implemented in a computer proof assistant. Although such a formalization is not part of this book, much of the material presented here was actually done first in the fully formalized setting inside a proof assistant, and only later “unformalized” to arrive at the presentation you find before you — a remarkable inversion of the usual state of affairs in formalized mathematics. Each of the above-named individuals contributed something to the Special Year — and so to this book — in the form of ideas, words, or deeds. The spirit of collaboration that prevailed throughout the year was truly extraordinary. Special thanks are due to the Institute for Advanced Study, without which this book would obviously never have come to be. It proved to be an ideal setting for the creation of this new branch of mathematics: stimulating, congenial, and supportive. May some trace of this unique atmosphere linger in the pages of this book, and in the future development of this new field of study. The Univalent Foundations Program Institute for Advanced Study Princeton, April 2013
Contents Introduction
1
I
Foundations
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Type theory 1.1 Type theory versus set theory . . . . 1.2 Function types . . . . . . . . . . . . . 1.3 Universes and families . . . . . . . . 1.4 Dependent function types (Π-types) 1.5 Product types . . . . . . . . . . . . . 1.6 Dependent pair types (Σ-types) . . . 1.7 Coproduct types . . . . . . . . . . . . 1.8 The type of booleans . . . . . . . . . 1.9 The natural numbers . . . . . . . . . 1.10 Pattern matching and recursion . . . 1.11 Propositions as types . . . . . . . . . 1.12 Identity types . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . .
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Homotopy type theory 2.1 Types are higher groupoids . . . . . . . . . . . 2.2 Functions are functors . . . . . . . . . . . . . . 2.3 Type families are fibrations . . . . . . . . . . . 2.4 Homotopies and equivalences . . . . . . . . . . 2.5 The higher groupoid structure of type formers 2.6 Cartesian product types . . . . . . . . . . . . . 2.7 Σ-types . . . . . . . . . . . . . . . . . . . . . . . 2.8 The unit type . . . . . . . . . . . . . . . . . . . . 2.9 Π-types and the function extensionality axiom 2.10 Universes and the univalence axiom . . . . . . 2.11 Identity type . . . . . . . . . . . . . . . . . . . .
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Contents 2.12 Coproducts . . . . . . . . . . . . 2.13 Natural numbers . . . . . . . . 2.14 Example: equality of structures 2.15 Universal properties . . . . . . Notes . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . .
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Sets and logic 3.1 Sets and n-types . . . . . . . . . . . 3.2 Propositions as types? . . . . . . . 3.3 Mere propositions . . . . . . . . . . 3.4 Classical vs. intuitionistic logic . . 3.5 Subsets and propositional resizing 3.6 The logic of mere propositions . . 3.7 Propositional truncation . . . . . . 3.8 The axiom of choice . . . . . . . . . 3.9 The principle of unique choice . . . 3.10 When are propositions truncated? 3.11 Contractibility . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . .
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Equivalences 4.1 Quasi-inverses . . . . . . . . . . . . . . . . . 4.2 Half adjoint equivalences . . . . . . . . . . 4.3 Bi-invertible maps . . . . . . . . . . . . . . . 4.4 Contractible fibers . . . . . . . . . . . . . . . 4.5 On the definition of equivalences . . . . . . 4.6 Surjections and embeddings . . . . . . . . . 4.7 Closure properties of equivalences . . . . . 4.8 The object classifier . . . . . . . . . . . . . . 4.9 Univalence implies function extensionality Notes . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . .
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Induction 5.1 Introduction to inductive types . . . . . . . 5.2 Uniqueness of inductive types . . . . . . . . 5.3 W-types . . . . . . . . . . . . . . . . . . . . . 5.4 Inductive types are initial algebras . . . . . 5.5 Homotopy-inductive types . . . . . . . . . 5.6 The general syntax of inductive definitions 5.7 Generalizations of inductive types . . . . . 5.8 Identity types and identity systems . . . . .
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Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 6
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Higher inductive types 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 6.2 Induction principles and dependent paths . . . . . 6.3 The interval . . . . . . . . . . . . . . . . . . . . . . 6.4 Circles and spheres . . . . . . . . . . . . . . . . . . 6.5 Suspensions . . . . . . . . . . . . . . . . . . . . . . 6.6 Cell complexes . . . . . . . . . . . . . . . . . . . . . 6.7 Hubs and spokes . . . . . . . . . . . . . . . . . . . 6.8 Pushouts . . . . . . . . . . . . . . . . . . . . . . . . 6.9 Truncations . . . . . . . . . . . . . . . . . . . . . . 6.10 Quotients . . . . . . . . . . . . . . . . . . . . . . . . 6.11 Algebra . . . . . . . . . . . . . . . . . . . . . . . . . 6.12 The flattening lemma . . . . . . . . . . . . . . . . . 6.13 The general syntax of higher inductive definitions Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Mathematics Homotopy theory 8.1 π1 (S1 ) . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Connectedness of suspensions . . . . . . . . . . . . 8.3 πk≤n of an n-connected space and πk
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Contents 8.10 Additional Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
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Category theory 9.1 Categories and precategories . 9.2 Functors and transformations . 9.3 Adjunctions . . . . . . . . . . . 9.4 Equivalences . . . . . . . . . . . 9.5 The Yoneda lemma . . . . . . . 9.6 Strict categories . . . . . . . . . 9.7 †-categories . . . . . . . . . . . 9.8 The structure identity principle 9.9 The Rezk completion . . . . . . Notes . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . .
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11 Real numbers 11.1 The field of rational numbers . . . . . . . . 11.2 Dedekind reals . . . . . . . . . . . . . . . . 11.3 Cauchy reals . . . . . . . . . . . . . . . . . . 11.4 Comparison of Cauchy and Dedekind reals 11.5 Compactness of the interval . . . . . . . . . 11.6 The surreal numbers . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . .
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10 Set theory 10.1 The category of sets . . . . 10.2 Cardinal numbers . . . . . 10.3 Ordinal numbers . . . . . 10.4 Classical well-orderings . 10.5 The cumulative hierarchy Notes . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . .
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Appendix A Formal type theory A.1 The first presentation . . A.2 The second presentation A.3 Homotopy type theory . A.4 Basic metatheory . . . .
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Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434
Bibliography
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Index of symbols
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Index
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Introduction Homotopy type theory is a new branch of mathematics that combines aspects of several different fields in a surprising way. It is based on a recently discovered connection between homotopy theory and type theory. Homotopy theory is an outgrowth of algebraic topology and homological algebra, with relationships to higher category theory; while type theory is a branch of mathematical logic and theoretical computer science. Although the connections between the two are currently the focus of intense investigation, it is increasingly clear that they are just the beginning of a subject that will take more time and more hard work to fully understand. It touches on topics as seemingly distant as the homotopy groups of spheres, the algorithms for type checking, and the definition of weak ∞-groupoids. Homotopy type theory also brings new ideas into the very foundation of mathematics. On the one hand, there is Voevodsky’s subtle and beautiful univalence axiom. The univalence axiom implies, in particular, that isomorphic structures can be identified, a principle that mathematicians have been happily using on workdays, despite its incompatibility with the “official” doctrines of conventional foundations. On the other hand, we have higher inductive types, which provide direct, logical descriptions of some of the basic spaces and constructions of homotopy theory: spheres, cylinders, truncations, localizations, etc. Both ideas are impossible to capture directly in classical set-theoretic foundations, but when combined in homotopy type theory, they permit an entirely new kind of “logic of homotopy types”. This suggests a new conception of foundations of mathematics, with intrinsic homotopical content, an “invariant” conception of the objects of mathematics — and convenient machine implementations, which can serve as a practical aid to the working mathematician. This is the Univalent Foundations program. The present book is intended as a first systematic exposition of the basics of univalent foundations, and a collection of examples of this new style of reasoning — but without requiring the reader to know or learn any formal logic, or to use any computer proof assistant. We emphasize that homotopy type theory is a young field, and univalent foundations is very much a work in progress. This book should be regarded as a “snapshot” of the state of the field at the time it was written, rather than a polished exposition of an established edifice. As we will discuss briefly later, there are many aspects of homotopy type theory that are not yet fully understood — but as of this writing, its broad outlines seem clear enough. The ultimate theory will probably not look exactly like the one described in this book, but it will surely be at least as capable and powerful; we therefore believe that univalent foundations will eventually become a viable alternative to set theory as the “implicit foundation” for the unformalized mathematics done by most mathematicians.
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I NTRODUCTION
Type theory Type theory was originally invented by Bertrand Russell [Rus08], as a device for blocking the paradoxes in the logical foundations of mathematics that were under investigation at the time. It was later developed as a rigorous formal system in its own right (under the name “λ-calculus”) by Alonzo Church [Chu33, Chu40, Chu41]. Although it is not generally regarded as the foundation for classical mathematics, set theory being more customary, type theory still has numerous applications, especially in computer science and the theory of programming languages [Pie02]. ¨ [ML98, ML75, ML82, ML84], among others, developed a “predicative” modificaPer Martin-Lof tion of Church’s type system, which is now usually called dependent, constructive, intuitionistic, or simply Martin-L¨of type theory. This is the basis of the system that we consider here; it was originally intended as a rigorous framework for the formalization of constructive mathematics. In what follows, we will often use “type theory” to refer specifically to this system and similar ones, although type theory as a subject is much broader (see [Som10, KLN04] for the history of type theory). In type theory, unlike set theory, objects are classified using a primitive notion of type, similar to the data-types used in programming languages. These elaborately structured types can be used to express detailed specifications of the objects classified, giving rise to principles of reasoning about these objects. To take a very simple example, the objects of a product type A × B are known to be of the form ( a, b), and so one automatically knows how to construct them and how to decompose them. Similarly, an object of function type A → B can be acquired from an object of type B parametrized by objects of type A, and can be evaluated at an argument of type A. This rigidly predictable behavior of all objects (as opposed to set theory’s more liberal formation principles, allowing inhomogeneous sets) is one aspect of type theory that has led to its extensive use in verifying the correctness of computer programs. The clear reasoning principles associated with the construction of types also form the basis of modern computer proof assistants, which are used for formalizing mathematics and verifying the correctness of formalized proofs. We return to this aspect of type theory below. One problem in understanding type theory from a mathematical point of view, however, has always been that the basic concept of type is unlike that of set in ways that have been hard to make precise. We believe that the new idea of regarding types, not as strange sets (perhaps constructed without using classical logic), but as spaces, viewed from the perspective of homotopy theory, is a significant step forward. In particular, it solves the problem of understanding how the notion of equality of elements of a type differs from that of elements of a set. In homotopy theory one is concerned with spaces and continuous mappings between them, up to homotopy. A homotopy between a pair of continuous maps f : X → Y and g : X → Y is a continuous map H : X × [0, 1] → Y satisfying H ( x, 0) = f ( x ) and H ( x, 1) = g( x ). The homotopy H may be thought of as a “continuous deformation” of f into g. The spaces X and Y are said to be homotopy equivalent, X ' Y, if there are continuous maps going back and forth, the composites of which are homotopical to the respective identity mappings, i.e., if they are isomorphic “up to homotopy”. Homotopy equivalent spaces have the same algebraic invariants (e.g., homology, or the fundamental group), and are said to have the same homotopy type.
3
Homotopy type theory Homotopy type theory (HoTT) interprets type theory from a homotopical perspective. In homotopy type theory, we regard the types as “spaces” (as studied in homotopy theory) or higher groupoids, and the logical constructions (such as the product A × B) as homotopy-invariant constructions on these spaces. In this way, we are able to manipulate spaces directly without first having to develop point-set topology (or any combinatorial replacement for it, such as the theory of simplicial sets). To briefly explain this perspective, consider first the basic concept of type theory, namely that the term a is of type A, which is written: a : A. This expression is traditionally thought of as akin to: “a is an element of the set A”. However, in homotopy type theory we think of it instead as: “a is a point of the space A”. Similarly, every function f : A → B in type theory is regarded as a continuous map from the space A to the space B. We should stress that these “spaces” are treated purely homotopically, not topologically. For instance, there is no notion of “open subset” of a type or of “convergence” of a sequence of elements of a type. We only have “homotopical” notions, such as paths between points and homotopies between paths, which also make sense in other models of homotopy theory (such as simplicial sets). Thus, it would be more accurate to say that we treat types as ∞-groupoids; this is a name for the “invariant objects” of homotopy theory which can be presented by topological spaces, simplicial sets, or any other model for homotopy theory. However, it is convenient to sometimes use topological words such as “space” and “path”, as long as we remember that other topological concepts are not applicable. (It is tempting to also use the phrase homotopy type for these objects, suggesting the dual interpretation of “a type (as in type theory) viewed homotopically” and “a space considered from the point of view of homotopy theory”. The latter is a bit different from the classical meaning of “homotopy type” as an equivalence class of spaces modulo homotopy equivalence, although it does preserve the meaning of phrases such as “these two spaces have the same homotopy type”.) The idea of interpreting types as structured objects, rather than sets, has a long pedigree, and is known to clarify various mysterious aspects of type theory. For instance, interpreting types as sheaves helps explain the intuitionistic nature of type-theoretic logic, while interpreting them as partial equivalence relations or “domains” helps explain its computational aspects. It also implies that we can use type-theoretic reasoning to study the structured objects, leading to the rich field of categorical logic. The homotopical interpretation fits this same pattern: it clarifies the nature of identity (or equality) in type theory, and allows us to use type-theoretic reasoning in the study of homotopy theory. The key new idea of the homotopy interpretation is that the logical notion of identity a = b of two objects a, b : A of the same type A can be understood as the existence of a path p : a ; b from
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I NTRODUCTION
point a to point b in the space A. This also means that two functions f , g : A → B can be identified if they are homotopic, since a homotopy is just a (continuous) family of paths p x : f ( x ) ; g( x ) in B, one for each x : A. In type theory, for every type A there is a (formerly somewhat mysterious) type Id A of identifications of two objects of A; in homotopy type theory, this is just the path space A I of all continuous maps I → A from the unit interval. In this way, a term p : Id A ( a, b) represents a path p : a ; b in A. The idea of homotopy type theory arose around 2006 in independent work by Awodey and Warren [AW09] and Voevodsky [Voe06], but it was inspired by Hofmann and Streicher’s earlier groupoid interpretation [HS98]. Indeed, higher-dimensional category theory (particularly the theory of weak ∞-groupoids) is now known to be intimately connected to homotopy theory, as proposed by Grothendieck and now being studied intensely by mathematicians of both sorts. The original semantic models of Awodey–Warren and Voevodsky use well-known notions and techniques from homotopy theory which are now also in use in higher category theory, such as Quillen model categories and Kan simplicial sets. Voevodsky recognized that the simplicial interpretation of type theory satisfies a further crucial property, dubbed univalence, which had not previously been considered in type theory (although Church’s principle of extensionality for propositions turns out to be a very special case of it). Adding univalence to type theory in the form of a new axiom has far-reaching consequences, many of which are natural, simplifying and compelling. The univalence axiom also further strengthens the homotopical view of type theory, since it holds in the simplicial model and other related models, while failing under the view of types as sets.
Univalent foundations Very briefly, the basic idea of the univalence axiom can be explained as follows. In type theory, one can have a universe U , the terms of which are themselves types, A : U , etc. Those types that are terms of U are commonly called small types. Like any type, U has an identity type IdU , which expresses the identity relation A = B between small types. Thinking of types as spaces, U is a space, the points of which are spaces; to understand its identity type, we must ask, what is a path p : A ; B between spaces in U ? The univalence axiom says that such paths correspond to homotopy equivalences A ' B, (roughly) as explained above. A bit more precisely, given any (small) types A and B, in addition to the primitive type IdU ( A, B) of identifications of A with B, there is the defined type Equiv( A, B) of equivalences from A to B. Since the identity map on any object is an equivalence, there is a canonical map, IdU ( A, B) → Equiv( A, B). The univalence axiom states that this map is itself an equivalence. At the risk of oversimplifying, we can state this succinctly as follows: Univalence Axiom: ( A = B) ' ( A ' B). In other words, identity is equivalent to equivalence. In particular, one may say that “equivalent types are identical”. However, this phrase is somewhat misleading, since it may sound like a sort of “skeletality” condition which collapses the notion of equivalence to coincide with identity,
5 whereas in fact univalence is about expanding the notion of identity so as to coincide with the (unchanged) notion of equivalence. From the homotopical point of view, univalence implies that spaces of the same homotopy type are connected by a path in the universe U , in accord with the intuition of a classifying space for (small) spaces. From the logical point of view, however, it is a radically new idea: it says that isomorphic things can be identified! Mathematicians are of course used to identifying isomorphic structures in practice, but they generally do so by “abuse of notation”, or some other informal device, knowing that the objects involved are not “really” identical. But in this new foundational scheme, such structures can be formally identified, in the logical sense that every property or construction involving one also applies to the other. Indeed, the identification is now made explicit, and properties and constructions can be systematically transported along it. Moreover, the different ways in which such identifications may be made themselves form a structure that one can (and should!) take into account. Thus in sum, for points A and B of the universe U (i.e., small types), the univalence axiom identifies the following three notions: • (logical) an identification p : A = B of A and B • (topological) a path p : A ; B from A to B in U
• (homotopical) an equivalence p : A ' B between A and B.
Higher inductive types One of the classical advantages of type theory is its simple and effective techniques for working with inductively defined structures. The simplest nontrivial inductively defined structure is the natural numbers, which is inductively generated by zero and the successor function. From this statement one can algorithmically extract the principle of mathematical induction, which characterizes the natural numbers. More general inductive definitions encompass lists and wellfounded trees of all sorts, each of which is characterized by a corresponding “induction principle”. This includes most data structures used in certain programming languages; hence the usefulness of type theory in formal reasoning about the latter. If conceived in a very general sense, inductive definitions also include examples such as a disjoint union A + B, which may be regarded as “inductively” generated by the two injections A → A + B and B → A + B. The “induction principle” in this case is “proof by case analysis”, which characterizes the disjoint union. In homotopy theory, it is natural to consider also “inductively defined spaces” which are generated not merely by a collection of points, but also by collections of paths and higher paths. Classically, such spaces are called CW complexes. For instance, the circle S1 is generated by a single point and a single path from that point to itself. Similarly, the 2-sphere S2 is generated by a single point b and a single two-dimensional path from the constant path at b to itself, while the torus T 2 is generated by a single point, two paths p and q from that point to itself, and a two-dimensional path from p q to q p. By using the identification of paths with identities in homotopy type theory, these sort of “inductively defined spaces” can be characterized in type theory by “induction principles”, entirely analogously to classical examples such as the natural numbers and the disjoint union. The
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I NTRODUCTION
resulting higher inductive types give a direct “logical” way to reason about familiar spaces such as spheres, which (in combination with univalence) can be used to perform familiar arguments from homotopy theory, such as calculating homotopy groups of spheres, in a purely formal way. The resulting proofs are a marriage of classical homotopy-theoretic ideas with classical typetheoretic ones, yielding new insight into both disciplines. Moreover, this is only the tip of the iceberg: many abstract constructions from homotopy theory, such as homotopy colimits, suspensions, Postnikov towers, localization, completion, and spectrification, can also be expressed as higher inductive types. Many of these are classically constructed using Quillen’s “small object argument”, which can be regarded as a finite way of algorithmically describing an infinite CW complex presentation of a space, just as “zero and successor” is a finite algorithmic description of the infinite set of natural numbers. Spaces produced by the small object argument are infamously complicated and difficult to understand; the typetheoretic approach is potentially much simpler, bypassing the need for any explicit construction by giving direct access to the appropriate “induction principle”. Thus, the combination of univalence and higher inductive types suggests the possibility of a revolution, of sorts, in the practice of homotopy theory.
Sets in univalent foundations We have claimed that univalent foundations can eventually serve as a foundation for “all” of mathematics, but so far we have discussed only homotopy theory. Of course, there are many specific examples of the use of type theory without the new homotopy type theory features to formalize mathematics, such as the recent formalization of the Feit–Thompson odd-order theorem in C OQ [GAA+ 13]. But the traditional view is that mathematics is founded on set theory, in the sense that all mathematical objects and constructions can be coded into a theory such as Zermelo–Fraenkel set theory (ZF). However, it is well-established by now that for most mathematics outside of set theory proper, the intricate hierarchical membership structure of sets in ZF is really unnecessary: a more “structural” theory, such as Lawvere’s Elementary Theory of the Category of Sets [Law05], suffices. In univalent foundations, the basic objects are “homotopy types” rather than sets, but we can define a class of types which behave like sets. Homotopically, these can be thought of as spaces in which every connected component is contractible, i.e. those which are homotopy equivalent to a discrete space. It is a theorem that the category of such “sets” satisfies Lawvere’s axioms (or related ones, depending on the details of the theory). Thus, any sort of mathematics that can be represented in an ETCS-like theory (which, experience suggests, is essentially all of mathematics) can equally well be represented in univalent foundations. This supports the claim that univalent foundations is at least as good as existing foundations of mathematics. A mathematician working in univalent foundations can build structures out of sets in a familiar way, with more general homotopy types waiting in the foundational background until there is need of them. For this reason, most of the applications in this book have been chosen to be areas where univalent foundations has something new to contribute that distinguishes it from existing foundational systems.
7 Unsurprisingly, homotopy theory and category theory are two of these, but perhaps less obvious is that univalent foundations has something new and interesting to offer even in subjects such as set theory and real analysis. For instance, the univalence axiom allows us to identify isomorphic structures, while higher inductive types allow direct descriptions of objects by their universal properties. Thus we can generally avoid resorting to arbitrarily chosen representatives or transfinite iterative constructions. In fact, even the objects of study in ZF set theory can be characterized, inside the sets of univalent foundations, by such an inductive universal property.
Informal type theory One difficulty often encountered by the classical mathematician when faced with learning about type theory is that it is usually presented as a fully or partially formalized deductive system. This style, which is very useful for proof-theoretic investigations, is not particularly convenient for use in applied, informal reasoning. Nor is it even familiar to most working mathematicians, even those who might be interested in foundations of mathematics. One objective of the present work is to develop an informal style of doing mathematics in univalent foundations that is at once rigorous and precise, but is also closer to the language and style of presentation of everyday mathematics. In present-day mathematics, one usually constructs and reasons about mathematical objects in a way that could in principle, one presumes, be formalized in a system of elementary set theory, such as ZFC — at least given enough ingenuity and patience. For the most part, one does not even need to be aware of this possibility, since it largely coincides with the condition that a proof be “fully rigorous” (in the sense that all mathematicians have come to understand intuitively through education and experience). But one does need to learn to be careful about a few aspects of “informal set theory”: the use of collections too large or inchoate to be sets; the axiom of choice and its equivalents; even (for undergraduates) the method of proof by contradiction; and so on. Adopting a new foundational system such as homotopy type theory as the implicit formal basis of informal reasoning will require adjusting some of one’s instincts and practices. The present text is intended to serve as an example of this “new kind of mathematics”, which is still informal, but could now in principle be formalized in homotopy type theory, rather than ZFC, again given enough ingenuity and patience. It is worth emphasizing that, in this new system, such formalization can have real practical benefits. The formal system of type theory is suited to computer systems and has been implemented in existing proof assistants. A proof assistant is a computer program which guides the user in construction of a fully formal proof, only allowing valid steps of reasoning. It also provides some degree of automation, can search libraries for existing theorems, and can even extract numerical algorithms from the resulting (constructive) proofs. We believe that this aspect of the univalent foundations program distinguishes it from other approaches to foundations, potentially providing a new practical utility for the working mathematician. Indeed, proof assistants based on older type theories have already been used to formalize substantial mathematical proofs, such as the four-color theorem and the Feit–Thompson theorem. Computer implementations of univalent foundations are presently works in progress (like the theory itself). However, even its currently available implementations (which are mostly small modifications to existing proof assistants such as C OQ and A GDA) have already demon-
8
I NTRODUCTION
strated their worth, not only in the formalization of known proofs, but in the discovery of new ones. Indeed, many of the proofs described in this book were actually first done in a fully formalized form in a proof assistant, and are only now being “unformalized” for the first time — a reversal of the usual relation between formal and informal mathematics. One can imagine a not-too-distant future when it will be possible for mathematicians to verify the correctness of their own papers by working within the system of univalent foundations, formalized in a proof assistant, and that doing so will become as natural as typesetting their own papers in TEX. In principle, this could be equally true for any other foundational system, but we believe it to be more practically attainable using univalent foundations, as witnessed by the present work and its formal counterpart.
Constructivity One of the most striking differences between classical foundations and type theory is the idea of proof relevance, according to which mathematical statements, and even their proofs, become first-class mathematical objects. In type theory, we represent mathematical statements by types, which can be regarded simultaneously as both mathematical constructions and mathematical assertions, a conception also known as propositions as types. Accordingly, we can regard a term a : A as both an element of the type A (or in homotopy type theory, a point of the space A), and at the same time, a proof of the proposition A. To take an example, suppose we have sets A and B (discrete spaces), and consider the statement “A is isomorphic to B”. In type theory, this can be rendered as: Iso( A, B) :≡ ∑ g ( f ( x )) = x × f ( g ( y )) = y . ∏(x:A) ∏(y:B) ∑ ( f :A→ B) ( g:B→ A)
Reading the type constructors Σ, Π, × here as “there exists”, “for all”, and “and” respectively yields the usual formulation of “A and B are isomorphic”; on the other hand, reading them as sums and products yields the type of all isomorphisms between A and B! To prove that A and B are isomorphic, one constructs a proof p : Iso( A, B), which is therefore the same as constructing an isomorphism between A and B, i.e., exhibiting a pair of functions f , g together with proofs that their composites are the respective identity maps. The latter proofs, in turn, are nothing but homotopies of the appropriate sorts. In this way, proving a proposition is the same as constructing an element of some particular type. In particular, to prove a statement of the form “A and B” is just to prove A and to prove B, i.e., to give an element of the type A × B. And to prove that A implies B is just to find an element of A → B, i.e. a function from A to B (determining a mapping of proofs of A to proofs of B). The logic of propositions-as-types is flexible and supports many variations, such as using only a subclass of types to represent propositions. In homotopy type theory, there are natural such subclasses arising from the fact that the system of all types, just like spaces in classical homotopy theory, is “stratified” according to the dimensions in which their higher homotopy structure exists or collapses. In particular, Voevodsky has found a purely type-theoretic definition of homotopy n-types, corresponding to spaces with no nontrivial homotopy information above dimension n. (The 0-types are the “sets” mentioned previously as satisfying Lawvere’s
9 axioms.) Moreover, with higher inductive types, we can universally “truncate” a type into an ntype; in classical homotopy theory this would be its nth Postnikov section. Particularly important for logic is the case of homotopy (−1)-types, which we call mere propositions. Classically, every (−1)-type is empty or contractible; we interpret these possibilities as the truth values “false” and “true” respectively. Using all types as propositions yields a “constructive” conception of logic (for more on which, see [Kol32, TvD88a, TvD88b]), which gives type theory its good computational character. For instance, every proof that something exists carries with it enough information to actually find such an object; and from a proof that “A or B” holds, one can extract either a proof that A holds or one that B holds. Thus, from every proof we can automatically extract an algorithm; this can be very useful in applications to computer programming. However, this logic does not faithfully represent certain important classical principles of reasoning, such as the axiom of choice and the law of excluded middle. For these we need to use the “(−1)-truncated” logic, in which only the homotopy (−1)-types represent propositions; and under this interpretation, the system is fully compatible with classical mathematics. Homotopy type theory is thus compatible with both constructive and classical conceptions of logic, and many more besides. More specifically, consider on one hand the axiom of choice: “if for every x : A there exists a y : B such that R( x, y), there is a function f : A → B such that for all x : A we have R( x, f ( x )).” The pure propositions-as-types notion of “there exists” is strong enough to make this statement simply provable — yet it does not have all the consequences of the usual axiom of choice. However, in (−1)-truncated logic, this statement is not automatically true, but is a strong assumption with the same sorts of consequences as its counterpart in classical set theory. On the other hand, consider the law of excluded middle: “for all A, either A or not A.” Interpreting this in the pure propositions-as-types logic yields a statement that is inconsistent with the univalence axiom. For since proving “A” means exhibiting an element of it, this assumption would give a uniform way of selecting an element from every nonempty type — a sort of Hilbertian choice operator. Univalence implies that the element of A selected by such a choice operator must be invariant under all self-equivalences of A, since these are identified with self-identities and every operation must respect identity; but clearly some types have automorphisms with no fixed points, e.g. we can swap the elements of a two-element type. However, the “(−1)-truncated law of excluded middle”, though also not automatically true, may consistently be assumed with most of the same consequences as in classical mathematics. In other words, while the pure propositions-as-types logic is “constructive” in the strong algorithmic sense mentioned above, the default (−1)-truncated logic is “constructive” in a different sense (namely, that of the logic formalized by Heyting under the name “intuitionistic”); and to the latter we may freely add the axioms of choice and excluded middle to obtain a logic that may be called “classical”. Thus, the homotopical perspective reveals that classical and constructive logic can coexist, as endpoints of a spectrum of different systems, with an infinite number of possibilities in between (the homotopy n-types for −1 < n < ∞). We may speak of “LEMn ” and “ACn ”, with AC∞ being provable and LEM∞ inconsistent with univalence, while AC−1 and LEM−1 are the versions familiar to classical mathematicians (hence in most cases it is appropriate to assume the subscript (−1) when none is given). Indeed, one can even have useful systems in
10
I NTRODUCTION
which only certain types satisfy such further “classical” principles, while types in general remain “constructive”. It is worth emphasizing that univalent foundations does not require the use of constructive or intuitionistic logic. Most of classical mathematics which depends on the law of excluded middle and the axiom of choice can be performed in univalent foundations, simply by assuming that these two principles hold (in their proper, (−1)-truncated, form). However, type theory does encourage avoiding these principles when they are unnecessary, for several reasons. First of all, every mathematician knows that a theorem is more powerful when proven using fewer assumptions, since it applies to more examples. The situation with AC and LEM is no different: type theory admits many interesting “nonstandard” models, such as in sheaf toposes, where classicality principles such as AC and LEM tend to fail. Homotopy type theory admits similar models in higher toposes, such as are studied in [TV02, Rez05, Lur09]. Thus, if we avoid using these principles, the theorems we prove will be valid internally to all such models. Secondly, one of the additional virtues of type theory is its computable character. In addition to being a foundation for mathematics, type theory is a formal theory of computation, and can be treated as a powerful programming language. From this perspective, the rules of the system cannot be chosen arbitrarily the way set-theoretic axioms can: there must be a harmony between them which allows all proofs to be “executed” as programs. We do not yet fully understand the new principles introduced by homotopy type theory, such as univalence and higher inductive types, from this point of view, but the basic outlines are emerging; see, for example, [LH12]. It has been known for a long time, however, that principles such as AC and LEM are fundamentally antithetical to computability, since they assert baldly that certain things exist without giving any way to compute them. Thus, avoiding them is necessary to maintain the character of type theory as a theory of computation. Fortunately, constructive reasoning is not as hard as it may seem. In some cases, simply by rephrasing some definitions, a theorem can be made constructive and its proof more elegant. Moreover, in univalent foundations this seems to happen more often. For instance: (i) In set-theoretic foundations, at various points in homotopy theory and category theory one needs the axiom of choice to perform transfinite constructions. But with higher inductive types, we can encode these constructions directly and constructively. In particular, none of the “synthetic” homotopy theory in Chapter 8 requires LEM or AC. (ii) In set-theoretic foundations, the statement “every fully faithful and essentially surjective functor is an equivalence of categories” is equivalent to the axiom of choice. But with the univalence axiom, it is just true; see Chapter 9. (iii) In set theory, various circumlocutions are required to obtain notions of “cardinal number” and “ordinal number” which canonically represent isomorphism classes of sets and well-ordered sets, respectively — possibly involving the axiom of choice or the axiom of foundation. But with univalence and higher inductive types, we can obtain such representatives directly by truncating the universe; see Chapter 10. (iv) In set-theoretic foundations, the definition of the real numbers as equivalence classes of Cauchy sequences requires either the law of excluded middle or the axiom of (countable) choice to be well-behaved. But with higher inductive types, we can give a version of this
11 definition which is well-behaved and avoids any choice principles; see Chapter 11. Of course, these simplifications could as well be taken as evidence that the new methods will not, ultimately, prove to be really constructive. However, we emphasize again that the reader does not have to care, or worry, about constructivity in order to read this book. The point is that in all of the above examples, the version of the theory we give has independent advantages, whether or not LEM and AC are assumed to be available. Constructivity, if attained, will be an added bonus. Given this discussion of adding new principles such as univalence, higher inductive types, AC, and LEM, one may wonder whether the resulting system remains consistent. (One of the original virtues of type theory, relative to set theory, was that it can be seen to be consistent by proof-theoretic means). As with any foundational system, consistency is a relative question: “consistent with respect to what?” The short answer is that all of the constructions and axioms considered in this book have a model in the category of Kan complexes, due to Voevodsky [KLV12] (see [LS13b] for higher inductive types). Thus, they are known to be consistent relative to ZFC (with as many inaccessible cardinals as we need nested univalent universes). Giving a more traditionally type-theoretic account of this consistency is work in progress (see, e.g., [LH12, BCH13]). We summarize the different points of view of the type-theoretic operations in Table 1. Types
Logic
Sets
Homotopy
A a:A B( x ) b( x ) : B( x ) 0, 1 A+B A×B A→B ∑(x:A) B( x )
proposition proof predicate conditional proof ⊥, > A∨B A∧B A⇒B ∃ x:A B( x )
set element family of sets family of elements ∅, {∅} disjoint union set of pairs set of functions disjoint sum
space point fibration section ∅, ∗ coproduct product space function space total space
∏(x:A) B( x ) Id A
∀ x:A B( x ) equality =
product
space of sections
{ ( x, x ) | x ∈ A }
path space A I
Table 1: Comparing points of view on type-theoretic operations
Open problems For those interested in contributing to this new branch of mathematics, it may be encouraging to know that there are many interesting open questions. Perhaps the most pressing of them is the “constructivity” of the Univalence Axiom, posed by Voevodsky in [Voe12]. The basic system of type theory follows the structure of Gentzen’s
12
I NTRODUCTION
natural deduction. Logical connectives are defined by their introduction rules, and have elimination rules justified by computation rules. Following this pattern, and using Tait’s computabil¨ ity method, originally designed to analyse Godel’s Dialectica interpretation, one can show the property of normalization for type theory. This in turn implies important properties such as decidability of type-checking (a crucial property since type-checking corresponds to proof-checking, and one can argue that we should be able to “recognize a proof when we see one”), and the so-called “canonicity property” that any closed term of the type of natural numbers reduces to a numeral. This last property, and the uniform structure of introduction/elimination rules, are lost when one extends type theory with an axiom, such as the axiom of function extensionality, or the univalence axiom. Voevodsky has formulated a precise mathematical conjecture connected to this question of canonicity for type theory extended with the axiom of Univalence: given a closed term of the type of natural numbers, is it always possible to find a numeral and a proof that this term is equal to this numeral, where this proof of equality may itself use the univalence axiom? More generally, an important issue is whether it is possible to provide a constructive justification of the univalence axiom. What about if one adds other homotopically motivated constructions, like higher inductive types? These questions remain open at the present time, although methods are currently being developed to try to find answers. Another basic issue is the difficulty of working with types, such as the natural numbers, that are essentially sets (i.e., discrete spaces), containing only trivial paths. At present, homotopy type theory can really only characterize spaces up to homotopy equivalence, which means that these “discrete spaces” may only be homotopy equivalent to discrete spaces. Type-theoretically, this means there are many paths that are equal to reflexivity, but not judgmentally equal to it (see §1.1 for the meaning of “judgmentally”). While this homotopy-invariance has advantages, these “meaningless” identity terms do introduce needless complications into arguments and constructions, so it would be convenient to have a systematic way of eliminating or collapsing them. A more specialized, but no less important, problem is the relation between homotopy type theory and the research on higher toposes currently happening at the intersection of higher category theory and homotopy theory. There is a growing conviction among those familiar with both subjects that they are intimately connected. For instance, the notion of a univalent universe should coincide with that of an object classifier, while higher inductive types should be an “elementary” reflection of local presentability. More generally, homotopy type theory should be the “internal language” of (∞, 1)-toposes, just as intuitionistic higher-order logic is the internal language of ordinary 1-toposes. Despite this general consensus, however, details remain to be worked out — in particular, questions of coherence and strictness remain to be addressed — and doing so will undoubtedly lead to further insights into both concepts. But by far the largest field of work to be done is in the ongoing formalization of everyday mathematics in this new system. Recent successes in formalizing some facts from basic homotopy theory and category theory have been encouraging; some of these are described in Chapters 8 and 9. Obviously, however, much work remains to be done. The homotopy type theory community maintains a web site and group blog at http:// homotopytypetheory.org, as well as a discussion email list. Newcomers are always welcome!
13
How to read this book This book is divided into two parts. Part I, “Foundations”, develops the fundamental concepts of homotopy type theory. This is the mathematical foundation on which the development of specific subjects is built, and which is required for the understanding of the univalent foundations approach. To a programmer, this is “library code”. Since univalent foundations is a new and different kind of mathematics, its basic notions take some getting used to; thus Part I is fairly extensive. Part II, “Mathematics”, consists of four chapters that build on the basic notions of Part I to exhibit some of the new things we can do with univalent foundations in four different areas of mathematics: homotopy theory (Chapter 8), category theory (Chapter 9), set theory (Chapter 10), and real analysis (Chapter 11). The chapters in Part II are more or less independent of each other, although occasionally one will use a lemma proven in another. A reader who wants to seriously understand univalent foundations, and be able to work in it, will eventually have to read and understand most of Part I. However, a reader who just wants to get a taste of univalent foundations and what it can do may understandably balk at having to work through over 200 pages before getting to the “meat” in Part II. Fortunately, not all of Part I is necessary in order to read the chapters in Part II. Each chapter in Part II begins with a brief overview of its subject, what univalent foundations has to contribute to it, and the necessary background from Part I, so the courageous reader can turn immediately to the appropriate chapter for their favorite subject. For those who want to understand one or more chapters in Part II more deeply than this, but are not ready to read all of Part I, we provide here a brief summary of Part I, with remarks about which parts are necessary for which chapters in Part II. Chapter 1 is about the basic notions of type theory, prior to any homotopical interpretation. A ¨ type theory can quickly skim it to pick up the particulars reader who is familiar with Martin-Lof of the theory we are using. However, readers without experience in type theory will need to read Chapter 1, as there are many subtle differences between type theory and other foundations such as set theory. Chapter 2 introduces the homotopical viewpoint on type theory, along with the basic notions supporting this view, and describes the homotopical behavior of each component of the type theory from Chapter 1. It also introduces the univalence axiom (§2.10) — the first of the two basic innovations of homotopy type theory. Thus, it is quite basic and we encourage everyone to read it, especially §§2.1–2.4. Chapter 3 describes how we represent logic in homotopy type theory, and its connection to classical logic as well as to constructive and intuitionistic logic. Here we define the law of excluded middle, the axiom of choice, and the axiom of propositional resizing (although, for the most part, we do not need to assume any of these in the rest of the book), as well as the propositional truncation which is essential for representing traditional logic. This chapter is essential background for Chapters 10 and 11, less important for Chapter 9, and not so necessary for Chapter 8. Chapters 4 and 5 study two special topics in detail: equivalences (and related notions) and generalized inductive definitions. While these are important subjects in their own rights and provide a deeper understanding of homotopy type theory, for the most part they are not necessary for Part II. Only a few lemmas from Chapter 4 are used here and there, while the general
14
I NTRODUCTION
discussions in §§5.1, 5.6 and 5.7 are helpful for providing the intuition required for Chapter 6. The generalized sorts of inductive definition discussed in §5.7 are also used in a few places in Chapters 10 and 11. Chapter 6 introduces the second basic innovation of homotopy type theory — higher inductive types — with many examples. Higher inductive types are the primary object of study in Chapter 8, and some particular ones play important roles in Chapters 10 and 11. They are not so necessary for Chapter 9, although one example is used in §9.9. Finally, Chapter 7 discusses homotopy n-types and related notions such as n-connected types. These notions are important for Chapter 8, but not so important in the rest of Part II, although the case n = −1 of some of the lemmas are used in §10.1. This completes Part I. As mentioned above, Part II consists of four largely unrelated chapters, each describing what univalent foundations has to offer to a particular subject. Of the chapters in Part II, Chapter 8 (Homotopy theory) is perhaps the most radical. Univalent foundations has a very different “synthetic” approach to homotopy theory in which homotopy types are the basic objects (namely, the types) rather than being constructed using topological spaces or some other set-theoretic model. This enables new styles of proof for classical theorems in algebraic topology, of which we present a sampling, from π1 (S1 ) = Z to the Freudenthal suspension theorem. In Chapter 9 (Category theory), we develop some basic (1-)category theory, adhering to the principle of the univalence axiom that equality is isomorphism. This has the pleasant effect of ensuring that all definitions and constructions are automatically invariant under equivalence of categories: indeed, equivalent categories are equal just as equivalent types are equal. (It also has connections to higher category theory and higher topos theory.) Chapter 10 (Set theory) studies sets in univalent foundations. The category of sets has its usual properties, hence provides a foundation for any mathematics that doesn’t need homotopical or higher-categorical structures. We also observe that univalence makes cardinal and ordinal numbers a bit more pleasant, and that higher inductive types yield a cumulative hierarchy satisfying the usual axioms of Zermelo–Fraenkel set theory. In Chapter 11 (Real numbers), we summarize the construction of Dedekind real numbers, and then observe that higher inductive types allow a definition of Cauchy real numbers that avoids some associated problems in constructive mathematics. Then we sketch a similar approach to Conway’s surreal numbers. Each chapter in this book ends with a Notes section, which collects historical comments, references to the literature, and attributions of results, to the extent possible. We have also included Exercises at the end of each chapter, to assist the reader in gaining familiarity with doing mathematics in univalent foundations. Finally, recall that this book was written as a massively collaborative effort by a large number of people. We have done our best to achieve consistency in terminology and notation, and to put the mathematics in a linear sequence that flows logically, but it is very likely that some imperfections remain. We ask the reader’s forgiveness for any such infelicities, and welcome suggestions for improvement of the next edition.
PART I
F OUNDATIONS
Chapter 1
Type theory 1.1
Type theory versus set theory
Homotopy type theory is (among other things) a foundational language for mathematics, i.e., an alternative to Zermelo–Fraenkel set theory. However, it behaves differently from set theory in several important ways, and that can take some getting used to. Explaining these differences carefully requires us to be more formal here than we will be in the rest of the book. As stated in the introduction, our goal is to write type theory informally; but for a mathematician accustomed to set theory, more precision at the beginning can help avoid some common misconceptions and mistakes. We note that a set-theoretic foundation has two “layers”: the deductive system of first-order logic, and, formulated inside this system, the axioms of a particular theory, such as ZFC. Thus, set theory is not only about sets, but rather about the interplay between sets (the objects of the second layer) and propositions (the objects of the first layer). By contrast, type theory is its own deductive system: it need not be formulated inside any superstructure, such as first-order logic. Instead of the two basic notions of set theory, sets and propositions, type theory has one basic notion: types. Propositions (statements which we can prove, disprove, assume, negate, and so on1 ) are identified with particular types, via the correspondence shown in Table 1 on page 11. Thus, the mathematical activity of proving a theorem is identified with a special case of the mathematical activity of constructing an object—in this case, an inhabitant of a type that represents a proposition. This leads us to another difference between type theory and set theory, but to explain it we must say a little about deductive systems in general. Informally, a deductive system is a collection of rules for deriving things called judgments. If we think of a deductive system as a formal game, then the judgments are the “positions” in the game which we reach by following the game rules. We can also think of a deductive system as a sort of algebraic theory, in which case the judgments are the elements (like the elements of a group) and the deductive rules are 1 Confusingly, it is also a common practice (dating back to Euclid) to use the word “proposition” synonymously with “theorem”. We will confine ourselves to the logician’s usage, according to which a proposition is a statement susceptible to proof, whereas a theorem (or “lemma” or “corollary”) is such a statement that has been proven. Thus “0 = 1” and its negation “¬(0 = 1)” are both propositions, but only the latter is a theorem.
18
C HAPTER 1. T YPE THEORY
the operations (like the group multiplication). From a logical point of view, the judgments can be considered to be the “external” statements, living in the metatheory, as opposed to the “internal” statements of the theory itself. In the deductive system of first-order logic (on which set theory is based), there is only one kind of judgment: that a given proposition has a proof. That is, each proposition A gives rise to a judgment “A has a proof”, and all judgments are of this form. A rule of first-order logic such as “from A and B infer A ∧ B” is actually a rule of “proof construction” which says that given the judgments “A has a proof” and “B has a proof”, we may deduce that “A ∧ B has a proof”. Note that the judgment “A has a proof” exists at a different level from the proposition A itself, which is an internal statement of the theory. The basic judgment of type theory, analogous to “A has a proof”, is written “a : A” and pronounced as “the term a has type A”, or more loosely “a is an element of A” (or, in homotopy type theory, “a is a point of A”). When A is a type representing a proposition, then a may be called a witness to the provability of A, or evidence of the truth of A (or even a proof of A, but we will try to avoid this confusing terminology). In this case, the judgment a : A is derivable in type theory (for some a) precisely when the analogous judgment “A has a proof” is derivable in first-order logic (modulo differences in the axioms assumed and in the encoding of mathematics, as we will discuss throughout the book). On the other hand, if the type A is being treated more like a set than like a proposition (although as we will see, the distinction can become blurry), then “a : A” may be regarded as analogous to the set-theoretic statement “a ∈ A”. However, there is an essential difference in that “a : A” is a judgment whereas “a ∈ A” is a proposition. In particular, when working internally in type theory, we cannot make statements such as “if a : A then it is not the case that b : B”, nor can we “disprove” the judgment “a : A”. A good way to think about this is that in set theory, “membership” is a relation which may or may not hold between two pre-existing objects “a” and “A”, while in type theory we cannot talk about an element “a” in isolation: every element by its very nature is an element of some type, and that type is (generally speaking) uniquely determined. Thus, when we say informally “let x be a natural number”, in set theory this is shorthand for “let x be a thing and assume that x ∈ N”, whereas in type theory “let x : N” is an atomic statement: we cannot introduce a variable without specifying its type. At first glance, this may seem an uncomfortable restriction, but it is arguably closer to the intuitive mathematical meaning of “let x be a natural number”. In practice, it seems that whenever we actually need “a ∈ A” to be a proposition rather than a judgment, there is always an ambient set B of which a is known to be an element and A is known to be a subset. This situation is also easy to represent in type theory, by taking a to be an element of the type B, and A to be a predicate on B; see §3.5. A last difference between type theory and set theory is the treatment of equality. The familiar notion of equality in mathematics is a proposition: e.g. we can disprove an equality or assume an equality as a hypothesis. Since in type theory, propositions are types, this means that equality is a type: for elements a, b : A (that is, both a : A and b : A) we have a type “a = A b”. (In homotopy type theory, of course, this equality proposition can behave in unfamiliar ways: see §1.12 and Chapter 2, and the rest of the book). When a = A b is inhabited, we say that a and b are
1.1 T YPE THEORY VERSUS SET THEORY
19
(propositionally) equal. However, in type theory there is also a need for an equality judgment, existing at the same level as the judgment “x : A”. This is called judgmental equality or definitional equality, and we write it as a ≡ b : A or simply a ≡ b. It is helpful to think of this as meaning “equal by definition”. For instance, if we define a function f : N → N by the equation f ( x ) = x2 , then the expression f (3) is equal to 32 by definition. Inside the theory, it does not make sense to negate or assume an equality-by-definition; we cannot say “if x is equal to y by definition, then z is not equal to w by definition”. Whether or not two expressions are equal by definition is just a matter of expanding out the definitions; in particular, it is algorithmically decidable (though the algorithm is necessarily meta-theoretic, not internal to the theory). As type theory becomes more complicated, judgmental equality can get more subtle than this, but it is a good intuition to start from. Alternatively, if we regard a deductive system as an algebraic theory, then judgmental equality is simply the equality in that theory, analogous to the equality between elements of a group—the only potential for confusion is that there is also an object inside the deductive system of type theory (namely the type “a = b”) which behaves internally as a notion of “equality”. The reason we want a judgmental notion of equality is so that it can control the other form of judgment, “a : A”. For instance, suppose we have given a proof that 32 = 9, i.e. we have derived the judgment p : (32 = 9) for some p. Then the same witness p ought to count as a proof that f (3) = 9, since f (3) is 32 by definition. The best way to represent this is with a rule saying that given the judgments a : A and A ≡ B, we may derive the judgment a : B. Thus, for us, type theory will be a deductive system based on two forms of judgment: Judgment
Meaning
a:A a≡b:A
“a is an object of type A” “a and b are definitionally equal objects of type A”
When introducing a definitional equality, i.e., defining one thing to be equal to another, we will use the symbol “:≡”. Thus, the above definition of the function f would be written as f ( x ) :≡ x2 . Because judgments cannot be put together into more complicated statements, the symbols “:” and “≡” bind more loosely than anything else.2 Thus, for instance, “p : x = y” should be parsed as “p : ( x = y)”, which makes sense since “x = y” is a type, and not as “( p : x ) = y”, which is senseless since “p : x” is a judgment and cannot be equal to anything. Similarly, “A ≡ x = y” can only be parsed as “A ≡ ( x = y)”, although in extreme cases such as this, one ought to add parentheses anyway to aid reading comprehension. Moreover, later on we will fall into the common notation of chaining together equalities — e.g. writing a = b = c = d to mean “a = b and b = c and c = d, hence a = d” — and we will also include judgmental equalities in such chains. Context usually suffices to make the intent clear. This is perhaps also an appropriate place to mention that the common mathematical notation “ f : A → B”, expressing the fact that f is a function from A to B, can be regarded as a typing 2 In formalized type theory, commas and turnstiles can bind even more loosely. For instance, x : A, y : B ` c : C is parsed as (( x : A), (y : B)) ` (c : C ). However, in this book we refrain from such notation until Appendix A.
20
C HAPTER 1. T YPE THEORY
judgment, since we use “A → B” as notation for the type of functions from A to B (as is standard practice in type theory; see §1.4). Judgments may depend on assumptions of the form x : A, where x is a variable and A is a type. For example, we may construct an object m + n : N under the assumptions that m, n : N. Another example is that assuming A is a type, x, y : A, and p : x = A y, we may construct an element p−1 : y = A x. The collection of all such assumptions is called the context; from a topological point of view it may be thought of as a “parameter space”. In fact, technically the context must be an ordered list of assumptions, since later assumptions may depend on previous ones: the assumption x : A can only be made after the assumptions of any variables appearing in the type A. If the type A in an assumption x : A represents a proposition, then the assumption is a type-theoretic version of a hypothesis: we assume that the proposition A holds. When types are regarded as propositions, we may omit the names of their proofs. Thus, in the second example above we may instead say that assuming x = A y, we can prove y = A x. However, since we are doing “proof-relevant” mathematics, we will frequently refer back to proofs as objects. In the example above, for instance, we may want to establish that p−1 together with the proofs of transitivity and reflexivity behave like a groupoid; see Chapter 2. Note that under this meaning of the word assumption, we can assume a propositional equality (by assuming a variable p : x = y), but we cannot assume a judgmental equality x ≡ y, since it is not a type that can have an element. However, we can do something else which looks kind of like assuming a judgmental equality: if we have a type or an element which involves a variable x : A, then we can substitute any particular element a : A for x to obtain a more specific type or element. We will sometimes use language like “now assume x ≡ a” to refer to this process of substitution, even though it is not an assumption in the technical sense introduced above. By the same token, we cannot prove a judgmental equality either, since it is not a type in which we can exhibit a witness. Nevertheless, we will sometimes state judgmental equalities as part of a theorem, e.g. “there exists f : A → B such that f ( x ) ≡ y”. This should be regarded as the making of two separate judgments: first we make the judgment f : A → B for some element f , then we make the additional judgment that f ( x ) ≡ y. In the rest of this chapter, we attempt to give an informal presentation of type theory, sufficient for the purposes of this book; we give a more formal account in Appendix A. Aside from some fairly obvious rules (such as the fact that judgmentally equal things can always be substituted for each other), the rules of type theory can be grouped into type formers. Each type former consists of a way to construct types (possibly making use of previously constructed types), together with rules for the construction and behavior of elements of that type. In most cases, these rules follow a fairly predictable pattern, but we will not attempt to make this precise here; see however the beginning of §1.5 and also Chapter 5. An important aspect of the type theory presented in this chapter is that it consists entirely of rules, without any axioms. In the description of deductive systems in terms of judgments, the rules are what allow us to conclude one judgment from a collection of others, while the axioms are the judgments we are given at the outset. If we think of a deductive system as a formal game, then the rules are the rules of the game, while the axioms are the starting position. And if we think of a deductive system as an algebraic theory, then the rules are the operations of the theory,
1.2 F UNCTION TYPES
21
while the axioms are the generators for some particular free model of that theory. In set theory, the only rules are the rules of first-order logic (such as the rule allowing us to deduce “A ∧ B has a proof” from “A has a proof” and “B has a proof”): all the information about the behavior of sets is contained in the axioms. By contrast, in type theory, it is usually the rules which contain all the information, with no axioms being necessary. For instance, in §1.5 we will see that there is a rule allowing us to deduce the judgment “( a, b) : A × B” from “a : A” and “b : B”, whereas in set theory the analogous statement would be (a consequence of) the pairing axiom. The advantage of formulating type theory using only rules is that rules are “procedural”. In particular, this property is what makes possible (though it does not automatically ensure) the good computational properties of type theory, such as “canonicity”. However, while this style works for traditional type theories, we do not yet understand how to formulate everything we need for homotopy type theory in this way. In particular, in §§2.9 and 2.10 and Chapter 6 we will have to augment the rules of type theory presented in this chapter by introducing additional axioms, notably the univalence axiom. In this chapter, however, we confine ourselves to a traditional rule-based type theory.
1.2
Function types
Given types A and B, we can construct the type A → B of functions with domain A and codomain B. We also sometimes refer to functions as maps. Unlike in set theory, functions are not defined as functional relations; rather they are a primitive concept in type theory. We explain the function type by prescribing what we can do with functions, how to construct them and what equalities they induce. Given a function f : A → B and an element of the domain a : A, we can apply the function to obtain an element of the codomain B, denoted f ( a) and called the value of f at a. It is common in type theory to omit the parentheses and denote f ( a) simply by f a, and we will sometimes do this as well. But how can we construct elements of A → B? There are two equivalent ways: either by direct definition or by using λ-abstraction. Introducing a function by definition means that we introduce a function by giving it a name — let’s say, f — and saying we define f : A → B by giving an equation f ( x ) :≡ Φ (1.2.1) where x is a variable and Φ is an expression which may use x. In order for this to be valid, we have to check that Φ : B assuming x : A. Now we can compute f ( a) by replacing the variable x in Φ with a. As an example, consider the function f : N → N which is defined by f ( x ) :≡ x + x. (We will define N and + in §1.9.) Then f (2) is judgmentally equal to 2 + 2. If we don’t want to introduce a name for the function, we can use λ-abstraction. Given an expression Φ of type B which may use x : A, as above, we write λ( x : A). Φ to indicate the same function defined by (1.2.1). Thus, we have
(λ( x : A). Φ) : A → B.
22
C HAPTER 1. T YPE THEORY
For the example in the previous paragraph, we have the typing judgment
(λ( x : N). x + x ) : N → N. As another example, for any types A and B and any element y : B, we have a constant function (λ( x : A). y) : A → B. We generally omit the type of the variable x in a λ-abstraction and write λx. Φ, since the typing x : A is inferable from the judgment that the function λx. Φ has type A → B. By convention, the “scope” of the variable binding “λx. ” is the entire rest of the expression, unless delimited with parentheses. Thus, for instance, λx. x + x should be parsed as λx. ( x + x ), not as (λx. x ) + x (which would, in this case, be ill-typed anyway). Another equivalent notation is
( x 7→ Φ) : A → B. We may also sometimes use a blank “–” in the expression Φ in place of a variable, to denote an implicit λ-abstraction. For instance, g( x, – ) is another way to write λy. g( x, y). Now a λ-abstraction is a function, so we can apply it to an argument a : A. We then have the following computation rule3 , which is a definitional equality:
(λx. Φ)( a) ≡ Φ0 where Φ0 is the expression Φ in which all occurrences of x have been replaced by a. Continuing the above example, we have (λx. x + x )(2) ≡ 2 + 2. Note that from any function f : A → B, we can construct a lambda abstraction function λx. f ( x ). Since this is by definition “the function that applies f to its argument” we consider it to be definitionally equal to f :4 f ≡ (λx. f ( x )). This equality is the uniqueness principle for function types, because it shows that f is uniquely determined by its values. The introduction of functions by definitions with explicit parameters can be reduced to simple definitions by using λ-abstraction: i.e., we can read a definition of f : A → B by f ( x ) :≡ Φ as f :≡ λx. Φ. When doing calculations involving variables, we have to be careful when replacing a variable with an expression that also involves variables, because we want to preserve the binding structure of expressions. By the binding structure we mean the invisible link generated by binders 3 Use 4 Use
of this equality is often referred to as β-conversion or β-reduction. of this equality is often referred to as η-conversion or η-expansion.
1.2 F UNCTION TYPES
23
such as λ, Π and Σ (the latter we are going to meet soon) between the place where the variable is introduced and where it is used. As an example, consider f : N → (N → N) defined as f ( x ) :≡ λy. x + y. Now if we have assumed somewhere that y : N, then what is f (y)? It would be wrong to just naively replace x by y everywhere in the expression “λy. x + y” defining f ( x ), obtaining λy. y + y, because this means that y gets captured. Previously, the substituted y was referring to our assumption, but now it is referring to the argument of the λ-abstraction. Hence, this naive substitution would destroy the binding structure, allowing us to perform calculations which are semantically unsound. But what is f (y) in this example? Note that bound (or “dummy”) variables such as y in the expression λy. x + y have only a local meaning, and can be consistently replaced by any other variable, preserving the binding structure. Indeed, λy. x + y is declared to be judgmentally equal5 to λz. x + z. It follows that f (y) is judgmentally equal to λz. y + z, and that answers our question. (Instead of z, any variable distinct from y could have been used, yielding an equal result.) Of course, this should all be familiar to any mathematician: it is the same phenomenon as R2 R 2 dt R 2 ds the fact that if f ( x ) :≡ 1 xdt −t , then f ( t ) is not 1 t−t but rather 1 t−s . A λ-abstraction binds a dummy variable in exactly the same way that an integral does. We have seen how to define functions in one variable. One way to define functions in several variables would be to use the cartesian product, which will be introduced later; a function with parameters A and B and results in C would be given the type f : A × B → C. However, there is another choice that avoids using product types, which is called currying (after the mathematician Haskell Curry). The idea of currying is to represent a function of two inputs a : A and b : B as a function which takes one input a : A and returns another function, which then takes a second input b : B and returns the result. That is, we consider two-variable functions to belong to an iterated function type, f : A → ( B → C ). We may also write this without the parentheses, as f : A → B → C, with associativity to the right as the default convention. Then given a : A and b : B, we can apply f to a and then apply the result to b, obtaining f ( a)(b) : C. To avoid the proliferation of parentheses, we allow ourselves to write f ( a)(b) as f ( a, b) even though there are no products involved. When omitting parentheses around function arguments entirely, we write f a b for ( f a) b, with the default associativity now being to the left so that f is applied to its arguments in the correct order. Our notation for definitions with explicit parameters extends to this situation: we can define a named function f : A → B → C by giving an equation f ( x, y) :≡ Φ where Φ : C assuming x : A and y : B. Using λ-abstraction this corresponds to f :≡ λx. λy. Φ, 5 Use
of this equality is often referred to as α-conversion.
24
C HAPTER 1. T YPE THEORY
which may also be written as f :≡ x 7→ y 7→ Φ. We can also implicitly abstract over multiple variables by writing multiple blanks, e.g. g( –, – ) means λx. λy. g( x, y). Currying a function of three or more arguments is a straightforward extension of what we have just described.
1.3
Universes and families
So far, we have been using the expression “A is a type” informally. We are going to make this more precise by introducing universes. A universe is a type whose elements are types. As in naive set theory, we might wish for a universe of all types U∞ including itself (that is, with U∞ : U∞ ). However, as in set theory, this is unsound, i.e. we can deduce from it that every type, including the empty type representing the proposition False (see §1.7), is inhabited. For instance, using a representation of sets as trees, we can directly encode Russell’s paradox [Coq92]. To avoid the paradox we introduce a hierarchy of universes
U0 : U1 : U2 : · · · where every universe Ui is an element of the next universe Ui+1 . Moreover, we assume that our universes are cumulative, that is that all the elements of the ith universe are also elements of the (i + 1)st universe, i.e. if A : Ui then also A : Ui+1 . This is convenient, but has the slightly unpleasant consequence that elements no longer have unique types, and is a bit tricky in other ways that need not concern us here; see the Notes. When we say that A is a type, we mean that it inhabits some universe Ui . We usually want to avoid mentioning the level i explicitly, and just assume that levels can be assigned in a consistent way; thus we may write A : U omitting the level. This way we can even write U : U , which can be read as Ui : Ui+1 , having left the indices implicit. Writing universes in this style is referred to as typical ambiguity. It is convenient but a bit dangerous, since it allows us to write valid-looking proofs that reproduce the paradoxes of self-reference. If there is any doubt about whether an argument is correct, the way to check it is to try to assign levels consistently to all universes appearing in it. When some universe U is assumed, we may refer to types belonging to U as small types. To model a collection of types varying over a given type A, we use functions B : A → U whose codomain is a universe. These functions are called families of types (or sometimes dependent types); they correspond to families of sets as used in set theory. An example of a type family is the family of finite sets Fin : N → U , where Fin(n) is a type with exactly n elements. (We cannot define the family Fin yet — indeed, we have not even introduced its domain N yet — but we will be able to soon; see Exercise 1.9.) We may denote the elements of Fin(n) by 0n , 1n , . . . , (n − 1)n , with subscripts to emphasize that the elements of Fin(n) are different from those of Fin(m) if n is different from m, and all are different from the ordinary natural numbers (which we will introduce in §1.9). A more trivial (but very important) example of a type family is the constant type family at a type B : U , which is of course the constant function (λ( x : A). B) : A → U .
1.4 D EPENDENT FUNCTION TYPES (Π- TYPES )
25
As a non-example, in our version of type theory there is no type family “λ(i : N). Ui ”. Indeed, there is no universe large enough to be its codomain. Moreover, we do not even identify the indices i of the universes Ui with the natural numbers N of type theory (the latter to be introduced in §1.9).
1.4
Dependent function types (Π-types)
In type theory we often use a more general version of function types, called a Π-type or dependent function type. The elements of a Π-type are functions whose codomain type can vary depending on the element of the domain to which the function is applied, called dependent functions. The name “Π-type” is used because this type can also be regarded as the cartesian product over a given type. Given a type A : U and a family B : A → U , we may construct the type of dependent functions ∏(x:A) B( x ) : U . There are many alternative notations for this type, such as
∏
∏(x:A) B( x )
B( x )
∏ ( x : A ), B ( x ).
( x:A)
If B is a constant family, then the dependent product type is the ordinary function type: ∏(x:A) B ≡ ( A → B). Indeed, all the constructions of Π-types are generalizations of the corresponding constructions on ordinary function types. We can introduce dependent functions by explicit definitions: to define f : ∏(x:A) B( x ), where f is the name of a dependent function to be defined, we need an expression Φ : B( x ) possibly involving the variable x : A, and we write f ( x ) :≡ Φ
for x : A.
Alternatively, we can use λ-abstraction λx. Φ :
∏ B ( x ).
(1.4.1)
x:A
As with non-dependent functions, we can apply a dependent function f : ∏(x:A) B( x ) to an argument a : A to obtain an element f ( a) : B( a). The equalities are the same as for the ordinary function type, i.e. we have the computation rule given a : A we have f ( a) ≡ Φ0 and (λx. Φ)( a) ≡ Φ0 , where Φ0 is obtained by replacing all occurrences of x in Φ by a (avoiding variable capture, as always). Similarly, we have the uniqueness principle f ≡ (λx. f ( x )) for any f : ∏(x:A) B( x ). As an example, recall from §1.3 that there is a type family Fin : N → U whose values are the standard finite sets, with elements 0n , 1n , . . . , (n − 1)n : Fin(n). There is then a dependent function fmax : ∏(n:N) Fin(n + 1) which returns the “largest” element of each nonempty finite type, fmax(n) :≡ nn+1 . As was the case for Fin itself, we cannot define fmax yet, but we will be able to soon; see Exercise 1.9.
26
C HAPTER 1. T YPE THEORY
Another important class of dependent function types, which we can define now, are functions which are polymorphic over a given universe. A polymorphic function is one which takes a type as one of its arguments, and then acts on elements of that type (or other types constructed from it). An example is the polymorphic identity function id : ∏( A:U ) A → A, which we define by id :≡ λ( A : U ). λ( x : A). x. We sometimes write some arguments of a dependent function as subscripts. For instance, we might equivalently define the polymorphic identity function by id A ( x ) :≡ x. Moreover, if an argument can be inferred from context, we may omit it altogether. For instance, if a : A, then writing id ( a) is unambiguous, since id must mean id A in order for it to be applicable to a. Another, less trivial, example of a polymorphic function is the “swap” operation that switches the order of the arguments of a (curried) two-argument function: swap :
∏ ∏ ∏
( A → B → C) → (B → A → C)
( A:U ) ( B:U ) (C:U )
We can define this by swap( A, B, C, g) :≡ λb. λa. g( a)(b). We might also equivalently write the type arguments as subscripts: swap A,B,C ( g)(b, a) :≡ g( a, b). Note that as we did for ordinary functions, we use currying to define dependent functions with several arguments (such as swap). However, in the dependent case the second domain may depend on the first one, and the codomain may depend on both. That is, given A : U and type families B : A → U and C : ∏(x:A) B( x ) → U , we may construct the type ∏(x:A) ∏(y:B(x)) C ( x, y) of functions with two arguments. (Like λ-abstractions, Πs automatically scope over the rest of the expression unless delimited; thus C : ∏(x:A) B( x ) → U means C : ∏(x:A) ( B( x ) → U ).) In the case when B is constant and equal to A, we may condense the notation and write ∏(x,y:A) ; for instance, the type of swap could also be written as swap :
∏
( A → B → C ) → ( B → A → C ).
A,B,C:U
Finally, given f : ∏(x:A) ∏(y:B(x)) C ( x, y) and arguments a : A and b : B( a), we have f ( a)(b) : C ( a, b), which, as before, we write as f ( a, b) : C ( a, b).
1.5
Product types
Given types A, B : U we introduce the type A × B : U , which we call their cartesian product. We also introduce a nullary product type, called the unit type 1 : U . We intend the elements of A × B to be pairs ( a, b) : A × B, where a : A and b : B, and the only element of 1 to be some particular object ? : 1. However, unlike in set theory, where we define ordered pairs to be particular sets and then collect them all together into the cartesian product, in type theory, ordered pairs are a primitive concept, as are functions. Remark 1.5.1. There is a general pattern for introduction of a new kind of type in type theory, and because products are our second example following this pattern,6 it is worth emphasizing 6 The
description of universes above is an exception.
1.5 P RODUCT TYPES
27
the general form: To specify a type, we specify: (i) how to form new types of this kind, via formation rules. (For example, we can form the function type A → B when A is a type and when B is a type. We can form the dependent function type ∏(x:A) B( x ) when A is a type and B( x ) is a type for x : A.) (ii) how to construct elements of that type. These are called the type’s constructors or introduction rules. (For example, a function type has one constructor, λ-abstraction. Recall that a direct definition like f ( x ) :≡ 2x can equivalently be phrased as a λ-abstraction f :≡ λx. 2x.) (iii) how to use elements of that type. These are called the type’s eliminators or elimination rules. (For example, the function type has one eliminator, namely function application.) (iv) a computation rule7 , which expresses how an eliminator acts on a constructor. (For example, for functions, the computation rule states that (λx. Φ)( a) is judgmentally equal to the substitution of a for x in Φ.) (v) an optional uniqueness principle8 , which expresses uniqueness of maps into or out of that type. For some types, the uniqueness principle characterizes maps into the type, by stating that every element of the type is uniquely determined by the results of applying eliminators to it, and can be reconstructed from those results by applying a constructor— thus expressing how constructors act on eliminators, dually to the computation rule. (For example, for functions, the uniqueness principle says that any function f is judgmentally equal to the “expanded” function λx. f ( x ), and thus is uniquely determined by its values.) For other types, the uniqueness principle says that every map (function) from that type is uniquely determined by some data. (An example is the coproduct type introduced in §1.7, whose uniqueness principle is mentioned in §2.15.) When the uniqueness principle is not taken as a rule of judgmental equality, it is often nevertheless provable as a propositional equality from the other rules for the type. In this case we call it a propositional uniqueness principle. (In later chapters we will also occasionally encounter propositional computation rules.) The inference rules in §A.2 are organized and named accordingly; see, for example, §A.2.4, where each possibility is realized. The way to construct pairs is obvious: given a : A and b : B, we may form ( a, b) : A × B. Similarly, there is a unique way to construct elements of
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Constructivism (philosophy of mathematics)
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In the philosophy of mathematics, constructivism asserts that it is necessary to find (or "construct") a mathematical object to prove that it exists. In classical mathematics, one can prove the existence of a mathematical object without "finding" that object explicitly, by assuming its non-existence and then deriving a contradiction from that assumption. This proof by contradiction is not constructively valid. The constructive viewpoint involves a verificational interpretation of the existential quantifier, which is at odds with its classical interpretation.
There are many forms of constructivism.[1] These include the program of intuitionism founded by Brouwer, the finitism of Hilbert and Bernays, the constructive recursive mathematics of Shanin and Markov, and Bishop's program of constructive analysis. Constructivism also includes the study of constructive set theories such as CZF and the study of topos theory.
Constructivism is often identified with intuitionism, although intuitionism is only one constructivist program. Intuitionism maintains that the foundations of mathematics lie in the individual mathematician's intuition, thereby making mathematics into an intrinsically subjective activity.[2] Other forms of constructivism are not based on this viewpoint of intuition, and are compatible with an objective viewpoint on mathematics.
Constructive mathematics
Much constructive mathematics uses intuitionistic logic, which is essentially classical logic without the law of the excluded middle. This law states that, for any proposition, either that proposition is true or its negation is. This is not to say that the law of the excluded middle is denied entirely; special cases of the law will be provable. It is just that the general law is not assumed as an axiom. The law of non-contradiction (which states that contradictory statements cannot both at the same time be true) is still valid.
For instance, in Heyting arithmetic, one can prove that for any proposition p that does not contain quantifiers, \( \forall x,y,z,\ldots \in \mathbb{N} : p \vee \neg p \) is a theorem (where x, y, z ... are the free variables in the proposition p). In this sense, propositions restricted to the finite are still regarded as being either true or false, as they are in classical mathematics, but this bivalence does not extend to propositions that refer to infinite collections.
In fact, L.E.J. Brouwer, founder of the intuitionist school, viewed the law of the excluded middle as abstracted from finite experience, and then applied to the infinite without justification. For instance, Goldbach's conjecture is the assertion that every even number (greater than 2) is the sum of two prime numbers. It is possible to test for any particular even number whether or not it is the sum of two primes (for instance by exhaustive search), so any one of them is either the sum of two primes or it is not. And so far, every one thus tested has in fact been the sum of two primes.
But there is no known proof that all of them are so, nor any known proof that not all of them are so. Thus to Brouwer, we are not justified in asserting "either Goldbach's conjecture is true, or it is not." And while the conjecture may one day be solved, the argument applies to similar unsolved problems; to Brouwer, the law of the excluded middle was tantamount to assuming that every mathematical problem has a solution.
With the omission of the law of the excluded middle as an axiom, the remaining logical system has an existence property that classical logic does not have: whenever \( \exists_{x\in X} P(x) \) is proven constructively, then in fact \( P(a) \) is proven constructively for (at least) one particular \( a\in X, \) often called a witness. Thus the proof of the existence of a mathematical object is tied to the possibility of its construction.
Example from real analysis
In classical real analysis, one way to define a real number is as an equivalence class of Cauchy sequences of rational numbers.
In constructive mathematics, one way to construct a real number is as a function Æ that takes a positive integer n and outputs a rational Æ(n), together with a function g that takes a positive integer n and outputs a positive integer g(n) such that
\( \forall n\ \forall i,j \ge g(n)\quad |f(i) - f(j)| \le {1 \over n} \)
so that as n increases, the values of Æ(n) get closer and closer together. We can use Æ and g together to compute as close a rational approximation as we like to the real number they represent.
Under this definition, a simple representation of the real number e is:
\( f(n) = \sum_{i=0}^n {1 \over i!}, \quad g(n) = n. \)
This definition corresponds to the classical definition using Cauchy sequences, except with a constructive twist: for a classical Cauchy sequence, it is required that, for any given distance, there exists (in a classical sense) a member in the sequence after which all members are closer together than that distance. In the constructive version, it is required that, for any given distance, it is possible to actually specify a point in the sequence where this happens (this required specification is often called the modulus of convergence). In fact, the standard constructive interpretation of the mathematical statement
\( \forall n : \exists m : \forall i,j \ge m: |f(i) - f(j)| \le {1 \over n} \)
is precisely the existence of the function computing the modulus of convergence. Thus the difference between the two definitions of real numbers can be thought of as the difference in the interpretation of the statement "for all... there exists..."
This then opens the question as to what sort of function from a countable set to a countable set, such as f and g above, can actually be constructed. Different versions of constructivism diverge on this point. Constructions can be defined as broadly as free choice sequences, which is the intuitionistic view, or as narrowly as algorithms (or more technically, the computable functions), or even left unspecified. If, for instance, the algorithmic view is taken, then the reals as constructed here are essentially what classically would be called the computable numbers.
Cardinality
To take the algorithmic interpretation above would seem at odds with classical notions of cardinality. By enumerating algorithms, we can show classically that the computable numbers are countable. And yet Cantor's diagonal argument shows that real numbers have higher cardinality. Furthermore, the diagonal argument seems perfectly constructive. To identify the real numbers with the computable numbers would then be a contradiction.
And in fact, Cantor's diagonal argument is constructive, in the sense that given a bijection between the real numbers and natural numbers, one constructs a real number that doesn't fit, and thereby proves a contradiction. We can indeed enumerate algorithms to construct a function T, about which we initially assume that it is a function from the natural numbers onto the reals. But, to each algorithm, there may or may not correspond a real number, as the algorithm may fail to satisfy the constraints, or even be non-terminating (T is a partial function), so this fails to produce the required bijection. In short, one who takes the view that real numbers are (individually) effectively computable interprets Cantor's result as showing that the real numbers (collectively) are not recursively enumerable.
Still, one might expect that since T is a partial function from the natural numbers onto the real numbers, that therefore the real numbers are no more than countable. And, since every natural number can be trivially represented as a real number, therefore the real numbers are no less than countable. They are, therefore exactly countable. However this reasoning is not constructive, as it still does not construct the required bijection. The classical theorem proving the existence of a bijection in such circumstances, namely the CantorâBernsteinâSchroeder theorem, is non-constructive. It has recently been shown that the CantorâBernsteinâSchroeder theorem implies the law of the excluded middle, hence there can be no constructive proof of the theorem.[3]
Axiom of choice
The status of the axiom of choice in constructive mathematics is complicated by the different approaches of different constructivist programs. One trivial meaning of "constructive", used informally by mathematicians, is "provable in ZF set theory without the axiom of choice." However, proponents of more limited forms of constructive mathematics would assert that ZF itself is not a constructive system.
In intuitionistic theories of type theory (especially higher-type arithmetic), many forms of the axiom of choice are permitted. For example, the axiom AC11 can be paraphrased to say that for any relation R on the set of real numbers, if you have proved that for each real number x there is a real number y such that R(x,y) holds, then there is actually a function F such that R(x,F(x)) holds for all real numbers. Similar choice principles are accepted for all finite types. The motivation for accepting these seemingly nonconstructive principles is the intuitionistic understanding of the proof that "for each real number x there is a real number y such that R(x,y) holds". According to the BHK interpretation, this proof itself is essentially the function F that is desired. The choice principles that intuitionists accept do not imply the law of the excluded middle.
However, in certain axiom systems for constructive set theory, the axiom of choice does imply the law of the excluded middle (in the presence of other axioms), as shown by the Diaconescu-Goodman-Myhill theorem. Some constructive set theories include weaker forms of the axiom of choice, such as the axiom of dependent choice in Myhill's set theory.
Measure theory
Classical measure theory is fundamentally non-constructive, since the classical definition of Lebesgue measure does not describe any way to compute the measure of a set or the integral of a function. In fact, if one thinks of a function just as a rule that "inputs a real number and outputs a real number" then there cannot be any algorithm to compute the integral of a function, since any algorithm would only be able to call finitely many values of the function at a time, and finitely many values are not enough to compute the integral to any nontrivial accuracy. The solution to this conundrum, carried out first in Bishop's 1967 book, is to consider only functions that are written as the pointwise limit of continuous functions (with known modulus of continuity), with information about the rate of convergence. An advantage of constructivizing measure theory is that if one can prove that a set is constructively of full measure, then there is an algorithm for finding a point in that set (again see Bishop's book). For example, this approach can be used to construct a real number that is normal to every base.[citation needed]
The place of constructivism in mathematics
Traditionally, some mathematicians have been suspicious, if not antagonistic, towards mathematical constructivism, largely because of limitations they believed it to pose for constructive analysis. These views were forcefully expressed by David Hilbert in 1928, when he wrote in Grundlagen der Mathematik, "Taking the principle of excluded middle from the mathematician would be the same, say, as proscribing the telescope to the astronomer or to the boxer the use of his fists".[4]
Errett Bishop, in his 1967 work Foundations of Constructive Analysis, worked to dispel these fears by developing a great deal of traditional analysis in a constructive framework.
Even though most mathematicians do not accept the constructivist's thesis that only mathematics done based on constructive methods is sound, constructive methods are increasingly of interest on non-ideological grounds. For example, constructive proofs in analysis may ensure witness extraction, in such a way that working within the constraints of the constructive methods may make finding witnesses to theories easier than using classical methods. Applications for constructive mathematics have also been found in typed lambda calculi, topos theory and categorical logic, which are notable subjects in foundational mathematics and computer science. In algebra, for such entities as topoi and Hopf algebras, the structure supports an internal language that is a constructive theory; working within the constraints of that language is often more intuitive and flexible than working externally by such means as reasoning about the set of possible concrete algebras and their homomorphisms.
Physicist Lee Smolin writes in Three Roads to Quantum Gravity that topos theory is "the right form of logic for cosmology" (page 30) and "In its first forms it was called 'intuitionistic logic'" (page 31). "In this kind of logic, the statements an observer can make about the universe are divided into at least three groups: those that we can judge to be true, those that we can judge to be false and those whose truth we cannot decide upon at the present time" (page 28).
Mathematicians who have made major contributions to constructivism
Leopold Kronecker (old constructivism, semi-intuitionism)
L. E. J. Brouwer (founder of intuitionism)
A. A. Markov (forefather of Russian school of constructivism)
Arend Heyting (formalized intuitionistic logic and theories)
Per Martin-Löf (founder of constructive type theories)
Errett Bishop (promoted a version of constructivism claimed to be consistent with classical mathematics)
Paul Lorenzen (developed constructive analysis)
Branches
Constructive logic
Constructive type theory
Constructive analysis
Constructive non-standard analysis
See also
Computability theory
Constructive proof
Finitism
Game semantics
Intuitionism
Intuitionistic type theory
Finitist set theory
Notes
Troelstra 1977a:974
Troelstra 1977b:1
Pradic, Pierre; Brown, Chad E. (2019-04-19). "Cantor-Bernstein implies Excluded Middle". arXiv:1904.09193 [math.LO].
Stanford Encyclopedia of Philosophy: Constructive Mathematics.
References
Solomon Feferman (1997), Relationships between Constructive, Predicative and Classical Systems of Analysis, http://math.stanford.edu/~feferman/papers/relationships.pdf.
A. S. Troelstra (1977a), "Aspects of constructive mathematics", Handbook of Mathematical Logic, pp. 973â1052.
A. S. Troelstra (1977b), Choice sequences, Oxford Logic Guides. ISBN 0-19-853163-X
A. S. Troelstra (1991), "A History of Constructivism in the 20th Century", University of Amsterdam, ITLI Prepublication Series ML-91-05, https://web.archive.org/web/20060209210015/http://staff.science.uva.nl/~anne/hhhist.pdf,
H. M. Edwards (2005), Essays in Constructive Mathematics, Springer-Verlag, 2005, ISBN 0-387-21978-1
Douglas Bridges, Fred Richman, "Varieties of Constructive Mathematics", 1987.
Michael J. Beeson, "Foundations of constructive mathematics: metamathematical studies", 1985.
Anne Sjerp Troelstra, Dirk van Dalen, "Constructivism in Mathematics: An Introduction, Volume 1", 1988
Anne Sjerp Troelstra, Dirk van Dalen, "Constructivism in Mathematics: An Introduction, Volume 2", 1988
External links
"Constructive Mathematics". Internet Encyclopedia of Philosophy.
Stanford Encyclopedia of Philosophy entry
vte
Philosophical logic
Critical thinking and
informal logic
Analysis Ambiguity Argument Belief Bias Credibility Evidence Explanation Explanatory power Fact Fallacy Inquiry Opinion Parsimony (Occam's razor) Premise Propaganda Prudence Reasoning Relevance Rhetoric Rigor Vagueness
Theories of deduction
Constructivism Dialetheism Fictionalism Finitism Formalism Intuitionism Logical atomism Logicism Nominalism Platonic realism Pragmatism Realism
Undergraduate Texts in Mathematics
Graduate Texts in Mathematics
Graduate Studies in Mathematics
Mathematics Encyclopedia
World
Index
Hellenica World - Scientific Library
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Fernando Zalamea - Academia.edu
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In his recent book “Peirce and the Threat of Nominalism”, Paul Forster (2011) argued how Peirce understood the nominalism. Forster also explained that the nominalist move of individualising concepts for collections could cost denial of properties of true continua. Peirce showed some vibrant issues both in mathematics and in metaphysics, as for example, the classic one of universals. That work, however, is still incomplete; needs to be clarified how these ideas bind with Peirce’s doctrine of ‘scholastic realism’. Peirce’s own theory of continuity might be the answer: I trace the origins of this doctrine in the theory of multitude. I show how the theory of continuity frees analysis from constraints of nominalist theories of reality. Peirce’s theory of multitude, instead, can be derived with mathematics: By drawing in the work of the ways of abstraction in diagrammatic reasoning made by Sun Jo Shin (2010) and in continuum theories by Cathy Legg (2010) I show the conceptual device of “diagrammatic reasoning” as a plausible pragmatic tool to represent continua and make sense of Peirce’s metaphysics. The analysis of continuity is a perfect example of how the method of diagrammatic reasoning helps to unblock the road of philosophical inquiry as well as clarify other problems as for example, the applicability of Mathematics. I conclude explaining that general concepts define ‘continua’. The properties of true continua are not reducible to individuals they comprise. Continua are the universals of metaphysics; they are intelligible and necessary to ground the science of inquiry.
In the course of his philosophic career, Charles Peirce made repeated attempts to construct mathematical definitions of the commonsense or experimental notion of'continuity'. In what I will label his Final Definition of Continuity, however, Peirce abandoned the attempt to achieve mathematical definition and assigned the analysis of continuity to an otherwise unnamed extra-mathematical science. In this paper, I identify the Final Definition, attempt to define its terms, and suggest that it belongs to Peirce's emergent semiotics of vagueness. I argue, further, that it marks the transformation of Peirce's synechism. Before the time of his Final Definition, Peirce adopted a theory of continuity as a foundational principle of metaphysics and assumed this principle might be formalized in a mathematics of continuity. After the Final Definition, Peirce abandoned his foundationalism in favor of what he called a critical common-sensism. This is the claim that philosophy (and with it, logic) derives its norms from the observation of actual cognitive practices and that continuity is a distinguishing mark of actual as opposed to merely possible or imagined practices.
Abstract: Contemporary discussions around the foundations of mathematics are traditionally stepping into two opposite standstowards the reality of mathematical objects or structures: some of them are plainly nominalistic, denying the existence of theobjects of mathematical inquiries; the opposite stand is the extreme Platonism that defends an account of a realm of mathemat-ical entities where they exist. Both solutions find serious problems in accounting for the applicability and continuity betweenformal theoretical inquiry and the applicability of mathematics in our best scientific theories. In this scenario, Mathematicalstructuralism offers a halfway through overcoming the metaphysical scruples of the nominalist and the extreme realism of thePlatonist. At the same time enables us a criterion to find a continuum between theory and applicability through an accountof diagrammatic reasoning into the mathematical inquiries. My aim is to show how Peirce’s plea for diagrammatic reasoningfulfils the problem of our access to these structures by an account of mathematical true continua. Keywords : Applicability of Mathematics, Mathematical Structuralism, Peirce’s diagrammatic reasonin
describes Euclid’s procedure in proving theorems. Euclid first presents his theorem in general terms and then translates it into singular terms. Peirce pays attention to the fact that the generality of the statement is not lost by that move. The next step is construction, which is followed by demonstration. Finally the ergo-sentence repeats the original general proposition. Peirce lays much emphasis on the distinction between corollarial and theorematic reasoning in geometry. He takes an argument to be corollarial if no auxiliary construction is needed. For Peirce, construction is “the principal theoric step ” of the demonstration. Peirce also stresses that it is the observation of diagrams that is essential to all reasoning and that even if no auxiliary constructions are made, there is always the step from a general to a singular statement in deductive reasoning; that means introducing a kind of diagram to reasoning. This paper seeks to argue for two theses. One is that the way of ...
La lógica experimenta actualmente un crecimiento vertiginoso. Acompañada de un impulso inter y transdisciplinario, se despliega como un saber que transforma la naturaleza de las nociones de epistemolog&iacure;a y semántica. Es hora de que los filósofos se confronten a los problemas y a las nuevas preguntas esbozadas por estos avances. En verdad es manifiesto que hay ya algunos emprendimientos importantes en ese sentido. Emprendimientos que contribuyen en la exploración de nuevas direcciones del rol de la lógica como organon. El objetivo principal de nuestra colección es precisamente promover la publicación y traducción de obras en lengua castellana sobre lógica, filosofía de la lógica, filosofía de la ciencia, epistemología y semántica de la lenguas naturales. Esperamos que un tal foro de publicación fomentará la consolidación de vínculos ya existentes entre las disciplinas mencionadas y la construcción de nuevos lazos inter-y transdisciplinarios. Logic is nowadays growing at a breathtaking pace with a trans-and interdisciplinary élan which is transforming the very nature of the notion of epistemology and semantics. The time is ripe for philosophers to take up the challenges resulting from this impulse and indeed there are clear signals that many philosophers have already responded by taking upon themselves the task of reinstating logic in a new version of its old role of organon.The aim of this series is to promote the publication in Spanish of worksin logic, philosophy of logic, philosophy of science, epistemology and semantic for natural languag. This publication forum should foster the consolidation of existing links between those disciplines and the building of new inter and transdisplinary ones. The Volume La lógica como herramienta de la razón has been published by the UK publisher College Publications. Their Spanish language series Cuadernos de Logica, Epistemologia y Lenguaje, distinguishes itself by promoting and supporting researche in the field of Logic, where logic takes anew the role of an Organon, prescribed in Antiquity by Aristotle. Nowadays; this role of logic is deployed in its inter-and transdisciplianry character. Two other salient characteristics of this collection that I would like to point out, is as, already mentioned, that it publishes original work in Spanish and the second is copyright policy of all CP titles that allows authors to retain full rights for their own work. Prof. Atocha Aliseda (Universidad Autonoma de Mexico) We congratulate, moreover (the Spanish-speaking scientific community) should congratulate ourselves for the launching of a collection that promotes the publication in Spanish of work in this field, " Prof Arancha San Ginés (Universidad de Granada) Extracted from the Radio interview with Atocha Aliseda (Universidad Autonoma de Mexico), on the occassion of the publication in 2014 of her book La lógica como herramienta de la razón. Directores de la Serie: Shahid Rahman (Université de Lille3-STL:UMR 8163-CNRS) Juan Redmond (Universidad de Valparaíso, STL-UMR8163 Lille3) Editor Asistente: Rodrigo López Orellana (Universidad de Valparaíso/Universidad de Salamanca)
Within the most recent philosophical discussions on the fascinating and controversial issue of the continuity/difference between human beings and natural world, there is a widespread tendency to maintain an idea of natural continuity which calls forth the notion of universe-life provided by Giordano Bruno. According to this notion, the distinction between Mind and World or their qualitative difference are rejected, and the necessity of natural conditions to the evolution of the specific human phenomenon is claimed . This philosophic view shares J. Dewey’s criticism of traditional philosophical «false antithesis» and therefore calls for a deep attention to the logical-semantic categories implied by the continuity argument. The anti-dichotomic commitment gives light on the naturalistic vein of the classical pragmatism and its relationship with Darwin’s theory of evolution. This essay analyzes Peirce’s theory of continuum and its strict connection with the fallibilistic criterion of knowledge, emphasizing peirceian suggestion to consider the differences between phenomena in terms of dynamic expressions of Nature and not as ontological breaks. This approach is compared with W. James’ metaphysical hypothesis of neutral monism, and also with H. Putnam non-reductionist naturalism that get back to classical pragmatism, in particular to James’s thought, thanks to its characteristic realistic claim, its emphasis on the epistemic value of social and practical dimension, and its anti-dichotomic commitment.
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https://www.cambridge.org/core/books/basic-proof-theory/928508F797214A017D245A1FB67CCCD9
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Basic Proof Theory
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Cambridge Core - Programming Languages and Applied Logic - Basic Proof Theory
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https://archive.org/stream/StudiesInLogicAndTheFoundationsOfMathematics121AnneS.TroelstraDirkVanDalenConstr/%2528Studies%2520in%2520Logic%2520and%2520the%2520Foundations%2520of%2520Mathematics%2520121%2529%2520Anne%2520S.%2520Troelstra%252C%2520Dirk%2520van%2520Dalen-Constructivism%2520in%2520mathematics_%2520An%2520introduction.%2520Volume%25201-N_djvu.txt
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( Studies In Logic And The Foundations Of Mathematics 121) Anne S. Troelstra, Dirk Van Dalen Constructivism In Mathematics An Introduction. Volume 1 N : Free Download, Borrow, and Streaming : Internet
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(Studies in Logic and the Foundations of Mathematics 121) Anne S. Troelstra, Dirk van Dalen-Constructivism in mathematics_ An introduction. Volume 1-N
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Introduction - The Institute for Logic, Language and Computation
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History of Maths 1900 To The Present
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History Of Maths 1900 To The Present - Free ebook download as PDF File (.pdf), Text File (.txt) or read book online for free. The Big Book of Mathematics, Principles, Theories, and Things. Part IV
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Justification Logic: Reasoning with Reasons 1108661106, 9781108661102
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Classical logic is concerned, loosely, with the behaviour of truths. Epistemic logic similarly is about the behaviour of...
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Citation preview
Justification Logic Classical logic is concerned, loosely, with the behavior of truths. Epistemic logic similarly is about the behavior of known or believed truths. Justification logic is a theory of reasoning that enables the tracking of evidence for statements and therefore provides a logical framework for the reliability of assertions. This book, the first in the area, is a systematic account of the subject, progressing from modal logic through to the establishment of an arithmetic interpretation of intuitionistic logic. The presentation is mathematically rigorous but in a style that will appeal to readers from a wide variety of areas to which the theory applies. These include mathematical logic, artificial intelligence, computer science, philosophical logic and epistemology, linguistics, and game theory.
s e r g e i a rt e m ov is Distinguished Professor at the City University of New York. He is a specialist in mathematical logic, logic in computer science, control theory, epistemology, and game theory. He is credited with solving long-standing problems in constructive logic that had been left open by G¨odel and Kolmogorov since the 1930s. He has pioneered studies in the logic of proofs and justifications that render a new, evidence-based theory of knowledge and belief. The most recent focus of his interests is epistemic foundations of game theory. m e lv i n f i t t i n g is Professor Emeritus at the City University of New York. He has written or edited a dozen books and has worked in intensional logic, semantics for logic programming, theory of truth, and tableau systems for nonclassical logics. In 2012 he received the Herbrand Award from the Conference on Automated Deduction. He was on the faculty of the City University of New York from 1969 to his retirement in 2013, at Lehman College, and at the Graduate Center, where he was in the Departments of Mathematics, Computer Science, and Philosophy.
CAMBRIDGE TRACTS IN MATHEMATICS
GENERAL EDITORS ´ W. FULTON, F. KIRWAN, B. BOLLOBAS, P. SARNAK, B. SIMON, B. TOTARO A complete list of books in the series can be found at www.cambridge.org/mathematics. Recent titles include the following:
181. 182. 183. 184. 185. 186. 187. 188. 189. 190. 191. 192. 193. 194. 195. 196. 197. 198. 199. 200. 201. 202. 203. 204. 205. 206. 207. 208. 209. 210. 211. 212. 213. 214. 215.
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Justification Logic Reasoning with Reasons S E R G E I A RT E M OV Graduate Center, City University of New York M E LV I N F I T T I N G Graduate Center, City University of New York
University Printing House, Cambridge CB2 8BS, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia 314–321, 3rd Floor, Plot 3, Splendor Forum, Jasola District Centre, New Delhi – 110025, India 79 Anson Road, #06–04/06, Singapore 079906 Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning, and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781108424912 DOI: 10.1017/9781108348034 © Sergei Artemov and Melvin Fitting 2019 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2019 Printed and bound in Great Britain by Clays Ltd, Elcograf S.p.A. A catalogue record for this publication is available from the British Library. Library of Congress Cataloging-in-Publication Data Names: Artemov, S. N., author. | Fitting, Melvin, 1942- author. Title: Justification logic : reasoning with reasons / Sergei Artemov (Graduate Center, City University of New York), Melvin Fitting (Graduate Center, City University of New York). Description: Cambridge ; New York, NY : Cambridge University Press, 2019. | Series: Cambridge tracts in mathematics ; 216 | Includes bibliographical references and index. Identifiers: LCCN 2018058431 | ISBN 9781108424912 (hardback : alk. paper) Subjects: LCSH: Logic, Symbolic and mathematical. | Inquiry (Theory of knowledge) | Science–Theory reduction. | Reasoning. Classification: LCC QA9 .A78 2019 | DDC 511.3–dc23 LC record available at https://lccn.loc.gov/2018058431 ISBN 978-1-108-42491-2 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.
To our wives, Lena and Roma.
Contents
Introduction 1 What Is This Book About? 2 What Is Not in This Book?
page x xii xvii
1
Why Justification Logic? 1.1 Epistemic Tradition 1.2 Mathematical Logic Tradition 1.3 Hyperintensionality 1.4 Awareness 1.5 Paraconsistency
1 1 4 8 9 10
2
The Basics of Justification Logic 2.1 Modal Logics 2.2 Beginning Justification Logics 2.3 J0 , the Simplest Justification Logic 2.4 Justification Logics in General 2.5 Fundamental Properties of Justification Logics 2.6 The First Justification Logics 2.7 A Handful of Less Common Justification Logics
11 11 12 14 15 20 23 27
3
The Ontology of Justifications 3.1 Generic Logical Semantics of Justifications 3.2 Models for J0 and J 3.3 Basic Models for Positive and Negative Introspection 3.4 Adding Factivity: Mkrtychev Models 3.5 Basic and Mkrtychev Models for the Logic of Proofs LP 3.6 The Inevitability of Possible Worlds: Modular Models 3.7 Connecting Justifications, Belief, and Knowledge 3.8 History and Commentary
31 31 36 38 39 42 42 45 46
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4
Fitting Models 4.1 Modal Possible World Semantics 4.2 Fitting Models 4.3 Soundness Examples 4.4 Canonical Models and Completeness 4.5 Completeness Examples 4.6 Formulating Justification Logics
48 48 49 52 60 65 72
5
Sequents and Tableaus 5.1 Background 5.2 Classical Sequents 5.3 Sequents for S4 5.4 Sequent Soundness, Completeness, and More 5.5 Classical Semantic Tableaus 5.6 Modal Tableaus for K 5.7 Other Modal Tableau Systems 5.8 Tableaus and Annotated Formulas 5.9 Changing the Tableau Representation
75 75 76 79 81 84 90 91 93 95
6
Realization – How It Began 6.1 The Logic LP 6.2 Realization for LP 6.3 Comments
100 100 103 108
7
Realization – Generalized 7.1 What We Do Here 7.2 Counterparts 7.3 Realizations 7.4 Quasi-Realizations 7.5 Substitution 7.6 Quasi-Realizations to Realizations 7.7 Proving Realization Constructively 7.8 Tableau to Quasi-Realization Algorithm 7.9 Tableau to Quasi-Realization Algorithm Correctness 7.10 An Illustrative Example 7.11 Realizations, Nonconstructively 7.12 Putting Things Together 7.13 A Brief Realization History
110 110 112 113 116 118 120 126 128 131 133 135 138 139
8
The Range of Realization 8.1 Some Examples We Already Discussed 8.2 Geach Logics 8.3 Technical Results
141 141 142 144
Contents 8.4 8.5 8.6 8.7 8.8
Geach Justification Logics Axiomatically Geach Justification Logics Semantically Soundness, Completeness, and Realization A Concrete S4.2/JT4.2 Example Why Cut-Free Is Needed
ix 147 149 150 152 155
9
Arithmetical Completeness and BHK Semantics 9.1 Arithmetical Semantics of the Logic of Proofs 9.2 A Constructive Canonical Model for the Logic of Proofs 9.3 Arithmetical Completeness of the Logic of Proofs 9.4 BHK Semantics 9.5 Self-Referentiality of Justifications
158 158 161 165 174 179
10
Quantifiers in Justification Logic 10.1 Free Variables in Proofs 10.2 Realization of FOS4 in FOLP 10.3 Possible World Semantics for FOLP 10.4 Arithmetical Semantics for FOLP
181 182 186 191 212
11
Going Past Modal Logic 11.1 Modeling Awareness 11.2 Precise Models 11.3 Justification Awareness Models 11.4 The Russell Scenario as a JAM 11.5 Kripke Models and Master Justification 11.6 Conclusion References Index
222 223 225 226 228 231 233 234 244
Introduction
Why is this thus? What is the reason of this thusness?1
Modal operators are commonly understood to qualify the truth status of a proposition: necessary truth, proved truth, known truth, believed truth, and so on. The ubiquitous possible world semantics for it characterizes things in universal terms: X is true in some state if X is true in all accessible states, where various conditions on accessibility are used to distinguish one modal logic from another. Then (X → Y) → (X → Y) is valid, no matter what conditions are imposed, by a simple and direct argument using universal quantification. Suppose both (X → Y) and X are true at an arbitrary state. Then both X and X → Y are true at all accessible states, whatever “accessible” may mean. By the usual understanding of →, Y is true at all accessible states too, and so Y is true at the arbitrary state we began with. Although arguments like these have a strictly formal nature and are studied as modal model theory, they also give us some insights into our informal, everyday use of modalities. Still, something is lacking. Suppose we think of as epistemic, and to emphasize this we use K instead of for the time being. For some particular X, if you assert the colloquial counterpart of KX, that is, if you say you know X, and I ask you why you know X, you would never tell me that it is because X is true in all states epistemically compatible with this one. You would, instead, give me some sort of explicit reason: “I have a mathematical proof of X,” or “I read X in the encyclopedia,” or “I observed that X is the case.” If I asked you why K(X → Y) → (KX → KY) is valid you would probably say something like “I could use my reason for X and combine it with my reason for X → Y, and infer Y.” This, in effect, would be your reason for Y, given that you had reasons for X and for X → Y. 1
Charles Farrar Browne (1834–1867) was an American humorist who wrote under the pen name Artemus Ward. He was a favorite writer of Abraham Lincoln, who would read his articles to his Cabinet. This quote is from a piece called Moses the Sassy, Ward (1861).
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xi
Notice that this neatly avoids the logical omniscience problem: that we know all the consequences of what we know. It replaces logical omniscience with the more acceptable claim that there are reasons for the consequences of what we know, based on the reasons for what we know, but reasons for consequences are more complicated things. In our example, the reason for Y has some structure to it. It combines reasons for X, reasons for X → Y, and inference as a kind of operation on reasons. We will see more examples of this sort; in fact, we have just seen a fundamental paradigm. In place of a modal operator, , justification logics have a family of justification terms, informally intended to represent reasons, or justifications. Instead of X we will see t:X, where t is a justification term and the formula is read “X is so for reason t,” or more briefly, “t justifies X.” At a minimum, justification terms are built up from justification variables, standing for arbitrary justifications. They are built up using a set of operations that, again at a minimum, contains a binary operation ·. For example, x · (y · x) is a justification term, where x and y are justification variables. The informal understanding of · is that t · u justifies Y provided t justifies an implication with Y as its consequent, and u justifies the antecedent. In justification logics the counterpart of (X → Y) → (X → Y) is t:(X → Y) → (u:X → [t · u]:Y) where, as we will often do, we have added square brackets to enhance readability. Note that this exactly embodies the informal explanation we gave in the previous paragraph for the validity of K(X → Y) → (KX → KY). That is, Y has a justification built from justifications for X and for X → Y using an inference that amounts to a modus ponens application—we can think of the · operation as an abstract representation of this inference. Other behaviors of modal operators, X → X for instance, will require operators in addition to ·, and appropriate postulated behavior, in order to produce justification logics that correspond to modal logics in which X → X is valid. Examples, general methods for doing this, and what it means to “correspond” all will be discussed during the course of this book. One more important point. Suppose X and Y are equivalent formulas, that is, we have X ↔ Y. Then in any normal modal logic we will also have X ↔ Y. Let us interpret the modal operator epistemically again, and write KX ↔ KY. In fact, KX ↔ KY, when read in the usual epistemic way, can sometimes be quite an absurd assertion. Consider some astronomically complicated tautology X of classical propositional logic. Because it is a tautology, it is equivalent
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to P ∨ ¬P, which we may take for Y. Y is hardly astronomically complicated. However, because X ↔ Y, we will have KX ↔ KY. Clearly, we know Y essentially by inspection and hence KY holds, while KX on the other hand will involve an astronomical amount of work just to read it, let alone to verify it. Informally we see that, while both X and Y are tautologies, and so both are knowable in principle, any justification we might give for knowing one, combined with quite a lot of formula manipulation, can give us some justification for knowing the other. The two justifications may not, indeed will not, be the same. One is simple, the other very complex. Modal logic is about propositions. Propositions are, in a sense, the content of formulas. Propositions are not syntactical objects. “It’s good to be the king” and “Being the king is good” express the same proposition, but not in the same way. Justifications apply to formulas. Equivalent formulas determine the same proposition, but can be quite different as formulas. Syntax must play a fundamental role for us, and you will see that it does, even in our semantics. Consider one more very simple example. A → (A∧ A) is an obvious tautology. We might expect KA → K(A ∧ A). But we should not expect t:A → t:(A ∧ A). If t does, in fact, justify A, a justification of A ∧ A may involve t, but also should involve facts about the redundancy of repetition; t by itself cannot be expected to suffice. Modal logics can express, more or less accurately, how various modal operators behave. This behavior is captured axiomatically by proofs, or semantically using possible world reasoning. These sorts of justifications for modal operator behavior are not within a modal logic, but are outside constructs. Justification logics, on the other hand, can represent the whys and wherefores of modal behavior quite directly, and from within the formal language itself. We will see that most standard modal logics have justification counterparts that can be used to give a fine-grained, internal analysis of modal behavior. Perhaps, this will help make clear why we used the quotation we did at the beginning of this Introduction.
1 What Is This Book About? How did justification logics originate? It is an interesting story, with revealing changes of direction along the way. Going back to the days when G¨odel was a young logician, there was a dream of finding a provability interpretation for intuitionistic logic. As part of his work on that project, in G¨odel (1933), G¨odel showed that one could abstract some of the key features of provability and make a propositional modal logic using them. Then, remarkably but
Introduction
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naturally, one could embed propositional intuitionistic logic into the resulting system. C. I. Lewis had pioneered the modern formal study of modal logics (Lewis, 1918; Lewis and Langford, 1932), and G¨odel observed that his system was equivalent to the Lewis system S4. All modern axiomatizations of modal logics follow the lines pioneered in G¨odel’s note, while Lewis’s original formulation is rarely seen today. G¨odel showed that propositional intuitionistic logic embedded into S4 using a mapping that inserted in front of every subformula. In effect, intuitionistic logic could be understood using classical logic plus an abstract notion of provability: a propositional formula X is an intuitionistic theorem if and only if the result of applying G¨odel’s mapping is a theorem of S4. (This story is somewhat simplified. There are several versions of the G¨odel translation—we have used the simplest one to describe. And G¨odel did not use the symbol but rather an operator Bew, short for beweisbar, or provability in the German language. None of this affects our main points.) Unfortunately, the story breaks off at this point because G¨odel also noted that S4 does not behave like formal provability (e.g., in arithmetic), by using the methods he had pioneered in his work on incompleteness. Specifically, S4 validates X → X, so in particular we have ⊥ → ⊥ (where ⊥ is falsehood). This is equivalent to ¬⊥, which is thus provable in S4. If we had an embedding of S4 into formal arithmetic under which corresponded to G¨odel’s arithmetic formula representing provability, we would be able to prove in arithmetic that falsehood was not provable. That is, we would be able to show provability of consistency, violating G¨odel’s second incompleteness theorem. So, work on an arithmetic semantics for propositional intuitionistic logic paused for a while. Although it did not solve the problem of a provability semantics for intuitionistic logic, an important modal/arithmetic connection was eventually worked out. One can define a modal logic by requiring that its validities are those that correspond to arithmetic validities when reading as G¨odel’s provability formula. It was shown in Solovay (1976) that this was a modal logic already known in the literature, though as noted earlier, it is not S4. Today, the logic is called GL, standing for G¨odel–L¨ob logic. GL is like S4 except that the T axiom X → X, an essential part of S4, is replaced by a modal formula abstractly representing L¨ob’s theorem: (X → X) → X. S4 and GL are quite different logics. By now the project for finding an arithmetic interpretation of intuitionistic logic had reached an impasse. Intuitionistic logic embedded into S4, but S4 did not embed into formal arithmetic. GL embedded into formal arithmetic, but the G¨odel translation does not embed intuitionistic logic into GL.
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In his work on incompleteness for Peano arithmetic, G¨odel gave a formula Bew(x, y)
that represents the relation: x is the G¨odel number of a proof of a formula with G¨odel number y. Then, a formal version of provability is ∃xBew(x, y) which expresses that there is a proof of (the formula whose G¨odel number is) y. If this formula is what corresponds to in an embedding from a modal language to Peano arithmetic, we get the logic GL. But in a lecture in 1938 G¨odel pointed out that we might work with explicit proof representatives instead of with provability (G¨odel, 1938). That is, instead of using an embedding translating every occurrence of by ∃xBew(x, y), we might associate with each occurrence of some formal term t that somehow represents a particular proof, allowing different occurrences of to be associated with different terms t. Then in the modal embedding, we could make the occurrence of associated with t correspond to Bew(ptq, y), where ptq is a G¨odel number for t. For each occurrence of we would need to find some appropriate term t, and then each occurrence of would be translated into arithmetic differently. The existential quantifier in ∃xBew(x, y) has been replaced with a meta-existential quantifier, outside the formal language. We provide an explicit proof term, rather than just asserting that one exists. G¨odel believed that this approach should lead to a provability embedding of S4 into Peano arithmetic. G¨odel’s proposal was not published until 1995 when Volume 3 of his collected works appeared. By this time the idea of using a modal-like language with explicit representatives for proofs had been rediscovered independently by Sergei Artemov, see Artemov (1995, 2001). The logic that Artemov created was called LP, which stood for logic of proofs. It was the first example of a justification logic. What are now called justification terms were called proof terms in LP. Crucially, Artemov showed LP filled the gap between modal S4 and Peano arithmetic. The connection with S4 is primarily embodied in a Realization Theorem, which has since been shown to hold for a wide range of justification logic, modal logic pairs. It will be extensively examined in this book. The connection between LP and formal arithmetic is Artemov’s Arithmetic Completeness Theorem, which also will be examined in this book. Its range is primarily limited to the original justification logic, LP, and a few close relatives. This should not be surprising, though. G¨odel’s motivation for his formulation of S4 was that should embody properties of a formal arithmetic proof predicate. This connection with arithmetic provability is not present for almost all modal
Introduction
xv
logics and is consequently also missing for corresponding justification logics, when they exist. Nonetheless, the venerable goal of finding a provability interpretation for propositional intuitionistic logic had been attained. The G¨odel translation embeds propositional intuitionistic logic into the modal logic S4. The Realization Theorem establishes an embedding of S4 into the justification logic LP. And the Arithmetic Completeness Theorem shows that LP embeds into formal arithmetic. It was recognized from the very beginning that the connection between S4 and LP could be weakened to sublogics of S4 and LP. Thus, there were justification logic counterparts for the standard modal logics, K, K4, T, and a few others. These justification logics had arithmetic connections because they were sublogics of LP. The use of proof term was replaced with justification term. Although the connection with arithmetic was weaker than it had been with LP, justification terms still had the role of supplying explicit justifications for epistemically necessary statements. One can consult Artemov (2008) and Artemov and Fitting (2012) for survey treatments, though the present book includes the material found there. Almost all of the early work on justification logics was proof-theoretically based. Realization theorems were shown constructively, making use of a sequent calculus. The existence of an algorithm to compute what are called realizers is important, but this proof-theoretic approach limits the field to those logics known to have sequent calculus proof systems. For a time it was hoped that various extensions of sequent and tableau calculi would be useful and, to some extent, this has been the case. The most optimistic version of this hope was expressed in Artemov (2001) quite directly, “Gabbay’s Labelled Deductive Systems, Gabbay (1994), may serve as a natural framework for LP.” Unfortunately this seems to have been too optimistic. While the formats had similarities, the goals were different, and the machinery did not interact well. A semantics for LP and its near relatives, not based on arithmetic provability, was introduced in Mkrtychev (1997) and is discussed in Chapter 3. (A constructive version of the canonical model for LP with a completeness theorem can be found already in Artemov (1995).) Mkrtychev’s semantics did not use possible worlds and had a strong syntactic flavor. Possible worlds were added to the mix in Fitting (2005), producing something that potentially applied much more broadly than the earlier semantics. This is the subject of Chapter 4. Using this possible world semantics, a nonconstructive, semantic-based, proof of realization was given. It was now possible to avoid the use of a sequent calculus, though the algorithmic nature of realization was lost. More recently, a semantics with a very simple structure was created, Artemov’s basic semantics (Artemov, 2012). It is presented in Chapter 3. Its machinery is almost minimal
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for the purpose. In this book, we will use possible world semantics to establish very general realization results, but basic models will often be used when we simply want to show some formula fails to be a theorem. Though its significance was not properly realized at the time, in 2005 the subject broadened when a justification logic counterpart of S5 was introduced in Pacuit (2005) and Rubtsova (2006a, b), with a connecting realization theorem. There was no arithmetical interpretation for this justification logic. Also there is no sequent calculus for S5 of the standard kind, so the proof given for realization was nonconstructive, using a version of the semantics from Fitting (2005). The semantics needed some modification to what is called its evidence function, and this turned out to have a greater impact than was first realized. Eventually constructive proofs connecting S5 and its justification counterpart were found. These made use of cut-free proof systems that were not exactly standard sequent calculi. Still, the door to a larger room was beginning to open. Out of the early studies of the logics of proofs and its variants a general logical framework for reasoning about epistemic justification at large naturally emerged, and the name, Justification Logic, was introduced (cf. Artemov, 2008). Justification Logic is based on justification assertions, t:F, that are read t is a justification for F, with a broader understanding of the word justification going beyond just mathematical proofs. The notion of justification, which has been an essential component of epistemic studies since Plato, had been conspicuously absent in the mathematical models of knowledge within the epistemic logic framework. The Justification Logic framework fills in this void. In Fitting (2016a) the subject expanded abruptly. Using nonconstructive semantic methods it was shown that the family of modal logics having justification counterparts is infinite. The justification phenomenon is not the relatively narrow one it first seemed to be. While that work was nonconstructive, there are now cut-free proof systems of various kinds for a broader range of modal logics than was once the case, and these have been used successfully to create realization algorithms, in Kuznets and Goetschi (2012), for instance. It may be that the very general proof methodologies of Fitting (2015) and especially Negri (2005) and Negri and von Plato (2001) will extend the constructive range still further, perhaps even to the infinite family that nonconstructive methods are known to work for. This is active current work. Work on quantified justification logics exists, but the subject is considerably behind its propositional counterpart. An important feature of justification logics is that they can, in a very precise sense, internalize their own proofs. Doing this for axioms is generally simple. Rules of inference are more of a problem. Earlier we discussed a justification formula as a simple, representative exam-
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ple: t:(X → Y) → (u:X → [t · u]:Y). This, in effect, internalizes the axiomatic modus ponens rule. The central problem in developing quantified justification logics was how to internalize the rule of universal generalization. It turned out that the key was the clear separation between two roles played by individual variables. On the one hand, they are formal symbols, and one can simply infer ∀xϕ(x) from a proof of ϕ(x). On the other hand, they can be thought of as open for substitution, that is, throughout a proof one can replace free occurrences of x with a term t to produce a new proof (subject to appropriate freeness of substitution conditions, of course). These two roles for variables are actually incompatible. It was the introduction of specific machinery to keep track of which role a variable occurrence had that made possible the internalization of proofs, and thus a quantified justification logic. An axiomatic version of first-order LP was introduced in Artemov and Yavorskaya (Sidon) (2011) and a possible world semantics for it in Fitting (2011a, 2014b). A connection with formal arithmetic was established. There is a constructive proof of a Realization Theorem, connecting first-order LP with firstorder S4. Unlike propositionally, no nonconstructive proof is currently known The possible world semantics includes the familiar monotonicity condition on world domains. It is likely that all this can be extended to a much broader range of quantified modal logics than just first-order S4, provided monotonicity is appropriate. A move to constant domain models, to quantified S5 in particular, has been made, and a semantics, but not yet a Realization Theorem, can be found in Fitting and Salvatore (2018). Much involving quantification is still uncharted territory. This book will cover the whole range of topics just described. It will not do so in the historical order that was followed in this Introduction, but will make use of the clearer understanding that has emerged from study of the subject thus far. We will finish with the current state of affairs, standing on the edge of unknown lands. We hope to prepare some of you for the journey, should you choose to explore further on your own.
2 What Is Not in This Book? There are several historical works and pivotal developments in justification logic that will not be covered in the book due to natural limitations, and in this section we will mention them briefly. We are confident that other books and surveys will do justice to these works in more detail. Apart from G¨odel’s lecture, G¨odel (1938), which remained unpublished
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until 1995 and thus could not influence development in this area, the first results and publications on the logic of proofs are dated 1992: a technical report, Artemov and Straßen (1992), based on work done in January of 1992 in Bern, and a conference presentation of this work at CSL’92 published in Springer Lecture Notes in Computer Science as Artemov and Straßen (1993a). In this work, the basic logic of proofs was presented: it had proof variables, and the format t is a proof of F, but without operations on proofs. However, it already had the first installment of the fixed-point arithmetical completeness construction together with an observation that, unlike provability logic, the logic of proofs cannot be limited to one standard proof predicate “from the textbook” or to any single-conclusion proof predicate. This line was further developed in Artemov and Straßen (1993b), where the logic of single-conclusion proof predicates (without operations on proofs) was studied. This work introduced the unification axiom, which captures singleconclusioness by propositional tools. After the full-scale logic of proofs with operations had been discovered (Artemov, 1995), the logic of single-conclusion proofs with operations was axiomatized in V. Krupski (1997, 2001). A similar technique was used recently to characterize so-called sharp single-conclusion justification models in Krupski (2018). Another research direction pursued after the papers on the basic logic of proofs was to combine provability and explicit proofs. Such a combination, with new provability principles, was given in Artemov (1994). Despite its title, this paper did not introduce what is known now as The Logic of Proofs, but rather a fusion of the provability logic GL and the basic logic of proofs without operations, but with new arithmetical principles combining proofs and provability and an arithmetical completeness theorem. After the logic of proofs paper (Artemov, 1995), the full-scale logic of provability and proofs (with operations), LPP, was axiomatized and proved arithmetically complete in Sidon (1997) and Yavorskaya (Sidon) (2001). A leaner logic combining provability and explicit proofs, GLA, was introduced and proved arithmetically complete in Nogina (2006, 2014b). Unlike LPP, the logic GLA did not use additional operations on proofs other than those inherited from LP. Later, GLA was used to find a complete classification of reflection principles in arithmetic that involve provability and explicit proofs (Nogina, 2014a). The first publication of the full-scale logic of proofs with operations, LP, which became the first justification logic in the modern sense, was Artemov (1995). This paper contains all the results needed to complete G¨odel’s program of characterizing intuitionistic propositional logic IPC and its BHK semantics via proofs in classical arithmetic: internalization, the realization theorem for S4 in LP, arithmetical semantics for LP, and the arithmetical completeness the-
Introduction
xix
orem. It took six years for the corresponding journal paper to appear: Artemov (2001). In Goris (2008), the completeness of LP for the semantics of proofs in Peano arithmetic was extended to the semantics of proofs in Buss’s bounded arithmetic S12 . In view of applications in epistemology, this result shows that explicit knowledge in the propositional framework can be made computationally feasible. Kuznets and Studer (2016) extend the arithmetical interpretation of LP from the original finite constant specifications to a wide class of constant specifications, including some infinite ones. In particular, this “weak” arithmetical interpretation captures the full logic of proofs LP with the total constant specification. Decidability of LP (with the total constant specification) was also established in Mkrtychev (1997), and this opened the door to decidability and complexity studies in justification logics using model-theoretic and other means. Among the milestones are complexity estimates in Kuznets (2000), Brezhnev and Kuznets (2006), Krupski (2006a), Milnikel (2007), Buss and Kuznets (2012), and Achilleos (2014a). The arithmetical provability semantics for the Logic of Proofs, LP, naturally generalizes to a first-order version with conventional quantifiers and to a version with quantifiers over proofs. In both cases, axiomatizability questions were answered negatively in Artemov and Yavorskaya (2001) and Yavorsky (2001). A natural and manageable first-order version of the logic of proofs, FOLP, has been studied in Artemov and Yavorskaya (Sidon) (2011), Fitting (2014a), and Fitting and Salvatore (2018) and will be covered in Chapter 10. Originally, the logic of proofs was formulated as a Hilbert-style axiomatic system, but this has gradually broadened. Early attempts were tableau based (which could equivalently be presented using sequent calculus machinery). Tableaus generally are analytic, meaning that everything entering into a proof is a subformula of what is being proved. This was problematic for attempts at LP tableaus because of the presence of the · operation, which represented an application of modus ponens, a rule that is decidedly not analytic. Successful tableau systems, though not analytic, for LP and closely related logics can be found in Fitting (2003, 2005) and Renne (2004, 2006). The analyticity problem was overcome in Ghari (2014, 2016a). Broader proof systems have been investigated: hypersequents in Kurokawa (2009, 2012), prefixed tableaus in Kurokawa (2013), and labeled deductive systems in Ghari (2017). Indeed some of this has led to new realization results (Artemov, 1995, 2001, 2002, 2006; Artemov and Bonelli, 2007; Ghari, 2012; Kurokawa, 2012). Finding a computational reading of justification logics has been a natural research goal. There were several attempts to use the ideas of LP for building a lambda-calculus with internalization, cf. Alt and Artemov (2001), Artemov
xx
Introduction
(2002), Artemov and Bonelli (2007), Pouliasis and Primiero (2014), and others. Corresponding combinatory logic systems with internalization were studied in Artemov (2004), Krupski (2006b), and Shamkanov (2011). These and other studies can serve as a ground for further applications in typed programming languages. A version of the logic of proofs with a built-in verification predicate was considered in Protopopescu (2016a, b). The aforementioned intuition that justification logic naturally avoids the logical omniscience problem has been formalized and studied in Artemov and Kuznets (2006, 2009, 2014). The key idea there was to view logical omniscience as a proof complexity problem: The logical omniscience defect occurs if an epistemic system assumes knowledge of propositions, which have no feasible proofs. Through this prism, standard modal logics are logically omniscient (modulo some common complexity assumptions), and justification logics are not logically omniscient. The ability of justification logic to track proof complexity via time bounds led to another formal definition of logical omniscience in Wang (2011a) with the same conclusion: Justification logic keeps logical omniscience under control. Shortly after the first paper on the logic of proofs, it became clear that the logical tools developed are capable of evidence tracking in a general setting and as such can be useful in epistemic logic. Perhaps, the first formal work in this direction was Artemov et al. (1999), in which modal logic S5 was equivalently modified and supplied with an LP-style explicit counterpart. Applications to epistemology have benefited greatly from Fitting semantics, which connected justification logics to mainstream epistemology via possible worlds models. In addition to applications discussed in this book, we would like to mention some other influential work. Game semantics of justification logic was studied in Renne (2008) and dynamic epistemic logic with justifications in Renne (2008) and Baltag et al. (2014). In Sedl´ar (2013), Fitting semantics for justification models was elaborated to a special case of the models of general awareness. Multiagent justification logic and common knowledge has been studied in Artemov (2006), Antonakos (2007), Yavorskaya (Sidon) (2008), Bucheli et al. (2010, 2011), Bucheli (2012), Antonakos (2013), and Achilleos (2014b, 2015a, b). In Dean and Kurokawa (2010), justification logic was used for the analysis of Knower and Knowability paradoxes. A fast-growing and promising area is probabilistic justification logic, cf. Milnikel (2014), Artemov (2016b), Kokkinis et al. (2016), Ghari (2016b), and Lurie (2018).
Introduction
xxi
We are deeply indebted to all contributors to the exciting justification logic project, without whom there would not be this book. Very special thanks to our devoted readers for their sharp eyes and their useful comments: Vladimir Krupski, Vincent Alexis Peluce, and Tatiana Yavorskaya (Sidon).
I think there is no sense in forming an opinion when there is no evidence to form it on. If you build a person without any bones in him he may look fair enough to the eye, but he will be limber and cannot stand up; and I consider that evidence is the bones of an opinion.2
2
Mark Twain (1835–1910). The quote is from his last novel, Personal Recollections of Joan of Arc, Twain (1896).
1 Why Justification Logic?
The formal details of justification logic will be presented starting with the next chapter, but first we give some background and motivation for why the subject was developed in the first place. We will see that it addresses, or at least partially addresses, many of the fundamental problems that have been found in epistemic logic over the years. We will also see in more detail how it relates to our understanding of intuitionistic logic. And finally, we will see how it can be used to mitigate some well-known issues that have arisen in philosophical investigations.
1.1 Epistemic Tradition The properties of knowledge and belief have been a subject for formal logic at least since von Wright and Hintikka (Hintikka, 1962; von Wright, 1951). Knowledge and belief are both treated as modalities in a way that is now very familiar—Epistemic Logic. But of the celebrated three criteria for knowledge (usually attributed to Plato), justified, true, belief, Gettier (1963); Hendricks (2005), epistemic modal logic really works with only two of them. Possible worlds and indistinguishability model belief—one believes what is so under all circumstances thought possible. Factivity brings a trueness component into play—if something is not so in the actual world it cannot be known, only believed. But there is no representation for the justification condition. Nonetheless, the modal approach has been remarkably successful in permitting the development of rich mathematical theory and applications (Fagin et al., 1995; van Ditmarsch et al., 2007). Still, it is not the whole picture. The modal approach to the logic of knowledge is, in a sense, built around the universal quantifier: X is known in a situation if X is true in all situations indistinguishable from that one. Justifications, on the other hand, bring an ex1
2
Why Justification Logic?
istential quantifier into the picture: X is known in a situation if there exists a justification for X in that situation. This universal/existential dichotomy is a familiar one to logicians—in formal logics there exists a proof for a formula X if and only if X is true in all models for the logic. One thinks of models as inherently nonconstructive, and proofs as constructive things. One will not go far wrong in thinking of justifications in general as much like mathematical proofs. Indeed, the first justification logic was explicitly designed to capture mathematical proofs in arithmetic, something that will be discussed later. In justification logic, in addition to the category of formulas, there is a second category of justifications. Justifications are formal terms, built up from constants and variables using various operation symbols. Constants represent justifications for commonly accepted truths—axioms. Variables denote unspecified justifications. Different justification logics differ on which operations are allowed (and also in other ways too). If t is a justification term and X is a formula, t:X is a formula, and is intended to be read t is a justification for X. One operation, common to all justification logics, is application, written like multiplication. The idea is, if s is a justification for A → B and t is a justification for A, then [s · t] is a justification for B.1 That is, the validity of the following is generally assumed s:(A → B) → (t:A → [s · t]:B).
(1.1)
This is the explicit version of the usual distributivity of knowledge operators, and modal operators generally, across implication K(A → B) → (KA → KB).
(1.2)
How adequately does the traditional modal form (1.2) embody epistemic closure? We argue that it does so poorly! In the classical logic context, (1.2) only claims that it is impossible to have both K(A → B) and KA true, but KB false. However, because (1.2), unlike (1.1), does not specify dependencies between K(A → B), KA, and KB, the purely modal formulation leaves room for a counterexample. The distinction between (1.1) and (1.2) can be exploited in a discussion of the paradigmatic Red Barn Example of Goldman and Kripke; here is a simplified version of the story taken from Dretske (2005). 1
For better readability brackets will be used in terms, “[,]”, and parentheses in formulas, “(,).” Both will be avoided when it is safe.
1.1 Epistemic Tradition
3
Suppose I am driving through a neighborhood in which, unbeknownst to me, papiermˆach´e barns are scattered, and I see that the object in front of me is a barn. Because I have barn-before-me percepts, I believe that the object in front of me is a barn. Our intuitions suggest that I fail to know barn. But now suppose that the neighborhood has no fake red barns, and I also notice that the object in front of me is red, so I know a red barn is there. This juxtaposition, being a red barn, which I know, entails there being a barn, which I do not, “is an embarrassment.”
In the first formalization of the Red Barn Example, logical derivation will be performed in a basic modal logic in which is interpreted as the “belief” modality. Then some of the occurrences of will be externally interpreted as a knowledge modality K according to the problem’s description. Let B be the sentence “the object in front of me is a barn,” and let R be the sentence “the object in front of me is red.” (1) B, “I believe that the object in front of me is a barn.” At the metalevel, by the problem description, this is not knowledge, and we cannot claim KB. (2) (B ∧ R), “I believe that the object in front of me is a red barn.” At the metalevel, this is actually knowledge, e.g., K(B ∧ R) holds. (3) (B∧R → B), a knowledge assertion of a logical axiom. This is obviously knowledge, i.e., K(B ∧ R → B). Within this formalization, it appears that epistemic closure in its modal form (1.2) is violated: K(B∧R), and K(B∧R → B) hold, whereas, by (1), we cannot claim KB. The modal language here does not seem to help resolving this issue. Next consider the Red Barn Example in justification logic where t:F is interpreted as “I believe F for reason t.” Let u be a specific individual justification for belief that B, and v for belief that B ∧ R. In addition, let a be a justification for the logical truth B ∧ R → B. Then the list of assumptions is (i) u:B, “u is a reason to believe that the object in front of me is a barn”; (ii) v:(B ∧ R), “v is a reason to believe that the object in front of me is a red barn”; (iii) a:(B ∧ R → B). On the metalevel, the problem description states that (ii) and (iii) are cases of knowledge, and not merely belief, whereas (i) is belief, which is not knowledge. Here is how the formal reasoning goes: (iv) a:(B ∧ R → B) → (v:(B ∧ R) → [a·v]:B), by principle (1.1); (v) v:(B ∧ R) → [a·v]:B, from 3 and 4, by propositional logic; (vi) [a·v]:B, from 2 and 5, by propositional logic.
4
Why Justification Logic?
Notice that conclusion (vi) is [a · v]:B, and not u:B; epistemic closure holds. By reasoning in justification logic it was concluded that [a·v]:B is a case of knowledge, i.e., “I know B for reason a · v.” The fact that u:B is not a case of knowledge does not spoil the closure principle because the latter claims knowledge specifically for [a·v]:B. Hence after observing a red fac¸ade, I indeed know B, but this knowledge has nothing to do with (i), which remains a case of belief rather than of knowledge. The justification logic formalization represents the situation fairly. Tracking justifications represents the structure of the Red Barn Example in a way that is not captured by traditional epistemic modal tools. The justification logic formalization models what seems to be happening in such a case; closure of knowledge under logical entailment is maintained even though “barn” is not perceptually known. One could devise a formalization of the Red Barn Example in a bimodal language with distinct modalities for knowledge and belief. However, it seems that such a resolution must involve reproducing justification tracking arguments in a way that obscures, rather than reveals, the truth. Such a bimodal formalization would distinguish u:B from [a · v]:B not because they have different reasons (which reflects the true epistemic structure of the problem), but rather because the former is labeled “belief” and the latter “knowledge.” But what if one needs to keep track of a larger number of different unrelated reasons? By introducing a multiplicity of distinct modalities and then imposing various assumptions governing the interrelationships between these modalities, one would essentially end up with a reformulation of the language of justification logic itself (with distinct terms replaced by distinct modalities). This suggests that there may not be a satisfactory “halfway point” between a modal language and the language of justification logic, at least inasmuch as one tries to capture the essential structure of examples involving the deductive nature of knowledge.
1.2 Mathematical Logic Tradition According to Brouwer, truth in constructive (intuitionistic) mathematics means the existence of a proof, cf. Troelstra and van Dalen (1988). In 1931–34, Heyting and Kolmogorov gave an informal description of the intended proof-based semantics for intuitionistic logic (Kolmogoroff, 1932; Heyting, 1934), which is now referred to as the Brouwer–Heyting–Kolmogorov (BHK) semantics. According to the BHK conditions, a formula is “true” if it hasa proof. Further-
1.2 Mathematical Logic Tradition
5
more, a proof of a compound statement is connected to proofs of its components in the following way: • a proof of A ∧ B consists of a proof of proposition A and a proof of proposition B, • a proof of A∨B is given by presenting either a proof of A or a proof of B, • a proof of A → B is a construction transforming proofs of A into proofs of B, • falsehood ⊥ is a proposition, which has no proof; ¬A is shorthand for A → ⊥. This provides a remarkably useful informal way of understanding what is and what is not intuitionistically acceptable. For instance, consider the classical tautology (P ∨ Q) ↔ (P ∨ (Q ∧ ¬P)), where we understand ↔ as mutual implication. And we understand ¬P as P → ⊥, so that a proof of ¬P would amount to a construction converting any proof of P into a proof of ⊥. Because ⊥ has no proof, this amounts to a proof that P has no proof—a refutation of P. According to BHK semantics the implication from right to left in (P ∨ Q) ↔ (P ∨ (Q ∧ ¬P)) should be intuitionistically valid, by the following argument. Given a proof of P ∨ (Q ∧ ¬P) it must be that we are given a proof of one of the disjuncts. If it is P, we have a proof of one of P ∨ Q. If it is Q ∧ ¬P, we have proofs of both conjuncts, hence a proof of Q, and hence again a proof of one of P ∨ Q. Thus we may convert a proof of P ∨ (Q ∧ ¬P) into a proof of P ∨ Q. On the other hand, (P ∨ Q) → (P ∨ (Q ∧ ¬P)) is not intuitionistically valid according to the BHK ideas. Suppose we are given a proof of P ∨ Q. If we have a proof of the disjunct P, we have a proof of P ∨ Q. But if we have a proof of Q, there is no reason to suppose we have a refutation of P, and so we cannot conclude we have a proof of Q ∧ ¬P, and things stop here. Kolmogorov explicitly suggested that the proof-like objects in his interpretation (“problem solutions”) came from classical mathematics (Kolmogoroff, 1932). Indeed, from a foundational point of view this reflects Kolmogorov’s and G¨odel’s goal to define intuitionism within classical mathematics. From this standpoint, intuitionistic mathematics is not a substitute for classical mathematics, but helps to determine what is constructive in the latter. The fundamental value of the BHK semantics for the justification logic project is that informally but unambiguously BHK suggests treating justifications, here mathematical proofs, as objects with operations. In G¨odel (1933), G¨odel took the first step toward developing a rigorous proof-based semantics for intuitionism. G¨odel considered the classical modal logic S4 to be a calculus describing properties of provability:
6
Why Justification Logic?
(1) (2) (3) (4)
Axioms and rules of classical propositional logic, (F → G) → (F → G), F → F, F → F, `F . (5) Rule of necessitation: ` F
Based on Brouwer’s understanding of logical truth as provability, G¨odel defined a translation tr(F) of the propositional formula F in the intuitionistic language into the language of classical modal logic: tr(F) is obtained by prefixing every subformula of F with the provability modality . Informally speaking, when the usual procedure of determining classical truth of a formula is applied to tr(F), it will test the provability (not the truth) of each of F’s subformulas, in agreement with Brouwer’s ideas. From G¨odel’s results and the McKinseyTarski work on topological semantics for modal logic (McKinsey and Tarski, 1948), it follows that the translation tr(F) provides a proper embedding of the Intuitionistic Propositional Calculus, IPC, into S4, i.e., an embedding of intuitionistic logic into classical logic extended by the provability operator. IPC ` F
⇔
S4 ` tr(F).
(1.3)
Conceptually, this defines IPC in S4. Still, G¨odel’s original goal of defining intuitionistic logic in terms of classical provability was not reached because the connection of S4 to the usual mathematical notion of provability was not established. Moreover, G¨odel noted that the straightforward idea of interpreting modality F as F is provable in a given formal system T contradicted his second incompleteness theorem. Indeed, (F → F) can be derived in S4 by the rule of necessitation from the axiom F → F. On the other hand, interpreting modality as the predicate of formal provability in theory T and F as contradiction converts this formula into a false statement that the consistency of T is internally provable in T . The situation after G¨odel (1933) can be described by the following figure where “X ,→ Y” should be read as “X is interpreted in Y”: IPC ,→ S4 ,→ ? ,→ CLASSICAL PROOFS.
In a public lecture in Vienna in 1938, G¨odel observed that using the format of explicit proofs t is a proof of F
(1.4)
can help in interpreting his provability calculus S4 (G¨odel, 1938). Unfortunately, G¨odel (1938) remained unpublished until 1995, by which time the
1.2 Mathematical Logic Tradition
7
G¨odelian logic of explicit proofs had already been rediscovered, axiomatized as the Logic of Proofs LP, and supplied with completeness theorems connecting it to both S4 and classical proofs (Artemov, 1995, 2001). The Logic of Proofs LP became the first in the justification logic family. Proof terms in LP are nothing but BHK terms understood as classical proofs. With LP, propositional intuitionistic logic received the desired rigorous BHK semantics: IPC ,→ S4 ,→ LP ,→ CLASSICAL PROOFS .
Several well-known mathematical notions that appeared prior to justification logic have sometimes been perceived as related to the BHK idea: Kleene realizability (Troelstra, 1998), Curry–Howard isomorphism (Girard et al., 1989; Troelstra and Schwichtenberg, 1996), Kreisel–Goodman theory of constructions (Goodman, 1970; Kreisel, 1962, 1965), just to name a few. These interpretations have been very instrumental for understanding intuitionistic logic, though none of them qualifies as the BHK semantics. Kleene realizability revealed a fundamental computational content of formal intuitionistic derivations; however it is still quite different from the intended BHK semantics. Kleene realizers are computational programs rather than proofs. The predicate “r realizes F” is not decidable, which leads to some serious deviations from intuitionistic logic. Kleene realizability is not adequate for the intuitionistic propositional calculus IPC. There are realizable propositional formulas not derivable in IPC (Rose, 1953).2 The Curry–Howard isomorphism transliterates natural derivations in IPC to typed λ-terms, thus providing a generic functional reading for logical derivations. However, the foundational value of this interpretation is limited because, as proof objects, Curry–Howard λ-terms denote nothing but derivations in IPC itself and thus yield a circular provability semantics for the latter. An attempt to formalize the BHK semantics directly was made by Kreisel in his theory of constructions (Kreisel, 1962, 1965). The original variant of the theory was inconsistent; difficulties already occurred at the propositional level. In Goodman (1970) this was fixed by introducing a stratification of constructions into levels, which ruined the BHK character of this semantics. In particular, a proof of A → B was no longer a construction that could be applied to any proof of A. 2
Kleene himself denied any connection of his realizability with the BHK interpretation.
8
Why Justification Logic?
1.3 Hyperintensionality Justification logic offers a formal framework for hyperintensionality. The hyperintensional paradox was formulated in Cresswell (1975). It is well known that it seems possible to have a situation in which there are two propositions p and q which are logically equivalent and yet are such that a person may believe the one but not the other. If we regard a proposition as a set of possible worlds then two logically equivalent propositions will be identical, and so if “x believes that” is a genuine sentential functor, the situation described in the opening sentence could not arise. I call this the paradox of hyperintensional contexts. Hyperintensional contexts are simply contexts which do not respect logical equivalence.
Starting with Cresswell himself, several ways of dealing with this have been proposed. Generally, these involve adding more layers to familiar possible world approaches so that some way of distinguishing between logically equivalent sentences is available. Cresswell suggested that the syntactic form of sentences be taken into account. Justification logic, in effect, does this through its mechanism for handling justifications for sentences. Thus justification logic addresses some of the central issues of hyperintensionality but, as a bonus, we automatically have an appropriate proof theory, model theory, complexity estimates, and a broad variety of applications. A good example of a hyperintensional context is the informal language used by mathematicians conversing with each other. Typically when a mathematician says he or she knows something, the understanding is that a proof is at hand, but this kind of knowledge is essentially hyperintensional. For instance Fermat’s Last Theorem, FLT, is logically equivalent to 0 = 0 because both are provable and hence denote the same proposition, as this is understood in modal logic. However, the context of proofs distinguishes them immediately because a proof of 0 = 0 is not necessarily a proof of FLT, and vice versa. To formalize mathematical speech, the justification logic LP is a natural choice because t:X was designed to have characteristics of “t is a proof of X.” The fact that propositions X and Y are equivalent in LP, that LP ` X ↔ Y, does not warrant the equivalence of the corresponding justification assertions, and typically t:X and t:Y are not equivalent, t:X 6↔ t:Y. Indeed, as we will see, this is the case for every justification logic. Going further LP, and justification logic in general, is not only sufficiently refined to distinguish justification assertions for logically equivalent sentences, but it also provides flexible machinery to connect justifications of equivalent sentences and hence to maintain constructive closure properties desirable for a logic system. For example, let X and Y be provably equivalent, i.e., there is a proof u of X ↔ Y, and so u:(X ↔ Y) is provable in LP. Suppose also
1.4 Awareness
9
that v is a proof of X, and so v:X. It has already been mentioned that this does not mean v is a proof of Y—this is a hyperintensional context. However within the framework of justification logic, building on the proofs of X and of X ↔ Y, we can construct a proof term f (u, v), which represents the proof of Y and so f (u, v):Y is provable. In this respect, justification logic goes beyond Cresswell’s expectations: Logically equivalent sentences display different but constructively controlled epistemic behavior.
1.4 Awareness The logical omniscience problem is that in epistemic logics all tautologies are known and knowledge is closed under consequence, both of which are unreasonable. In Fagin and Halpern (1988) a simple mechanism for avoiding the problems was introduced. One adds to the usual Kripke model structure an awareness function A indicating for each world which formulas the agent is aware of at this world. Then a formula is taken to be known at a possible world u if (1) the formula is true at all worlds accessible from u (the Kripkean condition for knowledge) and (2) the agent is aware of the formula at u. The awareness function A can serve as a practical tool for blocking knowledge of an arbitrary set of formulas. However, as logical structures, awareness models exhibit abnormal behavior due to the lack of natural closure properties. For example, the agent can know A ∧ A but be unaware of A and hence not know it. Fitting models for justification logic, presented in Chapter 4, use a forcing definition reminiscent of the one from awareness models: For any given justification t, the justification assertion t:F holds at world u iff (1) F holds at all worlds v accessible from u and (2) t is an admissible evidence for F at u, u ∈ E(s, F), read as “u is a possible world at which s is relevant evidence for F.” The principal difference is that postulated operations on justifications relate to natural closure conditions on admissible evidence functions E in justification logic models. Indeed, this idea has been explored in Sedl´ar (2013), which works with the language of LP and thinks of it as a multiagent modal logic, and taking justification terms as agents (more properly, actions of agents). This shows that justification logic models absorb the usual epistemic themes of awareness, group agency, and dynamics in a natural way.
10
Why Justification Logic?
1.5 Paraconsistency Justification logic offers a well-principled approach to paraconsistency, which looks for noncollapsing logical ways of dealing with contradictory sets of assumptions, e.g., {A, ¬A}. The following obvious observation shows how to convert any set of assumptions Γ = {A1 , A2 , A3 , . . .} into a logically consistent set of sentences while maintaining all the intrinsic structure of Γ. Informally, instead of (perhaps inconsistently) assuming that Γ holds, we assume only that each sentence A from Γ has a justification, i.e., ~x : Γ = {x1:A1 , x2:A2 , x3:A3 , . . .}. It is easy to see that for each Γ, the set ~x:Γ is consistent in what will be our basic justification logic J. For example, for Γ = {A, ¬A}, ~x : Γ = {x1:A, x2:¬A}, states that x1 is a justification for A and x2 is a justification for ¬A. Within justification logic J in which no factivity (or even consistency) of justifications is assumed, the set of assumptions {x1:A, x2:¬A} is consistent, unlike the original set of assumptions {A, ¬A}. There is nothing paraconsistent, magical, or artificial in reasoning from ~x:Γ in justification logic J. In practical terms, this means we gain the ability to effectively reason about inconsistent data sets, keeping track of justifications and their dependencies, with the natural possibility to draw meaningful conclusions even when some assumed justifications from ~x:Γ become compromised and should be discharged.
2 The Basics of Justification Logic
In this chapter we discuss matters of syntax and axiomatics. All material is propositional, and will be so until Chapter 10. Justification logics are closely related to modal logics, so we start briefly with them in order to fix the basic notation. And just as normal modal logics all extend a single simplest example, K, all justification logics extend a single simplest example, J0 . We will begin our discussion with modal logics, then we will discuss the justification logic J0 in detail, and finally we will extend things to the most common and bestknown justification logics. A much broader family of justification logics will be discussed in Chapter 8.
2.1 Modal Logics All propositional formulas throughout this book are built up from a countable family of propositional variables. We use P, Q, . . . as propositional variables, with subscripts if necessary, and we follow the usual convention that these are all distinct. As our main propositional connective we have implication, →. We have negation, ¬, which we will take as primitive, or defined using the propositional constant ⊥ representing falsehood, as convenient and appropriate at the time. We also use conjunction, ∧, disjunction, ∨, and equivalence, ↔, and these too may be primitive or defined depending on circumstances. We omit outer parentheses in formulas when it will do no harm. We usually have a single modal necessity operator. It will generally be represented by though in epistemic contexts it may be represented by K. A dual operator representing possibility, ♦, is a defined operator and actually plays little role here. There is much work on epistemic logics with multiple agents, and there is some study of justification counterparts for them. When 11
12
The Basics of Justification Logic
discussing these and their connections with modal logics, we will subscript the modal operators just described. To date, no justification logic corresponding to a nonnormal modal logic has been introduced, so only normal modal logics will appear here. A normal modal logic is a set of modal formulas that contains all tautologies and all formulas of the form (X → Y) → (X → Y) and is closed under uniform substitution, modus ponens, and necessitation (if X is present, so is X). The smallest normal modal logic is K; it is a subset of all normal modal logics. The logic K has a standard axiom system. Axioms are all tautologies (or enough of them) and all formulas of the form (X → Y) → (X → Y). Rules are Modus Ponens X, X → Y ⇒ Y and Necessitation X ⇒ X. We are not actually interested in the vast collection of normal modal logics, but only in those for which a Hilbert system exists, having an axiomatization using a finite set of axiom schemes. In practice, this means adding axiom schemes to the axiomatization for K just given. We assume everybody knows axiom systems like T, K4, S4, and so on. We will refer to such logics as axiomatically formulated. Of course semantics plays a big role in modal logics, but we postpone a discussion for the time being.
2.2 Beginning Justification Logics Justification logics, syntactically, are like modal logics except that justification terms take the place of . Justification terms are intended to represent reasons or justifications for formulas. They have structure that encodes reasoning that has gone into them. We begin our formal presentation here. Definition 2.1 (Justification Term) up as follows.
The set Tm of justification terms is built
(1) There is a set of justification variables, x, y, . . . , x1 , y1 , . . . . Every justification variable is a justification term. (2) There is a set of justification constants, a, b, . . . , a1 , b1 , . . . . Every justification constant is a justification term. (3) There are binary operation symbols, + and ·. If u and v are justification terms, so are (u + v) and (u · v). (4) There may be additional function symbols, f , g, . . . , f1 , g1 , . . . , of various arities. Which ones are present depends on the logic in question. If f is an nplace justification function symbol of the logic, and t1 , . . . , tn are justification terms, f (t1 , . . . , tn ) is a justification term.
2.2 Beginning Justification Logics
13
Neither + nor · is assumed to be commutative or associative, and there is no distributive law. We do, however, allow ourselves the notational convenience of omitting parentheses with multiple occurrences of ·, assuming associativity to the left. Thus, for instance, a · b · c · d is short for (((a · b) · c) · d). We make the same assumption concerning +, though it actually plays a much lesser role. Also we will generally assume that · binds more strongly than +, writing a·b+c instead of (a · b) + c for instance. Definition 2.2 (Justification Formula) The set of justification formulas, Fm, is built in the usual recursive way, as follows. (1) There is a set Var of propositional variables, P, Q, . . . , P1 , Q1 , . . . (these are also known as propositional letters). Every propositional variable is a justification formula. (2) ⊥ (falsehood) is a justification formula. (3) If X and Y are justification formulas, so is (X → Y). (4) If t is a justification term and X is a justification formula, then t:X is a justification formula. We will sometimes use other propositional connectives, ∧, ∨, ↔, which we can think of as defined connectives, or primitive as convenient. Outer parentheses may be omitted in formulas if no confusion will result. If justification term t has a complex structure we generally will write [t]:X, using square brackets as a visual aid. Square brackets have no theoretical significance. In a modal formula, is supposed to express that something is necessary, or known, or obligatory, or some such thing, but it does not say why. A justification term encodes this missing information; it provides the why absent from modal formulas. This is what their structure is for. Justification variables stand for arbitrary justification terms, and substitution for them is examined beginning with Definition 2.17. Justification constants stand for reasons that are not further analyzed—typically they are reasons for axioms. Their role is discussed in more detail once constant specifications are introduced, in Definition 10.32. The · operation corresponds to modus ponens. If X → Y is so for reason s and X is so for reason t, then Y is so for reason s · t. (Reasons are not unique—Y may be true for other reasons too.) The + operation is a kind of weakening. If X is so for either reason s or reason t, then s + t is also a reason for X. Other operations on justification terms, if present, correspond to features peculiar to particular modal logics and will be discussed as they come up.
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The Basics of Justification Logic
2.3 J0 , the Simplest Justification Logic As we will see, there are some justification logics having a version of a necessitation rule; there are others that do not. Some justification logics are closed under substitution of formulas for propositional variables, others are not. Allowing such a range of behavior is essential to enable us to capture and study the interactions of important features of modal logics that are sometimes hidden from us. But one consequence is, there is no good justification analog of the family of normal modal logics. Still, all justification logics have a common core, which we call J0 , and it is a kind of analog of the weakest normal modal logic, K, even though there is nothing structural we can point to as determining a “normal” justification logic apart from giving an axiomatization. In this section we present J0 axiomatically; subsequently we discuss what must be added to get the general family of justification logics. Definition 2.3 (Justification logic J0 ) The language of J0 has no justification function symbols beyond the basic two binary ones + and ·. The axiom schemes are as follows.
Classical: All tautologies (or enough of them) Application: All formulas of the form s:(X → Y) → (t:X → [s · t]:Y) Sum: All formulas of the forms s:X → [s + t]:X and t:X → [s + t]:X The only J0 rule of inference is Modus Ponens, X, X → Y ⇒ Y. J0 is a very weak justification logic. It is, for instance, incapable of proving that any formula has a justification, see Section 3.2. Reasoning in J0 is analogous to reasoning in the modal logic K without a necessitation rule! What we can do in J0 is derive interesting facts about justifications provided we make explicit what other formulas we would need to have justifications for. We give an example to illustrate this. To help bring out the points we want to make, if (X1 ∧ . . . ∧ Xn ) → Y is provable in J0 we may write X1 , . . . , Xn `J0 Y. Order of formulas and placement of parentheses in the conjunction of the Xi don’t matter because we have classical logic to work with. In modal K, a common first example of a theorem is (X ∧Y) → (X ∧Y). Here is the closest we can come to this in J0 . Our presentation is very much abbreviated.
2.4 Justification Logics in General
15
Example 2.4 Assume u, v, and w are justification variables. 1. 2. 3. 4. 5.
u:((X ∧ Y) → X) → (w:(X ∧ Y) → [u · w]:X) v:((X ∧ Y) → Y) → (w:(X ∧ Y) → [v · w]:Y) (u:((X ∧ Y) → X) ∧ v:((X ∧ Y) → y)) → (w:(X ∧ Y) → [u · w]:X) (u:((X ∧ Y) → X) ∧ v:((X ∧ Y) → y)) → (w:(X ∧ Y) → [v · w]:Y) (u:((X ∧ Y) → X) ∧ v:((X ∧ Y) → y)) → (w:(X ∧ Y) → ([u · w]:X ∧ [v · w]:Y))
Application Axiom Application Axiom Classical Logic on 1, 2 Classical Logic on 1, 2 Classical Logic on 3, 4
So we have shown that u:((X ∧ Y) → X), v:((X ∧ Y) → Y) `J0 (w:(X ∧ Y) → ([u · w]:X ∧ [v · w]:Y)) which we can read as an analog of (X ∧Y) → (X ∧Y) as follows. In J0 , for any w there are justification terms t1 and t2 such that w:(X ∧ Y) → (t1:X ∧ t2:Y), provided we have justifications for the tautologies (X ∧ Y) → X and (X ∧ Y) → Y. Note that t1 = u · w and t2 = v · w are different terms. But, making use of the Sum Axiom scheme, these can be brought together as t1 + t2 = u · w + v · w. It is important to understand that justifications, when they exist, are not unique.
2.4 Justification Logics in General The core justification logic J0 is extended to form other justification logics using two quite different types of machinery. First, one can add new operations on justification terms, besides the basic + and ·, along with axiom schemes governing their use, similar to Sum and Application. This is directly analogous to the way axiom schemes are added to K to create other modal logics. Second, one can specify which truths of logic we assume we have justifications for. This is related to the roles u:((X ∧ Y) → X) and v:((X ∧ Y) → Y) play in Example 2.4. We devote most of this section to the second kind of extension. It is, in fact, the intended role for justification constants that, up to now, have not been used for anything special. For the time being let us assume we have a justification logic resulting from the addition of function symbols and axiom schemes to J0 . The details don’t matter for now, but it should be understood that our axioms may go beyond those for J0 . Axioms of justification logics, like axioms generally, are simply assumed and are not analyzed further. The role of justification constant symbols is to
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The Basics of Justification Logic
serve as reasons or justifications for axioms. If A is an axiom, we can simply announce that constant symbol c plays the role of a justification for it. It may be that some axioms are assumed to have such justifications, but not necessarily all. Suppose we look at Example 2.4 again, and suppose we have decided that (X ∧ Y) → X is an axiom for which we have a specific justification, let us say the constant symbol c plays this role. Similarly let us say the constant symbol d represents a justification for (X ∧ Y) → Y. Examining the derivation given in Example 2.4, it is easy to see that if we replace the variable u throughout by c, and the variable v throughout by d we still have a derivation, but one of c:((X ∧ Y) → X), d:((X ∧ Y) → Y) `J0 (w:(X ∧ Y) → ([c · w]:X ∧ [d · w]:Y)). If we add c:((X ∧ Y) → X) and d:((X ∧ Y) → Y) to our axioms for J0 , we can simply prove the formula (w:(X ∧ Y) → ([c · w]:X ∧ [d · w]:Y)). Roughly speaking, a constant specification tells us what axioms we have justifications for and which constants justify these axioms. As we just saw, we can use a constant specification as a source of additional axioms. But there is an important complication. If A is an axiom and constant symbol c justifies it, c:A conceptually also acts like an axiom, and it too may have its own justification. Then a constant symbol, say d, could come in here too, as a justification for c:A, and thus we might want to assume d:c:A. This repeats further, of course. For many purposes exact details don’t matter much, so how constants are used, and for what purposes, is turned into a kind of parameter of our logics, called a constant specification. Definition 2.5 (Constant Specification) A constant specification CS for a given justification logic is a set of formulas meeting the following conditions. (1) Members of CS are of the form cn:cn−1: . . . c1:A where n > 0, A is an axiom of JL, and each ci is a constant symbol. (2) If cn:cn−1: . . . c1:A is in CS where n > 1, then cn−1: . . . c1:A is in CS too. Thus CS contains all intermediate specifications for whatever it contains. One reason why constant specifications are treated as parameters can be discovered through a close look at Definition 2.3. It does not really provide an axiomatization for J0 , but rather a scheme for axiomatizations. The axioms called Classical in that definition are not fully specified, and in common practice many classical logic axiomatizations are in use. Any set sufficient to derive all tautologies will do. Then many different axiomatizations for J0 would meet the required conditions, and similarly for any justification logic extending J0
2.4 Justification Logics in General
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as well. Because constants are supposed to be associated with axioms, a variety of constant specifications come up naturally. And because details like this often matter very little, treating constant specifications as a parameter is quite reasonable. Definition 2.6 (Logic of Justifications with a Constant Specification) Let JL be a justification logic, resulting from the addition of function symbols to the language of J0 and corresponding axiom schemes to those of J0 . Let CS be a constant specification for JL. Then JL(CS) is the logic JL with members of CS added as axioms (not axiom schemes), still with modus ponens as the only rule of inference. Constant specifications allow for great flexibility. A constant specification could associate many constants with a single axiom, or none at all. Allowing for many could be of use in tracking where particular pieces of reasoning come from. Allowing none might be appropriate in dealing with axioms that have some simple form, say X → X, but where the size of X is astronomical. Or again we might want to use the same constant for all instances of a particular axiom schema, or we might want to keep the instances clearly distinguishable. If details don’t matter at all for some particular purpose, we might want to associate a single constant symbol with every axiom, no matter what the form. Such a constant would simply be a record that a formula is an axiom, without going into particulars. Some conditions on constant specifications have shown themselves to be of special interest and have been given names. Here is a list of the most common. There are others. Definition 2.7 (Constant Specification Conditions) Let CS be a constant specification for a justification logic JL. The following requirements may be placed on CS. Empty: CS = ∅. This amounts to working with JL itself. Epistemically one can think of it as appropriate for the reasoning of a completely skeptical agent. Finite: CS is a finite set of formulas. This is fully representative because any specific derivation in a Justification Logic will be finite and so will involve only a finite set of constants. Schematic: If A and B are both instances of the same axiom scheme, c:A ∈ CS if and only if c:B ∈ CS, for every constant symbol c. Total: For each axiom A of JL and any constants c1 , c2 , . . . , cn we have cn:cn−1: . . . c1:A ∈ CS.
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The Basics of Justification Logic
Axiomatically Appropriate: For every axiom A and for every n > 0 there are constant symbols ci so that cn:cn−1: . . . c1:A ∈ CS. The working of justification axiom systems is specified as follows. Definition 2.8 (Consequence) Suppose JL is a justification logic, CS is a constant specification for JL, S is an arbitrary set of formulas (not schemes), and X is a single formula. By S `JL(CS) X we mean there is a finite sequence of formulas, ending with X, in which each formula is either a instance of an axiom scheme of JL, a member of CS, a member of S , or follows from earlier formulas by modus ponens. If {Y1 , . . . , Yk } `JL(CS) X we will simplify notation and write Y1 , . . . , Yk `JL(CS) X. If ∅ `JL(CS) X we just write `JL(CS) X, or sometimes even JL(CS) ` X. When presenting examples of axiomatic derivations using a constant speciCS fication CS, we will write c + X as a suggestive way of saying that c:X ∈ CS, and we will say “c justifies X”. We conclude this section with some examples of theorems of justification logics. For these we work with JL(CS) where JL is any justification logic and CS is any constant specification for it that is axiomatically appropriate. We assume JL has been axiomatized taking all tautologies as axioms, though taking “enough” would give similar results once we have Theorem 2.14. Example 2.9 P → P is a theorem of any normal modal logic. It has more than one proof. We could simply note that it is an instance of a tautology, X → X. Or we could begin with P → P, a simpler instance of this tautology, apply necessitation getting (P → P), and then use the K axiom (P → P) → (P → P) and modus ponens to conclude P → P. While these are different modal derivations, the result is the same. But when we mimic the steps in JL(CS), they lead to different results. Let t be an arbitrary justification term. Then t:P → t:P is a theorem of JL(CS) because it is an instance of a tautology. But also P → P is an instance of a tautology and so, because JL(CS) is assumed axiomatically appropriate, the constant specification assigns some constant to it; say c:(P → P) ∈ CS. Because c:(P → P) → (t:P → [c · t]:P) is an axiom, t:P → [c · t]:P follows by modus ponens. In justification logic, instead of a single formula P → P with two proofs we have two different theorems that contain traces of their proofs. Both t:P → t:P and t:P → [c · t]:P say that if there is a reason for P, then there is a reason for P, but they give us different reasons. One of the first things everybody shows axiomatically when studying modal
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logic is that (P ∧ Q) ↔ (P ∧ Q) is provable in K, and thus is provable in every axiom system for a normal modal logic. But the argument from left to right is quite different from the argument from right to left. Because justification theorems contain traces of their proofs, we should not expect a single justification analog of this modal equivalence, but rather separate results for the left–right implication and for the right–left implication. Example 2.10 Here is a justification derivation corresponding to the usual modal argument for (P ∧ Q) → (P ∧ Q). 1. 2. 3. 4. 5. 6. 7. 8. 9.
(P ∧ Q) → P c:((P ∧ Q) → P) c:((P ∧ Q) → P) → (t:(P ∧ Q) → [c · t]:P) t:(P ∧ Q) → [c · t]:P (P ∧ Q) → Q d:((P ∧ Q) → Q) d:((P ∧ Q) → Q) → (t:(P ∧ Q) → [d · t]:Q) t:(P ∧ Q) → [d · t]:Q t:(P ∧ Q) → ([c · t]:P ∧ [d · t]:Q)
tautology cons spec Application Axiom mod pon on 2, 3 tautology cons spec Application Axiom mod pon on 6, 7 class log on 4, 8
CS Then t:(P ∧ Q) → ([c · t]:P ∧ [d · t]:Q) is a theorem of JL(CS) where c + CS ((P ∧ Q) → P) and d + ((P ∧ Q) → Q).
Example 2.11 A justification counterpart of the modal theorem (P∧Q) → (P ∧ Q) follows. 1. 2. 3. 4. 5. 6.
P → (Q → (P ∧ Q)) c:(P → (Q → (P ∧ Q))) c:(P → (Q → (P ∧ Q))) → (t:P → [c · t]:(Q → (P ∧ Q))) t:P → [c · t]:(Q → (P ∧ Q)) [c · t]:(Q → (P ∧ Q)) → (u:Q → [c · t · u]:(P ∧ Q)) (t:P ∧ u:Q) → [c · t · u]:(P ∧ Q)
tautology cons spec Application Axiom mod pon on 2, 3 Application Axiom class log on 4, 5
CS So (t:P ∧ u:Q) → [c · t · u]:(P ∧ Q) is a theorem of JL(CS) where c + (P → (Q → (P ∧ Q))).
Our final example illustrates the use of +, which has not come up so far. It is for handling situations where there is more than one explanation needed for something, as in a proof by cases. At first glance this seems rather minor, but + turns out to play a vital role when we come to realization results. Example 2.12 (X ∨ Y) → (X ∨ Y) is a theorem of K with an elementary
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The Basics of Justification Logic
proof that we omit. Let us construct a counterpart in JL(CS), still assuming that CS is axiomatically appropriate and all tautologies are axioms. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.
X → (X ∨ Y) c:(X → (X ∨ Y)) c:(X → (X ∨ Y)) → (t:X → [c · t]:(X ∨ Y)) t:X → [c · t]:(X ∨ Y) Y → (X ∨ Y) d:(Y → (X ∨ Y)) d:(Y → (X ∨ Y)) → (u:Y → [d · u]:(X ∨ Y)) u:Y → [d · u]:(X ∨ Y) [c · t]:(X ∨ Y) → [c · t + d · u]:(X ∨ Y) [d · u]:(X ∨ Y) → [c · t + d · u]:(X ∨ Y) t:X → [c · t + d · u]:(X ∨ Y) u:Y → [c · t + d · u]:(X ∨ Y) (t:X ∨ u:Y) → [c · t + d · u]:(X ∨ Y)
tautology cons spec Application Axiom mod pon on 2, 3 tautology cons spec Application Axiom mod pon on 6, 7 Sum Axiom Sum Axiom clas log on 4, 9 clas log on 8, 10 class log on 11, 12
The consequents of 4 and 8 both provide reasons for X ∨ Y, but the reasons are different. We have used + to combine them, getting a justification analog of (X ∨ Y) → (X ∨ Y).
2.5 Fundamental Properties of Justification Logics All justification logics have certain common and useful properties. Some features are identical with those of classical logic; others have twists that are special to justification logics. This section is devoted to ones we will use over and over. Throughout this section let JL be a justification logic and CS be a constant specification for it. Because the only rule of inference is modus ponens the classical proof of the deduction theorem applies. We thus have S , X `JL(CS) Y if and only if S `JL(CS) X → Y. Because formal proofs are finite we have compactness that, combined with the deduction theorem, tells us: S `JL(CS) X if and only if `JL(CS) Y1 → (Y2 → . . . → (Yn → X) . . .) for some Y1 , Y2 , . . . , Yn ∈ S . These are exactly like their classical counterparts. Furthermore, the following serves as a replacement for the modal Necessitation Rule. Definition 2.13 (Internalization) JL has the internalization property relative to constant specification CS provided, if `JL(CS) X then for some justification term t, `JL(CS) t:X. In addition we say that JL has the strong internalization
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property if t contains no justification variables and no justification operation or function symbols except ·. That is, t is built up from justification constants using only ·. Theorem 2.14 If CS is an axiomatically appropriate constant specification for JL then JL has the strong internalization property relative to CS. Proof By induction on proof length. Suppose `JL(CS) X and the result is known for formulas with shorter proofs. If X is an axiom of JL or a member of CS, there is a justification constant c such that c:X is in CS, and so c:X is provable. If X follows from earlier proof lines by modus ponens from Y → X and Y then, by the induction hypothesis, `JL(CS) s:(Y → X) and `JL(CS) t:Y for some s, t containing no justification variables, and with · as the only function symbol. Using the J0 Application Axiom s:(Y → X) → (t:Y → [s · t]:X) and modus ponens, we get `JL(CS) [s · t]:X. If X is provable using an axiomatically appropriate constant specification so is t:X, and the term t constructed in the preceding proof actually internalizes the steps of the axiomatic proof of X, hence the name internalization. Of course different proofs of X will produce different justification terms. Here is an extremely simple example, but one that is already sufficient to illustrate this point. Example 2.15 Assume JL is a justification logic, CS is an axiomatically appropriate constant specification for it, and all tautologies are axioms of JL. CS P → P is a tautology so c + (P → P) for some c. Then c:(P → P) is a theorem, and we have the justification term c internalizing a proof of P → P. Here is a more roundabout proof of P → P, giving us a more complicated internalizing term. Following the method in the proof of Theorem 2.14, we construct the internalization simultaneously. 1. 2. 3. 4. 5.
(P → (P → P)) → ((P → P) → (P → P)) P → (P → P) (P → P) → (P → P) P→P P→P
tautology tautology mod pon on 1, 2 tautology mod pon on 3, 4
d (cons spec) e (cons spec) d·e c (cons spec) d·e·c
This time we get a justification term d · e · c, or more properly (d · e) · c, CS internalizing a proof of P → P, where c + ((P → (P → P)) → ((P → P) → CS CS (P → P))), e + (P → (P → P)), and c + (P → P). The problem of finding a “simplest” justification term is related to the problem of finding the “simplest” proof of a provable formula. It is not entirely clear what this actually means.
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The Basics of Justification Logic
Corollary 2.16 (Lifting Lemma) Suppose JL is a justification logic that has the internalization property relative to CS (in particular, if CS is axiomatically appropriate). If X1 , . . . , Xn `JL(CS) Y then for any justification terms t1 , . . . , tn there is a justification term u so that t1:X1 , . . . , tn:Xn `JL(CS) u:Y. Proof The proof is by induction on n. If n = 0 this is simply the definition of Internalization. Suppose the result is known for n, and we have X1 , . . . , Xn , Xn+1 `JL(CS) Y. We show that for any t1 , . . . , tn , tn+1 there is some u so that t1:X1 , . . . , tn:Xn , tn+1: Xn+1 `JL(CS) u:Y. Using the deduction theorem, X1 , . . . , Xn `JL(CS) (Xn+1 → Y). By the induction hypothesis, for some v we have t1:X1 , . . . , tn:Xn `JL(CS) v:(Xn+1 → Y). Now v:(Xn+1 → Y) → (tn+1:Xn+1 → [v · tn+1 ]:Y) is an axiom hence t1:X1 , . . ., tn:Xn `JL(CS) (tn+1:Xn+1 → [v · tn+1 ]:Y). By modus ponens, t1:X1 , . . . , tn:Xn , tn+1:Xn+1 `JL(CS) [v · tn+1 ]:Y, so take u to be v · tn+1 . Next we move on to the role of justification variables. We said earlier, rather informally, that variables stood for arbitrary justification terms. In order to make this somewhat more precise, we need to introduce substitution. Definition 2.17 (Substitution) A substitution is a function σ mapping some set of justification variables to justification terms, with no variable in the domain of σ mapping to itself. We are only interested in substitutions with finite domain. If the domain of σ is {x1 , . . . , xn }, and each xi maps to justification term ti , it is standard to represent this substitution by (x1 /t1 , . . . , xn /tn ), or sometimes as (~x/~t). For a justification formula X the result of applying a substitution σ is denoted Xσ; likewise tσ is the result of applying substitution σ to justification term t. Substitutions map axioms of a justification logic into axioms (because axiomatization is by schemes), and they preserve modus ponens applications. But one must be careful because the role of constants changes with a substitution. Suppose CS is a constant specification, A is an axiom, and c:A is added to a proof where this addition is authorized by CS. Because axiomatization is by schemes Aσ is also an axiom, but if we add c:Aσ to a proof this may no longer meet constant specification CS. A new constant specification, call it (CS)σ, can be computed from the original one: put c:Aσ ∈ (CS)σ just in case c:A ∈ CS, for any c. If CS was axiomatically appropriate, CS ∪ (CS)σ will also be. So, if X is provable using an axiomatically appropriate constant specification CS, the same will be true for Xσ, not using the original constant specification but rather using CS ∪ (CS)σ. But this is more detail than we generally need to care about. The following suffices for much of our purposes.
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Theorem 2.18 (Substitution Closure) Suppose JL is a justification logic and X is provable in JL using some (axiomatically appropriate) constant specification. Then for any substitution σ, Xσ is also provable in JL using some (axiomatically appropriate) constant specification. We introduce some special notation that suppresses details of constant specifications when we don’t need to care about these details. Definition 2.19 Let JL be a justification logic. We write `JL X as short for: there is some axiomatically appropriate constant specification CS so that `JL(CS) X. Theorem 2.20 Let JL be a justification logic. (1) If `JL X then `JL Xσ for any substitution σ. (2) If `JL X and `JL X → Y then `JL Y. Proof Item (1) is directly from Theorem 2.18. For item (2), suppose `JL X and `JL X → Y. Then there are axiomatically appropriate constant specifications CS1 and CS2 so that `JL(CS1 ) X and `JL(CS2 ) X → Y. Now CS1 ∪ CS2 will also be an axiomatically appropriate constant specification and `JL(CS1 ∪CS2 ) X and `JL(CS1 ∪CS2 ) X → Y, so `JL(CS1 ∪CS2 ) Y and hence `JL Y. In fact, it is easy to check that `JL X if and only if `JL(T) X, where T is the total constant specification. This gives an alternate, and easier, characterization.
2.6 The First Justification Logics In this section and the next we present a number of specific examples of justification logics. We have tried to be systematic in naming these justification logics. Of course modal logic is not entirely consistent in this respect, and justification logic inherits some of its quirks, but we have tried to minimize anomalies. Naming Conventions: It is common to name modal logics by stringing axiom names after K; for instance KT, K4, and so on, with K itself as the simplest case. When we have justification logic counterparts for such modal logics, we will use the same name except with a substitution of J for K; for instance JT, J4, and so on. There is a problem here because a modal logic generally has more than one justification counterpart (if it has any). We will specify which one we have in mind. Formally, JT, J4, and so on result from the addition of axiom schemes, justification function symbols, and a constant specification to
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The Basics of Justification Logic
J0 . When details of a constant specification matter, we will write things like JT(CS), J4(CS), and so on, making the constant specification explicit. We will rarely refer to J0 again because its definition does not actually allow for a constant specification. From now on we will use J for J0 extended with some constant specification, and we will write J(CS) when explicitness is called for. Note that J0 can be thought of as J(∅), where ∅ is the empty constant
specification. The general subject of justification logics evolved from the aforementioned G¨odel–Artemov project, which embeds intuitionistic logic into the modal logic S4, which in turn embeds into the justification logic known as LP (for logic of proofs). It is with LP and its standard sublogics that we are concerned in this section. These are the best-known justification logics, just as K, T (or sometimes KT), S4 (or sometimes KT4), and a few others are the best-known modal logics. For the time being the notion of a justification logic being a counterpart of a modal logic will be an intuitive one. A proper definition will be given in Section 7.2. With two exceptions, the justification logics examined here arise by adding additional operations to the + and · common to all justification logics. The first exception involves factivity, with which we begin. Factivity for modal logics is represented by the axiom scheme X → X. If we think of the necessity operator epistemically, this would be written KX → X. It asserts that if X is known, then X is so. The justification counterpart is the following axiom scheme. Factivity t:X → X Factivity is a strong assumption: justifications cannot be wrong. Nonetheless, if the justification is a mathematical proof, factivity is something mathematicians are generally convinced of. If we think of knowledge as justified, true belief, factivity is built in. Philosophers generally understand justified, true belief to be inherent in knowledge, but not sufficient, see Gettier (1963). The modal axiom scheme X → X is called T. The weakest normal modal logic including all instances of this scheme is KT, sometimes abbreviated simply as T. We use JT for J plus Factivity and, as noted earlier, we use JT(CS) when a specific constant specification is needed. Note that the languages of JT and J are the same. There is one more such example, after which additional operation symbols must be brought in. Consistency is an important special case of Factivity. Modally it can be represented in several ways. One can assume the axiom scheme X → ♦X. In
2.6 The First Justification Logics
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any normal modal logic this turns out to coincide with assuming ¬⊥ (where ⊥ represents falsehood), or equivalently ⊥ → ⊥, which is a very special instance of X → X. If one thinks of as representing provability, ¬⊥ says falsehood is not provable—consistency. Suppose one thinks of deontically, so that X is read that X is obligatory, or perhaps that it is obligatory to bring about a state in which X. Then X → ♦X, or equivalently X → ¬¬X says that if X is obligatory, then ¬X isn’t—a plausible condition on obligations. It is because of this interesting deontic reading that any of the equivalent versions is commonly called D, standing for deontic. Any of these has a justification counterpart. We adopt the following version. Consistency t:⊥ → ⊥ JD is J plus Consistency. Note that JT extends JD.
Positive Introspection is a common assumption about an agent’s knowledge: If an agent knows something, the agent knows that it is known; an agent can introspect about the knowledge he or she possesses. In logics of knowledge it is formulated as KX → KKX. If one understands as representing provability in formal arithmetic, it is possible to prove that a proof is correct: X → X. To formulate a justification logic counterpart, Artemov introduced a one-place function symbol on justification terms, denoted ! and written in prefix position. The intuitive idea is that if t is a justification of something, !t is a justification that t is, indeed, such a justification. Note that the basic language of justification logics has been extended, and this must be reflected in any constant speci
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History Of Maths 1900 To The Present - Free ebook download as PDF File (.pdf), Text File (.txt) or read book online for free. The Big Book of Mathematics, Principles, Theories, and Things. Part IV
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Sadly, we have to inform you that Anne Troelstra, after a short illness, suddenly died on March 7th, aged 79. A world-renowned eminent researcher, a supportive colleague, a teacher who ...
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Sadly, we have to inform you that Anne Troelstra, after a short illness, suddenly died on March 7th, aged 79. A world-renowned eminent researcher, a supportive colleague, a teacher who trained many students to become important scholars, is no longer with us. Beginning in 1957 as a mathematics student, he remained at the University of Amsterdam all his life, except for a number of visiting professorships. He rose quickly to take his place as the successor of Brouwer and Heyting. As a full professor from 1970, he was the universally recognized authority on intuitionism and constructivism in general, leaving behind a number of books that will remain landmarks for many years to come. Retiring in 2000, he made a further name for himself as an author on natural history travel narratives. Anne was regularly seen at the ILLC until this year. His impressive personality, always intensely occupied with his present interests, will be greatly missed.
Funeral announcement card
Anne Sjerp Troelstra (August 10, 1939–March 7, 2019)
Anne Troelstra was born on August 10th, 1939, in Maartensdijk. In 1957, he enrolled as a student of mathematics at the University of Amsterdam – and eventually his interests converged on intuitionism with Arend Heyting as his advisor. Students he was close to include Olga Bakker and E.W. Beth’s students Dick de Jongh and Hans Kamp. With Dick de Jongh, he even wrote a pioneering paper on intuitionistic propositional logic, published in 1966, that contained the first definition of the central notion of a p-morphism, as well as the simplest form of the duality between Heyting algebras and relational frames. After obtaining his master’s degree in 1964, Anne at once became an assistant professor, according to a custom of the time. It took him just two years from there to finish his dissertation, supervised by Heyting. Besides intuitionism, a main interest of Heyting was geometry and perhaps not accidentally Anne’s PhD thesis was a study of intuitionistic topology. This subject made him aware of the role of continuity in intuitionistic mathematics, a concept that was to play an important part in his research in the years to come, in many different forms.
Anne then obtained a scholarship to Stanford to visit Georg Kreisel, and spent the academic year 1966-7 there, with Olga whom he had married the year before. Anne sharpened and modified Kreisel’s ideas on choice sequences, and together they created formalizations of analysis resulting in a large article where the typically intuitionistic concept of a lawless sequence of numbers that successfully evades description by any fixed law, reached its final form. In August 1968, Anne played a central role in the famous Buffalo Conference on Intuitionism and Proof Theory, a meeting of all important logicians of the day with an interest in constructivism. He gave a series of ten lectures there, which turned into his first book, published in 1969, that contained the core of his seminal ideas on intuitionistic formal systems and their meta-mathematical investigation. Back home in 1968, he became a lector (associate professor) in 1968, and a full professor in 1970. Further early recognition was to follow. In 1976, he became a member of the Dutch Royal Academy of Sciences.
The meta-mathematics of intuitionistic systems was a chaotic jumble of results when Anne entered it. Here he showed his greatest strength: creating order in a vast and diverse area. In 1973, the order was there in his book Metamathe-matical Investigation of Intuitionistic Arithmetic and Analysis. Especially striking are the clarification of the properties of different models, various types of realizability, and functional interpretations. The last chapters on special topics were written by Jeff Zucker, Craig Smorynski, and Bill Howard, but the lion’s share had been written by Anne, editor and architect of the whole. This book, known in the community as ‘Springer 344’ still functions as a landmark for serious researchers in the subject. Over time, this work developed into the much larger Constructivism in Mathematics, in two volumes co-authored with Dirk van Dalen, published in 1988, the standard text on constructivism right until today.
Of course, Anne also published in depth on special topics. A central notion in the study of intuitionistic formal systems is realizability, introduced by Stephen Kleene in the 1940’s. Anne’s thorough studies of the subject resulted in a long article in the proceedings of the Second Scandinavian Logic Symposium of 1971. The interest remained with him for life. In 1998, a chapter on realizability by his hand came out in the Handbook of Proof Theory. Characteristically, Anne’s text had been finished a few years before, faithfully meeting his deadline, but delays by other authors kept him updating, somewhat grumblingly, with all new results in the area. What he published had to be the complete state of the art.
Other topics pursued in depth by Anne, through the 70s, 80s and 90s are the history of intuitionism and the philosophical basis of the theory of choice sequences. In his important 1977 book Choice Sequences, he proved lawless sequences to be essentially just a figure of speech by an elimination theorem, showing how statements about lawless sequences can be expressed in a theory containing only lawlike sequences. But he did stress that the notion of lawless sequence still serves a purpose as a clear notion derived by informal rigor.
Moving beyond intuitionism proper over the years, Anne broadened his scope to proof theory in general and wrote two more books which again created new order in diverse fields. In 1992, a textbook Lectures on Linear Logic came out proposing improved formats for a then still only partially understood new paradigm. He contributed the majority of the chapters in a book with Helmut Schwichtenberg called Basic Proof Theory, 1996, that is still a standard resource.
An important part of Anne’s life were his PhD students, of whom he supervised 17, and with many of whom he maintained a close relationship. His first PhD student Daniel Leivant finished a thesis in 1975 on the meta-mathematics of intuitionistic arithmetic, and later made his career in computer science. Initially a scarce commodity, in the 1980s, the number of PhD students increased, and Anne’s students wrote on a broad variety of topics, such as intuitionistic meta-mathematics, combinatory algebra, category theory, Martin-Löf type theory, bounded arithmetic, linear logic, and provability logic. Many of these topics reflected the introduction by Anne, often in a close collaboration with Dirk van Dalen in Utrecht, of new topics on the Dutch scene. These students then carried the torch further by themselves. For instance, Ieke Moerdijk became an international leader at the interface in topos theory and logic and category theory generally, while Jaap van Oosten became a worldwide authority on realizability. In the Netherlands alone, four of Anne’s students have become full professors, in mathematics, computer science, AI and philosophy. But Anne was an dedicated teacher at all academic levels, whose precision, clarity and scholarship influenced generations of students in Amsterdam.
Anne retired in 2000, but not to rest. All his life, he had a deep interest in natural history and a wide knowledge of the plants of the Netherlands and abroad. His annual linocuts of plants discovered on his travels in Europe were famous. To those on a walk while listening to him, what looked like an ordinary city street to the untrained eye would turn into a rich landscape of flora, history, and natural wonders. This very year 2019, an article by him will appear on new species of blackberries, his special interest. Anne also made a further name for himself as an author on natural history travel narratives, chronicling the exotic characters and adventures of the past denied to the average academic of today. His major Bibliography of Natural History Narratives was published in 2016.
Anne will be missed in the first place because he will no longer be there to tell us what he thinks about a question that you may have about constructivism. You always knew that you would get a completely honest answer from somebody who knew all the issues and had already thought much further than you. But it is just as much the personal qualities that will be missed. Anne was a very special, and to some, occasionally intimidating person: penetrating, honest, critical, ironic, sharp at times, but always open to arguments and unfailingly supportive of his students and colleagues. He will be deeply missed by all.
Our thoughts go out to his wife Olga and to his daughters Willemien and Ine.
Johan van Benthem
Dick de Jongh
Prof. Troelstra's archive is available at the the Noord-Hollands Archief. The index of the archive is filed as X-2003-01 in ILLC's Technical Notes Series. A list of publications and his CV are available here and here.
When Anne and Olga came to California in 1966 a lifelong friendship began, eventually bringing them and their teenage children to visit Yiannis and me and our teenage children in Greece. We admired Willemien and Ine's beautiful travel journals, learned from Olga the restorative power of tea with bread and good Dutch cheese, and were amazed by the family's botanical erudition. Anne taught me to recognize Phlomis fruticosa, Malva sylvestris and other common Greek wildflowers. In later years he and Olga visited us for botanical adventures and good fellowship, and we visited them in Muiderberg, where Anne guided us on forest walks and longer excursions of which vivid memories remain. We exchanged books and handcraft, cards and photos, looking forward to our next meeting.
At conferences, and in papers, books and correspondence, I have learned more about intuitionistic logic, arithmetic and analysis from Anne than from anyone else. He was a firm but kind critic, encouraging even modest advances by others, always giving fair credit. He took seriously his position as Brouwer's and Heyting's successor at the University of Amsterdam and once confessed to feeling exhausted by the responsibility to understand every new development concerning intuitionism. At retirement he donated much of his own mathematical library to the University of Athens, where he was a visiting lecturer for the graduate program in logic. His generous spirit has enriched innumerable lives. We miss Anne's presence but his essence lives on in books and papers and letters and in all our memories.
Joan Rand Moschovakis
Anne Troelstra
Around 1977, I was a master's student under Dirk van Dalen and Henk Barendregt in Utrecht. We students attended the `baby seminar' under supervision of Jan Willem Klop. That year's baby seminar was concerned with a book that was simply called 344. We students found the book somewhat hard to study, since it gave many details but little motivation. Nevertheless, we learned a lot from the book. The book was written by Anne Troelstra.
From 344, I learned, among many other things, about Heyting Arithmetic and Kleene Realizability. In my career, I returned again and again to Anne's presentation and I used the things I learned in my work. For example, Anne characterized the theory of Kleene realizability as Heyting Arithmetic plus an extended version of Church's Thesis. My paper on the Completeness Principle can be viewed as an answer to the question: what happens if we do the same thing Anne did when Kleene Realizability is replaced by the provability translation?
I met Anne for the first time in 1977/78 when he visited Dirk van Dalen at the Mathematical Institute in Utrecht (now the Freudenthal Building). Even if Anne looked, at first sight, somewhat stern, he turned out to be friendly and accessible. He was always prepared to answer questions and to discuss problems. At a certain point, I started calling Anne `Anne'. Dirk van Dalen noted this and asked `Did professor Troelstra specifically invite you to call him `Anne'?' No, but it felt appropriate. After that I avoided, for some time, any form of address for Anne and then returned to `Anne'. There was never any sign that Anne disapproved of `Anne'.
Anne did not care much for worldly fame. For, him the main motivation was understanding a thing in all possible detail. This was a defining property of his personality. We see it both in his work in logic and in his work on botanical travel stories.
I did not see Anne very often in the last years, but, now and then, we attended the same meeting. It was always good seeing him and talking to him. It is hard to understand that now this is not possible anymore.
Albert Visser
In my formative years around 1975 in the logic school of Dirk van Dalen and Henk Barendregt, at the Mathematical Institute, Boedapestlaan 6 in Utrecht, I was happily sharing a room with Albert Visser, adjacent to the rooms of Dirk and Henk and at the
same fourth floor corridor as the room of our friend Jeff Zucker. We had a well-attended Intercity Colloquium, bi-weekly taking place alternatingly at the institutes in Amsterdam and Utrecht. Anne Troelstra, Dick de Jongh, Roel de Vrijer were regular participants,
together with visitors in those years such as Walter van Stigt, David Isles and Craig Smorynski, and many more short or long term visiting friends and colleagues from abroad. As a junior Ph.D-student under supervision of Dirk and Henk my (too) difficult assignment was to assist a group of newcoming students, including Albert, in digesting and mastering various chapters of Anne’s famous Springer LNM 344, shortly known as ‘344’.
The encounter with 344 did not influence me as deeply as would have been desirable, but in my subsequent development to a theoretical computer scientist I did greatly profit from the treatment in 344 of important recursion theory theorems such as the ones of Myhill-Shepherdson and Kreisel-Lacombe-Schoenfield. They were instrumental in a later paper (1982) by Jan Bergstra and me about parametrized data types, continuing Jan’s well-known series of papers together with John Tucker from Swansea, analyzing the theory of abstract data types.
I used to drive our Utrecht group, Dirk, Henk, Jeff, Albert in my small car to the Amsterdam sessions of the Intercity seminar, and dually, drive the Amsterdam participants, Anne, Dick, Roel after the Utrecht sessions back to Utrecht Central Station. I remember Anne’s stimulating comments, when I had made good progress with my PhD-thesis, about term rewriting systems and lambda calculus. I remember Anne as a true scholar and a gentleman. He was an example, a role model for junior logicians and computer scientists. At the event of his emeritate, I thanked him for his continual inspiration. I remember Anne’s facial expression, somewhat amused and ironic.
Later, the past fifteen years, I encountered Anne many times in the meetings of the Section Mathematics of the Academy. His life and work will be for many of us, in logic and computer science, an ever-lasting inspiration and enrichment.
Jan Willem Klop
I first saw Anne in action when he taught "Analysis II", the major stumbling block for beginning mathematics students. It was whispered in our group that he was very clever, having become an Associate Professor at a very early age, but it would be saying too much that the field of Analysis came sparklingly alive. What did come alive was my image of Anne, he looked very much then like he looked all through his life: serious, sharp, erudite, and with his technical subjects at his fingertips. Later on, I took his all courses on intuitionism, even though I chose the path of model theory eventually, having tasted the sinful delights of getting mathematical results without constructivist proofs.
Anne, characteristically, never held my choice against me, nor did he object to my broader activism in philosophy, linguistics, computation, and even further areas over time. Anne was clearly a mathematical logician from the heartland, but he saw pursuing wider frontiers of logic as good for the whole field, rather than as a threat to established rank and dogma.
I owe a lot to Anne. He helped me at a crucial stage of my dissertation project in 1976, he had confidence in me when I was appointed as his collega proximus in 1986 (an honor that I am still vividly aware of after all these years), and in the formative time of the ILLC, he was quietly but persistently supportive, even though governance and organizational activism were not among his favorites. For many years, our offices were side by side, and our contacts happened daily. That does not mean Anne never criticized his next-door neighbor, sometimes with the aid of a list of points on his whiteboard, but always for good reasons: and in his turn, he was always open to arguments, and able to change his mind.
Familiarity may breed contempt, as is said, but it can also breed respect. Over the years, I came to appreciate Anne's qualities as a researcher and as a person more and more. I also admired his starting a new life after retirement, rather than following the inertia that people call 'still going strong in one's field'. With Anne gone, my world in Amsterdam looks reduced: it has lost a dimension.
Johan van Benthem
Arriving in September 1961 in Amsterdam for a master study with E.W. Beth I soon found myself in contact with one of Heyting's students: Anne Troelstra. Since our subjects were close we participated in a number of the same classes. He was meticulous, neat, always ahead of deadlines, everything I was not. After a while, I had to recognize he was very clever as well. We shared a love of intuitionistic logic, and looked at the basics together. Soon we found the 14 equivalence classes of formulas with only p, q and implication. More seriously, we delved into my subject, the theory of the models later called Kripke models, and established some important results. He stayed here for a PhD with Heyting, I left for one in the U.S. When I returned to take up a position at the UvA, he had already a solid reputation. It was always a safe feeling that he was there. When I had gone back again to the U.S. for two years of teaching, and on wanting to return found it was less easy than I had expected to find a position again in the Netherlands he was able to ease me back into the UvA. We both worked in intuitionism but from different angles. When we discussed issues in that area he often surprised me, suddenly showing insights I didn't suspect. This remained so even when his main interest had shifted from logic to natural history. Life will be different now that he is no longer there.
Dick de Jongh
I first met Anne Troelstra at the Mathematisches Forschungsinstitut Oberwolfach. It was at the time when he just had finished his marvellous lecture notes volume entitled "Metamathematical Investigations of Intuitionistic Arithmetic and Analysis". This impressive piece of work quickly became the standard source of knowledge for at least a generation of mathematics students with an interest in the logical foundations of constructive mathematics. In fact, at the time it was the most-read volume of the whole series of Springer Lecture Notes in Mathematics at the Mathematical Institute Library of LMU Munich. Later it was extended to the almost encyclopedic two-volume book "Constructivism in Mathematics", which he wrote together with Dirk van Dalen, and later again I had the great pleasure to be his coauthor in the book "Basic Proof Theory", with appeared in two editions around the turn of the century. For many years we also cooperated in organizing the regular workshop on Mathematical Logic at the Mathematisches Forschungsinstitut Oberwolfach.
In all these years he strongly impressed me by the clarity and originality of his mathematical work, and also his ability to relate it to the vast literature of his field. He was an absolutely honest person, who always insisted to give other researchers the deserved credit for their work. Apart from mathematics he had many other interests, which he pursued with similar quality and endurance as his mathematical work.
I will very much miss him as a dear colleague and friend.
Helmut Schwichtenberg
One of the great pleasures of the academic life is that occasionally you meet (and become friends with) people who look different, talk strangely and generally have nothing in common with you except math---in this case; and so it was that Joan and I met and became friends with Anne and Olga back in the sixties, when we were all very young. Joan had a lot to talk about with Anne, of course, but I, too, always enjoyed the scientific exchanges with him: he was a tolerant intuitionist, and I had started in constructive mathematics, reading Heyting's little book in '57 or '58, probably before Anne. (I lost my constructive faith in Grad School but like many lapsed Catholics, I never lost my respect for the faithful or lingering guilt for my dropping out.)
There have been many trips over the years---some with children---by the Troelstras to Greece and by us to the Netherlands, most recently three years ago. Many happy memories, mostly of talk about plants.
A very vivid one (some years back) is of a walk in Parnitha, the tallest of the four mountains that surround Athens. It was chilly (Spring or Fall most likely) and Anne was in his element, latin names of species pouring out of him; until he turned silent and said softly that he had seen more species that morning than exist in Holland---the single, greatest praise of the Greek countryside I ever heard.
We will miss him.
Yiannis Moschovakis
Professor Troelstra has been one of my main scientific mentors and supporters when I was young and needed support the most. He not only invited me to my first Oberwolfach Meeting in 1990 but also to my first talk abroad (the Intercity Logic Seminar between Amsterdam and Utrecht) again in 1990.
He was co-referee for both my master and my PhD theses and no scientific work had a greater influence on me than his Springer LNM 344 "Metamathematical Investigation of Intuitionistic Arithmetic and Analysis" from 1973 which is only book I had to buy twice since the first copy disintegrated due to its intensive use. There was a time when I remembered the page numbers on which certain facts were presented in this book.
He examplified for me the ideal scholar. I will always remember him in the highest regards and will be forever grateful to him.
Ulrich Kohlenbach
In the last quarter of the 20th century, Anne Troelstra, with my assistance, organized an Amsterdam-Münster logic contact between our two institutions: Every second year, a few Amsterdam logicians under Anne's leadership would visit the logic institute at Münster, some of them as well as some Münster logicians would present their recent research, and every other year, it would be the other way around. For many young logicians from Münster -possibly also from Amsterdam- this was their first encounter with the international logic scene, and it certainly influenced and improved the work on intuitionism and constructive mathematics at the Münster institute considerably.
This contact was also a basis for a deep friendship between the Troelstra and the Diller family, resulting in a number of visits to their respective homes at Muiderberg and Münster. We, the Dillers at Münster, are deeply moved by the sudden and unexpected death of Anne, and our feelings of sympathy are particularly with Olga.
Justus Diller
It must have been in the second year of my studying maths, in the spring of 1983, when I first met Professor Troelstra when I attended a logic seminar. We read Dana Scott's Lectures on a Mathematical Theory of Computation, and I was immediately tasked with presenting Chapter 3.
Half a year later, I was student-assistent under him. I helped with bibliographical research for the Omega-bibliography, of which Troelstra and Diller compiled the part on Constructivism.
Troelstra could be rather rigid when it came to social conventions. Dutch, like many other continental European languages, has a polite/formal form and a familiar form (different ways to say "you"). In those days, it was still customary to address a student in the polite form, whereas for a colleague one would of course use the familiar form. When I did my bibliographical chores, I was a colleague and was talked to in the familiar form. When, after a discussion about this work, I asked him whether I could ask a question about his course in lambda calculus, which I was following, he switched at once to the polite form.
I accompanied him on trips to Muenster in Germany. Once a year, a delegation of teachers, postdocs, PhD students and undergraduates went for a short weekend-seminar to the department of Diller and Pohlers (and once a year, the Muensteraner reciprocated with a visit to Amsterdam). Once when we passed a sculpture of a nymph-like person, her naked torso rising out of a rough substance, he commented: "the artist probably thought: if I reveal more of her body, it can only lead to disappointment". He had definitely a sense of humour.
Maybe once a year, he invited the whole logic group to his place for dinner. At the end of the evening he would put a large crate, full with books, on the floor, and everyone was invited to take what (s)he liked.
In the fall of 1986 I had been working on my master thesis, but my assistentship had ended and I had several jobs, so the thesis work had been put on hold. Troelstra was not in the habit of telling you what he thought of your work, and naturally I assumed I would do my master's some time and then find a position outside academia. But I was in for a surprise, when early february 1987 he was on the phone. "You should finish your thesis and get your master diploma, for you can start by March 1 as PhD student." It was not a question. At the ceremony for my master diploma (my parents were present) he called me a "rough diamond", which was sort of a compliment, I suppose. The emphasis was on "rough".
As a PhD student I of course got to know Troelstra better as a mathematician. I did my first steps in topos theory, following work by Martin Hyland. Although he had not worked in this field, he always held it in some kind of timorous esteem, although he stressed the importance of applications. Abstraction had its limits. He had lots of criticisms, mainly on my style of writing ("too succinct", "too terse"); later I realized that he taught me a lot during this period. Reading my manuscripts was sometimes a strain for him, and he complained that supervising me would shorten his life by a year. I came to appreciate how he could "feel" a result, even if he was not familiar with the formal details.
Again, he was not going to give you any compliments so after 3 years of work on my thesis I was just dead certain that it had been a total failure. I would leave academia without a thesis. But again there was a surprise: "you should do some research in the library for an overview of the field. And gather your results, for it is about time to start writing up".
One aspect of doing a PhD with Troelstra was that, starting right after the ceremony, one should no longer use the polite form with him and call him "Anne". This took some getting used to, but I managed eventually.
In the northern Dutch province of Friesland (which is not where Anne was born or grew up, but is, I presume, where his ancestors came from) the name "Anne" is a common boys' name. A funny story was told at the memorial conference which Ieke Moerdijk, Harold Schellinx and I organized to celebrate Anne's 60th birthday: when he first met Dana Scott, he told him that he had thought Dana was a girl. He probably didn't reflect on how his own name might sound to an English speaker.
I am grateful to have been able to study under a scholar of Anne's calibre. His staunch honesty will remain a beacon for me the rest of my life.
Jaap van Oosten
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Full Text (PDF) - The Institute for Logic, Language and Computation ...
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1 INDEX OF THE TROELSTRA ARCHIVE Introduction This index was mainly prepared in 2000 <strong>and</strong> 2001. 1 <strong>The</strong> archive itself was stored at the Mathematical <strong>Institute</strong> of the University of Amsterdam, afterwards (2001) brought to the Rijksarchief in Noord-Holl<strong>and</strong>. <strong>The</strong> length of the archive is circa 2.5 meters. <strong>The</strong> material in the archive has been divided into a number of major categories, distinguished by capitals in the numbering of items, e.g. B (scientific correspondence), C (courses given by Troelstra, seminars <strong>and</strong> colloquia conducted by him), P (all materials relating to his publications) etc. Usually these major categories are again subdivided. <strong>The</strong> description of the contents is not carried out with the same amount of detail <strong>for</strong> each (sub-) category. A lot of material in this index was already set up by A.S. Troelstra himself (especially publications, lectures <strong>and</strong> congresses). P. van Ulsen, Amsterdam, 2001. Curriculum vitae of Anne Sjerp Troelstra 2 10-VIII-1939 Born at Maartensdijk (Utrecht), the Netherl<strong>and</strong>s. 1951-1957 Secondary school: Lorentz Lyceum (gymnasium beta), Eindhoven. 1-IX-1957 Enrollment as a student at the University of Amsterdam. 25-III-1964 Passed doctoraalexamen in Mathematics, cum laude. (At that time a bit more than a M.Sc.) 1-IV-1964 Appointed ‘wetenschappelijk medewerker’ (approximately, assistant professor) at the Department of Mathematics of the University of Amsterdam. 15-VI-1966 Doctorate (Ph.D) in mathematics on the thesis ‘Intuitionistic general topology’ (thesis adviser Prof. Dr. A. Heyting). University of Amsterdam. 1-IX-1966 till 1-IX-1967. On leave as a visiting scholar at Stan<strong>for</strong>d University (departments of Mathematics <strong>and</strong> Philosophy) with a stipend from the Netherl<strong>and</strong>s Organization <strong>for</strong> the Advancement of Research (then ZWO, now called NWO). VIII-1968 Gave a series of ten lectures on Intuitionism at the Summer School on Proof <strong>The</strong>ory <strong>and</strong> Intuitionism at SUNY, Buffalo, New York. 1-IX-1968 Appointed ‘lector’ (associate professor) in mathematics at the University of Amsterdam. 1-IX-1970 Appointed ‘gewoon hoogleraar’ (full professor) in pure mathematics <strong>and</strong> foundations of mathematics at the University of Amsterdam. 4-VI-1976 Elected member of the Royal Dutch Academy of Sciences. 16-II-1996 Elected corresponding member of the Bavarian Academy of Sciences. 15-XI-1996 Received the F.L. Bauer–Prize of the ‘Bund der Freunde der Technischen Universität München’, <strong>for</strong> internationally outst<strong>and</strong>ing contributions to 1 <strong>The</strong> way Troelstra set up the Heyting Archive (now part of the Rijksarchief Noord-Holl<strong>and</strong> in Haarlem) <strong>and</strong> its index was a model <strong>for</strong> this archive <strong>and</strong> index. <strong>The</strong> Library of the Faculty of Sciences (Faculteit Natuurwetenschappen, Wiskunde en In<strong>for</strong>matica) of the University of Amsterdam, <strong>and</strong> especially the chief librarian H. Harmsen, gave me the opportunity to complete this index. 2 Data from the postscript file of A.S. Troelstra ( http://turing.wins.uva.nl/~anne/) . See also A.S. Troelstra, Looking back, ILLC magazine 2, (July 2000).
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Constructivism (philosophy of mathematics)
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https://en.wikipedia.org/wiki/Constructivism_(philosophy_of_mathematics)
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Mathematical viewpoint that existence proofs must be constructive
This article is about the view in the philosophy of mathematics. For other uses of the term, see Constructivism.
In the philosophy of mathematics, constructivism asserts that it is necessary to find (or "construct") a specific example of a mathematical object in order to prove that an example exists. Contrastingly, in classical mathematics, one can prove the existence of a mathematical object without "finding" that object explicitly, by assuming its non-existence and then deriving a contradiction from that assumption. Such a proof by contradiction might be called non-constructive, and a constructivist might reject it. The constructive viewpoint involves a verificational interpretation of the existential quantifier, which is at odds with its classical interpretation.
There are many forms of constructivism. These include the program of intuitionism founded by Brouwer, the finitism of Hilbert and Bernays, the constructive recursive mathematics of Shanin and Markov, and Bishop's program of constructive analysis. Constructivism also includes the study of constructive set theories such as CZF and the study of philosophy.
Constructivism is often identified with intuitionism, although intuitionism is only one of the constructivist programs. Intuitionism maintains that the foundations of mathematics lie in the individual mathematician's intuition, thereby making mathematics into an intrinsically subjective activity. Other forms of constructivism are not based on this viewpoint of intuition, and are compatible with an objective view point on maths
Constructive mathematics
[edit]
Much constructive mathematics uses intuitionistic logic, which is essentially classical logic without the law of the excluded middle. This law states that, for any proposition, either that proposition is true or its negation is. This is not to say that the law of the excluded middle is denied entirely; special cases of the law will be provable. It is just that the general law is not assumed as an axiom. The law of non-contradiction (which states that contradictory statements cannot both be true at the same time) is still valid.
For instance, in Heyting arithmetic, one can prove that for any proposition p that does not contain quantifiers, ∀ x , y , z , … ∈ N : p ∨ ¬ p {\displaystyle \forall x,y,z,\ldots \in \mathbb {N} :p\vee \neg p} is a theorem (where x, y, z ... are the free variables in the proposition p). In this sense, propositions restricted to the finite are still regarded as being either true or false, as they are in classical mathematics, but this bivalence does not extend to propositions that refer to infinite collections.
In fact, L. E. J. Brouwer, founder of the intuitionist school, viewed the law of the excluded middle as abstracted from finite experience, and then applied to the infinite without justification. For instance, Goldbach's conjecture is the assertion that every even number greater than 2 is the sum of two prime numbers. It is possible to test for any particular even number whether it is the sum of two primes (for instance by exhaustive search), so any one of them is either the sum of two primes or it is not. And so far, every one thus tested has in fact been the sum of two primes.
But there is no known proof that all of them are so, nor any known proof that not all of them are so; nor is it even known whether either a proof or a disproof of Goldbach's conjecture must exist (the conjecture may be undecidable in traditional ZF set theory). Thus to Brouwer, we are not justified in asserting "either Goldbach's conjecture is true, or it is not." And while the conjecture may one day be solved, the argument applies to similar unsolved problems. To Brouwer, the law of the excluded middle is tantamount to assuming that every mathematical problem has a solution.
With the omission of the law of the excluded middle as an axiom, the remaining logical system has an existence property that classical logic does not have: whenever ∃ x ∈ X P ( x ) {\displaystyle \exists _{x\in X}P(x)} is proven constructively, then in fact P ( a ) {\displaystyle P(a)} is proven constructively for (at least) one particular a ∈ X {\displaystyle a\in X} , often called a witness. Thus the proof of the existence of a mathematical object is tied to the possibility of its construction.
Example from real analysis
[edit]
In classical real analysis, one way to define a real number is as an equivalence class of Cauchy sequences of rational numbers.
In constructive mathematics, one way to construct a real number is as a function ƒ that takes a positive integer n {\displaystyle n} and outputs a rational ƒ(n), together with a function g that takes a positive integer n and outputs a positive integer g(n) such that
∀ n ∀ i , j ≥ g ( n ) | f ( i ) − f ( j ) | ≤ 1 n {\displaystyle \forall n\ \forall i,j\geq g(n)\quad |f(i)-f(j)|\leq {1 \over n}}
so that as n increases, the values of ƒ(n) get closer and closer together. We can use ƒ and g together to compute as close a rational approximation as we like to the real number they represent.
Under this definition, a simple representation of the real number e is:
f ( n ) = ∑ i = 0 n 1 i ! , g ( n ) = n . {\displaystyle f(n)=\sum _{i=0}^{n}{1 \over i!},\quad g(n)=n.}
This definition corresponds to the classical definition using Cauchy sequences, except with a constructive twist: for a classical Cauchy sequence, it is required that, for any given distance, there exists (in a classical sense) a member in the sequence after which all members are closer together than that distance. In the constructive version, it is required that, for any given distance, it is possible to actually specify a point in the sequence where this happens (this required specification is often called the modulus of convergence). In fact, the standard constructive interpretation of the mathematical statement
∀ n : ∃ m : ∀ i , j ≥ m : | f ( i ) − f ( j ) | ≤ 1 n {\displaystyle \forall n:\exists m:\forall i,j\geq m:|f(i)-f(j)|\leq {1 \over n}}
is precisely the existence of the function computing the modulus of convergence. Thus the difference between the two definitions of real numbers can be thought of as the difference in the interpretation of the statement "for all... there exists..."
This then opens the question as to what sort of function from a countable set to a countable set, such as f and g above, can actually be constructed. Different versions of constructivism diverge on this point. Constructions can be defined as broadly as free choice sequences, which is the intuitionistic view, or as narrowly as algorithms (or more technically, the computable functions), or even left unspecified. If, for instance, the algorithmic view is taken, then the reals as constructed here are essentially what classically would be called the computable numbers.
Cardinality
[edit]
To take the algorithmic interpretation above would seem at odds with classical notions of cardinality. By enumerating algorithms, we can show that the computable numbers are classically countable. And yet Cantor's diagonal argument here shows that real numbers have uncountable cardinality. To identify the real numbers with the computable numbers would then be a contradiction. Furthermore, the diagonal argument seems perfectly constructive.
Indeed Cantor's diagonal argument can be presented constructively, in the sense that given a bijection between the natural numbers and real numbers, one constructs a real number not in the functions range, and thereby establishes a contradiction. One can enumerate algorithms to construct a function T, about which we initially assume that it is a function from the natural numbers onto the reals. But, to each algorithm, there may or may not correspond a real number, as the algorithm may fail to satisfy the constraints, or even be non-terminating (T is a partial function), so this fails to produce the required bijection. In short, one who takes the view that real numbers are (individually) effectively computable interprets Cantor's result as showing that the real numbers (collectively) are not recursively enumerable.
Still, one might expect that since T is a partial function from the natural numbers onto the real numbers, that therefore the real numbers are no more than countable. And, since every natural number can be trivially represented as a real number, therefore the real numbers are no less than countable. They are, therefore exactly countable. However this reasoning is not constructive, as it still does not construct the required bijection. The classical theorem proving the existence of a bijection in such circumstances, namely the Cantor–Bernstein–Schroeder theorem, is non-constructive. It has recently been shown that the Cantor–Bernstein–Schroeder theorem implies the law of the excluded middle, hence there can be no constructive proof of the theorem.
Axiom of choice
[edit]
The status of the axiom of choice in constructive mathematics is complicated by the different approaches of different constructivist programs. One trivial meaning of "constructive", used informally by mathematicians, is "provable in ZF set theory without the axiom of choice." However, proponents of more limited forms of constructive mathematics would assert that ZF itself is not a constructive system.
In intuitionistic theories of type theory (especially higher-type arithmetic), many forms of the axiom of choice are permitted. For example, the axiom AC11 can be paraphrased to say that for any relation R on the set of real numbers, if you have proved that for each real number x there is a real number y such that R(x,y) holds, then there is actually a function F such that R(x,F(x)) holds for all real numbers. Similar choice principles are accepted for all finite types. The motivation for accepting these seemingly nonconstructive principles is the intuitionistic understanding of the proof that "for each real number x there is a real number y such that R(x,y) holds". According to the BHK interpretation, this proof itself is essentially the function F that is desired. The choice principles that intuitionists accept do not imply the law of the excluded middle.
However, in certain axiom systems for constructive set theory, the axiom of choice does imply the law of the excluded middle (in the presence of other axioms), as shown by the Diaconescu-Goodman-Myhill theorem. Some constructive set theories include weaker forms of the axiom of choice, such as the axiom of dependent choice in Myhill's set theory.
Measure theory
[edit]
Classical measure theory is fundamentally non-constructive, since the classical definition of Lebesgue measure does not describe any way how to compute the measure of a set or the integral of a function. In fact, if one thinks of a function just as a rule that "inputs a real number and outputs a real number" then there cannot be any algorithm to compute the integral of a function, since any algorithm would only be able to call finitely many values of the function at a time, and finitely many values are not enough to compute the integral to any nontrivial accuracy. The solution to this conundrum, carried out first in Bishop (1967), is to consider only functions that are written as the pointwise limit of continuous functions (with known modulus of continuity), with information about the rate of convergence. An advantage of constructivizing measure theory is that if one can prove that a set is constructively of full measure, then there is an algorithm for finding a point in that set (again see Bishop (1967)).
The place of constructivism in mathematics
[edit]
Traditionally, some mathematicians have been suspicious, if not antagonistic, towards mathematical constructivism, largely because of limitations they believed it to pose for constructive analysis. These views were forcefully expressed by David Hilbert in 1928, when he wrote in Grundlagen der Mathematik, "Taking the principle of excluded middle from the mathematician would be the same, say, as proscribing the telescope to the astronomer or to the boxer the use of his fists".[5]
Errett Bishop, in his 1967 work Foundations of Constructive Analysis, worked to dispel these fears by developing a great deal of traditional analysis in a constructive framework.
Even though most mathematicians do not accept the constructivist's thesis that only mathematics done based on constructive methods is sound, constructive methods are increasingly of interest on non-ideological grounds. For example, constructive proofs in analysis may ensure witness extraction, in such a way that working within the constraints of the constructive methods may make finding witnesses to theories easier than using classical methods. Applications for constructive mathematics have also been found in typed lambda calculi, topos theory and categorical logic, which are notable subjects in foundational mathematics and computer science. In algebra, for such entities as topoi and Hopf algebras, the structure supports an internal language that is a constructive theory; working within the constraints of that language is often more intuitive and flexible than working externally by such means as reasoning about the set of possible concrete algebras and their homomorphisms.
Physicist Lee Smolin writes in Three Roads to Quantum Gravity that topos theory is "the right form of logic for cosmology" (page 30) and "In its first forms it was called 'intuitionistic logic'" (page 31). "In this kind of logic, the statements an observer can make about the universe are divided into at least three groups: those that we can judge to be true, those that we can judge to be false and those whose truth we cannot decide upon at the present time" (page 28).
Mathematicians who have made major contributions to constructivism
[edit]
Leopold Kronecker (old constructivism, semi-intuitionism)
L. E. J. Brouwer (founder of intuitionism)
A. A. Markov (forefather of Russian school of constructivism)
Arend Heyting (formalized intuitionistic logic and theories)
Per Martin-Löf (founder of constructive type theories)
Errett Bishop (promoted a version of constructivism claimed to be consistent with classical mathematics)
Paul Lorenzen (developed constructive analysis)
Martin Hyland (discovered the effective topos in realizability)
Branches
[edit]
Constructive logic
Constructive set theory
Constructive type theory
Constructive analysis
Constructive non-standard analysis
See also
[edit]
Computability theory – Study of computable functions and Turing degrees
Constructive proof – Method of proof in mathematics
Finitism – Philosophy of mathematics that accepts the existence only of finite mathematical objects
Game semantics – approach to formal semanticsPages displaying wikidata descriptions as a fallback
Inhabited set – Property of sets used in constructive mathematics
Intuitionism – Approach in philosophy of mathematics and logic
Intuitionistic type theory – Alternative foundation of mathematics
Notes
[edit]
References
[edit]
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----JAN BURSE BARRED FROM comp.lang.prolog ---- RFD
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] |
[] |
[] |
[
""
] | null |
[] | null |
en
|
//www.gstatic.com/images/branding/product/1x/groups_32dp.png
|
https://groups.google.com/g/comp.lang.prolog/c/V8jXGy7iQlk
|
Stalker Kibo Analbuttfuckmanure John J. Veysey, Scott Stanley, Sethuraman Panchanathan, F. Fleming Crim, Dorothy E Aronson, Brian Stone, James S Ulvestad, Rebecca Lynn Keiser, Vernon D. Ross, Lloyd Whitman, are you paying Kibo to stalk sci.math, sci.physics???
Jan Burse is thoroughly insane-- for he goes -- out of his way, just to attack people.
Is this Jan Burse posting undercover as std World, for Kibo Parry M is a 28 year stalker and Jan Burse a 20 year stalker. Stalkers are insane people, and a shame governments have not cleaned up stalkers from the Internet, yet. Check the posting time of this-- for probably it is Jan Burse posting in early morning for it is too late in Boston.
On Tuesday, July 20, 2021 at 9:42:51 PM UTC-5, Michael Moroney wrote:
> "Hi, I pound male rectums".
>
On Tuesday, July 20, 2021 at 10:00:24 PM UTC-5, Michael Moroney wrote:
>AnalButtfuckManure"
If indeed, NSF Dr. Panchanathan hired the stalker Kibo Parry Moroney going back to 1993 and thereabouts, was a huge mistake, for Usenet is self policing. The govt hiring a paid stalker eventually turns to ruin, for although a nitwit like Kibo Parry can tell a crank and moron. The nitwit Kibo Parry cannot tell a genuine scientist doing science outside the box. So in this sense, Dr. Panchanathan should resign from the NSF.
On Sunday, July 18, 2021 at 6:16:15 AM UTC-5, Michael Moroney wrote:
>"AnalButtfuckManure"
Sethuraman Panchanathan, F. Fleming Crim, Dorothy E Aronson, Brian Stone, James S Ulvestad, Rebecca Lynn Keiser, Vernon D. Ross, Lloyd Whitman, John J. Veysey, Scott Stanley, are you paying Kibo to stalk sci.math, sci.physics???
Is NSF Dr. Panchanathan paying Kibo Parry M more money to stalk sci.math, sci.physics, than the top 5 officials at NSF combined and 5X the amount of pay of MIT professors who teach calculus in classrooms. Same question for Canada's stalker, how much pay from government for his bully tactics
Does NSF pay kibo Parry to stalk more than the combined salaries of Sethuraman Panchanathan, F. Fleming Crim, Dorothy E Aronson, Brian Stone, James S Ulvestad, Rebecca Lynn Keiser, Vernon D. Ross, Lloyd Whitman, John J. Veysey, Scott Stanley.
Kibo Parry M on Philip J. Hanlon Dartmouth College president
On Saturday, July 17, 2021 at 4:05:34 PM UTC-5, Michael Moroney wrote:
> 🐜 of Math and 🐛 of
> fails at math and science:
Kibo, is it because he can never do a geometry proof of calculus, nor can Dr. Panchanathan of National Science Foundation, NSF who possibly is paying kibo more money to stalk for 28 years than what Dartmouth pays professors to actually teach calculus in classrooms.
..
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; ` ;'.. ..-'' ' ' I am Kibo Parry Moroney, the grand failure of science with my 938 is 12% short of 945, and my ellipse is a conic when it never was, and my idiocy of thinking geothermal is not radioactivity but is recycled solar fossil. I stalk on Internet because NSF pays a million dollars and is 5 times the salary of those professors stuck with actually teaching science and all I do is attack dog style in sci.math, sci.physics. And Barry Shein loves to whisper in my ear how 10 OR 2 =12 with AND as subtraction
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---quoting Wikipedia ---
Controversy
Many government and university installations blocked, threatened to block, or attempted to shut-down The World's Internet connection until Software Tool & Die was eventually granted permission by the National Science Foundation to provide public Internet access on "an experimental basis."
--- end quote ---
NATIONAL SCIENCE FOUNDATION
Dr. Panchanathan , present day
France Anne Cordova
Subra Suresh
Arden Lee Bement Jr.
Rita R. Colwell
Neal Francis Lane
John Howard Gibbons 1993
Barry Shein, kibo parry std world
Jim Frost, Joe "Spike" Ilacqua
World's First Geometry Proof of Fundamental Theorem of Calculus// Math proof series, book 2 Kindle Edition
by Archimedes Plutonium (Author)
Last revision was 19May2021. This is AP's 11th published book of science.
Preface:
Actually my title is too modest, for the proof that lies within this book makes it the World's First Valid Proof of Fundamental Theorem of Calculus, for in my modesty, I just wanted to emphasis that calculus was geometry and needed a geometry proof. Not being modest, there has never been a valid proof of FTC until AP's 2015 proof. This also implies that only a geometry proof of FTC constitutes a valid proof of FTC.
Calculus needs a geometry proof of Fundamental Theorem of Calculus. But none could ever be obtained in Old Math so long as they had a huge mass of mistakes, errors, fakes and con-artist trickery such as the "limit analysis". To give a Geometry Proof of Fundamental Theorem of Calculus requires math be cleaned-up and cleaned-out of most of math's mistakes and errors. So in a sense, a Geometry FTC proof is a exercise in Consistency of all of Mathematics. In order to prove a FTC geometry proof, requires throwing out the error filled mess of Old Math. Can the Reals be the true numbers of mathematics if the Reals cannot deliver a Geometry proof of FTC? Can the functions that are not polynomial functions allow us to give a Geometry proof of FTC? Can a Coordinate System in 2D have 4 quadrants and still give a Geometry proof of FTC? Can a equation of mathematics with a number that is _not a positive decimal Grid Number_ all alone on the right side of the equation, at all times, allow us to give a Geometry proof of the FTC?
Cover Picture: Is my hand written, one page geometry proof of the Fundamental Theorem of Calculus, the world's first geometry proof of FTC, 2013-2015, by AP.
Length: 137 pages
Product details
ASIN : B07PQTNHMY
Publication date : March 14, 2019
Language : English
File size : 1307 KB
Text-to-Speech : Enabled
Screen Reader : Supported
Enhanced typesetting : Enabled
X-Ray : Not Enabled
Word Wise : Not Enabled
Print length : 137 pages
Lending : Enabled
Amazon Best Sellers Rank: #128,729 Paid in Kindle Store (See Top 100 Paid in Kindle Store)
#2 in 45-Minute Science & Math Short Reads
#134 in Calculus (Books)
#20 in Calculus (Kindle Store)
On Saturday, July 17, 2021 at 11:20:58 AM UTC-5, Archimedes Plutonium wrote:
> Dr. Panchanathan's NSF govt spammer Kibo Parry M on why Weinberg, Glashow, Higgs fail physics with their never asking the question which is the atom's true real electron? Is it the muon, stuck inside a proton torus of 840MeV doing the Faraday law and 0.5MeV particle is Dirac's magnetic monopole?
> On Saturday, July 17, 2021 at 8:36:13 AM UTC-5, Michael Moroney wrote:
> > fails at math and science:
> HISTORY OF THE PROTON MASS and the 945 MeV //Atom Totality series, book 3 Kindle Edition
> by Archimedes Plutonium (Author)
>
> In 2016-2017, AP discovered that the real proton has a mass of 840 MeV, not 938. The real electron was actually the muon and the muon stays inside the proton that forms a proton torus of 8 rings and with the muon as bar magnet is a Faraday Law producing magnetic monopoles. So this book is all about why researchers of physics and engineers keep getting the number 938MeV when they should be getting the number 840 MeV + 105 MeV = 945 MeV.
>
> Cover Picture is a proton torus of 8 rings with a muon of 1 ring inside the proton torus, doing the Faraday Law and producing magnetic monopoles.
> Length: 17 pages
>
> Product details
> • Publication Date : December 18, 2019
> • Word Wise : Enabled
> • Print Length : 17 pages
> • File Size : 698 KB
> • ASIN : B082WYGVNG
> • Language: : English
> • Text-to-Speech : Not enabled
> • Enhanced Typesetting : Enabled
> • Screen Reader : Supported
> • X-Ray : Not Enabled
> • Lending : Enabled
>
> #1-4, 105th published book
>
> Atom Geometry is Torus Geometry // Atom Totality series, book 4 Kindle Edition
> by Archimedes Plutonium (Author)
>
> Since all atoms are doing the Faraday Law inside them, of their thrusting muon into a proton coil in the shape of a geometry torus, then the torus is the geometry of each and every atom. But then we must explain the neutrons since the muon and proton are doing Faraday's Law, then the neutron needs to be explained in terms of this proton torus with muon inside, all three shaped as rings. The muon is a single ring and each proton is 8 rings. The neutron is shaped like a plate and is solid not hollow. The explanation of a neutron is that of a capacitor storing what the proton-muon rings produce in electricity. Where would the neutron parallel plates be located? I argue in this text that the neutron plates when fully grown from 1 eV until 945MeV are like two parallel plate capacitors where each neutron is part of one plate, like two pieces of bread with the proton-muon torus being a hamburger patty.
>
> Cover Picture: I assembled two atoms in this picture where the proton torus with a band of muons inside traveling around and around the proton torus producing electricity. And the pie-plates represent neutrons as parallel-plate capacitors.
> Length: 39 pages
>
> Product details
> • Publication Date : March 24, 2020
> • Word Wise : Not Enabled
> • ASIN : B086BGSNXN
> • Print Length : 39 pages
> • File Size : 935 KB
> • Language: : English
> • Text-to-Speech : Not enabled
> • Screen Reader : Supported
> • X-Ray : Not Enabled
> • Enhanced Typesetting : Enabled
> • Lending : Enabled
> Amazon Best Sellers Rank: #1,656,820 Paid in Kindle Store (See Top 100 Paid in Kindle Store)
> #6413 in Mathematics (Kindle Store)
> #315 in One-Hour Science & Math Short Reads
> #4953 in Physics (Kindle Store)
>
>
> #1-5, 112th published book
>
> New Perspective on Psi^2 in the Schrodinger Equation in a Atom Totality Universe// Atom Totality series, book 5
> Kindle Edition
> by Archimedes Plutonium (Author)
>
> I first heard of the Schrodinger equation in college chemistry class. We never actually did any problem solving with the equation, and we were only told about it. Then taking physics my next year in college and after I bought the Feynman Lectures on Physics, just for fun for side reading, three volume set did I learn what this Schrodinger equation and the Psi^2 wavefunction was about. I am not going to teach the mathematics of the Schrodinger equation and the math calculations of the Psi or Psi^2 in this book, but leave that up to the reader or student to do that from Feynman's Lectures on Physics. The purpose of this book is to give a new and different interpretation of what Psi^2 is, what Psi^2 means. Correct interpretation of physics experiments and observations turns out to be one of the most difficult tasks in all of physics.
>
> Cover Picture: a photograph taken of me in 1993, after the discovery of Plutonium Atom Totality, and I was 43 years old then, on a wintery hill of New Hampshire. It is nice that Feynman wrote a physics textbook series, for I am very much benefitting from his wisdom. If he had not done that, getting organized in physics by writing textbooks, I would not be writing this book. And I would not have discovered the true meaning of the Fine Structure Constant, for it was Feynman who showed us that FSC is really 0.0854, not that of 0.0072. All because 0.0854 is Psi, and Psi^2 is 0.0072.
> Length: 20 pages
>
> Product details
> • ASIN : B0875SVDC7
> • Publication date : April 15, 2020
> • Language: : English
> • File size : 1134 KB
> • Text-to-Speech : Enabled
> • Screen Reader : Supported
> • Enhanced typesetting : Enabled
> • X-Ray : Not Enabled
> • Word Wise : Enabled
> • Print length : 20 pages
> • Lending : Enabled
> • Best Sellers Rank: #240,066 in Kindle Store (See Top 100 in Kindle Store)
> ◦ #5 in 30-Minute Science & Math Short Reads
> ◦ #65 in General Chemistry & Reference
> ◦ #481 in Physics (Kindle Store)
>
> #1-6, 135th published book
>
> QED in Atom Totality theory where proton is a 8 ring torus and electron = muon inside proton doing Faraday Law// Atom Totality series, book 6 Kindle Edition
> by Archimedes Plutonium (Author)
> Since the real true electron of atoms is the muon and is a one ring bar magnet thrusting through the 8 ring torus of a proton, we need a whole entire new model of the hydrogen atom. Because the Bohr model with the 0.5MeV particle jumping orbitals as the explanation of Spectral Lines is all wrong. In this vacuum of explaining spectral line physics, comes the AP Model which simply states that the hydrogen atom creates Spectral lines because at any one instant of time 4 of the 8 proton rings is "in view" and the electricity coming from those 4 view rings creates spectral line physics.
>
> Cover Picture: Is a imitation of the 8 ring proton torus, with my fingers holding on the proton ring that has the muon ring perpendicular and in the equatorial plane of the proton rings, thrusting through. This muon ring is the same size as the 8 proton rings making 9 x 105MeV = 945MeV of energy. The muon ring has to be perpendicular and lie on the equator of the proton torus. Surrounding the proton-torus would be neutrons as skin or coating cover and act as capacitors in storing the electricity produced by the proton+muon.
>
>
> Product details
> • ASIN : B08K47K5BB
> • Publication date : September 25, 2020
> • Language : English
> • File size : 587 KB
> • Text-to-Speech : Enabled
> • Screen Reader : Supported
> • Enhanced typesetting : Enabled
> • X-Ray : Not Enabled
> • Word Wise : Not Enabled
> • Print length : 25 pages
> • Lending : Enabled
> • Best Sellers Rank: #291,001 in Kindle Store (See Top 100 in Kindle Store)
> ◦ #13 in 45-Minute Science & Math Short Reads
> ◦ #52 in General Chemistry & Reference
> ◦ #334 in General Chemistry
>
>
>
> #1-7, 138th published book
> The true NUCLEUS of Atoms are inner toruses moving around in circles of a larger outer torus// Rutherford, Geiger, Marsden Experiment revisited // Atom Totality Series, book 7 Kindle Edition
> by Archimedes Plutonium (Author)
>
> The geometry of Atoms of the Table of Chemical Elements is torus geometry. We know this to be true for the torus geometry forms the maximum electricity production when using the Faraday Law. We see this in Old Physics with their tokamak toruses attempting to make fusion, by accelerating particles of the highest possible acceleration for the torus is that geometry. But the torus is the geometry not only of maximum acceleration but of maximum electrical generation by having a speeding bar magnet go around and around inside a torus== the Faraday law, where the torus rings are the copper closed wire loop. The protons of atoms are 8 loops of rings in a torus geometry, and the electron of atoms is the muon as bar magnet, almost the same size as the proton loops but small enough to fit inside proton loops. It is torus geometry that we investigate the geometry of all atoms.
> Length: 41 pages
>
> Product details
> • Publication Date : October 9, 2020
> • File Size : 828 KB
> • Word Wise : Not Enabled
> • Print Length : 41 pages
> • ASIN : B08KZT5TCD
> • Language: : English
> • Text-to-Speech : Not enabled
> • Enhanced Typesetting : Enabled
> • Screen Reader : Supported
> • X-Ray : Not Enabled
> • Lending : Enabled
>
> #1-8, 1st published book
>
> Atom Totality Universe, 8th edition, 2017// A history log book: Atom Totality Series book 8 Kindle Edition
> by Archimedes Plutonium (Author)
>
>
> Last revision 7Apr2021. This was AP's first published science book.
>
> Advisory: This is a difficult book to read and is AP's research log book of the Atom Totality in 2016-2017. I want to keep it for its history value. AP advises all readers wanting to know the Plutonium Atom Totality theory to go to the 9th edition that is the latest up to date account of this theory. The reason AP wants to keep the 8th edition is because of Historical Value, for in this book, while writing it, caused the discovery of the real electron is the muon of atoms. The real proton of atoms is 840MeV and not the 938MeV that most books claim. The particle discovered by JJ Thomson in 1897 thinking he discovered the electron of atoms was actually the Dirac magnetic monopole at 0.5MeV. This discovery changes every, every science that uses atoms and electricity and magnetism, in other words, every science.
>
> Foreward:
> I wrote the 8th edition of Atom Totality and near the end of writing it in 2017, I had my second greatest physics discovery. I learned the real electron of atoms was the muon at 105MeV and not the tiny 0.5MeV particle that J.J.Thomson found in 1897. So I desperately tried to include that discovery in my 8th edition and it is quite plain to see for I tried to write paragraphs after each chapter saying as much. I knew in 2017, that it was a great discovery, changing all the hard sciences, and reframing and restructuring all the hard sciences.
> Length: 632 pages
>
>
> Product details
> File Size: 1132 KB
> Print Length: 632 pages
> Publication Date: March 11, 2019
> Sold by: Amazon Digital Services LLC
> Language: English
> ASIN: B07PLP9NDR
> Text-to-Speech: Enabled
> X-Ray: Not Enabled
> Word Wise: Enabled
> Lending: Enabled
> Screen Reader: Supported
> Enhanced Typesetting: Enabled
> Amazon Best Sellers Rank: #578,229 Paid in Kindle Store (See Top 100 Paid in Kindle Store)
> #1610 in Physics (Kindle Store)
> #8526 in Physics (Books)
> #18851 in Biological Sciences (Books)
>
> #2-1, 137th published book
>
> Introduction to AP's TEACHING TRUE PHYSICS// Physics textbook series, book 1 Kindle Edition
> by Archimedes Plutonium (Author)
>
>
>
> #1 New Release in Electromagnetic Theory
>
> This will be AP's 137th published book on science. And the number 137 is special to me for it is the number of QED, Quantum Electrodynamics as the inverse fine structure constant. I can always remember 137 as that special constant of physics and so I can remember where Teaching True Physics was started by me.
>
> Time has come for the world to have the authoritative textbooks for all of High School and College education. Written by the leading physics expert of the time. The last such was Feynman in the 1960s with Feynman Lectures on Physics. The time before was Maxwell in 1860s with his books and Encyclopedia Britannica editorship. The time is ripe in 2020 for the new authoritative texts on physics. It will be started in 2020 which is 60 years after Feynman. In the future, I request the physics community updates the premier physics textbook series at least every 30 years. For we can see that pattern of 30 years approximately from Faraday in 1830 to Maxwell in 1860 to Planck and Rutherford in about 1900, to Dirac in 1930 to Feynman in 1960 and finally to AP in 1990 and 2020. So much happens in physics after 30 years, that we need the revisions to take place in a timely manner. But also, as we move to Internet publishing such as Amazon's Kindle, we can see that updates can take place very fast, as editing can be a ongoing monthly or yearly activity. I for one keep constantly updating all my published books, at least I try to.
>
> Feynman was the best to make the last authoritative textbook series for his concentration was QED, Quantum Electrodynamics, the pinnacle peak of physics during the 20th century. Of course the Atom Totality theory took over after 1990 and all of physics; for all sciences are under the Atom Totality theory.
> And as QED was the pinnacle peak before 1990, the new pinnacle peak is the Atom Totality theory. The Atom Totality theory is the advancement of QED, for the Atom Totality theory primal axiom says -- All is Atom, and atoms are nothing but Electricity and Magnetism.
> Length: 64 pages
>
> Product details
> • File Size : 790 KB
> • Publication Date : October 5, 2020
> • Word Wise : Enabled
> • Print Length : 64 pages
> • Text-to-Speech : Not enabled
> • Screen Reader : Supported
> • Enhanced Typesetting : Enabled
> • X-Ray : Not Enabled
> • Language: : English
> • ASIN : B08KS4YGWY
> • Lending : Enabled
> • Best Sellers Rank: #430,602 in Kindle Store (See Top 100 in Kindle Store)
> ◦ #39 in Electromagnetic Theory
> ◦ #73 in Electromagnetism (Kindle Store)
> ◦ #74 in 90-Minute Science & Math Short Reads
>
> #2-2, 145th published book
>
>
> TEACHING TRUE PHYSICS//Junior High School// Physics textbook series, book 2
> Kindle Edition
> by Archimedes Plutonium (Author)
>
> What I am doing is clearing the field of physics, clearing it of all the silly mistakes and errors and beliefs that clutter up physics. Clearing it of its fraud and fakeries and con-artistry. I thought of doing these textbooks starting with Senior year High School, wherein I myself started learning physics. But because of so much fraud and fakery in physics education, I believe we have to drop down to Junior year High School to make a drastic and dramatic emphasis on fakery and con-artistry that so much pervades science and physics in particular. So that we have two years in High School to learn physics. And discard the nonsense of physics brainwash that Old Physics filled the halls and corridors of education.
>
> Product details
> • ASIN : B08PC99JJB
> • Publication date : November 29, 2020
> • Language: : English
> • File size : 682 KB
> • Text-to-Speech : Enabled
> • Screen Reader : Supported
> • Enhanced typesetting : Enabled
> • X-Ray : Not Enabled
> • Word Wise : Enabled
> • Print length : 78 pages
> • Lending : Enabled
> • Best Sellers Rank: #185,995 in Kindle Store (See Top 100 in Kindle Store)
> ◦ #42 in Two-Hour Science & Math Short Reads
> ◦ #344 in Physics (Kindle Store)
> ◦ #2,160 in Physics (Books)
>
> #2-3, 146th published book
>
> TEACHING TRUE PHYSICS// Senior High School// Physics textbook series, book 3
> Kindle Edition
> by Archimedes Plutonium (Author)
>
> Books in this series are.
> Introduction to AP's TEACHING TRUE PHYSICS// Physics textbook series, book 1
> TEACHING TRUE PHYSICS High School junior year, book 2
> TEACHING TRUE PHYSICS High School senior year, book 3
> TEACHING TRUE PHYSICS 1st year college, book 4
> TEACHING TRUE PHYSICS Sophomore college, book 5
> TEACHING TRUE PHYSICS Junior college, book 6
> TEACHING TRUE PHYSICS Senior college, book 7
>
> Preface: I believe that in knowing the history of a science is knowing half of that science. And that if you are amiss of knowing the history behind a science, you have only a partial understanding of the concepts and ideas behind the science. I further believe it is easier to teach a science by teaching its history than any other means of teaching. So for senior year High School, I believe physics history is the best way of teaching physics. And in later years of physics courses, we can always pick up on details. So I devote this senior year High School physics to a history of physics, but only true physics. And there are few books written on the history of physics, so I chose Asimov's The History of Physics, 1966 as the template book for this textbook.
>
> Product details
> • ASIN : B08RK33T8V
> • Publication date : December 28, 2020
> • Language: : English
> • File size : 917 KB
> • Text-to-Speech : Enabled
> • Screen Reader : Supported
> • Enhanced typesetting : Enabled
> • X-Ray : Not Enabled
> • Word Wise : Enabled
> • Print length : 114 pages
> • Lending : Enabled
>
>
> #3-1, 2nd published book
>
> True Chemistry: Chemistry Series, book 1 Kindle Edition
> by Archimedes Plutonium (Author)
>
> Physics and chemistry made a mistake in 1897 for they thought that J.J. Thomson's small particle of 0.5MeV was the electron of atoms. By 2017, Archimedes Plutonium discovered that the rest mass of 940 for neutron and proton was really 9 x 105MeV with a small sigma-error. Meaning that the real proton is 840MeV, real electron is 105 MeV= muon, and that little particle Thomson discovered was in fact the Dirac magnetic monopole. Dirac circa 1930s was looking for a magnetic monopole, and sadly, Dirac passed away before 2017, because if he had lived to 2017, he would have seen his long sought for magnetic monopole which is everywhere.
>
> Cover picture: shows 3 isomers of CO2 and the O2 molecule.
>
> Length: 1150 pages
>
>
> Product details
> • File Size : 2167 KB
> • ASIN : B07PLVMMSZ
> • Publication Date : March 11, 2019
> • Word Wise : Enabled
> • Print Length : 1150 pages
> • Language: : English
> • Text-to-Speech : Not enabled
> • Enhanced Typesetting : Enabled
> • X-Ray : Not Enabled
> • Lending : Enabled
> Amazon Best Sellers Rank: #590,212 Paid in Kindle Store (See Top 100 Paid in Kindle Store)
> #181 in General Chemistry & Reference
> #1324 in General Chemistry
> #1656 in Physics (Kindle Store)
y z
| /
| /
|/______ x
More people reading and viewing AP's newsgroup than viewing sci.math, sci.physics. So AP has decided to put all NEW WORK, to his newsgroup. And there is little wonder because in AP's newsgroups, there is only solid pure science going on, not a gang of hate spewing misfits blighting the skies.
In sci.math, sci.physics there is only stalking hate spew along with Police Drag Net Spam of no value and other than hate spew there is Police drag net spam day and night.
I re-opened the old newsgroup PAU of 1990s and there one can read my recent posts without the hassle of stalkers and spammers, Police Drag Net Spam that floods each and every day, book and solution manual spammers, off-topic-misfits, front-page-hogs, churning imbeciles, stalking mockers, suppression-bullies, and demonizers. And the taxpayer funded hate spew stalkers who ad hominem you day and night on every one of your posts.
There is no discussion of science in sci.math or sci.physics, just one long line of hate spewing stalkers followed up with Police Drag Net Spam (easy to spot-- very offtopic-- with hate charged content). And countries using sci.physics & sci.math as propaganda platforms, such as tampering in elections with their mind-rot.
Read my recent posts in peace and quiet.
https://groups.google.com/forum/?hl=en#!forum/plutonium-atom-universe
Archimedes Plutonium
Archimedes Plutonium's profile photo
Archimedes Plutonium
10:14 PM (2 minutes ago)
to
Stalker Kibo Analbuttfuckmanure F. Fleming Crim, Dorothy E Aronson, Brian Stone, James S Ulvestad, Rebecca Lynn Keiser, Vernon D. Ross, Lloyd Whitman, John J. Veysey, Scott Stanley, Sethuraman Panchanathan are you paying Kibo to stalk sci.math,
In the 1990s Univ Waterloo (correct me if wrong) hosted the archives of Usenet sci.math, sci.physics. And almost daily they posted a FAQ. In the FAQ was a broad warning to students that all manner of posts are in sci.math and sci.physics and that you should examine each post with a "grain of salt" or be highly speculative of what you read. Then around 1993, started this Stalking program of jerks being paid by governments to perpetually stalk posters. Then by end of 1990s, you never saw the FAQ, never saw the warnings. But all that remained was hyperactive goon clod paid stalkers.
It was a bad choice to make, to stop the FAQ and replace it with paid goon-squad stalkers like Kibo Parry Moroney or Dan Christensen and their allies. Bad choice because everyone can recognize cranks and crackpots, but hardly anyone can recognize people who think outside of the box.
How much is Dr. Panchanathan responsible for the 28 year long nonstop goon stalker Kibo Parry Moroney? Perhaps 0%, perhaps 50% perhaps 100%. And this is probably the reason the Wikipedia quote says-- > Many government and university installations blocked, threatened to block, or attempted to shut-down The World's Internet connection until Software Tool & Die was eventually granted permission by the National Science Foundation to provide public Internet access on "an experimental basis."
> --- end quote ---
I remember the 1990s and many a professor and educator bulking at this idea of paying stalkers like Kibo Parry or like Dan Christensen, some saying that if students cannot recognize cranks and crackpots they never belonged in science in the first place.
So, what worked well was the idea of a Daily FAQ doing the warning, a daily FAQ warning newcomers and students that much of what they read here is full of error. But not the crazy idea of paid for stalkers, because, as I said, a stalker cannot identify a real scientist who is thinking outside the box.
And one of the great reasons Usenet is mostly dead-- is because it is crisscrossed by mindless stalkers being paid handsomely for their banal stalking posts.
So, well, if you want to constantly stalk AP, then you can anticipate a constant reply of below.
Jan Burse complaining no-one at ETH asks the question, which is atom's real electron, is it the muon stuck inside a 840MeV proton torus doing the Faraday, or, is it the 0.5MeV particle that AP calls the Dirac magnetic monopole. (See books below).
On Tuesday, August 3, 2021 at 11:33:50 AM UTC-5, burs...@gmail.com wrote:
>brain farto, still up to stealing everybodies time.
HISTORY OF THE PROTON MASS and the 945 MeV //Atom Totality series, book 3 Kindle Edition
by Archimedes Plutonium (Author)
In 2016-2017, AP discovered that the real proton has a mass of 840 MeV, not 938. The real electron was actually the muon and the muon stays inside the proton that forms a proton torus of 8 rings and with the muon as bar magnet is a Faraday Law producing magnetic monopoles. So this book is all about why researchers of physics and engineers keep getting the number 938MeV when they should be getting the number 840 MeV + 105 MeV = 945 MeV.
Cover Picture is a proton torus of 8 rings with a muon of 1 ring inside the proton torus, doing the Faraday Law and producing magnetic monopoles.
Length: 17 pages
Product details
• Publication Date : December 18, 2019
• Word Wise : Enabled
• Print Length : 17 pages
• File Size : 698 KB
• ASIN : B082WYGVNG
• Language: : English
• Text-to-Speech : Not enabled
• Enhanced Typesetting : Enabled
• Screen Reader : Supported
• X-Ray : Not Enabled
• Lending : Enabled
#1-4, 105th published book
Atom Geometry is Torus Geometry // Atom Totality series, book 4 Kindle Edition
by Archimedes Plutonium (Author)
Since all atoms are doing the Faraday Law inside them, of their thrusting muon into a proton coil in the shape of a geometry torus, then the torus is the geometry of each and every atom. But then we must explain the neutrons since the muon and proton are doing Faraday's Law, then the neutron needs to be explained in terms of this proton torus with muon inside, all three shaped as rings. The muon is a single ring and each proton is 8 rings. The neutron is shaped like a plate and is solid not hollow. The explanation of a neutron is that of a capacitor storing what the proton-muon rings produce in electricity. Where would the neutron parallel plates be located? I argue in this text that the neutron plates when fully grown from 1 eV until 945MeV are like two parallel plate capacitors where each neutron is part of one plate, like two pieces of bread with the proton-muon torus being a hamburger patty.
Cover Picture: I assembled two atoms in this picture where the proton torus with a band of muons inside traveling around and around the proton torus producing electricity. And the pie-plates represent neutrons as parallel-plate capacitors.
Length: 39 pages
Product details
• Publication Date : March 24, 2020
• Word Wise : Not Enabled
• ASIN : B086BGSNXN
• Print Length : 39 pages
• File Size : 935 KB
• Language: : English
• Text-to-Speech : Not enabled
• Screen Reader : Supported
• X-Ray : Not Enabled
• Enhanced Typesetting : Enabled
• Lending : Enabled
Amazon Best Sellers Rank: #1,656,820 Paid in Kindle Store (See Top 100 Paid in Kindle Store)
#6413 in Mathematics (Kindle Store)
#315 in One-Hour Science & Math Short Reads
#4953 in Physics (Kindle Store)
#1-5, 112th published book
New Perspective on Psi^2 in the Schrodinger Equation in a Atom Totality Universe// Atom Totality series, book 5
Kindle Edition
by Archimedes Plutonium (Author)
I first heard of the Schrodinger equation in college chemistry class. We never actually did any problem solving with the equation, and we were only told about it. Then taking physics my next year in college and after I bought the Feynman Lectures on Physics, just for fun for side reading, three volume set did I learn what this Schrodinger equation and the Psi^2 wavefunction was about. I am not going to teach the mathematics of the Schrodinger equation and the math calculations of the Psi or Psi^2 in this book, but leave that up to the reader or student to do that from Feynman's Lectures on Physics. The purpose of this book is to give a new and different interpretation of what Psi^2 is, what Psi^2 means. Correct interpretation of physics experiments and observations turns out to be one of the most difficult tasks in all of physics.
Cover Picture: a photograph taken of me in 1993, after the discovery of Plutonium Atom Totality, and I was 43 years old then, on a wintery hill of New Hampshire. It is nice that Feynman wrote a physics textbook series, for I am very much benefitting from his wisdom. If he had not done that, getting organized in physics by writing textbooks, I would not be writing this book. And I would not have discovered the true meaning of the Fine Structure Constant, for it was Feynman who showed us that FSC is really 0.0854, not that of 0.0072. All because 0.0854 is Psi, and Psi^2 is 0.0072.
Length: 20 pages
Product details
• ASIN : B0875SVDC7
• Publication date : April 15, 2020
• Language: : English
• File size : 1134 KB
• Text-to-Speech : Enabled
• Screen Reader : Supported
• Enhanced typesetting : Enabled
• X-Ray : Not Enabled
• Word Wise : Enabled
• Print length : 20 pages
• Lending : Enabled
• Best Sellers Rank: #240,066 in Kindle Store (See Top 100 in Kindle Store)
◦ #5 in 30-Minute Science & Math Short Reads
◦ #65 in General Chemistry & Reference
◦ #481 in Physics (Kindle Store)
#1-6, 135th published book
QED in Atom Totality theory where proton is a 8 ring torus and electron = muon inside proton doing Faraday Law// Atom Totality series, book 6 Kindle Edition
by Archimedes Plutonium (Author)
Since the real true electron of atoms is the muon and is a one ring bar magnet thrusting through the 8 ring torus of a proton, we need a whole entire new model of the hydrogen atom. Because the Bohr model with the 0.5MeV particle jumping orbitals as the explanation of Spectral Lines is all wrong. In this vacuum of explaining spectral line physics, comes the AP Model which simply states that the hydrogen atom creates Spectral lines because at any one instant of time 4 of the 8 proton rings is "in view" and the electricity coming from those 4 view rings creates spectral line physics.
Cover Picture: Is a imitation of the 8 ring proton torus, with my fingers holding on the proton ring that has the muon ring perpendicular and in the equatorial plane of the proton rings, thrusting through. This muon ring is the same size as the 8 proton rings making 9 x 105MeV = 945MeV of energy. The muon ring has to be perpendicular and lie on the equator of the proton torus. Surrounding the proton-torus would be neutrons as skin or coating cover and act as capacitors in storing the electricity produced by the proton+muon.
Product details
• ASIN : B08K47K5BB
• Publication date : September 25, 2020
• Language : English
• File size : 587 KB
• Text-to-Speech : Enabled
• Screen Reader : Supported
• Enhanced typesetting : Enabled
• X-Ray : Not Enabled
• Word Wise : Not Enabled
• Print length : 25 pages
• Lending : Enabled
• Best Sellers Rank: #291,001 in Kindle Store (See Top 100 in Kindle Store)
◦ #13 in 45-Minute Science & Math Short Reads
◦ #52 in General Chemistry & Reference
◦ #334 in General Chemistry
#1-7, 138th published book
The true NUCLEUS of Atoms are inner toruses moving around in circles of a larger outer torus// Rutherford, Geiger, Marsden Experiment revisited // Atom Totality Series, book 7 Kindle Edition
by Archimedes Plutonium (Author)
The geometry of Atoms of the Table of Chemical Elements is torus geometry. We know this to be true for the torus geometry forms the maximum electricity production when using the Faraday Law. We see this in Old Physics with their tokamak toruses attempting to make fusion, by accelerating particles of the highest possible acceleration for the torus is that geometry. But the torus is the geometry not only of maximum acceleration but of maximum electrical generation by having a speeding bar magnet go around and around inside a torus== the Faraday law, where the torus rings are the copper closed wire loop. The protons of atoms are 8 loops of rings in a torus geometry, and the electron of atoms is the muon as bar magnet, almost the same size as the proton loops but small enough to fit inside proton loops. It is torus geometry that we investigate the geometry of all atoms.
Length: 41 pages
Product details
• Publication Date : October 9, 2020
• File Size : 828 KB
• Word Wise : Not Enabled
• Print Length : 41 pages
• ASIN : B08KZT5TCD
• Language: : English
• Text-to-Speech : Not enabled
• Enhanced Typesetting : Enabled
• Screen Reader : Supported
• X-Ray : Not Enabled
• Lending : Enabled
#1-8, 1st published book
Atom Totality Universe, 8th edition, 2017// A history log book: Atom Totality Series book 8 Kindle Edition
by Archimedes Plutonium (Author)
Last revision 7Apr2021. This was AP's first published science book.
Advisory: This is a difficult book to read and is AP's research log book of the Atom Totality in 2016-2017. I want to keep it for its history value. AP advises all readers wanting to know the Plutonium Atom Totality theory to go to the 9th edition that is the latest up to date account of this theory. The reason AP wants to keep the 8th edition is because of Historical Value, for in this book, while writing it, caused the discovery of the real electron is the muon of atoms. The real proton of atoms is 840MeV and not the 938MeV that most books claim. The particle discovered by JJ Thomson in 1897 thinking he discovered the electron of atoms was actually the Dirac magnetic monopole at 0.5MeV. This discovery changes every, every science that uses atoms and electricity and magnetism, in other words, every science.
Foreward:
I wrote the 8th edition of Atom Totality and near the end of writing it in 2017, I had my second greatest physics discovery. I learned the real electron of atoms was the muon at 105MeV and not the tiny 0.5MeV particle that J.J.Thomson found in 1897. So I desperately tried to include that discovery in my 8th edition and it is quite plain to see for I tried to write paragraphs after each chapter saying as much. I knew in 2017, that it was a great discovery, changing all the hard sciences, and reframing and restructuring all the hard sciences.
Length: 632 pages
Product details
File Size: 1132 KB
Print Length: 632 pages
Publication Date: March 11, 2019
Sold by: Amazon Digital Services LLC
Language: English
ASIN: B07PLP9NDR
Text-to-Speech: Enabled 
X-Ray: Not Enabled 
Word Wise: Enabled
Lending: Enabled
Screen Reader: Supported 
Enhanced Typesetting: Enabled 
Amazon Best Sellers Rank: #578,229 Paid in Kindle Store (See Top 100 Paid in Kindle Store)
#1610 in Physics (Kindle Store)
#8526 in Physics (Books)
#18851 in Biological Sciences (Books)
#2-1, 137th published book
Introduction to AP's TEACHING TRUE PHYSICS// Physics textbook series, book 1 Kindle Edition
by Archimedes Plutonium (Author)
#1 New Release in Electromagnetic Theory
This will be AP's 137th published book on science. And the number 137 is special to me for it is the number of QED, Quantum Electrodynamics as the inverse fine structure constant. I can always remember 137 as that special constant of physics and so I can remember where Teaching True Physics was started by me.
Time has come for the world to have the authoritative textbooks for all of High School and College education. Written by the leading physics expert of the time. The last such was Feynman in the 1960s with Feynman Lectures on Physics. The time before was Maxwell in 1860s with his books and Encyclopedia Britannica editorship. The time is ripe in 2020 for the new authoritative texts on physics. It will be started in 2020 which is 60 years after Feynman. In the future, I request the physics community updates the premier physics textbook series at least every 30 years. For we can see that pattern of 30 years approximately from Faraday in 1830 to Maxwell in 1860 to Planck and Rutherford in about 1900, to Dirac in 1930 to Feynman in 1960 and finally to AP in 1990 and 2020. So much happens in physics after 30 years, that we need the revisions to take place in a timely manner. But also, as we move to Internet publishing such as Amazon's Kindle, we can see that updates can take place very fast, as editing can be a ongoing monthly or yearly activity. I for one keep constantly updating all my published books, at least I try to.
Feynman was the best to make the last authoritative textbook series for his concentration was QED, Quantum Electrodynamics, the pinnacle peak of physics during the 20th century. Of course the Atom Totality theory took over after 1990 and all of physics; for all sciences are under the Atom Totality theory.
And as QED was the pinnacle peak before 1990, the new pinnacle peak is the Atom Totality theory. The Atom Totality theory is the advancement of QED, for the Atom Totality theory primal axiom says -- All is Atom, and atoms are nothing but Electricity and Magnetism.
Length: 64 pages
Product details
• File Size : 790 KB
• Publication Date : October 5, 2020
• Word Wise : Enabled
• Print Length : 64 pages
• Text-to-Speech : Not enabled
• Screen Reader : Supported
• Enhanced Typesetting : Enabled
• X-Ray : Not Enabled
• Language: : English
• ASIN : B08KS4YGWY
• Lending : Enabled
• Best Sellers Rank: #430,602 in Kindle Store (See Top 100 in Kindle Store)
◦ #39 in Electromagnetic Theory
◦ #73 in Electromagnetism (Kindle Store)
◦ #74 in 90-Minute Science & Math Short Reads
#2-2, 145th published book
TEACHING TRUE PHYSICS//Junior High School// Physics textbook series, book 2
Kindle Edition
by Archimedes Plutonium (Author)
What I am doing is clearing the field of physics, clearing it of all the silly mistakes and errors and beliefs that clutter up physics. Clearing it of its fraud and fakeries and con-artistry. I thought of doing these textbooks starting with Senior year High School, wherein I myself started learning physics. But because of so much fraud and fakery in physics education, I believe we have to drop down to Junior year High School to make a drastic and dramatic emphasis on fakery and con-artistry that so much pervades science and physics in particular. So that we have two years in High School to learn physics. And discard the nonsense of physics brainwash that Old Physics filled the halls and corridors of education.
Product details
• ASIN : B08PC99JJB
• Publication date : November 29, 2020
• Language: : English
• File size : 682 KB
• Text-to-Speech : Enabled
• Screen Reader : Supported
• Enhanced typesetting : Enabled
• X-Ray : Not Enabled
• Word Wise : Enabled
• Print length : 78 pages
• Lending : Enabled
• Best Sellers Rank: #185,995 in Kindle Store (See Top 100 in Kindle Store)
◦ #42 in Two-Hour Science & Math Short Reads
◦ #344 in Physics (Kindle Store)
◦ #2,160 in Physics (Books)
#2-3, 146th published book
TEACHING TRUE PHYSICS// Senior High School// Physics textbook series, book 3
Kindle Edition
by Archimedes Plutonium (Author)
Books in this series are.
Introduction to AP's TEACHING TRUE PHYSICS// Physics textbook series, book 1
TEACHING TRUE PHYSICS High School junior year, book 2
TEACHING TRUE PHYSICS High School senior year, book 3
TEACHING TRUE PHYSICS 1st year college, book 4
TEACHING TRUE PHYSICS Sophomore college, book 5
TEACHING TRUE PHYSICS Junior college, book 6
TEACHING TRUE PHYSICS Senior college, book 7
Preface: I believe that in knowing the history of a science is knowing half of that science. And that if you are amiss of knowing the history behind a science, you have only a partial understanding of the concepts and ideas behind the science. I further believe it is easier to teach a science by teaching its history than any other means of teaching. So for senior year High School, I believe physics history is the best way of teaching physics. And in later years of physics courses, we can always pick up on details. So I devote this senior year High School physics to a history of physics, but only true physics. And there are few books written on the history of physics, so I chose Asimov's The History of Physics, 1966 as the template book for this textbook.
Product details
• ASIN : B08RK33T8V
• Publication date : December 28, 2020
• Language: : English
• File size : 917 KB
• Text-to-Speech : Enabled
• Screen Reader : Supported
• Enhanced typesetting : Enabled
• X-Ray : Not Enabled
• Word Wise : Enabled
• Print length : 114 pages
• Lending : Enabled
#3-1, 2nd published book
True Chemistry: Chemistry Series, book 1 Kindle Edition
by Archimedes Plutonium (Author)
Physics and chemistry made a mistake in 1897 for they thought that J.J. Thomson's small particle of 0.5MeV was the electron of atoms. By 2017, Archimedes Plutonium discovered that the rest mass of 940 for neutron and proton was really 9 x 105MeV with a small sigma-error. Meaning that the real proton is 840MeV, real electron is 105 MeV= muon, and that little particle Thomson discovered was in fact the Dirac magnetic monopole. Dirac circa 1930s was looking for a magnetic monopole, and sadly, Dirac passed away before 2017, because if he had lived to 2017, he would have seen his long sought for magnetic monopole which is everywhere.
Cover picture: shows 3 isomers of CO2 and the O2 molecule.
Length: 1150 pages
Product details
• File Size : 2167 KB
• ASIN : B07PLVMMSZ
• Publication Date : March 11, 2019
• Word Wise : Enabled
• Print Length : 1150 pages
• Language: : English
• Text-to-Speech : Not enabled
• Enhanced Typesetting : Enabled
• X-Ray : Not Enabled
• Lending : Enabled
Amazon Best Sellers Rank: #590,212 Paid in Kindle Store (See Top 100 Paid in Kindle Store)
#181 in General Chemistry & Reference
#1324 in General Chemistry
#1656 in Physics (Kindle Store)
Dan Christensen says he is a super rectum looking for victims James Leech, Arthur B. McDonald, Linda Hasenfratz, Rose M Patten, for none can do a geometry proof of Fundamental Theorem of Calculus and still teaching the mindless 2 OR 1 = 3 with AND as subtraction.
On Tuesday, August 3, 2021 at 9:39:06 AM UTC-5, Dan Christensen wrote:
>I am Super Rectum
> STUDENTS BEWARE: Don't be a victim of
On Friday, May 28, 2021 at 1:25:16 PM UTC-5, Quantum Bubbles wrote:
> "Are you the same Archimedes Plutonium being mentioned on this old blog post?:
Re: *Fire the entire Univ Western Ontario math dept/ still teaching that the contradictory sine graph as sinusoid when it is really semicircle
by Dan Christensen Nov 21, 2017,
Re: 81,045-Student victims of Rose M. Patten Univ Toronto from stalker Dan Christensen teaching 10 OR 2 = 12 with AND as subtraction, never a geometry proof of Fundamental Theorem of Calculus Univ Toronto, physics, Gordon F. West, Michael B. Walker
by Frank Cassa 12Apr2021 7:00 AM
Re: 77,233 Student victims of Lawrence Bacow's Harvard from stalker Kibo Parry Moroney with his 938 is 12% short 945, his 10 OR 4 = 14 with AND as subtraction, and his mindless belief real electron = 0.5MeV when true electron is muon
11:57 AM 10Apr2021
by Wayne Decarlo
Re: 7,744-Student victims of Linda Hasenfratz Univ Western Ontario from stalker Dan Christensen teaching 10 OR 2 = 12 with AND as subtraction, never a geometry proof of Fundamental Theorem of Calculus Chancellor Linda Hasenfratz President Alan Shepard
11:53 AM 10Apr2021
by Wayne Decarlo
Re: 102,852-Student victims of Dominic Barton, Univ Waterloo from stalker Dan Christensen teaching 10 OR 2 = 12 with AND as subtraction, never a geometry proof of Fundamental Theorem of Calculus Dominic Barton, President Feridun Hamdullahpur physics
by konyberg Apr 15, 2021, 3:09:41 PM
Re: 176,232-Student Victims of Michael Meighen McGill Univ by Dan Christensen teaching 10 OR 2 = 12 with AND as subtraction, never a geometry proof of Fundamental Theorem of Calculus... 0.5MeV electron when in truth it is the muon as the real electron
by Dan Christensen Jul 2, 2021, 9:47:42 AM
Re: 135,568 Student victims Queen's Univ. James Leech, Arthur B. McDonald by Dan Christensen teaching 10 OR 2 = 12 with AND as subtraction, never a geometry proof of Fundamental Theorem of Calculus-- his mindless electron =0.5MeV when real electron of
May 10, 2021
by Professor Wordsmith
Re: 135,566 Student victims Queen's Univ. James Leech, Arthur B. McDonald by Dan Christensen teaching 10 OR 2 = 12 with AND as subtraction, never a geometry proof of Fundamental Theorem of Calculus-- his mindless electron =0.5MeV when real electron o
May 10, 2021
by Michael Moroney
On Tuesday, August 3, 2021 at 2:40:20 AM UTC-5, Mostowski Collapse wrote:
> How about steeling everybodies time with posting nonsense.
>
> LoL
NSF fraud waste abuse of taxpayer money $100 per stalker post-- Champagne,Hewitt,Crago
On Saturday, July 31, kibo Parry M. stalked
NSF fraud waste abuse of taxpayer money $100 per stalker post--
USA--NSF Dr. Panchanathan, F. Fleming Crim, Dorothy E Aronson, Brian Stone, James S Ulvestad, Rebecca Lynn Keiser, Vernon D. Ross, Lloyd Whitman, John J. Veysey, Scott Stanley
USA NSF---Sethuraman Panchanathan, F. Fleming Crim, Dorothy E Aronson, Brian Stone, James S Ulvestad, Rebecca Lynn Keiser, Vernon D. Ross, Lloyd Whitman, John J. Veysey, Scott Stanley
USA dept Educ, Cindy Marten
Canada's NSF-- Francois-Philippe Champagne, Ted Hewitt, Martha Crago, Frederic Bouchard, Cinthia Duclos, Normand Labrie
---quoting Wikipedia ---
Controversy
Many government and university installations blocked, threatened to block, or attempted to shut-down The World's Internet connection until Software Tool & Die was eventually granted permission by the National Science Foundation to provide public Internet access on "an experimental basis."
--- end quote ---
NATIONAL SCIENCE FOUNDATION
Dr. Panchanathan , present day
France Anne Cordova
Subra Suresh
Arden Lee Bement Jr.
Rita R. Colwell
Neal Francis Lane
John Howard Gibbons 1993
Barry Shein, kibo parry std world
Jim Frost, Joe "Spike" Ilacqua
AP restores FAQs to sci.math, sci.physics instead of the God awful mistake of NSF hiring stalkers.
An easy solution, USA govt, eliminate the license of std World isp.
Stalker Kibo subhuman, Dr. Panchanathan, F. Fleming Crim, Dorothy E Aronson, Brian Stone, James S Ulvestad, Rebecca Lynn Keiser, Vernon D. Ross, Lloyd Whitman, John J. Veysey, Scott Stanley, are you paying Kibo to stalk sci.math, sci.physics??? The Wikipedia quote below indicates you are paying him to stalk.
On Wednesday, July 21, 2021 at 11:12:06 AM UTC-5, Michael Moroney wrote:
>"subhuman"
On Friday, July 9, 2021 at 10:01:43 AM UTC-5, Michael Moroney wrote:
>"Splitter-Splatter-Fart-Shittee"
On Sunday, July 18, 2021 at 6:16:15 AM UTC-5, Michael Moroney wrote:
>"AnalButtfuckManure"
On Wednesday, July 21, 2021 at 10:25:22 PM UTC-5, Michael Moroney wrote:
> fails at math and science:
If indeed, NSF Dr. Panchanathan hired the stalker Kibo Parry Moroney going back to 1993 and thereabouts, was a huge mistake, for Usenet is self policing. The govt hiring a paid stalker eventually turns to ruin, for although a nitwit like Kibo Parry can tell a crank and moron. The nitwit Kibo Parry cannot tell a genuine scientist doing science outside the box. So in this sense, Dr. Panchanathan should resign from the NSF.
Sethuraman Panchanathan, F. Fleming Crim, Dorothy E Aronson, Brian Stone, James S Ulvestad, Rebecca Lynn Keiser, Vernon D. Ross, Lloyd Whitman, John J. Veysey, Scott Stanley, are you paying Kibo to stalk sci.math, sci.physics???
Is NSF Dr. Panchanathan paying Kibo Parry M more money to stalk sci.math, sci.physics, than the top 5 officials at NSF combined and 5X the amount of pay of MIT professors who teach calculus in classrooms. Same question for Canada's stalker, how much pay from government for his bully tactics
Does NSF pay kibo Parry to stalk more than the combined salaries of Sethuraman Panchanathan, F. Fleming Crim, Dorothy E Aronson, Brian Stone, James S Ulvestad, Rebecca Lynn Keiser, Vernon D. Ross, Lloyd Whitman, John J. Veysey, Scott Stanley.
Kibo Parry M on Philip J. Hanlon Dartmouth College president
On Saturday, July 17, 2021 at 4:05:34 PM UTC-5, Michael Moroney wrote:
> 🐜 of Math and 🐛 of
World's First Geometry Proof of Fundamental Theorem of Calculus// Math proof series, book 2 Kindle Edition
by Archimedes Plutonium (Author)
Last revision was 19May2021. This is AP's 11th published book of science.
Preface:
Actually my title is too modest, for the proof that lies within this book makes it the World's First Valid Proof of Fundamental Theorem of Calculus, for in my modesty, I just wanted to emphasis that calculus was geometry and needed a geometry proof. Not being modest, there has never been a valid proof of FTC until AP's 2015 proof. This also implies that only a geometry proof of FTC constitutes a valid proof of FTC.
Calculus needs a geometry proof of Fundamental Theorem of Calculus. But none could ever be obtained in Old Math so long as they had a huge mass of mistakes, errors, fakes and con-artist trickery such as the "limit analysis". To give a Geometry Proof of Fundamental Theorem of Calculus requires math be cleaned-up and cleaned-out of most of math's mistakes and errors. So in a sense, a Geometry FTC proof is a exercise in Consistency of all of Mathematics. In order to prove a FTC geometry proof, requires throwing out the error filled mess of Old Math. Can the Reals be the true numbers of mathematics if the Reals cannot deliver a Geometry proof of FTC? Can the functions that are not polynomial functions allow us to give a Geometry proof of FTC? Can a Coordinate System in 2D have 4 quadrants and still give a Geometry proof of FTC? Can a equation of mathematics with a number that is _not a positive decimal Grid Number_ all alone on the right side of the equation, at all times, allow us to give a Geometry proof of the FTC?
Cover Picture: Is my hand written, one page geometry proof of the Fundamental Theorem of Calculus, the world's first geometry proof of FTC, 2013-2015, by AP.
Length: 137 pages
Product details
ASIN : B07PQTNHMY
Publication date : March 14, 2019
Language : English
File size : 1307 KB
Text-to-Speech : Enabled
Screen Reader : Supported
Enhanced typesetting : Enabled
X-Ray : Not Enabled
Word Wise : Not Enabled
Print length : 137 pages
Lending : Enabled
Amazon Best Sellers Rank: #128,729 Paid in Kindle Store (See Top 100 Paid in Kindle Store)
#2 in 45-Minute Science & Math Short Reads
#134 in Calculus (Books)
#20 in Calculus (Kindle Store)
On Saturday, July 17, 2021 at 11:20:58 AM UTC-5, Archimedes Plutonium wrote:
> Dr. Panchanathan's NSF govt spammer Kibo Parry M on why Weinberg, Glashow, Higgs fail physics with their never asking the question which is the atom's true real electron? Is it the muon, stuck inside a proton torus of 840MeV doing the Faraday law and 0.5MeV particle is Dirac's magnetic monopole?
> On Saturday, July 17, 2021 at 8:36:13 AM UTC-5, Michael Moroney wrote:
> > fails at math and science:
> Introduction to AP's TEACHING TRUE PHYSICS// Physics textbook series, book 1
y z
| /
| /
|/______ x
More people reading and viewing AP's newsgroup than viewing sci.math, sci.physics. So AP has decided to put all NEW WORK, to his newsgroup. And there is little wonder because in AP's newsgroups, there is only solid pure science going on, not a gang of hate spewing misfits blighting the skies.
In sci.math, sci.physics there is only stalking hate spew along with Police Drag Net Spam of no value and other than hate spew there is Police drag net spam day and night.
I re-opened the old newsgroup PAU of 1990s and there one can read my recent posts without the hassle of stalkers and spammers, Police Drag Net Spam that floods each and every day, book and solution manual spammers, off-topic-misfits, front-page-hogs, churning imbeciles, stalking mockers, suppression-bullies, and demonizers. And the taxpayer funded hate spew stalkers who ad hominem you day and night on every one of your posts.
There is no discussion of science in sci.math or sci.physics, just one long line of hate spewing stalkers followed up with Police Drag Net Spam (easy to spot-- very offtopic-- with hate charged content). And countries using sci.physics & sci.math as propaganda platforms, such as tampering in elections with their mind-rot.
Read my recent posts in peace and quiet.
https://groups.google.com/forum/?hl=en#!forum/plutonium-atom-universe
Archimedes Plutonium
In the 1990s Univ Waterloo (correct me if wrong) hosted the archives of Usenet sci.math, sci.physics. And almost daily they posted a FAQ. In the FAQ was a broad warning to students that all manner of posts are in sci.math and sci.physics and that you should examine each post with a "grain of salt" or be highly speculative of what you read. Then around 1993, started this Stalking program of jerks being paid by governments to perpetually stalk posters. Then by end of 1990s, you never saw the FAQ, never saw the warnings. But all that remained was hyperactive goon clod paid stalkers.
It was a bad choice to make, to stop the FAQ and replace it with paid goon-squad stalkers like Kibo Parry Moroney or Dan Christensen and their allies. Bad choice because everyone can recognize cranks and crackpots, but hardly anyone can recognize people who think outside of the box.
How much is Dr. Panchanathan responsible for the 28 year long nonstop goon stalker Kibo Parry Moroney? Perhaps 0%, perhaps 50% perhaps 100%. And this is probably the reason the Wikipedia quote says-- > Many government and university installations blocked, threatened to block, or attempted to shut-down The World's Internet connection until Software Tool & Die was eventually granted permission by the National Science Foundation to provide public Internet access on "an experimental basis."
> --- end quote ---
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1 INDEX OF THE TROELSTRA ARCHIVE Introduction This index was mainly prepared in 2000 <strong>and</strong> 2001. 1 <strong>The</strong> archive itself was stored at the Mathematical <strong>Institute</strong> of the University of Amsterdam, afterwards (2001) brought to the Rijksarchief in Noord-Holl<strong>and</strong>. <strong>The</strong> length of the archive is circa 2.5 meters. <strong>The</strong> material in the archive has been divided into a number of major categories, distinguished by capitals in the numbering of items, e.g. B (scientific correspondence), C (courses given by Troelstra, seminars <strong>and</strong> colloquia conducted by him), P (all materials relating to his publications) etc. Usually these major categories are again subdivided. <strong>The</strong> description of the contents is not carried out with the same amount of detail <strong>for</strong> each (sub-) category. A lot of material in this index was already set up by A.S. Troelstra himself (especially publications, lectures <strong>and</strong> congresses). P. van Ulsen, Amsterdam, 2001. Curriculum vitae of Anne Sjerp Troelstra 2 10-VIII-1939 Born at Maartensdijk (Utrecht), the Netherl<strong>and</strong>s. 1951-1957 Secondary school: Lorentz Lyceum (gymnasium beta), Eindhoven. 1-IX-1957 Enrollment as a student at the University of Amsterdam. 25-III-1964 Passed doctoraalexamen in Mathematics, cum laude. (At that time a bit more than a M.Sc.) 1-IV-1964 Appointed ‘wetenschappelijk medewerker’ (approximately, assistant professor) at the Department of Mathematics of the University of Amsterdam. 15-VI-1966 Doctorate (Ph.D) in mathematics on the thesis ‘Intuitionistic general topology’ (thesis adviser Prof. Dr. A. Heyting). University of Amsterdam. 1-IX-1966 till 1-IX-1967. On leave as a visiting scholar at Stan<strong>for</strong>d University (departments of Mathematics <strong>and</strong> Philosophy) with a stipend from the Netherl<strong>and</strong>s Organization <strong>for</strong> the Advancement of Research (then ZWO, now called NWO). VIII-1968 Gave a series of ten lectures on Intuitionism at the Summer School on Proof <strong>The</strong>ory <strong>and</strong> Intuitionism at SUNY, Buffalo, New York. 1-IX-1968 Appointed ‘lector’ (associate professor) in mathematics at the University of Amsterdam. 1-IX-1970 Appointed ‘gewoon hoogleraar’ (full professor) in pure mathematics <strong>and</strong> foundations of mathematics at the University of Amsterdam. 4-VI-1976 Elected member of the Royal Dutch Academy of Sciences. 16-II-1996 Elected corresponding member of the Bavarian Academy of Sciences. 15-XI-1996 Received the F.L. Bauer–Prize of the ‘Bund der Freunde der Technischen Universität München’, <strong>for</strong> internationally outst<strong>and</strong>ing contributions to 1 <strong>The</strong> way Troelstra set up the Heyting Archive (now part of the Rijksarchief Noord-Holl<strong>and</strong> in Haarlem) <strong>and</strong> its index was a model <strong>for</strong> this archive <strong>and</strong> index. <strong>The</strong> Library of the Faculty of Sciences (Faculteit Natuurwetenschappen, Wiskunde en In<strong>for</strong>matica) of the University of Amsterdam, <strong>and</strong> especially the chief librarian H. Harmsen, gave me the opportunity to complete this index. 2 Data from the postscript file of A.S. Troelstra ( http://turing.wins.uva.nl/~anne/) . See also A.S. Troelstra, Looking back, ILLC magazine 2, (July 2000).
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A logician is a person who studies logic. Some famous logicians are listed below in English alphabetical transliteration order (by surname).
This is a dynamic list and may never be able to satisfy particular standards for completeness. You can help by adding missing items with reliable sources.
Peter Abelard (France, 1079–1142)
Wilhelm Ackermann (Germany, 1896–1962)
Sergei Adian (Russia/Soviet Union/Armenia, 1931–2020)
Rodolphus Agricola (Germany, 1443/1444–1485)
Kazimierz Ajdukiewicz (Poland, 1890–1963)
Alcuin (England/France, c. 735–804)
Alan Ross Anderson (US, 1924–1972)
Peter B. Andrews (US, born 1938)
Thomas Aquinas (Italy/France, 1225–1274)
Lennart Åqvist (Sweden, born 1932)
Aristotle (Greece, 384–322 BCE)
Heiric of Auxerre (France, 841–876)
Bahmanyār (Iran, died 1067)
Jayanta Bhatta (India, 850–910)
Alexander Bain (UK, 1818–1903)
Yehoshua Bar-Hillel (Israel, 1915–1975)
Ruth Barcan Marcus (US, 1921–2012)
Henk Barendregt (Netherlands, born 1947)
Jon Barwise (US, 1942–2000)
James Earl Baumgartner (US, 1943–2011)
John Lane Bell (UK and Canada, born 1945)
Nuel Belnap (US, born 1931)
Paul Benacerraf (US, born 1931)
Jean Paul Van Bendegem (Belgium, born 1953)
Johan van Benthem (Netherlands, born 1949)
Paul Bernays (Switzerland, 1888–1977)
Evert Willem Beth (Netherlands, 1908–1964)
Jean-Yves Béziau (Switzerland, born 1965)
Józef Maria Bocheński (Poland, 1902–1995)
Boethius (Rome/Ostrogothic Kingdom, c. 480–524/525)
Bernard Bolzano (Austrian Empire, 1781–1848)
Andrea Bonomi (Italy, born 1940)
George Boole (England/Ireland, 1815–1864)
George Boolos (US, 1940–1996)
Nicolas Bourbaki (pseudonym used by a group of French mathematicians, 20th century)
Thomas Bradwardine (England, c. 1290–26 August 1349)
Richard Brinkley (England, died c. 1379)
Luitzen Egbertus Jan Brouwer (Netherlands, 1881–1966)
Alan Richard Bundy (UK, born 1947)
Franco Burgersdijk (Netherlands, 1590–1629)
Jean Buridan (France, c. 1300–post 1358)
Walter Burley (England, c. 1275–1344/5)
Chanakya (India, Mouryan Empire, 371–285 BC)
Georg Ferdinand Cantor (Germany, 1845–1918)
Rudolf Carnap (Germany, 1891–1970)
Lewis Carroll (UK, 1832–1898)
Categoriae decem (Latin, fifth century)
Gregory Chaitin (Argentina/US, born 1947)
Chrysippus (Greece, c. 280 BC – c. 207 BC)
Alonzo Church (US, 1903–1995)
Leon Chwistek (Poland, 1884–1944)
Gordon H. Clark (US, 1902–1985)
Paul Joseph Cohen (US, 1934–2007)
Conimbricenses, name by which Jesuits of the University of Coimbra (Portugal) were known (1591–1606)
S. Barry Cooper (UK, 1943–2015)
Jack Copeland (UK, born 1950)
Thierry Coquand (France, born 1961)
John Corcoran (US, 1937–2021)
Newton da Costa (Brazil, 1929–2024)
William Craig (US, 1918–2016)
Haskell Curry (US, 1900–1982)
Tadeusz Czeżowski (Poland, 1889–1981)
Dirk van Dalen (Netherlands, born 1932)
Martin Davis (US, 1928–2023)
Augustus De Morgan (UK, 1806–1871)
René Descartes (France, 1596–1650)
Dharmakirti (India, c. 7th century)
Dignāga (India, fl. 5th century)
Diodorus Cronus (Greece, 4th–3rd century BC)
Martin Dorp (Netherlands, c. 1485–1525)
John Dumbleton (England, died c. 1349)
Michael A. E. Dummett (UK, 1925–2011)
Jon Michael Dunn (US, 1941–2021)
Samuel Eilenberg (Poland, 1913–1998)
Alexander Esenin-Volpin (Russia, 1924–2016)
John Etchemendy (US, born 1952)
Leonhard Euler (Switzerland, 1707–1783)
Solomon Feferman (US, 1928–2016)
Richard Ferrybridge (England, 14th century)
Hartry Field (US, born 1946)
Kit Fine (US, born 1946)
Melvin Fitting (US, born 1942)
Graeme Forbes (Scotland, 20th century)
Matthew Foreman (US, born 1957)
Michael Fourman (UK, born 1950)
Roland Fraïssé (France, 1920–2008)
Abraham Fraenkel (Germany, 1891–1965)
Gottlob Frege (Germany, 1848–1925)
Harvey Friedman (US, born 1948)
Dov Gabbay (UK, born 1945)
Haim Gaifman (US, born 1934)
L. T. F. Gamut (collective pseudonym used by a group of Dutch logicians, fl. 1980s–1990s)
Robin Gandy (UK, 1919–1995)
Sol Garfunkel (US, born 1943)
Garlandus Compotista (France, c. 11th century)
Akṣapāda Gautama, author of Nyāya Sūtras and founder of Nyaya school of Hindu philosophy (India, c. 6th century BC to 2nd century CE)
Gangesha Upadhyaya, author of Tattvacintāmaṇi (A Thought-Jewel of Truth) and founder of Navya-Nyāya (India, c. 14th century CE)
Peter Geach (UK, 1916–2013)
Gerhard Gentzen (Germany, 1909–1945)
Joseph Diaz Gergonne (France, 1771–1859)
Gilbert de la Porrée (France, 1070–1154)
Jean-Yves Girard (France, born 1947)
Kurt Gödel (Austria, US, 1906–1978)
Reuben Louis Goodstein (England, 1912–1985)
Valentin Goranko (Bulgaria/Sweden, born 1959)
Siegfried Gottwald (Germany, 1943–2015)
Jeroen Groenendijk (Netherlands, born 1949)
Susan Haack (UK, born 1945)
Petr Hájek (Czech Republic, 1941–2016)
Leo Harrington (US, born 1946)
Robert S. Hartman (Germany/US, 1910–1973)
Georg Wilhelm Friedrich Hegel (Germany, 1770–1831)
Jean Van Heijenoort (France/US, 1912–1986)
Leon Henkin (US, 1921–2006)
Jacques Herbrand (France, 1908–1931)
Arend Heyting (Netherlands, 1898–1980)
David Hilbert (Germany, 1862–1943)
Jaakko Hintikka (Finland, 1929–2015)
Alfred Horn (US, 1918–2001)
William Alvin Howard (US, born 1926)
Ehud Hrushovski (Israel, born 1959)
Gérard Huet (France, born 1947)
Ibn Taymiyyah (Turkey, 1263–1328 CE)
Marsilius of Inghen (Netherlands/France/Germany, 1330/1340–1396)
Giorgi Japaridze (Georgia, 20th century)
Stanisław Jaśkowski (Poland, 1906–1965)
Richard Jeffrey (US, 1926–2002)
Ronald Jensen (US, Europe, born 1936)
William Stanley Jevons (England, 1835–1882)
John of St. Thomas/John Poinsot (Portugal/Spain, 1589–1644)
William Ernest Johnson (UK, 1858–1931)
Dick de Jongh (Netherlands, born 1939)
Bjarni Jónsson (Iceland, 1920–2016)
Philip Jourdain (UK, 1879–1919)
Joachim Jungius (Germany, 1587–1657)
Jñanasrimitra (India, 10th century)
David Kaplan (US, born 1933)
Alexander S. Kechris (US, born 1946)
Howard Jerome Keisler (US, born 1936)
Ahmed Raza Khan (India, 1856–1921)
Richard Kilvington (England, c. 1305–1361)
Robert Kilwardby (England, c. 1215–1279)
Stephen Cole Kleene (US, 1909–1994)
Tadeusz Kotarbiński (Poland, 1886–1981)
Robert Kowalski (US, UK, born 1941)
Georg Kreisel (Austria/Britain/US, 1923–2015)
Saul Kripke (US, 1940–2022)
Leopold Kronecker (Germany, 1823–1891)
Kenneth Kunen (US, 1943–2020)
Christine Ladd-Franklin (US, 1847–1930)
Joachim Lambek (Canada, 1922–2014)
Johann Heinrich Lambert (France/Germany, 1728–1777)
Karel Lambert (US, born 1928)
Gottfried Wilhelm Leibniz (Germany, 1646–1716)
Stanisław Leśniewski (Poland, 1886–1939)
Clarence Irving Lewis (US, 1883–1964)
David Kellogg Lewis (US, 1941–2001)
Adolf Lindenbaum (Poland, 1904–1941)
Per Lindström (Sweden, 1936–2009)
Ramon Llull (Spain, 1232–1315)
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John Stuart Mill (England, 1806–1873)
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Sara Negri (Italy/Finland, born 1967)
Edward Nelson (US, 1932–2014)
John von Neumann (Hungary, US, 1903–1957)
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Paul of Venice (Italy, 1368–1428)
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Juan Luis Vives (Spain, 1493–1540)
Pakṣilasvāmin Vātsyāyana wrote the first known commentary on Goutama's Nyaya Sutras (5th century CE)
Hao Wang (China/US, 1921–1995)
Isaac Watts (England, 1674–1748)
Richard Whately (England, 1787–1863)
Alfred North Whitehead (UK, 1861–1947)
Ludwig Wittgenstein (Austria, UK, 1889–1951)
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History of Constructivism in The 20 Century
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History of Constructivism in the 20 century - Free download as PDF File (.pdf), Text File (.txt) or read online for free. This document provides a history of constructivism in the 20th century. It discusses several trends within constructivism including finitism, which rejects abstract concepts and quantifiers and interprets existence claims constructively; intuitionism, which insists mathematical objects are mental constructions; and Bishop's constructivism. It outlines the key figures that developed these trends like Kronecker, Skolem, Hilbert, Bernays, and Bishop. The document also discusses related areas like metamathematics, predicativism, and actualism, which critiques finitism for not accounting for large numbers.
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passfault/wordlists/wordlists/nlLongTail.txt at master · OWASP/passfault
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OWASP Passfault evaluates passwords and enforces password policy in a completely different way. - passfault/wordlists/wordlists/nlLongTail.txt at master · OWASP/passfault
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Justification Logic: Reasoning with Reasons 1108661106, 9781108661102
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Classical logic is concerned, loosely, with the behaviour of truths. Epistemic logic similarly is about the behaviour of...
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Citation preview
Justification Logic Classical logic is concerned, loosely, with the behavior of truths. Epistemic logic similarly is about the behavior of known or believed truths. Justification logic is a theory of reasoning that enables the tracking of evidence for statements and therefore provides a logical framework for the reliability of assertions. This book, the first in the area, is a systematic account of the subject, progressing from modal logic through to the establishment of an arithmetic interpretation of intuitionistic logic. The presentation is mathematically rigorous but in a style that will appeal to readers from a wide variety of areas to which the theory applies. These include mathematical logic, artificial intelligence, computer science, philosophical logic and epistemology, linguistics, and game theory.
s e r g e i a rt e m ov is Distinguished Professor at the City University of New York. He is a specialist in mathematical logic, logic in computer science, control theory, epistemology, and game theory. He is credited with solving long-standing problems in constructive logic that had been left open by G¨odel and Kolmogorov since the 1930s. He has pioneered studies in the logic of proofs and justifications that render a new, evidence-based theory of knowledge and belief. The most recent focus of his interests is epistemic foundations of game theory. m e lv i n f i t t i n g is Professor Emeritus at the City University of New York. He has written or edited a dozen books and has worked in intensional logic, semantics for logic programming, theory of truth, and tableau systems for nonclassical logics. In 2012 he received the Herbrand Award from the Conference on Automated Deduction. He was on the faculty of the City University of New York from 1969 to his retirement in 2013, at Lehman College, and at the Graduate Center, where he was in the Departments of Mathematics, Computer Science, and Philosophy.
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Justification Logic Reasoning with Reasons S E R G E I A RT E M OV Graduate Center, City University of New York M E LV I N F I T T I N G Graduate Center, City University of New York
University Printing House, Cambridge CB2 8BS, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia 314–321, 3rd Floor, Plot 3, Splendor Forum, Jasola District Centre, New Delhi – 110025, India 79 Anson Road, #06–04/06, Singapore 079906 Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning, and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781108424912 DOI: 10.1017/9781108348034 © Sergei Artemov and Melvin Fitting 2019 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2019 Printed and bound in Great Britain by Clays Ltd, Elcograf S.p.A. A catalogue record for this publication is available from the British Library. Library of Congress Cataloging-in-Publication Data Names: Artemov, S. N., author. | Fitting, Melvin, 1942- author. Title: Justification logic : reasoning with reasons / Sergei Artemov (Graduate Center, City University of New York), Melvin Fitting (Graduate Center, City University of New York). Description: Cambridge ; New York, NY : Cambridge University Press, 2019. | Series: Cambridge tracts in mathematics ; 216 | Includes bibliographical references and index. Identifiers: LCCN 2018058431 | ISBN 9781108424912 (hardback : alk. paper) Subjects: LCSH: Logic, Symbolic and mathematical. | Inquiry (Theory of knowledge) | Science–Theory reduction. | Reasoning. Classification: LCC QA9 .A78 2019 | DDC 511.3–dc23 LC record available at https://lccn.loc.gov/2018058431 ISBN 978-1-108-42491-2 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.
To our wives, Lena and Roma.
Contents
Introduction 1 What Is This Book About? 2 What Is Not in This Book?
page x xii xvii
1
Why Justification Logic? 1.1 Epistemic Tradition 1.2 Mathematical Logic Tradition 1.3 Hyperintensionality 1.4 Awareness 1.5 Paraconsistency
1 1 4 8 9 10
2
The Basics of Justification Logic 2.1 Modal Logics 2.2 Beginning Justification Logics 2.3 J0 , the Simplest Justification Logic 2.4 Justification Logics in General 2.5 Fundamental Properties of Justification Logics 2.6 The First Justification Logics 2.7 A Handful of Less Common Justification Logics
11 11 12 14 15 20 23 27
3
The Ontology of Justifications 3.1 Generic Logical Semantics of Justifications 3.2 Models for J0 and J 3.3 Basic Models for Positive and Negative Introspection 3.4 Adding Factivity: Mkrtychev Models 3.5 Basic and Mkrtychev Models for the Logic of Proofs LP 3.6 The Inevitability of Possible Worlds: Modular Models 3.7 Connecting Justifications, Belief, and Knowledge 3.8 History and Commentary
31 31 36 38 39 42 42 45 46
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Contents
4
Fitting Models 4.1 Modal Possible World Semantics 4.2 Fitting Models 4.3 Soundness Examples 4.4 Canonical Models and Completeness 4.5 Completeness Examples 4.6 Formulating Justification Logics
48 48 49 52 60 65 72
5
Sequents and Tableaus 5.1 Background 5.2 Classical Sequents 5.3 Sequents for S4 5.4 Sequent Soundness, Completeness, and More 5.5 Classical Semantic Tableaus 5.6 Modal Tableaus for K 5.7 Other Modal Tableau Systems 5.8 Tableaus and Annotated Formulas 5.9 Changing the Tableau Representation
75 75 76 79 81 84 90 91 93 95
6
Realization – How It Began 6.1 The Logic LP 6.2 Realization for LP 6.3 Comments
100 100 103 108
7
Realization – Generalized 7.1 What We Do Here 7.2 Counterparts 7.3 Realizations 7.4 Quasi-Realizations 7.5 Substitution 7.6 Quasi-Realizations to Realizations 7.7 Proving Realization Constructively 7.8 Tableau to Quasi-Realization Algorithm 7.9 Tableau to Quasi-Realization Algorithm Correctness 7.10 An Illustrative Example 7.11 Realizations, Nonconstructively 7.12 Putting Things Together 7.13 A Brief Realization History
110 110 112 113 116 118 120 126 128 131 133 135 138 139
8
The Range of Realization 8.1 Some Examples We Already Discussed 8.2 Geach Logics 8.3 Technical Results
141 141 142 144
Contents 8.4 8.5 8.6 8.7 8.8
Geach Justification Logics Axiomatically Geach Justification Logics Semantically Soundness, Completeness, and Realization A Concrete S4.2/JT4.2 Example Why Cut-Free Is Needed
ix 147 149 150 152 155
9
Arithmetical Completeness and BHK Semantics 9.1 Arithmetical Semantics of the Logic of Proofs 9.2 A Constructive Canonical Model for the Logic of Proofs 9.3 Arithmetical Completeness of the Logic of Proofs 9.4 BHK Semantics 9.5 Self-Referentiality of Justifications
158 158 161 165 174 179
10
Quantifiers in Justification Logic 10.1 Free Variables in Proofs 10.2 Realization of FOS4 in FOLP 10.3 Possible World Semantics for FOLP 10.4 Arithmetical Semantics for FOLP
181 182 186 191 212
11
Going Past Modal Logic 11.1 Modeling Awareness 11.2 Precise Models 11.3 Justification Awareness Models 11.4 The Russell Scenario as a JAM 11.5 Kripke Models and Master Justification 11.6 Conclusion References Index
222 223 225 226 228 231 233 234 244
Introduction
Why is this thus? What is the reason of this thusness?1
Modal operators are commonly understood to qualify the truth status of a proposition: necessary truth, proved truth, known truth, believed truth, and so on. The ubiquitous possible world semantics for it characterizes things in universal terms: X is true in some state if X is true in all accessible states, where various conditions on accessibility are used to distinguish one modal logic from another. Then (X → Y) → (X → Y) is valid, no matter what conditions are imposed, by a simple and direct argument using universal quantification. Suppose both (X → Y) and X are true at an arbitrary state. Then both X and X → Y are true at all accessible states, whatever “accessible” may mean. By the usual understanding of →, Y is true at all accessible states too, and so Y is true at the arbitrary state we began with. Although arguments like these have a strictly formal nature and are studied as modal model theory, they also give us some insights into our informal, everyday use of modalities. Still, something is lacking. Suppose we think of as epistemic, and to emphasize this we use K instead of for the time being. For some particular X, if you assert the colloquial counterpart of KX, that is, if you say you know X, and I ask you why you know X, you would never tell me that it is because X is true in all states epistemically compatible with this one. You would, instead, give me some sort of explicit reason: “I have a mathematical proof of X,” or “I read X in the encyclopedia,” or “I observed that X is the case.” If I asked you why K(X → Y) → (KX → KY) is valid you would probably say something like “I could use my reason for X and combine it with my reason for X → Y, and infer Y.” This, in effect, would be your reason for Y, given that you had reasons for X and for X → Y. 1
Charles Farrar Browne (1834–1867) was an American humorist who wrote under the pen name Artemus Ward. He was a favorite writer of Abraham Lincoln, who would read his articles to his Cabinet. This quote is from a piece called Moses the Sassy, Ward (1861).
x
Introduction
xi
Notice that this neatly avoids the logical omniscience problem: that we know all the consequences of what we know. It replaces logical omniscience with the more acceptable claim that there are reasons for the consequences of what we know, based on the reasons for what we know, but reasons for consequences are more complicated things. In our example, the reason for Y has some structure to it. It combines reasons for X, reasons for X → Y, and inference as a kind of operation on reasons. We will see more examples of this sort; in fact, we have just seen a fundamental paradigm. In place of a modal operator, , justification logics have a family of justification terms, informally intended to represent reasons, or justifications. Instead of X we will see t:X, where t is a justification term and the formula is read “X is so for reason t,” or more briefly, “t justifies X.” At a minimum, justification terms are built up from justification variables, standing for arbitrary justifications. They are built up using a set of operations that, again at a minimum, contains a binary operation ·. For example, x · (y · x) is a justification term, where x and y are justification variables. The informal understanding of · is that t · u justifies Y provided t justifies an implication with Y as its consequent, and u justifies the antecedent. In justification logics the counterpart of (X → Y) → (X → Y) is t:(X → Y) → (u:X → [t · u]:Y) where, as we will often do, we have added square brackets to enhance readability. Note that this exactly embodies the informal explanation we gave in the previous paragraph for the validity of K(X → Y) → (KX → KY). That is, Y has a justification built from justifications for X and for X → Y using an inference that amounts to a modus ponens application—we can think of the · operation as an abstract representation of this inference. Other behaviors of modal operators, X → X for instance, will require operators in addition to ·, and appropriate postulated behavior, in order to produce justification logics that correspond to modal logics in which X → X is valid. Examples, general methods for doing this, and what it means to “correspond” all will be discussed during the course of this book. One more important point. Suppose X and Y are equivalent formulas, that is, we have X ↔ Y. Then in any normal modal logic we will also have X ↔ Y. Let us interpret the modal operator epistemically again, and write KX ↔ KY. In fact, KX ↔ KY, when read in the usual epistemic way, can sometimes be quite an absurd assertion. Consider some astronomically complicated tautology X of classical propositional logic. Because it is a tautology, it is equivalent
xii
Introduction
to P ∨ ¬P, which we may take for Y. Y is hardly astronomically complicated. However, because X ↔ Y, we will have KX ↔ KY. Clearly, we know Y essentially by inspection and hence KY holds, while KX on the other hand will involve an astronomical amount of work just to read it, let alone to verify it. Informally we see that, while both X and Y are tautologies, and so both are knowable in principle, any justification we might give for knowing one, combined with quite a lot of formula manipulation, can give us some justification for knowing the other. The two justifications may not, indeed will not, be the same. One is simple, the other very complex. Modal logic is about propositions. Propositions are, in a sense, the content of formulas. Propositions are not syntactical objects. “It’s good to be the king” and “Being the king is good” express the same proposition, but not in the same way. Justifications apply to formulas. Equivalent formulas determine the same proposition, but can be quite different as formulas. Syntax must play a fundamental role for us, and you will see that it does, even in our semantics. Consider one more very simple example. A → (A∧ A) is an obvious tautology. We might expect KA → K(A ∧ A). But we should not expect t:A → t:(A ∧ A). If t does, in fact, justify A, a justification of A ∧ A may involve t, but also should involve facts about the redundancy of repetition; t by itself cannot be expected to suffice. Modal logics can express, more or less accurately, how various modal operators behave. This behavior is captured axiomatically by proofs, or semantically using possible world reasoning. These sorts of justifications for modal operator behavior are not within a modal logic, but are outside constructs. Justification logics, on the other hand, can represent the whys and wherefores of modal behavior quite directly, and from within the formal language itself. We will see that most standard modal logics have justification counterparts that can be used to give a fine-grained, internal analysis of modal behavior. Perhaps, this will help make clear why we used the quotation we did at the beginning of this Introduction.
1 What Is This Book About? How did justification logics originate? It is an interesting story, with revealing changes of direction along the way. Going back to the days when G¨odel was a young logician, there was a dream of finding a provability interpretation for intuitionistic logic. As part of his work on that project, in G¨odel (1933), G¨odel showed that one could abstract some of the key features of provability and make a propositional modal logic using them. Then, remarkably but
Introduction
xiii
naturally, one could embed propositional intuitionistic logic into the resulting system. C. I. Lewis had pioneered the modern formal study of modal logics (Lewis, 1918; Lewis and Langford, 1932), and G¨odel observed that his system was equivalent to the Lewis system S4. All modern axiomatizations of modal logics follow the lines pioneered in G¨odel’s note, while Lewis’s original formulation is rarely seen today. G¨odel showed that propositional intuitionistic logic embedded into S4 using a mapping that inserted in front of every subformula. In effect, intuitionistic logic could be understood using classical logic plus an abstract notion of provability: a propositional formula X is an intuitionistic theorem if and only if the result of applying G¨odel’s mapping is a theorem of S4. (This story is somewhat simplified. There are several versions of the G¨odel translation—we have used the simplest one to describe. And G¨odel did not use the symbol but rather an operator Bew, short for beweisbar, or provability in the German language. None of this affects our main points.) Unfortunately, the story breaks off at this point because G¨odel also noted that S4 does not behave like formal provability (e.g., in arithmetic), by using the methods he had pioneered in his work on incompleteness. Specifically, S4 validates X → X, so in particular we have ⊥ → ⊥ (where ⊥ is falsehood). This is equivalent to ¬⊥, which is thus provable in S4. If we had an embedding of S4 into formal arithmetic under which corresponded to G¨odel’s arithmetic formula representing provability, we would be able to prove in arithmetic that falsehood was not provable. That is, we would be able to show provability of consistency, violating G¨odel’s second incompleteness theorem. So, work on an arithmetic semantics for propositional intuitionistic logic paused for a while. Although it did not solve the problem of a provability semantics for intuitionistic logic, an important modal/arithmetic connection was eventually worked out. One can define a modal logic by requiring that its validities are those that correspond to arithmetic validities when reading as G¨odel’s provability formula. It was shown in Solovay (1976) that this was a modal logic already known in the literature, though as noted earlier, it is not S4. Today, the logic is called GL, standing for G¨odel–L¨ob logic. GL is like S4 except that the T axiom X → X, an essential part of S4, is replaced by a modal formula abstractly representing L¨ob’s theorem: (X → X) → X. S4 and GL are quite different logics. By now the project for finding an arithmetic interpretation of intuitionistic logic had reached an impasse. Intuitionistic logic embedded into S4, but S4 did not embed into formal arithmetic. GL embedded into formal arithmetic, but the G¨odel translation does not embed intuitionistic logic into GL.
xiv
Introduction
In his work on incompleteness for Peano arithmetic, G¨odel gave a formula Bew(x, y)
that represents the relation: x is the G¨odel number of a proof of a formula with G¨odel number y. Then, a formal version of provability is ∃xBew(x, y) which expresses that there is a proof of (the formula whose G¨odel number is) y. If this formula is what corresponds to in an embedding from a modal language to Peano arithmetic, we get the logic GL. But in a lecture in 1938 G¨odel pointed out that we might work with explicit proof representatives instead of with provability (G¨odel, 1938). That is, instead of using an embedding translating every occurrence of by ∃xBew(x, y), we might associate with each occurrence of some formal term t that somehow represents a particular proof, allowing different occurrences of to be associated with different terms t. Then in the modal embedding, we could make the occurrence of associated with t correspond to Bew(ptq, y), where ptq is a G¨odel number for t. For each occurrence of we would need to find some appropriate term t, and then each occurrence of would be translated into arithmetic differently. The existential quantifier in ∃xBew(x, y) has been replaced with a meta-existential quantifier, outside the formal language. We provide an explicit proof term, rather than just asserting that one exists. G¨odel believed that this approach should lead to a provability embedding of S4 into Peano arithmetic. G¨odel’s proposal was not published until 1995 when Volume 3 of his collected works appeared. By this time the idea of using a modal-like language with explicit representatives for proofs had been rediscovered independently by Sergei Artemov, see Artemov (1995, 2001). The logic that Artemov created was called LP, which stood for logic of proofs. It was the first example of a justification logic. What are now called justification terms were called proof terms in LP. Crucially, Artemov showed LP filled the gap between modal S4 and Peano arithmetic. The connection with S4 is primarily embodied in a Realization Theorem, which has since been shown to hold for a wide range of justification logic, modal logic pairs. It will be extensively examined in this book. The connection between LP and formal arithmetic is Artemov’s Arithmetic Completeness Theorem, which also will be examined in this book. Its range is primarily limited to the original justification logic, LP, and a few close relatives. This should not be surprising, though. G¨odel’s motivation for his formulation of S4 was that should embody properties of a formal arithmetic proof predicate. This connection with arithmetic provability is not present for almost all modal
Introduction
xv
logics and is consequently also missing for corresponding justification logics, when they exist. Nonetheless, the venerable goal of finding a provability interpretation for propositional intuitionistic logic had been attained. The G¨odel translation embeds propositional intuitionistic logic into the modal logic S4. The Realization Theorem establishes an embedding of S4 into the justification logic LP. And the Arithmetic Completeness Theorem shows that LP embeds into formal arithmetic. It was recognized from the very beginning that the connection between S4 and LP could be weakened to sublogics of S4 and LP. Thus, there were justification logic counterparts for the standard modal logics, K, K4, T, and a few others. These justification logics had arithmetic connections because they were sublogics of LP. The use of proof term was replaced with justification term. Although the connection with arithmetic was weaker than it had been with LP, justification terms still had the role of supplying explicit justifications for epistemically necessary statements. One can consult Artemov (2008) and Artemov and Fitting (2012) for survey treatments, though the present book includes the material found there. Almost all of the early work on justification logics was proof-theoretically based. Realization theorems were shown constructively, making use of a sequent calculus. The existence of an algorithm to compute what are called realizers is important, but this proof-theoretic approach limits the field to those logics known to have sequent calculus proof systems. For a time it was hoped that various extensions of sequent and tableau calculi would be useful and, to some extent, this has been the case. The most optimistic version of this hope was expressed in Artemov (2001) quite directly, “Gabbay’s Labelled Deductive Systems, Gabbay (1994), may serve as a natural framework for LP.” Unfortunately this seems to have been too optimistic. While the formats had similarities, the goals were different, and the machinery did not interact well. A semantics for LP and its near relatives, not based on arithmetic provability, was introduced in Mkrtychev (1997) and is discussed in Chapter 3. (A constructive version of the canonical model for LP with a completeness theorem can be found already in Artemov (1995).) Mkrtychev’s semantics did not use possible worlds and had a strong syntactic flavor. Possible worlds were added to the mix in Fitting (2005), producing something that potentially applied much more broadly than the earlier semantics. This is the subject of Chapter 4. Using this possible world semantics, a nonconstructive, semantic-based, proof of realization was given. It was now possible to avoid the use of a sequent calculus, though the algorithmic nature of realization was lost. More recently, a semantics with a very simple structure was created, Artemov’s basic semantics (Artemov, 2012). It is presented in Chapter 3. Its machinery is almost minimal
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Introduction
for the purpose. In this book, we will use possible world semantics to establish very general realization results, but basic models will often be used when we simply want to show some formula fails to be a theorem. Though its significance was not properly realized at the time, in 2005 the subject broadened when a justification logic counterpart of S5 was introduced in Pacuit (2005) and Rubtsova (2006a, b), with a connecting realization theorem. There was no arithmetical interpretation for this justification logic. Also there is no sequent calculus for S5 of the standard kind, so the proof given for realization was nonconstructive, using a version of the semantics from Fitting (2005). The semantics needed some modification to what is called its evidence function, and this turned out to have a greater impact than was first realized. Eventually constructive proofs connecting S5 and its justification counterpart were found. These made use of cut-free proof systems that were not exactly standard sequent calculi. Still, the door to a larger room was beginning to open. Out of the early studies of the logics of proofs and its variants a general logical framework for reasoning about epistemic justification at large naturally emerged, and the name, Justification Logic, was introduced (cf. Artemov, 2008). Justification Logic is based on justification assertions, t:F, that are read t is a justification for F, with a broader understanding of the word justification going beyond just mathematical proofs. The notion of justification, which has been an essential component of epistemic studies since Plato, had been conspicuously absent in the mathematical models of knowledge within the epistemic logic framework. The Justification Logic framework fills in this void. In Fitting (2016a) the subject expanded abruptly. Using nonconstructive semantic methods it was shown that the family of modal logics having justification counterparts is infinite. The justification phenomenon is not the relatively narrow one it first seemed to be. While that work was nonconstructive, there are now cut-free proof systems of various kinds for a broader range of modal logics than was once the case, and these have been used successfully to create realization algorithms, in Kuznets and Goetschi (2012), for instance. It may be that the very general proof methodologies of Fitting (2015) and especially Negri (2005) and Negri and von Plato (2001) will extend the constructive range still further, perhaps even to the infinite family that nonconstructive methods are known to work for. This is active current work. Work on quantified justification logics exists, but the subject is considerably behind its propositional counterpart. An important feature of justification logics is that they can, in a very precise sense, internalize their own proofs. Doing this for axioms is generally simple. Rules of inference are more of a problem. Earlier we discussed a justification formula as a simple, representative exam-
Introduction
xvii
ple: t:(X → Y) → (u:X → [t · u]:Y). This, in effect, internalizes the axiomatic modus ponens rule. The central problem in developing quantified justification logics was how to internalize the rule of universal generalization. It turned out that the key was the clear separation between two roles played by individual variables. On the one hand, they are formal symbols, and one can simply infer ∀xϕ(x) from a proof of ϕ(x). On the other hand, they can be thought of as open for substitution, that is, throughout a proof one can replace free occurrences of x with a term t to produce a new proof (subject to appropriate freeness of substitution conditions, of course). These two roles for variables are actually incompatible. It was the introduction of specific machinery to keep track of which role a variable occurrence had that made possible the internalization of proofs, and thus a quantified justification logic. An axiomatic version of first-order LP was introduced in Artemov and Yavorskaya (Sidon) (2011) and a possible world semantics for it in Fitting (2011a, 2014b). A connection with formal arithmetic was established. There is a constructive proof of a Realization Theorem, connecting first-order LP with firstorder S4. Unlike propositionally, no nonconstructive proof is currently known The possible world semantics includes the familiar monotonicity condition on world domains. It is likely that all this can be extended to a much broader range of quantified modal logics than just first-order S4, provided monotonicity is appropriate. A move to constant domain models, to quantified S5 in particular, has been made, and a semantics, but not yet a Realization Theorem, can be found in Fitting and Salvatore (2018). Much involving quantification is still uncharted territory. This book will cover the whole range of topics just described. It will not do so in the historical order that was followed in this Introduction, but will make use of the clearer understanding that has emerged from study of the subject thus far. We will finish with the current state of affairs, standing on the edge of unknown lands. We hope to prepare some of you for the journey, should you choose to explore further on your own.
2 What Is Not in This Book? There are several historical works and pivotal developments in justification logic that will not be covered in the book due to natural limitations, and in this section we will mention them briefly. We are confident that other books and surveys will do justice to these works in more detail. Apart from G¨odel’s lecture, G¨odel (1938), which remained unpublished
xviii
Introduction
until 1995 and thus could not influence development in this area, the first results and publications on the logic of proofs are dated 1992: a technical report, Artemov and Straßen (1992), based on work done in January of 1992 in Bern, and a conference presentation of this work at CSL’92 published in Springer Lecture Notes in Computer Science as Artemov and Straßen (1993a). In this work, the basic logic of proofs was presented: it had proof variables, and the format t is a proof of F, but without operations on proofs. However, it already had the first installment of the fixed-point arithmetical completeness construction together with an observation that, unlike provability logic, the logic of proofs cannot be limited to one standard proof predicate “from the textbook” or to any single-conclusion proof predicate. This line was further developed in Artemov and Straßen (1993b), where the logic of single-conclusion proof predicates (without operations on proofs) was studied. This work introduced the unification axiom, which captures singleconclusioness by propositional tools. After the full-scale logic of proofs with operations had been discovered (Artemov, 1995), the logic of single-conclusion proofs with operations was axiomatized in V. Krupski (1997, 2001). A similar technique was used recently to characterize so-called sharp single-conclusion justification models in Krupski (2018). Another research direction pursued after the papers on the basic logic of proofs was to combine provability and explicit proofs. Such a combination, with new provability principles, was given in Artemov (1994). Despite its title, this paper did not introduce what is known now as The Logic of Proofs, but rather a fusion of the provability logic GL and the basic logic of proofs without operations, but with new arithmetical principles combining proofs and provability and an arithmetical completeness theorem. After the logic of proofs paper (Artemov, 1995), the full-scale logic of provability and proofs (with operations), LPP, was axiomatized and proved arithmetically complete in Sidon (1997) and Yavorskaya (Sidon) (2001). A leaner logic combining provability and explicit proofs, GLA, was introduced and proved arithmetically complete in Nogina (2006, 2014b). Unlike LPP, the logic GLA did not use additional operations on proofs other than those inherited from LP. Later, GLA was used to find a complete classification of reflection principles in arithmetic that involve provability and explicit proofs (Nogina, 2014a). The first publication of the full-scale logic of proofs with operations, LP, which became the first justification logic in the modern sense, was Artemov (1995). This paper contains all the results needed to complete G¨odel’s program of characterizing intuitionistic propositional logic IPC and its BHK semantics via proofs in classical arithmetic: internalization, the realization theorem for S4 in LP, arithmetical semantics for LP, and the arithmetical completeness the-
Introduction
xix
orem. It took six years for the corresponding journal paper to appear: Artemov (2001). In Goris (2008), the completeness of LP for the semantics of proofs in Peano arithmetic was extended to the semantics of proofs in Buss’s bounded arithmetic S12 . In view of applications in epistemology, this result shows that explicit knowledge in the propositional framework can be made computationally feasible. Kuznets and Studer (2016) extend the arithmetical interpretation of LP from the original finite constant specifications to a wide class of constant specifications, including some infinite ones. In particular, this “weak” arithmetical interpretation captures the full logic of proofs LP with the total constant specification. Decidability of LP (with the total constant specification) was also established in Mkrtychev (1997), and this opened the door to decidability and complexity studies in justification logics using model-theoretic and other means. Among the milestones are complexity estimates in Kuznets (2000), Brezhnev and Kuznets (2006), Krupski (2006a), Milnikel (2007), Buss and Kuznets (2012), and Achilleos (2014a). The arithmetical provability semantics for the Logic of Proofs, LP, naturally generalizes to a first-order version with conventional quantifiers and to a version with quantifiers over proofs. In both cases, axiomatizability questions were answered negatively in Artemov and Yavorskaya (2001) and Yavorsky (2001). A natural and manageable first-order version of the logic of proofs, FOLP, has been studied in Artemov and Yavorskaya (Sidon) (2011), Fitting (2014a), and Fitting and Salvatore (2018) and will be covered in Chapter 10. Originally, the logic of proofs was formulated as a Hilbert-style axiomatic system, but this has gradually broadened. Early attempts were tableau based (which could equivalently be presented using sequent calculus machinery). Tableaus generally are analytic, meaning that everything entering into a proof is a subformula of what is being proved. This was problematic for attempts at LP tableaus because of the presence of the · operation, which represented an application of modus ponens, a rule that is decidedly not analytic. Successful tableau systems, though not analytic, for LP and closely related logics can be found in Fitting (2003, 2005) and Renne (2004, 2006). The analyticity problem was overcome in Ghari (2014, 2016a). Broader proof systems have been investigated: hypersequents in Kurokawa (2009, 2012), prefixed tableaus in Kurokawa (2013), and labeled deductive systems in Ghari (2017). Indeed some of this has led to new realization results (Artemov, 1995, 2001, 2002, 2006; Artemov and Bonelli, 2007; Ghari, 2012; Kurokawa, 2012). Finding a computational reading of justification logics has been a natural research goal. There were several attempts to use the ideas of LP for building a lambda-calculus with internalization, cf. Alt and Artemov (2001), Artemov
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Introduction
(2002), Artemov and Bonelli (2007), Pouliasis and Primiero (2014), and others. Corresponding combinatory logic systems with internalization were studied in Artemov (2004), Krupski (2006b), and Shamkanov (2011). These and other studies can serve as a ground for further applications in typed programming languages. A version of the logic of proofs with a built-in verification predicate was considered in Protopopescu (2016a, b). The aforementioned intuition that justification logic naturally avoids the logical omniscience problem has been formalized and studied in Artemov and Kuznets (2006, 2009, 2014). The key idea there was to view logical omniscience as a proof complexity problem: The logical omniscience defect occurs if an epistemic system assumes knowledge of propositions, which have no feasible proofs. Through this prism, standard modal logics are logically omniscient (modulo some common complexity assumptions), and justification logics are not logically omniscient. The ability of justification logic to track proof complexity via time bounds led to another formal definition of logical omniscience in Wang (2011a) with the same conclusion: Justification logic keeps logical omniscience under control. Shortly after the first paper on the logic of proofs, it became clear that the logical tools developed are capable of evidence tracking in a general setting and as such can be useful in epistemic logic. Perhaps, the first formal work in this direction was Artemov et al. (1999), in which modal logic S5 was equivalently modified and supplied with an LP-style explicit counterpart. Applications to epistemology have benefited greatly from Fitting semantics, which connected justification logics to mainstream epistemology via possible worlds models. In addition to applications discussed in this book, we would like to mention some other influential work. Game semantics of justification logic was studied in Renne (2008) and dynamic epistemic logic with justifications in Renne (2008) and Baltag et al. (2014). In Sedl´ar (2013), Fitting semantics for justification models was elaborated to a special case of the models of general awareness. Multiagent justification logic and common knowledge has been studied in Artemov (2006), Antonakos (2007), Yavorskaya (Sidon) (2008), Bucheli et al. (2010, 2011), Bucheli (2012), Antonakos (2013), and Achilleos (2014b, 2015a, b). In Dean and Kurokawa (2010), justification logic was used for the analysis of Knower and Knowability paradoxes. A fast-growing and promising area is probabilistic justification logic, cf. Milnikel (2014), Artemov (2016b), Kokkinis et al. (2016), Ghari (2016b), and Lurie (2018).
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We are deeply indebted to all contributors to the exciting justification logic project, without whom there would not be this book. Very special thanks to our devoted readers for their sharp eyes and their useful comments: Vladimir Krupski, Vincent Alexis Peluce, and Tatiana Yavorskaya (Sidon).
I think there is no sense in forming an opinion when there is no evidence to form it on. If you build a person without any bones in him he may look fair enough to the eye, but he will be limber and cannot stand up; and I consider that evidence is the bones of an opinion.2
2
Mark Twain (1835–1910). The quote is from his last novel, Personal Recollections of Joan of Arc, Twain (1896).
1 Why Justification Logic?
The formal details of justification logic will be presented starting with the next chapter, but first we give some background and motivation for why the subject was developed in the first place. We will see that it addresses, or at least partially addresses, many of the fundamental problems that have been found in epistemic logic over the years. We will also see in more detail how it relates to our understanding of intuitionistic logic. And finally, we will see how it can be used to mitigate some well-known issues that have arisen in philosophical investigations.
1.1 Epistemic Tradition The properties of knowledge and belief have been a subject for formal logic at least since von Wright and Hintikka (Hintikka, 1962; von Wright, 1951). Knowledge and belief are both treated as modalities in a way that is now very familiar—Epistemic Logic. But of the celebrated three criteria for knowledge (usually attributed to Plato), justified, true, belief, Gettier (1963); Hendricks (2005), epistemic modal logic really works with only two of them. Possible worlds and indistinguishability model belief—one believes what is so under all circumstances thought possible. Factivity brings a trueness component into play—if something is not so in the actual world it cannot be known, only believed. But there is no representation for the justification condition. Nonetheless, the modal approach has been remarkably successful in permitting the development of rich mathematical theory and applications (Fagin et al., 1995; van Ditmarsch et al., 2007). Still, it is not the whole picture. The modal approach to the logic of knowledge is, in a sense, built around the universal quantifier: X is known in a situation if X is true in all situations indistinguishable from that one. Justifications, on the other hand, bring an ex1
2
Why Justification Logic?
istential quantifier into the picture: X is known in a situation if there exists a justification for X in that situation. This universal/existential dichotomy is a familiar one to logicians—in formal logics there exists a proof for a formula X if and only if X is true in all models for the logic. One thinks of models as inherently nonconstructive, and proofs as constructive things. One will not go far wrong in thinking of justifications in general as much like mathematical proofs. Indeed, the first justification logic was explicitly designed to capture mathematical proofs in arithmetic, something that will be discussed later. In justification logic, in addition to the category of formulas, there is a second category of justifications. Justifications are formal terms, built up from constants and variables using various operation symbols. Constants represent justifications for commonly accepted truths—axioms. Variables denote unspecified justifications. Different justification logics differ on which operations are allowed (and also in other ways too). If t is a justification term and X is a formula, t:X is a formula, and is intended to be read t is a justification for X. One operation, common to all justification logics, is application, written like multiplication. The idea is, if s is a justification for A → B and t is a justification for A, then [s · t] is a justification for B.1 That is, the validity of the following is generally assumed s:(A → B) → (t:A → [s · t]:B).
(1.1)
This is the explicit version of the usual distributivity of knowledge operators, and modal operators generally, across implication K(A → B) → (KA → KB).
(1.2)
How adequately does the traditional modal form (1.2) embody epistemic closure? We argue that it does so poorly! In the classical logic context, (1.2) only claims that it is impossible to have both K(A → B) and KA true, but KB false. However, because (1.2), unlike (1.1), does not specify dependencies between K(A → B), KA, and KB, the purely modal formulation leaves room for a counterexample. The distinction between (1.1) and (1.2) can be exploited in a discussion of the paradigmatic Red Barn Example of Goldman and Kripke; here is a simplified version of the story taken from Dretske (2005). 1
For better readability brackets will be used in terms, “[,]”, and parentheses in formulas, “(,).” Both will be avoided when it is safe.
1.1 Epistemic Tradition
3
Suppose I am driving through a neighborhood in which, unbeknownst to me, papiermˆach´e barns are scattered, and I see that the object in front of me is a barn. Because I have barn-before-me percepts, I believe that the object in front of me is a barn. Our intuitions suggest that I fail to know barn. But now suppose that the neighborhood has no fake red barns, and I also notice that the object in front of me is red, so I know a red barn is there. This juxtaposition, being a red barn, which I know, entails there being a barn, which I do not, “is an embarrassment.”
In the first formalization of the Red Barn Example, logical derivation will be performed in a basic modal logic in which is interpreted as the “belief” modality. Then some of the occurrences of will be externally interpreted as a knowledge modality K according to the problem’s description. Let B be the sentence “the object in front of me is a barn,” and let R be the sentence “the object in front of me is red.” (1) B, “I believe that the object in front of me is a barn.” At the metalevel, by the problem description, this is not knowledge, and we cannot claim KB. (2) (B ∧ R), “I believe that the object in front of me is a red barn.” At the metalevel, this is actually knowledge, e.g., K(B ∧ R) holds. (3) (B∧R → B), a knowledge assertion of a logical axiom. This is obviously knowledge, i.e., K(B ∧ R → B). Within this formalization, it appears that epistemic closure in its modal form (1.2) is violated: K(B∧R), and K(B∧R → B) hold, whereas, by (1), we cannot claim KB. The modal language here does not seem to help resolving this issue. Next consider the Red Barn Example in justification logic where t:F is interpreted as “I believe F for reason t.” Let u be a specific individual justification for belief that B, and v for belief that B ∧ R. In addition, let a be a justification for the logical truth B ∧ R → B. Then the list of assumptions is (i) u:B, “u is a reason to believe that the object in front of me is a barn”; (ii) v:(B ∧ R), “v is a reason to believe that the object in front of me is a red barn”; (iii) a:(B ∧ R → B). On the metalevel, the problem description states that (ii) and (iii) are cases of knowledge, and not merely belief, whereas (i) is belief, which is not knowledge. Here is how the formal reasoning goes: (iv) a:(B ∧ R → B) → (v:(B ∧ R) → [a·v]:B), by principle (1.1); (v) v:(B ∧ R) → [a·v]:B, from 3 and 4, by propositional logic; (vi) [a·v]:B, from 2 and 5, by propositional logic.
4
Why Justification Logic?
Notice that conclusion (vi) is [a · v]:B, and not u:B; epistemic closure holds. By reasoning in justification logic it was concluded that [a·v]:B is a case of knowledge, i.e., “I know B for reason a · v.” The fact that u:B is not a case of knowledge does not spoil the closure principle because the latter claims knowledge specifically for [a·v]:B. Hence after observing a red fac¸ade, I indeed know B, but this knowledge has nothing to do with (i), which remains a case of belief rather than of knowledge. The justification logic formalization represents the situation fairly. Tracking justifications represents the structure of the Red Barn Example in a way that is not captured by traditional epistemic modal tools. The justification logic formalization models what seems to be happening in such a case; closure of knowledge under logical entailment is maintained even though “barn” is not perceptually known. One could devise a formalization of the Red Barn Example in a bimodal language with distinct modalities for knowledge and belief. However, it seems that such a resolution must involve reproducing justification tracking arguments in a way that obscures, rather than reveals, the truth. Such a bimodal formalization would distinguish u:B from [a · v]:B not because they have different reasons (which reflects the true epistemic structure of the problem), but rather because the former is labeled “belief” and the latter “knowledge.” But what if one needs to keep track of a larger number of different unrelated reasons? By introducing a multiplicity of distinct modalities and then imposing various assumptions governing the interrelationships between these modalities, one would essentially end up with a reformulation of the language of justification logic itself (with distinct terms replaced by distinct modalities). This suggests that there may not be a satisfactory “halfway point” between a modal language and the language of justification logic, at least inasmuch as one tries to capture the essential structure of examples involving the deductive nature of knowledge.
1.2 Mathematical Logic Tradition According to Brouwer, truth in constructive (intuitionistic) mathematics means the existence of a proof, cf. Troelstra and van Dalen (1988). In 1931–34, Heyting and Kolmogorov gave an informal description of the intended proof-based semantics for intuitionistic logic (Kolmogoroff, 1932; Heyting, 1934), which is now referred to as the Brouwer–Heyting–Kolmogorov (BHK) semantics. According to the BHK conditions, a formula is “true” if it hasa proof. Further-
1.2 Mathematical Logic Tradition
5
more, a proof of a compound statement is connected to proofs of its components in the following way: • a proof of A ∧ B consists of a proof of proposition A and a proof of proposition B, • a proof of A∨B is given by presenting either a proof of A or a proof of B, • a proof of A → B is a construction transforming proofs of A into proofs of B, • falsehood ⊥ is a proposition, which has no proof; ¬A is shorthand for A → ⊥. This provides a remarkably useful informal way of understanding what is and what is not intuitionistically acceptable. For instance, consider the classical tautology (P ∨ Q) ↔ (P ∨ (Q ∧ ¬P)), where we understand ↔ as mutual implication. And we understand ¬P as P → ⊥, so that a proof of ¬P would amount to a construction converting any proof of P into a proof of ⊥. Because ⊥ has no proof, this amounts to a proof that P has no proof—a refutation of P. According to BHK semantics the implication from right to left in (P ∨ Q) ↔ (P ∨ (Q ∧ ¬P)) should be intuitionistically valid, by the following argument. Given a proof of P ∨ (Q ∧ ¬P) it must be that we are given a proof of one of the disjuncts. If it is P, we have a proof of one of P ∨ Q. If it is Q ∧ ¬P, we have proofs of both conjuncts, hence a proof of Q, and hence again a proof of one of P ∨ Q. Thus we may convert a proof of P ∨ (Q ∧ ¬P) into a proof of P ∨ Q. On the other hand, (P ∨ Q) → (P ∨ (Q ∧ ¬P)) is not intuitionistically valid according to the BHK ideas. Suppose we are given a proof of P ∨ Q. If we have a proof of the disjunct P, we have a proof of P ∨ Q. But if we have a proof of Q, there is no reason to suppose we have a refutation of P, and so we cannot conclude we have a proof of Q ∧ ¬P, and things stop here. Kolmogorov explicitly suggested that the proof-like objects in his interpretation (“problem solutions”) came from classical mathematics (Kolmogoroff, 1932). Indeed, from a foundational point of view this reflects Kolmogorov’s and G¨odel’s goal to define intuitionism within classical mathematics. From this standpoint, intuitionistic mathematics is not a substitute for classical mathematics, but helps to determine what is constructive in the latter. The fundamental value of the BHK semantics for the justification logic project is that informally but unambiguously BHK suggests treating justifications, here mathematical proofs, as objects with operations. In G¨odel (1933), G¨odel took the first step toward developing a rigorous proof-based semantics for intuitionism. G¨odel considered the classical modal logic S4 to be a calculus describing properties of provability:
6
Why Justification Logic?
(1) (2) (3) (4)
Axioms and rules of classical propositional logic, (F → G) → (F → G), F → F, F → F, `F . (5) Rule of necessitation: ` F
Based on Brouwer’s understanding of logical truth as provability, G¨odel defined a translation tr(F) of the propositional formula F in the intuitionistic language into the language of classical modal logic: tr(F) is obtained by prefixing every subformula of F with the provability modality . Informally speaking, when the usual procedure of determining classical truth of a formula is applied to tr(F), it will test the provability (not the truth) of each of F’s subformulas, in agreement with Brouwer’s ideas. From G¨odel’s results and the McKinseyTarski work on topological semantics for modal logic (McKinsey and Tarski, 1948), it follows that the translation tr(F) provides a proper embedding of the Intuitionistic Propositional Calculus, IPC, into S4, i.e., an embedding of intuitionistic logic into classical logic extended by the provability operator. IPC ` F
⇔
S4 ` tr(F).
(1.3)
Conceptually, this defines IPC in S4. Still, G¨odel’s original goal of defining intuitionistic logic in terms of classical provability was not reached because the connection of S4 to the usual mathematical notion of provability was not established. Moreover, G¨odel noted that the straightforward idea of interpreting modality F as F is provable in a given formal system T contradicted his second incompleteness theorem. Indeed, (F → F) can be derived in S4 by the rule of necessitation from the axiom F → F. On the other hand, interpreting modality as the predicate of formal provability in theory T and F as contradiction converts this formula into a false statement that the consistency of T is internally provable in T . The situation after G¨odel (1933) can be described by the following figure where “X ,→ Y” should be read as “X is interpreted in Y”: IPC ,→ S4 ,→ ? ,→ CLASSICAL PROOFS.
In a public lecture in Vienna in 1938, G¨odel observed that using the format of explicit proofs t is a proof of F
(1.4)
can help in interpreting his provability calculus S4 (G¨odel, 1938). Unfortunately, G¨odel (1938) remained unpublished until 1995, by which time the
1.2 Mathematical Logic Tradition
7
G¨odelian logic of explicit proofs had already been rediscovered, axiomatized as the Logic of Proofs LP, and supplied with completeness theorems connecting it to both S4 and classical proofs (Artemov, 1995, 2001). The Logic of Proofs LP became the first in the justification logic family. Proof terms in LP are nothing but BHK terms understood as classical proofs. With LP, propositional intuitionistic logic received the desired rigorous BHK semantics: IPC ,→ S4 ,→ LP ,→ CLASSICAL PROOFS .
Several well-known mathematical notions that appeared prior to justification logic have sometimes been perceived as related to the BHK idea: Kleene realizability (Troelstra, 1998), Curry–Howard isomorphism (Girard et al., 1989; Troelstra and Schwichtenberg, 1996), Kreisel–Goodman theory of constructions (Goodman, 1970; Kreisel, 1962, 1965), just to name a few. These interpretations have been very instrumental for understanding intuitionistic logic, though none of them qualifies as the BHK semantics. Kleene realizability revealed a fundamental computational content of formal intuitionistic derivations; however it is still quite different from the intended BHK semantics. Kleene realizers are computational programs rather than proofs. The predicate “r realizes F” is not decidable, which leads to some serious deviations from intuitionistic logic. Kleene realizability is not adequate for the intuitionistic propositional calculus IPC. There are realizable propositional formulas not derivable in IPC (Rose, 1953).2 The Curry–Howard isomorphism transliterates natural derivations in IPC to typed λ-terms, thus providing a generic functional reading for logical derivations. However, the foundational value of this interpretation is limited because, as proof objects, Curry–Howard λ-terms denote nothing but derivations in IPC itself and thus yield a circular provability semantics for the latter. An attempt to formalize the BHK semantics directly was made by Kreisel in his theory of constructions (Kreisel, 1962, 1965). The original variant of the theory was inconsistent; difficulties already occurred at the propositional level. In Goodman (1970) this was fixed by introducing a stratification of constructions into levels, which ruined the BHK character of this semantics. In particular, a proof of A → B was no longer a construction that could be applied to any proof of A. 2
Kleene himself denied any connection of his realizability with the BHK interpretation.
8
Why Justification Logic?
1.3 Hyperintensionality Justification logic offers a formal framework for hyperintensionality. The hyperintensional paradox was formulated in Cresswell (1975). It is well known that it seems possible to have a situation in which there are two propositions p and q which are logically equivalent and yet are such that a person may believe the one but not the other. If we regard a proposition as a set of possible worlds then two logically equivalent propositions will be identical, and so if “x believes that” is a genuine sentential functor, the situation described in the opening sentence could not arise. I call this the paradox of hyperintensional contexts. Hyperintensional contexts are simply contexts which do not respect logical equivalence.
Starting with Cresswell himself, several ways of dealing with this have been proposed. Generally, these involve adding more layers to familiar possible world approaches so that some way of distinguishing between logically equivalent sentences is available. Cresswell suggested that the syntactic form of sentences be taken into account. Justification logic, in effect, does this through its mechanism for handling justifications for sentences. Thus justification logic addresses some of the central issues of hyperintensionality but, as a bonus, we automatically have an appropriate proof theory, model theory, complexity estimates, and a broad variety of applications. A good example of a hyperintensional context is the informal language used by mathematicians conversing with each other. Typically when a mathematician says he or she knows something, the understanding is that a proof is at hand, but this kind of knowledge is essentially hyperintensional. For instance Fermat’s Last Theorem, FLT, is logically equivalent to 0 = 0 because both are provable and hence denote the same proposition, as this is understood in modal logic. However, the context of proofs distinguishes them immediately because a proof of 0 = 0 is not necessarily a proof of FLT, and vice versa. To formalize mathematical speech, the justification logic LP is a natural choice because t:X was designed to have characteristics of “t is a proof of X.” The fact that propositions X and Y are equivalent in LP, that LP ` X ↔ Y, does not warrant the equivalence of the corresponding justification assertions, and typically t:X and t:Y are not equivalent, t:X 6↔ t:Y. Indeed, as we will see, this is the case for every justification logic. Going further LP, and justification logic in general, is not only sufficiently refined to distinguish justification assertions for logically equivalent sentences, but it also provides flexible machinery to connect justifications of equivalent sentences and hence to maintain constructive closure properties desirable for a logic system. For example, let X and Y be provably equivalent, i.e., there is a proof u of X ↔ Y, and so u:(X ↔ Y) is provable in LP. Suppose also
1.4 Awareness
9
that v is a proof of X, and so v:X. It has already been mentioned that this does not mean v is a proof of Y—this is a hyperintensional context. However within the framework of justification logic, building on the proofs of X and of X ↔ Y, we can construct a proof term f (u, v), which represents the proof of Y and so f (u, v):Y is provable. In this respect, justification logic goes beyond Cresswell’s expectations: Logically equivalent sentences display different but constructively controlled epistemic behavior.
1.4 Awareness The logical omniscience problem is that in epistemic logics all tautologies are known and knowledge is closed under consequence, both of which are unreasonable. In Fagin and Halpern (1988) a simple mechanism for avoiding the problems was introduced. One adds to the usual Kripke model structure an awareness function A indicating for each world which formulas the agent is aware of at this world. Then a formula is taken to be known at a possible world u if (1) the formula is true at all worlds accessible from u (the Kripkean condition for knowledge) and (2) the agent is aware of the formula at u. The awareness function A can serve as a practical tool for blocking knowledge of an arbitrary set of formulas. However, as logical structures, awareness models exhibit abnormal behavior due to the lack of natural closure properties. For example, the agent can know A ∧ A but be unaware of A and hence not know it. Fitting models for justification logic, presented in Chapter 4, use a forcing definition reminiscent of the one from awareness models: For any given justification t, the justification assertion t:F holds at world u iff (1) F holds at all worlds v accessible from u and (2) t is an admissible evidence for F at u, u ∈ E(s, F), read as “u is a possible world at which s is relevant evidence for F.” The principal difference is that postulated operations on justifications relate to natural closure conditions on admissible evidence functions E in justification logic models. Indeed, this idea has been explored in Sedl´ar (2013), which works with the language of LP and thinks of it as a multiagent modal logic, and taking justification terms as agents (more properly, actions of agents). This shows that justification logic models absorb the usual epistemic themes of awareness, group agency, and dynamics in a natural way.
10
Why Justification Logic?
1.5 Paraconsistency Justification logic offers a well-principled approach to paraconsistency, which looks for noncollapsing logical ways of dealing with contradictory sets of assumptions, e.g., {A, ¬A}. The following obvious observation shows how to convert any set of assumptions Γ = {A1 , A2 , A3 , . . .} into a logically consistent set of sentences while maintaining all the intrinsic structure of Γ. Informally, instead of (perhaps inconsistently) assuming that Γ holds, we assume only that each sentence A from Γ has a justification, i.e., ~x : Γ = {x1:A1 , x2:A2 , x3:A3 , . . .}. It is easy to see that for each Γ, the set ~x:Γ is consistent in what will be our basic justification logic J. For example, for Γ = {A, ¬A}, ~x : Γ = {x1:A, x2:¬A}, states that x1 is a justification for A and x2 is a justification for ¬A. Within justification logic J in which no factivity (or even consistency) of justifications is assumed, the set of assumptions {x1:A, x2:¬A} is consistent, unlike the original set of assumptions {A, ¬A}. There is nothing paraconsistent, magical, or artificial in reasoning from ~x:Γ in justification logic J. In practical terms, this means we gain the ability to effectively reason about inconsistent data sets, keeping track of justifications and their dependencies, with the natural possibility to draw meaningful conclusions even when some assumed justifications from ~x:Γ become compromised and should be discharged.
2 The Basics of Justification Logic
In this chapter we discuss matters of syntax and axiomatics. All material is propositional, and will be so until Chapter 10. Justification logics are closely related to modal logics, so we start briefly with them in order to fix the basic notation. And just as normal modal logics all extend a single simplest example, K, all justification logics extend a single simplest example, J0 . We will begin our discussion with modal logics, then we will discuss the justification logic J0 in detail, and finally we will extend things to the most common and bestknown justification logics. A much broader family of justification logics will be discussed in Chapter 8.
2.1 Modal Logics All propositional formulas throughout this book are built up from a countable family of propositional variables. We use P, Q, . . . as propositional variables, with subscripts if necessary, and we follow the usual convention that these are all distinct. As our main propositional connective we have implication, →. We have negation, ¬, which we will take as primitive, or defined using the propositional constant ⊥ representing falsehood, as convenient and appropriate at the time. We also use conjunction, ∧, disjunction, ∨, and equivalence, ↔, and these too may be primitive or defined depending on circumstances. We omit outer parentheses in formulas when it will do no harm. We usually have a single modal necessity operator. It will generally be represented by though in epistemic contexts it may be represented by K. A dual operator representing possibility, ♦, is a defined operator and actually plays little role here. There is much work on epistemic logics with multiple agents, and there is some study of justification counterparts for them. When 11
12
The Basics of Justification Logic
discussing these and their connections with modal logics, we will subscript the modal operators just described. To date, no justification logic corresponding to a nonnormal modal logic has been introduced, so only normal modal logics will appear here. A normal modal logic is a set of modal formulas that contains all tautologies and all formulas of the form (X → Y) → (X → Y) and is closed under uniform substitution, modus ponens, and necessitation (if X is present, so is X). The smallest normal modal logic is K; it is a subset of all normal modal logics. The logic K has a standard axiom system. Axioms are all tautologies (or enough of them) and all formulas of the form (X → Y) → (X → Y). Rules are Modus Ponens X, X → Y ⇒ Y and Necessitation X ⇒ X. We are not actually interested in the vast collection of normal modal logics, but only in those for which a Hilbert system exists, having an axiomatization using a finite set of axiom schemes. In practice, this means adding axiom schemes to the axiomatization for K just given. We assume everybody knows axiom systems like T, K4, S4, and so on. We will refer to such logics as axiomatically formulated. Of course semantics plays a big role in modal logics, but we postpone a discussion for the time being.
2.2 Beginning Justification Logics Justification logics, syntactically, are like modal logics except that justification terms take the place of . Justification terms are intended to represent reasons or justifications for formulas. They have structure that encodes reasoning that has gone into them. We begin our formal presentation here. Definition 2.1 (Justification Term) up as follows.
The set Tm of justification terms is built
(1) There is a set of justification variables, x, y, . . . , x1 , y1 , . . . . Every justification variable is a justification term. (2) There is a set of justification constants, a, b, . . . , a1 , b1 , . . . . Every justification constant is a justification term. (3) There are binary operation symbols, + and ·. If u and v are justification terms, so are (u + v) and (u · v). (4) There may be additional function symbols, f , g, . . . , f1 , g1 , . . . , of various arities. Which ones are present depends on the logic in question. If f is an nplace justification function symbol of the logic, and t1 , . . . , tn are justification terms, f (t1 , . . . , tn ) is a justification term.
2.2 Beginning Justification Logics
13
Neither + nor · is assumed to be commutative or associative, and there is no distributive law. We do, however, allow ourselves the notational convenience of omitting parentheses with multiple occurrences of ·, assuming associativity to the left. Thus, for instance, a · b · c · d is short for (((a · b) · c) · d). We make the same assumption concerning +, though it actually plays a much lesser role. Also we will generally assume that · binds more strongly than +, writing a·b+c instead of (a · b) + c for instance. Definition 2.2 (Justification Formula) The set of justification formulas, Fm, is built in the usual recursive way, as follows. (1) There is a set Var of propositional variables, P, Q, . . . , P1 , Q1 , . . . (these are also known as propositional letters). Every propositional variable is a justification formula. (2) ⊥ (falsehood) is a justification formula. (3) If X and Y are justification formulas, so is (X → Y). (4) If t is a justification term and X is a justification formula, then t:X is a justification formula. We will sometimes use other propositional connectives, ∧, ∨, ↔, which we can think of as defined connectives, or primitive as convenient. Outer parentheses may be omitted in formulas if no confusion will result. If justification term t has a complex structure we generally will write [t]:X, using square brackets as a visual aid. Square brackets have no theoretical significance. In a modal formula, is supposed to express that something is necessary, or known, or obligatory, or some such thing, but it does not say why. A justification term encodes this missing information; it provides the why absent from modal formulas. This is what their structure is for. Justification variables stand for arbitrary justification terms, and substitution for them is examined beginning with Definition 2.17. Justification constants stand for reasons that are not further analyzed—typically they are reasons for axioms. Their role is discussed in more detail once constant specifications are introduced, in Definition 10.32. The · operation corresponds to modus ponens. If X → Y is so for reason s and X is so for reason t, then Y is so for reason s · t. (Reasons are not unique—Y may be true for other reasons too.) The + operation is a kind of weakening. If X is so for either reason s or reason t, then s + t is also a reason for X. Other operations on justification terms, if present, correspond to features peculiar to particular modal logics and will be discussed as they come up.
14
The Basics of Justification Logic
2.3 J0 , the Simplest Justification Logic As we will see, there are some justification logics having a version of a necessitation rule; there are others that do not. Some justification logics are closed under substitution of formulas for propositional variables, others are not. Allowing such a range of behavior is essential to enable us to capture and study the interactions of important features of modal logics that are sometimes hidden from us. But one consequence is, there is no good justification analog of the family of normal modal logics. Still, all justification logics have a common core, which we call J0 , and it is a kind of analog of the weakest normal modal logic, K, even though there is nothing structural we can point to as determining a “normal” justification logic apart from giving an axiomatization. In this section we present J0 axiomatically; subsequently we discuss what must be added to get the general family of justification logics. Definition 2.3 (Justification logic J0 ) The language of J0 has no justification function symbols beyond the basic two binary ones + and ·. The axiom schemes are as follows.
Classical: All tautologies (or enough of them) Application: All formulas of the form s:(X → Y) → (t:X → [s · t]:Y) Sum: All formulas of the forms s:X → [s + t]:X and t:X → [s + t]:X The only J0 rule of inference is Modus Ponens, X, X → Y ⇒ Y. J0 is a very weak justification logic. It is, for instance, incapable of proving that any formula has a justification, see Section 3.2. Reasoning in J0 is analogous to reasoning in the modal logic K without a necessitation rule! What we can do in J0 is derive interesting facts about justifications provided we make explicit what other formulas we would need to have justifications for. We give an example to illustrate this. To help bring out the points we want to make, if (X1 ∧ . . . ∧ Xn ) → Y is provable in J0 we may write X1 , . . . , Xn `J0 Y. Order of formulas and placement of parentheses in the conjunction of the Xi don’t matter because we have classical logic to work with. In modal K, a common first example of a theorem is (X ∧Y) → (X ∧Y). Here is the closest we can come to this in J0 . Our presentation is very much abbreviated.
2.4 Justification Logics in General
15
Example 2.4 Assume u, v, and w are justification variables. 1. 2. 3. 4. 5.
u:((X ∧ Y) → X) → (w:(X ∧ Y) → [u · w]:X) v:((X ∧ Y) → Y) → (w:(X ∧ Y) → [v · w]:Y) (u:((X ∧ Y) → X) ∧ v:((X ∧ Y) → y)) → (w:(X ∧ Y) → [u · w]:X) (u:((X ∧ Y) → X) ∧ v:((X ∧ Y) → y)) → (w:(X ∧ Y) → [v · w]:Y) (u:((X ∧ Y) → X) ∧ v:((X ∧ Y) → y)) → (w:(X ∧ Y) → ([u · w]:X ∧ [v · w]:Y))
Application Axiom Application Axiom Classical Logic on 1, 2 Classical Logic on 1, 2 Classical Logic on 3, 4
So we have shown that u:((X ∧ Y) → X), v:((X ∧ Y) → Y) `J0 (w:(X ∧ Y) → ([u · w]:X ∧ [v · w]:Y)) which we can read as an analog of (X ∧Y) → (X ∧Y) as follows. In J0 , for any w there are justification terms t1 and t2 such that w:(X ∧ Y) → (t1:X ∧ t2:Y), provided we have justifications for the tautologies (X ∧ Y) → X and (X ∧ Y) → Y. Note that t1 = u · w and t2 = v · w are different terms. But, making use of the Sum Axiom scheme, these can be brought together as t1 + t2 = u · w + v · w. It is important to understand that justifications, when they exist, are not unique.
2.4 Justification Logics in General The core justification logic J0 is extended to form other justification logics using two quite different types of machinery. First, one can add new operations on justification terms, besides the basic + and ·, along with axiom schemes governing their use, similar to Sum and Application. This is directly analogous to the way axiom schemes are added to K to create other modal logics. Second, one can specify which truths of logic we assume we have justifications for. This is related to the roles u:((X ∧ Y) → X) and v:((X ∧ Y) → Y) play in Example 2.4. We devote most of this section to the second kind of extension. It is, in fact, the intended role for justification constants that, up to now, have not been used for anything special. For the time being let us assume we have a justification logic resulting from the addition of function symbols and axiom schemes to J0 . The details don’t matter for now, but it should be understood that our axioms may go beyond those for J0 . Axioms of justification logics, like axioms generally, are simply assumed and are not analyzed further. The role of justification constant symbols is to
16
The Basics of Justification Logic
serve as reasons or justifications for axioms. If A is an axiom, we can simply announce that constant symbol c plays the role of a justification for it. It may be that some axioms are assumed to have such justifications, but not necessarily all. Suppose we look at Example 2.4 again, and suppose we have decided that (X ∧ Y) → X is an axiom for which we have a specific justification, let us say the constant symbol c plays this role. Similarly let us say the constant symbol d represents a justification for (X ∧ Y) → Y. Examining the derivation given in Example 2.4, it is easy to see that if we replace the variable u throughout by c, and the variable v throughout by d we still have a derivation, but one of c:((X ∧ Y) → X), d:((X ∧ Y) → Y) `J0 (w:(X ∧ Y) → ([c · w]:X ∧ [d · w]:Y)). If we add c:((X ∧ Y) → X) and d:((X ∧ Y) → Y) to our axioms for J0 , we can simply prove the formula (w:(X ∧ Y) → ([c · w]:X ∧ [d · w]:Y)). Roughly speaking, a constant specification tells us what axioms we have justifications for and which constants justify these axioms. As we just saw, we can use a constant specification as a source of additional axioms. But there is an important complication. If A is an axiom and constant symbol c justifies it, c:A conceptually also acts like an axiom, and it too may have its own justification. Then a constant symbol, say d, could come in here too, as a justification for c:A, and thus we might want to assume d:c:A. This repeats further, of course. For many purposes exact details don’t matter much, so how constants are used, and for what purposes, is turned into a kind of parameter of our logics, called a constant specification. Definition 2.5 (Constant Specification) A constant specification CS for a given justification logic is a set of formulas meeting the following conditions. (1) Members of CS are of the form cn:cn−1: . . . c1:A where n > 0, A is an axiom of JL, and each ci is a constant symbol. (2) If cn:cn−1: . . . c1:A is in CS where n > 1, then cn−1: . . . c1:A is in CS too. Thus CS contains all intermediate specifications for whatever it contains. One reason why constant specifications are treated as parameters can be discovered through a close look at Definition 2.3. It does not really provide an axiomatization for J0 , but rather a scheme for axiomatizations. The axioms called Classical in that definition are not fully specified, and in common practice many classical logic axiomatizations are in use. Any set sufficient to derive all tautologies will do. Then many different axiomatizations for J0 would meet the required conditions, and similarly for any justification logic extending J0
2.4 Justification Logics in General
17
as well. Because constants are supposed to be associated with axioms, a variety of constant specifications come up naturally. And because details like this often matter very little, treating constant specifications as a parameter is quite reasonable. Definition 2.6 (Logic of Justifications with a Constant Specification) Let JL be a justification logic, resulting from the addition of function symbols to the language of J0 and corresponding axiom schemes to those of J0 . Let CS be a constant specification for JL. Then JL(CS) is the logic JL with members of CS added as axioms (not axiom schemes), still with modus ponens as the only rule of inference. Constant specifications allow for great flexibility. A constant specification could associate many constants with a single axiom, or none at all. Allowing for many could be of use in tracking where particular pieces of reasoning come from. Allowing none might be appropriate in dealing with axioms that have some simple form, say X → X, but where the size of X is astronomical. Or again we might want to use the same constant for all instances of a particular axiom schema, or we might want to keep the instances clearly distinguishable. If details don’t matter at all for some particular purpose, we might want to associate a single constant symbol with every axiom, no matter what the form. Such a constant would simply be a record that a formula is an axiom, without going into particulars. Some conditions on constant specifications have shown themselves to be of special interest and have been given names. Here is a list of the most common. There are others. Definition 2.7 (Constant Specification Conditions) Let CS be a constant specification for a justification logic JL. The following requirements may be placed on CS. Empty: CS = ∅. This amounts to working with JL itself. Epistemically one can think of it as appropriate for the reasoning of a completely skeptical agent. Finite: CS is a finite set of formulas. This is fully representative because any specific derivation in a Justification Logic will be finite and so will involve only a finite set of constants. Schematic: If A and B are both instances of the same axiom scheme, c:A ∈ CS if and only if c:B ∈ CS, for every constant symbol c. Total: For each axiom A of JL and any constants c1 , c2 , . . . , cn we have cn:cn−1: . . . c1:A ∈ CS.
18
The Basics of Justification Logic
Axiomatically Appropriate: For every axiom A and for every n > 0 there are constant symbols ci so that cn:cn−1: . . . c1:A ∈ CS. The working of justification axiom systems is specified as follows. Definition 2.8 (Consequence) Suppose JL is a justification logic, CS is a constant specification for JL, S is an arbitrary set of formulas (not schemes), and X is a single formula. By S `JL(CS) X we mean there is a finite sequence of formulas, ending with X, in which each formula is either a instance of an axiom scheme of JL, a member of CS, a member of S , or follows from earlier formulas by modus ponens. If {Y1 , . . . , Yk } `JL(CS) X we will simplify notation and write Y1 , . . . , Yk `JL(CS) X. If ∅ `JL(CS) X we just write `JL(CS) X, or sometimes even JL(CS) ` X. When presenting examples of axiomatic derivations using a constant speciCS fication CS, we will write c + X as a suggestive way of saying that c:X ∈ CS, and we will say “c justifies X”. We conclude this section with some examples of theorems of justification logics. For these we work with JL(CS) where JL is any justification logic and CS is any constant specification for it that is axiomatically appropriate. We assume JL has been axiomatized taking all tautologies as axioms, though taking “enough” would give similar results once we have Theorem 2.14. Example 2.9 P → P is a theorem of any normal modal logic. It has more than one proof. We could simply note that it is an instance of a tautology, X → X. Or we could begin with P → P, a simpler instance of this tautology, apply necessitation getting (P → P), and then use the K axiom (P → P) → (P → P) and modus ponens to conclude P → P. While these are different modal derivations, the result is the same. But when we mimic the steps in JL(CS), they lead to different results. Let t be an arbitrary justification term. Then t:P → t:P is a theorem of JL(CS) because it is an instance of a tautology. But also P → P is an instance of a tautology and so, because JL(CS) is assumed axiomatically appropriate, the constant specification assigns some constant to it; say c:(P → P) ∈ CS. Because c:(P → P) → (t:P → [c · t]:P) is an axiom, t:P → [c · t]:P follows by modus ponens. In justification logic, instead of a single formula P → P with two proofs we have two different theorems that contain traces of their proofs. Both t:P → t:P and t:P → [c · t]:P say that if there is a reason for P, then there is a reason for P, but they give us different reasons. One of the first things everybody shows axiomatically when studying modal
2.4 Justification Logics in General
19
logic is that (P ∧ Q) ↔ (P ∧ Q) is provable in K, and thus is provable in every axiom system for a normal modal logic. But the argument from left to right is quite different from the argument from right to left. Because justification theorems contain traces of their proofs, we should not expect a single justification analog of this modal equivalence, but rather separate results for the left–right implication and for the right–left implication. Example 2.10 Here is a justification derivation corresponding to the usual modal argument for (P ∧ Q) → (P ∧ Q). 1. 2. 3. 4. 5. 6. 7. 8. 9.
(P ∧ Q) → P c:((P ∧ Q) → P) c:((P ∧ Q) → P) → (t:(P ∧ Q) → [c · t]:P) t:(P ∧ Q) → [c · t]:P (P ∧ Q) → Q d:((P ∧ Q) → Q) d:((P ∧ Q) → Q) → (t:(P ∧ Q) → [d · t]:Q) t:(P ∧ Q) → [d · t]:Q t:(P ∧ Q) → ([c · t]:P ∧ [d · t]:Q)
tautology cons spec Application Axiom mod pon on 2, 3 tautology cons spec Application Axiom mod pon on 6, 7 class log on 4, 8
CS Then t:(P ∧ Q) → ([c · t]:P ∧ [d · t]:Q) is a theorem of JL(CS) where c + CS ((P ∧ Q) → P) and d + ((P ∧ Q) → Q).
Example 2.11 A justification counterpart of the modal theorem (P∧Q) → (P ∧ Q) follows. 1. 2. 3. 4. 5. 6.
P → (Q → (P ∧ Q)) c:(P → (Q → (P ∧ Q))) c:(P → (Q → (P ∧ Q))) → (t:P → [c · t]:(Q → (P ∧ Q))) t:P → [c · t]:(Q → (P ∧ Q)) [c · t]:(Q → (P ∧ Q)) → (u:Q → [c · t · u]:(P ∧ Q)) (t:P ∧ u:Q) → [c · t · u]:(P ∧ Q)
tautology cons spec Application Axiom mod pon on 2, 3 Application Axiom class log on 4, 5
CS So (t:P ∧ u:Q) → [c · t · u]:(P ∧ Q) is a theorem of JL(CS) where c + (P → (Q → (P ∧ Q))).
Our final example illustrates the use of +, which has not come up so far. It is for handling situations where there is more than one explanation needed for something, as in a proof by cases. At first glance this seems rather minor, but + turns out to play a vital role when we come to realization results. Example 2.12 (X ∨ Y) → (X ∨ Y) is a theorem of K with an elementary
20
The Basics of Justification Logic
proof that we omit. Let us construct a counterpart in JL(CS), still assuming that CS is axiomatically appropriate and all tautologies are axioms. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.
X → (X ∨ Y) c:(X → (X ∨ Y)) c:(X → (X ∨ Y)) → (t:X → [c · t]:(X ∨ Y)) t:X → [c · t]:(X ∨ Y) Y → (X ∨ Y) d:(Y → (X ∨ Y)) d:(Y → (X ∨ Y)) → (u:Y → [d · u]:(X ∨ Y)) u:Y → [d · u]:(X ∨ Y) [c · t]:(X ∨ Y) → [c · t + d · u]:(X ∨ Y) [d · u]:(X ∨ Y) → [c · t + d · u]:(X ∨ Y) t:X → [c · t + d · u]:(X ∨ Y) u:Y → [c · t + d · u]:(X ∨ Y) (t:X ∨ u:Y) → [c · t + d · u]:(X ∨ Y)
tautology cons spec Application Axiom mod pon on 2, 3 tautology cons spec Application Axiom mod pon on 6, 7 Sum Axiom Sum Axiom clas log on 4, 9 clas log on 8, 10 class log on 11, 12
The consequents of 4 and 8 both provide reasons for X ∨ Y, but the reasons are different. We have used + to combine them, getting a justification analog of (X ∨ Y) → (X ∨ Y).
2.5 Fundamental Properties of Justification Logics All justification logics have certain common and useful properties. Some features are identical with those of classical logic; others have twists that are special to justification logics. This section is devoted to ones we will use over and over. Throughout this section let JL be a justification logic and CS be a constant specification for it. Because the only rule of inference is modus ponens the classical proof of the deduction theorem applies. We thus have S , X `JL(CS) Y if and only if S `JL(CS) X → Y. Because formal proofs are finite we have compactness that, combined with the deduction theorem, tells us: S `JL(CS) X if and only if `JL(CS) Y1 → (Y2 → . . . → (Yn → X) . . .) for some Y1 , Y2 , . . . , Yn ∈ S . These are exactly like their classical counterparts. Furthermore, the following serves as a replacement for the modal Necessitation Rule. Definition 2.13 (Internalization) JL has the internalization property relative to constant specification CS provided, if `JL(CS) X then for some justification term t, `JL(CS) t:X. In addition we say that JL has the strong internalization
2.5 Fundamental Properties of Justification Logics
21
property if t contains no justification variables and no justification operation or function symbols except ·. That is, t is built up from justification constants using only ·. Theorem 2.14 If CS is an axiomatically appropriate constant specification for JL then JL has the strong internalization property relative to CS. Proof By induction on proof length. Suppose `JL(CS) X and the result is known for formulas with shorter proofs. If X is an axiom of JL or a member of CS, there is a justification constant c such that c:X is in CS, and so c:X is provable. If X follows from earlier proof lines by modus ponens from Y → X and Y then, by the induction hypothesis, `JL(CS) s:(Y → X) and `JL(CS) t:Y for some s, t containing no justification variables, and with · as the only function symbol. Using the J0 Application Axiom s:(Y → X) → (t:Y → [s · t]:X) and modus ponens, we get `JL(CS) [s · t]:X. If X is provable using an axiomatically appropriate constant specification so is t:X, and the term t constructed in the preceding proof actually internalizes the steps of the axiomatic proof of X, hence the name internalization. Of course different proofs of X will produce different justification terms. Here is an extremely simple example, but one that is already sufficient to illustrate this point. Example 2.15 Assume JL is a justification logic, CS is an axiomatically appropriate constant specification for it, and all tautologies are axioms of JL. CS P → P is a tautology so c + (P → P) for some c. Then c:(P → P) is a theorem, and we have the justification term c internalizing a proof of P → P. Here is a more roundabout proof of P → P, giving us a more complicated internalizing term. Following the method in the proof of Theorem 2.14, we construct the internalization simultaneously. 1. 2. 3. 4. 5.
(P → (P → P)) → ((P → P) → (P → P)) P → (P → P) (P → P) → (P → P) P→P P→P
tautology tautology mod pon on 1, 2 tautology mod pon on 3, 4
d (cons spec) e (cons spec) d·e c (cons spec) d·e·c
This time we get a justification term d · e · c, or more properly (d · e) · c, CS internalizing a proof of P → P, where c + ((P → (P → P)) → ((P → P) → CS CS (P → P))), e + (P → (P → P)), and c + (P → P). The problem of finding a “simplest” justification term is related to the problem of finding the “simplest” proof of a provable formula. It is not entirely clear what this actually means.
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The Basics of Justification Logic
Corollary 2.16 (Lifting Lemma) Suppose JL is a justification logic that has the internalization property relative to CS (in particular, if CS is axiomatically appropriate). If X1 , . . . , Xn `JL(CS) Y then for any justification terms t1 , . . . , tn there is a justification term u so that t1:X1 , . . . , tn:Xn `JL(CS) u:Y. Proof The proof is by induction on n. If n = 0 this is simply the definition of Internalization. Suppose the result is known for n, and we have X1 , . . . , Xn , Xn+1 `JL(CS) Y. We show that for any t1 , . . . , tn , tn+1 there is some u so that t1:X1 , . . . , tn:Xn , tn+1: Xn+1 `JL(CS) u:Y. Using the deduction theorem, X1 , . . . , Xn `JL(CS) (Xn+1 → Y). By the induction hypothesis, for some v we have t1:X1 , . . . , tn:Xn `JL(CS) v:(Xn+1 → Y). Now v:(Xn+1 → Y) → (tn+1:Xn+1 → [v · tn+1 ]:Y) is an axiom hence t1:X1 , . . ., tn:Xn `JL(CS) (tn+1:Xn+1 → [v · tn+1 ]:Y). By modus ponens, t1:X1 , . . . , tn:Xn , tn+1:Xn+1 `JL(CS) [v · tn+1 ]:Y, so take u to be v · tn+1 . Next we move on to the role of justification variables. We said earlier, rather informally, that variables stood for arbitrary justification terms. In order to make this somewhat more precise, we need to introduce substitution. Definition 2.17 (Substitution) A substitution is a function σ mapping some set of justification variables to justification terms, with no variable in the domain of σ mapping to itself. We are only interested in substitutions with finite domain. If the domain of σ is {x1 , . . . , xn }, and each xi maps to justification term ti , it is standard to represent this substitution by (x1 /t1 , . . . , xn /tn ), or sometimes as (~x/~t). For a justification formula X the result of applying a substitution σ is denoted Xσ; likewise tσ is the result of applying substitution σ to justification term t. Substitutions map axioms of a justification logic into axioms (because axiomatization is by schemes), and they preserve modus ponens applications. But one must be careful because the role of constants changes with a substitution. Suppose CS is a constant specification, A is an axiom, and c:A is added to a proof where this addition is authorized by CS. Because axiomatization is by schemes Aσ is also an axiom, but if we add c:Aσ to a proof this may no longer meet constant specification CS. A new constant specification, call it (CS)σ, can be computed from the original one: put c:Aσ ∈ (CS)σ just in case c:A ∈ CS, for any c. If CS was axiomatically appropriate, CS ∪ (CS)σ will also be. So, if X is provable using an axiomatically appropriate constant specification CS, the same will be true for Xσ, not using the original constant specification but rather using CS ∪ (CS)σ. But this is more detail than we generally need to care about. The following suffices for much of our purposes.
2.6 The First Justification Logics
23
Theorem 2.18 (Substitution Closure) Suppose JL is a justification logic and X is provable in JL using some (axiomatically appropriate) constant specification. Then for any substitution σ, Xσ is also provable in JL using some (axiomatically appropriate) constant specification. We introduce some special notation that suppresses details of constant specifications when we don’t need to care about these details. Definition 2.19 Let JL be a justification logic. We write `JL X as short for: there is some axiomatically appropriate constant specification CS so that `JL(CS) X. Theorem 2.20 Let JL be a justification logic. (1) If `JL X then `JL Xσ for any substitution σ. (2) If `JL X and `JL X → Y then `JL Y. Proof Item (1) is directly from Theorem 2.18. For item (2), suppose `JL X and `JL X → Y. Then there are axiomatically appropriate constant specifications CS1 and CS2 so that `JL(CS1 ) X and `JL(CS2 ) X → Y. Now CS1 ∪ CS2 will also be an axiomatically appropriate constant specification and `JL(CS1 ∪CS2 ) X and `JL(CS1 ∪CS2 ) X → Y, so `JL(CS1 ∪CS2 ) Y and hence `JL Y. In fact, it is easy to check that `JL X if and only if `JL(T) X, where T is the total constant specification. This gives an alternate, and easier, characterization.
2.6 The First Justification Logics In this section and the next we present a number of specific examples of justification logics. We have tried to be systematic in naming these justification logics. Of course modal logic is not entirely consistent in this respect, and justification logic inherits some of its quirks, but we have tried to minimize anomalies. Naming Conventions: It is common to name modal logics by stringing axiom names after K; for instance KT, K4, and so on, with K itself as the simplest case. When we have justification logic counterparts for such modal logics, we will use the same name except with a substitution of J for K; for instance JT, J4, and so on. There is a problem here because a modal logic generally has more than one justification counterpart (if it has any). We will specify which one we have in mind. Formally, JT, J4, and so on result from the addition of axiom schemes, justification function symbols, and a constant specification to
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The Basics of Justification Logic
J0 . When details of a constant specification matter, we will write things like JT(CS), J4(CS), and so on, making the constant specification explicit. We will rarely refer to J0 again because its definition does not actually allow for a constant specification. From now on we will use J for J0 extended with some constant specification, and we will write J(CS) when explicitness is called for. Note that J0 can be thought of as J(∅), where ∅ is the empty constant
specification. The general subject of justification logics evolved from the aforementioned G¨odel–Artemov project, which embeds intuitionistic logic into the modal logic S4, which in turn embeds into the justification logic known as LP (for logic of proofs). It is with LP and its standard sublogics that we are concerned in this section. These are the best-known justification logics, just as K, T (or sometimes KT), S4 (or sometimes KT4), and a few others are the best-known modal logics. For the time being the notion of a justification logic being a counterpart of a modal logic will be an intuitive one. A proper definition will be given in Section 7.2. With two exceptions, the justification logics examined here arise by adding additional operations to the + and · common to all justification logics. The first exception involves factivity, with which we begin. Factivity for modal logics is represented by the axiom scheme X → X. If we think of the necessity operator epistemically, this would be written KX → X. It asserts that if X is known, then X is so. The justification counterpart is the following axiom scheme. Factivity t:X → X Factivity is a strong assumption: justifications cannot be wrong. Nonetheless, if the justification is a mathematical proof, factivity is something mathematicians are generally convinced of. If we think of knowledge as justified, true belief, factivity is built in. Philosophers generally understand justified, true belief to be inherent in knowledge, but not sufficient, see Gettier (1963). The modal axiom scheme X → X is called T. The weakest normal modal logic including all instances of this scheme is KT, sometimes abbreviated simply as T. We use JT for J plus Factivity and, as noted earlier, we use JT(CS) when a specific constant specification is needed. Note that the languages of JT and J are the same. There is one more such example, after which additional operation symbols must be brought in. Consistency is an important special case of Factivity. Modally it can be represented in several ways. One can assume the axiom scheme X → ♦X. In
2.6 The First Justification Logics
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any normal modal logic this turns out to coincide with assuming ¬⊥ (where ⊥ represents falsehood), or equivalently ⊥ → ⊥, which is a very special instance of X → X. If one thinks of as representing provability, ¬⊥ says falsehood is not provable—consistency. Suppose one thinks of deontically, so that X is read that X is obligatory, or perhaps that it is obligatory to bring about a state in which X. Then X → ♦X, or equivalently X → ¬¬X says that if X is obligatory, then ¬X isn’t—a plausible condition on obligations. It is because of this interesting deontic reading that any of the equivalent versions is commonly called D, standing for deontic. Any of these has a justification counterpart. We adopt the following version. Consistency t:⊥ → ⊥ JD is J plus Consistency. Note that JT extends JD.
Positive Introspection is a common assumption about an agent’s knowledge: If an agent knows something, the agent knows that it is known; an agent can introspect about the knowledge he or she possesses. In logics of knowledge it is formulated as KX → KKX. If one understands as representing provability in formal arithmetic, it is possible to prove that a proof is correct: X → X. To formulate a justification logic counterpart, Artemov introduced a one-place function symbol on justification terms, denoted ! and written in prefix position. The intuitive idea is that if t is a justification of something, !t is a justification that t is, indeed, such a justification. Note that the basic language of justification logics has been extended, and this must be reflected in any constant speci
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Subfield of mathematics
For Quine's theory sometimes called "Mathematical Logic", see New Foundations. For other uses, see Logic (disambiguation).
Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power. However, it can also include uses of logic to characterize correct mathematical reasoning or to establish foundations of mathematics.
Since its inception, mathematical logic has both contributed to and been motivated by the study of foundations of mathematics. This study began in the late 19th century with the development of axiomatic frameworks for geometry, arithmetic, and analysis. In the early 20th century it was shaped by David Hilbert's program to prove the consistency of foundational theories. Results of Kurt Gödel, Gerhard Gentzen, and others provided partial resolution to the program, and clarified the issues involved in proving consistency. Work in set theory showed that almost all ordinary mathematics can be formalized in terms of sets, although there are some theorems that cannot be proven in common axiom systems for set theory. Contemporary work in the foundations of mathematics often focuses on establishing which parts of mathematics can be formalized in particular formal systems (as in reverse mathematics) rather than trying to find theories in which all of mathematics can be developed.
Subfields and scope
[edit]
The Handbook of Mathematical Logic in 1977 makes a rough division of contemporary mathematical logic into four areas:
set theory
model theory
recursion theory, and
proof theory and constructive mathematics (considered as parts of a single area).
Additionally, sometimes the field of computational complexity theory is also included as part of mathematical logic.[2] Each area has a distinct focus, although many techniques and results are shared among multiple areas. The borderlines amongst these fields, and the lines separating mathematical logic and other fields of mathematics, are not always sharp. Gödel's incompleteness theorem marks not only a milestone in recursion theory and proof theory, but has also led to Löb's theorem in modal logic. The method of forcing is employed in set theory, model theory, and recursion theory, as well as in the study of intuitionistic mathematics.
The mathematical field of category theory uses many formal axiomatic methods, and includes the study of categorical logic, but category theory is not ordinarily considered a subfield of mathematical logic. Because of its applicability in diverse fields of mathematics, mathematicians including Saunders Mac Lane have proposed category theory as a foundational system for mathematics, independent of set theory. These foundations use toposes, which resemble generalized models of set theory that may employ classical or nonclassical logic.
History
[edit]
Mathematical logic emerged in the mid-19th century as a subfield of mathematics, reflecting the confluence of two traditions: formal philosophical logic and mathematics. Mathematical logic, also called 'logistic', 'symbolic logic', the 'algebra of logic', and, more recently, simply 'formal logic', is the set of logical theories elaborated in the course of the nineteenth century with the aid of an artificial notation and a rigorously deductive method. Before this emergence, logic was studied with rhetoric, with calculationes, through the syllogism, and with philosophy. The first half of the 20th century saw an explosion of fundamental results, accompanied by vigorous debate over the foundations of mathematics.
Early history
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Further information: History of logic
Theories of logic were developed in many cultures in history, including China, India, Greece and the Islamic world. Greek methods, particularly Aristotelian logic (or term logic) as found in the Organon, found wide application and acceptance in Western science and mathematics for millennia. The Stoics, especially Chrysippus, began the development of predicate logic. In 18th-century Europe, attempts to treat the operations of formal logic in a symbolic or algebraic way had been made by philosophical mathematicians including Leibniz and Lambert, but their labors remained isolated and little known.
19th century
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In the middle of the nineteenth century, George Boole and then Augustus De Morgan presented systematic mathematical treatments of logic. Their work, building on work by algebraists such as George Peacock, extended the traditional Aristotelian doctrine of logic into a sufficient framework for the study of foundations of mathematics. In 1847. Vatroslav Bertić made substantial work on algebraization of logic, independently from Boole.[8] Charles Sanders Peirce later built upon the work of Boole to develop a logical system for relations and quantifiers, which he published in several papers from 1870 to 1885.
Gottlob Frege presented an independent development of logic with quantifiers in his Begriffsschrift, published in 1879, a work generally considered as marking a turning point in the history of logic. Frege's work remained obscure, however, until Bertrand Russell began to promote it near the turn of the century. The two-dimensional notation Frege developed was never widely adopted and is unused in contemporary texts.
From 1890 to 1905, Ernst Schröder published Vorlesungen über die Algebra der Logik in three volumes. This work summarized and extended the work of Boole, De Morgan, and Peirce, and was a comprehensive reference to symbolic logic as it was understood at the end of the 19th century.
Foundational theories
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Concerns that mathematics had not been built on a proper foundation led to the development of axiomatic systems for fundamental areas of mathematics such as arithmetic, analysis, and geometry.
In logic, the term arithmetic refers to the theory of the natural numbers. Giuseppe Peano published a set of axioms for arithmetic that came to bear his name (Peano axioms), using a variation of the logical system of Boole and Schröder but adding quantifiers. Peano was unaware of Frege's work at the time. Around the same time Richard Dedekind showed that the natural numbers are uniquely characterized by their induction properties. Dedekind proposed a different characterization, which lacked the formal logical character of Peano's axioms. Dedekind's work, however, proved theorems inaccessible in Peano's system, including the uniqueness of the set of natural numbers (up to isomorphism) and the recursive definitions of addition and multiplication from the successor function and mathematical induction.
In the mid-19th century, flaws in Euclid's axioms for geometry became known. In addition to the independence of the parallel postulate, established by Nikolai Lobachevsky in 1826, mathematicians discovered that certain theorems taken for granted by Euclid were not in fact provable from his axioms. Among these is the theorem that a line contains at least two points, or that circles of the same radius whose centers are separated by that radius must intersect. Hilbert developed a complete set of axioms for geometry, building on previous work by Pasch. The success in axiomatizing geometry motivated Hilbert to seek complete axiomatizations of other areas of mathematics, such as the natural numbers and the real line. This would prove to be a major area of research in the first half of the 20th century.
The 19th century saw great advances in the theory of real analysis, including theories of convergence of functions and Fourier series. Mathematicians such as Karl Weierstrass began to construct functions that stretched intuition, such as nowhere-differentiable continuous functions. Previous conceptions of a function as a rule for computation, or a smooth graph, were no longer adequate. Weierstrass began to advocate the arithmetization of analysis, which sought to axiomatize analysis using properties of the natural numbers. The modern (ε, δ)-definition of limit and continuous functions was already developed by Bolzano in 1817, but remained relatively unknown. Cauchy in 1821 defined continuity in terms of infinitesimals (see Cours d'Analyse, page 34). In 1858, Dedekind proposed a definition of the real numbers in terms of Dedekind cuts of rational numbers, a definition still employed in contemporary texts.
Georg Cantor developed the fundamental concepts of infinite set theory. His early results developed the theory of cardinality and proved that the reals and the natural numbers have different cardinalities. Over the next twenty years, Cantor developed a theory of transfinite numbers in a series of publications. In 1891, he published a new proof of the uncountability of the real numbers that introduced the diagonal argument, and used this method to prove Cantor's theorem that no set can have the same cardinality as its powerset. Cantor believed that every set could be well-ordered, but was unable to produce a proof for this result, leaving it as an open problem in 1895.
20th century
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In the early decades of the 20th century, the main areas of study were set theory and formal logic. The discovery of paradoxes in informal set theory caused some to wonder whether mathematics itself is inconsistent, and to look for proofs of consistency.
In 1900, Hilbert posed a famous list of 23 problems for the next century. The first two of these were to resolve the continuum hypothesis and prove the consistency of elementary arithmetic, respectively; the tenth was to produce a method that could decide whether a multivariate polynomial equation over the integers has a solution. Subsequent work to resolve these problems shaped the direction of mathematical logic, as did the effort to resolve Hilbert's Entscheidungsproblem, posed in 1928. This problem asked for a procedure that would decide, given a formalized mathematical statement, whether the statement is true or false.
Set theory and paradoxes
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Ernst Zermelo gave a proof that every set could be well-ordered, a result Georg Cantor had been unable to obtain. To achieve the proof, Zermelo introduced the axiom of choice, which drew heated debate and research among mathematicians and the pioneers of set theory. The immediate criticism of the method led Zermelo to publish a second exposition of his result, directly addressing criticisms of his proof. This paper led to the general acceptance of the axiom of choice in the mathematics community.
Skepticism about the axiom of choice was reinforced by recently discovered paradoxes in naive set theory. Cesare Burali-Forti was the first to state a paradox: the Burali-Forti paradox shows that the collection of all ordinal numbers cannot form a set. Very soon thereafter, Bertrand Russell discovered Russell's paradox in 1901, and Jules Richard discovered Richard's paradox.
Zermelo provided the first set of axioms for set theory. These axioms, together with the additional axiom of replacement proposed by Abraham Fraenkel, are now called Zermelo–Fraenkel set theory (ZF). Zermelo's axioms incorporated the principle of limitation of size to avoid Russell's paradox.
In 1910, the first volume of Principia Mathematica by Russell and Alfred North Whitehead was published. This seminal work developed the theory of functions and cardinality in a completely formal framework of type theory, which Russell and Whitehead developed in an effort to avoid the paradoxes. Principia Mathematica is considered one of the most influential works of the 20th century, although the framework of type theory did not prove popular as a foundational theory for mathematics.
Fraenkel proved that the axiom of choice cannot be proved from the axioms of Zermelo's set theory with urelements. Later work by Paul Cohen showed that the addition of urelements is not needed, and the axiom of choice is unprovable in ZF. Cohen's proof developed the method of forcing, which is now an important tool for establishing independence results in set theory.[27]
Symbolic logic
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Leopold Löwenheim and Thoralf Skolem obtained the Löwenheim–Skolem theorem, which says that first-order logic cannot control the cardinalities of infinite structures. Skolem realized that this theorem would apply to first-order formalizations of set theory, and that it implies any such formalization has a countable model. This counterintuitive fact became known as Skolem's paradox.
In his doctoral thesis, Kurt Gödel proved the completeness theorem, which establishes a correspondence between syntax and semantics in first-order logic. Gödel used the completeness theorem to prove the compactness theorem, demonstrating the finitary nature of first-order logical consequence. These results helped establish first-order logic as the dominant logic used by mathematicians.
In 1931, Gödel published On Formally Undecidable Propositions of Principia Mathematica and Related Systems, which proved the incompleteness (in a different meaning of the word) of all sufficiently strong, effective first-order theories. This result, known as Gödel's incompleteness theorem, establishes severe limitations on axiomatic foundations for mathematics, striking a strong blow to Hilbert's program. It showed the impossibility of providing a consistency proof of arithmetic within any formal theory of arithmetic. Hilbert, however, did not acknowledge the importance of the incompleteness theorem for some time.[a]
Gödel's theorem shows that a consistency proof of any sufficiently strong, effective axiom system cannot be obtained in the system itself, if the system is consistent, nor in any weaker system. This leaves open the possibility of consistency proofs that cannot be formalized within the system they consider. Gentzen proved the consistency of arithmetic using a finitistic system together with a principle of transfinite induction. Gentzen's result introduced the ideas of cut elimination and proof-theoretic ordinals, which became key tools in proof theory. Gödel gave a different consistency proof, which reduces the consistency of classical arithmetic to that of intuitionistic arithmetic in higher types.
The first textbook on symbolic logic for the layman was written by Lewis Carroll,[33] author of Alice's Adventures in Wonderland, in 1896.
Beginnings of the other branches
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Alfred Tarski developed the basics of model theory.
Beginning in 1935, a group of prominent mathematicians collaborated under the pseudonym Nicolas Bourbaki to publish Éléments de mathématique, a series of encyclopedic mathematics texts. These texts, written in an austere and axiomatic style, emphasized rigorous presentation and set-theoretic foundations. Terminology coined by these texts, such as the words bijection, injection, and surjection, and the set-theoretic foundations the texts employed, were widely adopted throughout mathematics.
The study of computability came to be known as recursion theory or computability theory, because early formalizations by Gödel and Kleene relied on recursive definitions of functions.[b] When these definitions were shown equivalent to Turing's formalization involving Turing machines, it became clear that a new concept – the computable function – had been discovered, and that this definition was robust enough to admit numerous independent characterizations. In his work on the incompleteness theorems in 1931, Gödel lacked a rigorous concept of an effective formal system; he immediately realized that the new definitions of computability could be used for this purpose, allowing him to state the incompleteness theorems in generality that could only be implied in the original paper.
Numerous results in recursion theory were obtained in the 1940s by Stephen Cole Kleene and Emil Leon Post. Kleene introduced the concepts of relative computability, foreshadowed by Turing, and the arithmetical hierarchy. Kleene later generalized recursion theory to higher-order functionals. Kleene and Georg Kreisel studied formal versions of intuitionistic mathematics, particularly in the context of proof theory.
Formal logical systems
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At its core, mathematical logic deals with mathematical concepts expressed using formal logical systems. These systems, though they differ in many details, share the common property of considering only expressions in a fixed formal language. The systems of propositional logic and first-order logic are the most widely studied today, because of their applicability to foundations of mathematics and because of their desirable proof-theoretic properties.[c] Stronger classical logics such as second-order logic or infinitary logic are also studied, along with Non-classical logics such as intuitionistic logic.
First-order logic
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Main article: First-order logic
First-order logic is a particular formal system of logic. Its syntax involves only finite expressions as well-formed formulas, while its semantics are characterized by the limitation of all quantifiers to a fixed domain of discourse.
Early results from formal logic established limitations of first-order logic. The Löwenheim–Skolem theorem (1919) showed that if a set of sentences in a countable first-order language has an infinite model then it has at least one model of each infinite cardinality. This shows that it is impossible for a set of first-order axioms to characterize the natural numbers, the real numbers, or any other infinite structure up to isomorphism. As the goal of early foundational studies was to produce axiomatic theories for all parts of mathematics, this limitation was particularly stark.
Gödel's completeness theorem established the equivalence between semantic and syntactic definitions of logical consequence in first-order logic. It shows that if a particular sentence is true in every model that satisfies a particular set of axioms, then there must be a finite deduction of the sentence from the axioms. The compactness theorem first appeared as a lemma in Gödel's proof of the completeness theorem, and it took many years before logicians grasped its significance and began to apply it routinely. It says that a set of sentences has a model if and only if every finite subset has a model, or in other words that an inconsistent set of formulas must have a finite inconsistent subset. The completeness and compactness theorems allow for sophisticated analysis of logical consequence in first-order logic and the development of model theory, and they are a key reason for the prominence of first-order logic in mathematics.
Gödel's incompleteness theorems establish additional limits on first-order axiomatizations. The first incompleteness theorem states that for any consistent, effectively given (defined below) logical system that is capable of interpreting arithmetic, there exists a statement that is true (in the sense that it holds for the natural numbers) but not provable within that logical system (and which indeed may fail in some non-standard models of arithmetic which may be consistent with the logical system). For example, in every logical system capable of expressing the Peano axioms, the Gödel sentence holds for the natural numbers but cannot be proved.
Here a logical system is said to be effectively given if it is possible to decide, given any formula in the language of the system, whether the formula is an axiom, and one which can express the Peano axioms is called "sufficiently strong." When applied to first-order logic, the first incompleteness theorem implies that any sufficiently strong, consistent, effective first-order theory has models that are not elementarily equivalent, a stronger limitation than the one established by the Löwenheim–Skolem theorem. The second incompleteness theorem states that no sufficiently strong, consistent, effective axiom system for arithmetic can prove its own consistency, which has been interpreted to show that Hilbert's program cannot be reached.
Other classical logics
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Many logics besides first-order logic are studied. These include infinitary logics, which allow for formulas to provide an infinite amount of information, and higher-order logics, which include a portion of set theory directly in their semantics.
The most well studied infinitary logic is L ω 1 , ω {\displaystyle L_{\omega _{1},\omega }} . In this logic, quantifiers may only be nested to finite depths, as in first-order logic, but formulas may have finite or countably infinite conjunctions and disjunctions within them. Thus, for example, it is possible to say that an object is a whole number using a formula of L ω 1 , ω {\displaystyle L_{\omega _{1},\omega }} such as
( x = 0 ) ∨ ( x = 1 ) ∨ ( x = 2 ) ∨ ⋯ . {\displaystyle (x=0)\lor (x=1)\lor (x=2)\lor \cdots .}
Higher-order logics allow for quantification not only of elements of the domain of discourse, but subsets of the domain of discourse, sets of such subsets, and other objects of higher type. The semantics are defined so that, rather than having a separate domain for each higher-type quantifier to range over, the quantifiers instead range over all objects of the appropriate type. The logics studied before the development of first-order logic, for example Frege's logic, had similar set-theoretic aspects. Although higher-order logics are more expressive, allowing complete axiomatizations of structures such as the natural numbers, they do not satisfy analogues of the completeness and compactness theorems from first-order logic, and are thus less amenable to proof-theoretic analysis.
Another type of logics are fixed-point logics that allow inductive definitions, like one writes for primitive recursive functions.
One can formally define an extension of first-order logic — a notion which encompasses all logics in this section because they behave like first-order logic in certain fundamental ways, but does not encompass all logics in general, e.g. it does not encompass intuitionistic, modal or fuzzy logic.
Lindström's theorem implies that the only extension of first-order logic satisfying both the compactness theorem and the downward Löwenheim–Skolem theorem is first-order logic.
Nonclassical and modal logic
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Main article: Non-classical logic
Modal logics include additional modal operators, such as an operator which states that a particular formula is not only true, but necessarily true. Although modal logic is not often used to axiomatize mathematics, it has been used to study the properties of first-order provability and set-theoretic forcing.
Intuitionistic logic was developed by Heyting to study Brouwer's program of intuitionism, in which Brouwer himself avoided formalization. Intuitionistic logic specifically does not include the law of the excluded middle, which states that each sentence is either true or its negation is true. Kleene's work with the proof theory of intuitionistic logic showed that constructive information can be recovered from intuitionistic proofs. For example, any provably total function in intuitionistic arithmetic is computable; this is not true in classical theories of arithmetic such as Peano arithmetic.
Algebraic logic
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Algebraic logic uses the methods of abstract algebra to study the semantics of formal logics. A fundamental example is the use of Boolean algebras to represent truth values in classical propositional logic, and the use of Heyting algebras to represent truth values in intuitionistic propositional logic. Stronger logics, such as first-order logic and higher-order logic, are studied using more complicated algebraic structures such as cylindric algebras.
Set theory
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Main article: Set theory
Set theory is the study of sets, which are abstract collections of objects. Many of the basic notions, such as ordinal and cardinal numbers, were developed informally by Cantor before formal axiomatizations of set theory were developed. The first such axiomatization, due to Zermelo, was extended slightly to become Zermelo–Fraenkel set theory (ZF), which is now the most widely used foundational theory for mathematics.
Other formalizations of set theory have been proposed, including von Neumann–Bernays–Gödel set theory (NBG), Morse–Kelley set theory (MK), and New Foundations (NF). Of these, ZF, NBG, and MK are similar in describing a cumulative hierarchy of sets. New Foundations takes a different approach; it allows objects such as the set of all sets at the cost of restrictions on its set-existence axioms. The system of Kripke–Platek set theory is closely related to generalized recursion theory.
Two famous statements in set theory are the axiom of choice and the continuum hypothesis. The axiom of choice, first stated by Zermelo, was proved independent of ZF by Fraenkel, but has come to be widely accepted by mathematicians. It states that given a collection of nonempty sets there is a single set C that contains exactly one element from each set in the collection. The set C is said to "choose" one element from each set in the collection. While the ability to make such a choice is considered obvious by some, since each set in the collection is nonempty, the lack of a general, concrete rule by which the choice can be made renders the axiom nonconstructive. Stefan Banach and Alfred Tarski showed that the axiom of choice can be used to decompose a solid ball into a finite number of pieces which can then be rearranged, with no scaling, to make two solid balls of the original size. This theorem, known as the Banach–Tarski paradox, is one of many counterintuitive results of the axiom of choice.
The continuum hypothesis, first proposed as a conjecture by Cantor, was listed by David Hilbert as one of his 23 problems in 1900. Gödel showed that the continuum hypothesis cannot be disproven from the axioms of Zermelo–Fraenkel set theory (with or without the axiom of choice), by developing the constructible universe of set theory in which the continuum hypothesis must hold. In 1963, Paul Cohen showed that the continuum hypothesis cannot be proven from the axioms of Zermelo–Fraenkel set theory. This independence result did not completely settle Hilbert's question, however, as it is possible that new axioms for set theory could resolve the hypothesis. Recent work along these lines has been conducted by W. Hugh Woodin, although its importance is not yet clear.
Contemporary research in set theory includes the study of large cardinals and determinacy. Large cardinals are cardinal numbers with particular properties so strong that the existence of such cardinals cannot be proved in ZFC. The existence of the smallest large cardinal typically studied, an inaccessible cardinal, already implies the consistency of ZFC. Despite the fact that large cardinals have extremely high cardinality, their existence has many ramifications for the structure of the real line. Determinacy refers to the possible existence of winning strategies for certain two-player games (the games are said to be determined). The existence of these strategies implies structural properties of the real line and other Polish spaces.
Model theory
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Main article: Model theory
Model theory studies the models of various formal theories. Here a theory is a set of formulas in a particular formal logic and signature, while a model is a structure that gives a concrete interpretation of the theory. Model theory is closely related to universal algebra and algebraic geometry, although the methods of model theory focus more on logical considerations than those fields.
The set of all models of a particular theory is called an elementary class; classical model theory seeks to determine the properties of models in a particular elementary class, or determine whether certain classes of structures form elementary classes.
The method of quantifier elimination can be used to show that definable sets in particular theories cannot be too complicated. Tarski established quantifier elimination for real-closed fields, a result which also shows the theory of the field of real numbers is decidable. He also noted that his methods were equally applicable to algebraically closed fields of arbitrary characteristic. A modern subfield developing from this is concerned with o-minimal structures.
Morley's categoricity theorem, proved by Michael D. Morley, states that if a first-order theory in a countable language is categorical in some uncountable cardinality, i.e. all models of this cardinality are isomorphic, then it is categorical in all uncountable cardinalities.
A trivial consequence of the continuum hypothesis is that a complete theory with less than continuum many nonisomorphic countable models can have only countably many. Vaught's conjecture, named after Robert Lawson Vaught, says that this is true even independently of the continuum hypothesis. Many special cases of this conjecture have been established.
Recursion theory
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Main article: Recursion theory
Recursion theory, also called computability theory, studies the properties of computable functions and the Turing degrees, which divide the uncomputable functions into sets that have the same level of uncomputability. Recursion theory also includes the study of generalized computability and definability. Recursion theory grew from the work of Rózsa Péter, Alonzo Church and Alan Turing in the 1930s, which was greatly extended by Kleene and Post in the 1940s.
Classical recursion theory focuses on the computability of functions from the natural numbers to the natural numbers. The fundamental results establish a robust, canonical class of computable functions with numerous independent, equivalent characterizations using Turing machines, λ calculus, and other systems. More advanced results concern the structure of the Turing degrees and the lattice of recursively enumerable sets.
Generalized recursion theory extends the ideas of recursion theory to computations that are no longer necessarily finite. It includes the study of computability in higher types as well as areas such as hyperarithmetical theory and α-recursion theory.
Contemporary research in recursion theory includes the study of applications such as algorithmic randomness, computable model theory, and reverse mathematics, as well as new results in pure recursion theory.
Algorithmically unsolvable problems
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An important subfield of recursion theory studies algorithmic unsolvability; a decision problem or function problem is algorithmically unsolvable if there is no possible computable algorithm that returns the correct answer for all legal inputs to the problem. The first results about unsolvability, obtained independently by Church and Turing in 1936, showed that the Entscheidungsproblem is algorithmically unsolvable. Turing proved this by establishing the unsolvability of the halting problem, a result with far-ranging implications in both recursion theory and computer science.
There are many known examples of undecidable problems from ordinary mathematics. The word problem for groups was proved algorithmically unsolvable by Pyotr Novikov in 1955 and independently by W. Boone in 1959. The busy beaver problem, developed by Tibor Radó in 1962, is another well-known example.
Hilbert's tenth problem asked for an algorithm to determine whether a multivariate polynomial equation with integer coefficients has a solution in the integers. Partial progress was made by Julia Robinson, Martin Davis and Hilary Putnam. The algorithmic unsolvability of the problem was proved by Yuri Matiyasevich in 1970.
Proof theory and constructive mathematics
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Main article: Proof theory
Proof theory is the study of formal proofs in various logical deduction systems. These proofs are represented as formal mathematical objects, facilitating their analysis by mathematical techniques. Several deduction systems are commonly considered, including Hilbert-style deduction systems, systems of natural deduction, and the sequent calculus developed by Gentzen.
The study of constructive mathematics, in the context of mathematical logic, includes the study of systems in non-classical logic such as intuitionistic logic, as well as the study of predicative systems. An early proponent of predicativism was Hermann Weyl, who showed it is possible to develop a large part of real analysis using only predicative methods.
Because proofs are entirely finitary, whereas truth in a structure is not, it is common for work in constructive mathematics to emphasize provability. The relationship between provability in classical (or nonconstructive) systems and provability in intuitionistic (or constructive, respectively) systems is of particular interest. Results such as the Gödel–Gentzen negative translation show that it is possible to embed (or translate) classical logic into intuitionistic logic, allowing some properties about intuitionistic proofs to be transferred back to classical proofs.
Recent developments in proof theory include the study of proof mining by Ulrich Kohlenbach and the study of proof-theoretic ordinals by Michael Rathjen.
Applications
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"Mathematical logic has been successfully applied not only to mathematics and its foundations (G. Frege, B. Russell, D. Hilbert, P. Bernays, H. Scholz, R. Carnap, S. Lesniewski, T. Skolem), but also to physics (R. Carnap, A. Dittrich, B. Russell, C. E. Shannon, A. N. Whitehead, H. Reichenbach, P. Fevrier), to biology (J. H. Woodger, A. Tarski), to psychology (F. B. Fitch, C. G. Hempel), to law and morals (K. Menger, U. Klug, P. Oppenheim), to economics (J. Neumann, O. Morgenstern), to practical questions (E. C. Berkeley, E. Stamm), and even to metaphysics (J. [Jan] Salamucha, H. Scholz, J. M. Bochenski). Its applications to the history of logic have proven extremely fruitful (J. Lukasiewicz, H. Scholz, B. Mates, A. Becker, E. Moody, J. Salamucha, K. Duerr, Z. Jordan, P. Boehner, J. M. Bochenski, S. [Stanislaw] T. Schayer, D. Ingalls)." "Applications have also been made to theology (F. Drewnowski, J. Salamucha, I. Thomas)."
Connections with computer science
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Main article: Logic in computer science
The study of computability theory in computer science is closely related to the study of computability in mathematical logic. There is a difference of emphasis, however. Computer scientists often focus on concrete programming languages and feasible computability, while researchers in mathematical logic often focus on computability as a theoretical concept and on noncomputability.
The theory of semantics of programming languages is related to model theory, as is program verification (in particular, model checking). The Curry–Howard correspondence between proofs and programs relates to proof theory, especially intuitionistic logic. Formal calculi such as the lambda calculus and combinatory logic are now studied as idealized programming languages.
Computer science also contributes to mathematics by developing techniques for the automatic checking or even finding of proofs, such as automated theorem proving and logic programming.
Descriptive complexity theory relates logics to computational complexity. The first significant result in this area, Fagin's theorem (1974) established that NP is precisely the set of languages expressible by sentences of existential second-order logic.
Foundations of mathematics
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Main article: Foundations of mathematics
In the 19th century, mathematicians became aware of logical gaps and inconsistencies in their field. It was shown that Euclid's axioms for geometry, which had been taught for centuries as an example of the axiomatic method, were incomplete. The use of infinitesimals, and the very definition of function, came into question in analysis, as pathological examples such as Weierstrass' nowhere-differentiable continuous function were discovered.
Cantor's study of arbitrary infinite sets also drew criticism. Leopold Kronecker famously stated "God made the integers; all else is the work of man," endorsing a return to the study of finite, concrete objects in mathematics. Although Kronecker's argument was carried forward by constructivists in the 20th century, the mathematical community as a whole rejected them. David Hilbert argued in favor of the study of the infinite, saying "No one shall expel us from the Paradise that Cantor has created."
Mathematicians began to search for axiom systems that could be used to formalize large parts of mathematics. In addition to removing ambiguity from previously naive terms such as function, it was hoped that this axiomatization would allow for consistency proofs. In the 19th century, the main method of proving the consistency of a set of axioms was to provide a model for it. Thus, for example, non-Euclidean geometry can be proved consistent by defining point to mean a point on a fixed sphere and line to mean a great circle on the sphere. The resulting structure, a model of elliptic geometry, satisfies the axioms of plane geometry except the parallel postulate.
With the development of formal logic, Hilbert asked whether it would be possible to prove that an axiom system is consistent by analyzing the structure of possible proofs in the system, and showing through this analysis that it is impossible to prove a contradiction. This idea led to the study of proof theory. Moreover, Hilbert proposed that the analysis should be entirely concrete, using the term finitary to refer to the methods he would allow but not precisely defining them. This project, known as Hilbert's program, was seriously affected by Gödel's incompleteness theorems, which show that the consistency of formal theories of arithmetic cannot be established using methods formalizable in those theories. Gentzen showed that it is possible to produce a proof of the consistency of arithmetic in a finitary system augmented with axioms of transfinite induction, and the techniques he developed to do so were seminal in proof theory.
A second thread in the history of foundations of mathematics involves nonclassical logics and constructive mathematics. The study of constructive mathematics includes many different programs with various definitions of constructive. At the most accommodating end, proofs in ZF set theory that do not use the axiom of choice are called constructive by many mathematicians. More limited versions of constructivism limit themselves to natural numbers, number-theoretic functions, and sets of natural numbers (which can be used to represent real numbers, facilitating the study of mathematical analysis). A common idea is that a concrete means of computing the values of the function must be known before the function itself can be said to exist.
In the early 20th century, Luitzen Egbertus Jan Brouwer founded intuitionism as a part of philosophy of mathematics. This philosophy, poorly understood at first, stated that in order for a mathematical statement to be true to a mathematician, that person must be able to intuit the statement, to not only believe its truth but understand the reason for its truth. A consequence of this definition of truth was the rejection of the law of the excluded middle, for there are statements that, according to Brouwer, could not be claimed to be true while their negations also could not be claimed true. Brouwer's philosophy was influential, and the cause of bitter disputes among prominent mathematicians. Kleene and Kreisel would later study formalized versions of intuitionistic logic (Brouwer rejected formalization, and presented his work in unformalized natural language). With the advent of the BHK interpretation and Kripke models, intuitionism became easier to reconcile with classical mathematics.
See also
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Philosophy portal
Mathematics portal
Argument
Informal logic
Universal logic
Knowledge representation and reasoning
Logic
List of computability and complexity topics
List of first-order theories
List of logic symbols
List of mathematical logic topics
List of set theory topics
Mereology
Propositional calculus
Well-formed formula
Notes
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References
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Undergraduate texts
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Walicki, Michał (2011). Introduction to Mathematical Logic. Singapore: World Scientific Publishing. ISBN 9789814343879.
Boolos, George; Burgess, John; Jeffrey, Richard (2002). Computability and Logic (4th ed.). Cambridge University Press. ISBN 9780521007580.
Crossley, J.N.; Ash, C.J.; Brickhill, C.J.; Stillwell, J.C.; Williams, N.H. (1972). What is mathematical logic?. London, Oxford, New York City: Oxford University Press. ISBN 9780198880875. Zbl 0251.02001.
Enderton, Herbert (2001). A mathematical introduction to logic (2nd ed.). Boston MA: Academic Press. ISBN 978-0-12-238452-3.
Fisher, Alec (1982). Formal Number Theory and Computability: A Workbook. (suitable as a first course for independent study) (1st ed.). Oxford University Press. ISBN 978-0-19-853188-3.
Hamilton, A.G. (1988). Logic for Mathematicians (2nd ed.). Cambridge University Press. ISBN 978-0-521-36865-0.
Ebbinghaus, H.-D.; Flum, J.; Thomas, W. (1994). Mathematical Logic (2nd ed.). New York City: Springer. ISBN 9780387942582.
Katz, Robert (1964). Axiomatic Analysis. Boston MA: D. C. Heath and Company.
Mendelson, Elliott (1997). Introduction to Mathematical Logic (4th ed.). London: Chapman & Hall. ISBN 978-0-412-80830-2.
Rautenberg, Wolfgang (2010). A Concise Introduction to Mathematical Logic (3rd ed.). New York City: Springer. doi:10.1007/978-1-4419-1221-3. ISBN 9781441912206.
Schwichtenberg, Helmut (2003–2004). Mathematical Logic (PDF). Munich: Mathematisches Institut der Universität München .
Shawn Hedman, A first course in logic: an introduction to model theory, proof theory, computability, and complexity, Oxford University Press, 2004, ISBN 0-19-852981-3. Covers logics in close relation with computability theory and complexity theory
van Dalen, Dirk (2013). Logic and Structure. Universitext. Berlin: Springer. doi:10.1007/978-1-4471-4558-5. ISBN 978-1-4471-4557-8.
Graduate texts
[edit]
Hinman, Peter G. (2005). Fundamentals of mathematical logic. A K Peters, Ltd. ISBN 1-56881-262-0.
Andrews, Peter B. (2002). An Introduction to Mathematical Logic and Type Theory: To Truth Through Proof (2nd ed.). Boston: Kluwer Academic Publishers. ISBN 978-1-4020-0763-7.
Barwise, Jon, ed. (1989). Handbook of Mathematical Logic. Studies in Logic and the Foundations of Mathematics. Amsterdam: Elsevier. ISBN 9780444863881.
Hodges, Wilfrid (1997). A shorter model theory. Cambridge University Press. ISBN 9780521587136.
Jech, Thomas (2003). Set Theory: Millennium Edition. Springer Monographs in Mathematics. Berlin, New York: Springer. ISBN 9783540440857.
Kleene, Stephen Cole.(1952), Introduction to Metamathematics. New York: Van Nostrand. (Ishi Press: 2009 reprint).
Kleene, Stephen Cole. (1967), Mathematical Logic. John Wiley. Dover reprint, 2002. ISBN 0-486-42533-9.
Shoenfield, Joseph R. (2001) [1967]. Mathematical Logic (2nd ed.). A K Peters. ISBN 9781568811352.
Troelstra, Anne Sjerp; Schwichtenberg, Helmut (2000). Basic Proof Theory. Cambridge Tracts in Theoretical Computer Science (2nd ed.). Cambridge University Press. ISBN 978-0-521-77911-1.
Research papers, monographs, texts, and surveys
[edit]
Augusto, Luis M. (2017). Logical consequences. Theory and applications: An introduction. London: College Publications. ISBN 978-1-84890-236-7.
Boehner, Philotheus (1950). Medieval Logic. Manchester.
Cohen, Paul J. (1966). Set Theory and the Continuum Hypothesis. Menlo Park CA: W. A. Benjamin.
Cohen, Paul J. (2008) [1966]. Set theory and the continuum hypothesis. Mineola NY: Dover Publications. ISBN 9780486469218.
J.D. Sneed, The Logical Structure of Mathematical Physics. Reidel, Dordrecht, 1971 (revised edition 1979).
Davis, Martin (1973). "Hilbert's tenth problem is unsolvable". The American Mathematical Monthly. 80 (3): 233–269. doi:10.2307/2318447. JSTOR 2318447.
Reprinted as an appendix in Martin Davis (1985). Computability and Unsolvability. Dover. ISBN 9780486614717.
Felscher, Walter (2000). "Bolzano, Cauchy, Epsilon, Delta". The American Mathematical Monthly. 107 (9): 844–862. doi:10.2307/2695743. JSTOR 2695743.
Ferreirós, José (2001). "The Road to Modern Logic-An Interpretation" (PDF). Bulletin of Symbolic Logic. 7 (4): 441–484. doi:10.2307/2687794. hdl:11441/38373. JSTOR 2687794. S2CID 43258676.
Hamkins, Joel David; Löwe, Benedikt (2007). "The modal logic of forcing". Transactions of the American Mathematical Society. 360 (4): 1793–1818. arXiv:math/0509616. doi:10.1090/s0002-9947-07-04297-3. S2CID 14724471.
Katz, Victor J. (1998). A History of Mathematics. Addison–Wesley. ISBN 9780321016188.
Morley, Michael (1965). "Categoricity in Power". Transactions of the American Mathematical Society. 114 (2): 514–538. doi:10.2307/1994188. JSTOR 1994188.
Soare, Robert I. (1996). "Computability and recursion". Bulletin of Symbolic Logic. 2 (3): 284–321. CiteSeerX 10.1.1.35.5803. doi:10.2307/420992. JSTOR 420992. S2CID 5894394.
Solovay, Robert M. (1976). "Provability Interpretations of Modal Logic". Israel Journal of Mathematics. 25 (3–4): 287–304. doi:10.1007/BF02757006. S2CID 121226261.
Woodin, W. Hugh (2001). "The Continuum Hypothesis, Part I" (PDF). Notices of the American Mathematical Society. 48 (6).
Classical papers, texts, and collections
[edit]
Banach, Stefan; Tarski, Alfred (1924). "Sur la décomposition des ensembles de points en parties respectivement congruentes" (PDF). Fundamenta Mathematicae (in French). 6: 244–277. doi:10.4064/fm-6-1-244-277.
Bochenski, Jozef Maria, ed. (1959). A Precis of Mathematical Logic. Synthese Library, Vol. 1. Translated by Otto Bird. Dordrecht: Springer. doi:10.1007/978-94-017-0592-9. ISBN 9789048183296.
Burali-Forti, Cesare (1897). A question on transfinite numbers. Reprinted in van Heijenoort 1976, pp. 104–111
Cantor, Georg (1874). "Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen" (PDF). Journal für die Reine und Angewandte Mathematik. 1874 (77): 258–262. doi:10.1515/crll.1874.77.258. S2CID 199545885. Carroll, Lewis (1896). Symbolic Logic. Kessinger Legacy Reprints. ISBN 9781163444955.
Dedekind, Richard (1872). Stetigkeit und irrationale Zahlen (in German). English translation as: "Consistency and irrational numbers".
Dedekind, Richard (1888). Was sind und was sollen die Zahlen?. Two English translations:
1963 (1901). Essays on the Theory of Numbers. Beman, W. W., ed. and trans. Dover.
1996. In From Kant to Hilbert: A Source Book in the Foundations of Mathematics, 2 vols, Ewald, William B., ed., Oxford University Press: 787–832.
Fraenkel, Abraham A. (1922). "Der Begriff 'definit' und die Unabhängigkeit des Auswahlsaxioms". Sitzungsberichte der Preussischen Akademie der Wissenschaften, Physikalisch-mathematische Klasse (in German). pp. 253–257. Reprinted in English translation as "The notion of 'definite' and the independence of the axiom of choice" in van Heijenoort 1976, pp. 284–289.
Frege, Gottlob (1879), Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens. Halle a. S.: Louis Nebert. Translation: Concept Script, a formal language of pure thought modelled upon that of arithmetic, by S. Bauer-Mengelberg in van Heijenoort 1976.
Frege, Gottlob (1884), Die Grundlagen der Arithmetik: eine logisch-mathematische Untersuchung über den Begriff der Zahl. Breslau: W. Koebner. Translation: J. L. Austin, 1974. The Foundations of Arithmetic: A logico-mathematical enquiry into the concept of number, 2nd ed. Blackwell.
Gentzen, Gerhard (1936). "Die Widerspruchsfreiheit der reinen Zahlentheorie". Mathematische Annalen. 112: 132–213. doi:10.1007/BF01565428. S2CID 122719892. Reprinted in English translation in Gentzen's Collected works, M. E. Szabo, ed., North-Holland, Amsterdam, 1969.
Gödel, Kurt (1929). Über die Vollständigkeit des Logikkalküls [Completeness of the logical calculus]. doctoral dissertation. University Of Vienna.
Gödel, Kurt (1930). "Die Vollständigkeit der Axiome des logischen Funktionen-kalküls" [The completeness of the axioms of the calculus of logical functions]. Monatshefte für Mathematik und Physik (in German). 37: 349–360. doi:10.1007/BF01696781. S2CID 123343522.
Gödel, Kurt (1931). "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I" [On Formally Undecidable Propositions of Principia Mathematica and Related Systems]. Monatshefte für Mathematik und Physik (in German). 38 (1): 173–198. doi:10.1007/BF01700692. S2CID 197663120.
Gödel, Kurt (1958). "Über eine bisher noch nicht benützte Erweiterung des finiten Standpunktes". Dialectica (in German). 12 (3–4): 280–287. doi:10.1111/j.1746-8361.1958.tb01464.x. Reprinted in English translation in Gödel's Collected Works, vol II, Solomon Feferman et al., eds. Oxford University Press, 1993.
van Heijenoort, Jean, ed. (1976) [1967]. From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931 (3rd ed.). Cambridge MA: Harvard University Press. ISBN 9780674324497. (pbk.).
Hilbert, David (1899). Grundlagen der Geometrie (in German). Leipzig: Teubner. English 1902 edition (The Foundations of Geometry) republished 1980, Open Court, Chicago.
Hilbert, David (1929). "Probleme der Grundlegung der Mathematik". Mathematische Annalen. 102: 1–9. doi:10.1007/BF01782335. S2CID 122870563. Lecture given at the International Congress of Mathematicians, 3 September 1928. Published in English translation as "The Grounding of Elementary Number Theory", in Mancosu 1998, pp. 266–273.
Hilbert, David; Bernays, Paul (1934). Grundlagen der Mathematik. I. Die Grundlehren der mathematischen Wissenschaften. Vol. 40. Berlin, New York City: Springer. ISBN 9783540041344. JFM 60.0017.02. MR 0237246.
Kleene, Stephen Cole (1943). "Recursive Predicates and Quantifiers". Transactions of the American Mathematical Society. 53 (1): 41–73. doi:10.2307/1990131. JSTOR 1990131.
Lobachevsky, Nikolai (1840). Geometrishe Untersuchungen zur Theorie der Parellellinien (in German). Reprinted in English translation as Robert Bonola, ed. (1955). "Geometric Investigations on the Theory of Parallel Lines". Non-Euclidean Geometry. Dover. ISBN 0-486-60027-0.
Löwenheim, Leopold (1915). "Über Möglichkeiten im Relativkalkül". Mathematische Annalen (in German). 76 (4): 447–470. doi:10.1007/BF01458217. ISSN 0025-5831. S2CID 116581304. Translated as "On possibilities in the calculus of relatives" in Jean van Heijenoort (1967). A Source Book in Mathematical Logic, 1879–1931. Harvard Univ. Press. pp. 228–251.
Mancosu, Paolo, ed. (1998). From Brouwer to Hilbert. The Debate on the Foundations of Mathematics in the 1920s. Oxford University Press.
Pasch, Moritz (1882). Vorlesungen über neuere Geometrie.
Peano, Giuseppe (1889). Arithmetices principia, nova methodo exposita (in Lithuanian). Excerpt reprinted in English translation as "The principles of arithmetic, presented by a new method"in van Heijenoort 1976, pp. 83–97.
Richard, Jules (1905). "Les principes des mathématiques et le problème des ensembles". Revue Générale des Sciences Pures et Appliquées (in French). 16: 541. Reprinted in English translation as "The principles of mathematics and the problems of sets" in van Heijenoort 1976, pp. 142–144.
Skolem, Thoralf (1920). "Logisch-kombinatorische Untersuchungen über die Erfüllbarkeit oder Beweisbarkeit mathematischer Sätze nebst einem Theoreme über dichte Mengen". Videnskapsselskapet Skrifter, I. Matematisk-naturvidenskabelig Klasse (in German). 6: 1–36.
Soare, Robert Irving (22 December 2011). "Computability Theory and Applications: The Art of Classical Computability" (PDF). Department of Mathematics. University of Chicago . Swineshead, Richard (1498). Calculationes Suiseth Anglici (in Lithuanian). Papie: Per Franciscum Gyrardengum.
Tarski, Alfred (1948). A decision method for elementary algebra and geometry. Santa Monica CA: RAND Corporation.
Turing, Alan M. (1939). "Systems of Logic Based on Ordinals". Proceedings of the London Mathematical Society. 45 (2): 161–228. doi:10.1112/plms/s2-45.1.161. hdl:21.11116/0000-0001-91CE-3.
Weyl, Hermann (1918). Das Kontinuum. Kritische Untersuchungen über die Grund lagen der Analysis (in German). Leipzig.
Zermelo, Ernst (1904). "Beweis, daß jede Menge wohlgeordnet werden kann". Mathematische Annalen (in German). 59 (4): 514–516. doi:10.1007/BF01445300. S2CID 124189935. Reprinted in English translation as "Proof that every set can be well-ordered" in van Heijenoort 1976, pp. 139–141.
Zermelo, Ernst (1908a). "Neuer Beweis für die Möglichkeit einer Wohlordnung". Mathematische Annalen (in German). 65: 107–128. doi:10.1007/BF01450054. ISSN 0025-5831. S2CID 119924143. Reprinted in English translation as "A new proof of the possibility of a well-ordering" in van Heijenoort 1976, pp. 183–198.
Zermelo, Ernst (1908b). "Untersuchungen über die Grundlagen der Mengenlehre". Mathematische Annalen. 65 (2): 261–281. doi:10.1007/BF01449999. S2CID 120085563.
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Famous Mathematicians from the Netherlands
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2013-12-19T00:00:00
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List of notable or famous mathematicians from the Netherlands, with bios and photos, including the top mathematicians born in the Netherlands and even some ...
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en
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/img/icons/touch-icon-iphone.png
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Ranker
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https://www.ranker.com/list/famous-mathematicians-from-netherlands/reference
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List of notable or famous mathematicians from the Netherlands, with bios and photos, including the top mathematicians born in the Netherlands and even some popular mathematicians who immigrated to the Netherlands. If you're trying to find out the names of famous Dutch mathematicians then this list is the perfect resource for you. These mathematicians are among the most prominent in their field, and information about each well-known mathematician from the Netherlands is included when available.
Johan de Witt and Tjalling Koopmans are only the beginning of the people on this list.
This historic mathematicians from the Netherlands list can help answer the questions "Who are some Dutch mathematicians of note?" and "Who are the most famous mathematicians from the Netherlands?" These prominent mathematicians of the Netherlands may or may not be currently alive, but what they all have in common is that they're all respected Dutch mathematicians.
Use this list of renowned Dutch mathematicians to discover some new mathematicians that you aren't familiar with. Don't forget to share this list by clicking one of the social media icons at the top or bottom of the page. {#nodes}
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7589
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https://www.academia.edu/47115957/Intuitionistic_completeness_of_first_order_logic
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Intuitionistic completeness of first-order logic
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2021-04-20T00:00:00
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Intuitionistic completeness of first-order logic
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https://www.academia.edu/47115957/Intuitionistic_completeness_of_first_order_logic
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Abstract: In 1933 Godel Introduced an axiomatic system, currently known as S4, for a logic of an absolute provability. The problem of finding a fair probability model for S4 was left open. In the current paper we demonstrate how the Intuitionistic propositional logic Int can be directly realized by proof polynomials. It is shown that Int is complete with respect to this proof realizability.
Let T be any of the three canonical truth theories CT− (Compositional truth without extra induction), FS− (Friedman–Sheard truth without extra induction), and KF− (Kripke–Feferman truth without extra induction), where the base theory of T is PA (Peano arithmetic). We show that T is feasibly reducible to PA, i.e., there is a polynomial time computable function f such that for any proof π of an arithmetical sentence φ in T , f(π) is a proof of φ in PA. In particular, T has at most polynomial speed-up over PA, in sharp contrast to the situation for T [B] for finitely axiomatizable base theories B.
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1 INDEX OF THE TROELSTRA ARCHIVE Introduction This index was mainly prepared in 2000 <strong>and</strong> 2001. 1 <strong>The</strong> archive itself was stored at the Mathematical <strong>Institute</strong> of the University of Amsterdam, afterwards (2001) brought to the Rijksarchief in Noord-Holl<strong>and</strong>. <strong>The</strong> length of the archive is circa 2.5 meters. <strong>The</strong> material in the archive has been divided into a number of major categories, distinguished by capitals in the numbering of items, e.g. B (scientific correspondence), C (courses given by Troelstra, seminars <strong>and</strong> colloquia conducted by him), P (all materials relating to his publications) etc. Usually these major categories are again subdivided. <strong>The</strong> description of the contents is not carried out with the same amount of detail <strong>for</strong> each (sub-) category. A lot of material in this index was already set up by A.S. Troelstra himself (especially publications, lectures <strong>and</strong> congresses). P. van Ulsen, Amsterdam, 2001. Curriculum vitae of Anne Sjerp Troelstra 2 10-VIII-1939 Born at Maartensdijk (Utrecht), the Netherl<strong>and</strong>s. 1951-1957 Secondary school: Lorentz Lyceum (gymnasium beta), Eindhoven. 1-IX-1957 Enrollment as a student at the University of Amsterdam. 25-III-1964 Passed doctoraalexamen in Mathematics, cum laude. (At that time a bit more than a M.Sc.) 1-IV-1964 Appointed ‘wetenschappelijk medewerker’ (approximately, assistant professor) at the Department of Mathematics of the University of Amsterdam. 15-VI-1966 Doctorate (Ph.D) in mathematics on the thesis ‘Intuitionistic general topology’ (thesis adviser Prof. Dr. A. Heyting). University of Amsterdam. 1-IX-1966 till 1-IX-1967. On leave as a visiting scholar at Stan<strong>for</strong>d University (departments of Mathematics <strong>and</strong> Philosophy) with a stipend from the Netherl<strong>and</strong>s Organization <strong>for</strong> the Advancement of Research (then ZWO, now called NWO). VIII-1968 Gave a series of ten lectures on Intuitionism at the Summer School on Proof <strong>The</strong>ory <strong>and</strong> Intuitionism at SUNY, Buffalo, New York. 1-IX-1968 Appointed ‘lector’ (associate professor) in mathematics at the University of Amsterdam. 1-IX-1970 Appointed ‘gewoon hoogleraar’ (full professor) in pure mathematics <strong>and</strong> foundations of mathematics at the University of Amsterdam. 4-VI-1976 Elected member of the Royal Dutch Academy of Sciences. 16-II-1996 Elected corresponding member of the Bavarian Academy of Sciences. 15-XI-1996 Received the F.L. Bauer–Prize of the ‘Bund der Freunde der Technischen Universität München’, <strong>for</strong> internationally outst<strong>and</strong>ing contributions to 1 <strong>The</strong> way Troelstra set up the Heyting Archive (now part of the Rijksarchief Noord-Holl<strong>and</strong> in Haarlem) <strong>and</strong> its index was a model <strong>for</strong> this archive <strong>and</strong> index. <strong>The</strong> Library of the Faculty of Sciences (Faculteit Natuurwetenschappen, Wiskunde en In<strong>for</strong>matica) of the University of Amsterdam, <strong>and</strong> especially the chief librarian H. Harmsen, gave me the opportunity to complete this index. 2 Data from the postscript file of A.S. Troelstra ( http://turing.wins.uva.nl/~anne/) . See also A.S. Troelstra, Looking back, ILLC magazine 2, (July 2000).
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https://www.scribd.com/document/438223923/History-of-Constructivism-in-the-20-century
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History of Constructivism in The 20 Century
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History of Constructivism in the 20 century - Free download as PDF File (.pdf), Text File (.txt) or read online for free. This document provides a history of constructivism in the 20th century. It discusses several trends within constructivism including finitism, which rejects abstract concepts and quantifiers and interprets existence claims constructively; intuitionism, which insists mathematical objects are mental constructions; and Bishop's constructivism. It outlines the key figures that developed these trends like Kronecker, Skolem, Hilbert, Bernays, and Bishop. The document also discusses related areas like metamathematics, predicativism, and actualism, which critiques finitism for not accounting for large numbers.
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History of Constructivism in The 20 Century
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https://dokumen.pub/download/justification-logic-reasoning-with-reasons-1108661106-9781108661102.html
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Justification Logic: Reasoning with Reasons 1108661106, 9781108661102
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Classical logic is concerned, loosely, with the behaviour of truths. Epistemic logic similarly is about the behaviour of...
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dokumen.pub
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Citation preview
Justification Logic Classical logic is concerned, loosely, with the behavior of truths. Epistemic logic similarly is about the behavior of known or believed truths. Justification logic is a theory of reasoning that enables the tracking of evidence for statements and therefore provides a logical framework for the reliability of assertions. This book, the first in the area, is a systematic account of the subject, progressing from modal logic through to the establishment of an arithmetic interpretation of intuitionistic logic. The presentation is mathematically rigorous but in a style that will appeal to readers from a wide variety of areas to which the theory applies. These include mathematical logic, artificial intelligence, computer science, philosophical logic and epistemology, linguistics, and game theory.
s e r g e i a rt e m ov is Distinguished Professor at the City University of New York. He is a specialist in mathematical logic, logic in computer science, control theory, epistemology, and game theory. He is credited with solving long-standing problems in constructive logic that had been left open by G¨odel and Kolmogorov since the 1930s. He has pioneered studies in the logic of proofs and justifications that render a new, evidence-based theory of knowledge and belief. The most recent focus of his interests is epistemic foundations of game theory. m e lv i n f i t t i n g is Professor Emeritus at the City University of New York. He has written or edited a dozen books and has worked in intensional logic, semantics for logic programming, theory of truth, and tableau systems for nonclassical logics. In 2012 he received the Herbrand Award from the Conference on Automated Deduction. He was on the faculty of the City University of New York from 1969 to his retirement in 2013, at Lehman College, and at the Graduate Center, where he was in the Departments of Mathematics, Computer Science, and Philosophy.
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Justification Logic Reasoning with Reasons S E R G E I A RT E M OV Graduate Center, City University of New York M E LV I N F I T T I N G Graduate Center, City University of New York
University Printing House, Cambridge CB2 8BS, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia 314–321, 3rd Floor, Plot 3, Splendor Forum, Jasola District Centre, New Delhi – 110025, India 79 Anson Road, #06–04/06, Singapore 079906 Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning, and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781108424912 DOI: 10.1017/9781108348034 © Sergei Artemov and Melvin Fitting 2019 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2019 Printed and bound in Great Britain by Clays Ltd, Elcograf S.p.A. A catalogue record for this publication is available from the British Library. Library of Congress Cataloging-in-Publication Data Names: Artemov, S. N., author. | Fitting, Melvin, 1942- author. Title: Justification logic : reasoning with reasons / Sergei Artemov (Graduate Center, City University of New York), Melvin Fitting (Graduate Center, City University of New York). Description: Cambridge ; New York, NY : Cambridge University Press, 2019. | Series: Cambridge tracts in mathematics ; 216 | Includes bibliographical references and index. Identifiers: LCCN 2018058431 | ISBN 9781108424912 (hardback : alk. paper) Subjects: LCSH: Logic, Symbolic and mathematical. | Inquiry (Theory of knowledge) | Science–Theory reduction. | Reasoning. Classification: LCC QA9 .A78 2019 | DDC 511.3–dc23 LC record available at https://lccn.loc.gov/2018058431 ISBN 978-1-108-42491-2 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.
To our wives, Lena and Roma.
Contents
Introduction 1 What Is This Book About? 2 What Is Not in This Book?
page x xii xvii
1
Why Justification Logic? 1.1 Epistemic Tradition 1.2 Mathematical Logic Tradition 1.3 Hyperintensionality 1.4 Awareness 1.5 Paraconsistency
1 1 4 8 9 10
2
The Basics of Justification Logic 2.1 Modal Logics 2.2 Beginning Justification Logics 2.3 J0 , the Simplest Justification Logic 2.4 Justification Logics in General 2.5 Fundamental Properties of Justification Logics 2.6 The First Justification Logics 2.7 A Handful of Less Common Justification Logics
11 11 12 14 15 20 23 27
3
The Ontology of Justifications 3.1 Generic Logical Semantics of Justifications 3.2 Models for J0 and J 3.3 Basic Models for Positive and Negative Introspection 3.4 Adding Factivity: Mkrtychev Models 3.5 Basic and Mkrtychev Models for the Logic of Proofs LP 3.6 The Inevitability of Possible Worlds: Modular Models 3.7 Connecting Justifications, Belief, and Knowledge 3.8 History and Commentary
31 31 36 38 39 42 42 45 46
vii
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Contents
4
Fitting Models 4.1 Modal Possible World Semantics 4.2 Fitting Models 4.3 Soundness Examples 4.4 Canonical Models and Completeness 4.5 Completeness Examples 4.6 Formulating Justification Logics
48 48 49 52 60 65 72
5
Sequents and Tableaus 5.1 Background 5.2 Classical Sequents 5.3 Sequents for S4 5.4 Sequent Soundness, Completeness, and More 5.5 Classical Semantic Tableaus 5.6 Modal Tableaus for K 5.7 Other Modal Tableau Systems 5.8 Tableaus and Annotated Formulas 5.9 Changing the Tableau Representation
75 75 76 79 81 84 90 91 93 95
6
Realization – How It Began 6.1 The Logic LP 6.2 Realization for LP 6.3 Comments
100 100 103 108
7
Realization – Generalized 7.1 What We Do Here 7.2 Counterparts 7.3 Realizations 7.4 Quasi-Realizations 7.5 Substitution 7.6 Quasi-Realizations to Realizations 7.7 Proving Realization Constructively 7.8 Tableau to Quasi-Realization Algorithm 7.9 Tableau to Quasi-Realization Algorithm Correctness 7.10 An Illustrative Example 7.11 Realizations, Nonconstructively 7.12 Putting Things Together 7.13 A Brief Realization History
110 110 112 113 116 118 120 126 128 131 133 135 138 139
8
The Range of Realization 8.1 Some Examples We Already Discussed 8.2 Geach Logics 8.3 Technical Results
141 141 142 144
Contents 8.4 8.5 8.6 8.7 8.8
Geach Justification Logics Axiomatically Geach Justification Logics Semantically Soundness, Completeness, and Realization A Concrete S4.2/JT4.2 Example Why Cut-Free Is Needed
ix 147 149 150 152 155
9
Arithmetical Completeness and BHK Semantics 9.1 Arithmetical Semantics of the Logic of Proofs 9.2 A Constructive Canonical Model for the Logic of Proofs 9.3 Arithmetical Completeness of the Logic of Proofs 9.4 BHK Semantics 9.5 Self-Referentiality of Justifications
158 158 161 165 174 179
10
Quantifiers in Justification Logic 10.1 Free Variables in Proofs 10.2 Realization of FOS4 in FOLP 10.3 Possible World Semantics for FOLP 10.4 Arithmetical Semantics for FOLP
181 182 186 191 212
11
Going Past Modal Logic 11.1 Modeling Awareness 11.2 Precise Models 11.3 Justification Awareness Models 11.4 The Russell Scenario as a JAM 11.5 Kripke Models and Master Justification 11.6 Conclusion References Index
222 223 225 226 228 231 233 234 244
Introduction
Why is this thus? What is the reason of this thusness?1
Modal operators are commonly understood to qualify the truth status of a proposition: necessary truth, proved truth, known truth, believed truth, and so on. The ubiquitous possible world semantics for it characterizes things in universal terms: X is true in some state if X is true in all accessible states, where various conditions on accessibility are used to distinguish one modal logic from another. Then (X → Y) → (X → Y) is valid, no matter what conditions are imposed, by a simple and direct argument using universal quantification. Suppose both (X → Y) and X are true at an arbitrary state. Then both X and X → Y are true at all accessible states, whatever “accessible” may mean. By the usual understanding of →, Y is true at all accessible states too, and so Y is true at the arbitrary state we began with. Although arguments like these have a strictly formal nature and are studied as modal model theory, they also give us some insights into our informal, everyday use of modalities. Still, something is lacking. Suppose we think of as epistemic, and to emphasize this we use K instead of for the time being. For some particular X, if you assert the colloquial counterpart of KX, that is, if you say you know X, and I ask you why you know X, you would never tell me that it is because X is true in all states epistemically compatible with this one. You would, instead, give me some sort of explicit reason: “I have a mathematical proof of X,” or “I read X in the encyclopedia,” or “I observed that X is the case.” If I asked you why K(X → Y) → (KX → KY) is valid you would probably say something like “I could use my reason for X and combine it with my reason for X → Y, and infer Y.” This, in effect, would be your reason for Y, given that you had reasons for X and for X → Y. 1
Charles Farrar Browne (1834–1867) was an American humorist who wrote under the pen name Artemus Ward. He was a favorite writer of Abraham Lincoln, who would read his articles to his Cabinet. This quote is from a piece called Moses the Sassy, Ward (1861).
x
Introduction
xi
Notice that this neatly avoids the logical omniscience problem: that we know all the consequences of what we know. It replaces logical omniscience with the more acceptable claim that there are reasons for the consequences of what we know, based on the reasons for what we know, but reasons for consequences are more complicated things. In our example, the reason for Y has some structure to it. It combines reasons for X, reasons for X → Y, and inference as a kind of operation on reasons. We will see more examples of this sort; in fact, we have just seen a fundamental paradigm. In place of a modal operator, , justification logics have a family of justification terms, informally intended to represent reasons, or justifications. Instead of X we will see t:X, where t is a justification term and the formula is read “X is so for reason t,” or more briefly, “t justifies X.” At a minimum, justification terms are built up from justification variables, standing for arbitrary justifications. They are built up using a set of operations that, again at a minimum, contains a binary operation ·. For example, x · (y · x) is a justification term, where x and y are justification variables. The informal understanding of · is that t · u justifies Y provided t justifies an implication with Y as its consequent, and u justifies the antecedent. In justification logics the counterpart of (X → Y) → (X → Y) is t:(X → Y) → (u:X → [t · u]:Y) where, as we will often do, we have added square brackets to enhance readability. Note that this exactly embodies the informal explanation we gave in the previous paragraph for the validity of K(X → Y) → (KX → KY). That is, Y has a justification built from justifications for X and for X → Y using an inference that amounts to a modus ponens application—we can think of the · operation as an abstract representation of this inference. Other behaviors of modal operators, X → X for instance, will require operators in addition to ·, and appropriate postulated behavior, in order to produce justification logics that correspond to modal logics in which X → X is valid. Examples, general methods for doing this, and what it means to “correspond” all will be discussed during the course of this book. One more important point. Suppose X and Y are equivalent formulas, that is, we have X ↔ Y. Then in any normal modal logic we will also have X ↔ Y. Let us interpret the modal operator epistemically again, and write KX ↔ KY. In fact, KX ↔ KY, when read in the usual epistemic way, can sometimes be quite an absurd assertion. Consider some astronomically complicated tautology X of classical propositional logic. Because it is a tautology, it is equivalent
xii
Introduction
to P ∨ ¬P, which we may take for Y. Y is hardly astronomically complicated. However, because X ↔ Y, we will have KX ↔ KY. Clearly, we know Y essentially by inspection and hence KY holds, while KX on the other hand will involve an astronomical amount of work just to read it, let alone to verify it. Informally we see that, while both X and Y are tautologies, and so both are knowable in principle, any justification we might give for knowing one, combined with quite a lot of formula manipulation, can give us some justification for knowing the other. The two justifications may not, indeed will not, be the same. One is simple, the other very complex. Modal logic is about propositions. Propositions are, in a sense, the content of formulas. Propositions are not syntactical objects. “It’s good to be the king” and “Being the king is good” express the same proposition, but not in the same way. Justifications apply to formulas. Equivalent formulas determine the same proposition, but can be quite different as formulas. Syntax must play a fundamental role for us, and you will see that it does, even in our semantics. Consider one more very simple example. A → (A∧ A) is an obvious tautology. We might expect KA → K(A ∧ A). But we should not expect t:A → t:(A ∧ A). If t does, in fact, justify A, a justification of A ∧ A may involve t, but also should involve facts about the redundancy of repetition; t by itself cannot be expected to suffice. Modal logics can express, more or less accurately, how various modal operators behave. This behavior is captured axiomatically by proofs, or semantically using possible world reasoning. These sorts of justifications for modal operator behavior are not within a modal logic, but are outside constructs. Justification logics, on the other hand, can represent the whys and wherefores of modal behavior quite directly, and from within the formal language itself. We will see that most standard modal logics have justification counterparts that can be used to give a fine-grained, internal analysis of modal behavior. Perhaps, this will help make clear why we used the quotation we did at the beginning of this Introduction.
1 What Is This Book About? How did justification logics originate? It is an interesting story, with revealing changes of direction along the way. Going back to the days when G¨odel was a young logician, there was a dream of finding a provability interpretation for intuitionistic logic. As part of his work on that project, in G¨odel (1933), G¨odel showed that one could abstract some of the key features of provability and make a propositional modal logic using them. Then, remarkably but
Introduction
xiii
naturally, one could embed propositional intuitionistic logic into the resulting system. C. I. Lewis had pioneered the modern formal study of modal logics (Lewis, 1918; Lewis and Langford, 1932), and G¨odel observed that his system was equivalent to the Lewis system S4. All modern axiomatizations of modal logics follow the lines pioneered in G¨odel’s note, while Lewis’s original formulation is rarely seen today. G¨odel showed that propositional intuitionistic logic embedded into S4 using a mapping that inserted in front of every subformula. In effect, intuitionistic logic could be understood using classical logic plus an abstract notion of provability: a propositional formula X is an intuitionistic theorem if and only if the result of applying G¨odel’s mapping is a theorem of S4. (This story is somewhat simplified. There are several versions of the G¨odel translation—we have used the simplest one to describe. And G¨odel did not use the symbol but rather an operator Bew, short for beweisbar, or provability in the German language. None of this affects our main points.) Unfortunately, the story breaks off at this point because G¨odel also noted that S4 does not behave like formal provability (e.g., in arithmetic), by using the methods he had pioneered in his work on incompleteness. Specifically, S4 validates X → X, so in particular we have ⊥ → ⊥ (where ⊥ is falsehood). This is equivalent to ¬⊥, which is thus provable in S4. If we had an embedding of S4 into formal arithmetic under which corresponded to G¨odel’s arithmetic formula representing provability, we would be able to prove in arithmetic that falsehood was not provable. That is, we would be able to show provability of consistency, violating G¨odel’s second incompleteness theorem. So, work on an arithmetic semantics for propositional intuitionistic logic paused for a while. Although it did not solve the problem of a provability semantics for intuitionistic logic, an important modal/arithmetic connection was eventually worked out. One can define a modal logic by requiring that its validities are those that correspond to arithmetic validities when reading as G¨odel’s provability formula. It was shown in Solovay (1976) that this was a modal logic already known in the literature, though as noted earlier, it is not S4. Today, the logic is called GL, standing for G¨odel–L¨ob logic. GL is like S4 except that the T axiom X → X, an essential part of S4, is replaced by a modal formula abstractly representing L¨ob’s theorem: (X → X) → X. S4 and GL are quite different logics. By now the project for finding an arithmetic interpretation of intuitionistic logic had reached an impasse. Intuitionistic logic embedded into S4, but S4 did not embed into formal arithmetic. GL embedded into formal arithmetic, but the G¨odel translation does not embed intuitionistic logic into GL.
xiv
Introduction
In his work on incompleteness for Peano arithmetic, G¨odel gave a formula Bew(x, y)
that represents the relation: x is the G¨odel number of a proof of a formula with G¨odel number y. Then, a formal version of provability is ∃xBew(x, y) which expresses that there is a proof of (the formula whose G¨odel number is) y. If this formula is what corresponds to in an embedding from a modal language to Peano arithmetic, we get the logic GL. But in a lecture in 1938 G¨odel pointed out that we might work with explicit proof representatives instead of with provability (G¨odel, 1938). That is, instead of using an embedding translating every occurrence of by ∃xBew(x, y), we might associate with each occurrence of some formal term t that somehow represents a particular proof, allowing different occurrences of to be associated with different terms t. Then in the modal embedding, we could make the occurrence of associated with t correspond to Bew(ptq, y), where ptq is a G¨odel number for t. For each occurrence of we would need to find some appropriate term t, and then each occurrence of would be translated into arithmetic differently. The existential quantifier in ∃xBew(x, y) has been replaced with a meta-existential quantifier, outside the formal language. We provide an explicit proof term, rather than just asserting that one exists. G¨odel believed that this approach should lead to a provability embedding of S4 into Peano arithmetic. G¨odel’s proposal was not published until 1995 when Volume 3 of his collected works appeared. By this time the idea of using a modal-like language with explicit representatives for proofs had been rediscovered independently by Sergei Artemov, see Artemov (1995, 2001). The logic that Artemov created was called LP, which stood for logic of proofs. It was the first example of a justification logic. What are now called justification terms were called proof terms in LP. Crucially, Artemov showed LP filled the gap between modal S4 and Peano arithmetic. The connection with S4 is primarily embodied in a Realization Theorem, which has since been shown to hold for a wide range of justification logic, modal logic pairs. It will be extensively examined in this book. The connection between LP and formal arithmetic is Artemov’s Arithmetic Completeness Theorem, which also will be examined in this book. Its range is primarily limited to the original justification logic, LP, and a few close relatives. This should not be surprising, though. G¨odel’s motivation for his formulation of S4 was that should embody properties of a formal arithmetic proof predicate. This connection with arithmetic provability is not present for almost all modal
Introduction
xv
logics and is consequently also missing for corresponding justification logics, when they exist. Nonetheless, the venerable goal of finding a provability interpretation for propositional intuitionistic logic had been attained. The G¨odel translation embeds propositional intuitionistic logic into the modal logic S4. The Realization Theorem establishes an embedding of S4 into the justification logic LP. And the Arithmetic Completeness Theorem shows that LP embeds into formal arithmetic. It was recognized from the very beginning that the connection between S4 and LP could be weakened to sublogics of S4 and LP. Thus, there were justification logic counterparts for the standard modal logics, K, K4, T, and a few others. These justification logics had arithmetic connections because they were sublogics of LP. The use of proof term was replaced with justification term. Although the connection with arithmetic was weaker than it had been with LP, justification terms still had the role of supplying explicit justifications for epistemically necessary statements. One can consult Artemov (2008) and Artemov and Fitting (2012) for survey treatments, though the present book includes the material found there. Almost all of the early work on justification logics was proof-theoretically based. Realization theorems were shown constructively, making use of a sequent calculus. The existence of an algorithm to compute what are called realizers is important, but this proof-theoretic approach limits the field to those logics known to have sequent calculus proof systems. For a time it was hoped that various extensions of sequent and tableau calculi would be useful and, to some extent, this has been the case. The most optimistic version of this hope was expressed in Artemov (2001) quite directly, “Gabbay’s Labelled Deductive Systems, Gabbay (1994), may serve as a natural framework for LP.” Unfortunately this seems to have been too optimistic. While the formats had similarities, the goals were different, and the machinery did not interact well. A semantics for LP and its near relatives, not based on arithmetic provability, was introduced in Mkrtychev (1997) and is discussed in Chapter 3. (A constructive version of the canonical model for LP with a completeness theorem can be found already in Artemov (1995).) Mkrtychev’s semantics did not use possible worlds and had a strong syntactic flavor. Possible worlds were added to the mix in Fitting (2005), producing something that potentially applied much more broadly than the earlier semantics. This is the subject of Chapter 4. Using this possible world semantics, a nonconstructive, semantic-based, proof of realization was given. It was now possible to avoid the use of a sequent calculus, though the algorithmic nature of realization was lost. More recently, a semantics with a very simple structure was created, Artemov’s basic semantics (Artemov, 2012). It is presented in Chapter 3. Its machinery is almost minimal
xvi
Introduction
for the purpose. In this book, we will use possible world semantics to establish very general realization results, but basic models will often be used when we simply want to show some formula fails to be a theorem. Though its significance was not properly realized at the time, in 2005 the subject broadened when a justification logic counterpart of S5 was introduced in Pacuit (2005) and Rubtsova (2006a, b), with a connecting realization theorem. There was no arithmetical interpretation for this justification logic. Also there is no sequent calculus for S5 of the standard kind, so the proof given for realization was nonconstructive, using a version of the semantics from Fitting (2005). The semantics needed some modification to what is called its evidence function, and this turned out to have a greater impact than was first realized. Eventually constructive proofs connecting S5 and its justification counterpart were found. These made use of cut-free proof systems that were not exactly standard sequent calculi. Still, the door to a larger room was beginning to open. Out of the early studies of the logics of proofs and its variants a general logical framework for reasoning about epistemic justification at large naturally emerged, and the name, Justification Logic, was introduced (cf. Artemov, 2008). Justification Logic is based on justification assertions, t:F, that are read t is a justification for F, with a broader understanding of the word justification going beyond just mathematical proofs. The notion of justification, which has been an essential component of epistemic studies since Plato, had been conspicuously absent in the mathematical models of knowledge within the epistemic logic framework. The Justification Logic framework fills in this void. In Fitting (2016a) the subject expanded abruptly. Using nonconstructive semantic methods it was shown that the family of modal logics having justification counterparts is infinite. The justification phenomenon is not the relatively narrow one it first seemed to be. While that work was nonconstructive, there are now cut-free proof systems of various kinds for a broader range of modal logics than was once the case, and these have been used successfully to create realization algorithms, in Kuznets and Goetschi (2012), for instance. It may be that the very general proof methodologies of Fitting (2015) and especially Negri (2005) and Negri and von Plato (2001) will extend the constructive range still further, perhaps even to the infinite family that nonconstructive methods are known to work for. This is active current work. Work on quantified justification logics exists, but the subject is considerably behind its propositional counterpart. An important feature of justification logics is that they can, in a very precise sense, internalize their own proofs. Doing this for axioms is generally simple. Rules of inference are more of a problem. Earlier we discussed a justification formula as a simple, representative exam-
Introduction
xvii
ple: t:(X → Y) → (u:X → [t · u]:Y). This, in effect, internalizes the axiomatic modus ponens rule. The central problem in developing quantified justification logics was how to internalize the rule of universal generalization. It turned out that the key was the clear separation between two roles played by individual variables. On the one hand, they are formal symbols, and one can simply infer ∀xϕ(x) from a proof of ϕ(x). On the other hand, they can be thought of as open for substitution, that is, throughout a proof one can replace free occurrences of x with a term t to produce a new proof (subject to appropriate freeness of substitution conditions, of course). These two roles for variables are actually incompatible. It was the introduction of specific machinery to keep track of which role a variable occurrence had that made possible the internalization of proofs, and thus a quantified justification logic. An axiomatic version of first-order LP was introduced in Artemov and Yavorskaya (Sidon) (2011) and a possible world semantics for it in Fitting (2011a, 2014b). A connection with formal arithmetic was established. There is a constructive proof of a Realization Theorem, connecting first-order LP with firstorder S4. Unlike propositionally, no nonconstructive proof is currently known The possible world semantics includes the familiar monotonicity condition on world domains. It is likely that all this can be extended to a much broader range of quantified modal logics than just first-order S4, provided monotonicity is appropriate. A move to constant domain models, to quantified S5 in particular, has been made, and a semantics, but not yet a Realization Theorem, can be found in Fitting and Salvatore (2018). Much involving quantification is still uncharted territory. This book will cover the whole range of topics just described. It will not do so in the historical order that was followed in this Introduction, but will make use of the clearer understanding that has emerged from study of the subject thus far. We will finish with the current state of affairs, standing on the edge of unknown lands. We hope to prepare some of you for the journey, should you choose to explore further on your own.
2 What Is Not in This Book? There are several historical works and pivotal developments in justification logic that will not be covered in the book due to natural limitations, and in this section we will mention them briefly. We are confident that other books and surveys will do justice to these works in more detail. Apart from G¨odel’s lecture, G¨odel (1938), which remained unpublished
xviii
Introduction
until 1995 and thus could not influence development in this area, the first results and publications on the logic of proofs are dated 1992: a technical report, Artemov and Straßen (1992), based on work done in January of 1992 in Bern, and a conference presentation of this work at CSL’92 published in Springer Lecture Notes in Computer Science as Artemov and Straßen (1993a). In this work, the basic logic of proofs was presented: it had proof variables, and the format t is a proof of F, but without operations on proofs. However, it already had the first installment of the fixed-point arithmetical completeness construction together with an observation that, unlike provability logic, the logic of proofs cannot be limited to one standard proof predicate “from the textbook” or to any single-conclusion proof predicate. This line was further developed in Artemov and Straßen (1993b), where the logic of single-conclusion proof predicates (without operations on proofs) was studied. This work introduced the unification axiom, which captures singleconclusioness by propositional tools. After the full-scale logic of proofs with operations had been discovered (Artemov, 1995), the logic of single-conclusion proofs with operations was axiomatized in V. Krupski (1997, 2001). A similar technique was used recently to characterize so-called sharp single-conclusion justification models in Krupski (2018). Another research direction pursued after the papers on the basic logic of proofs was to combine provability and explicit proofs. Such a combination, with new provability principles, was given in Artemov (1994). Despite its title, this paper did not introduce what is known now as The Logic of Proofs, but rather a fusion of the provability logic GL and the basic logic of proofs without operations, but with new arithmetical principles combining proofs and provability and an arithmetical completeness theorem. After the logic of proofs paper (Artemov, 1995), the full-scale logic of provability and proofs (with operations), LPP, was axiomatized and proved arithmetically complete in Sidon (1997) and Yavorskaya (Sidon) (2001). A leaner logic combining provability and explicit proofs, GLA, was introduced and proved arithmetically complete in Nogina (2006, 2014b). Unlike LPP, the logic GLA did not use additional operations on proofs other than those inherited from LP. Later, GLA was used to find a complete classification of reflection principles in arithmetic that involve provability and explicit proofs (Nogina, 2014a). The first publication of the full-scale logic of proofs with operations, LP, which became the first justification logic in the modern sense, was Artemov (1995). This paper contains all the results needed to complete G¨odel’s program of characterizing intuitionistic propositional logic IPC and its BHK semantics via proofs in classical arithmetic: internalization, the realization theorem for S4 in LP, arithmetical semantics for LP, and the arithmetical completeness the-
Introduction
xix
orem. It took six years for the corresponding journal paper to appear: Artemov (2001). In Goris (2008), the completeness of LP for the semantics of proofs in Peano arithmetic was extended to the semantics of proofs in Buss’s bounded arithmetic S12 . In view of applications in epistemology, this result shows that explicit knowledge in the propositional framework can be made computationally feasible. Kuznets and Studer (2016) extend the arithmetical interpretation of LP from the original finite constant specifications to a wide class of constant specifications, including some infinite ones. In particular, this “weak” arithmetical interpretation captures the full logic of proofs LP with the total constant specification. Decidability of LP (with the total constant specification) was also established in Mkrtychev (1997), and this opened the door to decidability and complexity studies in justification logics using model-theoretic and other means. Among the milestones are complexity estimates in Kuznets (2000), Brezhnev and Kuznets (2006), Krupski (2006a), Milnikel (2007), Buss and Kuznets (2012), and Achilleos (2014a). The arithmetical provability semantics for the Logic of Proofs, LP, naturally generalizes to a first-order version with conventional quantifiers and to a version with quantifiers over proofs. In both cases, axiomatizability questions were answered negatively in Artemov and Yavorskaya (2001) and Yavorsky (2001). A natural and manageable first-order version of the logic of proofs, FOLP, has been studied in Artemov and Yavorskaya (Sidon) (2011), Fitting (2014a), and Fitting and Salvatore (2018) and will be covered in Chapter 10. Originally, the logic of proofs was formulated as a Hilbert-style axiomatic system, but this has gradually broadened. Early attempts were tableau based (which could equivalently be presented using sequent calculus machinery). Tableaus generally are analytic, meaning that everything entering into a proof is a subformula of what is being proved. This was problematic for attempts at LP tableaus because of the presence of the · operation, which represented an application of modus ponens, a rule that is decidedly not analytic. Successful tableau systems, though not analytic, for LP and closely related logics can be found in Fitting (2003, 2005) and Renne (2004, 2006). The analyticity problem was overcome in Ghari (2014, 2016a). Broader proof systems have been investigated: hypersequents in Kurokawa (2009, 2012), prefixed tableaus in Kurokawa (2013), and labeled deductive systems in Ghari (2017). Indeed some of this has led to new realization results (Artemov, 1995, 2001, 2002, 2006; Artemov and Bonelli, 2007; Ghari, 2012; Kurokawa, 2012). Finding a computational reading of justification logics has been a natural research goal. There were several attempts to use the ideas of LP for building a lambda-calculus with internalization, cf. Alt and Artemov (2001), Artemov
xx
Introduction
(2002), Artemov and Bonelli (2007), Pouliasis and Primiero (2014), and others. Corresponding combinatory logic systems with internalization were studied in Artemov (2004), Krupski (2006b), and Shamkanov (2011). These and other studies can serve as a ground for further applications in typed programming languages. A version of the logic of proofs with a built-in verification predicate was considered in Protopopescu (2016a, b). The aforementioned intuition that justification logic naturally avoids the logical omniscience problem has been formalized and studied in Artemov and Kuznets (2006, 2009, 2014). The key idea there was to view logical omniscience as a proof complexity problem: The logical omniscience defect occurs if an epistemic system assumes knowledge of propositions, which have no feasible proofs. Through this prism, standard modal logics are logically omniscient (modulo some common complexity assumptions), and justification logics are not logically omniscient. The ability of justification logic to track proof complexity via time bounds led to another formal definition of logical omniscience in Wang (2011a) with the same conclusion: Justification logic keeps logical omniscience under control. Shortly after the first paper on the logic of proofs, it became clear that the logical tools developed are capable of evidence tracking in a general setting and as such can be useful in epistemic logic. Perhaps, the first formal work in this direction was Artemov et al. (1999), in which modal logic S5 was equivalently modified and supplied with an LP-style explicit counterpart. Applications to epistemology have benefited greatly from Fitting semantics, which connected justification logics to mainstream epistemology via possible worlds models. In addition to applications discussed in this book, we would like to mention some other influential work. Game semantics of justification logic was studied in Renne (2008) and dynamic epistemic logic with justifications in Renne (2008) and Baltag et al. (2014). In Sedl´ar (2013), Fitting semantics for justification models was elaborated to a special case of the models of general awareness. Multiagent justification logic and common knowledge has been studied in Artemov (2006), Antonakos (2007), Yavorskaya (Sidon) (2008), Bucheli et al. (2010, 2011), Bucheli (2012), Antonakos (2013), and Achilleos (2014b, 2015a, b). In Dean and Kurokawa (2010), justification logic was used for the analysis of Knower and Knowability paradoxes. A fast-growing and promising area is probabilistic justification logic, cf. Milnikel (2014), Artemov (2016b), Kokkinis et al. (2016), Ghari (2016b), and Lurie (2018).
Introduction
xxi
We are deeply indebted to all contributors to the exciting justification logic project, without whom there would not be this book. Very special thanks to our devoted readers for their sharp eyes and their useful comments: Vladimir Krupski, Vincent Alexis Peluce, and Tatiana Yavorskaya (Sidon).
I think there is no sense in forming an opinion when there is no evidence to form it on. If you build a person without any bones in him he may look fair enough to the eye, but he will be limber and cannot stand up; and I consider that evidence is the bones of an opinion.2
2
Mark Twain (1835–1910). The quote is from his last novel, Personal Recollections of Joan of Arc, Twain (1896).
1 Why Justification Logic?
The formal details of justification logic will be presented starting with the next chapter, but first we give some background and motivation for why the subject was developed in the first place. We will see that it addresses, or at least partially addresses, many of the fundamental problems that have been found in epistemic logic over the years. We will also see in more detail how it relates to our understanding of intuitionistic logic. And finally, we will see how it can be used to mitigate some well-known issues that have arisen in philosophical investigations.
1.1 Epistemic Tradition The properties of knowledge and belief have been a subject for formal logic at least since von Wright and Hintikka (Hintikka, 1962; von Wright, 1951). Knowledge and belief are both treated as modalities in a way that is now very familiar—Epistemic Logic. But of the celebrated three criteria for knowledge (usually attributed to Plato), justified, true, belief, Gettier (1963); Hendricks (2005), epistemic modal logic really works with only two of them. Possible worlds and indistinguishability model belief—one believes what is so under all circumstances thought possible. Factivity brings a trueness component into play—if something is not so in the actual world it cannot be known, only believed. But there is no representation for the justification condition. Nonetheless, the modal approach has been remarkably successful in permitting the development of rich mathematical theory and applications (Fagin et al., 1995; van Ditmarsch et al., 2007). Still, it is not the whole picture. The modal approach to the logic of knowledge is, in a sense, built around the universal quantifier: X is known in a situation if X is true in all situations indistinguishable from that one. Justifications, on the other hand, bring an ex1
2
Why Justification Logic?
istential quantifier into the picture: X is known in a situation if there exists a justification for X in that situation. This universal/existential dichotomy is a familiar one to logicians—in formal logics there exists a proof for a formula X if and only if X is true in all models for the logic. One thinks of models as inherently nonconstructive, and proofs as constructive things. One will not go far wrong in thinking of justifications in general as much like mathematical proofs. Indeed, the first justification logic was explicitly designed to capture mathematical proofs in arithmetic, something that will be discussed later. In justification logic, in addition to the category of formulas, there is a second category of justifications. Justifications are formal terms, built up from constants and variables using various operation symbols. Constants represent justifications for commonly accepted truths—axioms. Variables denote unspecified justifications. Different justification logics differ on which operations are allowed (and also in other ways too). If t is a justification term and X is a formula, t:X is a formula, and is intended to be read t is a justification for X. One operation, common to all justification logics, is application, written like multiplication. The idea is, if s is a justification for A → B and t is a justification for A, then [s · t] is a justification for B.1 That is, the validity of the following is generally assumed s:(A → B) → (t:A → [s · t]:B).
(1.1)
This is the explicit version of the usual distributivity of knowledge operators, and modal operators generally, across implication K(A → B) → (KA → KB).
(1.2)
How adequately does the traditional modal form (1.2) embody epistemic closure? We argue that it does so poorly! In the classical logic context, (1.2) only claims that it is impossible to have both K(A → B) and KA true, but KB false. However, because (1.2), unlike (1.1), does not specify dependencies between K(A → B), KA, and KB, the purely modal formulation leaves room for a counterexample. The distinction between (1.1) and (1.2) can be exploited in a discussion of the paradigmatic Red Barn Example of Goldman and Kripke; here is a simplified version of the story taken from Dretske (2005). 1
For better readability brackets will be used in terms, “[,]”, and parentheses in formulas, “(,).” Both will be avoided when it is safe.
1.1 Epistemic Tradition
3
Suppose I am driving through a neighborhood in which, unbeknownst to me, papiermˆach´e barns are scattered, and I see that the object in front of me is a barn. Because I have barn-before-me percepts, I believe that the object in front of me is a barn. Our intuitions suggest that I fail to know barn. But now suppose that the neighborhood has no fake red barns, and I also notice that the object in front of me is red, so I know a red barn is there. This juxtaposition, being a red barn, which I know, entails there being a barn, which I do not, “is an embarrassment.”
In the first formalization of the Red Barn Example, logical derivation will be performed in a basic modal logic in which is interpreted as the “belief” modality. Then some of the occurrences of will be externally interpreted as a knowledge modality K according to the problem’s description. Let B be the sentence “the object in front of me is a barn,” and let R be the sentence “the object in front of me is red.” (1) B, “I believe that the object in front of me is a barn.” At the metalevel, by the problem description, this is not knowledge, and we cannot claim KB. (2) (B ∧ R), “I believe that the object in front of me is a red barn.” At the metalevel, this is actually knowledge, e.g., K(B ∧ R) holds. (3) (B∧R → B), a knowledge assertion of a logical axiom. This is obviously knowledge, i.e., K(B ∧ R → B). Within this formalization, it appears that epistemic closure in its modal form (1.2) is violated: K(B∧R), and K(B∧R → B) hold, whereas, by (1), we cannot claim KB. The modal language here does not seem to help resolving this issue. Next consider the Red Barn Example in justification logic where t:F is interpreted as “I believe F for reason t.” Let u be a specific individual justification for belief that B, and v for belief that B ∧ R. In addition, let a be a justification for the logical truth B ∧ R → B. Then the list of assumptions is (i) u:B, “u is a reason to believe that the object in front of me is a barn”; (ii) v:(B ∧ R), “v is a reason to believe that the object in front of me is a red barn”; (iii) a:(B ∧ R → B). On the metalevel, the problem description states that (ii) and (iii) are cases of knowledge, and not merely belief, whereas (i) is belief, which is not knowledge. Here is how the formal reasoning goes: (iv) a:(B ∧ R → B) → (v:(B ∧ R) → [a·v]:B), by principle (1.1); (v) v:(B ∧ R) → [a·v]:B, from 3 and 4, by propositional logic; (vi) [a·v]:B, from 2 and 5, by propositional logic.
4
Why Justification Logic?
Notice that conclusion (vi) is [a · v]:B, and not u:B; epistemic closure holds. By reasoning in justification logic it was concluded that [a·v]:B is a case of knowledge, i.e., “I know B for reason a · v.” The fact that u:B is not a case of knowledge does not spoil the closure principle because the latter claims knowledge specifically for [a·v]:B. Hence after observing a red fac¸ade, I indeed know B, but this knowledge has nothing to do with (i), which remains a case of belief rather than of knowledge. The justification logic formalization represents the situation fairly. Tracking justifications represents the structure of the Red Barn Example in a way that is not captured by traditional epistemic modal tools. The justification logic formalization models what seems to be happening in such a case; closure of knowledge under logical entailment is maintained even though “barn” is not perceptually known. One could devise a formalization of the Red Barn Example in a bimodal language with distinct modalities for knowledge and belief. However, it seems that such a resolution must involve reproducing justification tracking arguments in a way that obscures, rather than reveals, the truth. Such a bimodal formalization would distinguish u:B from [a · v]:B not because they have different reasons (which reflects the true epistemic structure of the problem), but rather because the former is labeled “belief” and the latter “knowledge.” But what if one needs to keep track of a larger number of different unrelated reasons? By introducing a multiplicity of distinct modalities and then imposing various assumptions governing the interrelationships between these modalities, one would essentially end up with a reformulation of the language of justification logic itself (with distinct terms replaced by distinct modalities). This suggests that there may not be a satisfactory “halfway point” between a modal language and the language of justification logic, at least inasmuch as one tries to capture the essential structure of examples involving the deductive nature of knowledge.
1.2 Mathematical Logic Tradition According to Brouwer, truth in constructive (intuitionistic) mathematics means the existence of a proof, cf. Troelstra and van Dalen (1988). In 1931–34, Heyting and Kolmogorov gave an informal description of the intended proof-based semantics for intuitionistic logic (Kolmogoroff, 1932; Heyting, 1934), which is now referred to as the Brouwer–Heyting–Kolmogorov (BHK) semantics. According to the BHK conditions, a formula is “true” if it hasa proof. Further-
1.2 Mathematical Logic Tradition
5
more, a proof of a compound statement is connected to proofs of its components in the following way: • a proof of A ∧ B consists of a proof of proposition A and a proof of proposition B, • a proof of A∨B is given by presenting either a proof of A or a proof of B, • a proof of A → B is a construction transforming proofs of A into proofs of B, • falsehood ⊥ is a proposition, which has no proof; ¬A is shorthand for A → ⊥. This provides a remarkably useful informal way of understanding what is and what is not intuitionistically acceptable. For instance, consider the classical tautology (P ∨ Q) ↔ (P ∨ (Q ∧ ¬P)), where we understand ↔ as mutual implication. And we understand ¬P as P → ⊥, so that a proof of ¬P would amount to a construction converting any proof of P into a proof of ⊥. Because ⊥ has no proof, this amounts to a proof that P has no proof—a refutation of P. According to BHK semantics the implication from right to left in (P ∨ Q) ↔ (P ∨ (Q ∧ ¬P)) should be intuitionistically valid, by the following argument. Given a proof of P ∨ (Q ∧ ¬P) it must be that we are given a proof of one of the disjuncts. If it is P, we have a proof of one of P ∨ Q. If it is Q ∧ ¬P, we have proofs of both conjuncts, hence a proof of Q, and hence again a proof of one of P ∨ Q. Thus we may convert a proof of P ∨ (Q ∧ ¬P) into a proof of P ∨ Q. On the other hand, (P ∨ Q) → (P ∨ (Q ∧ ¬P)) is not intuitionistically valid according to the BHK ideas. Suppose we are given a proof of P ∨ Q. If we have a proof of the disjunct P, we have a proof of P ∨ Q. But if we have a proof of Q, there is no reason to suppose we have a refutation of P, and so we cannot conclude we have a proof of Q ∧ ¬P, and things stop here. Kolmogorov explicitly suggested that the proof-like objects in his interpretation (“problem solutions”) came from classical mathematics (Kolmogoroff, 1932). Indeed, from a foundational point of view this reflects Kolmogorov’s and G¨odel’s goal to define intuitionism within classical mathematics. From this standpoint, intuitionistic mathematics is not a substitute for classical mathematics, but helps to determine what is constructive in the latter. The fundamental value of the BHK semantics for the justification logic project is that informally but unambiguously BHK suggests treating justifications, here mathematical proofs, as objects with operations. In G¨odel (1933), G¨odel took the first step toward developing a rigorous proof-based semantics for intuitionism. G¨odel considered the classical modal logic S4 to be a calculus describing properties of provability:
6
Why Justification Logic?
(1) (2) (3) (4)
Axioms and rules of classical propositional logic, (F → G) → (F → G), F → F, F → F, `F . (5) Rule of necessitation: ` F
Based on Brouwer’s understanding of logical truth as provability, G¨odel defined a translation tr(F) of the propositional formula F in the intuitionistic language into the language of classical modal logic: tr(F) is obtained by prefixing every subformula of F with the provability modality . Informally speaking, when the usual procedure of determining classical truth of a formula is applied to tr(F), it will test the provability (not the truth) of each of F’s subformulas, in agreement with Brouwer’s ideas. From G¨odel’s results and the McKinseyTarski work on topological semantics for modal logic (McKinsey and Tarski, 1948), it follows that the translation tr(F) provides a proper embedding of the Intuitionistic Propositional Calculus, IPC, into S4, i.e., an embedding of intuitionistic logic into classical logic extended by the provability operator. IPC ` F
⇔
S4 ` tr(F).
(1.3)
Conceptually, this defines IPC in S4. Still, G¨odel’s original goal of defining intuitionistic logic in terms of classical provability was not reached because the connection of S4 to the usual mathematical notion of provability was not established. Moreover, G¨odel noted that the straightforward idea of interpreting modality F as F is provable in a given formal system T contradicted his second incompleteness theorem. Indeed, (F → F) can be derived in S4 by the rule of necessitation from the axiom F → F. On the other hand, interpreting modality as the predicate of formal provability in theory T and F as contradiction converts this formula into a false statement that the consistency of T is internally provable in T . The situation after G¨odel (1933) can be described by the following figure where “X ,→ Y” should be read as “X is interpreted in Y”: IPC ,→ S4 ,→ ? ,→ CLASSICAL PROOFS.
In a public lecture in Vienna in 1938, G¨odel observed that using the format of explicit proofs t is a proof of F
(1.4)
can help in interpreting his provability calculus S4 (G¨odel, 1938). Unfortunately, G¨odel (1938) remained unpublished until 1995, by which time the
1.2 Mathematical Logic Tradition
7
G¨odelian logic of explicit proofs had already been rediscovered, axiomatized as the Logic of Proofs LP, and supplied with completeness theorems connecting it to both S4 and classical proofs (Artemov, 1995, 2001). The Logic of Proofs LP became the first in the justification logic family. Proof terms in LP are nothing but BHK terms understood as classical proofs. With LP, propositional intuitionistic logic received the desired rigorous BHK semantics: IPC ,→ S4 ,→ LP ,→ CLASSICAL PROOFS .
Several well-known mathematical notions that appeared prior to justification logic have sometimes been perceived as related to the BHK idea: Kleene realizability (Troelstra, 1998), Curry–Howard isomorphism (Girard et al., 1989; Troelstra and Schwichtenberg, 1996), Kreisel–Goodman theory of constructions (Goodman, 1970; Kreisel, 1962, 1965), just to name a few. These interpretations have been very instrumental for understanding intuitionistic logic, though none of them qualifies as the BHK semantics. Kleene realizability revealed a fundamental computational content of formal intuitionistic derivations; however it is still quite different from the intended BHK semantics. Kleene realizers are computational programs rather than proofs. The predicate “r realizes F” is not decidable, which leads to some serious deviations from intuitionistic logic. Kleene realizability is not adequate for the intuitionistic propositional calculus IPC. There are realizable propositional formulas not derivable in IPC (Rose, 1953).2 The Curry–Howard isomorphism transliterates natural derivations in IPC to typed λ-terms, thus providing a generic functional reading for logical derivations. However, the foundational value of this interpretation is limited because, as proof objects, Curry–Howard λ-terms denote nothing but derivations in IPC itself and thus yield a circular provability semantics for the latter. An attempt to formalize the BHK semantics directly was made by Kreisel in his theory of constructions (Kreisel, 1962, 1965). The original variant of the theory was inconsistent; difficulties already occurred at the propositional level. In Goodman (1970) this was fixed by introducing a stratification of constructions into levels, which ruined the BHK character of this semantics. In particular, a proof of A → B was no longer a construction that could be applied to any proof of A. 2
Kleene himself denied any connection of his realizability with the BHK interpretation.
8
Why Justification Logic?
1.3 Hyperintensionality Justification logic offers a formal framework for hyperintensionality. The hyperintensional paradox was formulated in Cresswell (1975). It is well known that it seems possible to have a situation in which there are two propositions p and q which are logically equivalent and yet are such that a person may believe the one but not the other. If we regard a proposition as a set of possible worlds then two logically equivalent propositions will be identical, and so if “x believes that” is a genuine sentential functor, the situation described in the opening sentence could not arise. I call this the paradox of hyperintensional contexts. Hyperintensional contexts are simply contexts which do not respect logical equivalence.
Starting with Cresswell himself, several ways of dealing with this have been proposed. Generally, these involve adding more layers to familiar possible world approaches so that some way of distinguishing between logically equivalent sentences is available. Cresswell suggested that the syntactic form of sentences be taken into account. Justification logic, in effect, does this through its mechanism for handling justifications for sentences. Thus justification logic addresses some of the central issues of hyperintensionality but, as a bonus, we automatically have an appropriate proof theory, model theory, complexity estimates, and a broad variety of applications. A good example of a hyperintensional context is the informal language used by mathematicians conversing with each other. Typically when a mathematician says he or she knows something, the understanding is that a proof is at hand, but this kind of knowledge is essentially hyperintensional. For instance Fermat’s Last Theorem, FLT, is logically equivalent to 0 = 0 because both are provable and hence denote the same proposition, as this is understood in modal logic. However, the context of proofs distinguishes them immediately because a proof of 0 = 0 is not necessarily a proof of FLT, and vice versa. To formalize mathematical speech, the justification logic LP is a natural choice because t:X was designed to have characteristics of “t is a proof of X.” The fact that propositions X and Y are equivalent in LP, that LP ` X ↔ Y, does not warrant the equivalence of the corresponding justification assertions, and typically t:X and t:Y are not equivalent, t:X 6↔ t:Y. Indeed, as we will see, this is the case for every justification logic. Going further LP, and justification logic in general, is not only sufficiently refined to distinguish justification assertions for logically equivalent sentences, but it also provides flexible machinery to connect justifications of equivalent sentences and hence to maintain constructive closure properties desirable for a logic system. For example, let X and Y be provably equivalent, i.e., there is a proof u of X ↔ Y, and so u:(X ↔ Y) is provable in LP. Suppose also
1.4 Awareness
9
that v is a proof of X, and so v:X. It has already been mentioned that this does not mean v is a proof of Y—this is a hyperintensional context. However within the framework of justification logic, building on the proofs of X and of X ↔ Y, we can construct a proof term f (u, v), which represents the proof of Y and so f (u, v):Y is provable. In this respect, justification logic goes beyond Cresswell’s expectations: Logically equivalent sentences display different but constructively controlled epistemic behavior.
1.4 Awareness The logical omniscience problem is that in epistemic logics all tautologies are known and knowledge is closed under consequence, both of which are unreasonable. In Fagin and Halpern (1988) a simple mechanism for avoiding the problems was introduced. One adds to the usual Kripke model structure an awareness function A indicating for each world which formulas the agent is aware of at this world. Then a formula is taken to be known at a possible world u if (1) the formula is true at all worlds accessible from u (the Kripkean condition for knowledge) and (2) the agent is aware of the formula at u. The awareness function A can serve as a practical tool for blocking knowledge of an arbitrary set of formulas. However, as logical structures, awareness models exhibit abnormal behavior due to the lack of natural closure properties. For example, the agent can know A ∧ A but be unaware of A and hence not know it. Fitting models for justification logic, presented in Chapter 4, use a forcing definition reminiscent of the one from awareness models: For any given justification t, the justification assertion t:F holds at world u iff (1) F holds at all worlds v accessible from u and (2) t is an admissible evidence for F at u, u ∈ E(s, F), read as “u is a possible world at which s is relevant evidence for F.” The principal difference is that postulated operations on justifications relate to natural closure conditions on admissible evidence functions E in justification logic models. Indeed, this idea has been explored in Sedl´ar (2013), which works with the language of LP and thinks of it as a multiagent modal logic, and taking justification terms as agents (more properly, actions of agents). This shows that justification logic models absorb the usual epistemic themes of awareness, group agency, and dynamics in a natural way.
10
Why Justification Logic?
1.5 Paraconsistency Justification logic offers a well-principled approach to paraconsistency, which looks for noncollapsing logical ways of dealing with contradictory sets of assumptions, e.g., {A, ¬A}. The following obvious observation shows how to convert any set of assumptions Γ = {A1 , A2 , A3 , . . .} into a logically consistent set of sentences while maintaining all the intrinsic structure of Γ. Informally, instead of (perhaps inconsistently) assuming that Γ holds, we assume only that each sentence A from Γ has a justification, i.e., ~x : Γ = {x1:A1 , x2:A2 , x3:A3 , . . .}. It is easy to see that for each Γ, the set ~x:Γ is consistent in what will be our basic justification logic J. For example, for Γ = {A, ¬A}, ~x : Γ = {x1:A, x2:¬A}, states that x1 is a justification for A and x2 is a justification for ¬A. Within justification logic J in which no factivity (or even consistency) of justifications is assumed, the set of assumptions {x1:A, x2:¬A} is consistent, unlike the original set of assumptions {A, ¬A}. There is nothing paraconsistent, magical, or artificial in reasoning from ~x:Γ in justification logic J. In practical terms, this means we gain the ability to effectively reason about inconsistent data sets, keeping track of justifications and their dependencies, with the natural possibility to draw meaningful conclusions even when some assumed justifications from ~x:Γ become compromised and should be discharged.
2 The Basics of Justification Logic
In this chapter we discuss matters of syntax and axiomatics. All material is propositional, and will be so until Chapter 10. Justification logics are closely related to modal logics, so we start briefly with them in order to fix the basic notation. And just as normal modal logics all extend a single simplest example, K, all justification logics extend a single simplest example, J0 . We will begin our discussion with modal logics, then we will discuss the justification logic J0 in detail, and finally we will extend things to the most common and bestknown justification logics. A much broader family of justification logics will be discussed in Chapter 8.
2.1 Modal Logics All propositional formulas throughout this book are built up from a countable family of propositional variables. We use P, Q, . . . as propositional variables, with subscripts if necessary, and we follow the usual convention that these are all distinct. As our main propositional connective we have implication, →. We have negation, ¬, which we will take as primitive, or defined using the propositional constant ⊥ representing falsehood, as convenient and appropriate at the time. We also use conjunction, ∧, disjunction, ∨, and equivalence, ↔, and these too may be primitive or defined depending on circumstances. We omit outer parentheses in formulas when it will do no harm. We usually have a single modal necessity operator. It will generally be represented by though in epistemic contexts it may be represented by K. A dual operator representing possibility, ♦, is a defined operator and actually plays little role here. There is much work on epistemic logics with multiple agents, and there is some study of justification counterparts for them. When 11
12
The Basics of Justification Logic
discussing these and their connections with modal logics, we will subscript the modal operators just described. To date, no justification logic corresponding to a nonnormal modal logic has been introduced, so only normal modal logics will appear here. A normal modal logic is a set of modal formulas that contains all tautologies and all formulas of the form (X → Y) → (X → Y) and is closed under uniform substitution, modus ponens, and necessitation (if X is present, so is X). The smallest normal modal logic is K; it is a subset of all normal modal logics. The logic K has a standard axiom system. Axioms are all tautologies (or enough of them) and all formulas of the form (X → Y) → (X → Y). Rules are Modus Ponens X, X → Y ⇒ Y and Necessitation X ⇒ X. We are not actually interested in the vast collection of normal modal logics, but only in those for which a Hilbert system exists, having an axiomatization using a finite set of axiom schemes. In practice, this means adding axiom schemes to the axiomatization for K just given. We assume everybody knows axiom systems like T, K4, S4, and so on. We will refer to such logics as axiomatically formulated. Of course semantics plays a big role in modal logics, but we postpone a discussion for the time being.
2.2 Beginning Justification Logics Justification logics, syntactically, are like modal logics except that justification terms take the place of . Justification terms are intended to represent reasons or justifications for formulas. They have structure that encodes reasoning that has gone into them. We begin our formal presentation here. Definition 2.1 (Justification Term) up as follows.
The set Tm of justification terms is built
(1) There is a set of justification variables, x, y, . . . , x1 , y1 , . . . . Every justification variable is a justification term. (2) There is a set of justification constants, a, b, . . . , a1 , b1 , . . . . Every justification constant is a justification term. (3) There are binary operation symbols, + and ·. If u and v are justification terms, so are (u + v) and (u · v). (4) There may be additional function symbols, f , g, . . . , f1 , g1 , . . . , of various arities. Which ones are present depends on the logic in question. If f is an nplace justification function symbol of the logic, and t1 , . . . , tn are justification terms, f (t1 , . . . , tn ) is a justification term.
2.2 Beginning Justification Logics
13
Neither + nor · is assumed to be commutative or associative, and there is no distributive law. We do, however, allow ourselves the notational convenience of omitting parentheses with multiple occurrences of ·, assuming associativity to the left. Thus, for instance, a · b · c · d is short for (((a · b) · c) · d). We make the same assumption concerning +, though it actually plays a much lesser role. Also we will generally assume that · binds more strongly than +, writing a·b+c instead of (a · b) + c for instance. Definition 2.2 (Justification Formula) The set of justification formulas, Fm, is built in the usual recursive way, as follows. (1) There is a set Var of propositional variables, P, Q, . . . , P1 , Q1 , . . . (these are also known as propositional letters). Every propositional variable is a justification formula. (2) ⊥ (falsehood) is a justification formula. (3) If X and Y are justification formulas, so is (X → Y). (4) If t is a justification term and X is a justification formula, then t:X is a justification formula. We will sometimes use other propositional connectives, ∧, ∨, ↔, which we can think of as defined connectives, or primitive as convenient. Outer parentheses may be omitted in formulas if no confusion will result. If justification term t has a complex structure we generally will write [t]:X, using square brackets as a visual aid. Square brackets have no theoretical significance. In a modal formula, is supposed to express that something is necessary, or known, or obligatory, or some such thing, but it does not say why. A justification term encodes this missing information; it provides the why absent from modal formulas. This is what their structure is for. Justification variables stand for arbitrary justification terms, and substitution for them is examined beginning with Definition 2.17. Justification constants stand for reasons that are not further analyzed—typically they are reasons for axioms. Their role is discussed in more detail once constant specifications are introduced, in Definition 10.32. The · operation corresponds to modus ponens. If X → Y is so for reason s and X is so for reason t, then Y is so for reason s · t. (Reasons are not unique—Y may be true for other reasons too.) The + operation is a kind of weakening. If X is so for either reason s or reason t, then s + t is also a reason for X. Other operations on justification terms, if present, correspond to features peculiar to particular modal logics and will be discussed as they come up.
14
The Basics of Justification Logic
2.3 J0 , the Simplest Justification Logic As we will see, there are some justification logics having a version of a necessitation rule; there are others that do not. Some justification logics are closed under substitution of formulas for propositional variables, others are not. Allowing such a range of behavior is essential to enable us to capture and study the interactions of important features of modal logics that are sometimes hidden from us. But one consequence is, there is no good justification analog of the family of normal modal logics. Still, all justification logics have a common core, which we call J0 , and it is a kind of analog of the weakest normal modal logic, K, even though there is nothing structural we can point to as determining a “normal” justification logic apart from giving an axiomatization. In this section we present J0 axiomatically; subsequently we discuss what must be added to get the general family of justification logics. Definition 2.3 (Justification logic J0 ) The language of J0 has no justification function symbols beyond the basic two binary ones + and ·. The axiom schemes are as follows.
Classical: All tautologies (or enough of them) Application: All formulas of the form s:(X → Y) → (t:X → [s · t]:Y) Sum: All formulas of the forms s:X → [s + t]:X and t:X → [s + t]:X The only J0 rule of inference is Modus Ponens, X, X → Y ⇒ Y. J0 is a very weak justification logic. It is, for instance, incapable of proving that any formula has a justification, see Section 3.2. Reasoning in J0 is analogous to reasoning in the modal logic K without a necessitation rule! What we can do in J0 is derive interesting facts about justifications provided we make explicit what other formulas we would need to have justifications for. We give an example to illustrate this. To help bring out the points we want to make, if (X1 ∧ . . . ∧ Xn ) → Y is provable in J0 we may write X1 , . . . , Xn `J0 Y. Order of formulas and placement of parentheses in the conjunction of the Xi don’t matter because we have classical logic to work with. In modal K, a common first example of a theorem is (X ∧Y) → (X ∧Y). Here is the closest we can come to this in J0 . Our presentation is very much abbreviated.
2.4 Justification Logics in General
15
Example 2.4 Assume u, v, and w are justification variables. 1. 2. 3. 4. 5.
u:((X ∧ Y) → X) → (w:(X ∧ Y) → [u · w]:X) v:((X ∧ Y) → Y) → (w:(X ∧ Y) → [v · w]:Y) (u:((X ∧ Y) → X) ∧ v:((X ∧ Y) → y)) → (w:(X ∧ Y) → [u · w]:X) (u:((X ∧ Y) → X) ∧ v:((X ∧ Y) → y)) → (w:(X ∧ Y) → [v · w]:Y) (u:((X ∧ Y) → X) ∧ v:((X ∧ Y) → y)) → (w:(X ∧ Y) → ([u · w]:X ∧ [v · w]:Y))
Application Axiom Application Axiom Classical Logic on 1, 2 Classical Logic on 1, 2 Classical Logic on 3, 4
So we have shown that u:((X ∧ Y) → X), v:((X ∧ Y) → Y) `J0 (w:(X ∧ Y) → ([u · w]:X ∧ [v · w]:Y)) which we can read as an analog of (X ∧Y) → (X ∧Y) as follows. In J0 , for any w there are justification terms t1 and t2 such that w:(X ∧ Y) → (t1:X ∧ t2:Y), provided we have justifications for the tautologies (X ∧ Y) → X and (X ∧ Y) → Y. Note that t1 = u · w and t2 = v · w are different terms. But, making use of the Sum Axiom scheme, these can be brought together as t1 + t2 = u · w + v · w. It is important to understand that justifications, when they exist, are not unique.
2.4 Justification Logics in General The core justification logic J0 is extended to form other justification logics using two quite different types of machinery. First, one can add new operations on justification terms, besides the basic + and ·, along with axiom schemes governing their use, similar to Sum and Application. This is directly analogous to the way axiom schemes are added to K to create other modal logics. Second, one can specify which truths of logic we assume we have justifications for. This is related to the roles u:((X ∧ Y) → X) and v:((X ∧ Y) → Y) play in Example 2.4. We devote most of this section to the second kind of extension. It is, in fact, the intended role for justification constants that, up to now, have not been used for anything special. For the time being let us assume we have a justification logic resulting from the addition of function symbols and axiom schemes to J0 . The details don’t matter for now, but it should be understood that our axioms may go beyond those for J0 . Axioms of justification logics, like axioms generally, are simply assumed and are not analyzed further. The role of justification constant symbols is to
16
The Basics of Justification Logic
serve as reasons or justifications for axioms. If A is an axiom, we can simply announce that constant symbol c plays the role of a justification for it. It may be that some axioms are assumed to have such justifications, but not necessarily all. Suppose we look at Example 2.4 again, and suppose we have decided that (X ∧ Y) → X is an axiom for which we have a specific justification, let us say the constant symbol c plays this role. Similarly let us say the constant symbol d represents a justification for (X ∧ Y) → Y. Examining the derivation given in Example 2.4, it is easy to see that if we replace the variable u throughout by c, and the variable v throughout by d we still have a derivation, but one of c:((X ∧ Y) → X), d:((X ∧ Y) → Y) `J0 (w:(X ∧ Y) → ([c · w]:X ∧ [d · w]:Y)). If we add c:((X ∧ Y) → X) and d:((X ∧ Y) → Y) to our axioms for J0 , we can simply prove the formula (w:(X ∧ Y) → ([c · w]:X ∧ [d · w]:Y)). Roughly speaking, a constant specification tells us what axioms we have justifications for and which constants justify these axioms. As we just saw, we can use a constant specification as a source of additional axioms. But there is an important complication. If A is an axiom and constant symbol c justifies it, c:A conceptually also acts like an axiom, and it too may have its own justification. Then a constant symbol, say d, could come in here too, as a justification for c:A, and thus we might want to assume d:c:A. This repeats further, of course. For many purposes exact details don’t matter much, so how constants are used, and for what purposes, is turned into a kind of parameter of our logics, called a constant specification. Definition 2.5 (Constant Specification) A constant specification CS for a given justification logic is a set of formulas meeting the following conditions. (1) Members of CS are of the form cn:cn−1: . . . c1:A where n > 0, A is an axiom of JL, and each ci is a constant symbol. (2) If cn:cn−1: . . . c1:A is in CS where n > 1, then cn−1: . . . c1:A is in CS too. Thus CS contains all intermediate specifications for whatever it contains. One reason why constant specifications are treated as parameters can be discovered through a close look at Definition 2.3. It does not really provide an axiomatization for J0 , but rather a scheme for axiomatizations. The axioms called Classical in that definition are not fully specified, and in common practice many classical logic axiomatizations are in use. Any set sufficient to derive all tautologies will do. Then many different axiomatizations for J0 would meet the required conditions, and similarly for any justification logic extending J0
2.4 Justification Logics in General
17
as well. Because constants are supposed to be associated with axioms, a variety of constant specifications come up naturally. And because details like this often matter very little, treating constant specifications as a parameter is quite reasonable. Definition 2.6 (Logic of Justifications with a Constant Specification) Let JL be a justification logic, resulting from the addition of function symbols to the language of J0 and corresponding axiom schemes to those of J0 . Let CS be a constant specification for JL. Then JL(CS) is the logic JL with members of CS added as axioms (not axiom schemes), still with modus ponens as the only rule of inference. Constant specifications allow for great flexibility. A constant specification could associate many constants with a single axiom, or none at all. Allowing for many could be of use in tracking where particular pieces of reasoning come from. Allowing none might be appropriate in dealing with axioms that have some simple form, say X → X, but where the size of X is astronomical. Or again we might want to use the same constant for all instances of a particular axiom schema, or we might want to keep the instances clearly distinguishable. If details don’t matter at all for some particular purpose, we might want to associate a single constant symbol with every axiom, no matter what the form. Such a constant would simply be a record that a formula is an axiom, without going into particulars. Some conditions on constant specifications have shown themselves to be of special interest and have been given names. Here is a list of the most common. There are others. Definition 2.7 (Constant Specification Conditions) Let CS be a constant specification for a justification logic JL. The following requirements may be placed on CS. Empty: CS = ∅. This amounts to working with JL itself. Epistemically one can think of it as appropriate for the reasoning of a completely skeptical agent. Finite: CS is a finite set of formulas. This is fully representative because any specific derivation in a Justification Logic will be finite and so will involve only a finite set of constants. Schematic: If A and B are both instances of the same axiom scheme, c:A ∈ CS if and only if c:B ∈ CS, for every constant symbol c. Total: For each axiom A of JL and any constants c1 , c2 , . . . , cn we have cn:cn−1: . . . c1:A ∈ CS.
18
The Basics of Justification Logic
Axiomatically Appropriate: For every axiom A and for every n > 0 there are constant symbols ci so that cn:cn−1: . . . c1:A ∈ CS. The working of justification axiom systems is specified as follows. Definition 2.8 (Consequence) Suppose JL is a justification logic, CS is a constant specification for JL, S is an arbitrary set of formulas (not schemes), and X is a single formula. By S `JL(CS) X we mean there is a finite sequence of formulas, ending with X, in which each formula is either a instance of an axiom scheme of JL, a member of CS, a member of S , or follows from earlier formulas by modus ponens. If {Y1 , . . . , Yk } `JL(CS) X we will simplify notation and write Y1 , . . . , Yk `JL(CS) X. If ∅ `JL(CS) X we just write `JL(CS) X, or sometimes even JL(CS) ` X. When presenting examples of axiomatic derivations using a constant speciCS fication CS, we will write c + X as a suggestive way of saying that c:X ∈ CS, and we will say “c justifies X”. We conclude this section with some examples of theorems of justification logics. For these we work with JL(CS) where JL is any justification logic and CS is any constant specification for it that is axiomatically appropriate. We assume JL has been axiomatized taking all tautologies as axioms, though taking “enough” would give similar results once we have Theorem 2.14. Example 2.9 P → P is a theorem of any normal modal logic. It has more than one proof. We could simply note that it is an instance of a tautology, X → X. Or we could begin with P → P, a simpler instance of this tautology, apply necessitation getting (P → P), and then use the K axiom (P → P) → (P → P) and modus ponens to conclude P → P. While these are different modal derivations, the result is the same. But when we mimic the steps in JL(CS), they lead to different results. Let t be an arbitrary justification term. Then t:P → t:P is a theorem of JL(CS) because it is an instance of a tautology. But also P → P is an instance of a tautology and so, because JL(CS) is assumed axiomatically appropriate, the constant specification assigns some constant to it; say c:(P → P) ∈ CS. Because c:(P → P) → (t:P → [c · t]:P) is an axiom, t:P → [c · t]:P follows by modus ponens. In justification logic, instead of a single formula P → P with two proofs we have two different theorems that contain traces of their proofs. Both t:P → t:P and t:P → [c · t]:P say that if there is a reason for P, then there is a reason for P, but they give us different reasons. One of the first things everybody shows axiomatically when studying modal
2.4 Justification Logics in General
19
logic is that (P ∧ Q) ↔ (P ∧ Q) is provable in K, and thus is provable in every axiom system for a normal modal logic. But the argument from left to right is quite different from the argument from right to left. Because justification theorems contain traces of their proofs, we should not expect a single justification analog of this modal equivalence, but rather separate results for the left–right implication and for the right–left implication. Example 2.10 Here is a justification derivation corresponding to the usual modal argument for (P ∧ Q) → (P ∧ Q). 1. 2. 3. 4. 5. 6. 7. 8. 9.
(P ∧ Q) → P c:((P ∧ Q) → P) c:((P ∧ Q) → P) → (t:(P ∧ Q) → [c · t]:P) t:(P ∧ Q) → [c · t]:P (P ∧ Q) → Q d:((P ∧ Q) → Q) d:((P ∧ Q) → Q) → (t:(P ∧ Q) → [d · t]:Q) t:(P ∧ Q) → [d · t]:Q t:(P ∧ Q) → ([c · t]:P ∧ [d · t]:Q)
tautology cons spec Application Axiom mod pon on 2, 3 tautology cons spec Application Axiom mod pon on 6, 7 class log on 4, 8
CS Then t:(P ∧ Q) → ([c · t]:P ∧ [d · t]:Q) is a theorem of JL(CS) where c + CS ((P ∧ Q) → P) and d + ((P ∧ Q) → Q).
Example 2.11 A justification counterpart of the modal theorem (P∧Q) → (P ∧ Q) follows. 1. 2. 3. 4. 5. 6.
P → (Q → (P ∧ Q)) c:(P → (Q → (P ∧ Q))) c:(P → (Q → (P ∧ Q))) → (t:P → [c · t]:(Q → (P ∧ Q))) t:P → [c · t]:(Q → (P ∧ Q)) [c · t]:(Q → (P ∧ Q)) → (u:Q → [c · t · u]:(P ∧ Q)) (t:P ∧ u:Q) → [c · t · u]:(P ∧ Q)
tautology cons spec Application Axiom mod pon on 2, 3 Application Axiom class log on 4, 5
CS So (t:P ∧ u:Q) → [c · t · u]:(P ∧ Q) is a theorem of JL(CS) where c + (P → (Q → (P ∧ Q))).
Our final example illustrates the use of +, which has not come up so far. It is for handling situations where there is more than one explanation needed for something, as in a proof by cases. At first glance this seems rather minor, but + turns out to play a vital role when we come to realization results. Example 2.12 (X ∨ Y) → (X ∨ Y) is a theorem of K with an elementary
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The Basics of Justification Logic
proof that we omit. Let us construct a counterpart in JL(CS), still assuming that CS is axiomatically appropriate and all tautologies are axioms. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.
X → (X ∨ Y) c:(X → (X ∨ Y)) c:(X → (X ∨ Y)) → (t:X → [c · t]:(X ∨ Y)) t:X → [c · t]:(X ∨ Y) Y → (X ∨ Y) d:(Y → (X ∨ Y)) d:(Y → (X ∨ Y)) → (u:Y → [d · u]:(X ∨ Y)) u:Y → [d · u]:(X ∨ Y) [c · t]:(X ∨ Y) → [c · t + d · u]:(X ∨ Y) [d · u]:(X ∨ Y) → [c · t + d · u]:(X ∨ Y) t:X → [c · t + d · u]:(X ∨ Y) u:Y → [c · t + d · u]:(X ∨ Y) (t:X ∨ u:Y) → [c · t + d · u]:(X ∨ Y)
tautology cons spec Application Axiom mod pon on 2, 3 tautology cons spec Application Axiom mod pon on 6, 7 Sum Axiom Sum Axiom clas log on 4, 9 clas log on 8, 10 class log on 11, 12
The consequents of 4 and 8 both provide reasons for X ∨ Y, but the reasons are different. We have used + to combine them, getting a justification analog of (X ∨ Y) → (X ∨ Y).
2.5 Fundamental Properties of Justification Logics All justification logics have certain common and useful properties. Some features are identical with those of classical logic; others have twists that are special to justification logics. This section is devoted to ones we will use over and over. Throughout this section let JL be a justification logic and CS be a constant specification for it. Because the only rule of inference is modus ponens the classical proof of the deduction theorem applies. We thus have S , X `JL(CS) Y if and only if S `JL(CS) X → Y. Because formal proofs are finite we have compactness that, combined with the deduction theorem, tells us: S `JL(CS) X if and only if `JL(CS) Y1 → (Y2 → . . . → (Yn → X) . . .) for some Y1 , Y2 , . . . , Yn ∈ S . These are exactly like their classical counterparts. Furthermore, the following serves as a replacement for the modal Necessitation Rule. Definition 2.13 (Internalization) JL has the internalization property relative to constant specification CS provided, if `JL(CS) X then for some justification term t, `JL(CS) t:X. In addition we say that JL has the strong internalization
2.5 Fundamental Properties of Justification Logics
21
property if t contains no justification variables and no justification operation or function symbols except ·. That is, t is built up from justification constants using only ·. Theorem 2.14 If CS is an axiomatically appropriate constant specification for JL then JL has the strong internalization property relative to CS. Proof By induction on proof length. Suppose `JL(CS) X and the result is known for formulas with shorter proofs. If X is an axiom of JL or a member of CS, there is a justification constant c such that c:X is in CS, and so c:X is provable. If X follows from earlier proof lines by modus ponens from Y → X and Y then, by the induction hypothesis, `JL(CS) s:(Y → X) and `JL(CS) t:Y for some s, t containing no justification variables, and with · as the only function symbol. Using the J0 Application Axiom s:(Y → X) → (t:Y → [s · t]:X) and modus ponens, we get `JL(CS) [s · t]:X. If X is provable using an axiomatically appropriate constant specification so is t:X, and the term t constructed in the preceding proof actually internalizes the steps of the axiomatic proof of X, hence the name internalization. Of course different proofs of X will produce different justification terms. Here is an extremely simple example, but one that is already sufficient to illustrate this point. Example 2.15 Assume JL is a justification logic, CS is an axiomatically appropriate constant specification for it, and all tautologies are axioms of JL. CS P → P is a tautology so c + (P → P) for some c. Then c:(P → P) is a theorem, and we have the justification term c internalizing a proof of P → P. Here is a more roundabout proof of P → P, giving us a more complicated internalizing term. Following the method in the proof of Theorem 2.14, we construct the internalization simultaneously. 1. 2. 3. 4. 5.
(P → (P → P)) → ((P → P) → (P → P)) P → (P → P) (P → P) → (P → P) P→P P→P
tautology tautology mod pon on 1, 2 tautology mod pon on 3, 4
d (cons spec) e (cons spec) d·e c (cons spec) d·e·c
This time we get a justification term d · e · c, or more properly (d · e) · c, CS internalizing a proof of P → P, where c + ((P → (P → P)) → ((P → P) → CS CS (P → P))), e + (P → (P → P)), and c + (P → P). The problem of finding a “simplest” justification term is related to the problem of finding the “simplest” proof of a provable formula. It is not entirely clear what this actually means.
22
The Basics of Justification Logic
Corollary 2.16 (Lifting Lemma) Suppose JL is a justification logic that has the internalization property relative to CS (in particular, if CS is axiomatically appropriate). If X1 , . . . , Xn `JL(CS) Y then for any justification terms t1 , . . . , tn there is a justification term u so that t1:X1 , . . . , tn:Xn `JL(CS) u:Y. Proof The proof is by induction on n. If n = 0 this is simply the definition of Internalization. Suppose the result is known for n, and we have X1 , . . . , Xn , Xn+1 `JL(CS) Y. We show that for any t1 , . . . , tn , tn+1 there is some u so that t1:X1 , . . . , tn:Xn , tn+1: Xn+1 `JL(CS) u:Y. Using the deduction theorem, X1 , . . . , Xn `JL(CS) (Xn+1 → Y). By the induction hypothesis, for some v we have t1:X1 , . . . , tn:Xn `JL(CS) v:(Xn+1 → Y). Now v:(Xn+1 → Y) → (tn+1:Xn+1 → [v · tn+1 ]:Y) is an axiom hence t1:X1 , . . ., tn:Xn `JL(CS) (tn+1:Xn+1 → [v · tn+1 ]:Y). By modus ponens, t1:X1 , . . . , tn:Xn , tn+1:Xn+1 `JL(CS) [v · tn+1 ]:Y, so take u to be v · tn+1 . Next we move on to the role of justification variables. We said earlier, rather informally, that variables stood for arbitrary justification terms. In order to make this somewhat more precise, we need to introduce substitution. Definition 2.17 (Substitution) A substitution is a function σ mapping some set of justification variables to justification terms, with no variable in the domain of σ mapping to itself. We are only interested in substitutions with finite domain. If the domain of σ is {x1 , . . . , xn }, and each xi maps to justification term ti , it is standard to represent this substitution by (x1 /t1 , . . . , xn /tn ), or sometimes as (~x/~t). For a justification formula X the result of applying a substitution σ is denoted Xσ; likewise tσ is the result of applying substitution σ to justification term t. Substitutions map axioms of a justification logic into axioms (because axiomatization is by schemes), and they preserve modus ponens applications. But one must be careful because the role of constants changes with a substitution. Suppose CS is a constant specification, A is an axiom, and c:A is added to a proof where this addition is authorized by CS. Because axiomatization is by schemes Aσ is also an axiom, but if we add c:Aσ to a proof this may no longer meet constant specification CS. A new constant specification, call it (CS)σ, can be computed from the original one: put c:Aσ ∈ (CS)σ just in case c:A ∈ CS, for any c. If CS was axiomatically appropriate, CS ∪ (CS)σ will also be. So, if X is provable using an axiomatically appropriate constant specification CS, the same will be true for Xσ, not using the original constant specification but rather using CS ∪ (CS)σ. But this is more detail than we generally need to care about. The following suffices for much of our purposes.
2.6 The First Justification Logics
23
Theorem 2.18 (Substitution Closure) Suppose JL is a justification logic and X is provable in JL using some (axiomatically appropriate) constant specification. Then for any substitution σ, Xσ is also provable in JL using some (axiomatically appropriate) constant specification. We introduce some special notation that suppresses details of constant specifications when we don’t need to care about these details. Definition 2.19 Let JL be a justification logic. We write `JL X as short for: there is some axiomatically appropriate constant specification CS so that `JL(CS) X. Theorem 2.20 Let JL be a justification logic. (1) If `JL X then `JL Xσ for any substitution σ. (2) If `JL X and `JL X → Y then `JL Y. Proof Item (1) is directly from Theorem 2.18. For item (2), suppose `JL X and `JL X → Y. Then there are axiomatically appropriate constant specifications CS1 and CS2 so that `JL(CS1 ) X and `JL(CS2 ) X → Y. Now CS1 ∪ CS2 will also be an axiomatically appropriate constant specification and `JL(CS1 ∪CS2 ) X and `JL(CS1 ∪CS2 ) X → Y, so `JL(CS1 ∪CS2 ) Y and hence `JL Y. In fact, it is easy to check that `JL X if and only if `JL(T) X, where T is the total constant specification. This gives an alternate, and easier, characterization.
2.6 The First Justification Logics In this section and the next we present a number of specific examples of justification logics. We have tried to be systematic in naming these justification logics. Of course modal logic is not entirely consistent in this respect, and justification logic inherits some of its quirks, but we have tried to minimize anomalies. Naming Conventions: It is common to name modal logics by stringing axiom names after K; for instance KT, K4, and so on, with K itself as the simplest case. When we have justification logic counterparts for such modal logics, we will use the same name except with a substitution of J for K; for instance JT, J4, and so on. There is a problem here because a modal logic generally has more than one justification counterpart (if it has any). We will specify which one we have in mind. Formally, JT, J4, and so on result from the addition of axiom schemes, justification function symbols, and a constant specification to
24
The Basics of Justification Logic
J0 . When details of a constant specification matter, we will write things like JT(CS), J4(CS), and so on, making the constant specification explicit. We will rarely refer to J0 again because its definition does not actually allow for a constant specification. From now on we will use J for J0 extended with some constant specification, and we will write J(CS) when explicitness is called for. Note that J0 can be thought of as J(∅), where ∅ is the empty constant
specification. The general subject of justification logics evolved from the aforementioned G¨odel–Artemov project, which embeds intuitionistic logic into the modal logic S4, which in turn embeds into the justification logic known as LP (for logic of proofs). It is with LP and its standard sublogics that we are concerned in this section. These are the best-known justification logics, just as K, T (or sometimes KT), S4 (or sometimes KT4), and a few others are the best-known modal logics. For the time being the notion of a justification logic being a counterpart of a modal logic will be an intuitive one. A proper definition will be given in Section 7.2. With two exceptions, the justification logics examined here arise by adding additional operations to the + and · common to all justification logics. The first exception involves factivity, with which we begin. Factivity for modal logics is represented by the axiom scheme X → X. If we think of the necessity operator epistemically, this would be written KX → X. It asserts that if X is known, then X is so. The justification counterpart is the following axiom scheme. Factivity t:X → X Factivity is a strong assumption: justifications cannot be wrong. Nonetheless, if the justification is a mathematical proof, factivity is something mathematicians are generally convinced of. If we think of knowledge as justified, true belief, factivity is built in. Philosophers generally understand justified, true belief to be inherent in knowledge, but not sufficient, see Gettier (1963). The modal axiom scheme X → X is called T. The weakest normal modal logic including all instances of this scheme is KT, sometimes abbreviated simply as T. We use JT for J plus Factivity and, as noted earlier, we use JT(CS) when a specific constant specification is needed. Note that the languages of JT and J are the same. There is one more such example, after which additional operation symbols must be brought in. Consistency is an important special case of Factivity. Modally it can be represented in several ways. One can assume the axiom scheme X → ♦X. In
2.6 The First Justification Logics
25
any normal modal logic this turns out to coincide with assuming ¬⊥ (where ⊥ represents falsehood), or equivalently ⊥ → ⊥, which is a very special instance of X → X. If one thinks of as representing provability, ¬⊥ says falsehood is not provable—consistency. Suppose one thinks of deontically, so that X is read that X is obligatory, or perhaps that it is obligatory to bring about a state in which X. Then X → ♦X, or equivalently X → ¬¬X says that if X is obligatory, then ¬X isn’t—a plausible condition on obligations. It is because of this interesting deontic reading that any of the equivalent versions is commonly called D, standing for deontic. Any of these has a justification counterpart. We adopt the following version. Consistency t:⊥ → ⊥ JD is J plus Consistency. Note that JT extends JD.
Positive Introspection is a common assumption about an agent’s knowledge: If an agent knows something, the agent knows that it is known; an agent can introspect about the knowledge he or she possesses. In logics of knowledge it is formulated as KX → KKX. If one understands as representing provability in formal arithmetic, it is possible to prove that a proof is correct: X → X. To formulate a justification logic counterpart, Artemov introduced a one-place function symbol on justification terms, denoted ! and written in prefix position. The intuitive idea is that if t is a justification of something, !t is a justification that t is, indeed, such a justification. Note that the basic language of justification logics has been extended, and this must be reflected in any constant speci
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"Heather Wilson"
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2012-03-11T12:44:55+11:00
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Posts about M.S. Prabhu written by Heather Wilson
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https://cinemachaat.com/tag/m-s-prabhu/
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It took me a couple of films before I started to appreciate Suriya but after Pithamagan and Vaaranam Aayirami, I began to understand why so many people raved about him. The lovely Dolce recommended this film in a comment and after watching I am indeed a complete Suriya convert! Although the film is standard masala action fare with a paper-thin storyline, what makes it stand out are excellent performances from the lead actors and good well-rounded characterisations. In particular the scene-stealing Jagan Ayan is a surprise bonus in his role as a friend to Suriya’s character Deva.
The story follows a bad guys vs. good guys format although the good guys are smugglers and not exactly on the side of law and order. Suriya is Deva, an MSc graduate in computer engineering who works as a smuggler for his deceased father’s friend Dass (Prabhu). Despite his criminal activities, Dass has principles and refuses to smuggle drugs, preferring to deal in pirate DVD’s and diamonds. Now these seem to be at opposite ends of the smuggling scale to me, and I can’t imagine anyone being involved in both, but it’s not the most glaringly hard to swallow plot point, so it’s probably best not to dwell on it.
Dass is at the top of his game and apparently ranks as the number one smuggler in Chennai, a fact which does not go down well with his rival Kamalesh (Akashdeep Saigal). Kamalesh is a fairly pathetic villain, who has plenty of ambition but not much else going for him. Rather incongruously for a wannabe tough guy, he has very long hair, which he tosses back at every available opportunity and looks more like an aspiring supermodel than gangster. As far as criminal activities go he’s inept and bungling and, since Kamalesh looks like someone who wouldn’t manage to get an extra bottle of wine through customs let alone diamonds, his attempts to be top smuggler appear to be doomed to failure.
Chitti (Jagan Ayan) turns up as a hopeful member of the gang and after he does Dass a favour, is accepted into the group. He rapidly becomes an indispensable part of the team and Deva’s best friend, and the two have some excellent chemistry together. Jagan Ayan is brilliant as Chitti and I love the way his character is well developed and detailed for a non-hero role. Chitti has many shades of grey and this, along with the fact that his motivation is simply to make more money and enjoy life, makes him a more realistic character than expected from his first appearance.
Although Chitti’s character provides most of the comedy in the film, he does have a more serious part to play in the proceedings later and is just as good in the more dramatic moments. There isn’t a separate comedy track thankfully, and all of the comedy is integrated well into the main story. Deva also gets his fair share and this song features a number of ‘disguises’ worn by Deva – many of which are actually characters from his previous films. While I think he disproves the stereotype of heroes in drag and actually makes a passable woman, the long shaggy hair here is a definite no!
The other full time member of the group is Dilli (Karunas) who has a minor, but still vital, role to play and acts mainly as a driver for the others. There are a few other gang members who come and go, but the secret of Dass’s success seems to be in keeping his operation small and well hidden behind the front of a garbage disposal company. However in spite of all his precautions, the gang is continually raided by the police and Dass begins to suspect that one of the group is selling them out to Kamalesh.
The diamonds storyline means that the action shifts to ‘The Congo’ in Africa although the filming apparently took place in Tanzania and Namibia. It does make a change from various European locations at any rate and director K. V. Anand seems to have involved quite a few locals to good effect. Although all of the film is very well shot, this section in particular features some excellent cinematography from M. S Prabhu. It also includes possibly the best chase sequence I’ve seen in a Tamil film so far. After Deva collects the diamonds via an inexplicably convoluted system of torn banknotes and secret codes, they are stolen from his hotel room. He chases after the thief who has a very large circle of friends available who keep passing the diamonds to each other and keep Deva always just one step behind. The chase has a number of parkour-inspired sequences and is cleverly directed by South African stunt co-ordinator Franz Spilhaus to look fast, slick and very convincing.
After all the action, time for some romance. Chitti has a sister, Yamuna (Tamanna) and everyone seems just as baffled as I was that they are actually siblings.
They look nothing alike, and the difference is mentioned quite a few times in Yamuna’s introduction. However Yamuna does appear to have rather good taste in men, considering that her room has a number of Shah Rukh Khan pictures on the wall (I approve!) and of course she falls in love with Deva. She’s not a shrinking violet either and is quite happy to pounce on Deva at every available opportunity. And really, who can blame her!
The love story is fairly straight forward without any major obstacles although Chitti has some of the best lines as he teases his sister and friend about their relationship. Suriya and Tamanna make a sweet and reasonably credible couple even if they fall in love rather quickly and the romance only makes fleeting appearances in the second half of the film. Tamanna looks beautiful and her character has plenty of personality which she conveys by some excellent facial expressions. I really like Tamanna as an actress and she manages to be more than just the love interest, which is always an achievement in such a very hero-centric film.
Just before the interval the traitor in the gang is revealed and once the plot twist is exposed the rest of the film loses most of the suspense and tension and becomes just another action flick. At least until near the end, where everything picks up again until the rather OTT climax fight. The second half does tend to drag in parts and it’s not helped by the rather odd placement of the songs which mostly just disrupt the story. There is one terrible item song with Koena Mitra featuring a noticeable lack of dancing and dreadful lyrics which is used during a scene in a club and could very easily have been replaced with random dancing bodies for a more watchable effect. However a rather graphic depiction of the realities and consequences of becoming a drug mule is excellently done, and there are some great car chase sequences and explosions in the second half which almost make up for the meandering plot.
The film seems designed mainly to allow Suriya to show off his action hero persona and on that level it works well. He looks fit and capable and perfectly plays the action, romantic and comedy scenes, easily switching between the different moods and illustrating his versatility. The other characters are also well developed with both Chitti and Dass having plenty of input into the storyline and their presence also helps to define Deva’s character. Prabhas is excellent as Dass and injects a surprising amount of dignity into his role as a smuggler. The relationship between Dass and Deva is also nicely portrayed and there is genuine warmth between the characters. Renuka is good as Surya’s mother Kaveri and the other support actors all seem to fit well into their parts. Ponvannan also makes an appearance as Partiban, the harrassed Police officer in charge of customs at Chennai Airport who searches Deva on a fairly regular schedule. It’s really only Akashdeep Saigal who disappoints in both characterisation and dialogue.
Despite the unconvincing criminal Kamalesh, I really enjoy watching this film. There is plenty of action, good chemistry between Suriya and Tamanna, (although better between Suriya and Jagan) and the movie looks slick and polished. It just needs a snappier script and tighter story to make better use of the clever twists in the plot. Still well worth a watch if you are a Suriya fan or enjoy a mass action film which keeps the action coming. 3 ½ stars for the action, story and overall film but 5 stars for Suriya!
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Make Your Day
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https://www.indiatoday.in/movies/regional-cinema/story/samantha-ruth-prabhu-adds-disclaimer-to-podcast-video-after-spat-with-the-liver-doc-2573581-2024-07-30
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Samantha Ruth Prabhu adds disclaimer to podcast video after spat with The Liver Doc
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[
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"Samantha health podcast",
"The Liver doc",
"hydrogen peroxide nebulization",
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[
"Janani K"
] |
2024-07-30T00:00:00
|
Samantha Ruth Prabhu added a disclaimer to her health podcast video after her spat with Dr Cyriac Philips Abby aka The Liver Doc. Earlier, he slammed her for promoting hydrogen peroxide nebulization.
|
en
|
https://www.indiatoday.in/favicon.ico
|
India Today
|
https://www.indiatoday.in/movies/regional-cinema/story/samantha-ruth-prabhu-adds-disclaimer-to-podcast-video-after-spat-with-the-liver-doc-2573581-2024-07-30
|
Actor Samantha Ruth Prabhu, in a recent health podcast video with Dr David Jockers (natural medicine practitioner), spoke about discovering supplements and alternate medicine. She added a disclaimer to her podcast video asking people to consult their physician and care provider before taking them. She further mentioned that the video she posted is for 'informational purposes only'. This comes weeks after her online spat with hepatologist Sr Cyriac Philip Abby aka The Liver Doc over hydrogen peroxide nebulization.
advertisement
In the podcast, she said that she found Dr David Jockers when she researched functional nutrition. She also claimed that Jockers had been a friend to her from the United States.
Samantha has shared over 20 health podcasts over the last few weeks. However, what's different this time is the addition of a disclaimer.
Her disclaimer reads, "The information, including but not limited to, audio, text, graphics, images and other material contained in this episode are for informational purposes only. No material in this episode is intended to be a substitute for professional medical advice, diagnosis or treatment. Always seek the advice of your physician or other qualified health care provider with any questions you may have regarding a medical condition or treatment and before undertaking a new health care regimen, and never disregard professional medical advice or delay in seeking it because of something you have heard in this podcast (sic)."
Here's the post:
Earlier, Samantha spoke about how hydrogen peroxide and distilled water nebulization helped cure her cold. Her post led to a war of words with Dr Cyriac Philips Abby, who is fondly known as The Liver Doc on X.
Dr Philips debunked her claim and called her a 'health illiterate'. He criticised her harshly and stated that hydrogen peroxide nebulization would lead to irritation in the lungs and eventually damage the organ as well.
Samantha shared a three-page statement in which she said that her intention was to help people and not to harm others. In the podcast that was released recently, she added a disclaimer. This disclaimer wasn't a part of her podcast-related posts.
Here's her statement:
Samantha, after her Myositis (an auto-immune condition) diagnosis, found alternate medicine to help her. Over the years, she had been vocal about it as well.
Work-wise, Samantha Ruth Prabhu will be seen next in Prime Video's 'Citadel: Honey Bunny'. She also has a Telugu film titled 'Bangaram' in the pipeline.
|
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https://www.indiaforums.com/forum/topic/3336263
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en
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~Neethaane En Ponvasantha Reviews & Discussions ~
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2012-12-10T01:43:21+00:00
|
Neethaane En Ponvasantham Discussions Reviews First Day First Show Neethane En Ponvasantham (FDFS) Review
Neethane
En Ponvasantham
|
en
|
India Forums
|
https://www.indiaforums.com/forum/topic/3336263
|
First Day First Show Neethane En Ponvasantham (FDFS) Review
Neethane En Ponvasantham (NEP) is a love story focusing on the moments from Varun and Nithya's love story spanning 3 phases of their life. Neethane En Ponvasantham is directed by Gautham Vasudev Menon and jointly produced by Photon Kathaas and RS Infotainment. The song tracks by Ilayaraja are already well received.
A lot of things are riding on this movie. This is an important movie for many talented people including Jiiva who is looking for a hit to stabilize his stardom. Gautham Menon's last hit was Vinnaithaandi Varuvaya was almost 3 years ago. Fans are also hoping Ilayaraja's 919th movie will bring him back to working on commercial movies after disassociating himself with popular directors for most part of the last decade.
Story Moments
As Gautham Menon himself revealed in many interviews, there is nothing new about the story. It is just about the little moments in the lives of Varun (Jiiva) and Nithya (Samantha) that is expected to keep you glued to the screen. To keep audience engaged for almost 154 minutes hinging on just moments is not an easy task.
For these moments to sustain the interest, there needs to be at least 4 key things going for it -- relate-able situations, characterization, actor performance and dialogues. Gautham Menon probably gets latter 2 out of these 4 things right. Dialogues are convincing and relate-able in most situations. But, the situations may seem too ordinary and characterizations are not clear since the rationale for why characters behaves the way they do is not clear.
Screenplay and Direction
Gautham Menon decides to narrate the story in a linear fashion. He goes through 4 episodes of Varun and Nithya's life (children, high school, college and work). In all these situations, they break-up due to minor squabbles. It is very clear right from the beginning that the movie's journey is all about finding whether Varun and Nithya will live happily ever after when the credits roll. The story moves at slow place and intentionally at a lower gear for almost 3/4th of the movie. Towards the last 30 minutes, Gautham Menon takes full control over the proceedings and makes you root for the lead pairs intensely. However, it is not convincing how they lose touch completely without any withdrawal symptoms which would have given them the chance to get back together.
We will not reveal whether Varun and Nithya will come together in the end. Given Gautham Menon's track record of not letting the lead pairs live together, the viewers knows very well that the director will be ready to steer the movie in whichever direction he wants without commercial compulsions. The last few reels grabs you by surprise and score high points in mostly uneventful movie.
Jiiva, Samantha and Santhanam
The movie is carried only by the strengths of these 3 characters. Jiiva is as usual performs with any insecurity. Samantha dubs in her own voice and shines with her expressions. Ever-dependable Santhanam enlivens the proceedings whenever he shows up on screen. No doubt, Gautham Menon has extracted great performance from Jiiva and Samantha.
For a movie filled with sweet nothings, Jiiva and Samantha's conviction is commendable. Jiiva comes across as calm, composed, mature and practical. He lives by bullet-list of priorities and makes clear decisions using his brain reasoning out objectively. On the other hand, Samantha is emotional and makes her decision with her heart. She reacts to the situations at the spur-of-the-moment and later regrets her decisions. There are no villains. You hardly ever see the lead pairs' parents. Except for a few friends, there are not many characters in the movie.
Music and Technicalities
Ilaiyaraja has composed 8 tracks in total. Except for the 'pudikale maamu', rest of the songs come in multiple parts throughout the movie making it a full-fledged musical movie. The background tracks are also mostly borrowed from the interludes in the songs, which work well. Cinematography is subdued and employs long wide shots for critical scenes with mixed results.
The editing by Antony in the last 30 minutes is noteworthy when scenes reminiscing the past is well blended with flash-cuts. There are no fight situations and no foreign location songs. Not much work for art director with most of the scenes shot in candid locations. There are also no big sets erected for songs. Except for the opening title song and cultural scenes, dance choreographer doesn't have much work in this movie. Gautham has given the stunt department a rain-check.
Downers
The story drags and the moments are too pedestrian. The movie should be crisper by at least 20 minutes. For a movie riding on 3 characters, the characterization should be stronger. It is unclear why the director decided to shoot so many important scenes without any cuts and in wide range. This makes you think whether the dialogues were actually written after the shots were actually picturized.
The dubbing choices may come in for question. Samantha dubs in her own voice and may evoke mixed reactions from public. Varun's father, Ravi Raghavendar's distinctively recognizable voice, is not capitalized with his voice dubbed by a voice artist. Gautham Menon himself sings a lengthy portion of "Neethane en ponvasantham" song in college culturals when Varun first meets Nithya. Again, Gautham's voice is a distraction and takes the focus away from the beauty of the moment.
Bottomline
Most of the audience may find it difficult to get immersed in these undramatic moments, they will find the movie lengthy and pointless. For a section of audience who considers Vinnaithandi Varuvaya is an all-time classic, they will not complain about this movie.
Gautham Menon has taken a huge gamble by releasing a soft romance story in between a heavily crowded release season. The box office success of this movie is will dependent on the success of Kumki and how aggressively the long line-up of movies will try to grab "Neethane En Ponvasantham" screens from next week.
Rating - 3/5
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Samantha Akkineni's 'Neethane En Ponvasantham's title was inspired by 80's Tamil song?
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../favicon.ico
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https://www.republicworld.com/entertainment/samantha-akkinenis-neethane-en-ponvasanthams-title-was-inspired-by-80s-tamil-song
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http://kollywooddoctor.blogspot.com/2012/12/neethane-en-ponvasantham-you-are-my.html
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Movie Reviews and Stuff: Neethane En Ponvasantham (You Are My Golden Spring)
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[
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First hand reviews on latest Tamil movies; news, views, analysis and more on Movie Reviews and Stuff. Visit Kollywood Doctor's Blog now!
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http://kollywooddoctor.blogspot.com/favicon.ico
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http://kollywooddoctor.blogspot.com/2012/12/neethane-en-ponvasantham-you-are-my.html
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Check out the list of Upcoming Tamil Movies releasing in 2024 with release date, casts, genres, photos and trailers.
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https://www.91mobiles.com/
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Check out the list of Upcoming Tamil Movies releasing in 2024 with release date, casts, genres, photos and trailers.
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http://filmstudentjp.blogspot.com/2012/12/neethane-en-ponvasantham-review.html
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Film - Visualizations of Imagination and Emotion
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Official Poster Neethane En Ponvasantham, latest flick by Director Gautham Vasudev Menon sir after his Nadunissi Naaygal. Neeth...
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Samantha Ruth Prabhu On The Heart Of The Matter
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[
"The Heart of the Matter",
"SAMANTHA RUTH PRABHU"
] | null |
[
"Shilpa Dubey"
] |
2024-03-08T11:00:00+05:30
|
A spark that lights up a room. Sanmantha Ruth Prabhu gives a glimpse into what it takes to be a self-made superstar
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en
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https://www.femina.in/celebs/indian/samantha-ruth-prabhu-on-the-heart-of-the-matter-283649.html
|
Dress: MACH & MACH
Soft-spoken, yet articulate. Funny, with a laugh that resonates. And, that confidence. A spark that lights up a room. Samantha Ruth Prabhu gives us a glimpse into what it takes to be a self-made superstar in India
The year is 2005, or maybe 2006? Samantha Ruth Prabhu is sitting with her girlfriends from college. The barely-20-somethings are playing a game. Each of them takes a turn to say what they wish for in life. Someone hopes for a good boy, someone else, a good-looking husband. Samantha wants a house. “I think my biggest dream was to afford a house I could call my own,” she reminisces. “That was always the dream. Back then, it was more than enough; I wouldn’t have dared to dream bigger than that.” That’s exactly why she is not quite comfortable with this memory today. She wants her younger self to have wished for more, to have dreamt bigger, and to not have been afraid of her ambitions. “I’m not saying it will be easy, but you should not be afraid to dream big. You have every right to dream big. You are allowed to have the biggest ambitions. If you’re thinking of a little white house, change that dream. Dream of a mansion! Dream it all!” The camera captures her conviction as she appeals to young women everywhere to not settle for less. She arrived, warm and almost nonchalant, in a grey jumpsuit to our cover shoot set. She switched to siren mode in a hot second once in front of the camera, and didn’t take a break until it was pack-up time. Once she starts something, the lady sees it through.
What Samantha feels about her ambition from a different lifetime is reflected in her journey from Ye Maaya Chesave (2010) to Citadel India (2024). Last month, Samantha completed 14 years in the entertainment industry. In this period, she has stood tall through several twists and turns – from the peaks of success and award-winning performances to a life-threatening illness and a divorce. And she wouldn’t have life happen any other way. “All the highs and lows that I’ve experienced have brought me to this point in life,” she avers. “I have seen the highest highs and the lowest lows, and I wouldn’t change a thing. People might think that success defines you, but it is failure and loss that truly define you, and introduce you to the best version of yourself. These lows and these losses have made me a person I am truly proud of.”
Dress: Gaurav Gupta
In the initial years, she struck gold in the Tamil and Telugu film industries with back-to-back hits. In 2012, her films Neethaane En Ponvasantham and Eega brought her several accolades – including major awards. She also became only the second actor to win both the Filmfare Award for Best Actress – Telugu and the Best Actress – Tamil award in the same year. Samantha believes that the whirlwind of initial hits contributed immensely to her filmography, awards shelf, stars and destiny. “It wasn’t really hard work,” she says, adding, after a brief pause, “But, now, I can say I am truly proud of myself for having gone through some hardships and come out of them a stronger, better person. Today, I can say I’m prouder of this version of me than I’ve ever been.” Her recovery from the rare auto-immune disease myositis, Samantha tells us, has given her a more evolved perspective on life. Her profession, she adds, also makes her a better person. “The kind of appreciation and the rewarding experience that you get from being an actor – it makes you feel like you need to give back. It makes you believe that destiny has been kind to you, people have been kind to you, and it has taught me the importance of giving back,” she reiterates.
Lights, Camera, ACTION!
As we go to press, the world is waiting to get a glimpse of this strength of character in her portrayal of Honey in the Amazon epic action series Citadel India, which will release in India in April 2024. “I will never ever forget playing the role of Honey because showing up was a major task,” she reveals of her time in recovery from her debilitating condition. “Every day was an effort. When I completed the first season of Citadel India, I realised that playing Honey has been the most fulfilling role of my life. I’m really proud of what I was able to do with that role, given the circumstances. I think that it could be my best work yet.”
It is this same attitude of steely determination that Samantha has always approached her projects with. In Shakuntalam (2023), her character needed to be adorned in floral jewellery, the kind that gave her an allergic reaction, but her need for perfection as an artist meant that she shot the whole day in those flowers and even had a floral-shaped mark imprinted on her hand for a month because of the reaction.
The controlled enthusiasm in her voice makes you want to go and rewatch her movies right away. There’s something magnetic about the way she says things in a low-pitched, yet clear tone, and in the way she holds your gaze when she talks to you. Her command over Tamil, Telugu, Hindi and English only amplifies her wheelhouse and makes her a reigning star in the pan-India entertainment biz; she has been recognised as the most popular Indian female actor in 2022 and 2023, as per Ormax Media’s annual reports.
Dress and Shoes: MACH & MACH
“Jack Of All Trades”
Yet, Samantha tells us she has had to work twice as hard as others to master her skills and display them flawlessly on screen. “I am a Jack of all trades and master of none,” she laughs, before continuing thoughtfully, “While I work in Telugu, Hindi and Tamil films, I didn’t really grow up speaking these languages. I would have had an extra edge if I was able to think in those languages. I need to put in twice the hard work compared to someone who grew up speaking these languages, so it forces me to never get too relaxed in any scene, big or small. I’m always working hard before a scene and I hope that that shows,” she explains in a single breath. This commitment speaks once again to the fact that she loves her craft and has embraced the good and bad it has to offer.
In 2015, Samantha announced that she might quit the industry if she didn’t get to work on artistic projects. “My focus is always to put my heart and soul into bringing out a character that is honest and real,” she shares. “I don’t think so much about the business side of it. I think many of us make mistakes when we focus on getting the big numbers; we might not deliver the best quality work.”
Her nuanced acting and her ability to wear her characters as second skin speak volumes about her talent, which explains why parts started to be written for her as she began picking author-backed roles. That move led to female-centric performances. Theri (2016), Mersal (2017), Rangastahlam (2018), Mahanati (2018), U-Turn (2018), Oh! Baby (2019), Super Deluxe (2019), Majili (2019), The Family Man 2 (2021) and Yashoda (2022) are all testament to her well-deserved and magnificent rise to stardom – all on her own terms. And it has come from more than just love for the art and her innate talent, it is also thanks to her ability to connect with audiences, transcending linguistic and cultural boundaries. “I have always stayed honest to the portrayal of a role and to the telling of a story that has real emotions,” she agrees. “It has always worked in my favour. I think, now more than ever, I’m really focused on telling good stories.”
Lace Corset and Pants: Munique
The Beauty And The Battle
Her battle with myositis led Samantha to make the “hardest decision of her life” – to take a break from work. In retrospect, she thinks it was the best decision she has ever made, “because there’s no way I would have been able to continue working. The stress of work, coupled with the condition, is not the easiest thing to handle. I’m really glad that I gave myself time to recover; I had been working for 13 years straight.” She took a year off to travel, to heal, to be with her thoughts, to come back stronger.
Right now, Samantha feels stronger and more beautiful than ever – which she credits to the strength and confidence that brews inside of her. This new-found centre of Samantha’s core comprises sisterhood, holistic living and self-love. The road within is often the hardest, Samantha admits, and does not hide her rite of passage through the darkness. “I’ve had my fair share of self-loathing and really low confidence, but I’ve always strived to grow as a person,” she discloses. “With that growth came a deeper understanding of my insecurities and self-loathing. I was able to heal by addressing them – not by trying to fix them from the outside, but by fixing the inner trauma that needed more healing than any external quick fix.”
The biggest lesson from everything she has gone through is gratitude. “Anyone who goes through a difficult phase always comes out stronger; always consider yourself lucky for the tough times,” she says, “You are going to come out of it with so much knowledge. Without going through these experiences, people can only dream of being as strong as you are. Be grateful for the lessons you learn, and keep in mind that God only gives us problems that we can handle,” she affirms.
Breastplate: Misho; Trench Coat and Trousers: Hermès
When Life Gives You Lemons…
Internet trolls. Where do they not strike? During the period of her separation from her then-husband, actor Naga Chaitanya, leading up to their divorce and throughout her illness, there was a lot of chatter around everything she did. With her every move under scrutiny, Samantha was harrowed by the media. There are good and bad sides to everyone having an opinion, she states, but she strongly feels being an actor makes the experience worse. “Sometimes, being an actor makes your world very small and closed; there tends to be a group of ‘yes’ people around you who don’t usually tell you what is on their mind,” she explains. Social media allows Samantha a perspective on how the world perceives her. “I’m a big girl now; I know how to differentiate between slander and constructive criticism,” she asserts. “I keep my eyes and ears open for constructive criticism, which I think is important for me as an actor and as a human being.”
Destiny’s Woman
Born on April 28, Samantha is a true Taurean in many ways, from her fondness for earthy tones – both in her wardrobe and her cosy home – to her head-on approach to anything. Taureans are intellectual, analytical, practical and grounded, and their ability to stay patient and fight back when irked is well known. But, while Taureans are not known to enjoy change, Samantha’s personality defies the stars. She strongly believes that change is the key to evolution.
Dress: Gaurav Gupta
Her faith in a universal power is unwavering, but Samantha is open to changing her opinion if logical reasoning and empathy are brought into play. In an age when authenticity is more a filter than an acquired skill, Samantha claims that the authentic self should never be rigid. She believes that the true test of the self lies in a willingness to understand others and an openness to change with time. Rigidity in thought ticks her off. “I’m not rigid in my beliefs,” she points out. “I sometimes see how people judge others and say, ‘You said this before and now you’re changing your opinion.’ There’s nothing wrong with changing your opinion. We learn as we go along, and we should be allowed to change things that now seem wrong.” She further muses, “Even the strongest, the brightest, of us all succumb to certain ideas of success, certain ideas of how a ‘good woman’ should be, certain ideas of respectable choices. At the end of the day, you have to live with yourself.” she says. Whether you have a little white house or a comfortable mansion in Jubilee Hills, if the person you’ve become doesn’t make you happy, don’t settle. “Make decisions not based on societal conditioning, but on what you truly want for yourself.” This is something Samantha would tell her younger self today. We all should too.
Photographs: Taras Taraporvala
Styling: Mohit Rai
Art direction: Bendi Vishan
Makeup: Avni Rambhia
Hair: Rohit Bhatkar
Styling Assistance: Tarang Agarwal, Ruchi Krishna, Shubhi Kumar
Videography: Abhishek Trivedi
Hospitality Partner: Chholay
Also Read: Life As She Knows It
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On a rocky road
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[] |
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[
"Rajiv Vijayakar",
"R...Rajkumar",
"Prabhu Deva",
"shahid Kapoor",
"Sonakshi sinha"
] | null |
[
"DHNS"
] |
2013-12-07T16:42:56+05:30
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After a string of average releases and failures, actor Shahid Kapoor is hoping for a strong comeback with Prabhu Deva's ...
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Deccan Herald
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https://www.deccanherald.com/features/on-rocky-road-2298267
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A decade after he started out with Ishq Vishq (he was a child model and did several ads, of course), Shahid Kapoor is in the space where he is always praised for his work even if his films under-perform. Yes, he would like to change the second part, and hopes to do that with Prabhudheva’s R…Rajkumar, made by the maverick choreographer-filmmaker, who put on track the careers of Salman Khan and Akshay Kumar after lean phases.
In this film, where he is doing his first hardcore action and also romancing and dancing to chartbuster songs, Shahid has put in his all. As he puts it, “I enjoy the action genre. Beating up baddies is everyone’s dream, as is anything that you cannot do in real life. And I am excited about this movie. Prabhu sir is a stylised director, and with him, normal action too becomes stylised and larger than life. Unhonein to meri band bajaa di in terms of making me fulfill his vision.” And Shahid’s countenance breaks into a crinkly smile and a guffaw — two aspects that frequently embellish a 20-minute conversation that often goes off-track with casual banter.
Across genres
Working with Prabhudheva clearly seems to have been a life-changing experience for the actor, despite a past track record of eminence with names like Sooraj R Barjatya (Vivah), Imtiaz Ali (Jab We Met), Abbas-Mustan (36 China Town), Priyadarshan (Chup Chup Ke), Vishal Bhardwaj (Kaminey — a dual role), Rajkumar Santoshi (Phata Poster Nikhla Hero), Mahesh Manjrekar (Waah!...Life Ho To Aisi) and his father Pankaj Kapur (Mausam).
“He is a genius,” Shahid raves about his director. “He knows the common man’s fantasy and the tapori-mawwali space like no one else does. I am sure that he must have been a quiet, bottled-up child storing experiences, because he once told me, ‘I make films about real characters,’ even if he fulfills a larger-than-life fantasy in what they do on screen.”
About his modus operandi as a director, Shahid says, “A day before the shoot, he would introduce me to something completely new and make me rehearse it for hours. Prabhu sir is a man of few words, who would summon me and say, ‘Come! See!’ and ‘Yeah, come start!’ Since he knew that I was a trained dancer — though our styles could not be more different — he would push me to do things people would not expect from me, but never ever praise my work. And the same applied to his action sequences. Prabhu sir keeps reinventing his action by changing his action directors in each film and discussing shots with them for days.”
The actor grins and adds, “Prabhu sir even gave me some friendly advice: That after the film was over, I would be mentally and physically drained and should take a long holiday — and he was right. After 50 days of action, 16 days of songs and 30 days of talkie, I really wish I could take a break.”
Clearly, the actor is on an anticipatory high even if all his films in the last few years have been rejected at the box-office. Getting a tad sober when we mention his bad run post-Jab We Met in 2007, he says, “I need lots and lots of luck. I know that people have always said ‘Shahid was good but the film was bad,’ and I am thankful for their support, but things have to change. I am asked what went wrong with Phata Poster Nikhla Hero. My only thought is that maybe the second half got too serious.”
He goes on, “It’s like this: A film’s success is most important. Even in a cricket match, a player scoring 55 runs in a winning team gets more attention than a guy who scores 125 runs when his team loses.”
About his character and the mystery in the R… of the title, he wants the audience to wait until they watch the film. “After we could not use the title of Rambo Rajkumar, Prabhu sir decided to go along with this for a reason that will be clear when the film releases,” he grins. “As for Rajkumar, he is a very clear guy with two fundas as mentioned in the promos: pyar pyar pyar or maar maar maar.”
Role play
Admitting that while he had done some challenging roles in his career, like in Kaminey, Shahid says that they were a cakewalk compared to this one. “Such a character is much more difficult to essay,” he declares. “As an actor, you are tested at every level. The body language has to be perfect, as you are neither a cool dude nor someone who is like a larger-than-life star. For example, in keeping with my character, I was given this special vest to wear, which was so incredibly tacky and cheap that I wondered if anyone wore such things. And then while driving to Film City one day, I actually saw a chaiwalla boy wearing something identical. And yes, it actually helped me get the body language right.”
Shahid is generous in his praise for co-star Sonakshi Sinha, who he terms a “super actor and a warm human being.” Terming her one of the “few complete heroines we have today,” the actor recalls how she picked up the dance steps and remembered them so fast when he was fumbling. “That was the first impression she made on me, as it was the song, Saree ke fall sa that we first shot together.”
He is also affectionate about the song’s creator, his music composer Pritam. “With few exceptions, Pritam has composed music for all my films since Jab We Met, and irrespective of how the films turned out, he has been consistent in his music for me. We have known each other for ages. I remember he once gave me this ring to wear for luck, something that looked like an ugly emerald, and I told him, ‘Dude, that’s not cool,’ but he insisted I wore it.”
About on-screen antagonist Sonu Sood, he says warmly, “We were shooting when I had six to seven cuts on my leg and he had multiple fractures on his, so we were actually protective towards each other during our fight sequences so that we did not hurt each other,” he recalls. “Action is like dance, synchronised and with a strict rhythm that if accidentally missed can be dangerous. That is why we see professional fighters reverently touch the ground and perform a small puja before they begin work.”
Shahid is now working on Vishal Bhardwaj’s Haider. How does he plan to ensure that his films to come do not go wrong? “I think a filmmaker’s vision of the characters is crucial,” he replies thoughtfully. “As an actor, however, I cannot see the eventuality of the completed product and have to make a commitment on the limited information given to me and available at the time.”
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Ayan (film)
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2008-03-25T01:45:06+00:00
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en
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https://en.wikipedia.org/wiki/Ayan_(film)
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2009 film by K. V. Anand
Ayan
Official poster
Directed byK. V. AnandProduced byM. Saravanan
M. S. GuhanWritten bySubha (Dialogue)Screenplay by
K. V. Anand
Subha
Story by
K. V. Anand
Subha
StarringMusic byHarris JayarajCinematographyM. S. PrabhuEdited byAnthony Gonsalves
Production
company
Distributed by
Release date
3 April 2009 ( )
Running time
162 minutesCountryIndiaLanguageTamilBudget₹15 crore[2]Box office₹80 crore[2]
Ayan ( (help·info)) (English: Creator) is a 2009 Indian Tamil-language action film co-written and directed by K. V. Anand. The film, starring Suriya, Prabhu, Tamannaah, Akashdeep Saighal, Jagan, and Karunas, was produced by M. Saravanan and M. S. Guhan, distributed by Sun Pictures. The film score and soundtrack was composed by Harris Jayaraj, Edited by Anthony Gonsalvez, the film was filmed by M. S. Prabhu.[3]
The film's story revolves around Devaraj Velusamy, a youngster whose mother wants him to become a government official. He, on the other hand, works for a smuggling group run by Anthony Dass, who has been his mentor and has been looking after him since childhood. Conflict occurs when Deva's archrival, Kamalesh, opposes and tries to eliminate Dass from the smuggling business. Who wins in the conflict forms the climax of the story.
The film was launched in Chennai, while filming also took place in various locations out of India, including Namibia, Malaysia, Zanzibar and South Africa. It released on 3 April 2009 worldwide to positive reviews.[4][5] Ayan was declared as the solo blockbuster of the year in Tamil cinema,[6] collecting about ₹80 crore (US$11 million) worldwide.[2] It was also dubbed and released on 1 May 2009 in Telugu as Veedokkade, which was a box office success in Andhra Pradesh.[7][8][9] and also in Kerala.[2]
Plot[edit]
Devaraj Velusamy (Suriya) arrives at the Chennai International Airport after running an errand for Anthony Dass (Prabhu) of smuggling pre-release unlicensed movies on DVD. The two of them leave for their hideout and instruct Dilli (Karunas), a hearing-impaired assistant of Dass, to make copies of the DVD. Just when police crews arrive into their vicinity, Dass informs Deva and the rest to leave. Deva tells the others to leave while he quickly sets up the burning process. Deva also leaves the hideout just in time. However, the police officers arrive at the hideout, only to see the burning process of the DVDs complete. The police then seizes the DVDs and computers. The inspector, who was a friend of Dass, tells him that he can close the case if one of Dass's men agrees to the crime. Deva attempts to go, but another man, Chitti Babu (Jagan), who had no affiliation to the volunteers, takes the blame. Later, Chitti joins Dass's group and befriends Deva.
Dass is a diamond trafficker who smuggles diamonds from Africa. Deva's mother Kaveri (Renuka) does not appreciate his affairs with Dass, as she wants Deva, who holds a Master of Science degree, to have a government job. He occasionally visits home and her roadside grocery shop, only to get scoldings from his loving mother.
When Deva is invited to Chitti's house, he gets impatient and knocks on the door. As the door opens, Deva slips and falls on the bed, expecting to see Chitti but scaring his younger sister Yamuna (Tamannaah) instead. Just before he drives off, he curses himself for making such a stupid mistake. Yamuna catches this and accepts his apologies, then they slowly fall in love, and their relationship is accepted by Chitti.
Meanwhile, Kamalesh (Akashdeep Saighal), the arch-enemy and competitor in smuggling of Dass, slowly starts to try to foil Dass and his employees. A prank phone call from Chitti leads to a near-death experience for both Deva and Kaveri. Escaping alive and outraged, Deva goes and confronts Chitti, releasing his anger to the fullest extent. Chitti then reveals that he was Kamalesh's personal spy and that even though that their bosses were foes, he was happy to be a friend of Deva. Disgusted, he leaves, and Chitti to. At that moment, Yamuna arrives, hearing everything. Thus, this stir of events causes the interval of the movie.
After these events, Deva and Yamuna break up. In Malaysia, under circumstances, Chitti and Deva meet. Chitti, who was fed with drugs, falls prey to them and starts reacting. In a last attempt to rekindle their friendship, Deva attempts to save him. Chitti then explains that it was wrong of him to be a spy, but still admits that he was lucky to have Deva as a friend. He dies when Kamalesh's henchmen try to salvage the drugs by cutting him open. Heartbroken, Deva burns Chitti's body upon his request and returns to India.
When Deva arrives, Yamuna approaches him and asks the police officers to arrest him. However, she notices Chitti's phone, which he used to record a video before dying. Yamuna brings it out, which contains a recording of how Chitti died. She also sent the video to Dass. It is here that Deva did not kill Chitti but hid from Kamalesh's henchmen. Yamuna apologises to Deva, and the two come together again.
Later, circumstances force Deva to become a part of the security squad in the airport as he is accused by J. Parthiban IRS (Ponvannan), the Assistant Commissioner of the Income Tax Air Intelligence Unit of carrying drugs. He denies it, saying that the packet of drugs was given by a passenger's relative as she had forgotten it. The drugs are opened, and it is found that heroin is sealed in them. Deva then realises and explains that huge amount of drugs are being transported by distracting him with a small amount. The flight is delayed and all passengers are asked to take food from the canteen. There are a few who do not do so, and hence, their stomach is X-rayed, and a huge packet of heroin is found. Deva is then asked to help Parthiban and the customs for arresting Kamalesh.
Deva and Parthiban put a cab-like room outside Kamalesh's house, and with Yamuna's help, a transmitter is put in Kamalesh's study. Then, all his conversations with his clients are recorded. Kamalesh later finds out and attempts to kill them, which fails. He later finds out that idols made of heroin are compressed under high pressure to disguise their smell and properties. With this evidence, the income tax department comes to raid Kamalesh's house, but he acts quickly and removes everything. During a final attempt to arrest Kamalesh, the truth is forced out of his accountant, but Kamalesh kills him to avoid any witnesses. Later, Deva's house is almost burnt, and Dass is killed by Kamalesh, which infuriates Deva. Deva is then forced to go to Africa and confront Kamalesh.
When Deva returns to India, he is again inspected by the customs officer. Deva learns that it was his mother this time who exposed his smuggling mission. He has no choice but to surrender the diamonds, and so he does. Parthiban hands Deva to sign a form, which Deva hesitates. He then learns that is a government security job application form. Parthiban says that Deva's criminal mind is required for such a job in the customs. He then asks what had happened to Kamalesh, to which Deva replies that he had gone to the Congo to strike a business deal. Then, a flashback shows that when Kamalesh was returning from stealing the diamonds, his gang is killed by Deva, and he is pushed off the cliff.
The film ends on a happy note as Deva leaves the airport with his mother and Yamuna.
Cast[edit]
Production[edit]
Development[edit]
Three years after the release of his debut venture Kana Kandaen starring Srikanth and Gopika, cinematographer K. V. Anand expressed his desire to commence his second film as director. He and Subha discussed several plots and settled for "Ayan" because "it was not only different but had scope for entertainment".[10] It was later announced that Anand would be directing his next film, produced under the AVM Production banner, titled Ayan. The film was inaugurated at AVM Studios on 24 March 2008 with the presence of most of the unit members.[11] On the occasion, Anand announced the film's genre to be an action thriller interlaced with romance and comedy. He also suggested Ayan meant "outstanding", excellence" or can be the name for sun rays in five different languages. However, this was later doubted by a few critics, raising a point saying Ayan was not necessarily a Tamil language word. Despite the film's lyricist Vairamuthu's calls for the word to be a Tamil word, it was argued that Ayan was a nickname for the Hindu deity Lord Brahma. Since the word was then touted to be borrowed from the Sanskrit language, the film was prone to a title change, in light of Tamil Nadu's Entertainment Tax Exemption Act, which was passed in 2006.[12] A similar problem was faced by the producers of Aegan, which was also under production at the time. The controversy was later dropped. Ayan was announced to be predominantly set in various locations of both South Asia and Africa. It was made at a production cost of ₹ 200 million.[13]
Casting[edit]
K. V. Anand announced the film with both Suriya and Tamannaah to play the lead roles in the film.[14] Surya was initially expected to play the lead role in Anand's earlier film, but was not able to do so.[10][15] Anand expressed his thoughts during the film's inauguration that "Surya was apt for the title role" since the film's title meant "sun rays" and the name "Surya" refers to the sun. Ayan would also make Surya's second film with AVM Productions, after their previous partnership in Perazhagan. Furthermore, Surya had worked with Anand since Surya's debut in the 1997 film Nerukku Ner for which Anand was the film's photographer.[10][16] Surya's physique was key for his character, as he would be acting as a powerful and active youngster. During the film's launch, he announced that he would give the film his best, understanding the nature of the producers, who previously presented the big-budget film Sivaji. Tamanna, after starring in the Tamil films Kalloori (for which she was nominated for a Filmfare Award) and Padikathavan, was cast in Ayan. Prabhu was consecutively cast in Ayan in a pivotal role, as per his previous films, in which he played important supporting characters.[17] It was later announced that Anand would introduce a new actor to Tamil cinema in the film, who will be playing a negative role, which was later known to be Akashdeep Saighal, who predominantly works in Bollywood films.[18]
Apart from the film's cast, the film's crew consisted of Harris Jayaraj as the film composer along with Vairamuthu, Pa. Vijay and Na. Muthukumar as the lyricists. M. S. Prabhu was chosen as the film's cinematographer, who is a friend of Anand and worked with him under the guidance of P. C. Sriram.[19]
Filming[edit]
Nenje Nenje’ song was shot on the sand dunes of the Namib-Naukluft National Park, Dune 7 and Dead Vlei in Namibia, in freezing temperature. Songs were composed in Mauritius, where William Honk choreographed the car chase shots.[10] The stunt sequences were shot at Binny Mills with a huge set resembling an airport while another fight was shot at Cape Town, Africa.[20] The filming was also held at Kuala Lampur, Malaysia, Botswana, Zimbabwe and Zanzibar.[21][22]
Themes and influences[edit]
The film dealt with the concept of smuggling and custom officers. In order to prepare the script, Anand did a lot of research and is said to have spoken to a lot of custom officers to understand the modus operandi of smuggling.[23][24]
Soundtrack[edit]
The soundtrack album was composed by Harris Jayaraj. The Audio was launched on January 18, 2009.[25] The Times of India credited the popularity of the film to the popularity of songs. Critics were impressed with the album, with praise being dedicated to the entire soundtrack, most notably "Vizhi Moodi".[26] Due to the album's critical and commercial success Harris Jayaraj received his fifth Filmfare Award for Best Music Director, the Mirchi Award for Best Album of the Year and the Edison Award for Best Music Director.
AyanSoundtrack album by ReleasedJanuary 18, 2009Recorded2009GenreSoundtrackLength30:21LanguageTamilLabelAvm MuzikProducerHarris JayarajHarris Jayaraj chronology
Vaaranam Aayiram
(2008) Ayan
(2009) Aadhavan
(2009)
Track-List[27]No.TitleLyricsSinger(s)Length1."Pala Pala"Na. MuthukumarHariharan5:282."Nenje Nenje"VairamuthuHarish Raghavendra, Mahathi5:473."Honey Honey"Pa. VijaySayanora Philip, Devan Ekambaram5:224."Vizhi Moodi"Na. MuthukumarKarthik5:355."Oyaayiye Yaayiye"Pa. VijayBenny Dayal, Haricharan, Chinmayi5:386."Oh Super Nova"Na. MuthukumarKrish2:31Total length:30:21
VeedokkadeSoundtrack album by ReleasedMarch 10, 2009Recorded2009GenreSoundtrackLength30:31LanguageTelugu LabelAvm MuzikProducerHarris Jayaraj
All tracks are written by Bhuvanachandra.
Telugu Track-List[28]No.TitleSinger(s)Length1."Pela Pela"Hariharan5:242."Nene Nene Needhanne"Harish Raghavendra, Mahathi5:443."Honey Honey"Sayanora Philip, Devan Ekambaram5:214."Kallu Moosi Yochisthey"Karthik5:535."Oyaayiye Aayiye"Benny Dayal, Haricharan, Chinmayi5:386."Oh Super Nova"Krish2:31Total length:30:31
Release[edit]
The satellite rights of the film were secured by Sun TV. The film was given a "U/A" certificate by the Indian Censor Board. AVM Productions sold the film's distribution rights for Tamil Nadu to Sun Pictures for ₹ 200 million.[29]
Critical reception[edit]
Malathi Rangarajan of The Hindu wrote: "Ayan is Suriya’s show all the way. He bears the onus with a smile and the death defying stunts add to the robust image he aims to project".[30] Behindwoods wrote: "Ayan is fun. Just buy a huge bag of popcorn, a can of cola and have a blast! But do remember the first step about the logic".[31] Indiaglitz wrote: "Though the storyline is familiar and oft-seen in the past, the pacy narration and captivating visuals provide the necessary pep to the film".[32] [33]
Box-office[edit]
In Chennai alone, box office totals were reported as ₹74.3 (US$1.00) in theatrical revenue.[34] International distribution rights were sold to Ayngaran International. Ayan's revenue was US$1,046,027 in Malaysia[35] and £119,220 in the UK.[36] The film's Telugu version, Veedokkade, was sold to Hyderabad based producer, Bellamkonda Suresh.[37][38][39][40] [41][42] The final worldwide box office was around ₹65 crore (US$9.1 million).[2] The film ran over 100 days in Kerala box office.[2] the Telugu version Veedokkade ran over 100 Days in Andhra box office and ran over 201 days in Tamil Nadu.[2]
Awards[edit]
Ayan received the most nominations (12) in the Filmfare Awards South 2009 and in the 4th Annual Vijay Awards (18).
Award Category Nominee Result 2009 Filmfare Awards South Best Actor Suriya Nominated Best Director K.V. Anand Nominated Best Film Ayan Nominated Best Lyricist Na. Muthukumar
("Vizhi Moodi") Won Best Lyricist Vairamuthu
("Nenje Nenje") Nominated Best Male Playback Harish Raghavendra
("Nenje Nenje") Nominated Best Male Playback Karthik (singer)
("Vizhi Moodi") Nominated Best Music Director Harris Jayaraj Won Best Supporting Actor Prabhu Nominated Best Supporting Actor Jagan Nominated Best Supporting Actress Renuka Nominated Best Dance Choreographer Dinesh
("Pala Pala") Won 2009 Vijay Awards Best Actor Suriya Nominated Best Art Director Rajeevan Nominated Best Choreographer Dhinesh
("Pala Pala") Nominated Best Director K. V. Anand Nominated Favourite Director K. V. Anand Nominated Best Entertainer Suriya Won Favourite Hero Suriya Nominated Favourite Heroine Tamannaah Nominated Favourite Film AVM Productions Won Favourite Song Vizhi Mooodi Nominated Best Lyricist Vairamuthu
("Nenje Nenje") Nominated Best Male Playback Harish Raghavendra
("Nenje Nenje") Nominated Best VFX Compositor Srinivas Karthik Kotamraju
(EFX) Nominated Best Supporting Actor Jagan Nominated Best Comedian Jagan Nominated Best Stunt Director Kanal Kannan Nominated Best Costume Designer Nalini Sriram Nominated Icon of the Year Suriya Won 2009 Meera Isaiaruvi Tamil Music Awards Best Album of the Year Harris Jayaraj Won 2009 South Scope 2010 Awards Best Actor – Tamil Suriya Nominated Best Actress – Tamil Tamannaah Nominated Best Supporting Actor – Tamil Prabhu Won Best Comedian – Tamil Jagan Nominated Best Music Director – Tamil Harris Jayaraj Won Best Male Playback – Tamil Harish Raghavendra
(Nenje Nenje) Nominated Best Male Playback – Tamil Karthik
("Vizhi Moodi") Nominated Best Female Playback- Tamil Mahathi
("Nenje Nenje") Nominated Best Lyricist – Tamil Na. Muthukumar
("Vizhi Moodi") Won Best Cinematographer – Tamil M.S.Prabhu Nominated
References[edit]
[edit]
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https://www.sampspeak.in/2012/10/neethane-en-pon-vasantham-goes-viral.html
|
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"Sampath Speaking" - the thoughts of an Insurer from Thiruvallikkeni
|
[
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[
""
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[] | null |
A blog on Insurance, especially Marine Insurance; Sports more of Cricket and Current affairs involving SYMA, Triplicane, Tamil Nadu and India
|
https://www.sampspeak.in/favicon.ico
|
https://www.sampspeak.in/2012/10/neethane-en-pon-vasantham-goes-viral.html
|
Hi - this is Srinivasan Sampathkumar from Triplicane. I have a passion for Marine Insurance, Cricket and Temples especially - Sri Parthasarathi swami thirukKoyil, Thiruvallikkeni. From Sept 2009, I am posting my thoughts in this blog; From July 2010, my postings on Temples & Tamil are on my other blog titled "Kairavini Karayinile " (www.tamil.sampspeak.in) Nothing gives the author more happiness than comments & feedbacks on posts ~ look forward to hearing your views !
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https://www.timeslive.co.za/sunday-times/lifestyle/2013-02-24-moving-romantic-movie-is-a-feast-of-music/
|
en
|
Moving romantic movie is a feast of music
|
https://lh3.googleusercontent.com/TbUCrUbLTeGwmtppr2JNWzzdGHZ3jGPt78rzppWSyI3YnPs99HHkit3z98p80BbIv3hkXx1-b7isjQtUOTmu=s1000
|
https://lh3.googleusercontent.com/TbUCrUbLTeGwmtppr2JNWzzdGHZ3jGPt78rzppWSyI3YnPs99HHkit3z98p80BbIv3hkXx1-b7isjQtUOTmu=s1000
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2013-02-24T00:00:00
|
Goutham Vasudev Menon returns to the genre he shines in, romance, with Neethane en Ponvasantham. NEETHANE EN PONVASANTHAM (9/10)INTRODUCTIONJiiva and Samatha Ruth Prabhu play the lead roles, supported by Santhaanam, Ravi Prakash and Anupama Kumar.STORYNEETHANE en Ponvasantham tells the story of the love that grows between Varun (Jiiva) and Nithya (Prabhu).
|
en
|
/favicon.png
|
TimesLIVE
|
https://www.timeslive.co.za/sunday-times/lifestyle/2013-02-24-moving-romantic-movie-is-a-feast-of-music/
|
Goutham Vasudev Menon returns to the genre he shines in, romance, with Neethane en Ponvasantham.
NEETHANE EN PONVASANTHAM (9/10)
INTRODUCTION
Jiiva and Samatha Ruth Prabhu play the lead roles, supported by Santhaanam, Ravi Prakash and Anupama Kumar.
STORY
NEETHANE en Ponvasantham tells the story of the love that grows between Varun (Jiiva) and Nithya (Prabhu). There are neither villains nor cruel blows from fate, but their romance is a rocky road because of a variety of issues. These include personal ego, family and life commitments and, often, a simple failure to communicate.
Reshma Ghatala, who penned the story, takes simplicity and credibility to new levels, but what is a strength for some is a weakness for others.
PERFORMANCES
JIIVA handles his role brilliantly, especially with his ability to express emotions.
Santhaanam provides good support in a role that transcends the usual, but it is Prabhu whose performance lingers long after the film ends. Her powerful expressions dominate her interpretation of the role. She also chose to dub herself in the film and uses her slightly stammering Tamil to her advantage.
DIRECTION
MENON infuses sensibility into a well-narrated story and the result is an easy-paced film that is anything but boring. One gets into the minds and hearts of the characters and the result is a love story that deserves a place in the classics.
CINEMATOGRAPHY
CINEMATOGRAPHY by MS Prabhu and Om Prakash is, for the most part, brilliant. However, there were two scenes - one where Varun's family is introduced and the second in the long scene just before interval - where a single camera is used in an extra-long shot. It is possible that this is a new technique but the effect is unimpressive.
SCREENPLAY AND SCRIPTING
THE film has been criticised for its slow pace but I appreciated it because the characters are credible. It is also refreshing to have a love story without comedians, villains or any other sideshows. The main characters are villains to each other.
MUSIC
NOT enough can be said about the contribution by Isai Nyaani Illaiyaraja. Apart from the mesmerising songs, his background score lives up to his usual high standards. This is truly a musical feast.
OVERALL
LIKE Kaadhalil Sodhupuvadhu Yeppadi, whatever your experience in love, you will find something to warm your heart and move you to the core in Neethane en Ponvasantham.
Last week's questions and answers
Q: Name the two female lead actresses in Paandiyarajan's 'Aan Paavam'.
A: Seetha and Revathy.
Q: Who played the female lead in Kamalhassan's 100th film?
A: Madhavi.
Q: Who directed the hit film 'Vasantha Maaligai', which had Sivaji Ganesan and Vanishree in the lead roles?
A: KSPrakash Rao.
This week's questions
1. Who scored the music in Jiiva's debut film, 'Aasai Aasaiyaai'?
2. Who played the two male leads in the debut film of Goutham Vasudev Menon?
3. Which film features the hit song 'Madras Nalla Madras'?
|
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https://www.missmalini.com/2021/11/26/samantha-ruth-prabhu-signs-first-international-project-with-downton-abbey-director
|
en
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Samantha Ruth Prabhu Signs First International Project With Downton Abbey Director
|
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[] |
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[
""
] | null |
[
"Akash Bhatnagar"
] |
2021-11-26T00:00:00
|
Samantha Ruth Prabhu signs her first international project with Downtown Abbey director Philip John, titled The Arrangements Of Love, based on a novel by the same name
|
en
|
MissMalini
|
https://www.missmalini.com/2021/11/26/samantha-ruth-prabhu-signs-first-international-project-with-downton-abbey-director
|
Samantha Ruth Prabhu has been one of the biggest superstars in the South film industry for over a decade now. She is beautiful, talented, versatile and has a screen presence that is unmatched by anyone. This year, she took the Hindi film industry by storm with her impressive role as Raaji in The Family Man 2 and that was how I was introduced to her power on screen. Her aura and craft compelled me to revisit some of her old work in the Southern industry and my admiration for her just grew post that. And now, after making her presence felt in the Hindi industry, Samantha is set to go international with renowned Downtown Abbey director, Philip John.
According to a source, the actress had a meeting with Philip, earlier this month in Chennai which then eventually evolved into a concrete discussion about a collaboration. The project is a feature film titled The Arrangements Of Love, based on Timeri N Murari’s novel by the same name.
Taking to her Instagram, Samantha made this news official earlier today by sharing the announcement poster of the film along with a picture of herself with Philip.
Check out the post here:
An excited Samantha opened up to Variety about this big news in her career, while raving about her director Philip. The project also marks her reunion with Sunitha Tati, who is backing this film and had also backed her 2014 film, Oh! Baby.
Samantha said:
“A whole new world opens up for me today as I start my journey with ‘Arrangements of Love,’ which has such an endearing and personal story. I am excited to work with Philip John, whose projects I have closely followed for many years, being a big fan of ‘Downton Abbey.’ I am looking forward to collaborating with Sunitha once again and I hope for only more success than we previously had with ‘Oh! Baby’. My role is a complex character and it will be both a challenge and an opportunity for me to play it. I cannot wait to get on set.”
The Arrangements Of Love follows the story of a Welsh-Indian man on his quest to find his estranged father with an impromptu visit to his homeland. In that, Samantha plays a strong minded and funny force of nature 27-year-old, who is a progressive bisexual Tamil woman and runs her own detective agency, thus becoming part of the search. She has ultra-traditional parents, who want her to have an arranged marriage, and as much as she wants to please her parents, she also wants to be able to make her own life choices.
Samantha has an immense amount of talent that cannot be restrained and this project is just the beginning for her. I am sure that The Arrangements Of Love will lay the perfect groundwork for the actress’ flourishing career in the future. As an Indian, it fills me with pride whenever any talent from our country is recognised globally and I am really excited for Samantha as she embarks on this new journey, trying to make India proud.
|
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| 15
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https://en.wikipedia.org/wiki/Gautham_Vasudev_Menon
|
en
|
Gautham Vasudev Menon
|
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2006-05-01T09:37:02+00:00
|
en
|
/static/apple-touch/wikipedia.png
|
https://en.wikipedia.org/wiki/Gautham_Vasudev_Menon
|
Indian film director, film producer, screenwriter and actor
Gautham Vasudev Menon (born 25 February 1973) is an Indian film director, screenwriter, film producer and actor who predominantly works in Tamil film industry.[2] He has also directed Telugu and Hindi films that either simultaneously shot with or remakes of his own Tamil films. He has won two National Film Awards, three Nandi Awards and one Tamil Nadu State Film Award.
Many of his films have been both critically acclaimed and commercially successful, most notably his romantic films Minnale (2001), Vaaranam Aayiram (2008), Vinnaithaandi Varuvaayaa (2010), his cop action thrillers Kaakha Kaakha (2003), Vettaiyaadu Vilaiyaadu (2006), Yennai Arindhaal (2015) and his gangster drama Vendhu Thanindhathu Kaadu (2022). His 2008 Tamil film, Vaaranam Aayiram won the National Film Award for Best Feature Film in Tamil. Menon produces films through his film production company named Photon Kathaas. His production Thanga Meengal (2013) won the National Film Award for Best Feature Film in Tamil.[3]
Early life and education
[edit]
Gautham was born on 25 February 1973 in Ottapalam, a town in the Palakkad district of Kerala. His father Vasudeva Menon was a Malayali and his mother Uma was Tamil. Although born in Kerala, he grew up in Anna Nagar, Chennai.[4][5]
Menon did his schooling at the Madras Christian College Higher Secondary School.[6] He then earned a bachelor's degree in Mechanical Engineering from Mookambigai College of Engineering, Pudukkottai.[7][8]
Film career
[edit]
Early work, 2001
[edit]
Menon's time at university inspired him to write the lead roles of Minnale, Vaaranam Aayiram, Vinnaithaandi Varuvaayaa, Neethaane En Ponvasantham and Enai Noki Paayum Thota who were students in the same course.[9] During the period, he was inspired by films such as Dead Poets Society (1989) and Nayakan (1987) and expressed his desire to his parents to change his career path and become a filmmaker. His mother insisted that he become an ad filmmaker by shooting various commercials and he took an apprenticeship under filmmaker Rajiv Menon. He went on to work as an assistant director for Minsara Kanavu (1997), in which he also appeared in a cameo role.[10]
Menon launched a Tamil romance film O Lala in 2000 with the project eventually changing producers and titles into Minnale (2001) with Madhavan, who was at the beginning of his career, being signed on to portray the lead role.[11] About the making of the film, Menon revealed that he found it difficult as the team was new to the industry with only the editor of the film, Suresh Urs, being an experienced technician.[12] Menon came under further pressure when Madhavan insisted that the film's story was narrated to the actor's mentor, Mani Ratnam, to identify if the film was a positive career move. Despite initial reservations, Menon did so and Ratnam was unimpressed; however, Menon has since cited that he thought that Madhavan "felt sorry" and later agreed to continue with the project.[12] The film also featured Abbas and newcomer Reemma Sen in significant roles, whilst Menon introduced Harris Jayaraj as music composer with the film.[11] The film was advertised as a Valentine's Day release in 2001 and told the tale of a young man who falls in love with the girl engaged to his ex-college rival. Upon release it went on to become a large success commercially and won positive reviews from critics, with claims that the film had a lot of "lot of verve and vigour" and that it was "technically excellent".[13]
The success of the film led producer Vashu Bhagnani to sign him on to direct the Hindi language remake of the film, Rehnaa Hai Terre Dil Mein (2001), starring Madhavan alongside Dia Mirza and Saif Ali Khan. Menon was initially apprehensive but said it eventually took "half an hour" to agree, but against his intentions, the producer opted against retaining the technical crew of the original.[10] He changed a few elements, deleted certain scenes and added some more for the version. A critic felt that "the presentation is not absorbing" though stating that Menon "handled certain sequences with aplomb"; the film subsequently became a below-average box office performer.[14] The failure of the film left him disappointed, with Menon claiming in hindsight that the film lacked the simplicity of the original with the producer's intervention affecting proceedings.[15] Several years after release, the film later gained popularity through screenings on television and subsequently developed a cult following amongst young Hindi-speaking audiences.[16] In 2011, the producer of the film approached him to remake Rehnaa Hai Terre Dil Mein with the producer's son Jackky Bhagnani in the lead role, but Menon was uninterested with the offer.[17] Later on in 2001, it was reported that he was working on a film tentatively titled Iru Vizhi Unadhu, though the project did not develop into production.[18]
Police duology, 2003–06
[edit]
Gautham Menon returned in 2003 by directing the realistic police thriller Kaakha Kaakha (2003) starring Suriya, Jyothika and Jeevan. The film portrayed the personal life of a police officer and how his life is affected by gangsters, showing a different perspective of police in comparison to other Tamil films of the time.[15] Menon revealed that he was inspired to make the film after reading articles on how to encounter specialists who shoot gangsters and how their families get threatening calls in return, and initially approached Madhavan, Ajith Kumar and then Vikram for the role without success, with all three actors citing that they did not want to play a police officer. The lead actress Jyothika asked Menon to consider Suriya for the role, and he was subsequently selected after Menon saw his portrayal in Nandha.[19] He held a rehearsal of the script with the actors, a costume trial with Jyothika and then enrolled Suriya in a commando training school before beginning production, which he described as a "very planned shoot".[19] The film consequently opened to very positive reviews from critics on the way to becoming another success for Menon, with critics labeling it as a "career high film".[20] Furthermore, the film was described as "for action lovers who believe in logical storylines and deft treatment" with Menon being praised for his linear narrative screenplay.[21]
Menon subsequently remade the film in the Telugu language as Gharshana (2004) starring Venkatesh in Suriya's role. The film also featured actress Asin and Salim Baig in prominent roles and went on to earn commercial and critical acclaim with reviewers citing that "film redeems itself due to the technical excellence and masterful craft of Gautham", drawing comparisons of Menon with noted film makers Mani Ratnam and Ram Gopal Varma.[22] In July 2004, Menon also agreed terms to direct and produce another version of Kaakha Kaakha in Hindi with Sunny Deol in the lead role and revealed that the script was written five years ago with Deol in mind, but the film eventually failed to take off.[23] Producer Vipul Shah approached him to direct the Hindi version of the film in 2010 as Force with John Abraham and Genelia D'Souza, and Menon initially agreed before pulling out again.[24] Menon and the original producer, Dhanu, also floated an idea of an English-language version with a Chechnyan backdrop, though talks with a potential collaboration with Ashok Amritraj collapsed.[19] In 2018, Menon revealed that he had plans of making a sequel to Kaakha Kaakha with Suriya.[25]
He was then signed on to direct a venture starring Kamal Haasan and produced by Kaja Mohideen, and initially suggested a one-line story which went on to become Pachaikili Muthucharam for the collaboration.[19] Kamal Haasan wanted a different story and thus the investigative thriller film Vettaiyaadu Vilaiyaadu (2006), was written with Jyothika, Kamalinee Mukherjee, Prakash Raj, Daniel Balaji and Salim Baig added to the cast. The film told another episode from a police officer's life, with an Indian cop moved to America to investigate the case of psychotic serial killers before returning to pursue the chase in India. During the shooting, the unit ran into problems after the producer had attempted suicide and as a result, Kamal Haasan wanted to quit the project.[19] Menon subsequently convinced him to stay on as they had taken advance payments. He has since revealed that unlike Kamal Haasan's other films, the actor did not take particular control of the script or production of the film. The film however had gone through change from the original script with less emphasis on the antagonists than Menon had hoped and he also revealed that scenes for songs were forced in and shot without him.[19] The film released in August 2006 and went on to become his third successive hit film in Tamil and once again, he won rave reviews for his direction.[26][27] Menon later expressed interest in remaking the film in Hindi with Amitabh Bachchan in the lead role without the love angle, though the project fell through after discussions. In 2012, he re-began negotiations with producers to make a Hindi version of the film with Shahrukh Khan in the lead role.[28] He had stated his intent on making a trilogy of police episode films, with a possible third featuring Vikram in the lead role, before completing it in 2015 with Ajith Kumar in Yennai Arindhaal.[19]
Success, 2007–08
[edit]
His next project, Pachaikili Muthucharam (2007), based on the novel Derailed by James Siegel, featured Sarath Kumar and Jyothika and was released in February 2007. Initially, the lead role was offered to Kamal Haasan who passed the opportunity, while actors Cheran and Madhavan declined citing date and image problems respectively.[29] Menon met Sarath Kumar at an event where he cited he was looking to change his 'action' image and Menon subsequently cast him in the lead role.[29] During production, the film ran into further casting trouble with Simran dropping out her assigned role and was replaced by Shobana after another actress, Tabu, also rejected the role.[30] Shobana was also duly replaced by a newcomer, Andrea Jeremiah to portray the character of Kalyani in the film. The film was under production for over a year and coincided with the making of his previous film which was largely delayed. The film initially opened to positive reviews with a critic citing that Menon is "growing with each passing film. His style is distinctive, his vision clear, his team rallies around him and he manages to pull it off each time he attempts".[31][32] However the film became a financial failure for the producer, V. Ravichandran and in regard to the failure of the film, Menon went on to claim that Sarath Kumar was "wrong for the film" and that he tweaked the story to fit his image; he also claimed that his father's ailing health and consequent death a week before the release had left him mentally affected.[29] In mid-2007, Menon announced and began work on a youth-centric film titled Chennaiyil Oru Mazhaikaalam featuring Trisha and an ensemble of newcomers. Set in the backdrop of Chennai's booming IT industry, the team began its shoot in September 2007 and continued for thirty days but was later delayed and eventually shelved.[33] In 2011, he revealed that the film was dropped because he felt the actors "needed to be trained", and would consider restarting the project at a later stage.[34][35]
His next release, Vaaranam Aayiram (2008), saw him re-collaborate with Suriya, who played dual roles in the film. The film illustrates the theme of how a father often came across in his son's life as a hero and inspiration, and Menon dedicated the film to his late father who died in 2007.[36] The pre-production of the film, then titled Chennaiyil Oru Mazhaikaalam began in 2003, with Menon planning it as a romantic film with Suriya as a follow-up to their successful previous collaboration, Kaakha Kaakha.[37] Abhirami was signed and then dropped due to her height before a relatively new actress at the time Asin was selected to make her debut in Tamil films with the project. The first schedule of the film began in January 2004 in Visakhapatanam and consequently romantic scenes with Suriya and Asin were shot for ten days and then a photo shoot with the pair.[37] The film was subsequently stalled and was eventually relaunched with a new cast including Divya Spandana, Simran and Sameera Reddy in 2006 with Aascar Ravichandran stepping in as producer, who opted for a change of title. Menon has described the film as "autobiographical and a very personal story and if people didn't know, that 70% of this [the film] is from my life".[29] Throughout the film-making process, Menon improvised the script to pay homage to his late father by adding a family angle to the initial romantic script, with Suriya eventually playing dual roles. The film's production process became noted for the strain and the hard work that Suriya had gone through to portray the different roles with production taking nearly two years.[36] The film was released to a positive response, with critics heaping praise on Suriya's performance while claiming that the film was "just a feather in Gautam's hat" and that it was "a classic".[38] The film was made at a budget of 150 million rupees and became a commercial success, bringing in almost 220 million rupees worldwide.[36] It went on to become Menon's most appreciated work till date winning five Filmfare Awards, nine Vijay Awards and the National Film Award for Best Feature Film in Tamil for 2008 amongst other accolades. Post-release of the film, Menon had a public fallout with his regular music composer Harris Jayaraj and announced that they would no longer work together, though they later returned in 2015 for Yennai Arindhaal.[39] In late 2008, during the making of Vaaranam Aayiram, he had signed on with Sivaji Productions to direct Ajith Kumar and Sameera Reddy in an action film titled Surangani.[40] Menon later pulled out of the commitment citing that the producers were not willing to let him take his own time with scripting.[41]
Romance and experimentation, 2010–2014
[edit]
In 2010, Menon made a return to romantic genre after 2 years with the Tamil romantic film Vinnaithaandi Varuvaayaa (2010), starring Silambarasan and Trisha.[42] Originally planned as a Jessie with Mahesh Babu in the lead role, the actor's refusal prompted Menon to make the Tamil version first. The film explored the complicated relationship between a Hindu Tamil assistant director, Karthik, and a Christian Malayali girl, Jessie and their resultant emotional conflicts. The film featured music by A. R. Rahman in his first collaboration with Menon whilst cinematographer Manoj Paramahamsa was also selected to be a part of the technical crew. Menon cited that he was "a week away from starting the film with a newcomer" before his producer insisted they looked at Silambarasan, with Menon revealing that he was unimpressed with the actor's previous work.[12] The film was in production for close to a year and throughout the opening week of filming, promotional posters from classic Indian romantic films were released featuring the lead pair.[43] Prior to release, the film became the first Tamil project to have a music soundtrack premiere outside of India, with a successful launch at the BAFTA in London.[44] Upon release, the film achieved positive reviews, with several critics giving the film "classic" status, whilst also become a commercially successful venture.[45][46] Reviewers praised Menon citing that "credit for their perfect portrayal, of course, goes to Gautam Vasudev Menon. This is one director who's got the pulse of today's urban youth perfectly" and that "crafted a movie that will stay in our hearts for a long, long time."[46] Soon after the Tamil version began shoot, Menon decided to begin a Telugu version titled Ye Maaya Chesave (2010) and release it simultaneously, with featuring a fresh cast of Naga Chaitanya and debutante Samantha. Like the Tamil version, the film won critical acclaim and was given "classic" status from critics, as it went on to become among the most profitable Telugu films of 2010.[46][47][48] In 2016, he revealed that he had scripted a spin-off film from Vinnaithaandi Varuvaayaa titled Ondraga, where the character of Karthik's life would be followed eight years after the happenings of the previous film.[35]
He next began research and pre-production work on a 1920s period spy thriller titled Thuppariyum Anand in early 2010, with both Ajith Kumar and then Suriya considered for the lead roles, but the film failed to progress.[49][50] Menon had also made progress over the previous two years directing the psychological thriller Nadunisi Naaygal (2011) featuring his assistant and debutant Veera Bahu and Sameera Reddy. Menon claimed that the film was inspired by a true event from the US, while also claiming that a novel also helped form the story of the film.[12] During the making, he explicitly revealed that the film was for "the multiplex audience" and would face a limited release, citing that "it will not cater to all sections of the audience".[12] He promoted the film by presenting a chat show dubbed as Koffee with Gautham where he interviewed Bharathiraja and Silambarasan, both of whom had previously worked in such psychological thriller films with Sigappu Rojakkal and Manmadhan. The film, which was his first home production under Photon Kathaas and did not have a background score, told the story of a victim of child abuse and the havoc he causes to women, narrating the events of a particular day. The film opened to mixed reviews with one critic citing it as "above average" but warning that "don't go expecting a typical Gautham romantic film" and that it "is definitely not for the family audiences", while criticizing that "there are too many loopholes in the story, raising doubts about logic".[51] In contrast another critic dubbed it as an "unimpressive show by director Menon, as it is neither convincing nor appealing, despite having some engrossing moments".[52] A group of protesters held a protest outside Gautham's house on reason for misusing a goddess's name in his film and also showing explicit sex and violent scenes, claiming that it was against Tamil culture.[53] Soon after the release of the film, Menon began pre-production work on a television series featuring Parthiban in the lead role of a detective, but did not carry through with the idea after he failed to find financiers.[54]
Menon returned to Bollywood with the Hindi remake of Vinnaithaandi Varuvaayaa, titled Ekk Deewana Tha (2012), with Prateik Babbar and Amy Jackson.[55] Unlike the South Indian versions, the film opened to unanimously below average reviews, with critics noting the story "got lost in translation",[56] and became a box office failure.[57] Post-release, Menon admitted that he "got the casting wrong", and subsequently other Hindi films he had pre-planned were dropped.[58] During the period, Menon also began pre-production work on the first film of an action-adventure series of films titled Yohan starring Vijay in the title role. However, after a year of pre-production, the director shelved the film citing differences of opinion about the project.[59]
Menon's next releases were the romantic films Neethaane En Ponvasantham (2012) in Tamil and Yeto Vellipoyindhi Manasu (2012) in Telugu, both co-produced by Photon Kathaas. Jiiva and Nani played the lead roles in each version respectively, while Samantha was common in both films. Ilaiyaraaja was chosen as music composer for the film, which told the story of three stages in the life of a couple.[60][61] A third Hindi version Assi Nabbe Poorey Sau, was also shot simultaneously with Aditya Roy Kapoor playing the lead role, though the failure of Ek Deewana Tha saw production ultimately halted.[62][63] The films both opened to average reviews and collections, with critics noting Menon "falls into the trap every seasoned filmmaker dreads -- of repeating his own mandatory formula" though noting that the film has its "sparkling moments".[64][65] The response of the film prompted a legal tussle to ensue between Menon and the film's producer Elred Kumar, prompting the director to release an emotionally charged letter attempting to clear his name of any financial wrongdoing.[66] Menon was then briefly associated with the anthology film, X, helping partially direct a script written by Thiagarajan Kumararaja before opting out and being replaced by Nalan Kumarasamy.[67] He also began production work on a big-budgeted venture titled Dhruva Natchathiram, signing up an ensemble cast including Suriya, Trisha and Arun Vijay, with a series of posters issued and an official launch event being held. However, in October 2013, Suriya left the film citing Menon's lack of progress in developing the script and the film was subsequently dropped.[68] Later in early 2015, Menon restarted pre-production for the project with Vikram and Nayantara, but again was forced to postpone the film citing financial restraints.[69][70]
Return to action genre, 2015–present
[edit]
Following Suriya's withdrawal from Dhruva Natchathiram, Menon moved on to begin a romantic thriller film with Silambarasan and Pallavi Subhash in the lead role from November 2013. The film developed under the title Sattendru Maarathu Vaanilai and was shot for thirty days, before the film was put on hold as a result of Menon getting an offer from producer A. M. Rathnam to begin a film starring Ajith Kumar. Consequently, in April 2014, he began filming for Yennai Arindhaal (2015), the third instalment in his franchise of police films. He described Ajith's character Sathyadev as an "extension" of the protagonists from Kaakha Kaakha and Vettaiyaadu Vilaiyaadu, while Trisha, Anushka Shetty, Arun Vijay and Parvathy Nair were also selected to portray supporting roles.[71] The film saw him collaborate again with music composer Harris Jayaraj, for the first time since their spat in 2008, while writers Shridhar Raghavan and Thiagarajan Kumararaja were both also involved in the screen-writing process. Focussing on the story of a police officer's professional and personal life from the ages of thirteen to thirty-eight with the backdrop of tackling an organ-trafficking gang, Yennai Arindhaal opened to mixed to positive reviews in February 2015.[72][73] Critics from The Hindu wrote it "leaves you feeling like having gone back to a well-known play you have enjoyed a few times over", and that it is "a much-needed intervention in the Tamil commercial cinema space" while also "the most engaging of the three [police films]".[74] Reviewer Udhav Naig of The Hindu added that "Gautham wins as he has reconfigured, albeit not radically, the basic contours of a Tamil cop", and that he "has consistently improved on the character sketch in the last three films."[75] The film also performed well at the box office and gave Menon his first commercially successful Tamil film in five years.[76] Soon after the release, he began work on a sequel to the film and expressed his interest in approaching Ajith again to work together in the future.[77] Menon also worked as a singer in Radha Mohan's film Uppu Karuvaadu (2015).[78]
After Yennai Arindhaal, Menon resumed work on his film with Silambarasan under the new title of Achcham Yenbadhu Madamaiyada, with Manjima Mohan joining the cast to replace Pallavi Sharda. A Telugu version with Naga Chaitanya and Manjima was simultaneously shot under the title Sahasam Swasaga Sagipo, with Menon revealing that he hoped to finance the Tamil version through the salary he received from the Telugu film's producers.[79] The film developed slowly and was further delayed after Silambarasan refused to shoot for the film following a salary dispute during mid-2016. Menon also has Enai Noki Paayum Thota, a drama featuring Dhanush and Megha Akash in production, with the shoot began during February 2016.
Menon also has several proposed directorial projects in production. He began producing and directing the spy thriller Dhruva Natchathiram in late 2016 with Vikram leading an ensemble cast. Despite regular schedules throughout 2017, the film has been put on hold as Menon looks to raise funds. He has agreed terms to make an anthology film for Netflix, and another film titled Joshua: Adhiyayam Ondru for Vels Film International featuring Varun.[80][81][82] He has also completed a web series titled Queen based on the life of political leader Jayalalithaa, which features Ramya Krishnan in the lead role. The series is released on MX Player.[83][84] Menon also still aspires to complete his proposed multilingual film titled Ondraaga, which would be a spiritual sequel to Vinnaithaandi Varuvaayaa.[85][86][87]
Filmmaking style
[edit]
Menon has stated that he is largely inspired by the "depth and aesthetics" that are created by American films.[88] He usually makes the characters in his films by sport identical haircuts and urban casual wear and by speaking in English. His films are also known for their strong depiction of female characters, in contrast to other contemporary Tamil films which, according to journalist Sudhish Kamath, are "hero-worshipping star vehicles where the heroine is just a mere prop". Kamath also notes that several defining traits of Menon's films include liberal doses of English and restraint, the villains being "a seriously dangerous threat", his male protagonists being a "picture of grace and dignity" who are yet fallible, who love their fathers and are trying hard to be good men, who respect women and accept them for who they are.[89] The majority of Menon's police films feature a woman, typically the male lead's wife or lover, being fridged.[90] Menon stated that distributors and financiers often lay several limitations and constraints on his films, that such actions only drift his thoughts and make him feel like he is losing creative control.[88] Though his films are perceived as targeting mainly urban audiences, Menon feels they can be enjoyed by anyone.[91]
According to The Hindu's Udhav Naig, Menon's films are "regulated by a matrix of strong middle-class values", and also have biographical elements which, according to Menon, are inspired by his own life.[92] Menon prefers to write the climax of his films only after filming has significantly progressed, stating that though he has an idea about the climax, it always changes when the film starts shooting.[93] He also names his films after classic Tamil phrases and lines from Tamil film songs such as Achcham Yenbadhu Madamaiyada being named after the namesake song from Mannathi Mannan (1960) and Yennai Arindhaal being named after the song "Unnai Arindhaal" from Vettaikkaran (1964).[94] Menon dislikes watching dubbed versions of his own films, and claims his scripts have "a universal theme", citing this as the reason he chose to film Vinnaithaandi Varuvaaya in Telugu as Ye Maaya Chesave, rather than dub.[95] Menon also makes cameo appearances in the films he directs.[96] Menon's films notably feature voiceovers, either from the view of the protagonist or the antagonist.[97]
Personal life
[edit]
Gautham married Preethi Menon and they have 3 sons Arya, Dhruva and Adhya.[citation needed] Costume designer Uthara Menon is his sister, and she has worked on his films following Yennai Arindhaal (2015).[98]
Filmography
[edit]
As a director
[edit]
Note: He was credited as Gautham from 2001 to 2008.
List of films directed by Gautham Menon Year Title Language Producer Acting role Other 2001 Minnale Tamil No Flower man 2001 Rehnaa Hai Terre Dil Mein Hindi No Maddy's boss 2003 Kaakha Kaakha Tamil No Police officer Vasudevan Nair Dubbing artist of Jeevan 2004 Gharshana Telugu No No Dubbing artist of Salim Baig 2006 Vettaiyaadu Vilaiyaadu Tamil No Dancer in song "Manjal Veyil" Dubbing artist of Salim Baig 2007 Pachaikili Muthucharam No Bus passenger 2008 Vaaranam Aayiram No Informer 2010 Vinnaithaandi Varuvaayaa No Himself Ye Maaya Chesave Telugu No Actor 2011 Nadunisi Naaygal Tamil Yes No 2012 Ekk Deewana Tha Hindi Yes Actor Neethaane En Ponvasantham Tamil Yes No Yeto Vellipoyindhi Manasu Telugu Yes No 2015 Yennai Arindhaal Tamil No Police officer 2016 Achcham Yenbadhu Madamaiyada Yes Police officer Sahasam Swasaga Sagipo Telugu Yes Police officer 2019 Enai Noki Paayum Thota Tamil Yes No 2020 Putham Pudhu Kaalai No No Anthology film; segment: Avarum Naanum – Avalum Naanum 2021 Kutty Story No Aadhi Anthology film; segment: Ethirpara Mutham[99][100] 2022 Vendhu Thanindhathu Kaadu Part 1: The Kindling No No Also writer 2024 Joshua Imai Pol Kaakha No No TBA Dhruva Natchathiram: Chapter One – Yuddha Kaandam † Yes TBA
Television
[edit]
Year Title Director Writer Acting role Other 2019 Queen No No Sridhar Co-directed with Prasath Murugesan[101][102] 2020 Paava Kadhaigal No Yes Sathya Anthology series; segment: Vaanmagal 2021 Navarasa Yes No No Anthology series; segment: Guitar Kambi Mele Nindru[103]
Other roles
[edit]
List of films contributed to by Gautham Menon Year Title Language Producer As an actor Other Ref. 1997 Minsara Kanavu Tamil No Man in crowd Uncredited role 2010 Ratha Sarithiram Tamil No No Narrator 2011 Kandaen Tamil No No Audio rights owner [104] Veppam Tamil Yes No Dubbing artist for Muthukumar 2013 Kanna Laddu Thinna Aasaiya Tamil No Himself Thanga Meenkal Tamil Yes No 2014 Endrendrum Tamil No No Narrator 2015 Kaaval Tamil No No Narrator [105] Yatchan Tamil No Himself Courier Boy Kalyan Telugu Yes No Oru Naal Iravil Tamil No Himself Uppu Karuvaadu Tamil No No Playback singer for "Pudhu Oru Kadhavu" 2016 Zero Tamil No No Narrator Tamilselvanum Thaniyar Anjalum Tamil Yes No Meendum Oru Kadhal Kadhai Tamil No No Narrator 2017 Power Paandi Tamil No Himself 2018 Thaanaa Serndha Koottam Tamil No No Dubbing artist for Suresh Menon Naam Malayalam No Himself Goli Soda 2 Tamil No Raghavan Ee Nagaraniki Emaindhi Telugu No Himself Cameo appearance 2019 Puppy Tamil No No Playback singer for "Uyire Vaa" 2020 Oh My Kadavule Tamil No Himself Trance Malayalam No Solomon Kannum Kannum Kollaiyadithaal Tamil No DCP Prathap Chakravarthi Lock Up Tamil No No Narrator Theeviram Tamil No DCP Vikraman 2021 99 Songs Tamil No No Dialogue writer for the Tamil Version Kodiyil Oruvan Tamil No No Playback singer for song "Slum Anthem" Rudra Thandavam Tamil No Vathapirajan 3:33 Tamil No Paranormal investigator 2022 FIR Tamil No Ajay Dewan Selfie Tamil No Ravi Varma Don Tamil No Himself Sita Ramam Telugu No Major Selvan [106] 2023 Michael Telugu No Gurunath and Jai (dual role) Lovefully Yours Veda Malayalam No Sri Kumar Kartha [107] Pathu Thala Tamil No Naanjilaar Gunasekaran Viduthalai Part 1 Tamil No Sunil Menon Anuragam Malayalam No Shankar [108] Takkar Tamil No Recited part of the song "Nira" Ustaad Telugu No Joseph D’Souza Karumegangal Kalaigindrana Tamil No Mathagam Tamil No Thirumaaram IAS Disney+ Hotstar series Leo Tamil No Joshy Andrews Sesham Mike-il Fathima Malayalam No Shiva Narayanan Cameo appearance [109] 2024 Rathnam Tamil No Anbazhagan [110] Hit List Tamil No [111] NBK109 † Telugu No TBA Filming [112] Viduthalai 2 † Tamil No Sunil Menon Filming Varaaham Malayalam No Post-production [113] TBA Bazooka † Malayalam No Benjamin Joshua Post-production [114]
Music videos
[edit]
Year Title Role Actors Ref. 2010 Semmozhiyaana Thamizh Mozhiyaam Music video director Various actors 2018 Bodhai Kodhai Music video director Atharvaa, Aishwarya Rajesh [115] 2021 Allipoola Vennela Music video director Mekha Rajan, Anagha, Angelina [116] 2023 Muththa Pichchai Composer, director Santhosh, Urmi
Short films
[edit]
Year Title Director Producer Acting role Other 2020 Karthik Dial Seytha Yenn Yes Yes No Short film
Awards
[edit]
Awards
Honour by Government of Tamil Nadu - Kalaimamani (2021)
Vijay Award for Favourite Director for Vaaranam Aayiram (2008)
National Film Awards
National Film Award for Best Feature Film in Tamil for Vaaranam Aayiram (2008)
National Film Award for Best Feature Film in Tamil for Thanga Meengal (2013)
Filmfare Awards
Filmfare Award for Best Film – Tamil for Thanga Meengal (2013)
Nandi Awards
Nandi Award for Best Screenplay Writer for Ye Maaya Chesave (2010)
Nandi Special Jury Award for Ye Maaya Chesave (2010)[117]
Other Awards
Vijay Award for Best Film for Thanga Meengal (2013)
Rotary Club of Chennai – Honored for Creative Excellence (2010)
See also
[edit]
Gautham Vasudev Menon's unrealized projects
References
[edit]
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