fact stringlengths 9 24k | type stringclasses 2
values | library stringclasses 5
values | imports listlengths 0 0 | filename stringclasses 5
values | symbolic_name stringlengths 1 24 | docstring stringlengths 12 292k ⌀ |
|---|---|---|---|---|---|---|
jarri : |- ( ps -> ch ) | theorem | set | [] | set.mm | jarri | Inference associated with ~ jarr . Partial converse of ~ ja (the other partial converse being ~ jarli ). (Contributed by Wolf Lammen, 20-Sep-2013.) |
pm2.86d : |- ( ph -> ( ps -> ( ch -> th ) ) ) | theorem | set | [] | set.mm | pm2.86d | Deduction associated with ~ pm2.86 . (Contributed by NM, 29-Jun-1995.) (Proof shortened by Wolf Lammen, 3-Apr-2013.) |
pm2.86 : |- ( ( ( ph -> ps ) -> ( ph -> ch ) ) -> ( ph -> ( ps -> ch ) ) ) | theorem | set | [] | set.mm | pm2.86 | Converse of Axiom ~ ax-2 . Theorem *2.86 of [WhiteheadRussell] p. 108. (Contributed by NM, 25-Apr-1994.) (Proof shortened by Wolf Lammen, 3-Apr-2013.) |
pm2.86i : |- ( ph -> ( ps -> ch ) ) | theorem | set | [] | set.mm | pm2.86i | Inference associated with ~ pm2.86 . (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 3-Apr-2013.) |
loolin : |- ( ( ( ph -> ps ) -> ( ps -> ph ) ) -> ( ps -> ph ) ) | theorem | set | [] | set.mm | loolin | The Linearity Axiom of the infinite-valued sentential logic (L-infinity) of Lukasiewicz. See ~ loowoz for an alternate axiom. (Contributed by Mel L. O'Cat, 12-Aug-2004.) |
loowoz : |- ( ( ( ph -> ps ) -> ( ph -> ch ) ) -> ( ( ps -> ph ) -> ( ps -> ch ) ) ) | theorem | set | [] | set.mm | loowoz | An alternate for the Linearity Axiom of the infinite-valued sentential logic (L-infinity) of Lukasiewicz ~ loolin , due to Barbara Wozniakowska, _Reports on Mathematical Logic_ 10, 129-137 (1978). (Contributed by Mel L. O'Cat, 8-Aug-2004.) |
con4 : |- ( ( -. ph -> -. ps ) -> ( ps -> ph ) ) | theorem | set | [] | set.mm | con4 | =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Logical negation =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= This section makes our first use of the third axiom of propositional calculus, ~ ax-3 . It introduces logical negation. $) $( Alias for ~ ax-3 ... |
con4i : |- ( ps -> ph ) | theorem | set | [] | set.mm | con4i | null |
con4d : |- ( ph -> ( ch -> ps ) ) | theorem | set | [] | set.mm | con4d | $j usage 'con4i' avoids 'ax-1' 'ax-2'; $) $} ${ con4d.1 $e |- ( ph -> ( -. ps -> -. ch ) ) $. $( Deduction associated with ~ con4 . (Contributed by NM, 26-Mar-1995.) |
mt4 : |- ps | theorem | set | [] | set.mm | mt4 | The rule of modus tollens. Inference associated with ~ con4i . (Contributed by Wolf Lammen, 12-May-2013.) |
mt4d : |- ( ph -> ch ) | theorem | set | [] | set.mm | mt4d | Modus tollens deduction. Deduction form of ~ mt4 . (Contributed by NM, 9-Jun-2006.) |
mt4i : |- ( ph -> ps ) | theorem | set | [] | set.mm | mt4i | Modus tollens inference. (Contributed by Wolf Lammen, 12-May-2013.) |
pm2.21i : |- ( ph -> ps ) | theorem | set | [] | set.mm | pm2.21i | A contradiction implies anything. Inference associated with ~ pm2.21 . Its associated inference is ~ pm2.24ii . (Contributed by NM, 16-Sep-1993.) |
pm2.24ii : |- ps | theorem | set | [] | set.mm | pm2.24ii | A contradiction implies anything. Inference associated with ~ pm2.21i and ~ pm2.24i . (Contributed by NM, 27-Feb-2008.) |
pm2.21d : |- ( ph -> ( ps -> ch ) ) | theorem | set | [] | set.mm | pm2.21d | $j usage 'pm2.24ii' avoids 'ax-2'; $) $} ${ pm2.21d.1 $e |- ( ph -> -. ps ) $. $( A contradiction implies anything. Deduction associated with ~ pm2.21 . (Contributed by NM, 10-Feb-1996.) |
pm2.21ddALT : |- ( ph -> ch ) | theorem | set | [] | set.mm | pm2.21ddALT | Alternate proof of ~ pm2.21dd . (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
pm2.21 : |- ( -. ph -> ( ph -> ps ) ) | theorem | set | [] | set.mm | pm2.21 | null |
pm2.24 : |- ( ph -> ( -. ph -> ps ) ) | theorem | set | [] | set.mm | pm2.24 | Theorem *2.24 of [WhiteheadRussell] p. 104. Its associated inference is ~ pm2.24i . Commuted form of ~ pm2.21 . (Contributed by NM, 3-Jan-2005.) |
jarl : |- ( ( ( ph -> ps ) -> ch ) -> ( -. ph -> ch ) ) | theorem | set | [] | set.mm | jarl | Elimination of a nested antecedent. (Contributed by Wolf Lammen, 10-May-2013.) |
jarli : |- ( -. ph -> ch ) | theorem | set | [] | set.mm | jarli | Inference associated with ~ jarl . Partial converse of ~ ja (the other partial converse being ~ jarri ). (Contributed by Wolf Lammen, 4-Oct-2013.) |
pm2.18d : |- ( ph -> ps ) | theorem | set | [] | set.mm | pm2.18d | null |
pm2.18 : |- ( ( -. ph -> ph ) -> ph ) | theorem | set | [] | set.mm | pm2.18 | Clavius law, or "consequentia mirabilis" ("admirable consequence"). If a formula is implied by its negation, then it is true. Can be used in proofs by contradiction. Theorem *2.18 of [WhiteheadRussell] p. 103. See also the weak Clavius law ~ pm2.01 . (Contributed by NM, 29-Dec-1992.) (Proof shortened by Wolf Lammen, 17... |
pm2.18i : |- ph | theorem | set | [] | set.mm | pm2.18i | Inference associated with the Clavius law ~ pm2.18 . (Contributed by BJ, 30-Mar-2020.) |
notnotr : |- ( -. -. ph -> ph ) | theorem | set | [] | set.mm | notnotr | null |
notnotri : |- ph | theorem | set | [] | set.mm | notnotri | Inference associated with ~ notnotr . For a shorter proof using ~ ax-2 , see ~ notnotriALT . (Contributed by NM, 27-Feb-2008.) (Proof shortened by Wolf Lammen, 15-Jul-2021.) Remove dependency on ~ ax-2 . (Revised by Steven Nguyen, 27-Dec-2022.) |
notnotriALT : |- ph | theorem | set | [] | set.mm | notnotriALT | $j usage 'notnotri' avoids 'ax-2'; $) $( Alternate proof of ~ notnotri . The proof via ~ notnotr and ~ ax-mp also has three essential steps, but has a total number of steps equal to 8, instead of the present 7, because it has to construct the formula ` ph ` twice and the formula ` -. -. ph ` once, whereas the present p... |
notnotrd : |- ( ph -> ps ) | theorem | set | [] | set.mm | notnotrd | Deduction associated with ~ notnotr and ~ notnotri . Double negation elimination rule. A translation of the natural deduction rule ` -. -. ` C , ` _G |- -. -. ps => _G |- ps ` ; see ~ natded . This is Definition NNC in [Pfenning] p. 17. This rule is valid in classical logic (our logic), but not in intuitionistic logic.... |
con2d : |- ( ph -> ( ch -> -. ps ) ) | theorem | set | [] | set.mm | con2d | A contraposition deduction. (Contributed by NM, 19-Aug-1993.) |
con2 : |- ( ( ph -> -. ps ) -> ( ps -> -. ph ) ) | theorem | set | [] | set.mm | con2 | Contraposition. Theorem *2.03 of [WhiteheadRussell] p. 100. (Contributed by NM, 29-Dec-1992.) (Proof shortened by Wolf Lammen, 12-Feb-2013.) |
mt2d : |- ( ph -> -. ps ) | theorem | set | [] | set.mm | mt2d | Modus tollens deduction. (Contributed by NM, 4-Jul-1994.) |
mt2i : |- ( ph -> -. ps ) | theorem | set | [] | set.mm | mt2i | Modus tollens inference. (Contributed by NM, 26-Mar-1995.) (Proof shortened by Wolf Lammen, 15-Sep-2012.) |
nsyl3 : |- ( ch -> -. ph ) | theorem | set | [] | set.mm | nsyl3 | A negated syllogism inference. (Contributed by NM, 1-Dec-1995.) |
con2i : |- ( ps -> -. ph ) | theorem | set | [] | set.mm | con2i | A contraposition inference. Its associated inference is ~ mt2 . (Contributed by NM, 10-Jan-1993.) (Proof shortened by Mel L. O'Cat, 28-Nov-2008.) (Proof shortened by Wolf Lammen, 13-Jun-2013.) |
nsyl : |- ( ph -> -. ch ) | theorem | set | [] | set.mm | nsyl | A negated syllogism inference. (Contributed by NM, 31-Dec-1993.) (Proof shortened by Wolf Lammen, 2-Mar-2013.) |
nsyl2 : |- ( ph -> ch ) | theorem | set | [] | set.mm | nsyl2 | A negated syllogism inference. (Contributed by NM, 26-Jun-1994.) (Proof shortened by Wolf Lammen, 14-Nov-2023.) |
notnot : |- ( ph -> -. -. ph ) | theorem | set | [] | set.mm | notnot | Double negation introduction. Converse of ~ notnotr and one implication of ~ notnotb . Theorem *2.12 of [WhiteheadRussell] p. 101. This was the sixth axiom of Frege, specifically Proposition 41 of [Frege1879] p. 47. (Contributed by NM, 28-Dec-1992.) (Proof shortened by Wolf Lammen, 2-Mar-2013.) |
notnoti : |- -. -. ph | theorem | set | [] | set.mm | notnoti | Inference associated with ~ notnot . (Contributed by NM, 27-Feb-2008.) |
notnotd : |- ( ph -> -. -. ps ) | theorem | set | [] | set.mm | notnotd | Deduction associated with ~ notnot and ~ notnoti . (Contributed by Jarvin Udandy, 2-Sep-2016.) Avoid biconditional. (Revised by Wolf Lammen, 27-Mar-2021.) |
con1d : |- ( ph -> ( -. ch -> ps ) ) | theorem | set | [] | set.mm | con1d | A contraposition deduction. (Contributed by NM, 27-Dec-1992.) |
con1 : |- ( ( -. ph -> ps ) -> ( -. ps -> ph ) ) | theorem | set | [] | set.mm | con1 | Contraposition. Theorem *2.15 of [WhiteheadRussell] p. 102. Its associated inference is ~ con1i . (Contributed by NM, 29-Dec-1992.) (Proof shortened by Wolf Lammen, 12-Feb-2013.) |
con1i : |- ( -. ps -> ph ) | theorem | set | [] | set.mm | con1i | null |
mt3d : |- ( ph -> ps ) | theorem | set | [] | set.mm | mt3d | Modus tollens deduction. (Contributed by NM, 26-Mar-1995.) |
mt3i : |- ( ph -> ps ) | theorem | set | [] | set.mm | mt3i | Modus tollens inference. (Contributed by NM, 26-Mar-1995.) (Proof shortened by Wolf Lammen, 15-Sep-2012.) |
pm2.24i : |- ( -. ph -> ps ) | theorem | set | [] | set.mm | pm2.24i | null |
pm2.24d : |- ( ph -> ( -. ps -> ch ) ) | theorem | set | [] | set.mm | pm2.24d | Deduction form of ~ pm2.24 . (Contributed by NM, 30-Jan-2006.) |
con3d : |- ( ph -> ( -. ch -> -. ps ) ) | theorem | set | [] | set.mm | con3d | A contraposition deduction. Deduction form of ~ con3 . (Contributed by NM, 10-Jan-1993.) |
con3 : |- ( ( ph -> ps ) -> ( -. ps -> -. ph ) ) | theorem | set | [] | set.mm | con3 | Contraposition. Theorem *2.16 of [WhiteheadRussell] p. 103. This was the fourth axiom of Frege, specifically Proposition 28 of [Frege1879] p. 43. Its associated inference is ~ con3i . (Contributed by NM, 29-Dec-1992.) (Proof shortened by Wolf Lammen, 13-Feb-2013.) |
con3i : |- ( -. ps -> -. ph ) | theorem | set | [] | set.mm | con3i | A contraposition inference. Inference associated with ~ con3 . Its associated inference is ~ mto . (Contributed by NM, 3-Jan-1993.) (Proof shortened by Wolf Lammen, 20-Jun-2013.) |
con3rr3 : |- ( -. ch -> ( ph -> -. ps ) ) | theorem | set | [] | set.mm | con3rr3 | Rotate through consequent right. (Contributed by Wolf Lammen, 3-Nov-2013.) |
nsyld : |- ( ph -> ( ps -> -. ta ) ) | theorem | set | [] | set.mm | nsyld | A negated syllogism deduction. (Contributed by NM, 9-Apr-2005.) |
nsyli : |- ( ph -> ( th -> -. ps ) ) | theorem | set | [] | set.mm | nsyli | A negated syllogism inference. (Contributed by NM, 3-May-1994.) |
nsyl4 : |- ( -. ch -> ps ) | theorem | set | [] | set.mm | nsyl4 | A negated syllogism inference. (Contributed by NM, 15-Feb-1996.) |
nsyl5 : |- ( -. ps -> ch ) | theorem | set | [] | set.mm | nsyl5 | A negated syllogism inference. (Contributed by Wolf Lammen, 20-May-2024.) |
pm3.2im : |- ( ph -> ( ps -> -. ( ph -> -. ps ) ) ) | theorem | set | [] | set.mm | pm3.2im | Theorem *3.2 of [WhiteheadRussell] p. 111, expressed with primitive connectives (see ~ pm3.2 ). (Contributed by NM, 29-Dec-1992.) (Proof shortened by Josh Purinton, 29-Dec-2000.) |
jc : |- ( ph -> -. ( ps -> -. ch ) ) | theorem | set | [] | set.mm | jc | Deduction joining the consequents of two premises. A deduction associated with ~ pm3.2im . (Contributed by NM, 28-Dec-1992.) |
jcn : |- ( ph -> ( -. ps -> -. ( ph -> ps ) ) ) | theorem | set | [] | set.mm | jcn | Theorem joining the consequents of two premises. Theorem 8 of [Margaris] p. 60. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Josh Purinton, 29-Dec-2000.) |
jcnd : |- ( ph -> -. ( ps -> ch ) ) | theorem | set | [] | set.mm | jcnd | Deduction joining the consequents of two premises. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Proof shortened by Wolf Lammen, 10-Apr-2024.) |
impi : |- ( -. ( ph -> -. ps ) -> ch ) | theorem | set | [] | set.mm | impi | An importation inference. (Contributed by NM, 29-Dec-1992.) (Proof shortened by Wolf Lammen, 20-Jul-2013.) |
expi : |- ( ph -> ( ps -> ch ) ) | theorem | set | [] | set.mm | expi | An exportation inference. (Contributed by NM, 29-Dec-1992.) (Proof shortened by Mel L. O'Cat, 28-Nov-2008.) |
simprim : |- ( -. ( ph -> -. ps ) -> ps ) | theorem | set | [] | set.mm | simprim | Simplification. Similar to Theorem *3.27 (Simp) of [WhiteheadRussell] p. 112. (Contributed by NM, 3-Jan-1993.) (Proof shortened by Wolf Lammen, 13-Nov-2012.) |
simplim : |- ( -. ( ph -> ps ) -> ph ) | theorem | set | [] | set.mm | simplim | Simplification. Similar to Theorem *3.26 (Simp) of [WhiteheadRussell] p. 112. (Contributed by NM, 3-Jan-1993.) (Proof shortened by Wolf Lammen, 21-Jul-2012.) |
pm2.5g : |- ( -. ( ph -> ps ) -> ( -. ph -> ch ) ) | theorem | set | [] | set.mm | pm2.5g | General instance of Theorem *2.5 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 9-Oct-2012.) |
pm2.5 : |- ( -. ( ph -> ps ) -> ( -. ph -> ps ) ) | theorem | set | [] | set.mm | pm2.5 | Theorem *2.5 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) |
conax1 : |- ( -. ( ph -> ps ) -> -. ps ) | theorem | set | [] | set.mm | conax1 | Contrapositive of ~ ax-1 . (Contributed by BJ, 28-Oct-2023.) |
conax1k : |- ( -. ( ph -> ps ) -> ( ch -> -. ps ) ) | theorem | set | [] | set.mm | conax1k | Weakening of ~ conax1 . General instance of ~ pm2.51 and of ~ pm2.52 . (Contributed by BJ, 28-Oct-2023.) |
pm2.51 : |- ( -. ( ph -> ps ) -> ( ph -> -. ps ) ) | theorem | set | [] | set.mm | pm2.51 | Theorem *2.51 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) |
pm2.52 : |- ( -. ( ph -> ps ) -> ( -. ph -> -. ps ) ) | theorem | set | [] | set.mm | pm2.52 | Theorem *2.52 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 8-Oct-2012.) |
pm2.521g : |- ( -. ( ph -> ps ) -> ( ps -> ch ) ) | theorem | set | [] | set.mm | pm2.521g | A general instance of Theorem *2.521 of [WhiteheadRussell] p. 107. (Contributed by BJ, 28-Oct-2023.) |
pm2.521g2 : |- ( -. ( ph -> ps ) -> ( ch -> ph ) ) | theorem | set | [] | set.mm | pm2.521g2 | A general instance of Theorem *2.521 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 8-Oct-2012.) |
pm2.521 : |- ( -. ( ph -> ps ) -> ( ps -> ph ) ) | theorem | set | [] | set.mm | pm2.521 | Theorem *2.521 of [WhiteheadRussell] p. 107. Instance of ~ pm2.521g and of ~ pm2.521g2 . (Contributed by NM, 3-Jan-2005.) |
expt : |- ( ( -. ( ph -> -. ps ) -> ch ) -> ( ph -> ( ps -> ch ) ) ) | theorem | set | [] | set.mm | expt | Exportation theorem ~ pm3.3 (closed form of ~ ex ) expressed with primitive connectives. (Contributed by NM, 28-Dec-1992.) |
impt : |- ( ( ph -> ( ps -> ch ) ) -> ( -. ( ph -> -. ps ) -> ch ) ) | theorem | set | [] | set.mm | impt | Importation theorem ~ pm3.1 (closed form of ~ imp ) expressed with primitive connectives. (Contributed by NM, 25-Apr-1994.) (Proof shortened by Wolf Lammen, 20-Jul-2013.) |
pm2.61d : |- ( ph -> ch ) | theorem | set | [] | set.mm | pm2.61d | Deduction eliminating an antecedent. (Contributed by NM, 27-Apr-1994.) (Proof shortened by Wolf Lammen, 12-Sep-2013.) |
pm2.61d1 : |- ( ph -> ch ) | theorem | set | [] | set.mm | pm2.61d1 | Inference eliminating an antecedent. (Contributed by NM, 15-Jul-2005.) |
pm2.61d2 : |- ( ph -> ch ) | theorem | set | [] | set.mm | pm2.61d2 | Inference eliminating an antecedent. (Contributed by NM, 18-Aug-1993.) |
pm2.61i : |- ps | theorem | set | [] | set.mm | pm2.61i | Inference eliminating an antecedent. (Contributed by NM, 5-Apr-1994.) (Proof shortened by Wolf Lammen, 19-Nov-2023.) |
pm2.61ii : |- ch | theorem | set | [] | set.mm | pm2.61ii | Inference eliminating two antecedents. (Contributed by NM, 4-Jan-1993.) (Proof shortened by Josh Purinton, 29-Dec-2000.) |
pm2.61nii : |- ch | theorem | set | [] | set.mm | pm2.61nii | Inference eliminating two antecedents. (Contributed by NM, 13-Jul-2005.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 13-Nov-2012.) |
pm2.61iii : |- th | theorem | set | [] | set.mm | pm2.61iii | Inference eliminating three antecedents. (Contributed by NM, 2-Jan-2002.) (Proof shortened by Wolf Lammen, 22-Sep-2013.) |
ja : |- ( ( ph -> ps ) -> ch ) | theorem | set | [] | set.mm | ja | null |
jad : |- ( ph -> ( ( ps -> ch ) -> th ) ) | theorem | set | [] | set.mm | jad | Deduction form of ~ ja . (Contributed by Scott Fenton, 13-Dec-2010.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) |
pm2.01 : |- ( ( ph -> -. ph ) -> -. ph ) | theorem | set | [] | set.mm | pm2.01 | Weak Clavius law. If a formula implies its negation, then it is false. A form of "reductio ad absurdum", which can be used in proofs by contradiction. Theorem *2.01 of [WhiteheadRussell] p. 100. Provable in minimal calculus, contrary to the Clavius law ~ pm2.18 . (Contributed by NM, 18-Aug-1993.) (Proof shortened by Me... |
pm2.01i : |- -. ph | theorem | set | [] | set.mm | pm2.01i | Inference associated with the weak Clavius law ~ pm2.01 . (Contributed by BJ, 30-Mar-2020.) |
pm2.01d : |- ( ph -> -. ps ) | theorem | set | [] | set.mm | pm2.01d | Deduction based on reductio ad absurdum. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Wolf Lammen, 5-Mar-2013.) |
pm2.6 : |- ( ( -. ph -> ps ) -> ( ( ph -> ps ) -> ps ) ) | theorem | set | [] | set.mm | pm2.6 | Theorem *2.6 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) |
pm2.61 : |- ( ( ph -> ps ) -> ( ( -. ph -> ps ) -> ps ) ) | theorem | set | [] | set.mm | pm2.61 | Theorem *2.61 of [WhiteheadRussell] p. 107. Useful for eliminating an antecedent. (Contributed by NM, 4-Jan-1993.) (Proof shortened by Wolf Lammen, 22-Sep-2013.) |
pm2.65 : |- ( ( ph -> ps ) -> ( ( ph -> -. ps ) -> -. ph ) ) | theorem | set | [] | set.mm | pm2.65 | Theorem *2.65 of [WhiteheadRussell] p. 107. Proof by contradiction. (Contributed by NM, 21-Jun-1993.) (Proof shortened by Wolf Lammen, 8-Mar-2013.) |
pm2.65i : |- -. ph | theorem | set | [] | set.mm | pm2.65i | Inference for proof by contradiction. (Contributed by NM, 18-May-1994.) (Proof shortened by Wolf Lammen, 11-Sep-2013.) |
pm2.21dd : |- ( ph -> ch ) | theorem | set | [] | set.mm | pm2.21dd | A contradiction implies anything. Deduction from ~ pm2.21 . (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof shortened by Wolf Lammen, 22-Jul-2019.) |
pm2.65d : |- ( ph -> -. ps ) | theorem | set | [] | set.mm | pm2.65d | Deduction for proof by contradiction. (Contributed by NM, 26-Jun-1994.) (Proof shortened by Wolf Lammen, 26-May-2013.) |
mto : |- -. ph | theorem | set | [] | set.mm | mto | The rule of modus tollens. The rule says, "if ` ps ` is not true, and ` ph ` implies ` ps ` , then ` ph ` must also be not true". Modus tollens is short for "modus tollendo tollens", a Latin phrase that means "the mode that by denying denies" - remark in [Sanford] p. 39. It is also called denying the consequent. Modus ... |
mtod : |- ( ph -> -. ps ) | theorem | set | [] | set.mm | mtod | Modus tollens deduction. (Contributed by NM, 3-Apr-1994.) (Proof shortened by Wolf Lammen, 11-Sep-2013.) |
mtoi : |- ( ph -> -. ps ) | theorem | set | [] | set.mm | mtoi | Modus tollens inference. (Contributed by NM, 5-Jul-1994.) (Proof shortened by Wolf Lammen, 15-Sep-2012.) |
mt2 : |- -. ph | theorem | set | [] | set.mm | mt2 | A rule similar to modus tollens. Inference associated with ~ con2i . (Contributed by NM, 19-Aug-1993.) (Proof shortened by Wolf Lammen, 10-Sep-2013.) |
mt3 : |- ph | theorem | set | [] | set.mm | mt3 | A rule similar to modus tollens. Inference associated with ~ con1i . (Contributed by NM, 18-May-1994.) (Proof shortened by Wolf Lammen, 11-Sep-2013.) |
peirce : |- ( ( ( ph -> ps ) -> ph ) -> ph ) | theorem | set | [] | set.mm | peirce | null |
looinv : |- ( ( ( ph -> ps ) -> ps ) -> ( ( ps -> ph ) -> ph ) ) | theorem | set | [] | set.mm | looinv | The Inversion Axiom of the infinite-valued sentential logic (L-infinity) of Lukasiewicz. Using ~ dfor2 , we can see that this essentially expresses "disjunction commutes". Theorem *2.69 of [WhiteheadRussell] p. 108. It is a special instance of the axiom "Roll", see ~ peirceroll . (Contributed by NM, 12-Aug-2004.) |
bijust0 : |- -. ( ( ph -> ph ) -> -. ( ph -> ph ) ) | theorem | set | [] | set.mm | bijust0 | A self-implication (see ~ id ) does not imply its own negation. The justification theorem ~ bijust is one of its instances. (Contributed by NM, 11-May-1999.) (Proof shortened by Josh Purinton, 29-Dec-2000.) Extract ~ bijust0 from proof of ~ bijust . (Revised by BJ, 19-Mar-2020.) |
bijust : |- -. ( ( -. ( ( ph -> ps ) -> -. ( ps -> ph ) ) -> -. ( ( ph -> ps ) -> -. ( ps -> ph ) ) ) -> -. ( -. ( ( ph -> ps ) -> -. ( ps -> ph ) ) -> -. ( ( ph -> ps ) -> -. ( ps -> ph ) ) ) ) | theorem | set | [] | set.mm | bijust | null |
impbi : |- ( ( ph -> ps ) -> ( ( ps -> ph ) -> ( ph <-> ps ) ) ) | theorem | set | [] | set.mm | impbi | Define the biconditional (logical "iff" or "if and only if"), also called biimplication. Definition ~ df-bi in this section is our first definition, which introduces and defines the biconditional connective ` <-> ` . We define a wff of the form ` ( ph <-> ps ) ` as an abbreviation for ` -. ( ( ph -> ps ) -> -. ( ps -> ... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.