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Russell paradox Summary Russel's_paradox At the end of the 1890s, Georg Cantor – considered the founder of modern set theory – had already realized that his theory would lead to a contradiction, as he told Hilbert and Richard Dedekind by letter.According to the unrestricted comprehension principle, for any sufficiently... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Russell paradox Summary Russel's_paradox If R is not a member of itself, then its definition entails that it is a member of itself; yet, if it is a member of itself, then it is not a member of itself, since it is the set of all sets that are not members of themselves. The resulting contradiction is Russell's paradox. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Russell paradox Summary Russel's_paradox In symbols: Let R = { x ∣ x ∉ x } , then R ∈ R ⟺ R ∉ R {\displaystyle {\text{Let }}R=\{x\mid x\not \in x\}{\text{, then }}R\in R\iff R\not \in R} Russell also showed that a version of the paradox could be derived in the axiomatic system constructed by the German philosopher and ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Russell paradox Summary Russel's_paradox With the additional contributions of Abraham Fraenkel, Zermelo set theory developed into the now-standard Zermelo–Fraenkel set theory (commonly known as ZFC when including the axiom of choice). The main difference between Russell's and Zermelo's solution to the paradox is that Z... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Skolem arithmetic Summary Skolem_arithmetic In mathematical logic, Skolem arithmetic is the first-order theory of the natural numbers with multiplication, named in honor of Thoralf Skolem. The signature of Skolem arithmetic contains only the multiplication operation and equality, omitting the addition operation entirel... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Skolem arithmetic Summary Skolem_arithmetic Unlike Peano arithmetic, Skolem arithmetic is a decidable theory. This means it is possible to effectively determine, for any sentence in the language of Skolem arithmetic, whether that sentence is provable from the axioms of Skolem arithmetic. The asymptotic running-time com... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Type in type Summary System_U In mathematical logic, System U and System U− are pure type systems, i.e. special forms of a typed lambda calculus with an arbitrary number of sorts, axioms and rules (or dependencies between the sorts). They were both proved inconsistent by Jean-Yves Girard in 1972. This result led to the... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Tarski's high school algebra problem Summary Tarski's_high_school_algebra_problem In mathematical logic, Tarski's high school algebra problem was a question posed by Alfred Tarski. It asks whether there are identities involving addition, multiplication, and exponentiation over the positive integers that cannot be prove... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Boolean-valued model Summary Boolean-valued_model In mathematical logic, a Boolean-valued model is a generalization of the ordinary Tarskian notion of structure from model theory. In a Boolean-valued model, the truth values of propositions are not limited to "true" and "false", but instead take values in some fixed com... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Gödel encoding Summary Gödel_encoding In mathematical logic, a Gödel numbering is a function that assigns to each symbol and well-formed formula of some formal language a unique natural number, called its Gödel number. The concept was developed by Kurt Gödel for the proof of his incompleteness theorems. (Gödel 1931) A ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Herbrand interpretation Summary Herbrand_interpretation In mathematical logic, a Herbrand interpretation is an interpretation in which all constants and function symbols are assigned very simple meanings. Specifically, every constant is interpreted as itself, and every function symbol is interpreted as the function tha... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Herbrand interpretation Summary Herbrand_interpretation The importance of Herbrand interpretations is that, if any interpretation satisfies a given set of clauses S then there is a Herbrand interpretation that satisfies them. Moreover, Herbrand's theorem states that if S is unsatisfiable then there is a finite unsatisf... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hintikka set Summary Hintikka_set In mathematical logic, a Hintikka set is a set of logical formulas whose elements satisfy the following properties: An atom or its conjugate can appear in the set but not both, If a formula in the set has a main operator that is of "conjuctive-type", then its two operands appear in the... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Lindström quantifier Summary Lindström_quantifier In mathematical logic, a Lindström quantifier is a generalized polyadic quantifier. Lindström quantifiers generalize first-order quantifiers, such as the existential quantifier, the universal quantifier, and the counting quantifiers. They were introduced by Per Lindströ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Conservative extension Summary Non-conservative_extension In mathematical logic, a conservative extension is a supertheory of a theory which is often convenient for proving theorems, but proves no new theorems about the language of the original theory. Similarly, a non-conservative extension is a supertheory which is n... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Conservative extension Summary Non-conservative_extension Note that a conservative extension of a consistent theory is consistent. If it were not, then by the principle of explosion, every formula in the language of T 2 {\displaystyle T_{2}} would be a theorem of T 2 {\displaystyle T_{2}} , so every formula in the lang... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Conservative extension Summary Non-conservative_extension This can also be seen as a methodology for writing and structuring large theories: start with a theory, T 0 {\displaystyle T_{0}} , that is known (or assumed) to be consistent, and successively build conservative extensions T 1 {\displaystyle T_{1}} , T 2 {\disp... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Deduction theorem Summary Deduction_theorem In mathematical logic, a deduction theorem is a metatheorem that justifies doing conditional proofs from a hypothesis in systems that do not explicitly axiomatize that hypothesis, i.e. to prove an implication A → B, it is sufficient to assume A as an hypothesis and then proce... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Deduction theorem Summary Deduction_theorem In the special case where Δ {\displaystyle \Delta } is the empty set, the deduction theorem claim can be more compactly written as: A ⊢ B {\displaystyle A\vdash B} implies ⊢ A → B {\displaystyle \vdash A\to B} . The deduction theorem for predicate logic is similar, but comes ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Deduction theorem Summary Deduction_theorem The deduction theorem holds for all first-order theories with the usual deductive systems for first-order logic. However, there are first-order systems in which new inference rules are added for which the deduction theorem fails. Most notably, the deduction theorem fails to h... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Definable set Summary Definable_set In mathematical logic, a definable set is an n-ary relation on the domain of a structure whose elements satisfy some formula in the first-order language of that structure. A set can be defined with or without parameters, which are elements of the domain that can be referenced in the ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Gödel logic Summary Gödel_logic In mathematical logic, a first-order Gödel logic is a member of a family of finite- or infinite-valued logics in which the sets of truth values V are closed subsets of the unit interval containing both 0 and 1. Different such sets V in general determine different Gödel logics. The conce... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Decidability of first-order theories of the real numbers Summary Decidability_of_first-order_theories_of_the_real_numbers In mathematical logic, a first-order language of the real numbers is the set of all well-formed sentences of first-order logic that involve universal and existential quantifiers and logical combinat... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Decidability of first-order theories of the real numbers Summary Decidability_of_first-order_theories_of_the_real_numbers The theory of real closed fields is the theory in which the primitive operations are multiplication and addition; this implies that, in this theory, the only numbers that can be defined are the real... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Decidability of first-order theories of the real numbers Summary Decidability_of_first-order_theories_of_the_real_numbers Tarski's exponential function problem concerns the extension of this theory to another primitive operation, the exponential function. It is an open problem whether this theory is decidable, but if S... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Decidability of first-order theories of the real numbers Summary Decidability_of_first-order_theories_of_the_real_numbers Still, one can handle the undecidable case with functions such as sine by using algorithms that do not necessarily terminate always. In particular, one can design algorithms that are only required t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
One-place predicate Summary First-order_predicate In mathematical logic, a first-order predicate is a predicate that takes only individual(s) constants or variables as argument(s). Compare second-order predicate and higher-order predicate. This is not to be confused with a one-place predicate or monad, which is a predi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Formal calculation Summary Formal_calculation In mathematical logic, a formal calculation, or formal operation, is a calculation that is systematic but without a rigorous justification. It involves manipulating symbols in an expression using a generic substitution without proving that the necessary conditions hold. Ess... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Formal language theory Formal theories, systems, and proofs Formal_expression > Applications > Formal theories, systems, and proofs In mathematical logic, a formal theory is a set of sentences expressed in a formal language. A formal system (also called a logical calculus, or a logical system) consists of a formal lang... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Formal language theory Formal theories, systems, and proofs Formal_expression > Applications > Formal theories, systems, and proofs A formal system is used to derive one expression from one or more other expressions. Although a formal language can be identified with its formulas, a formal system cannot be likewise iden... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Formal language theory Formal theories, systems, and proofs Formal_expression > Applications > Formal theories, systems, and proofs A formal proof or derivation is a finite sequence of well-formed formulas (which may be interpreted as sentences, or propositions) each of which is an axiom or follows from the preceding f... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Mathematical theorem Theorems in logic Formal_theorem > Theorems in logic In mathematical logic, a formal theory is a set of sentences within a formal language. A sentence is a well-formed formula with no free variables. A sentence that is a member of a theory is one of its theorems, and the theory is the set of its th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Mathematical theorem Theorems in logic Formal_theorem > Theorems in logic Some accounts define a theory to be closed under the semantic consequence relation ( ⊨ {\displaystyle \models } ), while others define it to be closed under the syntactic consequence, or derivability relation ( ⊢ {\displaystyle \vdash } ). For a ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Mathematical theorem Theorems in logic Formal_theorem > Theorems in logic The closure of the empty set under the relation of logical consequence yields the set that contains just those sentences that are the theorems of the deductive system. In the broad sense in which the term is used within logic, a theorem does not ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Mathematical theorem Theorems in logic Formal_theorem > Theorems in logic The definition of theorems as sentences of a formal language is useful within proof theory, which is a branch of mathematics that studies the structure of formal proofs and the structure of provable formulas. It is also important in model theory,... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Mathematical theorem Theorems in logic Formal_theorem > Theorems in logic What makes formal theorems useful and interesting is that they may be interpreted as true propositions and their derivations may be interpreted as a proof of their truth. A theorem whose interpretation is a true statement about a formal system (a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Formula In mathematical logic Mathematical_formula > In mathematics > In mathematical logic In mathematical logic, a formula (often referred to as a well-formed formula) is an entity constructed using the symbols and formation rules of a given logical language. For example, in first-order logic, ∀ x ∀ y ( P ( f ( x ) )... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Negation normal form Summary Negation_normal_form In mathematical logic, a formula is in negation normal form (NNF) if the negation operator ( ¬ {\displaystyle \lnot } , not) is only applied to variables and the only other allowed Boolean operators are conjunction ( ∧ {\displaystyle \land } , and) and disjunction ( ∨ {... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Negation normal form Summary Negation_normal_form 204): A ⇒ B → ¬ A ∨ B A ⇔ B → ( ¬ A ∨ B ) ∧ ( A ∨ ¬ B ) ¬ ( A ∨ B ) → ¬ A ∧ ¬ B ¬ ( A ∧ B ) → ¬ A ∨ ¬ B ¬ ¬ A → A ¬ ∃ x A → ∀ x ¬ A ¬ ∀ x A → ∃ x ¬ A {\displaystyle {\begin{aligned}A\Rightarrow B&~\to ~\lnot A\lor B\\A\Leftrightarrow B&~\to ~(\lnot A\lor B)\land (A\lor ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Negation normal form Summary Negation_normal_form Repeated application of distributivity may exponentially increase the size of a formula. In the classical propositional logic, transformation to negation normal form does not impact computational properties: the satisfiability problem continues to be NP-complete, and th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Shoenfield absoluteness theorem Summary Absoluteness_(logic) In mathematical logic, a formula is said to be absolute to some class of structures (also called models), if it has the same truth value in each of the members of that class. One can also speak of absoluteness of a formula between two structures, if it is abs... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Shoenfield absoluteness theorem Summary Absoluteness_(logic) If the truth of a formula in a structure N implies its truth in each structure M extending N, the formula is upward absolute. Issues of absoluteness are particularly important in set theory and model theory, fields where multiple structures are considered sim... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Shoenfield absoluteness theorem Summary Absoluteness_(logic) In set theory, the issue of which properties of sets are absolute is well studied. The Shoenfield absoluteness theorem, due to Joseph Shoenfield (1961), establishes the absoluteness of a large class of formulas between a model of set theory and its constructi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Satisfiability problem Summary Satisfiability_problem In mathematical logic, a formula is satisfiable if it is true under some assignment of values to its variables. For example, the formula x + 3 = y {\displaystyle x+3=y} is satisfiable because it is true when x = 3 {\displaystyle x=3} and y = 6 {\displaystyle y=6} , ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Satisfiability problem Summary Satisfiability_problem Formally, satisfiability is studied with respect to a fixed logic defining the syntax of allowed symbols, such as first-order logic, second-order logic or propositional logic. Rather than being syntactic, however, satisfiability is a semantic property because it rel... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Satisfiability problem Summary Satisfiability_problem While this allows non-standard interpretations of symbols such as + {\displaystyle +} , one can restrict their meaning by providing additional axioms. The satisfiability modulo theories problem considers satisfiability of a formula with respect to a formal theory, w... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Satisfiability problem Summary Satisfiability_problem Satisfiability and validity are defined for a single formula, but can be generalized to an arbitrary theory or set of formulas: a theory is satisfiable if at least one interpretation makes every formula in the theory true, and valid if every formula is true in every... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Satisfiability problem Summary Satisfiability_problem This concept is closely related to the consistency of a theory, and in fact is equivalent to consistency for first-order logic, a result known as Gödel's completeness theorem. The negation of satisfiability is unsatisfiability, and the negation of validity is invali... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Satisfiability problem Summary Satisfiability_problem The problem of determining whether a formula in propositional logic is satisfiable is decidable, and is known as the Boolean satisfiability problem, or SAT. In general, the problem of determining whether a sentence of first-order logic is satisfiable is not decidabl... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Skolem function Summary Skolem_hull In mathematical logic, a formula of first-order logic is in Skolem normal form if it is in prenex normal form with only universal first-order quantifiers. Every first-order formula may be converted into Skolem normal form while not changing its satisfiability via a process called Sko... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Fragment (logic) Summary Fragment_(logics) In mathematical logic, a fragment of a logical language or theory is a subset of this logical language obtained by imposing syntactical restrictions on the language. Hence, the well-formed formulae of the fragment are a subset of those in the original logic. However, the seman... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Fragment (logic) Summary Fragment_(logics) An important problem in computational logic is to determine fragments of well-known logics such as first-order logic that are as expressive as possible yet are decidable or more strongly have low computational complexity. The field of descriptive complexity theory aims at esta... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Ground expression Summary Ground_sentence In mathematical logic, a ground term of a formal system is a term that does not contain any variables. Similarly, a ground formula is a formula that does not contain any variables. In first-order logic with identity with constant symbols a {\displaystyle a} and b {\displaystyle... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Judgment (mathematical logic) Summary Logical_assertion In mathematical logic, a judgment (or judgement) or assertion is a statement or enunciation in a metalanguage. For example, typical judgments in first-order logic would be that a string is a well-formed formula, or that a proposition is true. Similarly, a judgment... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Judgment (mathematical logic) Summary Logical_assertion Judgments are used in formalizing deduction systems: a logical axiom expresses a judgment, premises of a rule of inference are formed as a sequence of judgments, and their conclusion is a judgment as well (thus, hypotheses and conclusions of proofs are judgments).... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Judgment (mathematical logic) Summary Logical_assertion The same cannot be done with the other two deductions systems: as context is changed in some of their rules of inferences, they cannot be formalized so that hypothetical judgments could be avoided—not even if we want to use them just for proving derivability of ta... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Boolean literal Summary Literal_(mathematical_logic) In mathematical logic, a literal is an atomic formula (also known as an atom or prime formula) or its negation. The definition mostly appears in proof theory (of classical logic), e.g. in conjunctive normal form and the method of resolution. Literals can be divided i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Boolean literal Summary Literal_(mathematical_logic) In logics with double negation elimination (where ¬ ¬ x ≡ x {\displaystyle \lnot \lnot x\equiv x} ) the complementary literal or complement of a literal l {\displaystyle l} can be defined as the literal corresponding to the negation of l {\displaystyle l} . We can wr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Boolean literal Summary Literal_(mathematical_logic) Double negation elimination occurs in classical logics but not in intuitionistic logic. In the context of a formula in the conjunctive normal form, a literal is pure if the literal's complement does not appear in the formula. In Boolean functions, each separate occur... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Finite model property Summary Finite_model_property In mathematical logic, a logic L has the finite model property (fmp for short) if any non-theorem of L is falsified by some finite model of L. Another way of putting this is to say that L has the fmp if for every formula A of L, A is an L-theorem if and only if A is a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Soundness Logical systems Soundness > Use in mathematical logic > Logical systems In mathematical logic, a logical system has the soundness property if every formula that can be proved in the system is logically valid with respect to the semantics of the system. In most cases, this comes down to its rules having the pr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Soundness Logical systems Soundness > Use in mathematical logic > Logical systems , A n {\displaystyle A_{1},A_{2},...,A_{n}} of sentences in its language, if A 1 , A 2 , . . . | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Soundness Logical systems Soundness > Use in mathematical logic > Logical systems , A n ⊢ C {\displaystyle A_{1},A_{2},...,A_{n}\vdash C} , then A 1 , A 2 , . . . | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Soundness Logical systems Soundness > Use in mathematical logic > Logical systems , A n ⊨ C {\displaystyle A_{1},A_{2},...,A_{n}\models C} . In other words, a system is sound when all of its theorems are tautologies. Soundness is among the most fundamental properties of mathematical logic. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Soundness Logical systems Soundness > Use in mathematical logic > Logical systems The soundness property provides the initial reason for counting a logical system as desirable. The completeness property means that every validity (truth) is provable. Together they imply that all and only validities are provable. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Soundness Logical systems Soundness > Use in mathematical logic > Logical systems Most proofs of soundness are trivial. For example, in an axiomatic system, proof of soundness amounts to verifying the validity of the axioms and that the rules of inference preserve validity (or the weaker property, truth). If the system... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Epistemic structural realism Definition of structure Ontic_structural_realism > Definition of structure In mathematical logic, a mathematical structure is a standard concept. A mathematical structure is a set of abstract entities with relations between them. The natural numbers under arithmetic constitute a structure, ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Epistemic structural realism Definition of structure Ontic_structural_realism > Definition of structure Points in space and the real numbers under Euclidean geometry are another structure, with relations such as "the distance between point P1 and point P2 is real number R1"; equivalently, the "distance" relation includ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Non-standard number Summary Non-standard_model_of_arithmetic In mathematical logic, a non-standard model of arithmetic is a model of (first-order) Peano arithmetic that contains non-standard numbers. The term standard model of arithmetic refers to the standard natural numbers 0, 1, 2, …. The elements of any model of Pe... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Predicate symbol Summary Predicate_symbol In mathematical logic, a predicate variable is a predicate letter which functions as a "placeholder" for a relation (between terms), but which has not been specifically assigned any particular relation (or meaning). Common symbols for denoting predicate variables include capita... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Proof system Summary Proof_calculus In mathematical logic, a proof calculus or a proof system is built to prove statements. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Propositional variable Summary Propositional_variable In mathematical logic, a propositional variable (also called a sentential variable or sentential letter) is an input variable (that can either be true or false) of a truth function. Propositional variables are the basic building-blocks of propositional formulas, use... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Redundant proof Summary Redundant_proof In mathematical logic, a redundant proof is a proof that has a subset that is a shorter proof of the same result. In other words, a proof is redundant if it has more proof steps than are actually necessary to prove the result. Formally, a proof ψ {\displaystyle \psi } of κ {\disp... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Second-order predicate Summary Second-order_predicate In mathematical logic, a second-order predicate is a predicate that takes a first-order predicate as an argument. Compare higher-order predicate. The idea of second order predication was introduced by the German mathematician and philosopher Frege. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Second-order predicate Summary Second-order_predicate It is based on his idea that a predicate such as "is a philosopher" designates a concept, rather than an object. Sometimes a concept can itself be the subject of a proposition, such as in "There are no Bosnian philosophers". In this case, we are not saying anything ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Second-order predicate Summary Second-order_predicate Thus the predicate "is not satisfied" attributes something to the concept "is a Bosnian philosopher", and is thus a second-level predicate. This idea is the basis of Frege's theory of number. == References == | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Sentence (mathematical logic) Summary Sentence_(mathematical_logic) In mathematical logic, a sentence (or closed formula) of a predicate logic is a Boolean-valued well-formed formula with no free variables. A sentence can be viewed as expressing a proposition, something that must be true or false. The restriction of ha... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Sentence (mathematical logic) Summary Sentence_(mathematical_logic) Sentences are then built up out of atomic formulas by applying connectives and quantifiers. A set of sentences is called a theory; thus, individual sentences may be called theorems. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Sentence (mathematical logic) Summary Sentence_(mathematical_logic) To properly evaluate the truth (or falsehood) of a sentence, one must make reference to an interpretation of the theory. For first-order theories, interpretations are commonly called structures. Given a structure or interpretation, a sentence will have... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Sequent Summary Sequent In mathematical logic, a sequent is a very general kind of conditional assertion. A 1 , … , A m ⊢ B 1 , … , B n . {\displaystyle A_{1},\,\dots ,A_{m}\,\vdash \,B_{1},\,\dots ,B_{n}.} A sequent may have any number m of condition formulas Ai (called "antecedents") and any number n of asserted form... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Deductive closure Summary Deductive_closure In mathematical logic, a set T {\displaystyle {\mathcal {T}}} of logical formulae is deductively closed if it contains every formula φ {\displaystyle \varphi } that can be logically deduced from T {\displaystyle {\mathcal {T}}} , formally: if T ⊢ φ {\displaystyle {\mathcal {T... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Superintuitionistic logic Summary Superintuitionistic_logic In mathematical logic, a superintuitionistic logic is a propositional logic extending intuitionistic logic. Classical logic is the strongest consistent superintuitionistic logic; thus, consistent superintuitionistic logics are called intermediate logics (the l... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Tautological implication Summary Tautology_(logic) In mathematical logic, a tautology (from Greek: ταυτολογία) is a formula or assertion that is true in every possible interpretation. An example is "x=y or x≠y". Similarly, "either the ball is green, or the ball is not green" is always true, regardless of the colour of ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Tautological implication Summary Tautology_(logic) In logic, a formula is satisfiable if it is true under at least one interpretation, and thus a tautology is a formula whose negation is unsatisfiable. In other words, it cannot be false. It cannot be untrue. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Tautological implication Summary Tautology_(logic) Unsatisfiable statements, both through negation and affirmation, are known formally as contradictions. A formula that is neither a tautology nor a contradiction is said to be logically contingent. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Tautological implication Summary Tautology_(logic) Such a formula can be made either true or false based on the values assigned to its propositional variables. The double turnstile notation ⊨ S {\displaystyle \vDash S} is used to indicate that S is a tautology. Tautology is sometimes symbolized by "Vpq", and contradict... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Tautological implication Summary Tautology_(logic) The tee symbol ⊤ {\displaystyle \top } is sometimes used to denote an arbitrary tautology, with the dual symbol ⊥ {\displaystyle \bot } (falsum) representing an arbitrary contradiction; in any symbolism, a tautology may be substituted for the truth value "true", as sym... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Tautological implication Summary Tautology_(logic) Indeed, in propositional logic, there is no distinction between a tautology and a logically valid formula. In the context of predicate logic, many authors define a tautology to be a sentence that can be obtained by taking a tautology of propositional logic, and uniform... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Finite terms Summary Finite_terms In mathematical logic, a term denotes a mathematical object while a formula denotes a mathematical fact. In particular, terms appear as components of a formula. This is analogous to natural language, where a noun phrase refers to an object and a whole sentence refers to a fact. A first... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Theory (mathematical logic) Summary First-order_theory In mathematical logic, a theory (also called a formal theory) is a set of sentences in a formal language. In most scenarios a deductive system is first understood from context, after which an element ϕ ∈ T {\displaystyle \phi \in T} of a deductively closed theory T... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Extension by new constant and function names Summary Extension_by_new_constant_and_function_names In mathematical logic, a theory can be extended with new constants or function names under certain conditions with assurance that the extension will introduce no contradiction. Extension by definitions is perhaps the best-... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Extension by new constant and function names Summary Extension_by_new_constant_and_function_names Let T 1 {\displaystyle T_{1}} be a theory obtained from T {\displaystyle T} by extending its language with new constants a 1 , … , a m {\displaystyle a_{1},\ldots ,a_{m}} and adding a new axiom φ ( a 1 , … , a m ) {\displa... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Extension by new constant and function names Summary Extension_by_new_constant_and_function_names Then T 1 {\displaystyle T_{1}} is a conservative extension of T {\displaystyle T} , i.e. the theories T {\displaystyle T} and T 1 {\displaystyle T_{1}} prove the same theorems not involving the functional symbol f {\displa... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Existential theory of the reals Background Existential_theory_of_the_reals > Background In mathematical logic, a theory is a formal language consisting of a set of sentences written using a fixed set of symbols. The first-order theory of real closed fields has the following symbols: the constants 0 and 1, a countable c... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Existential theory of the reals Background Existential_theory_of_the_reals > Background However, there are other number systems that are not accurately described by these axioms; in particular, the theory defined in the same way for integers instead of real numbers is undecidable, even for existential sentences (Diopha... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Existential theory of the reals Background Existential_theory_of_the_reals > Background The set of n {\displaystyle n} -tuples of real numbers ( X 1 , … X n ) {\displaystyle (X_{1},\dots X_{n})} for which F ( X 1 , … X n ) {\displaystyle F(X_{1},\dots X_{n})} is true is called a semialgebraic set, so the decision probl... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Categorical (model theory) Summary Categorical_theory In mathematical logic, a theory is categorical if it has exactly one model (up to isomorphism). Such a theory can be viewed as defining its model, uniquely characterizing the model's structure. In first-order logic, only theories with a finite model can be categoric... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Categorical (model theory) Summary Categorical_theory Higher-order logic contains categorical theories with an infinite model. For example, the second-order Peano axioms are categorical, having a unique model whose domain is the set of natural numbers N . {\displaystyle \mathbb {N} .} | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Categorical (model theory) Summary Categorical_theory In model theory, the notion of a categorical theory is refined with respect to cardinality. A theory is κ-categorical (or categorical in κ) if it has exactly one model of cardinality κ up to isomorphism. Morley's categoricity theorem is a theorem of Michael D. Morle... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Complete theory Summary Maximal_consistent_set In mathematical logic, a theory is complete if it is consistent and for every closed formula in the theory's language, either that formula or its negation is provable. That is, for every sentence φ , {\displaystyle \varphi ,} the theory T {\displaystyle T} contains the sen... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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