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c_pb5wc8a2k1hw | 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 well-defined property, there is the set ... | Russell paradox |
c_8xdl6qdp343m | 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. | Russell paradox |
c_8shcdl88jyub | 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 mathematician Gottlob Frege, hence underm... | Russell paradox |
c_24fr61xpki7l | 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 Zermelo modified the axioms of set theory ... | Russell paradox |
c_pgdums46qct9 | 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 entirely. Skolem arithmetic is weaker than Peano ar... | Skolem arithmetic |
c_hqxzxreekb0x | 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 computational complexity of this decision probl... | Skolem arithmetic |
c_rsjcgpkt3mhb | 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 realization that Martin-Löf's... | Type in type |
c_6cr66rssvpgx | 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 proved using eleven axioms about these operations that are taught in high-school-level ... | Tarski's high school algebra problem |
c_4arh7iwwwbvq | 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 complete Boolean algebra. Boolean-valued models were ... | Boolean-valued model |
c_v7clo0j0xmtr | 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 Gödel numbering can be interpreted as ... | Gödel encoding |
c_ss5sft8jszck | 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 that applies it. The interpretation also defines predicate ... | Herbrand interpretation |
c_a4ims6q9wriw | 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 unsatisfiable set of ground instances from the Herbrand universe... | Herbrand interpretation |
c_bbuxdxdtkfdc | 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 set, If a formula in the set has ... | Hintikka set |
c_wcwj959lwbx1 | 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 Lindström in 1966. They were later studied for their appli... | Lindström quantifier |
c_lbxdnj45sp1j | 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 not conservative, and can prove more theorems than the orig... | Conservative extension |
c_lbv2i5d24ed5 | 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 language of T 1 {\displaystyle T_{1}} would be a theorem of T ... | Conservative extension |
c_wecdoeoll4uf | 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 {\displaystyle T_{2}} , ... of it. Recently, conservative extens... | Conservative extension |
c_cm90zav1a7w0 | 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 proceed to derive B. Deduction theorems exist for... | Deduction theorem |
c_2iry2fcw46bm | 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 with some extra constraints (that would for ... | Deduction theorem |
c_0ubuus9zhq4j | 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 hold in Birkhoff–von Neumann quantum logic, b... | Deduction theorem |
c_iaaaaa8gqx0i | 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 formula defining the relation. | Definable set |
c_1ywl0didin9b | 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 concept is named after Kurt Gödel. ==... | Gödel logic |
c_urdf98p735dl | 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 combinations of equalities and inequalities of expressions over real variables. The corresponding first-order theory is the set of... | Decidability of first-order theories of the real numbers |
c_shbv7s250t2i | 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 algebraic numbers. As proven by Tarski, this theory is decidable; see Tarski–Seidenberg theorem and Quantifier eliminatio... | Decidability of first-order theories of the real numbers |
c_3ugtd6040bd1 | 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 Schanuel's conjecture holds then the decidability of this theory would follow. In contrast, the extension of the theory of ... | Decidability of first-order theories of the real numbers |
c_v2cib6vcp53n | 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 to terminate for input formulas that are robust, that is, formulas whose satisfiability does not change if the formula is s... | Decidability of first-order theories of the real numbers |
c_pffx4h5xsa4e | 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 predicate that takes only one argument. For example, th... | One-place predicate |
c_nmotcwzb1xbk | 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. Essentially, it involves the form of an expressio... | Formal calculation |
c_le4clnew5ai0 | 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 language together with a deductive apparatus (also called a deductive system). The deductive apparatus may consist of a set of transform... | Formal language theory |
c_y8paf0szvldd | 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 identified by its theorems. Two formal systems F S {\displaystyle {\mathcal {FS}}} and F S ′ {\displaystyle {\mathcal {FS'}}} may have a... | Formal language theory |
c_20pfm4qlhdy8 | 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 formulas in the sequence by a rule of inference. The last sentence in the sequence is a theorem of a formal system. Formal proofs are... | Formal language theory |
c_l3btk23521m7 | 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 theorems. Usually a theory is understood to be closed under the relation of ... | Mathematical theorem |
c_jqce9caabim5 | 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 theory to be closed under a derivability relation, it must be associated w... | Mathematical theorem |
c_epn7rnrh48yr | 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 have to be true, since the theory that contains it may be unsound relative... | Mathematical theorem |
c_ykkxtcs12kdb | 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, which is concerned with the relationship between formal theories and stru... | Mathematical theorem |
c_gkykvo2j6x9n | 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 (as opposed to within a formal system) is called a metatheorem. Some importa... | Mathematical theorem |
c_zw6pi0mpifuc | 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 ) ) → ¬ ( P ( x ) → Q ( f ( y ) , x , z ) ) ) {\displaystyle \forall x\forall y(P(f(x))\rightar... | Formula |
c_4s0ezn60uo7n | 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 ( ∨ {\displaystyle \lor } , or). Negation normal form i... | Negation normal form |
c_04g6ddh4rh2t | 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 \lnot B)\\\lnot (A\lor B)&~\to ~\lnot A\land \lnot... | Negation normal form |
c_cmxecs3mqobe | 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 the validity problem continues to be co-NP-complete.... | Negation normal form |
c_8dyc3e73qpxw | 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 absolute to some class which contains both of them.. Theorems ab... | Shoenfield absoluteness theorem |
c_rcm5drq1mjuv | 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 simultaneously. In model theory, several basic results and defin... | Shoenfield absoluteness theorem |
c_fkkmzl55xvru | 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 constructible universe, with important methodological consequences. The... | Shoenfield absoluteness theorem |
c_u1l3jo6101b6 | 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} , while the formula x + 1 = x {\displaystyle x+1=x} is n... | Satisfiability problem |
c_d4r0s6ebgd61 | 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 relates to the meaning of the symbols, for example, the m... | Satisfiability problem |
c_dkdej0mtdf05 | 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, which is a (finite or infinite) set of axioms. | Satisfiability problem |
c_zlgnfgdyfom8 | 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 interpretation. For example, theories of arithmetic s... | Satisfiability problem |
c_6qw6yo1wfid7 | 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 invalidity. These four concepts are related to each other in... | Satisfiability problem |
c_ed9y01j5rcii | 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 decidable. In universal algebra, equational theory, and automa... | Satisfiability problem |
c_as50gzjb2c4i | 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 Skolemization (sometimes spelled Skolem... | Skolem function |
c_gp254l1t2640 | 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 semantics of the formulae in the fragment and in... | Fragment (logic) |
c_97p1jyjb2gwk | 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 establishing a link between logics and computat... | Fragment (logic) |
c_7ud2s40f8uo1 | 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 b} , the sentence Q ( a ) ∨ P ( b ) {\dis... | Ground expression |
c_1yggjo2fktbt | 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 may assert the occurrence of a free variable in an expr... | Judgment (mathematical logic) |
c_r4cgopy6agmq | 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). A characteristic feature of the variants of Hilbert-sty... | Judgment (mathematical logic) |
c_rgcks39pwh0v | 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 tautologies. This basic diversity among the various calcul... | Judgment (mathematical logic) |
c_ahpzrgqw6a0b | 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 into two types: A positive literal is just an atom (e.... | Boolean literal |
c_sx3schlrb0h4 | 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 write l ¯ {\displaystyle {\bar {l}}} to denote the comp... | Boolean literal |
c_0y1x1at0f6jb | 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 occurrence of a variable, either in inverse or uncomplemen... | Boolean literal |
c_inehskih06xm | 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 theorem of the theory of finite models of L. If L i... | Finite model property |
c_5g22ltidrj4d | 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 property of preserving truth. The converse of soundness is known as completeness. A ... | Soundness |
c_99nx6n6p7hfq | , A n {\displaystyle A_{1},A_{2},...,A_{n}} of sentences in its language, if A 1 , A 2 , . . . | Soundness |
c_d0oumuft7n5c | , A n ⊢ C {\displaystyle A_{1},A_{2},...,A_{n}\vdash C} , then A 1 , A 2 , . . . | Soundness |
c_8qiqxjuxzhxn | , 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. | Soundness |
c_mdmswmq9rpz0 | 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. | Soundness |
c_7v4yc3tpvfbb | 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 allows Hilbert-style deduction, it requires only verifying the validity of the ax... | Soundness |
c_icaoeqdccrjs | 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, with relations such as "is evenly divisible by" and "is greater than". Here the relation "is greater tha... | Epistemic structural realism |
c_pzrucyli1d8x | 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 includes the element (P1, P2, R1). Other structures include the Riemann space of general relativity and the Hi... | Epistemic structural realism |
c_v78o2s0i548z | 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 Peano arithmetic are linearly ordered and possess an initial se... | Non-standard number |
c_c2wzb25cf8rm | 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 capital roman letters such as P {\displaystyle P... | Predicate symbol |
c_xhmbr25q8op3 | In mathematical logic, a proof calculus or a proof system is built to prove statements. | Proof system |
c_d599ziknxdpj | 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, used in propositional logic and higher-order logics. | Propositional variable |
c_u16l64gpslj0 | 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 {\displaystyle \kappa } is considered redundan... | Redundant proof |
c_hyryrkj4bnph | 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. | Second-order predicate |
c_mjn77pz4bf90 | 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 of any Bosnian philosophers, but of the concept "is a ... | Second-order predicate |
c_enf2vlof6dcc | 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 == | Second-order predicate |
c_jr05y5bko18c | 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 having no free variables is needed to make sure that sentences can hav... | Sentence (mathematical logic) |
c_05adoibsr7l3 | 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. | Sentence (mathematical logic) |
c_hq9x7cj3o5m1 | 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 a fixed truth value. A theory is satisfiable when it is possible to... | Sentence (mathematical logic) |
c_rd24fwjuxmzm | 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 formulas Bj (called "succede... | Sequent |
c_e4evnd8geci1 | 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}}\vdash \varphi } always implies φ ∈ T {\di... | Deductive closure |
c_uvpptzkml91i | 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 logics are intermediate between intuitionistic logic and clas... | Superintuitionistic logic |
c_tl5ysanvrrbf | 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 the ball. The philosopher Ludwig Wittgenstein first... | Tautological implication |
c_gq0bi7fug6bz | 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. | Tautological implication |
c_lfrzjpl4p8f8 | 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. | Tautological implication |
c_8y83k3lbqo4s | 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 contradiction by "Opq". | Tautological implication |
c_xyqc1xh1pfqb | 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 symbolized, for instance, by "1".Tautologies are a key... | Tautological implication |
c_kszxefnuofcz | 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 uniformly replacing each propositional variable by a first... | Tautological implication |
c_r886cxom53ey | 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-order term is recursively constru... | Finite terms |
c_vq1aqnwwibdo | 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 {\displaystyle T} is then called a theorem of the theo... | Theory (mathematical logic) |
c_6a58hn5afjz4 | 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-known approach, but it requires unique existence of an object with the desired property. Addition ... | Extension by new constant and function names |
c_vj4kepmn579a | 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 ) {\displaystyle \varphi (a_{1},\ldots ,a_{m})} .Then T 1 {\displaystyle T_{1}} is a conservative extension ... | Extension by new constant and function names |
c_s9mxwt2dlnq5 | 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 {\displaystyle f} ). Shoenfield states the theorem in the form for a new function name, and constants are ... | Extension by new constant and function names |
c_nmnt06hj2n3x | 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 collection of variables X i {\displaystyle X_{i}} , the addition, subtraction, multiplica... | Existential theory of the reals |
c_wvaxrshrq8wh | 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 (Diophantine equations) by Matiyasevich's theorem.The existential theory of the reals is the fr... | Existential theory of the reals |
c_j2w40kpqgsyk | 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 problem for the existential theory of the reals can equivalently be rephrased as testing whet... | Existential theory of the reals |
c_zz58sn7tdq4h | 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 categorical. | Categorical (model theory) |
c_xd1xx3qoop13 | 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} .} | Categorical (model theory) |
c_kkpnezqtph4i | 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. Morley (1965) stating that if a first-order theory in a cou... | Categorical (model theory) |
c_hbru3lr5bxqi | 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 sentence or its negation but not both (that is, ei... | Complete theory |
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