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Qu'est-ce (qui) est first-order logic - définition

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First-order predicate calculus; First-order predicate logic; Predicate logic; Predicate calculus; First order logic; Predicate Calculus; First Order Logic; First order language; First-order language; Quantification theory; First order predicate calculus; Predicate logic (Philosophy); First order logic with equality; 1st order logic; First Order Language; FOPL; First order predicate logic; Polyadic predicate calculus; Predicate logic (philosophy); First-order logic with equality; First-Order Logic; First-order sentence; Quantification calculus; Satisfaction relation; Predicate Logic; Many-sorted first-order logic; First-order Peano arithmetic; FOPC; Lower Predicate Calculus; Tarskian semantics; Classical predicate logic; First-order-logic; Equational first-order logic; Semantics of first-order logic; Deductive systems for first-order logic

first-order logic

<language, logic> The language describing the truth of mathematical formulas. Formulas describe properties of terms and have a truth value. The following are atomic formulas: True False p(t1,..tn) where t1,..,tn are terms and p is a predicate. If F1, F2 and F3 are formulas and v is a variable then the following are compound formulas: F1 ^ F2 conjunction - true if both F1 and F2 are true, F1 V F2 disjunction - true if either or both are true, F1 => F2 implication - true if F1 is false or F2 is true, F1 is the antecedent, F2 is the consequent (sometimes written with a thin arrow), F1 <= F2 true if F1 is true or F2 is false, F1 == F2 true if F1 and F2 are both true or both false (normally written with a three line equivalence symbol) first-order logicF1 negation - true if f1 is false (normally written as a dash '-' with a shorter vertical line hanging from its right hand end). For all v . F universal quantification - true if F is true for all values of v (normally written with an inverted A). Exists v . F existential quantification - true if there exists some value of v for which F is true. (Normally written with a reversed E). The operators ^ V => <= == first-order logic are called connectives. "For all" and "Exists" are quantifiers whose scope is F. A term is a mathematical expression involving numbers, operators, functions and variables. The "order" of a logic specifies what entities "For all" and "Exists" may quantify over. First-order logic can only quantify over sets of atomic propositions. (E.g. For all p . p => p). Second-order logic can quantify over functions on propositions, and higher-order logic can quantify over any type of entity. The sets over which quantifiers operate are usually implicit but can be deduced from well-formedness constraints. In first-order logic quantifiers always range over ALL the elements of the domain of discourse. By contrast, second-order logic allows one to quantify over subsets. ["The Realm of First-Order Logic", Jon Barwise, Handbook of Mathematical Logic (Barwise, ed., North Holland, NYC, 1977)]. (2005-12-27)

First-order logic

First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over non-logical objects, and allows the use of sentences that contain variables, so that rather than propositions such as "Socrates is a man", one can have expressions in the form "there exists x such that x is Socrates and x is a man", where "there exists" is a quantifier, while x is a variable.

https://en.wikipedia.org/wiki/First-order_logic

predicate calculus

predicate logic

Wikipédia

First-order logic

First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over non-logical objects, and allows the use of sentences that contain variables, so that rather than propositions such as "Socrates is a man", one can have expressions in the form "there exists x such that x is Socrates and x is a man", where "there exists" is a quantifier, while x is a variable. This distinguishes it from propositional logic, which does not use quantifiers or relations; in this sense, propositional logic is the foundation of first-order logic.

A theory about a topic, such as set theory, a theory for groups, or a formal theory of arithmetic, is usually a first-order logic together with a specified domain of discourse (over which the quantified variables range), finitely many functions from that domain to itself, finitely many predicates defined on that domain, and a set of axioms believed to hold about them. Sometimes, "theory" is understood in a more formal sense as just a set of sentences in first-order logic.

The adjective "first-order" distinguishes first-order logic from higher-order logic, in which there are predicates having predicates or functions as arguments, or in which quantification over predicates or functions, or both, are permitted.^: 56 In first-order theories, predicates are often associated with sets. In interpreted higher-order theories, predicates may be interpreted as sets of sets.

There are many deductive systems for first-order logic which are both sound (i.e., all provable statements are true in all models) and complete (i.e. all statements which are true in all models are provable). Although the logical consequence relation is only semidecidable, much progress has been made in automated theorem proving in first-order logic. First-order logic also satisfies several metalogical theorems that make it amenable to analysis in proof theory, such as the Löwenheim–Skolem theorem and the compactness theorem.

First-order logic is the standard for the formalization of mathematics into axioms, and is studied in the foundations of mathematics. Peano arithmetic and Zermelo–Fraenkel set theory are axiomatizations of number theory and set theory, respectively, into first-order logic. No first-order theory, however, has the strength to uniquely describe a structure with an infinite domain, such as the natural numbers or the real line. Axiom systems that do fully describe these two structures (that is, categorical axiom systems) can be obtained in stronger logics such as second-order logic.

The foundations of first-order logic were developed independently by Gottlob Frege and Charles Sanders Peirce. For a history of first-order logic and how it came to dominate formal logic, see José Ferreirós (2001).

https://en.wikipedia.org/wiki/First-order logic