Predicatelogica

Predicatelogica, commonly called predicate logic or first-order logic, is a formal system that extends propositional logic by including variables, predicates and quantifiers. It allows statements about objects and their properties, relations among objects, and can express general and existential claims.

The basic vocabulary consists of constants, function symbols, and predicate symbols with fixed arities. Terms are constructed from constants and function symbols; atomic formulas are of the form P(t1,...,tn) where P is an n-ary predicate and t's are terms. Complex formulas are built from atomic formulas using logical connectives such as and, or, not, implies, and by means of quantifiers ∀ (for all) and ∃ (exists) that bind variables. A formula is closed if all variables are bound; otherwise it is free. Example: ∀x (Human(x) → Mortal(x)).

Semantics assign meaning via structures: a domain D, an interpretation of each predicate symbol as a relation on D, and an interpretation of function symbols as actual functions on D. Terms are interpreted as elements of D, and a formula is true or false in a structure under a variable assignment. The quantifiers are evaluated by varying the assignment over D. This makes first-order logic capable of expressing many mathematical and empirical statements.

First-order predicate logic was foundationally developed by Gottlob Frege and later formalized by others. The logic has sound and complete proof systems (Gödel's completeness theorem). It is, however, only semi-decidable; there is no algorithm that decides the truth of all first-order sentences. It is strictly more expressive than propositional logic but less than higher-order logics.

Predicate logic is the standard formal framework in mathematics, computer science, linguistics, and artificial intelligence for formal specification, theorem proving, knowledge representation, and formal semantics of natural language.