Model theory

Functional predicate

In formal logic and related branches of mathematics, a functional predicate, or function symbol, is a logical symbol that may be applied to an object term to produce another object term.Functional predicates are also sometimes called mappings, but that term has additional meanings in mathematics.In a model, a function symbol will be modelled by a function. Specifically, the symbol F in a formal language is a functional symbol if, given any symbol X representing an object in the language, F(X) is again a symbol representing an object in that language.In typed logic, F is a functional symbol with domain type T and codomain type U if, given any symbol X representing an object of type T, F(X) is a symbol representing an object of type U.One can similarly define function symbols of more than one variable, analogous to functions of more than one variable; a function symbol in zero variables is simply a constant symbol. Now consider a model of the formal language, with the types T and U modelled by sets [T] and [U] and each symbol X of type T modelled by an element [X] in [T].Then F can be modelled by the set which is simply a function with domain [T] and codomain [U].It is a requirement of a consistent model that [F(X)] = [F(Y)] whenever [X] = [Y]. (Wikipedia).

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Related pages

Zermelo–Fraenkel set theory | If and only if | Metalogic | Logical connective | Mathematics | Formal system | Function (mathematics) | Function composition | Set (mathematics) | Theorem | Formal language | Axiom schema of replacement | Codomain | Material conditional | Subtyping | Logical constant