Predicate (mathematical logic)

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Short description: Symbol representing a property or relation in logic

In logic, a predicate is a symbol that represents a property or a relation. For instance, in the first-order formula [math]\displaystyle{ P(a) }[/math], the symbol [math]\displaystyle{ P }[/math] is a predicate that applies to the individual constant [math]\displaystyle{ a }[/math]. Similarly, in the formula [math]\displaystyle{ R(a,b) }[/math], the symbol [math]\displaystyle{ R }[/math] is a predicate that applies to the individual constants [math]\displaystyle{ a }[/math] and [math]\displaystyle{ b }[/math].

In the semantics of logic, predicates are interpreted as relations. For instance, in a standard semantics for first-order logic, the formula [math]\displaystyle{ R(a,b) }[/math] would be true on an interpretation if the entities denoted by [math]\displaystyle{ a }[/math] and [math]\displaystyle{ b }[/math] stand in the relation denoted by [math]\displaystyle{ R }[/math]. Since predicates are non-logical symbols, they can denote different relations depending on the interpretation given to them. While first-order logic only includes predicates that apply to individual constants, other logics may allow predicates that apply to other predicates.

Predicates in different systems

A predicate is a statement or mathematical assertion that contains variables, sometimes referred to as predicate variables, and may be true or false depending on those variables’ value or values.

See also


  1. Lavrov, Igor Andreevich; Maksimova, Larisa (2003). Problems in Set Theory, Mathematical Logic, and the Theory of Algorithms. New York: Springer. p. 52. ISBN 0306477122. 

External links