Editing Predicate (logic) (section)

{{Short description|Symbol representing a property or relation in logic}}
{{Other uses|Predicate (disambiguation)#Logic}}
In [[mathematical logic|logic]], a '''predicate''' is a symbol that represents a property or a relation. For instance, in the [[first order logic|first-order formula]] <math>P(a)</math>, the symbol <math>P</math> is a predicate that applies to the [[individual constant]] <math>a</math>. Similarly, in the formula <math>R(a,b)</math>, the symbol <math>R</math> is a predicate that applies to the individual constants <math>a</math> and <math>b</math>. 

According to [[Gottlob Frege]], the meaning of a predicate is exactly a function from the domain of objects to the [[truth value]]s "true" and "false".

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