Predicate (mathematical logic)

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

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

In the semantics of logic, predicates are interpreted as relations. For instance, in a standard semantics for first-order logic, the formula ${\displaystyle R(a,b)}$ would be true on an interpretation if the entities denoted by ${\displaystyle a}$ and ${\displaystyle b}$ stand in the relation denoted by ${\displaystyle R}$. 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 objects, other logics may allow predicates that apply to collections of objects defined by other predicates.