# Predicate (mathematical logic)

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 $\displaystyle{ P(a) }$, 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 $\displaystyle{ b }$.

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 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.