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

In the semantics of logic, predicates are interpreted as relations. For instance, in a standard semantics for first-order logic, the formula $R(a,b)$ would be true on an interpretation if the entities denoted by $a$ and $b$ stand in the relation denoted by $R$ . 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.