Predicate (logic)

In logic, a predicate is a non-logical symbol that represents a property or a relation, though, formally, does not need to represent anything at all. For instance, in the first-order formula ${\displaystyle P(a)}$, the symbol ${\displaystyle P}$ is a predicate that applies to the individual constant ${\displaystyle a}$ which evaluates to either true or false. Similarly, in the formula ${\displaystyle R(a,b)}$, the symbol ${\displaystyle R}$ is a predicate that applies to the individual constants ${\displaystyle a}$ and ${\displaystyle b}$. Predicates are considered a primitive notion of first-order, and higher-order logic and are therefore not defined in terms of other more basic concepts.

The term derives from the grammatical term "predicate", meaning a word or phrase that represents a property or relation.

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.

Strictly speaking, a predicate does not need to be given any interpretation, so long as its syntactic properties are well-defined. For example, equality may be understood solely through its reflexive and substitution properties (cf. Equality (mathematics) § Axioms). Other properties can be derived from these, and they are sufficient for proving theorems in mathematics. Similarly, set membership can be understood solely through the axioms of Zermelo–Fraenkel set theory.