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

P

(

a

)

{\displaystyle P(a)}

, the symbol

P

{\displaystyle P}

is a predicate that applies to the individual constant

a

{\displaystyle a}

which evaluates to either true or false. Similarly, in the formula

R

(

a

,

b

)

{\displaystyle R(a,b)}

, the symbol

R

{\displaystyle R}

is a predicate that applies to the individual constants

a

{\displaystyle a}

and

b

{\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

R

(

a

,

b

)

{\displaystyle R(a,b)}

would be true on an interpretation if the entities denoted by

a

{\displaystyle a}

and

b

{\displaystyle b}

stand in the relation denoted by

R

{\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.