Predicate

Definition

Predicate

An -ary predicate on a set is a function

where is the set of input tuples , means true, and means false. The arity specifies how many arguments takes.

A unary predicate expresses a property of one object; a predicate with several arguments expresses a relation between objects. Equivalently, specifies the set of tuples for which it holds:

Predicate vs Function

A predicate is a truth-valued function. A general function may return elements of any codomain ; it need not return an element of .

The indicator function represents the same predicate numerically: when , and otherwise. Being a predicate does not imply being decidable.

Predicate Symbols

In first-order logic, a predicate symbol belongs to the syntax. An interpretation assigns it a relation on the universe. The symbol names the relation; it is not the relation itself.

Example

A property of one number

Let express evenness:

Thus and ; the corresponding indicator values are and .

A relation between two numbers

Let be given by exactly when . Then , whereas .