Predicate (logic)

Sometimes it is inconvenient or impossible to describe a set by listing all of its elements. Another useful way to define a set is by specifying a property that the elements of the set have in common. We use the notation P(x) to denote a sentence or statement P concerning the variable object x. The set defined by P(x) written {x | P(x)}, is just a collection of all the objects for which P is sensible and true.

For instance, {x | x is a positive integer less than 4} is the set {1,2,3}.

Thus, an element of {x | P(x)} is an object t for which the statement P(t) is true. Such a sentence P(x) is called a Predicate. P(x) is also called a propositional function, because each choice of x produces a proposition P(x) that is either true or false.

In formal semantics a predicate is an expression of the semantic type of sets. An equivalent formulation is that they are thought of as indicator functions of sets, i.e. functions from an entity to a truth value.

In First-order logic, a predicate can take the role as either a property or a relation between entities.