Predicate (logic)

From Wikipedia, the free encyclopedia

Please help improve this article or section by expanding it.
Further information might be found on the talk page or at requests for expansion. (January 2007)

It has been suggested that this article or section be merged with Predicate (mathematics). (Discuss)

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.