# Predicate (logic)

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. The notation "P(x)" is used 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.

ee also

* Set-builder notation makes use of predicates
* Free variables and bound variables

* [http://cs.odu.edu/~toida/nerzic/content/logic/pred_logic/predicate/pred_intro.html Introduction to predicates]

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