# Predicate (mathematical logic)

In mathematical logic, a predicate is commonly understood to be a Boolean-valued function P: X→ {true, false}, called a predicate on X. However, predicates have many different uses and interpretations in mathematics and logic, and their precise definition, meaning and use will vary from theory to theory. For example, when a theory defines the concept of a relation, a predicate simply becomes the characteristic function (otherwise known as the indicator function) of a relation. However, not all theories have relations, or are founded on set theory, and so one must be careful with the proper definition and semantic interpretation of a predicate.

## Simplified overview

Informally, a predicate, often denoted by capital roman letters such as ${\displaystyle P}$, ${\displaystyle Q}$ and ${\displaystyle R}$,[1] is a statement that may be true or false depending on the values of its variables.[2] It can be thought of as an operator or function, that returns a value that is either true or false depending on its input.[3][4] For example, predicates are sometimes used to indicate set membership: when talking about sets, it is sometimes inconvenient or impossible to describe a set by listing all of its elements. Thus, a predicate P(x) will be true or false, depending on whether x belongs to a set or not.

A predicate can be a proposition if the placeholder x is defined by domain or selection.

Predicates are also commonly used to talk about the properties of objects, by defining the set of all objects that have some property in common. For example, when P is a predicate on X, one might sometimes say P is a property of X. Similarly, the notation P(x) is used to denote a sentence or statement P concerning the variable object x. The set defined by P(x), also called the extension[5] of P, is written as {x | P(x)}, and is the set of objects for which P is true.

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

If t is an element of the set {x | P(x)}, then the statement P(t) is true.

Here, P(x) is referred to as the predicate, and x the placeholder of the proposition. Sometimes, P(x) is also called a (template in the role of) propositional function, as each choice of the placeholder x produces a proposition.

A simple form of predicate is a Boolean expression, in which case the inputs to the expression are themselves Boolean values, combined using Boolean operations. Similarly, a Boolean expression with inputs predicates is itself a more complex predicate.

## Formal definition

The precise semantic interpretation of an atomic formula and an atomic sentence will vary from theory to theory.