Definition:Propositional Function

Definition
A propositional function $P \paren {x_1, x_2, \ldots}$ is an operator which acts on the objects denoted by the object variables $x_1, x_2, \ldots$ in a particular universe to return a truth value which depends on:


 * $(1): \quad$ The values of $x_1, x_2, \ldots$
 * $(2): \quad$ The nature of $P$.

Also known as
In various contexts, the term propositional function may be given as:


 * condition
 * sentential function
 * formula
 * property; on, property refers to the everyday meaning of this word as a synonym of "feature", as can be seen on Definition:Property.
 * predicate: in this context it is compatible with Definition:Predicate.

The notation for indicating that $x$ has the property $P$ varies; the notation $P x$ can often be seen.

Also see
Compare with Definition:Predicate Symbol.

In the context of predicate logic:
 * $P \paren x$ is usually interpreted to mean "$x$ has the property $P$".


 * $P \paren {x, y}$ can often be interpreted to mean "$x$ has the relation $P$ to $y$".

A propositional function extends this concept, putting it in the context of determining whether $P \paren x$ is true or false, that is, whether $x$ has $P$ or not.