Definition:Propositional Function

Definition
A propositional function $P \left({x_1, x_2, \ldots}\right)$ 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:


 * The values of $x_1, x_2, \ldots$;
 * The nature of $P$.

Satisfaction
Let $P \left({x_1, x_2, \ldots, x_n}\right)$ be an $n$-ary propositional function.

If $a_1, a_2, \ldots, a_n$ are values which make $P \left({x_1, x_2, \ldots, x_n}\right)$ true, then the ordered tuple $\left({a_1, a_2, \ldots, a_n}\right)$ satisfies $P \left({x_1, x_2, \ldots, x_n}\right)$.

Examples
For example, let the universe be the set of all integers $\Z$.

Let $P \left({x}\right)$ be the propositional function defined as:
 * $x$ is even

Then we can insert particular values of $x \in \Z$, for example, as follows:

Thus $P \left({x}\right)$ is a unary propositional function (pronounced yoo-nary).

Let $P \left({x, y}\right)$ be the propositional function defined as:
 * $x$ is less than $y$

Then we can create the propositional statements:

Thus $P \left({x, y}\right)$ is a binary propositional function.

Let $P \left({x, y, z}\right)$ be the propositional function defined as:
 * $x$ is between $y$ and $z$.

Then:

Thus $P \left({x, y, z}\right)$ is a ternary propositional function.

Also see
Compare with Definition:Predicate Symbol.

In the context of predicate logic:
 * $P \left({x}\right)$ is usually interpreted to mean "$x$ has the property $P$".


 * $P \left({x, y}\right)$ 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 $P \left({x}\right)$ is true or false, that is, whether $x$ has $P$ or not.

Also known as
A propositional function is sometimes referred to as a condition.

It can also be referred to as a property; on, property refers to the everyday meaning of this word as a synonym of "feature", as can be seen on Definition:Property.

Some sources refer to a propositional function as a predicate, and in this context it is compatible with Definition:Predicate.