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 $P$.

Domain

The domain of $P \left({x_1, x_2, \ldots}\right)$ is the set of values that each object variable may take.

See domain for a definition of this term in a wider context.

Arity

The arity of a propositional function is the number of object variables associated with it.

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:

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle P \left({1}\right)\)	\(=\)	\(\displaystyle F\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle P \left({2}\right)\)	\(=\)	\(\displaystyle T\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle P \left({591}\right)\)	\(=\)	\(\displaystyle F\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

Thus $P \left({x, y}\right)$ 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:

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle P \left({1, 2}\right)\)	\(=\)	\(\displaystyle T\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle P \left({2, 1}\right)\)	\(=\)	\(\displaystyle F\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle P \left({3, 3}\right)\)	\(=\)	\(\displaystyle F\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

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:

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle P \left({1, 2, 3}\right)\)	\(=\)	\(\displaystyle F\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle P \left({2, 1, 3}\right)\)	\(=\)	\(\displaystyle T\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle P \left({5, 4, 3}\right)\)	\(=\)	\(\displaystyle F\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

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

Also see

Compare with 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.

Extending this notion further, a propositional function can be considered as being a boolean-valued function whose codomain is the set $\left\{{\text{True}, \text{False}}\right\}$.

Note

A propositional function is sometimes referred to as a condition, or a property.

Sources

• Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 4$
• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 3$
• H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability (1996): $\S 2.1$