Definition:Propositional Function

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Definition

A propositional function $\map P {x_1, x_2, \ldots}$ 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 $P$.


Satisfaction

Let $P \paren {x_1, x_2, \ldots, x_n}$ be an $n$-ary propositional function.

If $a_1, a_2, \ldots, a_n$ have values which make $P \paren {x_1, x_2, \ldots, x_n}$ true, then the ordered tuple $\paren {a_1, a_2, \ldots, a_n}$ satisfies $P \paren {x_1, x_2, \ldots, x_n}$.


Examples

Let the universe be the set of 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:

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

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

$\Box$


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

$x$ is less than $y$

Then we can create the propositional statements:

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

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

$\Box$


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

$x$ is between $y$ and $z$.

Then:

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

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

$\Box$


Also known as

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

  • condition
  • sentential function
  • formula
  • property; on $\mathsf{Pr} \infty \mathsf{fWiki}$, 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:

$\map P x$ is usually interpreted to mean: $x$ has the property $P$.
$\map P {x, y}$ 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 $\map P x$ is true or false, that is, whether $x$ has $P$ or not.


Sources