Definition:Propositional Function

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Definition

A propositional function $\map P {x_1, x_2, \ldots}$ is an operation 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 $\map P {x_1, x_2, \ldots, x_n}$ be an $n$-ary propositional function.

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


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.


Examples

Let the universe be the set of integers $\Z$.

Let $\map P x$ be the propositional function defined as:

$x$ is even

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

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

Thus $\map P x$ is a unary propositional function (pronounced yoo-nary).

$\Box$


Let $\map P {x, y}$ be the propositional function defined as:

$x$ is less than $y$

Then we can create the propositional statements:

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

Thus $\map P {x, y}$ is a binary propositional function .

$\Box$


Let $\map P {x, y, z}$ be the propositional function defined as:

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

Then:

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

Thus $\map P {x, y, z}$ is a ternary propositional function .

$\Box$


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