# Definition:Existential Statement

## Definition

An existential statement is one which expresses the existence of at least one object (in a particular universe of discourse) which has a particular property.

That is, a statement of the form:

$\exists x: P \paren x$

where:

$\exists$ is the existential quantifier
$P$ is a predicate symbol.

It means:

There exists at least one $x$ (in some given universe of discourse) which has the property $P$.

### Bound Variable

In the existential statement:

$\exists x: \map P x$

the symbol $x$ is a bound variable.

Thus, the meaning of $\exists x: \map P x$ does not change if $x$ is replaced by another symbol.

That is, $\exists x: \map P x$ means the same thing as $\exists y: \map P y$ or $\exists \alpha: \map P \alpha$. And so on.

## Conditionally Existential Statement

A conditionally existential statement is an existential statement which states the existence of an object fulfilling a certain propositional function dependent upon the existence of certain other objects.

### Absolutely Existential Statement

An absolutely existential statement is an existential statement which states the existence of an object without that existence being dependent upon other conditions.

## Also known as

An existential statement can also be referred to as a existential sentence, or more wordily, a sentence of an existential character.