# 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**.

## Also see

## Sources

- 1946: Alfred Tarski:
*Introduction to Logic and to the Methodology of Deductive Sciences*(2nd ed.) ... (previous) ... (next): $\S 1.3$: Universal and Existential Sentences - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 3$: Statements and conditions; quantifiers