# Definition:Existential Quantifier/Unique

## Definition

The symbol $\exists !$ denotes the existence of a unique object fulfilling a particular condition.

- $\exists ! x: \map P x$

means:

**There exists exactly one object $x$ such that $\map P x$ holds**

or:

**There exists one and only one $x$ such that $\map P x$ holds**.

This quantifier is called the **unique existential quantifier**.

### Definition 1

There exists a unique object $x$ such that $\map P x$, denoted $\exists ! x: \map P x$, if and only if:

- $\exists x : \paren {\map P x \land \forall y : \paren {\map P y \implies x = y} }$

In natural language, this means:

*There exists exactly one $x$ with the property $P$*

- is logically equivalent to:

### Definition 2

There exists a unique object $x$ such that $\map P x$, denoted $\exists ! x: \map P x$, if and only if:

- $\exists x : \forall y : \paren {\map P y \iff x = y}$

### Definition 3

There exists a unique object $x$ such that $\map P x$, denoted $\exists ! x: \map P x$, if and only if both:

- $\exists x : \map P x$

and:

- $\forall y : \forall z : \paren {\paren {\map P y \land \map P z} \implies y = z }$

The symbol $\exists !$ is a variant of the existential quantifier $\exists$: **there exists at least one**.

## Also denoted as

The symbol $\exists_1$ is also found for the same concept, being an instance of the exact existential quantifier $\exists_n$.

Some sources, for example 1972: Patrick Suppes: *Axiomatic Set Theory*, use $\operatorname E !$, which is idiosyncratic, considering the use in the same source of $\exists$ for the general existential quantifier.

## Also known as

Some sources refer to this as the **uniqueness quantifier**.

## Also see

- Results about
**the unique existential quantifier**can be found**here**.

## Sources

- 1972: A.G. Howson:
*A Handbook of Terms used in Algebra and Analysis*... (previous) ... (next): $\S 1$: Some mathematical language: Variables and quantifiers - 1972: Patrick Suppes:
*Axiomatic Set Theory*(2nd ed.) ... (previous) ... (next): $\S 1.2$ Logic and Notation