# 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:
There exists an $x$ such that $x$ has the property $P$, and for every $y$, $y$ has the property $P$ only if $x$ and $y$ are the same object.

### 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.