Definition:Unique

Definition

Suppose $A$ and $B$ are two objects whose definition is in terms of a given set of properties.

If it can be demonstrated that, in order for both $A$ and $B$ to fulfil those properties, it is necessary for $A$ to be equal to $B$, then $A$ (and indeed $B$) is unique.

Equivalently, there is one and only one, or exactly one, such object.

Thus, intuitively, an object is unique if there is precisely one such object.

Unique Existential Quantifier

In the language of predicate logic, uniqueness can be defined as follows:

Let $P \left({x}\right)$ be a propositional function and let $x$ and $y$ be objects.

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.

Formally:

$\exists ! x: \map P x \dashv \vdash \exists x: \map P x \land \forall y: \map P y \implies x = y$

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.

Also known as

Uniqueness can also be defined as:

An object $x$ is unique (in a given context) if and only if:
there exists at most one $x$
and:
there exists at least one $x$.

Thus the phrase at most and at least one can occasionally be seen to mean unique.

Such a definition can be a useful technique for proving uniqueness.