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.

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.


 * $\exists !x: P \left({x}\right) \dashv \vdash \exists x: P \left({x}\right) \land \forall y: P \left({y}\right) \implies x = y$

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) iff:
 * 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.