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.

Thus, intuitively, an object is unique if there is only one of it.

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.


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

means:


 * That there exists exactly one $x$ with the property $P$ is logically equivalent to saying that there exists an $x$ with the property $P$ such that $x$ and $y$ have the property $P$ if and only if $x$ and $y$ are the same object.