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:
- $\exists ! x: \map P x$
- $\exists ! x: \map P x \dashv \vdash \exists x: \map P x \land \forall y: \map P y \implies x = y$
In natural language, this means:
- is logically equivalent to:
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 $x$
- 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.
- 1946: Alfred Tarski: Introduction to Logic and to the Methodology of Deductive Sciences (2nd ed.): $\S 4.26$
- 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 6.10$