Uniqueness quantification

Logical property of being the one and only object satisfying a condition

In mathematics and logic, the term "uniqueness" refers to the property of being the one and only object satisfying a certain condition.[1][2] This sort of quantification is known as uniqueness quantification or unique existential quantification, and is often denoted with the symbols "!"[3] or "∃=1". For example, the formal statement

In general, both existence (there exists at least one object) and uniqueness (there exists at most one object) must be proven, in order to conclude that there exists exactly one object satisfying a said condition.

An equivalent definition that separates the notions of existence and uniqueness into two clauses, at the expense of brevity, is

The uniqueness quantification can be generalized into counting quantification (or numerical quantification[4]). This includes both quantification of the form "exactly k objects exist such that …" as well as "infinitely many objects exist such that …" and "only finitely many objects exist such that…". The first of these forms is expressible using ordinary quantifiers, but the latter two cannot be expressed in ordinary first-order logic.[5]

Uniqueness depends on a notion of equality. Loosening this to some coarser equivalence relation yields quantification of uniqueness up to that equivalence (under this framework, regular uniqueness is "uniqueness up to equality"). For example, many concepts in category theory are defined to be unique up to isomorphism.