Adjective

unique (comparable only in disputed usage)

1. Being the only one of its kind; unequaled, unparalleled or unmatched.
   • 1978, Jimmy Carter, Proclamation 4611:

     Admiralty Island contains unique resources of scientific interest which need protection to assure continued opportunities for study.

   • 2002, The American Practical Navigator:

     GPS assigns a unique C/A code and a unique P code to each satellite.

   • 1941, Allen v. Walt Disney:

     3. Both were written and published with the same unique chorus structure;

     4. Both compositions were written and published with the same unique harmonic structure;

   • 1920, Robert W. Lawson, Relativity: The Special and General Theory, translation of original by Albert Einstein:

     Perhaps the reader will wonder why we have placed our " beings " on a sphere rather than on another closed surface. But this choice has its justification in the fact that, of all closed surfaces, the sphere is unique in possessing the property that all points on it are equivalent.

2. Of a feature, such that only one holder has it.
3. (disputed) Of a rare quality.
4. (disputed) Unusual.

In mathematics and logic, the phrase "there is one and only one" is used to indicate that exactly one object with a certain property exists. In mathematical logic, this sort of quantification is known as uniqueness quantification or unique existential quantification.

Uniqueness quantification is often denoted with the symbols "∃!" or ∃₌₁". For example, the formal statement

may be read aloud as "there is exactly one natural number n such that n - 2 = 4".

Reduction to ordinary existential and universal quantification

Uniqueness quantification can be expressed in terms of the existential and universal quantifiers of predicate logic by defining the formula ∃!x P(x) to mean

.

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

.

Another equivalent definition with the advantage of brevity is

.