Uniqueness quantification

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For definition of the word unique outside of mathematics, see wiktionary:unique.

In predicate logic and technical fields that depend on it, uniqueness quantification, or unique existential quantification, is an attempt to formalise the notion of something being true for exactly one thing, or exactly one thing of a certain type.

Uniqueness quantification is a kind of quantification; more information about quantification in general is in the Quantification article. This article deals with the ideas peculiar to uniqueness quantification. A generalization of uniqueness quantification is counting quantification.

For example:

There is exactly one natural number x such that x - 2 = 4.

Symbolically, this can be written:

∃!x in N, x - 2 = 4

The symbol "∃!" is called the uniqueness quantifier, or unique existential quantifier. It is usually read "there exists one and only one", or "there exists a unique" (Several variations on the grammar for this symbol exist, as well as for how it's read.)

Uniqueness quantification is usually thought of as a combination of universal quantification ("for all", "∀"), existential quantification ("for some", "∃"), and equality ("equals", "="). Thus if P(x) is the predicate being quantified over (in our example above, P(x) is "x - 2 = 4"), then ∃!x, P(x) means:

(∃a, P(a) ∧ (∀b, P(b) → (a = b)))

In words:

For some a, P(a) and for all b, if P(b), then a equals b.

Or even more succinctly:

For some a such that P(a), for all b such that P(b), a equals b.

Here, a is the unique object such that P(a); it exists, and furthermore, if any other object b also satisfies P(b), then b must be that same unique object a.

The statement that exactly one x exists such that P(x) can also be seen as a logical conjunction of two weaker statements:

For at least one x, P(x); and
For at most one x, P(x).

The 1st of these is simply existential quantification; ∃x, P(x). The 2nd is uniqueness without existence, sometimes written !x, P(x). This is defined as:

∀a, ∀b, P(a) ∧ P(b) → a = b

The conjunction of these statements is logically equivalent to the single statement given earlier. But in practice, proving unique existence is often done by proving these two separate statements.