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. 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.
**See also:** one and only one. |