Quantifier (logic)

In mathematical logic, quantifiers are formal counterparts of natural-language adjectives like all, some, most, few, etc. which indicate the number of objects satisfying a given property. More precisely, a quantifier is an operator that specifies how many individuals in the domain of discourse satisfy an open formula. For instance, the universal quantifier${\displaystyle \forall }$ in the first-order formula${\displaystyle \forall x\;x\geq 0}$ expresses that all numbers in the domain are non-negative; this formula is true for the natural numbers domain, but false for the integer domain. On the other hand, the existential quantifier${\displaystyle \exists }$ in the formula ${\displaystyle \exists x\;x^{2}-5x+6=0}$ expresses that some numbers in the domain satisfy the given quadratic equation; indeed, both 2 and 3 do.

The most commonly used quantifiers are ${\displaystyle \forall }$ and ${\displaystyle \exists }$. Other quantifiers are only definable within second-order logic or higher-order logics. Quantifiers have been generalized beginning with the work of Andrzej Mostowski and Per Lindström.

In a first-order logic statement, quantifications in the same type (either universal quantifications or existential quantifications) can be exchanged without changing the meaning of the statement. In contrast, the exchange of quantifications in different types changes the meaning. As an example for the latter, the only difference in the definition of uniform continuity and (ordinary) continuity is the order of quantifications.