Quantifier (logic)

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In logic, 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 ${\displaystyle \forall xP(x)}$ expresses that everything in the domain satisfies the property denoted by ${\displaystyle P}$. On the other hand, the existential quantifier ${\displaystyle \exists }$ in the formula ${\displaystyle \exists xP(x)}$ expresses that there exists something in the domain which satisfies that property. A formula where a quantifier takes widest scope is called a quantified formula. A quantified formula must contain a bound variable and a subformula specifying a property of the referent of that variable.

The most commonly used quantifiers are ${\displaystyle \forall }$ and ${\displaystyle \exists }$. These quantifiers are standardly defined as duals; in classical logic, they are interdefinable using negation. They can also be used to define more complex quantifiers, as in the formula ${\displaystyle \neg \exists xP(x)}$ which expresses that nothing has the property ${\displaystyle P}$. Other quantifiers are only definable within second order logic or higher order logics. Quantifiers have been generalized beginning with the work of Mostowski and 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, while the exchange of quantifications in different types changes the meaning. As an example, the only difference in the definition of uniform continuity and (ordinary) continuity is the order of quantifications.

First order quantifiers approximate the meanings of some natural language quantifiers such as "some" and "all". However, many natural language quantifiers can only be analyzed in terms of generalized quantifiers.