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In [[logic]], '''quantification''' is a construct that specifies the quantity of specimens in the [[domain of discourse]] that satisfy an open formula.

== Description ==

For example, in [[arithmetic]], it allows the expression of the statement that every natural number has a successor.

A language element which generates a quantification (such as "every") is called a quantifier. The resulting expression is a quantified expression, it is said to be quantified over the predicate (such as "the natural number x has a successor") whose free variable is bound by the quantifier.

In [[formal languages]], quantification is a formula constructor that produces new formulas from old ones. The semantics of the language specifies how the constructor is interpreted.

Two fundamental kinds of quantification in predicate logic are universal quantification and existential quantification.

The traditional symbol for the universal quantifier "all" is "∀", a rotated letter "A", and for the existential quantifier "exists" is "∃", a rotated letter "E". These quantifiers have been generalized beginning with the work of [[Mostowski]] and [[Lindström]].

Quantification is used as well in [[Natural language|natural languages]]; examples of quantifiers in English are for all, for some, many, few, a lot, and no; see [[Quantifier (linguistics)]] for details.

== See also ==

* [[Formal languages]]
* [[Logic]]
* [[Generalized quantifier]] — a higher-order property used as standard semantics of quantified [[noun phrases]]
* [[Lindström quantifier]] — a generalized [[polyadic quantifier]]
* [[Quantifier (linguistics)]]
* [[Quantifier elimination]]

== External links ==

* [https://en.wikipedia.org/wiki/Quantifier_(logic) Quantifier (logic)]

[[Category:Logic]]
[[Category:Mathematical logic]]