Quantifier (logic)

In logic, quantification specifies the quantity of specimens in the domain of discourse that satisfy an open formula. The two most common quantifiers mean "for all" and "there exists". For example, in arithmetic, quantifiers allows one to say that the natural numbers go on for ever, by writing that for all n (where n is a natural number), there is another number (say, the successor of n) which is one bigger than n.

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 languages; examples of quantifiers in English are for all, for some, many, few, a lot, and no; see Quantifier (linguistics) for details.

Consider the following statement:

This has the appearance of an infinite conjunction of propositions. From the point of view of formal languages this is immediately a problem, since syntax rules are expected to generate finite objects. The example above is fortunate in that there is a procedure to generate all the conjuncts. However, if an assertion were to be made about every irrational number, there would be no way to enumerate all the conjuncts, since irrationals cannot be enumerated. A succinct formulation which avoids these problems uses universal quantification:

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