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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. 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 languages; examples of quantifiers in English are ''for all'', ''for some'', ''many'', ''few'', ''a lot'', and ''no''; see Quantifier (linguistics) for details. ==Mathematics== Consider the following statement: : 1 · 2 = 1 + 1, and 2 · 2 = 2 + 2, and 3 · 2 = 3 + 3, ..., and 100 · 2 = 100 + 100, and ..., etc. 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'': : For each natural number ''n'', ''n'' · 2 = ''n'' + ''n''. A similar analysis applies to the disjunction, : 1 is equal to 5 + 5, or 2 is equal to 5 + 5, or 3 is equal to 5 + 5, ... , or 100 is equal to 5 + 5, or ..., etc. which can be rephrased using ''existential quantification'': : For some natural number ''n'', ''n'' is equal to 5+5. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Quantifier (logic)」の詳細全文を読む スポンサード リンク
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