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In predicate logic, a universal quantification is a type of quantifier, a logical constant which is interpreted as "given any" or "for all". It expresses that a propositional function can be satisfied by every member of a domain of discourse. In other words, it is the predication of a property or relation to every member of the domain. It asserts that a predicate within the scope of a universal quantifier is true of every value of a predicate variable. It is usually denoted by the turned A (∀) logical operator symbol, which, when used together with a predicate variable, is called a universal quantifier ("∀x", "∀(x)", or sometimes by "(x)" alone). Universal quantification is distinct from ''existential'' quantification ("there exists"), which asserts that the property or relation holds only for at least one member of the domain. Quantification in general is covered in the article on quantification (logic). Symbols are encoded . == Basics == Suppose it is given that 2·0 = 0 + 0, and 2·1 = 1 + 1, and 2·2 = 2 + 2, etc. This would seem to be a logical conjunction because of the repeated use of "and". However, the "etc." cannot be interpreted as a conjunction in formal logic. Instead, the statement must be rephrased: For all natural numbers ''n'', 2·''n'' = ''n'' + ''n''. This is a single statement using universal quantification. This statement can be said to be more precise than the original one. While the "etc." informally includes natural numbers, and nothing more, this was not rigorously given. In the universal quantification, on the other hand, the natural numbers are mentioned explicitly. This particular example is true, because any natural number could be substituted for ''n'' and the statement "2·''n'' = ''n'' + ''n''" would be true. In contrast, For all natural numbers ''n'', 2·''n'' > 2 + ''n'' is false, because if ''n'' is substituted with, for instance, 1, the statement "2·1 > 2 + 1" is false. It is immaterial that "2·''n'' > 2 + ''n''" is true for ''most'' natural numbers ''n'': even the existence of a single counterexample is enough to prove the universal quantification false. On the other hand, for all composite numbers ''n'', 2·''n'' > 2 + ''n'' is true, because none of the counterexamples are composite numbers. This indicates the importance of the ''domain of discourse'', which specifies which values ''n'' can take.〔Further information on using domains of discourse with quantified statements can be found in the Quantification (logic) article.〕 In particular, note that if the domain of discourse is restricted to consist only of those objects that satisfy a certain predicate, then for universal quantification this requires a logical conditional. For example, For all composite numbers ''n'', 2·''n'' > 2 + ''n'' is logically equivalent to For all natural numbers ''n'', if ''n'' is composite, then 2·''n'' > 2 + ''n''. Here the "if ... then" construction indicates the logical conditional. 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「Universal quantification」の詳細全文を読む スポンサード リンク
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