|
In predicate logic, an existential quantification is a type of quantifier, a logical constant which is interpreted as "there exists", "there is at least one", or "for some". It is usually denoted by the turned E (∃) logical operator symbol, which, when used together with a predicate variable, is called an existential quantifier ("∃x" or "∃(x)"). Existential quantification is distinct from ''universal'' quantification ("for all"), which asserts that the property or relation holds for ''all'' members of the domain. Symbols are encoded and . == Basics == Consider a formula that states that some natural number multiplied by itself is 25. : 0·0 = 25, or 1·1 = 25, or 2·2 = 25, or 3·3 = 25, and so on. This would seem to be a logical disjunction because of the repeated use of "or". However, the "and so on" makes this impossible to integrate and to interpret as a disjunction in formal logic. Instead, the statement could be rephrased more formally as : For some natural number ''n'', ''n''·''n'' = 25. This is a single statement using existential quantification. This statement is more precise than the original one, as the phrase "and so on" does not necessarily include all natural numbers, and nothing more. Since the domain was not stated explicitly, the phrase could not be interpreted formally. In the quantified statement, on the other hand, the natural numbers are mentioned explicitly. This particular example is true, because 5 is a natural number, and when we substitute 5 for ''n'', we produce "5·5 = 25", which is true. It does not matter that "''n''·''n'' = 25" is only true for a single natural number, 5; even the existence of a single solution is enough to prove the existential quantification true. In contrast, "For some even number ''n'', ''n''·''n'' = 25" is false, because there are no even solutions. The ''domain of discourse'', which specifies which values the variable ''n'' is allowed to take, is therefore of critical importance in a statement's trueness or falseness. Logical conjunctions are used to restrict the domain of discourse to fulfill a given predicate. For example: : For some positive odd number ''n'', ''n''·''n'' = 25 is logically equivalent to : For some natural number ''n'', ''n'' is odd and ''n''·''n'' = 25. Here, "and" is the logical conjunction. In symbolic logic, "∃" (a backwards letter "E" in a sans-serif font) is used to indicate existential quantification.〔This symbol is also known as the ''existential operator''. It is sometimes represented with ''V''.〕 Thus, if ''P''(''a'', ''b'', ''c'') is the predicate "''a''·''b'' = c" and is the set of natural numbers, then : is the (true) statement : For some natural number ''n'', ''n''·''n'' = 25. Similarly, if ''Q''(''n'') is the predicate "''n'' is even", then : is the (false) statement : For some natural number ''n'', ''n'' is even and ''n''·''n'' = 25. In mathematics, the proof of a "some" statement may be achieved either by a constructive proof, which exhibits an object satisfying the "some" statement, or by a nonconstructive proof which shows that there must be such an object but without exhibiting one. 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「Existential quantification」の詳細全文を読む スポンサード リンク
|