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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Vacuous truth ： ウィキペディア英語版 Vacuous truth A vacuous truth is a statement that asserts that all members of the empty set have a certain property. For example, the statement "all cell phones in the room are turned off" may be true simply because there are no cell phones in the room. In this case, the statement "all cell phones in the room are turned ''on''" would also be vacuously true, as would the conjunction of the two: "all cell phones in the room are turned on ''and'' turned off". More formally, a relatively well-defined usage refers to a conditional statement with a false antecedent. One example of such a statement is "if Uluru is in France, then the Eiffel Tower is in Bolivia". Such statements are considered vacuous because the fact that the antecedent is false prevents using the statement to infer anything about the truth value of the consequent. They are true because a material conditional is defined to be true when the antecedent is false (regardless of whether the conclusion is true). In pure mathematics, vacuously true statements are not generally of interest by themselves, but they frequently arise as the base case of proofs by mathematical induction.〔.〕 This notion has relevance as well as in any other field which uses classical logic. Outside of mathematics, statements which can be characterized informally as vacuously true can be misleading. Such statements make reasonable assertions about qualified objects which do not actually exist. For example, a child might tell his or her parent "I ate every vegetable on my plate", when there were no vegetables on the child's plate to begin with. == Scope of the concept == A statement $S$ is "vacuously true" if it resembles the statement $P\; \backslash Rightarrow\; Q$, where $P$ is known to be false. Statements that can be reduced (with suitable transformations) to this basic form include the following universally quantified statements: * $\backslash forall\; x:\; P(x)\; \backslash Rightarrow\; Q(x)$, where it is the case that $\backslash forall\; x:\; \backslash neg\; P(x)$. * $\backslash forall\; x\; \backslash in\; A:\; Q(x)$, where the set $A$ is empty. * $\backslash forall\; \backslash xi:\; Q(\backslash xi)$, where the symbol $\backslash xi$ is restricted to a type that has no representatives. Vacuous truth most commonly appears in classical logic, which in particular is two-valued. However, vacuous truth also appears in, for example, intuitionistic logic in the same situations given above. Indeed, if $P$ is false, $P\; \backslash Rightarrow\; Q$ will yield vacuous truth in any logic that uses the material conditional; if $P$ is a necessary falsehood, then it will also yield vacuous truth under the strict conditional. Other non-classical logics (for example, relevance logic) may attempt to avoid vacuous truths by using alternative conditionals (for example, the counterfactual 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「Vacuous truth」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.