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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 Denying the consequent ： ウィキペディア英語版 Modus tollens In propositional logic, ''modus tollens''〔University of North Carolina, Philosophy Department, (Logic Glossary ). Accessdate on 31 October 2007.〕〔Copi and Cohen〕〔Hurley〕〔Moore and Parker〕 (or ''modus tollendo tollens'' and also denying the consequent)〔Sanford, David Hawley. 2003. ''If P, Then Q: Conditionals and the Foundations of Reasoning''. London, UK: Routledge: 39 "() tollens is always an abbreviation for modus tollendo tollens, the mood that by denying denies."〕 (Latin for "the way that denies by denying")〔Stone, Jon R. 1996. ''Latin for the Illiterati: Exorcizing the Ghosts of a Dead Language''. London, UK: Routledge: 60.〕 is a valid argument form and a rule of inference. It is an application of the general truth that if a statement is true, then so is its contra-positive. The first to explicitly describe the argument form ''modus tollens'' were the Stoics.〔("Stanford Encyclopedia of Philosophy: ''Ancient Logic: The Stoics''" )〕 The inference rule ''modus tollens'' validates the inference from $P$ implies $Q$ and the contradictory of $Q$ to the contradictory of $P$. The ''modus tollens'' rule can be stated formally as: :$\backslash frac$ where $P\; \backslash to\; Q$ stands for the statement "P implies Q". $\backslash neg\; Q$ stands for "it is not the case that Q" (or in brief "not Q"). Then, whenever "$P\; \backslash to\; Q$" and "$\backslash neg\; Q$" each appear by themselves as a line of a proof, then "$\backslash neg\; P$" can validly be placed on a subsequent line. The history of the inference rule ''modus tollens'' goes back to antiquity.〔Susanne Bobzien (2002). ("The Development of Modus Ponens in Antiquity" ), ''Phronesis'' 47.〕 ''Modus tollens'' is closely related to ''modus ponens''. There are two similar, but invalid, forms of argument: affirming the consequent and denying the antecedent. See also contraposition and proof by contrapositive. == Formal notation == The ''modus tollens'' rule may be written in sequent notation: :$P\backslash to\; Q,\; \backslash neg\; Q\; \backslash vdash\; \backslash neg\; P$ where $\backslash vdash$ is a metalogical symbol meaning that $\backslash neg\; P$ is a syntactic consequence of $P\; \backslash to\; Q$ and $\backslash neg\; Q$ in some logical system; or as the statement of a functional tautology or theorem of propositional logic: :$((P\; \backslash to\; Q)\; \backslash and\; \backslash neg\; Q)\; \backslash to\; \backslash neg\; P$ where $P$ and $Q$ are propositions expressed in some formal system; or including assumptions: :$\backslash frac$ though since the rule does not change the set of assumptions, this is not strictly necessary. More complex rewritings involving ''modus tollens'' are often seen, for instance in set theory: :$P\backslash subseteq\; Q$ :$x\backslash notin\; Q$ :$\backslash therefore\; x\backslash notin\; P$ ("P is a subset of Q. x is not in Q. Therefore, x is not in P.") Also in first-order predicate logic: :$\backslash forall\; x:~P(x)\; \backslash to\; Q(x)$ :$\backslash exists\; x:~\backslash neg\; Q(x)$ :$\backslash therefore\; \backslash exists\; x:~\backslash neg\; P(x)$ ("For all x if x is P then x is Q. There exists some x that is not Q. Therefore, there exists some x that is not P.") Strictly speaking these are not instances of ''modus tollens'', but they may be derived from ''modus tollens'' using a few extra steps. 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「Modus tollens」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.