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In propositional logic, ''modus ponendo ponens'' (Latin for "the way that affirms by affirming"; generally abbreviated to MP or ''modus ponens''〔Copi and Cohen〕〔Hurley〕〔Moore and Parker〕) or implication elimination is a valid, simple argument form and rule of inference.〔Enderton 2001:110〕 It can be summarized as "''P'' implies ''Q''; ''P'' is asserted to be true, so therefore ''Q'' must be true." The history of ''modus ponens'' goes back to antiquity.〔Susanne Bobzien (2002). The Development of Modus Ponens in Antiquity, ''Phronesis'' 47, No. 4, 2002.〕 While ''modus ponens'' is one of the most commonly used concepts in logic it must not be mistaken for a logical law; rather, it is one of the accepted mechanisms for the construction of deductive proofs that includes the "rule of definition" and the "rule of substitution".〔Alfred Tarski 1946:47. Also Enderton 2001:110ff.〕 ''Modus ponens'' allows one to eliminate a conditional statement from a logical proof or argument (the antecedents) and thereby not carry these antecedents forward in an ever-lengthening string of symbols; for this reason modus ponens is sometimes called the rule of detachment.〔Tarski 1946:47〕 Enderton, for example, observes that "modus ponens can produce shorter formulas from longer ones",〔Enderton 2001:111〕 and Russell observes that "the process of the inference cannot be reduced to symbols. Its sole record is the occurrence of ⊦q (consequent ) . . . an inference is the dropping of a true premise; it is the dissolution of an implication".〔Whitehead and Russell 1927:9〕 A justification for the "trust in inference is the belief that if the two former assertions (antecedents ) are not in error, the final assertion (consequent ) is not in error".〔Whitehead and Russell 1927:9〕 In other words: if one statement or proposition implies a second one, and the first statement or proposition is true, then the second one is also true. If ''P'' implies ''Q'' and ''P'' is true, then ''Q'' is true. An example is: :If it is raining, I will meet you at the theater. :It is raining. :Therefore, I will meet you at the theater. ''Modus ponens'' can be stated formally as: : where the rule is that whenever an instance of "''P'' → ''Q''" and "''P''" appear by themselves on lines of a logical proof, ''Q'' can validly be placed on a subsequent line; furthermore, the premise ''P'' and the implication "dissolves", their only trace being the symbol ''Q'' that is retained for use later e.g. in a more complex deduction. It is closely related to another valid form of argument, ''modus tollens''. Both have apparently similar but invalid forms such as affirming the consequent, denying the antecedent, and evidence of absence. Constructive dilemma is the disjunctive version of modus ponens. Hypothetical syllogism is closely related to modus ponens and sometimes thought of as "double modus ponens." == Formal notation == The ''modus ponens'' rule may be written in sequent notation: : where ⊢ is a metalogical symbol meaning that ''Q'' is a syntactic consequence of ''P'' → ''Q'' and ''P'' in some logical system; or as the statement of a truth-functional tautology or theorem of propositional logic: : where ''P'', and ''Q'' are propositions expressed in some formal system. 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 ponens」の詳細全文を読む スポンサード リンク
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