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In logic, Peirce's law is named after the philosopher and logician Charles Sanders Peirce. It was taken as an axiom in his first axiomatisation of propositional logic. It can be thought of as the law of excluded middle written in a form that involves only one sort of connective, namely implication. In propositional calculus, Peirce's law says that ((''P''→''Q'')→''P'')→''P''. Written out, this means that ''P'' must be true if there is a proposition ''Q'' such that the truth of ''P'' follows from the truth of "if ''P'' then ''Q''". In particular, when ''Q'' is taken to be a false formula, the law says that if ''P'' must be true whenever it implies falsity, then ''P'' is true. In this way Peirce's law implies the law of excluded middle. Peirce's law does not hold in intuitionistic logic or intermediate logics and cannot be deduced from the deduction theorem alone. Under the Curry–Howard isomorphism, Peirce's law is the type of continuation operators, e.g. call/cc in Scheme.〔(A Formulae-as-Types Notion of Control ) - Griffin defines K on page 3 as an equivalent to Scheme's call/cc and then discusses its type being the equivalent of Peirce's law at the end of section 5 on page 9.〕 ==History== Here is Peirce's own statement of the law: : A ''fifth icon'' is required for the principle of excluded middle and other propositions connected with it. One of the simplest formulae of this kind is: : This is hardly axiomatical. That it is true appears as follows. It can only be false by the final consequent ''x'' being false while its antecedent (''x'' → ''y'') → ''x'' is true. If this is true, either its consequent, ''x'', is true, when the whole formula would be true, or its antecedent ''x'' → ''y'' is false. But in the last case the antecedent of ''x'' → ''y'', that is ''x'', must be true. (Peirce, the ''Collected Papers'' 3.384). Peirce goes on to point out an immediate application of the law: : From the formula just given, we at once get: : where the ''a'' is used in such a sense that (''x'' → ''y'') → ''a'' means that from (''x'' → ''y'') every proposition follows. With that understanding, the formula states the principle of excluded middle, that from the falsity of the denial of ''x'' follows the truth of ''x''. (Peirce, the ''Collected Papers'' 3.384). Warning: ((''x''→''y'')→''a'')→''x'' is ''not'' a tautology. However, ()→() is a tautology. 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「Peirce's law」の詳細全文を読む スポンサード リンク
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