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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 Brouwer–Heyting–Kolmogorov interpretation ： ウィキペディア英語版 Brouwer–Heyting–Kolmogorov interpretation In mathematical logic, the Brouwer–Heyting–Kolmogorov interpretation, or BHK interpretation, of intuitionistic logic was proposed by L. E. J. Brouwer, Arend Heyting and independently by Andrey Kolmogorov. It is also sometimes called the realizability interpretation, because of the connection with the realizability theory of Stephen Kleene. == The interpretation == The interpretation states exactly what is intended to be a proof of a given formula. This is specified by induction on the structure of that formula: *A proof of $P\; \backslash wedge\; Q$ is a pair <''a'', ''b''> where ''a'' is a proof of ''P'' and ''b'' is a proof of ''Q''. *A proof of $P\; \backslash vee\; Q$ is a pair <''a'', ''b''> where ''a'' is 0 and ''b'' is a proof of ''P'', or ''a'' is 1 and ''b'' is a proof of ''Q''. *A proof of $P\; \backslash to\; Q$ is a function ''f'' that converts a proof of ''P'' into a proof of ''Q''. *A proof of $\backslash exists\; x\; \backslash in\; S\; :\; \backslash varphi(x)$ is a pair <''a'', ''b''> where ''a'' is an element of ''S'', and ''b'' is a proof of ''φ(a)''. *A proof of $\backslash forall\; x\; \backslash in\; S\; :\; \backslash varphi(x)$ is a function ''f'' that converts an element ''a'' of ''S'' into a proof of ''φ(a)''. *The formula $\backslash neg\; P$ is defined as $P\; \backslash to\; \backslash bot$, so a proof of it is a function ''f'' that converts a proof of ''P'' into a proof of $\backslash bot$. *There is no proof of $\backslash bot$ (the absurdity). The interpretation of a primitive proposition is supposed to be known from context. In the context of arithmetic, a proof of the formula ''s''=''t'' is a computation reducing the two terms to the same numeral. Kolmogorov followed the same lines but phrased his interpretation in terms of problems and solutions. To assert a formula is to claim to know a solution to the problem represented by that formula. For instance $P\; \backslash to\; Q$ is the problem of reducing ''Q'' to ''P''; to solve it requires a method to solve problem ''Q'' given a solution to problem ''P''. 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「Brouwer–Heyting–Kolmogorov interpretation」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.