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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 Interpretability logic ： ウィキペディア英語版 Interpretability logic Interpretability logics comprise a family of modal logics that extend provability logic to describe interpretability or various related metamathematical properties and relations such as weak interpretability, Π1-conservativity, cointerpretability, tolerance, cotolerance, and arithmetic complexities. Main contributors to the field are Alessandro Berarducci, Petr Hájek, Konstantin Ignatiev, Giorgi Japaridze, Franco Montagna, Vladimir Shavrukov, Rineke Verbrugge, Albert Visser, and Domenico Zambella. == Examples == * Logic ILM: The language of ILM extends that of classical propositional logic by adding the unary modal operator $\backslash Box$ and the binary modal operator $\backslash triangleright$ (as always, $\backslash Diamond\; p$ is defined as $\backslash neg\; \backslash Box\backslash neg\; p$). The arithmetical interpretation of $\backslash Box\; p$ is “$p$ is provable in Peano Arithmetic PA”, and $p\; \backslash triangleright\; q$ is understood as “$PA+q$ is interpretable in $PA+p$”. Axiom schemata: 1. All classical tautologies 2. $\backslash Box(p\; \backslash rightarrow\; q)\; \backslash rightarrow\; (\backslash Box\; p\; \backslash rightarrow\; \backslash Box\; q)$ 3. $\backslash Box(\backslash Box\; p\; \backslash rightarrow\; p)\; \backslash rightarrow\; \backslash Box\; p$ 4. $\backslash Box\; (p\; \backslash rightarrow\; q)\; \backslash rightarrow\; (p\; \backslash triangleright\; q)$ 5. $(p\; \backslash triangleright\; q)\backslash wedge\; (q\; \backslash triangleright\; r)\backslash rightarrow\; (p\backslash triangleright\; r)$ 6. $(p\; \backslash triangleright\; r)\backslash wedge\; (q\; \backslash triangleright\; r)\backslash rightarrow\; ((p\backslash vee\; q)\backslash triangleright\; r)$ 7. $(p\; \backslash triangleright\; q)\backslash rightarrow\; (\backslash Diamond\; p\backslash triangleright\backslash Diamond\; q)$ 8. $\backslash Diamond\; p\; \backslash triangleright\; p$ 9. $(p\; \backslash triangleright\; q)\backslash rightarrow((p\backslash wedge\backslash Box\; r)\backslash triangleright\; (q\backslash wedge\backslash Box\; r))$ Rules inference: 1. “From $p$ and $p\backslash rightarrow\; q$ conclude $q$” 2. “From $p$ conclude $\backslash Box\; p$”. The completeness of ILM with respect to its arithmetical interpretation was independently proven by Alessandro Berarducci and Vladimir Shavrukov. * Logic TOL: The language of TOL extends that of classical propositional logic by adding the modal operator $\backslash Diamond$ which is allowed to take any nonempty sequence of arguments. The arithmetical interpretation of $\backslash Diamond(\; p\_1,\backslash ldots,p\_n)$ is “$(PA+p\_1,\backslash ldots,PA+p\_n)$ is a tolerant sequence of theories”. Axioms (with $p,q$ standing for any formulas, $\backslash vec,\backslash vec$ for any sequences of formulas, and $\backslash Diamond()$ identified with ⊤): 1. All classical tautologies 2. $\backslash Diamond\; (\backslash vec,p,\backslash vec)\backslash rightarrow\; \backslash Diamond\; (\backslash vec,\; p\backslash wedge\backslash neg\; q,\backslash vec)\backslash vee\; \backslash Diamond\; (\backslash vec,\; q,\backslash vec)$ 3. $\backslash Diamond\; (p)\backslash rightarrow\; \backslash Diamond\; (p\backslash wedge\; \backslash neg\backslash Diamond\; (p))$ 4. $\backslash Diamond\; (\backslash vec,p,\backslash vec)\backslash rightarrow\; \backslash Diamond\; (\backslash vec,\backslash vec)$ 5. $\backslash Diamond\; (\backslash vec,p,\backslash vec)\backslash rightarrow\; \backslash Diamond\; (\backslash vec,p,p,\backslash vec)$ 6. $\backslash Diamond\; (p,\backslash Diamond(\backslash vec))\backslash rightarrow\; \backslash Diamond\; (p\backslash wedge\backslash Diamond(\backslash vec))$ 7. $\backslash Diamond\; (\backslash vec,\backslash Diamond(\backslash vec))\backslash rightarrow\; \backslash Diamond\; (\backslash vec,\backslash vec)$ Rules inference: 1. “From $p$ and $p\backslash rightarrow\; q$ conclude $q$” 2. “From $\backslash neg\; p$ conclude $\backslash neg\; \backslash Diamond(\; p)$”. The completeness of TOL with respect to its arithmetical interpretation was proven by Giorgi Japaridze. 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「Interpretability logic」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.