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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 Admissible rule ： ウィキペディア英語版 Admissible rule In logic, a rule of inference is admissible in a formal system if the set of theorems of the system does not change when that rule is added to the existing rules of the system. In other words, every formula that can be derived using that rule is already derivable without that rule, so, in a sense, it is redundant. The concept of an admissible rule was introduced by Paul Lorenzen (1955). ==Definitions== Admissibility has been systematically studied only in the case of structural rules in propositional non-classical logics, which we will describe next. Let a set of basic propositional connectives be fixed (for instance, $\backslash$ in the case of superintuitionistic logics, or $\backslash$ in the case of monomodal logics). Well-formed formulas are built freely using these connectives from a countably infinite set of propositional variables ''p''''n''. A substitution σ is a function from formulas to formulas which commutes with the connectives, i.e., :$\backslash sigma\; f(A\_1,\backslash dots,A\_n)=f(\backslash sigma\; A\_1,\backslash dots,\backslash sigma\; A\_n)$ for every connective ''f'', and formulas ''A''1, …, ''A''''n''. (We may also apply substitutions to sets Γ of formulas, making ) A Tarski-style consequence relation〔Blok & Pigozzi (1989), Kracht (2007)〕 is a relation $\backslash vdash$ between sets of formulas, and formulas, such that for all formulas ''A'', ''B'', and sets of formulas Γ, Δ. A consequence relation such that for all substitutions σ is called structural. (Note that the term "structural" as used here and below is unrelated to the notion of structural rules in sequent calculi.) A structural consequence relation is called a propositional logic. A formula ''A'' is a theorem of a logic $\backslash vdash$ if $\backslash varnothing\backslash vdash\; A$. For example, we identify a superintuitionistic logic ''L'' with its standard consequence relation $\backslash vdash\_L$ axiomatizable by modus ponens and axioms, and we identify a normal modal logic with its global consequence relation $\backslash vdash\_L$ axiomatized by modus ponens, necessitation, and axioms. A structural inference rule〔Rybakov (1997), Def. 1.1.3〕 (or just rule for short) is given by a pair (Γ,''B''), usually written as :$\backslash fracB\backslash qquad\backslash text\backslash qquad\; A\_1,\backslash dots,A\_n/B,$ where Γ =  is a finite set of formulas, and ''B'' is a formula. An instance of the rule is :$\backslash sigma\; A\_1,\backslash dots,\backslash sigma\; A\_n/\backslash sigma\; B$ for a substitution σ. The rule Γ/''B'' is derivable in $\backslash vdash$, if $\backslash Gamma\backslash vdash\; B$. It is admissible if for every instance of the rule, σ''B'' is a theorem whenever all formulas from σΓ are theorems.〔Rybakov (1997), Def. 1.7.2〕 In other words, a rule is admissible if, when added to the logic, does not lead to new theorems.〔(From de Jongh’s theorem to intuitionistic logic of proofs )〕 We also write $\backslash Gamma\backslash ,|\backslash !\backslash !\backslash !\backslash sim\; B$ if Γ/''B'' is admissible. (Note that $|\backslash !\backslash !\backslash !\backslash sim$ is a structural consequence relation on its own.) Every derivable rule is admissible, but not vice versa in general. A logic is structurally complete if every admissible rule is derivable, i.e., $=$.〔Rybakov (1997), Def. 1.7.7〕 In logics with a well-behaved conjunction connective (such as superintuitionistic or modal logics), a rule $A\_1,\backslash dots,A\_n/B$ is equivalent to $A\_1\backslash land\backslash dots\backslash land\; A\_n/B$ with respect to admissibility and derivability. It is therefore customary to only deal with unary rules ''A''/''B''. 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「Admissible rule」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.