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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, \ in the case of superintuitionistic logics, or \ 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.,
:\sigma f(A_1,\dots,A_n)=f(\sigma A_1,\dots,\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 \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 \vdash if \varnothing\vdash A.
For example, we identify a superintuitionistic logic ''L'' with its standard consequence relation \vdash_L axiomatizable by modus ponens and axioms, and we identify a normal modal logic with its global consequence relation \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
:\fracB\qquad\text\qquad A_1,\dots,A_n/B,
where Γ =  is a finite set of formulas, and ''B'' is a formula. An instance of the rule is
:\sigma A_1,\dots,\sigma A_n/\sigma B
for a substitution σ. The rule Γ/''B'' is derivable in \vdash, if \Gamma\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 \Gamma\,|\!\!\!\sim B if Γ/''B'' is admissible. (Note that |\!\!\!\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,\dots,A_n/B is equivalent to A_1\land\dots\land A_n/B with respect to admissibility and derivability. It is therefore customary to only deal with unary rules ''A''/''B''.

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