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Criteria for admissibility of rules of inference with parameters in the intuitionistic propositional calculus. (English. Russian original) Zbl 0743.03037
Math. USSR, Izv. 37, No. 3, 693-703 (1991); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 54, No. 6, 1331-1341 (1990).
This is a refinement of the author’s solution of a problem raised by H. Friedman: admissibility of rules with parameters in the intuitionistic propositional calculus is decidable. The rule $$A(x;p)/B(x;p)$$ with parameters $$p$$, where $$x=x_ 1,\dots,x_ m$$, $$p=p_ 1,\dots,p_ n$$, is admissible if for any formulas $$F_ 1,\dots,F_ m=F$$, derivability of $$A(F;p)$$ implies derivability of $$B(F;p)$$. If $$B$$=false, the rule is inadmissible exactly when the “logical equation” $$A(x;p)$$ is solvable for $$x$$. The proof uses reduction to finite intuitionistic Kripke models without detour via modal logic [the author, Algebra Logika 25, No. 2, 172-204 (1986; Zbl 0624.03007) and Ann. Pure Appl. Logic 50, No. 1, 71- 106 (1990; Zbl 0709.03009)].
Reviewer: G.Mints (Stanford)
##### MSC:
 03F50 Metamathematics of constructive systems
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