Geodesic
Criteria for admissibility of rules of inference with parameters in the intuituonistc propositional calculus
Izvestiya. Mathematics , Tome 37 (1991) no. 3, pp. 693-703.

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The author studies the problem of recognizing admissibility of rules of inference of a general form – rules of inference with parameters – in the intuitionistic propositional calculus. Algorithmic and semantic criteria are found for recognizing admissibility of such rules, based on special intuitionistic Kripke models. In particular, Friedman's problem generalized to rules of inference with parameters is solved. These results are applied to recognizing the solvability of logical equations and constructing their solutions. 
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V. V. Rybakov. Criteria for admissibility of rules of inference with parameters in the intuituonistc propositional calculus. Izvestiya. Mathematics , Tome 37 (1991) no. 3, pp. 693-703. http://geodesic.mathdoc.fr/item/IM2_1991_37_3_a11/

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