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Rules of inference with parameters for intuitionistic logic

Published online by Cambridge University Press:  12 March 2014

Vladimir V. Rybakov*
Affiliation:
Department of Mathematics, Krasnoyarsk State University, 660062 Krasnoyarsk, USSR

Abstract

An algorithm recognizing admissibility of inference rules in generalized form (rules of inference with parameters or metavariables) in the intuitionistic calculus H and, in particular, also in the usual form without parameters, is presented. This algorithm is obtained by means of special intuitionistic Kripke models, which are constructed for a given inference rule. Thus, in particular, the direct solution by intuitionistic techniques of Friedman's problem is found. As a corollary an algorithm for the recognition of the solvability of logical equations in H and for constructing some solutions for solvable equations is obtained. A semantic criterion for admissibility in H is constructed.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1992

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References

REFERENCES

[1]Mints, G. E., Derivability of admissible rules, Zapiski Nauchnykh Seminarov L0M1, vol. 32 (1972), pp. 8599; English translation, Journal of Soviet Mathematics, vol. 6(1976), no. 4, pp. 417-421.Google Scholar
[2]Rybakov, V. V., A criterion for the admissibility of inference rules in the modal system S4 and intuitionistic logic, Algebra i Logika, vol. 23 (1984), pp. 546572; English translation, Algebra and Logic, vol. 23(1984), pp. 369-384.Google Scholar
[3]Rybakov, V. V., Decidability of admissibility in the modal system Grzand in intuitionistic logic, Izvestiya Akademii Nauk SSSR Seriya Matematicheskaya, vol. 50 (1986), pp. 598616; English translation, Mathematics of the USSR Izvestiya, vol. 28 (1987), pp. 589-608.Google Scholar
[4]Rybakov, V. V., Equations in a free topo-Boolean algebra and the substitution problem, Doklady Akademii Nauk SSSR, vol. 287 (1986), pp. 554557; English translation, Soviet Mathematics Doklady, vol. 33 (1986), pp. 428-431.Google Scholar
[5]Rybakov, V. V., The elementary theories of free topo-Boolean and pseudo-Boolean algebras, Matematicheskie Zametki, vol. 37 (1985), pp. 797802; English translation, Mathematical Notes of the Academy of Sciences of the USSR, vol. 37 (1985), pp. 435-438.Google Scholar
[6]Tsitkin, A. I., On admissible rules of intuitionistic prepositional logic, Matematicheskiĭ Sbornik, vol. 102 (144) (1977), pp. 314323; English translation in Mathematics of the USSR Sbornik, vol. 31 (1977), pp. 279-288.Google Scholar
[7]Shekhtman, V. B., Rieger-Nishimura lattices, Doklady Akademii Nauk SSSR, vol. 241 (1978), pp. 12881291; English translation, Soviet Mathematics Doklady, vol. 19 (1978), pp. 1014-1018.Google Scholar
[8]Schütte, Kurt, Vollständige Systeme modaler und intuitionistischer Logik, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 42, Springer-Verlag, Berlin, 1968.CrossRefGoogle Scholar
[9]Ershov, Yu. L. and Goncharov, S. S. (editors), Logical notebook: unsolved questions of mathematical logic, Institute of Mathematics, Siberian Branch, Academy of Sciences of the USSR, Novosibirsk, 1986. (Russian)Google Scholar
[10]Friedman, Harvey, One hundred and two problems in mathematical logic, this Journal, vol. 40 (1975), pp. 113129.Google Scholar