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 Mat. Sb. (N.S.), 1985, Volume 128(170), Number 3(11), Pages 321–338 (Mi msb2162)

Bases of admissible rules of the modal system Grz and of intuitionistic logic

V. V. Rybakov

Abstract: It is proved that the free pseudoboolean algebra $F_\omega(\mathrm{Int})$ and the free topoboolean algebra $F_\omega(\mathrm{Grz})$ do not have bases of quasi-identities in a finite number of variables. A corollary is that the intuitionistic propositional logic $\mathrm{Int}$ and the modal system $\mathrm{Grz}$ do not have finite bases of admissible rules. Infinite recursive bases of quasi-identities are found for $F_\omega(\mathrm{Int})$ and $F_\omega(\mathrm{Grz})$. This implies that the problem of admissibility of rules in the logics $\mathrm{Grz}$ and $\mathrm{Int}$ is algorithmically decidable.
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English version:
Mathematics of the USSR-Sbornik, 1987, 56:2, 311–331

Bibliographic databases:

UDC: 510.6+512
MSC: Primary 03B45; Secondary 03F55

Citation: V. V. Rybakov, “Bases of admissible rules of the modal system Grz and of intuitionistic logic”, Mat. Sb. (N.S.), 128(170):3(11) (1985), 321–338; Math. USSR-Sb., 56:2 (1987), 311–331

Citation in format AMSBIB
\Bibitem{Ryb85} \by V.~V.~Rybakov \paper Bases of admissible rules of the modal system Grz and of intuitionistic logic \jour Mat. Sb. (N.S.) \yr 1985 \vol 128(170) \issue 3(11) \pages 321--338 \mathnet{http://mi.mathnet.ru/msb2162} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=815267} \zmath{https://zbmath.org/?q=an:0617.03007} \transl \jour Math. USSR-Sb. \yr 1987 \vol 56 \issue 2 \pages 311--331 \crossref{https://doi.org/10.1070/SM1987v056n02ABEH003038} 

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. V. V. Rybakov, “Admissibility of rules of inference, and logical equations, in modal logics axiomatizing provability”, Math. USSR-Izv., 36:2 (1991), 369–390
2. V.V. Rybakov, “Problems of substitution and admissibility in the modal system Grz and in intuitionistic propositional calculus”, Annals of Pure and Applied Logic, 50:1 (1990), 71
3. Rybakov V., “Admission of the Inherence Rules with Parameters for the Intuitionistic Logics and the Intuitionistic Kripke Models”, 312, no. 1, 1990, 42–45
4. Ronald Fagin, J.Y.. Halpern, M.Y.. Vardi, “What is an inference rule?”, J. symb. log, 57:03 (1992), 1018
5. Vladimir V. Rybakov, “Intermediate logics preserving admissible inference rules of heyting calculus”, MLQ-Math Log Quart, 39:1 (1993), 403
6. B. R. Fedorishin, “An explicit basis for the admissible inference rules in the Gödel–Löb logic $GL$”, Siberian Math. J., 48:2 (2007), 339–345
7. V. V. Rimatskii, “An explicit basis for admissible inference rules in table modal logics of width 2”, Algebra and Logic, 48:1 (2009), 72–86
8. V. V. Rimatskii, “Table admissible inference rules”, Algebra and Logic, 48:3 (2009), 228–236
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