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Logic and Logical Philosophy

The Kuznetsov-Gerčiu and Rieger-Nishimura logics
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The Kuznetsov-Gerčiu and Rieger-Nishimura logics

Authors

  • Guram Bezhanishvili New Mexico State University, Las Cruces
  • Nick Bezhanishvili University of Leicester
  • Dick de Jongh University of Amsterdam

DOI:

https://doi.org/10.12775/LLP.2008.006

Keywords

intermediate logics, Heyting algebras, finite model property

Abstract

We give a systematic method of constructing extensions of the Kuznetsov-Gerčiu logic KG without the finite model property (fmp for short), and show that there are continuum many such. We also introduce a new technique of gluing of cyclic intuitionistic descriptive frames and give a new simple proof of Gerčiu’s result [9, 8] that all extensions of the Rieger-Nishimura logic RN have the fmp. Moreover, we show that each extension of RN has the poly-size model property, thus improving on [9]. Furthermore, for each function f: \omega -> \omega, we construct an extension Lf of KG such that Lf has the fmp, but does not have the f-size model property. We also give a new simple proof of another result of Gerčiu [9] characterizing the only extension of KG that bounds the fmp for extensions of KG. We conclude the paper by proving that RN.KC = RN + (¬p \vee ¬¬p) is the only pre-locally tabular extension of KG, introduce the internal depth of an extension L of RN, and show that L is locally tabular if and only if the internal depth of L is finite.

Author Biographies

Guram Bezhanishvili, New Mexico State University, Las Cruces

Department of Mathematical Sciences

Nick Bezhanishvili, University of Leicester

Department of Computer Science

Dick de Jongh, University of Amsterdam

Institute for Logic, Language and Computation

References

Bezhanishvili, G., “Locally finite varieties”, Algebra Universalis 46 (2001), 531–548.

Bezhanishvili, G., and R. Grigolia, “Subalgebras and homomorphic images of the Rieger-Nishimura lattice”, pp. 9–16 in Proceedings of the Institute of Cybernetics, volume 1, Georgian Academy of Sciences, Tbilisi, 2000.

Bezhanishvili, N., Lattices of Intermediate and Cylindric Modal Logics, PhD thesis, ILLC, University of Amsterdam, 2006.

Chagrov, A., and M. Zakharyaschev, Modal Logic, Oxford University Press, 1997.

Citkin, A.I., “An example of prelocally tabular superintuitionistic propositional logic”, pp. 97–99 in Logic and Foundations of Mathematics, Vilnius, 1982.

Davey, B., and H. Priestley, Introduction to Lattices and Order, Cambridge University Press, 1990.

Esakia, L., “Gödel-Löb modal system – addendum”, pp. 77–79 in Proceedings of the third International Conference, Smirnov’s Readings, Moscow, 2001, Russian Academy of Sciences (Russian).

Gerčiu, V.Ja., “Correction to the article The finite approximability of superintuitionistic logics, Mat. Issled. 7 (1972), 1(23), 186–192”, Mat. Issled. 7 (1972), 3 (25), 278 (Russian).

Gerčiu, V.Ja., “The finite approximability of superintuitionistic logics”, Mat. Issled. 7 (1972), 1 (23), 186–192 (Russian).

Gerčiu, V.Ja., and A.V. Kuznetsov, “The finitely axiomatizable superintuitionistic logics”, Soviet Mathematics Doklady 11 (1970), 1654–1658.

Harrop, R., “On the existence of finite models and decision procedures for propositional calculi”, Proc. Cambridge Philos. Soc. 54 (1958), 1–13.

Jankov, V.A., “The construction of a sequence of strongly independent superintuitionistic propositional calculi”, Soviet Mathematics Doklady 9 (1968), 806–807.

Kracht, M., “Prefinitely axiomatizable modal and intermediate logics”, Mathematical Logic Quarterly, 39 (1993), 301–322.

Kuznetsov, A.V., and V.Ja. Gerčiu, “Superintuitionistic logics and finite approximability”, Soviet Mathematics Doklady, 11 (1970), 6, 1614–1619.

Kuznetsov, A.V., “Superintuitionistic logics”, Mat. Issled. 10 (1975), 2 (36), 150–158, 284–285 (Russian).

Kuznetsov, A.V., “Means for detection of nondeducibility and inexpressibility”, pp. 5–33 in Logical Inference (Moscow, 1974), Nauka, Moscow, 1979, (Russian).

Kuznetsov, A.V., “Algorithms, algebras and intuitionistic logic”, Mat. Issled. (98, Neklass. Logiki) 152 (1987), 10–14 (Russian).

Kuznetsov, A.V., and A.Yu. Muravitsky, “On superintuitionistic logics as fragments of proof logic extensions”, Studia Logica 45 (1986), 1, 77–99.

Mardaev, S.I., “The number of prelocal-tabular superintuitionistic propositional logics”, Algebra and Logic, 23 (1984), 1, 56–66.

Nishimura, I., “On formulas of one variable in intuitionistic propositional calculus”, Journal of Symbolic Logic 25 (1960), 327–331.

Rieger, L., “On the lattice theory of Brouwerian propositional logic”, Acta fac. rerum nat. Univ. Car. 189 (1949), 1–40.

Logic and Logical Philosophy

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Published

2008-06-19

How to Cite

1.
BEZHANISHVILI, Guram, BEZHANISHVILI, Nick and DE JONGH, Dick. The Kuznetsov-Gerčiu and Rieger-Nishimura logics. Logic and Logical Philosophy. Online. 19 June 2008. Vol. 17, no. 1-2, pp. 73-110. [Accessed 6 July 2025]. DOI 10.12775/LLP.2008.006.
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