The contribution of A.V. Kuznetsov to the theory of modal systems and structures

Alexei Y. Muravitsky

DOI: http://dx.doi.org/10.12775/LLP.2008.004

Abstract


We will outline the contributions of A.V. Kuznetsov to modal logic. In his research he focused mainly on semantic, i.e. algebraic, issues and lattices of extensions of particular modal logics, though his proof of the Full Conservativeness Theorem for the proof-intuitionistic logic KM (Theorem 17 below) is a gem of proof-theoretic art.

Keywords


intuitionistic propositional logic and its extensions; modal logic S4 and its extensions; algebraic semantics for modal logics (S4-, Grz-, GL-, KM-algebras); lattice of the extensions of a logic

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References


Blok, W.J., Varieties of Interior Algebras, doctoral dissertation, University of Amsterdam, 197

Czelakowski, J., Model-Theoretic Methods in Methodology of Propositional Calculi, Polish Academy of Sciences, Institute of Philosophy and Sociology, Warsaw, 1980.

Esakia, L.L., “On modal counterparts of superintuitionistic logics”, pages 135–136 in The Seventh All-Soviet Symposium on Logic and Methodology of Science, Abstracts, Kiev, 1976 (in Russian).

Esakia, L.L., “Diagonal constructions, Löb formula, and Cantor’s scattered spaces”, pages 178–179 in Modal and Intensional Logics, Abstracts, Institute of Philosophy, Academy of Sciences of USSR, Moscow, 1978 (in Russian).

Esakia, L.L., “The Modalized Heyting Calculus: A Conservative Modal Extension of the Intuitionistic Logic”, Journal of Applied Non-Classical Logic 16, 3–4 (2006), 51–69.

Dummett, M.A., “A propositional calculus with denumerable matrix”, Journal of Symbolic Logic 24, 2 (1959), 97–106.

Gödel, K., “Zum intuitionistischen Aussagenkalkül”, pages 222–225 in Kurt Gödel, Collected Works, vol. 1, Oxford University Press.

Gödel, K., “Eine Interpretation des Intuitionistische Aussagenkalkulus”, pages 301–302 in Kurt Gödel, Collected Works, vol. 1, Oxford University Press.

Grzegorczyk, A., “Some relational systems and associated topological spaces”, Fundamenta Mathematica 60 (1967), 223–231.

Jaśkowski, S., “Recherches sur le systéme de la logique intuitioniste”, in Internat. Congress Philos. Sci. 6 (1936), 58–61.

Kuznetsov, A.V., “On superintuitionistic logics”, pages 243–249 in Proceedings of the International Congress of Mathematics, Vancouver, 1975.

Kuznetsov, A.V., “Proof-intuitionistic logic”, pages 75–79 in Modal and Intensional Logics, Abstracts, Institute of Philosophy, Academy of Sciences of USSR, Moscow, 1978 (in Russian).

Kuznetsov, A.V., “Tools for detecting non-derivability or non-expressibility”, pages 5–23 in V.A. Smirnov (ed.), Logical Inference, Proceedings of the USSR Symposium on the Theory of Logical Inference, 1979, Nauka, Moscow (in Russian).

Kuznetsov, A.V., “On algebras of open sets”, in The 4 th Tiraspol Symposium on General Topology and Its Applications, Abstracts, Shtiintsa, Kishinev, 1979 (in Russian).

Kuznetsov, A.V., “Proof-intuitionistic propositional calculus”, Doklady Akad. Nauk SSSR 283, 1 (1985), 27–30. English translation: Soviet Mathematics – Doklady 32, 1 (1985), 27–30.

Kuznetsov, A.V., “Algorithms, algebras and intuitionistic logic”, Matematicheskie Isselovaniia, Neklassicheskie logiki, 98 (1987), 10–14 (in Russian).

Kuznetsov, A.V., and A.Yu. Muravitsky, “The logic of provability”, page 73 in The 4 th All-Union Conference on Mathematical Logic, Abstracts, 1976, Kishinev, Shtiintsa (in Russian).

Kuznetsov, A.V., and A.Yu. Muravitsky, “Magari algebras”, pages 105–106 in The 14 th Soviet Algebraic Conference, part 2, 1977, Novosibirsk (in Russian).

Kuznetsov, A.V., and A.Yu. Muravitsky, “Provability as modality”, pages 193–230 in Current Problems of Logic and Methodology of Science, Naukova Dumka, Kiev, 1980 (in Russian).

Kuznetsov, A.V., and A.Yu. Muravitsky, “On superintuitionistic logics as fragments of proof logic extensions”, Studia Logica 45 (1986), 76–99.

Magari, R., “The diagonalizable algebras”, Boll. Unione mat. Ital. 12 (1975), suppl. fasc. 3, 117–125.

Maksimova, L.L., “Modal logics of finite layers”, Algebra and Logic 14, 3 (1975), 304–319 (in Russian; there is an English translation).

Maksimova, L.L., and V.V. Rybakov, “On the lattice of normal modal logics”, Algebra and Logic 13, 2 (1974), 188–216 (in Russian; there is an English translation).

Maltsev, A.I., Algebraic Systems, Springer-Verlag, 1973.

McKenzie, R.N., G.F. McNulty, and W.F. Taylor, Algebras, Lattices, Varieties, vol. 1, Wadsworth & Brooks/Cole, 1987.

McKinsey, J.C.C., and A. Tarski, “Some theorems about the sentential calculi of Lewis and Heyting”, Journal of Symbolic Logic 13, 1 (1948), 1–15.

Mendelson, E., Introduction to Mathematical Logic, 4 th edition, Chapman & Hall, 1997.

Muravitsky, A.Yu., “Extensions of logic of provability”, Mathematical Notes 33, 5–6 (1983), 469–475.

Muravitsky, A.Yu., “Correspondence of proof-intuitionistic logic extensions to proof-logic extensions”, Doklady Akad. Nauk SSSR 281, 4 (1985), 789–793. English translation: Soviet Mathematics Doklady 31, 2 (1985), 345–348.

Muravitsky, A.Yu., “An algebraic proof of the separation property for an intuitionistic provability calculus”, Matem. Sbornik 131, 3 (173) (1986), 403–412. English translation: Math. SSSR Sbornik 59, 2 (1988), 397–406.

Muravitsky, A.Yu., “Embedding of extensions of Grzegorczyk logic into extensions of provability logic”, pages 75–80 in Intensional Logics and Logical Structure of Theories, Metsniereba Press, Tbilisi, 1988 (in Russian).

Muravitsky, A.Yu., “Correspondence of proof-intuitionistic logic extensions to proof-logic extensions”, pages 104–120 in Trudy Inst. Mat. (Novosibirsk), 12 (1989), Mat. Logika i Algoritm. Problemy (in Russian).

Muravitsky, A.Yu., “Magari and ?-pseudo-Boolean algebras”, Siberian Mathematical Journal 31, 4 (1990), 111–117.

Muravitsky, A.Yu., “The embedding theorem: its further developments and consequences. Part 1”, Notre Dame Journal of Formal Logic (to appear).

Novikov, P.S., Constructive Mathematical Logic from the Point of View of Classical Logic, Nauka, Moscow, 1977 (in Russian).

Segerberg, K., An Essay in Classical Modal Logic, doctoral dissertation, Uppsala Universitet, Uppsala, 1971, vol. 2.

Simonova, I.G., “The separation property for the provability-intuitionistic Calculus”, pages 121–133 in Matematicheskie Issledovaniya, no. 98, Neklassicheskie Logiki, Shtiintsa, 1987 (in Russian).

Simonova, I.G., “On the interpolation property for extensions of proofintuitionistic logic”, Mathematical Notes 47, 5–6 (1990), 483–490.

Solovay, R.M., “Provability interpretation of modal logic”, Israel Journal of Mathematics 25 (1976), 287–304.








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