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Canonical Formulas for Modal and Superintuitionistic Logics: A Short Outline

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Book cover Advances in Intensional Logic

Part of the book series: Applied Logic Series ((APLS,volume 7))

Abstract

Every time when I am asked what canonical formulas are, I have some difficulty. Strictly speaking, they are modal or intuitionistic formulas of a rather special form associated with finite frames. But I have never used them as formulas. As formulas they seem to be useless. All one needs to know about them is that they are.

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References

  1. J.G. Anderson. Superconstructive propositional calculi with extra axiom sche-mes containing one variable. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 18:113–130, 1972.

    Article  Google Scholar 

  2. J. van Benthem. Some kinds of modal completeness. Studia Logica, 39:125–141, 1980.

    Article  Google Scholar 

  3. J. van Benthem. Modal logic and classical logic, Bibliopolis, Naples, 1983.

    Google Scholar 

  4. J. van Benthem. Correspondence theory. In D. Gabbay and F. Guenthner, editors, Handbook of Philosophical Logic, volume 2, pages 167–247. Reidel, Dordrecht, 1984.

    Chapter  Google Scholar 

  5. J. van Benthem and I.L. Humberstone. Halldén completeness by gluing of Kripke frames. Notre Dame Journal of Formal Logic, 24:426–430, 1983.

    Article  Google Scholar 

  6. W.J. Blok. Varieties of interior algebras, PhD thesis, University of Amsterdam, 1976.

    Google Scholar 

  7. R.A. Bull. That all normal extensions of S4.3 have the finite model property. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 12:341–344, 1966.

    Article  Google Scholar 

  8. R.A. Bull and K. Segerberg. Basic modal logic. In D. Gabbay and F. Guenthner, editors, Handbook of Philosophical Logic, volume 2, pages 1–88, Reidel, Dordrecht, 1984.

    Chapter  Google Scholar 

  9. A.V. Chagrov. On the polynomial finite model property of modal and intermediate logics. In Mathematical Logic, Mathematical Linguistics and Algorithm Theory, pages 75–83, Kalinin State University, Kalinin, 1983. (Russian)

    Google Scholar 

  10. A.V. Chagrov. Varieties of logical matrices. Algebra and Logic, 24:426–489, 1985. (Russian)

    Article  Google Scholar 

  11. A.V. Chagrov. On the complexity of propositional logics. In Complexity Problems in Mathematical Logic, pages 80–90, Kalinin State University, Kalinin, 1985. (Russian)

    Google Scholar 

  12. A.V Chagrov. Undecidable properties of extensions of provability logic, parts I, II. Algebra and Logic, 29:350–367, 613–623, 1990. (Russian)

    Google Scholar 

  13. A.V. Chagrov and M.V. Zakharyaschev. The undecidability of the disjunction property of intermediate calculi. Preprint, Institute of Applied Mathematics, the USSR Academy of Sciences, no. 57, 1989. (Russian)

    Google Scholar 

  14. A.V. Chagrov and M.V. Zakharyaschev. An essay in complexity aspects of intermediate calculi. Proceedings of the Fourth Asian Logic Conference, pages 26–29, CSK Education Centre, Tokyo, 1990.

    Google Scholar 

  15. A.V. Chagrov and M.V. Zakharyaschev. The disjunction property of intermediate propositional logics. Studio. Logica, 50:189–216, 1991.

    Article  Google Scholar 

  16. A.V. Chagrov and M.V Zakharyaschev. Modal companions of intermediate propositional logics. Studio Logica, 51:49–82, 1992.

    Article  Google Scholar 

  17. A.V. Chagrov and M.V. Zakharyaschev. The undecidability of the disjunction property of propositional logics and other related problems. Journal of Symbolic Logic, 58:967–1002, 1993.

    Article  Google Scholar 

  18. A.V. Chagrov and M.V. Zakharyaschev. On the independent axiomatizability of modal and intermediate logics. Journal of Logic and Computation, 5:287–302, 1995.

    Article  Google Scholar 

  19. A.V. Chagrov and M.V Zakharyaschev. Sahlqvist formulas are not so elementary even above S4. In L. Csirmaz, D.M. Gabbay and M. de Rijke, editors, Logic Colloquium ‘92, pages 61–73, Studies in Logic, Language and Information, CSLI Publications/FoLLI, Stanford, CA, 1995.

    Google Scholar 

  20. L.A. Chagrova. On first-order definability of intuitionistic formulas with restrictions on occurrences of connectives. In Logical Methods for Constructing Effective Algorithms, pages 135–136, Kalinin State University, Kalinin, 1986. (Russian)

    Google Scholar 

  21. L.A. Chagrova. An undecidable problem in correspondence theory. Journal of Symbolic Logic, 56:1261–1272, 1991.

    Article  Google Scholar 

  22. D. van Dalen. Intuitionistic logic. In D. Gabbay and F. Guenthner, editors, Handbook of Philosophical Logic, volume 3, pages 225–339, Reidel, Dordrecht, 1986.

    Google Scholar 

  23. M.A.E. Dummett and L.J. Lemmon. Modal logics between S4 and S5. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 5:250–264, 1959.

    Article  Google Scholar 

  24. L.L. Esakia. On varieties of Grzegorczyk algebras. In Studies in Non-Classical Logics and Set Theory, pages 257–287, Nauka, Moscow, 1979. (Russian)

    Google Scholar 

  25. K. Fine. The logics containing S4.3. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 17:371–376, 1979.

    Article  Google Scholar 

  26. K. Fine. Logics containing S4 without the finite model property. In W. Hodges, editor, Conference in Mathematical Logic, London 1970, Lecture Notes in Mathematics, vol.255, pages 88–102, Springer-Verlag, Berlin, 1972.

    Google Scholar 

  27. K. Fine. An incomplete logic containing S4. Theoria, 40:23–29, 1974.

    Article  Google Scholar 

  28. K. Fine. An ascending chain of S4 logics. Theoria, 40:110–116, 1974.

    Article  Google Scholar 

  29. K. Fine. Logics containing K4. Part I. Journal of Symbolic Logic, 39:31–42, 1974.

    Article  Google Scholar 

  30. K. Fine. Some connections between elementary and modal logic. In S. Kanger, editor, Proceedings of the Third Scandinavian Logic Symposium, pages 15–31, North-Holland, Amsterdam, 1975.

    Chapter  Google Scholar 

  31. K. Fine. Logics containing K4. Part II. Journal of Symbolic Logic, 50:619–651, 1985.

    Article  Google Scholar 

  32. D.M. Gabbay. Semantical Investigations in Heyting’s Intuitionistic Logic. Reidel, Dordrecht, 1981.

    Google Scholar 

  33. K. Gödel. Eine Interpretation des intuitionistischen Aussagenkalküls. Ergebnisse eines mathematischen Kolloquiums, Heft 4, pages 39–40, 1933.

    Google Scholar 

  34. R.I. Goldblatt. Metamathematics of modal logic. Reports on Mathematical Logic, 6, pp. 41–77; 7, pp. 21–52.

    Google Scholar 

  35. R.I. Goldblatt. The McKinsey axiom is not canonical. Journal of Symbolic Logic, 56:554–562, 1991.

    Article  Google Scholar 

  36. A. Grzegorczyk. Some relational systems and the associated topological spaces. Fundamentu Mathematicae, 60:223–231, 1967.

    Google Scholar 

  37. T. Hosoi and H. Ono. Intermediate propositional logics (A survey). Journal of Tsuda College, 5:67–82, 1973.

    Google Scholar 

  38. G.E. Hughes and M.J. Cresswell A Companion to Modal Logic, Methuen, London, 1984.

    Google Scholar 

  39. V.A. Jankov. The relationship between deducibility in the intuitionistic propositional calculus and finite implicational structures. Soviet Mathematics Doklady, 4:1203–1204, 1963.

    Google Scholar 

  40. V.A. Jankov. Conjunctively indecomposable formulas in propositional calculi. Izvestija Academii Nauk SSSR, 33:13–38, 1969. (English translation: Mathematics of USSR, Izvestija, 3:17–35.)

    Google Scholar 

  41. M. Kracht. An almost general splitting theorem for modal logic. Studia Logica, 49:455–470, 1990.

    Article  Google Scholar 

  42. M. Kracht. Sometimes FMP is not preserved under joins. Manuscript, 1992.

    Google Scholar 

  43. M. Kracht. How completeness and correspondence theory got married. In M. de Rijke, editor, Diamonds and Defaults, pages 175–214, Kluwer Academic Publishers, Dordrecht, 1993.

    Google Scholar 

  44. M. Kracht. Splittings and the finite model property. Journal of Symbolic Logic, 58:139–157, 1993.

    Article  Google Scholar 

  45. S. Kripke. Semantical analysis of modal logic I: Normal propositional calculi. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 9:67–96, 1963.

    Article  Google Scholar 

  46. C. Kuratowski. Sur l’opération A de l Analysis Situs. Fundamentu Mathematicae, 3:182–199, 1922.

    Google Scholar 

  47. E.J. Lemmon. An Introduction to Modal Logic, in collaboration with D. Scott, Blackwell, Oxford, 1977.

    Google Scholar 

  48. Logic Notebook (1986), Novosibirsk, 1986. (Russian)

    Google Scholar 

  49. L.L. Maksimova. On maximal intermediate logics with the disjunction property. Studia Logica, 45:69–75, 1986.

    Article  Google Scholar 

  50. L.L. Maksimova and V.V. Rybakov. On the lattice of normal modal logics. Algebra and Logic, 13:188–216, 1974. (Russian)

    Article  Google Scholar 

  51. C.G. McKay. The decidability of certain intermediate logics. Journal of Symbolic Logic, 33:258–264, 1968.

    Article  Google Scholar 

  52. J.C.C. McKinsey and A. Tarski. On closed elements in closure algebras. Annals of Mathematics, 47:122–162, 1946.

    Article  Google Scholar 

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

    Article  Google Scholar 

  54. P. Minari. Intermediate logics with the same disjunctionless fragment as intuitionistic logic. Studio. Logica, 45:207–222, 1986.

    Article  Google Scholar 

  55. H. Ono and A. Nakamura. On the size of refutation Kripke models for some linear modal and tense logics. Studio Logica, 39:325–333, 1980.

    Article  Google Scholar 

  56. W. Rautenberg. Klassische und nichtklassische Aussagenlogik, Vieweg Verlag, Wiesbaden, 1979.

    Book  Google Scholar 

  57. RH. Rodenburg. Intuitionistic Correspondence Theory, PhD thesis, University of Amsterdam, 1986.

    Google Scholar 

  58. H. Sahlqvist. Completeness and correspondence in the first and second order semantics for modal logic. In S. Kanger, editor, Proceedings of the Third Scandinavian Logic Symposium, pages 110–143, North-Holland, Amsterdam, 1975.

    Chapter  Google Scholar 

  59. K. Segerberg. An Essay in Classical Modal Logic, Philosophical Studies, vol.13, University of Uppsala, 1971.

    Google Scholar 

  60. V.Ju. Shavrukov. On two extensions of the provability logic GL. Mathematical Sbornik, 181:240–255, 1990. (English translation: Mathematics of the USSR Sbornik, 69 255–270.)

    Google Scholar 

  61. V.B. Shehtman. On incomplete propositional logics. Soviet Mathematics Doklady, 235:542–545, 1977. (Russian)

    Google Scholar 

  62. V.B. Shehtman. Undecidable superintuitionistic calculus. Soviet Mathematics Doklady, 240:549–552, 1978. (Russian)

    Google Scholar 

  63. V.B. Shehtman. Topological models of propositional logics. Semiotics and Information Science, 15:74–98, 1980. (Russian)

    Google Scholar 

  64. V.B. Shehtman. Derived sets in Euclidean spaces and modal logic. Preprint X-90–05, University of Amsterdam, 1990.

    Google Scholar 

  65. T. Shimura. Kripke completeness of some intermediate predicate logics with the axiom of constant domain and a variant of canonical formulas. Studia Logica, 52:23–40, 1993.

    Article  Google Scholar 

  66. T. Shimura. On completeness of intermediate predicate logics with respect to Kripke semantics. Bulletin of the Section of Logic, 24:41–45, 1995.

    Google Scholar 

  67. S.K. Sobolev. On the finite approximability of superintuitionistic logics. Mathematical Sbornik, 102:289–301, 1977. (Russian)

    Google Scholar 

  68. S.K. Thomason. Noncompactness in propositional modal logic. Journal of Symbolic Logic, 37:716–720, 1972.

    Article  Google Scholar 

  69. S.K. Thomason. An incompleteness theorem in modal logic. Theoria, 40:30–34, 1974.

    Article  Google Scholar 

  70. F. Wolter. Lattices of Modal Logics. PhD thesis, FU Berlin, 1993.

    Google Scholar 

  71. F. Wolter. Properties of tense logics. Mathematical Logic Quarterly, in print.

    Google Scholar 

  72. F. Wolter. Tense logics without tense operators. Mathematical Logic Quarterly, 42:145–171, 1996.

    Article  Google Scholar 

  73. A. Wroński. Intermediate logics and the disjunction property. Reports on Mathematical Logic, 1:39–51, 1973.

    Google Scholar 

  74. M.V. Zakharyaschev. Certain classes of intermediate logics. Preprint, Institute of Applied Mathematics, the USSR Academy of Sciences, no. 160, 1981. (Russian)

    Google Scholar 

  75. M.V. Zakharyaschev. On intermediate logics. Soviet Mathematics Doklady, 27:274–277, 1983.

    Google Scholar 

  76. M.V. Zakharyaschev. Normal modal logics containing S4. Soviet Mathematics Doklady, 29:252–255, 1984.

    Google Scholar 

  77. M.V. Zakharyaschev. Syntax and Semantics of Superintuitionistic and Modal Logics. PhD thesis, Moscow, 1984. (Russian)

    Google Scholar 

  78. M.V. Zakharyaschev. On the disjunction property of intermediate and modal logics. Mathematical Zametki, 42:729–738, 1987. (English translation: Mathematical Notes, 42:901–905).

    Google Scholar 

  79. M.V. Zakharyaschev. Syntax and semantics of modal logics containing S4. Algebra and Logic, 27:408–428, 1988.

    Article  Google Scholar 

  80. M.V. Zakharyaschev. Syntax and semantics of intermediate logics. Algebra and Logic, 28:262–282, 1989.

    Article  Google Scholar 

  81. M.V. Zakharyaschev. Modal companions of intermediate logics: syntax, semantics and preservation theorems. Mathematical Sbornik, 180:1415–1427, 1989. (English translation: Mathematics of the USSR Sbornik, 68:277–289.)

    Google Scholar 

  82. M.V. Zakharyaschev. Canonical formulas for K4. Part I: Basic results. Journal of Symbolic Logic, 57:1377–1402, 1992.

    Article  Google Scholar 

  83. M.V. Zakharyaschev. A sufficient condition for the finite model property of modal logics above K4. Bulletin of the IGPL, 1:13–21, 1993.

    Article  Google Scholar 

  84. M.V. Zakharyaschev. A new solution to the problem of T. Hosoi and H. Ono. Notre Dame Journal of Formal Logic, 35:450–457, 1994.

    Article  Google Scholar 

  85. M.V. Zakharyaschev. Canonical formulas for K4. Part II: Cofinal subframe logics. Journal of Symbolic Logic, 61(2):421–449, 1996.

    Article  Google Scholar 

  86. M.V. Zakharyaschev. Canonical formulas for K4. Part III: The finite model property. Journal of Symbolic Logic, in print.

    Google Scholar 

  87. M.V. Zakharyaschev and A.V Alekseev. All finitely axiomatizable normal extensions of K4.3 are decidable. Mathematical Logic Quarterly, 41:15–23, 1995.

    Article  Google Scholar 

  88. M.V. Zakharyaschev and S.V. Popov. On the complexity of countermodels for intuitionistic calculus. Preprint, Institute of Applied Mathematics, the USSR Academy of Sciences, no. 45, 1980. (Russian)

    Google Scholar 

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  1. II. Mathematisches Institut, Freie Universität Berlin, Germany

    Michael Zakharyaschev

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  1. Department of Computer Science, University of Warwick, Coventry, UK

    Maarten de Rijke

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Zakharyaschev, M. (1997). Canonical Formulas for Modal and Superintuitionistic Logics: A Short Outline. In: de Rijke, M. (eds) Advances in Intensional Logic. Applied Logic Series, vol 7. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8879-9_8

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  • DOI: https://doi.org/10.1007/978-94-015-8879-9_8

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