Journal of Philosophical Logic

, Volume 27, Issue 4, pp 353–387 | Cite as

On Logics with Coimplication

  • Frank Wolter
Article

Abstract

This paper investigates (modal) extensions of Heyting–Brouwer logic, i.e., the logic which results when the dual of implication (alias coimplication) is added to the language of intuitionistic logic. We first develop matrix as well as Kripke style semantics for those logics. Then, by extending the Gödel-embedding of intuitionistic logic into S4 , it is shown that all (modal) extensions of Heyting–Brouwer logic can be embedded into tense logics (with additional modal operators). An extension of the Blok–Esakia-Theorem is proved for this embedding.

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REFERENCES

  1. 1.
    Balbes, R. and Dwinger, Ph., Distributive Lattices, University of Missouri Press, 1974.Google Scholar
  2. 2.
    van Benthem, J., The Logic of Time, Reidel, Dordrecht, 1983.Google Scholar
  3. 3.
    van Benthem, J., Temporal logic. In: Gabbay, Hogger, and Robinson (eds), Handbook of Logic in Artificial Intelligence and Logic Programming, Vol. 4, 1995, pp. 241–350.Google Scholar
  4. 4.
    Blok, W., Varieties of Interior Algebras, Dissertation, University of Amsterdam, 1976.Google Scholar
  5. 5.
    Blok, W. and Köhler, P., Algebraic semantics for quasi-classical modal logics, Journal of Symbolic Logic 48 (1983), 941–964.Google Scholar
  6. 6.
    Basic tense logic. In: D. Gabbay and F. Guenthner (eds), Handbook of Philosophical Logic, Vol. 2, 1984, pp. 89–133.Google Scholar
  7. 7.
    Bosic, M. and Došen, K., Models for normal intuitionistic modal logics, Studia Logica 43 (1984), 217–245.Google Scholar
  8. 8.
    Chagrov, A. V. and Zakharyaschev, M. V., Modal companions of intermediate propositional logics, Studia Logica 51 (1992), 49–82.Google Scholar
  9. 9.
    Chagrov, A. V. and Zakharyaschev, M. V., Modal and superintuitionistic logics, Oxford University Press, 1996. LOGISEG6.tex; 19/03/1998; 13:42; v.7; p.33Google Scholar
  10. 10.
    Dummett, M. and Lemmon, E., Modal logics between S4 and S5, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 5 (1959), 250–264.Google Scholar
  11. 11.
    Esakia, L., On Varieties of Grzegorczyk Algebras, in Studies in Nonclassical Logics and Set Theory, Moscow, Nauka, 1979, pp. 257–287.Google Scholar
  12. 12.
    Fine, K., Logics containing K4, Part I, Journal of Symbolic Logic 39 (1974), 229–237.Google Scholar
  13. 13.
    Fine, K., Logics containing K4, Part II, Journal of Symbolic Logic 50 (1985), 619–651.Google Scholar
  14. 14.
    Fine, K. and Schurz, G., Transfer theorems for stratified modal logics, in Proceedings of the Arthur Prior Memorial Conference, Christchurch, New Zealand, 1991.Google Scholar
  15. 15.
    Fischer Servi, G., Semantics for a class of intuitionistic modal calculi. In: M. L. Dalla Chiara (ed.), Italian Studies in the Philosophy of Science, Reidel, Dordrecht, 1980, pp. 59–72.Google Scholar
  16. 16.
    Fischer Servi, G., Axiomatizations for some Intuitionistic Modal Logics, Rend. Sem. Mat. Univers. Polit. 42 (1984), 179–194.Google Scholar
  17. 17.
    Font, J., Modality and possibility in some intuitionistic modal logics, Notre Dame Journal of Formal Logic 27 (1986), 533–546.Google Scholar
  18. 18.
    Gödel, K., Eine Interpretation des intuitionistischen Aussagenkalküls, Ergebnisse eines mathematischen Kolloquiums 6 (1933), 39–40.Google Scholar
  19. 19.
    Goldblatt, R., Metamathematics of Modal Logic, Reports on Mathematical Logic 6 (1976), 41–78, 7 (1976), 21–52.Google Scholar
  20. 20.
    Goldblatt, R., Logics of Time and Computation, Number 7 in CSLI Lecture Notes, CSLI, 1987.Google Scholar
  21. 21.
    Grzegorczyk, A., A philosophically plausible formal interpretation of intuitionistic logic, Indag. Math. 26, 596–601.Google Scholar
  22. 22.
    Köhler, R., A subdirectly irreducible double Heyting algebra which is not simple, Algebra Universalis 10 (1980), 189–194.Google Scholar
  23. 23.
    Kracht, M., Even more on the lattice of tense logics, Arch. Math. Logic 31 (1992), 243–257.Google Scholar
  24. 24.
    Kracht, M. and Wolter, F., Properties of independently axiomatizable bimodal logics, Journal of Symbolic Logic 56 (1991), 1469–1485.Google Scholar
  25. 25.
    Kripke, S., A semantical analysis of intuitionistic logic I. In: J. Crossley and M. Dummett (eds), Formal Systems and Recursive Functions, North-Holland, Amsterdam, 1965, pp. 92–129.Google Scholar
  26. 26.
    Makkai, M. and Reyes, G., Completeness results for intuitionistic and modal logics in a categorical setting, Annals of Pure ans Applied Logic 72 (1995), 25–101.Google Scholar
  27. 27.
    Maksimova, L. and Rybakov, V., Lattices of modal logics, Algebra and Logic 13 (1974), 105–122.Google Scholar
  28. 28.
    Ono, H., On some intuitionistic modal logics, Publ. Kyoto University 13 (1977), 687–722.Google Scholar
  29. 29.
    Rauszer, C., Semi-Boolean algebras and their applications to intuitionistic logic with dual operators, Fund. Math. 83 (1974), 219–249.Google Scholar
  30. 30.
    Rauszer, C., A formalization of propositional calculus of H-B logic, Studia Logica 33 (1974).Google Scholar
  31. 31.
    Rauszer, C., An algebraic and Kripke-style approach to a certain extension of intuitionistic logic, Dissertationes Mathematicae, vol. CLXVII, Warszawa, 1980.Google Scholar
  32. 32.
    Rautenberg, W., Klassische und Nichtklassische Aussagenlogik, Wiesbaden, 1979.Google Scholar
  33. 33.
    Segerberg, K., An Essay in Classical Modal Logic, Uppsala, 1971.Google Scholar
  34. 34.
    Segerberg, K., That every extension of S4.3 is normal. In: S. Kanger (ed.), Proceedings of the Third Scandinavian Logic Symposium, Amsterdam, 1976, pp. 194–196.Google Scholar
  35. 35.
    Thomason, S. K., Semantic analysis of tense logics, Journal of Symbolic Logic 37 (1972), 150–158.Google Scholar
  36. 36.
    Troelstra, A. and van Dalen, D., Constructivism in Mathematics, vol. I, North-Holland, Amsterdam, 1988.Google Scholar
  37. 37.
    Wojcicki, R., Theory of Logical Calculi, Dordrecht, 1988.Google Scholar
  38. 38.
    Wolter, F., The finite model property in tense logic, Journal of Symbolic Logic 60 (1995), 757–774.Google Scholar
  39. 39.
    Wolter, F., Superintuitionistic companions of classical modal logics, Studia Logica 58 (1997), 229–259.Google Scholar
  40. 40.
    Wolter, F., Completeness and decidability of tense logics closely related to logics containing K4, Journal of Symbolic Logic 62 (1997), 131–158.Google Scholar
  41. 41.
    Wolter, F. and Zakharyaschev, M., Intuitionistic modal logics as fragments of classical bimodal logics, in logic at work, Essays in honour of H. Rasiowa, forthcoming.Google Scholar
  42. 42.
    Wolter, F. and Zakharyaschev, M., On the relation between intuitionistic and classical modal logics, to appear in Algebra and Logic, 1996.Google Scholar
  43. 43.
    Zakharyaschev, M., Canonical Formulas for K4, Part II, to appear in Journal of Symbolic Logic, 1996.Google Scholar

Copyright information

© Kluwer Academic Publishers 1998

Authors and Affiliations

  • Frank Wolter
    • 1
  1. 1.School of Information ScienceJAISTTatsunokuchiJapan (e-mail

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