Correspondence, Canonicity, and Model Theory for Monotonic Modal Logics


We investigate the role of coalgebraic predicate logic, a logic for neighborhood frames first proposed by Chang, in the study of monotonic modal logics. We prove analogues of the Goldblatt–Thomason theorem and Fine’s canonicity theorem for classes of monotonic neighborhood frames closed under elementary equivalence in coalgebraic predicate logic. The elementary equivalence here can be relativized to the classes of monotonic, quasi-filter, augmented quasi-filter, filter, or augmented filter neighborhood frames, respectively. The original, Kripke-semantic versions of the theorems follow as a special case concerning the classes of augmented filter neighborhood frames.

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  1. 1.

    Alur, R., T. Henzinger, and O. Kupferman, Alternating-time temporal logic, in Journal of the ACM, IEEE Computer Society Press, 1997, pp. 100–109.

  2. 2.

    Blackburn, P., M. de Rijke, and Y. Venema, Modal Logic, Cambridge Tracts in Theoretical Computer Science, Cambridge University Press, New York, NY, USA, 2001.

  3. 3.

    Chang, C. C., Modal model theory, in A. R. D. Mathias, and H. Rogers, (eds.), Cambridge Summer School in Mathematical Logic, New York: Springer Verlag, 1973, pp. 599–617.

  4. 4.

    Chellas, B. F., Modal Logic: An Introduction, Cambridge University Press, 1980.

  5. 5.

    Došen, K., Duality between modal algebras and neighbourhood frames, Studia Logica 48(2): 219–234, 1989.

    Article  Google Scholar 

  6. 6.

    Fine, K., Some connections between elementary and modal logic, in Proceedings of the Third Scandinavian Logic Symposium, 1975.

  7. 7.

    Flum, J., and M. Ziegler, Topological Model Theory, Springer-Verlag, 1980.

    Google Scholar 

  8. 8.

    Gehrke, M., and J. Harding, Bounded lattice expansions, Journal of Algebra 238(1): 345–371, 2001.

    Article  Google Scholar 

  9. 9.

    Gehrke, M., J. Harding, and Y. Venema, Macneille completions and canonical extensions, Transactions of the American Mathematical Society 358(2): 573–590, 2006.

    Article  Google Scholar 

  10. 10.

    Gerson, M., A neighbourhood frame for T with no equivalent relational frame, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 22: 29–34, 1976.

    Article  Google Scholar 

  11. 11.

    Gerson, M. S., An extension of S4 complete for the neighbourhood semantics but incomplete for the relational semantics, Studia Logica: An International Journal for Symbolic Logic 34(4): 333–342, 1975.

    Article  Google Scholar 

  12. 12.

    Goldblatt, R. I., and S. K. Thomason, Axiomatic classes in propositional modal logic, in J. N. Crossley, (ed.), Algebra and Logic, Springer, Berlin, Heidelberg, 1975, pp. 163–173.

  13. 13.

    Hansen, H. H., Monotonic Modal Logics, Master’s thesis, University of Amsterdam, 2003.

  14. 14.

    Hansen, H. H., C. Kupke, and E. Pacuit, Neighbourhood structures: Bisimilarity and basic model theory, Logical Methods in Computer Science, Volume 5, Issue 2 2009.

  15. 15.

    Kurz, A., and J. Rosickỳ, The Goldblatt-Thomason theorem for coalgebras, in International Conference on Algebra and Coalgebra in Computer Science, Springer, 2007, pp. 342–355.

  16. 16.

    Litak, T., D. Pattinson, K. Sano, and L. Schröder, Model Theory and Proof Theory of Coalgebraic Predicate Logic, Logical Methods in Computer Science, 14(1) 2018.

  17. 17.

    Marker, D., Model Theory: An Introduction, Springer-Verlag New York, 2002.

    Google Scholar 

  18. 18.

    Pacuit, E., Neighborhood Semantics for Modal Logic, Short Textbooks in Logic, Springer International Publishing, Cham, 2017.

  19. 19.

    Palmigiano, A., S. Sourabh, and Z. Zhao, Sahlqvist theory for impossible worlds, Journal of Logic and Computation 27(3): 775–816, 2016.

    Google Scholar 

  20. 20.

    Parikh, R., The logic of games and its applications, in Annals of Discrete Mathematics, Elsevier, 1985, pp. 111–140.

  21. 21.

    Pauly, M., Logic for Social Software, Ph.D. thesis, University of Amsterdam, 2001.

  22. 22.

    Schröder, L., D. Pattinson, and T. Litak, A van Benthem/Rosen theorem for coalgebraic predicate logic, Journal of Logic and Computation 27(3): 749–773, 2017.

    Google Scholar 

  23. 23.

    ten Cate, B., D. Gabelaia, and D. Sustretov, Modal languages for topology: Expressivity and definability, Annals of Pure and Applied Logic 159(1): 146–170, 2009.

  24. 24.

    van Benthem, J. F. A. K., Modal Logic and Classical Logic, Distributed in the U.S.A. By Humanities Press, 1983.

  25. 25.

    van Benthem, J., Universal algebra and model theory: the DNA of modal logic, in Logic in Hungary, Janos Bolyai Mathematical Society, 2005.

  26. 26.

    van Benthem, J., and G. Bezhanishvili, Modal Logics of Space, Springer Netherlands, Dordrecht, 2007, pp. 217–298.

  27. 27.

    Venema, Y., Algebras and coalgebras, in Handbook of Modal Logic, chap. 6, Elsevier, 2007, pp. 331–426.

  28. 28.

    Ziegler, M., Chapter xv: Topological model theory, in J. Barwise, and S. Feferman, (eds.), Model-Theoretic Logics, vol. Volume 8 of Perspectives in Mathematical Logic, chap. XV, Springer-Verlag, New York, 1985, pp. 557–577.

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I wish to give special thanks to Wesley Holliday for his extensive and helpful comments and discussion. I also wish to thank Tadeusz Litak, Lutz Schröder, and Frederik Lauridsen for useful comments on earlier drafts. Also, I am indebted to the anonymous reviewers for their helpful comments. Finally, I gratefully acknowledge financial support from the Takenaka Scholarship Foundation.

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Correspondence to Kentarô Yamamoto.

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Yamamoto, K. Correspondence, Canonicity, and Model Theory for Monotonic Modal Logics. Stud Logica (2020).

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  • Modal logic
  • Canonicity
  • Fine’s theorem
  • Goldblatt–Thomason theorem
  • Neighborhood frames