Skip to main content
SpringerLink
Log in

On Modal Logics of Model-Theoretic Relations

  • Published:
Studia Logica Aims and scope Submit manuscript

Abstract

Given a class \(\mathcal {C}\) of models, a binary relation \(\mathcal {R}\) between models, and a model-theoretic language L, we consider the modal logic and the modal algebra of the theory of \(\mathcal {C}\) in L where the modal operator is interpreted via \(\mathcal {R}\). We discuss how modal theories of \(\mathcal {C}\) and \(\mathcal {R}\) depend on the model-theoretic language, their Kripke completeness, and expressibility of the modality inside L. We calculate such theories for the submodel and the quotient relations. We prove a downward Löwenheim–Skolem theorem for first-order language expanded with the modal operator for the extension relation between models.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Artemov, S. N., and L. D. Beklemishev, Provability logic, in Handbook of Philosophical Logic, 2nd edn., Kluwer, 2004, pp. 229–403.

  2. Barwise, J., and S. Feferman, Model-theoretic logics, Perspectives in mathematical logic, Springer-Verlag, 1985.

  3. Barwise, J., and J. van Benthem, Interpolation, preservation, and pebble games, Journal of Symbolic Logic 64(2): 881–903, 1999.

    Article  Google Scholar 

  4. Benerecetti, M., F. Mogavero, and A. Murano, Substructure temporal logic, in Proceedings of the 2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’13, IEEE Computer Society, Washington, DC, USA, 2013, pp. 368–377.

  5. Berarducci, A., The interpretability logic of Peano arithmetic, Journal of Symbolic Logic 55(3): 1059–1089, 1990.

    Article  Google Scholar 

  6. Blackburn, P., M. de Rijke, and Y. Venema, Modal Logic, vol. 53 of Cambridge Tracts in Theoretical Computer Science, Cambridge University Press, 2002.

  7. Block, A. C., and B. Löwe, Modal logics and multiverses, RIMS Kokyuroku 1949: 5–23, 2015.

    Google Scholar 

  8. Buss, S. R., The modal logic of pure provability, Notre Dame Journal of Formal Logic 1(2): 225–231, 1990.

    Article  Google Scholar 

  9. Chagrov, A., and M. Zakharyaschev, Modal Logic, vol. 35 of Oxford Logic Guides, Oxford University Press, 1997.

  10. Chang, C. C., and H. J. Keisler, Model Theory, Studies in Logic and the Foundations of Mathematics, 3rd edn., Elsevier Science, 1990.

  11. D’agostino, G., and M. Hollenberg, Logical questions concerning the \(\mu \)-calculus: Interpolation, Lyndon and Łoś–Tarski, Journal of Symbolic Logic 65(1): 310–332, 2000.

  12. Fine, B., A. Gaglione, A. Myasnikov, G. Rosenberger, and D. Spellman, The Elementary Theory of Groups, digital original edn., Expositions in Mathematics, De Gruyter, 2014.

  13. Goldblatt, R., Diodorean modality in Minkowski spacetime, Studia Logica 39: 219–236, 1980.

    Article  Google Scholar 

  14. Goldblatt, R., Logics of Time and Computation, 2 edn., no. 7 in CSLI Lecture Notes, Center for the Study of Language and Information, 1992.

  15. Hamkins, J. D., A simple maximality principle, Journal of Symbolic Logic 68(2): 527–550, 2003.

    Article  Google Scholar 

  16. Hamkins, J. D., The modal logic of arithmetic potentialism and the universal algorithm, ArXiv e-prints 2018, 1–35. arXiv:1801.04599.

  17. Hamkins, J. D., G. Leibman, and B. Löwe, Structural connections between a forcing class and its modal logic, Israel Journal of Mathematics 207: 617–651, 2015.

    Article  Google Scholar 

  18. Hamkins, J. D., and B. Löwe, The modal logic of forcing, Transactions of the American Mathematical Society 360(4): 1793–1817, 2007.

    Article  Google Scholar 

  19. Henk, P., Kripke models built from models of arithmetic, in M. Aher, D. Hole, E. Jeřábek, and C. Kupke (eds.), Logic, Language, and Computation, Springer, Berlin Heidelberg, 2015, pp. 157–174.

  20. Ignatiev, K. N., The provability logic for \({\Sigma }_{1}\)-interpolability, Annals of Pure and Applied Logic 64(1): 1–25, 1993.

    Article  Google Scholar 

  21. Inamdar, T., and B. Löwe, The modal logic of inner models, Journal of Symbolic Logic 81(1): 225–236, 2016.

    Article  Google Scholar 

  22. Mal’cev, A. I., Algebraic Systems, Die Grundlehren der mathematischen Wissenschaften 192, Springer-Verlag, Berlin Heidelberg, 1973.

    Google Scholar 

  23. Marker, D., Model Theory: An Introduction, Graduate Texts in Mathematics, Springer, New York, 2002.

    Google Scholar 

  24. Rosenstein, J. G., Linear Orderings, Academic Press, 1982.

  25. Saveliev, D. I., On first-order expressibility of satisfiability in submodels, in R. Iemhoff et al. (eds.), WoLLIC 2019, Lecture Notes in Computer Science 11541, 2019, pp. 584–593.

  26. Saveliev, D. I., and I. B. Shapirovsky, On modal logic of submodels, in 11th Advances in Modal Logic, Short Papers, 2016, pp. 115–119.

  27. Shavrukov, V. Yu., The logic of relative interpretability over Peano arithmetic, Tech. Rep. 4, Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, 1988. In Russian.

  28. Shavrukov, V. Yu., Subalgebras of diagonalizable algebras of theories containing arithmetic, Instytut Matematyczny Polskiej Akademii Nauk, 1993.

  29. Shehtman, V. B., Modal logics of domains on the real plane, Studia Logica 42(1): 63–80, 1983.

    Article  Google Scholar 

  30. Shehtman, V. B., Modal counterparts of Medvedev logic of finite problems are not finitely axiomatizable, Studia Logica 49(3): 365–385, 1990.

    Article  Google Scholar 

  31. Solovay, R. M., Provability interpretations of modal logic, Israel Journal of Mathematics 25(3): 287–304, 1976.

    Article  Google Scholar 

  32. van Benthem, J., Logical Dynamics of Information and Interaction, Cambridge University Press, New York, NY, USA, 2014.

  33. Veltman, F., Defaults in update semantics, Journal of Philosophical Logic 25(3): 221–261, 1996.

    Article  Google Scholar 

  34. Visser, A., An overview of interpretability logic, in M. Kracht, M. de Rijke, H. Wansing, and M. Zakharyaschev (eds.), Advances in Modal Logic 1, CSLI Publications, 1996, pp. 307–359.

  35. Visser, A., The interpretability of inconsistency: Feferman’s theorem and related results, Logic Group preprint series 318, 2014.

  36. Westerståhl, D., Quantifiers in formal and natural languages, in Handbook of Philosophical Logic, 2nd edn., vol. 14, Springer, 2007, pp. 234–339.

Download references

Acknowledgements

We are grateful to the reviewers for their multiple suggestions and questions on earlier versions of the paper. We are also grateful to Lev Beklemishev, Philip Kremer, Fedor Pakhomov, Vladimir Shavrukov, Valentin Shehtman, Albert Visser for their attention to our work, useful comments, and discussions. The work on this paper was supported by the Russian Science Foundation under Grant 16-11-10252 and carried out at Steklov Mathematical Institute of Russian Academy of Sciences.

Author information

Authors and Affiliations

  1. Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia

    Denis I. Saveliev & Ilya B. Shapirovsky

  2. Institute for Information Transmission Problems of Russian Academy of Sciences, Moscow, Russia

    Denis I. Saveliev & Ilya B. Shapirovsky

Authors
  1. Denis I. Saveliev
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ilya B. Shapirovsky
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ilya B. Shapirovsky.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Saveliev, D.I., Shapirovsky, I.B. On Modal Logics of Model-Theoretic Relations. Stud Logica 108, 989–1017 (2020). https://doi.org/10.1007/s11225-019-09885-y

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11225-019-09885-y

Keywords

Search

Navigation