Abstract
The present contribution shows that a Hilbert-style axiomatization for dynamic logic of relation changers is complete for the standard Kripke semantics not by a well-known rewriting technique but by the idea of an auxiliary semantics studied by van Benthem and Wang et al. A key insight of our auxiliary semantics for dynamic logic of relation changers can be described as: “relation changers are bounded morphisms.” Moreover, we demonstrate that this semantic insight can be used to provide a modular cut-free labelled sequent calculus for the logic in the sense that our calculus can be regarded as a natural expansion of a labelled sequent calculus of iteration-free propositional dynamic logic.
Similar content being viewed by others
References
Aucher, G., J. van Benthem, and D. Grossi, Modal logics of sabotage revisited, Journal of Logic and Computation 28(2):269–303, 2017.
Balbiani, P., V. Demange, and D. Galmiche, A sequent calculus with labels for PAL, Advances in Modal Logic 6, 2014.
Balbiani, P., H. van Ditmarsch, A. Herzig, and T. De Lima, Tableaux for public announcement logic, Journal of Logic and Computation 20(1):55–76, 2010.
Blackburn, P., M. De Rijke, and Y. Venema, Modal Logic, vol. 53, Cambridge University Press, 2002.
Bozzelli, L., H. van Ditmarsch, T. French, J. Hales, and S. Pinchinat, Refinement modal logic, Information and Computation 239:303–339, 2014.
Gerbrandy, J., and W. Groeneveld, Reasoning about information change, Journal of Logic, Language and Information 6(2):147–169, 1997.
Girard, P., J. Seligman, and F. Liu, General dynamic logic, Advances in Modal Logic 9:239–260, 2012.
Harel, D., D. Kozen, and J. Tiuryn, Dynamic logic, MIT Press, Cambridge, 2000.
Hatano, R., K. Sano, and S. Tojo, Cut free labelled sequent calculus for dynamic logic of relation changers, in S. C.-M. Yang, K. Y. Lee, and H. Ono, (eds.), Philosophical Logic: Current Trends in Asia, Springer Singapore, 2017, pp. 153–180.
Hoshi, T., Epistemic dynamics and protocol information, Stanford University, Stanford, 2009.
Kashima, R., Mathematical Logic, Asakura Publishing Co., Ltd, Tokyo, (in Japanese), 2009.
Kooi, B., and B. Renne, Arrow update logic, The Review of Symbolic Logic 4(4):536–559, 2011.
Liu, F., Reasoning about preference dynamics, vol. 354, Springer Science & Business Media, Berlin, 2011.
Ma, M., K. Sano, F. Schwarzentruber, and F. R. Velázquez-Quesada, Tableaux for non-normal public announcement logic, Logic and Its Applications 8923:132–145, 2015.
Ma, M., and J. Seligman, Algebraic semantics for dynamic dynamic logic, in W. van der Hoek, W. H. Holliday, and W. Wang, (eds.), Logic, Rationality, and Interaction, Springer Berlin, 2015, pp. 255–267.
Maffezioli, P., and A. Naibo, Proof theory of epistemic logic of programs, Logic and Logical Philosophy 23(3):301–328, 2013.
Maffezioli, P., and S. Negri, A Gentzen-style analysis of public announcement logic, in Proceedings of the International Workshop on Logic and Philosophy of Knowledge, Communication and Action, 2010, pp. 293–313.
Motoura, S., A general framework for dynamic epistemic logic: towards canonical correspondences, Journal of Applied Non-Classical Logics 27(1-2):50–89, 2017.
Negri, S., Proof analysis in modal logic, Journal of Philosophical Logic 34(5):507, 2005.
Negri, S., J. von Plato, and A. Ranta, Structural Proof Theory, Cambridge University Press, Cambridge, 2001.
Nomura, S., K. Sano, and S. Tojo, Revising a sequent calculus for public announcement logic, Structural Analysis of Non-classical Logics: The Proceedings of the Second Taiwan Philosophical Logic Colloquium (TPLC-2014), 2015, pp. 131–157.
Ono, H., and Y. Komori, Logics without the contraction rule, The Journal of Symbolic Logic 50(01):169–201, 1985.
Plaza, J., Logics of public communications, in M. L. Emrich, M. S. Pfeifer, M. Hadzikadic, and Z. W. Ras, (eds.), Proceedings of the 4th International Symposium on Methodologies for Intelligent Systems, 1989, pp. 201–216.
Segerberg, K., An Essay in Classical Modal Logic, vol. 13 of Filosofiska Studier, University of Uppsala, 1971.
Segerberg, K., Belief revision from the point of view of doxastic logic, Logic Journal of the IGPL 3(4):535–553, 1995.
Segerberg, K., Default logic as dynamic doxastic logic, Erkenntnis 50(2):333–352, 1999.
Seligman, J., Internalization: The Case of Hybrid Logics, Journal of Logic and Computation 11(5):671–689, 2001.
Troelstra, A. S., and H. Schwichtenberg, Basic proof theory, no. 43, 2nd edition, Cambridge University Press, Cambridge, 2000.
van Benthem, J., An Essay on Sabotage and Obstruction, Springer, Berlin, 2005, pp. 268–276.
van Benthem, J., Man Muss Immer Umkehren!, vol. 7 of Tributes, London: College Publications, 2008, pp. 53–66.
van Benthem, J., Two Logical Faces of Belief Revision, Springer Netherlands, Dordrecht, 2014, pp. 281–300.
van Benthem, J., J. Gerbrandy, T. Hoshi, and E. Pacuit, Merging frameworks for interaction, Journal of Philosophical Logic 38(5):491–526, 2009.
van Benthem, J., and F. Liu, Dynamic logic of preference upgrade, Journal of Applied Non-Classical Logics 17(2):157–182, 2007.
van Ditmarsch, H., W. van der Hoek, B. Kooi, and L. B. Kuijer, Arbitrary arrow update logic, Artificial Intelligence 242:80–106, 2017.
van Ditmarsch, H., W. van der Hoek, and B. P. Kooi, Dynamic epistemic logic, vol. 337, Springer, Berlin, 2007.
Wang, Y., and G. Aucher, An alternative axiomatization of del and its applications, in IJCAI International Joint Conference in Artificial Intelligence, 2013, pp. 1139–1146.
Wang, Y., and Q. Cao, On axiomatizations of public announcement logic, Synthese 190(1):103–134, 2013.
Yamada, T., Acts of commanding and changing obligations, in K. Inoue, K. Satoh, and F. Toni, (eds.), Computational Logic in Multi-Agent Systems, Springer, Heidelberg, 2007, pp. 1–19.
Acknowledgements
We would like to thank Johan van Benthem for his helpful and detailed comments to our draft. We also would like to thank Yanjing Wang for his comments and suggestions on a possible direction of further study at AWPL 2018. We would like to acknowledge anonymous reviewers for their helpful comments and suggestions on our manuscript. We are very grateful to editors and staff of Studia Logica for their kind and continuous supports. Both authors were partially supported by JSPS Core-to-Core Program (A. Advanced Research Networks). The work of the second author was also partially supported by JSPS KAKENHI Grant-in-Aid for Scientific Research (C) Grant Number 19K12113 and JSPS KAKENHI Grant-in-Aid for Scientific Research (B) Grant Number 17H02258.
Funding Funding was provided by Japan Society for the Promotion of Science (Core-to-Core Program A. Advanced Research Networks), Japan Society for the Promotion of Science (Grant No. 19K12113), and Japan Society for the Promotion of Science (Grant No. 17H02258).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Hatano, R., Sano, K. Recapturing Dynamic Logic of Relation Changers via Bounded Morphisms. Stud Logica 109, 95–124 (2021). https://doi.org/10.1007/s11225-020-09902-5
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11225-020-09902-5