Symmetric Contingency Logic with Unlimitedly Many Modalities

Abstract

The completeness of the axiomatization of contingency logic over symmetric frames has been thought of as a nontrivial job, the unimodal case of which cannot be generalized to the finitely multimodal case, which in turn cannot be generalized to the infinitely multimodal case. This paper deals with the completeness of symmetric contingency logic with unlimitedly many modalities, no matter whether the set of modalities is finite or infinite.

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

References

  1. 1.

    Costa-Leite, A. (2007). Interactions of metaphysical and epistemic concepts. Université de Neuchâtel: PhD thesis.

    Google Scholar 

  2. 2.

    Fan, J. (2018). A logic of temporal contingency without propositional constants. Under submission.

  3. 3.

    Fan, J., Wang, Y., van Ditmarsch, H. (2014). Almost necessary. In Advances in modal logic (Vol. 10, pp. 178–196).

  4. 4.

    Fan, J., Wang, Y., van Ditmarsch, H. (2015). Contingency and knowing whether. The Review of Symbolic Logic, 8(1), 75–107.

    Article  Google Scholar 

  5. 5.

    Humberstone, L. (1995). The logic of non-contingency. Notre Dame Journal of Formal Logic, 36(2), 214–229.

    Article  Google Scholar 

  6. 6.

    Kuhn, S. (1995). Minimal non-contingency logic. Notre Dame Journal of Formal Logic, 36(2), 230–234.

    Article  Google Scholar 

  7. 7.

    Liu, F., Seligman, J., Girard, P. (2014). Logical dynamics of belief change in the community. Synthese, 191(11), 2403–2431.

    Article  Google Scholar 

  8. 8.

    Montgomery, H., & Routley, R. (1966). Contingency and non-contingency bases for normal modal logics. Logique et Analyse, 9, 318–328.

    Google Scholar 

  9. 9.

    Steinsvold, C. (2008). A note on logics of ignorance and borders. Notre Dame Journal of Formal Logic, 49(4), 385–392.

    Article  Google Scholar 

  10. 10.

    van der Hoek, W., & Lomuscio, A. (2004). A logic for ignorance. Electronic Notes in Theoretical Computer Science, 85(2), 117–133.

    Article  Google Scholar 

  11. 11.

    Von Wright, G.H. (1951). Deontic logic. Mind, 60(237), 1–15.

    Article  Google Scholar 

  12. 12.

    Zolin, E. (1999). Completeness and definability in the logic of noncontingency. Notre Dame Journal of Formal Logic, 40(4), 533–547.

    Article  Google Scholar 

  13. 13.

    Zolin, E. (2001). Sequent logic of arithmetic decidability. Moscow University Mathematics Bulletin, 56(6), 22–27.

    Google Scholar 

Download references

Acknowledgements

This research is supported by the project 17CZX053 of National Social Science Fundation of China. We would like to acknowledge an anonymous referee for his/her insightful comments.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Jie Fan.

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

Verify currency and authenticity via CrossMark

Cite this article

Fan, J. Symmetric Contingency Logic with Unlimitedly Many Modalities. J Philos Logic 48, 851–866 (2019). https://doi.org/10.1007/s10992-018-09498-1

Download citation

Keywords

  • Contingency
  • Symmetric frames
  • Axiomatization
  • Completeness
  • Unlimitedly many modalities