Skip to main content

A Unified Logic for Contingency and Accident


As shown in Fan (Journal of Philosophical Logic, 48, 425–445, 2019), there are some similarities/resemblances between contingency and accident. Given this, one may naturally ask if we can unify the two operators to manifest all of their similarities/resemblances. In this article, instead of looking at the interactions between the two operators like in Fan (Journal of Philosophical Logic, 48, 425–445, 2019), we turn our attention to the resemblances between the two operators. We extend the unification method in Fan (Logic Journal of the IGPL, 2020) to the current setting. The main results include some model-theoretical ones, such as expressivity, frame definability, bisimulation, and some axiomatization ones.

This is a preview of subscription content, access via your institution.


  1. Bakhtiari, Z., van Ditmarsch, H., & Hansen, H. H. (2017). Neighbourhood contingency bisimulation. In Indian conference on logic and its applications (pp. 48–63). Berlin: Springer.

  2. Balbiani, P., & Fan, J. (2017). A complete axiomatization of Euclidean strong non-contingency logic. In Proceedings of 12th Tbilisi Symposium of Language, Logic and Computation (TbiLLC) (pp. 43–48). Lagodekhi.

  3. Costa-Leite, A. (2016). Interplays of knowledge and non-contingency. Logic and Logical Philosophy, 25(4), 521–534.

    Google Scholar 

  4. Cresswell, M. (1988). Necessity and contingency. Studia Logica, 47, 145–149.

    Article  Google Scholar 

  5. Fan, J. (2015). Logics of essence and accident. arXiv:1506.01872. unpublished manuscript.

  6. Fan, J. (2018). Neighborhood contingency logic: a new perspective. Studies in Logic, 11(4), 37–55.

    Google Scholar 

  7. Fan, J. (2019). Bimodal logics with contingency and accident. Journal of Philosophical Logic, 48, 425–445.

    Article  Google Scholar 

  8. Fan, J. (2019). A family of neighborhood contingency logics. Notre Dame Journal of Formal Logic, 60(4), 683–699.

    Article  Google Scholar 

  9. Fan, J. (2019). Symmetric contingency logic with unlimitedly many modalities. Journal of Philosophical Logic, 48, 851–866.

    Article  Google Scholar 

  10. Fan, J. (2020). A family of Kripke contingency logics. Theoria, 86(4), 482–499.

    Article  Google Scholar 

  11. Fan, J. (2020). Logics of (in)sane and (un)reliable beliefs. Logic Journal of the IGPL.

  12. Fan, J. (2021). Bimodal logic with contingency and accident: Bisimulation and axiomatizations. Logica Universalis, 15, 123–147.

    Article  Google Scholar 

  13. Fan, J., & van Ditmarsch, H. (2015). Neighborhood contingency logic. In M. Banerjee S. Krishna (Eds.) Logic and Its Application, volume 8923 of Lecture Notes in Computer Science (pp. 88–99). Springer.

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

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

    Article  Google Scholar 

  16. Gilbert, D. R., & Venturi, G. (2016). Reflexive-insensitive modal logics. The Review of Symbolic Logic, 9(1), 167–180.

    Article  Google Scholar 

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

    Article  Google Scholar 

  18. Humberstone, L. (2013). Zolin and Pizzi: Defining necessity from noncontingency. Erkenntnis, 78(6), 1275–1302.

    Article  Google Scholar 

  19. Humberstone, L. (2021). Propositional variables occurring exactly once in candidate modal axioms. Filosofiska Notiser, 8(1), 27–73.

    Google Scholar 

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

    Article  Google Scholar 

  21. Marcos, J. (2005). Logics of essence and accident. Bulletin of the Section of Logic, 34(1), 43–56.

    Google Scholar 

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

    Google Scholar 

  23. Montgomery, H., & Routley, R. (1968). Modal reduction axioms in extensions of S1. Logique et Analyse, 11, 492–501.

    Google Scholar 

  24. Montgomery, H., & Routley, R. (1968). Non-contingency axioms for S4 and S5. Logique et Analyse, 11(43), 422–424.

    Google Scholar 

  25. Montgomery, H., & Routley, R. (1969). Modalities in a sequence of normal non-contingency modal systems. Logique et Analyse, 12, 225–227.

    Google Scholar 

  26. Mortensen, C. (1976). A sequence of normal modal systems with non-contingency bases. Logique et Analyse, 19, 341–344.

    Google Scholar 

  27. Pizzi, C. (2007). Necessity and relative contingency. Studia Logica, 85(3), 395–410.

    Article  Google Scholar 

  28. Steinsvold, C. (2008). Completeness for various logics of essence and accident. Bulletin of the Section of Logic, 37(2), 93–101.

    Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  32. Zolin, E. (2001). Infinitary expressibility of necessity in terms of contingency. In Proceedings of the sixth ESSLLI student session (pp. 325–334).

Download references


This research is supported by the project 17CZX053 of National Social Science Foundation of China. The author would like to thank two anonymous referees of this journal for insightful comments on an earlier version of this paper.

Author information

Authors and Affiliations


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. A Unified Logic for Contingency and Accident. J Philos Logic 51, 693–720 (2022).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


  • Contingency
  • Accident
  • Expressivity
  • Frame definability
  • Bisimulation
  • Axiomatizations
  • Public announcements
  • Unification