Neighborhood Contingency Logic

  • Jie Fan
  • Hans van Ditmarsch
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8923)


A formula is contingent, if it is possibly true and possibly false; a formula is non-contingent, if it is not contingent, i.e., if it is necessarily true or necessarily false. In this paper, we propose a neighborhood semantics for contingency logic, in which the interpretation of the non-contingency operator is consistent with its philosophical intuition. Based on this semantics, we compare the relative expressivity of contingency logic and modal logic on various classes of neighborhood models, and investigate the frame definability of contingency logic. We present a decidable axiomatization for classical contingency logic (the obvious counterpart of classical modal logic), and demonstrate that for contingency logic, neighborhood semantics can be seen as an extension of Kripke semantics.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Areces, C., Figueira, D.: Which semantics for neighbourhood semantics? In: IJCAI, pp. 671–676 (2009)Google Scholar
  2. 2.
    Brogan, A.: Aristotle’s logic of statements about contingency. Mind 76(301), 49–61 (1967)CrossRefGoogle Scholar
  3. 3.
    Chellas, B.F.: Modal Logic: An Introduction. Cambridge University Press (1980)Google Scholar
  4. 4.
    Cresswell, M.: Necessity and contingency. Studia Logica 47, 145–149 (1988)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Fan, J., Wang, Y., van Ditmarsch, H.: Almost necessary. Advances in Modal Logic 10, 178–196 (2014)Google Scholar
  6. 6.
    Fan, J., Wang, Y., van Ditmarsch, H.: Contingency and knowing whether (to appear, 2014)Google Scholar
  7. 7.
    Hansen, H.H., Kupke, C., Pacuit, E.: Neighbourhood structures: Bisimilarity and basic model theory. Logical Methods in Computer Science 5(2), 1–38 (2009)CrossRefMathSciNetGoogle Scholar
  8. 8.
    van der Hoek, W., Lomuscio, A.: A logic for ignorance. Electronic Notes in Theoretical Computer Science 85(2), 117–133 (2004)CrossRefGoogle Scholar
  9. 9.
    Ma, M., Sano, K.: How to update neighborhood models. In: Grossi, D., Roy, O., Huang, H. (eds.) LORI. LNCS, vol. 8196, pp. 204–217. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  10. 10.
    Montague, R.: Universal grammar. Theoria 36, 373–398 (1970)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Montgomery, H., Routley, R.: Contingency and non-contingency bases for normal modal logics. Logique et Analyse 9, 318–328 (1966)zbMATHMathSciNetGoogle Scholar
  12. 12.
    Pacuit, E.: Neighborhood semantics for modal logic: An introduction. ESSLLI Lecture (2007),
  13. 13.
    Scott, D.: Advice on modal logic. In: Philosophical Problems in Logic: Some Recent Developments, pp. 143–173 (1970)Google Scholar
  14. 14.
    Steinsvold, C.: A note on logics of ignorance and borders. Notre Dame Journal of Formal Logic 49(4), 385–392 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Wheeler, G.: AGM belief revision in monotone modal logics. In: Proc. of LPAR 2010 (2010)Google Scholar
  16. 16.
    Zolin, E.: Completeness and definability in the logic of noncontingency. Notre Dame Journal of Formal Logic 40(4), 533–547 (1999)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Jie Fan
    • 1
  • Hans van Ditmarsch
    • 2
  1. 1.Department of PhilosophyPeking UniversityChina
  2. 2.LORIA, CNRSUniversité de LorraineFrance

Personalised recommendations