On the ‘in many cases’ Modality: Tableaux, Decidability, Complexity, Variants

  • Costas D. Koutras
  • Christos Moyzes
  • Christos Nomikos
  • Yorgos Zikos
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8445)


The modality ‘true in many cases’ is used to handle non-classical patterns of reasoning, like ‘probably φ is the case’ or ‘normally φ holds’. It is of interest in Knowledge Representation as it has found interesting applications in Epistemic Logic, ‘Typicality’ logics, and it also provides a foundation for defining ‘normality’ conditionals in Non-Monotonic Reasoning. In this paper we contribute to the study of this modality, providing results on the ‘majority logic’ Θ of V. Jauregui. The logic Θ captures a simple notion of ‘a large number of cases’, which has been independently introduced by K. Schlechta and appeared implicitly in earlier attempts to axiomatize the modality ‘probably φ’. We provide a tableaux proof procedure for the logic Θ and prove its soundness and completeness with respect to the class of neighborhood semantics modelling ‘large’ sets of alternative situations. The tableaux-based decision procedure allows us to prove that the satisfiability problem for Θ is NP-complete. We discuss a more natural notion of ‘large’ sets which accurately captures ‘clear majority’ and we prove that it can be also used, at the high cost however of destroying the finite model property for the resulting logic. Then, we show how to extend our results in the logic of complete majority spaces, suited for applications where either a proposition or its negation (but not both) are to be considered ‘true in many cases’, a notion useful in epistemic logic.


default modality majority modal logic tableaux proof procedure 


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  1. 1.
    Askounis, D., Koutras, C.D., Zikos, Y.: Knowledge means ‘All’, belief means ‘Most’. In: del Cerro, et al. (eds.) [6], pp. 41–53Google Scholar
  2. 2.
    Blackburn, P., de Rijke, M., Venema, Y.: Modal Logic. Cambridge Tracts in Theoretical Computer Science, vol. 53. Cambridge University Press (2001)Google Scholar
  3. 3.
    Boutilier, C.: Conditional logics of normality: A modal approach. Artificial Intelligence 68(1), 87–154 (1994)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Burgess, J.P.: Probability logic. J. Symb. Log. 34(2), 264–274 (1969)CrossRefMATHGoogle Scholar
  5. 5.
    Chellas, B.F.: Modal Logic, An Introduction. Cambridge University Press (1980)Google Scholar
  6. 6.
    del Cerro, L.F., Herzig, A., Mengin, J. (eds.): JELIA 2012. LNCS, vol. 7519. Springer, Heidelberg (2012)MATHGoogle Scholar
  7. 7.
    Dubois, D., Welty, C.A., Williams, M.-A.(eds.): Principles of Knowledge Representation and Reasoning: Proceedings of the Ninth International Conference (KR 2004), Whistler, Canada, June 2-5. AAAI Press (2004)Google Scholar
  8. 8.
    Fitting, M., Mendelsohn, R.L.: First-Order Modal Logic. Synthése Library, vol. 277. Kluwer Academic Publishers (1998)Google Scholar
  9. 9.
    Fitting, M.C.: Proof Methods for Modal and Intuitionistic Logics. D. Reidel Publishing Co., Dordrecht (1983)CrossRefMATHGoogle Scholar
  10. 10.
    Goldblatt, R.: Logics of Time and Computation, 2nd edn. CSLI Lecture Notes, vol. 7. Center for the Study of Language and Information. Stanford University (1992)Google Scholar
  11. 11.
    Herzig, A.: Modal probability, belief, and actions. Fundam. Inform. 57(2-4), 323–344 (2003)MATHMathSciNetGoogle Scholar
  12. 12.
    Hughes, G.E., Cresswell, M.J.: A New Introduction to Modal Logic. Routledge (1996)Google Scholar
  13. 13.
    Jauregui, V.: The ‘Majority’ and ‘by Default’ Modalities. In: Orgun, Thornton (eds.) [16], pp. 263–272Google Scholar
  14. 14.
    Jauregui, V.: Modalities, Conditionals and Nonmonotonic Reasoning. PhD thesis, Department of Computer Science and Engineering, University of New South Wales (2008)Google Scholar
  15. 15.
    Koutras, C.D., Moyzes, C., Zikos, Y.: A modal logic of Knowledge, Belief and Estimation. Technical Report (2013), http://users.uop.gr/~ckoutras/KMZ-KBE-Full.pdf
  16. 16.
    Orgun, M.A., Thornton, J. (eds.): AI 2007. LNCS (LNAI), vol. 4830. Springer, Heidelberg (2007)Google Scholar
  17. 17.
    Pacuit, E.: Neighborhood semantics for modal logic: An introduction. Course Notes for ESSLLI 2007 (2007)Google Scholar
  18. 18.
    Pacuit, E., Salame, S.: Majority logic. In: Dubois, et al. (eds.) [7], pp. 598–605Google Scholar
  19. 19.
    Salame, S.: Majority Logic and Majority Spaces in contrast with Ultrafilters. PhD thesis, Graduate Center, City University of New York (2006)Google Scholar
  20. 20.
    Schlechta, K.: Filters and partial orders. Logic Journal of the IGPL 5(5), 753–772 (1997)CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Segerberg, K.: An essay in Clasical Modal Logic. Filosofiska Studies, Uppsala (1971)Google Scholar
  22. 22.
    van der Hoek, W.: On the semantics of graded modalities. Journal of Applied Non-Classical Logics 2(1) (1992)Google Scholar
  23. 23.
    Vardi, M.: On the complexity of epistemic reasoning. In: Proceedings of the Fourth Annual Symposium on Logic in Computer Science, pp. 243–252. IEEE Press, Piscataway (1989)CrossRefGoogle Scholar
  24. 24.
    Zikos, Y.: Modal Epistemic Logics without Negative Introspection: Epistemic structures and extensions with estimation and information. PhD thesis, Graduate Programme in Logic, Algorithms & Computation (MPLA), Dept. of Mathematics, University of Athens. In: Greek (2012)Google Scholar

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© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Costas D. Koutras
    • 1
  • Christos Moyzes
    • 2
  • Christos Nomikos
    • 3
  • Yorgos Zikos
    • 2
  1. 1.Department of Computer Science and TechnologyUniversity of PeloponneseTripolisGreece
  2. 2.Graduate Programme in Logic, Algorithms and Computation (MPLA), Department of MathematicsUniversity of AthensIlissiaGreece
  3. 3.Department of Computer Science and EngineeringUniversity of IoanninaIoanninaGreece

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