A Note on Assumption-Completeness in Modal Logic

  • Jonathan A. Zvesper
  • Eric Pacuit
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6006)

Abstract

We study the notion of assumption-completeness, which is a property of belief models first introduced in [18]. In that paper it is considered a limitative result – of significance for game theory – if a given language does not have an assumption-complete belief model. We show that there are assumption-complete models for the basic modal language (Theorem 8).

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Apt, K.R., Zvesper, J.A.: Common Beliefs and Public Announcements in Strategic Games with Arbitrary Strategy Sets. Manuscript (2007); CoRR abs/0710.3536 Google Scholar
  2. 2.
    Areces, C., Blackburn, P., Marx, M.: Hybrid Logic is the Bounded Fragment of First Order Logic. In: de Queiroz, R., Carnielli, W. (eds.) WoLLIC 1999, pp. 33–50. Rio de Janeiro, Brazil (1999)Google Scholar
  3. 3.
    Aumann, R.J.: Agreeing to Disagree. Ann. Stat. 4(6), 1236–1239 (1976)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Baltag, A.: A Structural Theory of Sets. Ph.D. thesis, Indiana University (1998)Google Scholar
  5. 5.
    Baltag, A., Smets, S., Zvesper, J.A.: When All is Done but Not (Yet) Said: Dynamic Rationality in Extensive Games. In: van Benthem, J., Pacuit, E. (eds.) Proceedings of Workshop on Logic and Intelligent Interaction, ESSLLI (2008)Google Scholar
  6. 6.
    Battigalli, P., Bonanno, G.: Recent Results on Belief, Knowledge and the Epistemic Foundations of Game Theory. Res. Econ. 53, 149–225 (1999)CrossRefGoogle Scholar
  7. 7.
    Battigalli, P., Siniscalchi, M.: Strong Belief and Forward Induction Reasoning. J. Econ. Theory 106(2), 356–391 (2002)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    van Benthem, J.: Minimal Deontic Logics. Bull. Section Log. 8(1), 36–42 (1979)MATHGoogle Scholar
  9. 9.
    van Benthem, J.: Games in Dynamic Epistemic Logic. Bull. Econ. Res. 53(4), 219–248 (2001)CrossRefMathSciNetGoogle Scholar
  10. 10.
    van Benthem, J.: Rational Dynamics and Epistemic Logic in Games. Int. Game Theory Rev. 9(1), 13–45 (2007) (Erratum reprint 9(2), 377–409)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    van Benthem, J., van Otterloo, S., Roy, O.: Preference Logic, Conditionals, and Solution Concepts in Games. ILLC Publications PP-2005-28. Universiteit van Amsterdam (2005)Google Scholar
  12. 12.
    Blackburn, P., de Rijke, M., Venema, Y.: Modal Logic. Cambridge Tracts in Theoretical Computer Science, vol. 53. Cambridge University Press, Cambridge (2001)MATHGoogle Scholar
  13. 13.
    Board, O.: Dynamic Interactive Epistemology. Games Econ. Behav. 49, 49–80 (2002)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Bonanno, G.: Modal Logic and Game Theory: Two Alternative Approaches. Risk, Decision and Policy 7(3), 309–324 (2002)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Brandenburger, A.: On the Existence of a “Complete” Possibility Structure. In: Dimitri, N., Basili, M., Giboa, I. (eds.) Cognitive Processes and Economic Behavior, pp. 30–34. Routledge, London (2003)Google Scholar
  16. 16.
    Brandenburger, A.: The Power of Paradox: Some Recent Developments in Interactive Epistemology. Int. J. Game Theory 35(4), 465–492 (2007)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Brandenburger, A., Friedenberg, A., Keisler, H.J.: Admissibility in Games. Econometrica 76(2), 307–352 (2008)MATHMathSciNetGoogle Scholar
  18. 18.
    Brandenburger, A., Keisler, H.J.: An Impossibility Theorem on Beliefs in Games. Stud. Log. 84(2), 211–240 (2006)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    de Bruin, B.: Explaining Games: On the Logic of Game Theoretic Explanations. Ph.D. thesis, Universiteit van Amsterdam (2004); ILLC Publications DS-2004-03Google Scholar
  20. 20.
    ten Cate, B.: Model Theory for Extended Modal Languages. Ph.D. thesis, Universiteit van Amsterdam (2005); ILLC Publications DS-2005-01Google Scholar
  21. 21.
    Devlin, K.: The Joy of Sets: Fundamentals of Contemporary Set Theory. Undergraduate Texts in Mathematics. Springer, Heidelberg (1993)MATHGoogle Scholar
  22. 22.
    Feferman, S.: Persistent and Invariant Formulas for Outer Extensions. Compositio Math. 20, 29–52 (1968)MATHMathSciNetGoogle Scholar
  23. 23.
    Halpern, J.Y., Lakemeyer, G.: Multi-agent Only Knowing. J. Log. Comp. 11(1), 41–70 (2001)MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Halpern, J.Y., Moses, Y.: Characterizing Solution Concepts in Games Using Knowledge-based Programs. In: Veloso, M.M. (ed.) IJCAI 2007, pp. 1300–1307. Morgan Kaufmann, San Francisco (2007)Google Scholar
  25. 25.
    Harsanyi, J.C.: Games with Incompletete Information Played by ‘Bayesian’ Players. Part I: The Basic Model. Management Sci. 14(3), 159–182 (1967)MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Harsanyi, J.C.: Games with Incompletete Information Played by ‘Bayesian’ Players. Part II: Bayesian Equilibrium Points. Management Sci. 14(5), 320–334 (1968)MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Harsanyi, J.C.: Games with Incompletete Information Played by ‘Bayesian’ Players. Part III: The Basic Probability Distribution of the Game. Management Sci. 14(7), 486–502 (1968)MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Hintikka, J.: Knowledge and Belief: an Introduction to the Logic of the Two Notions. Cornell University Press (1962)Google Scholar
  29. 29.
    Humberstone, I.L.: The modal Logic of All and Only. Notre Dame J. Form. Log. 28, 177–188 (1987)MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Levesque, H.J.: All I Know: a Study in Autoepistemic Logic. Artif. Intell. 42(2-3), 263–309 (1990)MATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Mariotti, T., Meier, M., Piccione, M.: Hierarchies of Beliefs for Compact Possibility Models. J. Math. Econ. 41, 303–324 (2005)MATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Pacuit, E.: Understanding the Brandenburger-Keisler Paradox. Stud. Log. 86(3), 435–454 (2007)MATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Zvesper, J.A.: The Brandenburger-Keisler Paradox in Normal Modal Logics (2007) (manuscript)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Jonathan A. Zvesper
    • 1
  • Eric Pacuit
    • 2
  1. 1.Computing LaboratoryUniversity of OxfordOxfordUnited Kingdom
  2. 2.Center for Logic and Philosophy of ScienceUniversiteit van TilburgTilburgThe Netherlands

Personalised recommendations