Studia Logica

, Volume 86, Issue 3, pp 435–454 | Cite as

Understanding the Brandenburger-Keisler Paradox

  • Eric PacuitEmail author
Open Access


Adam Brandenburger and H. Jerome Keisler have recently discovered a two person Russell-style paradox. They show that the following configurations of beliefs is impossible: Ann believes that Bob assumes that Ann believes that Bob’s assumption is wrong. In [7] a modal logic interpretation of this paradox is proposed. The idea is to introduce two modal operators intended to represent the agents’ beliefs and assumptions. The goal of this paper is to take this analysis further and study this paradox from the point of view of a modal logician. In particular, we show that the paradox can be seen as a theorem of an appropriate hybrid logic.


Epistemic Foundations of Game Theory Epistemic Logic Belief Paradox 


  1. 1.
    Aumann R.(1999) ‘Interactive epistemology I: knowledge’. International Journal of Game Theory 28:263–300CrossRefGoogle Scholar
  2. 2.
    Blackburn, P., M. de Rijke, and Y. Venema, Modal Logic, Cambridge University Press, 2002.Google Scholar
  3. 3.
    Blackburn, P., and M. Marx, ‘Tableaux for quantified hybrid logic’, in Automated Reasoning with Analyctic Tableaux and Related Methods, 2002.Google Scholar
  4. 4.
    Blackburn, Patrick, Jerry Seligman (1995) ‘Hybrid languages’. Journal of Logic, Language and Information 4:251–272CrossRefGoogle Scholar
  5. 5.
    Bonanno, G., and P. Battigalli, ‘Recent results on belief, knowledge and the epistemic foundations of game theory’, Research in Economics 53 (2):149–225, June 1999.Google Scholar
  6. 6.
    Brandenburger, Adam, ‘The power of paradox: some recent developments in interactive epistomology’, International Journal of Game Theory, 2006 (forthcoming).Google Scholar
  7. 7.
    Brandenburger, Adam, and H. J. Keisler, ‘An impossibility theorem on beliefs in games, Studia Logica 84 (2):211–240, 2006, available at \(\tt{}\) , July 2004.Google Scholar
  8. 8.
    Chellas, Brian (1980) Modal Logic: An Introduction. Cambridge University Press, CambridgeGoogle Scholar
  9. 9.
    de Bruin, B., Explaining Games: On the Logic of Game Theoretic Explanations, PhD thesis, University of Amsterdam, 2004.Google Scholar
  10. 10.
    Fagin, R., J. Halpern, Y. Moses, and M. Vardi, Reasoning about Knowledge, The MIT Press, 1995.Google Scholar
  11. 11.
    Fitting, M., First-Order Logic and Automated Theorem Proving (second edition), Springer Verlag, 1996.Google Scholar
  12. 12.
    Fitting, M., and R. Mendelsohn, First-Order Modal Logic, Kluwer, 1998.Google Scholar
  13. 13.
    Goranko, V., ‘Modal definability in enriched languages’, Notre Dame Journal of Formal Logic 31, 1996.Google Scholar
  14. 14.
    Harsanyi, J. C., ‘Games with incompletete information played by bayesian players’ parts I-III, Management Sciences 14, 1967.Google Scholar
  15. 15.
    Hendricks, Vincent, Mainstream and Formal Epistemology, Cambridge University Press, 2006.Google Scholar
  16. 16.
    Humberstone, I. L., ‘The modal logic of ‘all and only", Notre Dame Journal of Formal Logic, 1987.Google Scholar
  17. 17.
    Mongin, P., M. O. L. Bacharach, L. A. Gerard-Varet, and H. S. Shin (eds.), Epistemic logic and the theory of games and decisions, Theory and Decision Library, Kluwer Academic Publishers, 1997.Google Scholar
  18. 18.
    Montague R.(1970) ‘Universal grammar’. Theoria 36:373–398CrossRefGoogle Scholar
  19. 19.
    Osborne, M., and A. Rubinstein, A Course in Game Theory, The MIT Press, 1994.Google Scholar
  20. 20.
    Passy S., Tinchev T.(1991) ‘An essay in combinatory dynamic logic’. Information and Computation 93:263–332CrossRefGoogle Scholar
  21. 21.
    Scott, Dana, ‘Advice in modal logic’, in K. Lambert, Philosophical Problems in Logic, 1970, pp. 143–173.Google Scholar
  22. 22.
    Stalnaker, Robert (1998) ‘Belief revision in games: forward and backward induction’. Mathematical Social Sciences (36):31–56CrossRefGoogle Scholar
  23. 23.
    vander Nat, A., ‘Beyond non-normal possible worlds’, Notre Dame Journal of Formal Logic 20, 1979.Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2007

Authors and Affiliations

  1. 1.Institute for Logic, Language and ComputationsUniversity of AmsterdamAmsterdamThe Netherlands

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