Journal of Philosophical Logic

, Volume 28, Issue 1, pp 61–79

Iterative and fixed point common belief

  • Aviad Heifetz

Abstract

We define infinitary extensions to classical epistemic logic systems, and add also a common belief modality, axiomatized in a finitary, fixed-point manner. In the infinitary K system, common belief turns to be provably equivalent to the conjunction of all the finite levels of mutual belief. In contrast, in the infinitary monotonic system, common belief implies every transfinite level of mutual belief but is never implied by it. We conclude that the fixed- point notion of common belief is more powerful than the iterative notion of common belief.

common knowledge common belief infinitary logic 

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REFERENCES

  1. Barwise, J. (1988), Three views of common knowledge, in: Proc. of the 2nd Conference on Theoretical Aspects of Reasoning about Knowledge, Morgan Kaufmann, California, pp. 365–379.Google Scholar
  2. Barwise, J. (1989), On the model theory of common knowledge, in: The Situation in Logic, CSLI Lecture Notes 17, pp. 201–220.Google Scholar
  3. Chellas, B. F. (1980), Modal Logic, an Introduction, Cambridge University Press, Cambridge.Google Scholar
  4. Fagin, R. (1994), A quantitative analysis of modal logic, J. Symbolic Logic 59, 209–252.Google Scholar
  5. Fagin, R., Halpern, J. Y., Moses, Y. and Vardi, M. (1995), Reasoning about Knowledge, MIT Press.Google Scholar
  6. Geanakoplos, J. (1995), Common knowledge, in: R. J. Aumann and S. Hart (eds.), Handbook of Game Theory with Economic Applications, Vol. II, Chapter 40, Elsevier– North-Holland.Google Scholar
  7. Halpern, J. Y. and Moses, Y. O. (1992), A guide to completeness and complexity for modal logics of knowledge and beliefs, Artif. Intell. 54, 319–379.Google Scholar
  8. Heifetz, A. (1997), Infinitary S5 epistemic logic, Math. Log. Quart. 43, 333–342. A preliminary version appeared in the Proceedings of the 5th Conference on Theoretical Aspects of Reasoning about Knowledge, Morgann Kaufmann, California, pp. 95–107.Google Scholar
  9. Heifetz, A. (1996), Common belief in monotonic epistemic logic, Mathematical Social Sciences 32, 109–123.Google Scholar
  10. Kaneko, M. and Nagashima, T. (1997), Axiomatic indefinability of common knowledge in finitary logics, in: M. Bacharach, L. A. Gerard-Varet, P. Mongin and H. Shin (eds.), Epistemic Logic and the Theory of Games and Decisions, Kluwer Academic Publishers.Google Scholar
  11. Karp, C. R. (1964), Languages with Expressions of Infinite Length, North-Holland Publ. Comp., Amsterdam.Google Scholar
  12. Lismont, L. (1992), La Connaissance Commune – Approches Modale, Ensembliste et Probabiliste, Ph.D. Dissertation, Université Catholique de Louvain.Google Scholar
  13. Lismont, L. (1993), La connaissance commune en logique modale, Mathematical Logic Quarterly 39, 115–130.Google Scholar
  14. Lismont, L. (1995), Common knowledge: Relating anti-founded situation semantics to modal logic neighborhood semantics, J. Logic, Language and Information 3, 285–302.Google Scholar
  15. Lismont, L. and Mongin, P. (1994), On the logic of common belief and common knowledge, Theory and Decision 37, 75–106.Google Scholar
  16. Lismont, L. and Mongin, P. (1995), Belief closure: A semantics of common knowledge for modal propositional logic, Mathematical Social Sciences 30, 127–153.Google Scholar
  17. Monderer, D. and Samet, D. (1989), Approximating common knowledge with common beliefs, Games and Economic Behavior 1, 170–190.Google Scholar

Copyright information

© Kluwer Academic Publishers 1999

Authors and Affiliations

  • Aviad Heifetz
    • 1
  1. 1.The School of EconomicsTel Aviv UniversityTel AvivIsrael

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