, 165:179 | Cite as

Probabilistic dynamic belief revision

  • Alexandru Baltag
  • Sonja Smets


We investigate the discrete (finite) case of the Popper–Renyi theory of conditional probability, introducing discrete conditional probabilistic models for knowledge and conditional belief, and comparing them with the more standard plausibility models. We also consider a related notion, that of safe belief, which is a weak (non-negatively introspective) type of “knowledge”. We develop a probabilistic version of this concept (“degree of safety”) and we analyze its role in games. We completely axiomatize the logic of conditional belief, knowledge and safe belief over conditional probabilistic models. We develop a theory of probabilistic dynamic belief revision, introducing probabilistic “action models” and proposing a notion of probabilistic update product, that comes together with appropriate reduction laws.


Belief revision Conditional belief Dynamic-epistemic logic Popper functions 


  1. Alchourrón C., Gärdenfors P., Makinson D. (1985) On the logic of theory change: Partial meet contraction and revision functions. Journal of Symbolic Logic 50: 510–530CrossRefGoogle Scholar
  2. Arlo-Costa H., Parikh R. (2005) Conditional probability and defeasible inference. Journal of Philosophical Logic 34: 97–119CrossRefGoogle Scholar
  3. Aucher, G. (2003). A combined system for update logic and belief revision. Master’s thesis, ILLC, University of Amsterdam, Amsterdam, The Netherlands.Google Scholar
  4. Aumann R. (1995) Backwards induction and common knowledge of rationality. Games and Economic Behavior 8: 6–19CrossRefGoogle Scholar
  5. Baltag A. (2002) A logic for suspicious players: Epistemic actions and belief updates in games. Bulletin of Economic Research 54(1): 1–46CrossRefGoogle Scholar
  6. Baltag, A., & Moss, L. (2004). Logics for epistemic programs. Synthese, 139, 165–224. Knowledge, Rationality & Action (pp. 1–60).Google Scholar
  7. Baltag, A., Moss, L., & Solecki, S. (1998). The logic of common knowledge, public announcements, and private suspicions. In I. Gilboa (Ed.), Proceedings of the 7th Conference on Theoretical Aspects of Rationality and Knowledge (TARK 98) (pp. 43–56).Google Scholar
  8. Baltag A., Smets S. (2006) Conditional doxastic models: A qualitative approach to dynamic belief revision. Electronic Notes in Theoretical Computer Science 165: 5–21CrossRefGoogle Scholar
  9. Baltag, A., & Smets, S. (2006b). Dynamic belief revision over multi-agent plausibility models. In W. van der Hoek & M. Wooldridge (Eds.), Proceedings of LOFT’06 (pp. 11–24). Liverpool.Google Scholar
  10. Baltag, A., & Smets, S. (2006c). The logic of conditional doxastic actions: A theory of dynamic multi-agent belief revision. In Proceedings of ESSLLI Workshop on Rationality and Knowledge.Google Scholar
  11. Baltag, A., & Smets, S. (2007). Probabilistic dynamic belief revision. In J. van Benthem, S. Ju, & F. Veltman (Eds.), Proceedings of LORI’07 (pp. 21–39). London: College Publications.Google Scholar
  12. Baltag A., Smets S. (2008) A qualitative theory of dynamic interactive belief revision. In: Bonanno G., van der Hoek W., Wooldridge M. (eds) Texts in logic and games (vol. 3). Amsterdam University Press, AmsterdamGoogle Scholar
  13. Board O. (2004) Dynamic interactive epistemology. Games and Economic Behaviour 49: 49–80CrossRefGoogle Scholar
  14. Boutilier C. (1995) On the revision of probabilistic belief states. Notre Dame Journal of Formal Logic 36(1): 158–183CrossRefGoogle Scholar
  15. Gärdenfors P. (1988) Knowledge in flux: Modeling the dynamics of epistemic states. Bradford Books, MIT Press, Cambridge, MAGoogle Scholar
  16. Gerbrandy, J. (1999). Dynamic epistemic logic. Logic, language and information (Vol. 2). Stanford: CSLI Publications, Stanford University.Google Scholar
  17. Halpern, J. (2001). Lexicographic probability, conditional probability, and nonstandard probability. In Proceedings of the Eighth Conference on Theoretical Aspects of Rationality and Knowledge (TARK 8) (pp. 17–30).Google Scholar
  18. Halpern J. (2003) Reasoning about uncertainty. MIT Press, Cambridge, MAGoogle Scholar
  19. Katsuno H., Mendelzon A. (1992) On the difference between updating a knowledge base and revising it. In: Gärdenfors P. (eds) Cambridge tracts in theoretical computer science. Cambridge University Press, Cambridge, pp 183–203Google Scholar
  20. Kooi B.P. (2003) Probabilistic dynamic epistemic logic. Journal of Logic, Language and Information 12: 381–408CrossRefGoogle Scholar
  21. Lehrer K. (1990) Theory of knowledge. Routledge, LondonGoogle Scholar
  22. Mihalache, D. (2007). Safe belief, rationality and backwards induction in games. Masters Thesis in Computer Science. Oxford: Oxford University.Google Scholar
  23. Pappas, G., Swain , M. (eds) (1978) Essays on knowledge and justification. Cornell University Press, Ithaca NYGoogle Scholar
  24. Popper, K. (1968). The logic of scientific discovery (revised Edition) (1st ed., 1934). London Hutchison.Google Scholar
  25. Renyi A. (1955) On a new axiomatic theory of probability. . Acta Mathematica Academiae Scientiarum Hungaricae 6: 285–335CrossRefGoogle Scholar
  26. Renyi A. (1964) Sur les espaces simples des Probabilites conditionnelles. Annales de L’institut Henri Poincare: Section B: Calcul des probabilities et statistique B B1: 3–21Google Scholar
  27. Stalnaker R. (1996) Knowledge, belief and counterfactual reasoning in games. Economics and Philosophy 12: 133–163CrossRefGoogle Scholar
  28. van Benthem J. (2003) Conditional probability meets update logic. Journal of Philosophical Logic 12(4): 409–421Google Scholar
  29. van Benthem, J. (2006). Dynamic logic of belief revision. Technical Report, ILLC, DARE electronic archive, University of Amsterdam. To appear in Journal of Applied Non-Classical Logics.Google Scholar
  30. van Benthem, J., Gerbrandy, J., & Kooi, B. (2006a). Dynamic update with probabilities. In W. van der Hoek & M. Wooldridge (Eds.), Proceedings of LOFT’06. Liverpool.Google Scholar
  31. van Benthem, J., & Liu, F. (2004). Dynamic logic of preference upgrade. Technical report. ILLC Research Report PP-2005-29.Google Scholar
  32. van Benthem J., van Eijck J., Kooi B. (2006) Logics of communication and change. Information and Computation 204(11): 1620–1662CrossRefGoogle Scholar
  33. van Ditmarsch H. (2005) Prolegomena to dynamic logic for belief revision. Synthese (Knowledge, Rationality & Action) 147: 229–275Google Scholar
  34. van Fraassen B. (1976) Representations of conditional probabilities. Journal of Philosophical Logic 5: 417–430Google Scholar
  35. van Fraassen B. (1995) Fine-grained opinion, probability, and the logic of full belief. Journal of Philosophical Logic 24: 349–377CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Oxford University Computing LaboratoryUniversity of OxfordOxfordUK
  2. 2.GPIUniversity of HertfordshireHatfieldUK
  3. 3.Center for Logic and Philosophy of ScienceVrije Universiteit BrusselBrusselsBelgium
  4. 4.IEGUniversity of OxfordOxfordUK

Personalised recommendations