Journal of Philosophical Logic

, Volume 48, Issue 5, pp 809–824 | Cite as

The Modal Logic of Bayesian Belief Revision

  • William Brown
  • Zalán GyenisEmail author
  • Miklós Rédei
Open Access


In Bayesian belief revision a Bayesian agent revises his prior belief by conditionalizing the prior on some evidence using Bayes’ rule. We define a hierarchy of modal logics that capture the logical features of Bayesian belief revision. Elements in the hierarchy are distinguished by the cardinality of the set of elementary propositions on which the agent’s prior is defined. Inclusions among the modal logics in the hierarchy are determined. By linking the modal logics in the hierarchy to the strongest modal companion of Medvedev’s logic of finite problems it is shown that the modal logic of belief revision determined by probabilities on a finite set of elementary propositions is not finitely axiomatizable.


Modal logic Bayesian inference Bayes learning Bayes logic Medvedev frames 



We are truly grateful to the anonymous referees: their careful reading of the manuscript and their helpful comments have greatly improved the paper. We would like to thank Prof. David Makinson for his comments on an early version of this paper. Research supported in part by the Hungarian Scientific Research Found (OTKA). Contract number: K 115593. Zalán Gyenis was supported by the Premium Postdoctoral Grant of the Hungarian Academy of Sciences.


  1. 1.
    Alchourrón, C., Gärdenfors, P., Makinson, D. (1985). On the logic of theory change: partial meet contraction and revision functions. The Journal of Symbolic Logic, 50, 510–530.CrossRefGoogle Scholar
  2. 2.
    Bacchus, F. (1990). Probabilistic belief logics. In Proceedings of European conference on artificial intelligence (ECAI-90) (pp. 59–64).Google Scholar
  3. 3.
    Blackburn, P., de Rijke, M., Venema, Y. (2002). Modal logic. Cambridge University Press.Google Scholar
  4. 4.
    Chagrov, A., & Zakharyaschev, M. (1997). Modal logic. Oxford: Clarendon Press.Google Scholar
  5. 5.
    Diaconis, P., & Zabell, S. (1983). Some alternatives to Bayes’ rule. Technical Report 205 Stanford University.Google Scholar
  6. 6.
    Gärdenfors, P. (1988). Knowledge in flux: modeling the dynamics of epistemic states. MIT Press.Google Scholar
  7. 7.
    Gyenis, Z. (2018). On the modal logic of Jeffrey conditionalization. Logica Universalis. To appear.Google Scholar
  8. 8.
    Gyenis, Z. (2018). Standard Bayes logic is not finitely axiomatizable. Submitted for publication.Google Scholar
  9. 9.
    Gyenis, Z., & Rédei, M. (2017). General properties of Bayesian learning as statistical inference determined by conditional expectations. The Review of Symbolic Logic, 10(4), 719–755.CrossRefGoogle Scholar
  10. 10.
    Jeffrey, R.C. (1965). The logic of decision, 1st Edn. Chicago: The University of Chicago Press.Google Scholar
  11. 11.
    Kolmogorov, A.N. (1933). Grundbegriffe der Wahrscheinlichkeitsrechnung. Berlin: Springer. English translation: Foundations of the Theory of Probability, (Chelsea New York, 1956).CrossRefGoogle Scholar
  12. 12.
    Łazarz, M. (2013). Characterization of Medvedev’s logic by means of Kubiński’s frames. Bulletin of the Section of Logic, 42, 83–90.Google Scholar
  13. 13.
    Maksimova, L., Shehtman, V., Skvortsov, D. (1979). The impossibility of a finite axiomatization of Medvedev’s logic of finitary problems. Soviet Math Dokl, 20(2), 394–398.Google Scholar
  14. 14.
    Medvedev, Yu.T. (1962). Finite problems. Soviet Mathematics, 3(1), 227–230.Google Scholar
  15. 15.
    Medvedev, Yu.T. (1963). Interpretation of logical formulas by means of finite problems and its relation to the realizability theory. Soviet Mathematics, 4(1), 180–183.Google Scholar
  16. 16.
    Nilsson, N.J. (1986). Probabilistic logic. Artificial Intelligence, 71–87.CrossRefGoogle Scholar
  17. 17.
    Shehtman, V. (1990). Modal counterparts of Medvedev logic of finite problems are not finitely axiomatizable. Studia Logica, 49, 365–385.CrossRefGoogle Scholar
  18. 18.
    van Benthem, J. (2007). Dynamic logic for belief revision. Journal of Applied Non-Classical Logics, 17(2), 129–155.CrossRefGoogle Scholar
  19. 19.
    van Ditmarsch, H., van Der Hoek, W., Kooi, B. (2007). Dynamic epistemic logic Vol. 1. New York: Springer.Google Scholar
  20. 20.
    Williamson, J. (2010). In defence of objective Bayesianism. Oxford: Oxford University Press.CrossRefGoogle Scholar

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© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Department of LogicEötvös Loránd UniversityBudapestHungary
  2. 2.Department of LogicJagiellonian UniversityKrakówPoland
  3. 3.Department of Philosophy, Logic and Scientific MethodLondon School of Economics and Political ScienceLondonUK

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