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The Modal Logic of Bayesian Belief Revision

Abstract

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.

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Acknowledgements

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.

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Correspondence to Zalán Gyenis.

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Brown, W., Gyenis, Z. & Rédei, M. The Modal Logic of Bayesian Belief Revision. J Philos Logic 48, 809–824 (2019). https://doi.org/10.1007/s10992-018-9495-9

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Keywords

  • Modal logic
  • Bayesian inference
  • Bayes learning
  • Bayes logic
  • Medvedev frames