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On the Modal Logic of Jeffrey Conditionalization

  • Zalán Gyenis
Open Access
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Abstract

We continue the investigations initiated in the recent papers (Brown et al. in The modal logic of Bayesian belief revision, 2017; Gyenis in Standard Bayes logic is not finitely axiomatizable, 2018) where Bayes logics have been introduced to study the general laws of Bayesian belief revision. In Bayesian belief revision a Bayesian agent revises (updates) his prior belief by conditionalizing the prior on some evidence using the Bayes rule. In this paper we take the more general Jeffrey formula as a conditioning device and study the corresponding modal logics that we call Jeffrey logics, focusing mainly on the countable case. The containment relations among these modal logics are determined and it is shown that the logic of Bayes and Jeffrey updating are very close. It is shown that the modal logic of belief revision determined by probabilities on a finite or countably infinite set of elementary propositions is not finitely axiomatizable. The significance of this result is that it clearly indicates that axiomatic approaches to belief revision might be severely limited.

Keywords

Modal logic Bayesian inference Bayes learning Bayes logic Jeffrey learning Jeffrey conditionalization 

Mathematics Subject Classification

Primary 03B42 03B45 Secondary 03A10 

Notes

Acknowledgements

The author is grateful to Miklós Rédei for all the pleasant conversations about this topic (and often about more important other topics). The author would like to acknowledge the Premium Postdoctoral Grant of the Hungarian Academy of Sciences hosted by the Logic Department at Eötvös Loránd University, and the Hungarian Scientific Research Found (OTKA), Contract No. K115593.

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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 (http://creativecommons.org/licenses/by/4.0/), 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 LogicJagiellonian UniversityKrakówPoland

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