On the Modal Logic of Jeffrey Conditionalization
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.
KeywordsModal logic Bayesian inference Bayes learning Bayes logic Jeffrey learning Jeffrey conditionalization
Mathematics Subject ClassificationPrimary 03B42 03B45 Secondary 03A10
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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