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Probabilities with Gaps and Gluts


Belnap-Dunn logic (BD), sometimes also known as First Degree Entailment, is a four-valued propositional logic that complements the classical truth values of True and False with two non-classical truth values Neither and Both. The latter two are to account for the possibility of the available information being incomplete or providing contradictory evidence. In this paper, we present a probabilistic extension of BD that permits agents to have probabilistic beliefs about the truth and falsity of a proposition. We provide a sound and complete axiomatization for the framework defined and also identify policies for conditionalization and aggregation. Concretely, we introduce four-valued equivalents of Bayes’ and Jeffrey updating and also suggest mechanisms for aggregating information from different sources.


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We dedicate this work to the memory of J. Michael Dunn. This work started from an inspiring talk by him and greatly benefited from his valuable and generous insights and comments throughout. Besides, we would like to thank Timothy Childers, Johannes Korbmacher, Olivier Roy, Frederik Van De Putte, an anonymous reviewer and the audience of the MCMP logic colloquium, the Amsterdam LIRA seminar, Advances in Philosophical Logic 2019 in Lublin, the Prague workshop on non-classical epistemic logics, the Tulips seminar in Utrecht, and the logic seminar at the University of Maryland for valuable feedback and suggestions. The work of OM was supported by the Czech Science Foundation. trough the project Reasoning with Graded Properties [GA18-00113S]. The work of DK and SRR was partially supported by Deutsche Forschungsgemeinschaft (DFG) and Agence Nationale de la Recherche (ANR) as part of the joint project Collective Attitude Formation [RO 4548/8-1], by DFG through the project From Shared Evidence to Group Attitudes [RO 4548/6-1], by DFG through the network grants Simulations of Social Scientific Inquiry [426833574] and Foundations, Applications and Theory of Inductive Logic [432308570], and by the National Science Foundation of China as part of the project Logics of Information Flow in Social Networks [17ZDA026].

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Correspondence to Dominik Klein.

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Klein, D., Majer, O. & Rafiee Rad, S. Probabilities with Gaps and Gluts. J Philos Logic 50, 1107–1141 (2021).

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  • Belnap-Dunn logic
  • First Degree Entailment
  • Non-standard probability theory
  • Probability theory
  • Bayes’ updating
  • Jeffrey updating
  • Probability aggregation