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On the Logic of Belief and Propositional Quantification

Abstract

We consider extending the modal logic KD45, commonly taken as the baseline system for belief, with propositional quantifiers that can be used to formalize natural language sentences such as “everything I believe is true” or “there is something that I neither believe nor disbelieve.” Our main results are axiomatizations of the logics with propositional quantifiers of natural classes of complete Boolean algebras with an operator (BAOs) validating KD45. Among them is the class of complete, atomic, and completely multiplicative BAOs validating KD45. Hence, by duality, we also cover the usual method of adding propositional quantifiers to normal modal logics by considering their classes of Kripke frames. In addition, we obtain decidability for all the concrete logics we discuss.

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Correspondence to Yifeng Ding.

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I thank Wesley Holliday for his numerous helpful suggestions and Fengkui Ju at the 2019 Modal Logic Workshop at Peking University for reminding me that 4 strengthens 4.

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Ding, Y. On the Logic of Belief and Propositional Quantification. J Philos Logic 50, 1143–1198 (2021). https://doi.org/10.1007/s10992-021-09595-8

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Keywords

  • Modal logic
  • Doxastic logic
  • Propositional quantifiers
  • Algebraic semantics
  • Pesudo-monadic algebras
  • Kripke incompleteness