Abstract
Plausibility models are Kripke models that agents use to reason about knowledge and belief, both of themselves and of each other. Such models are used to interpret the notions of conditional belief, degrees of belief, and safe belief. The logic of conditional belief contains that modality and also the knowledge modality, and similarly for the logic of degrees of belief and the logic of safe belief. With respect to these logics, plausibility models may contain too much information. A proper notion of bisimulation is required that characterises them. We define that notion of bisimulation and prove the required characterisations: on the class of image-finite and preimage-finite models (with respect to the plausibility relation), two pointed Kripke models are modally equivalent in either of the three logics, if and only if they are bisimilar. As a result, the information content of such a model can be similarly expressed in the logic of conditional belief, or the logic of degrees of belief, or that of safe belief. This, we found a surprising result. Still, that does not mean that the logics are equally expressive: the logics of conditional and degrees of belief are incomparable, the logics of degrees of belief and safe belief are incomparable, while the logic of safe belief is more expressive than the logic of conditional belief. In view of the result on bisimulation characterisation, this is an equally surprising result. We hope our insights may contribute to the growing community of formal epistemology and on the relation between qualitative and quantitative modelling.
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Notes
This notion of minimality is non-standard and taken from Baltag and Smets (2008). Usually a minimal element of a set is an element that is not greater than any other element.
Without the restriction to (pre)image-finite models, we were unable to prove the existence of a largest bisimulation. We leave this challenge open to future research(ers).
With our usage of \(\equiv \) it is clear from context whether we’re referring to modal equivalence, formulas or languages.
If we consider infinitary versions of the modalities in our logical languages, in other words, common knowledge and common belief modalities, we preserve the bisimulation characterisation results (for a more refined notion of bisimulation) but it is then to be expected that all three logics become equally expressive (oral communication by Tim French).
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Acknowledgments
We thank Giovanni Cina and Johannes Marti for productive exchanges of ideas following an ILLC seminar in 2015. We are also in dept to the anonymous reviewers for their thorough reading of the manuscript leading to many helpful comments and suggestions for revisions. Hans van Ditmarsch is also affiliated to IMSc (Institute of Mathematical Sciences), Chennai, as research associate. He acknowledges support from European Research Council grant EPS 313360. Preliminary versions of the results in this paper can be found in the PhD theses of Mikkel Birkegaard Andersen (2015, Chapter 4) and Martin Holm Jensen (2014, Chapter 5).
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Andersen, M.B., Bolander, T., van Ditmarsch, H. et al. Bisimulation and expressivity for conditional belief, degrees of belief, and safe belief. Synthese 194, 2447–2487 (2017). https://doi.org/10.1007/s11229-016-1060-x
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DOI: https://doi.org/10.1007/s11229-016-1060-x