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Plausibility models are Kripke models that agents use to reason aboutknowledge and belief, both of themselves and of each other. Such models areused to interpret the notions of conditional belief, degrees of belief, andsafe belief. The logic of conditional belief contains that modality and alsothe knowledge modality, and similarly for the logic of degrees of belief andthe logic of safe belief. With respect to these logics, plausibility models maycontain too much information. A proper notion of bisimulation is required thatcharacterises them. We define that notion of bisimulation and prove therequired characterisations: on the class of image-finite and preimage-finitemodels (with respect to the plausibility relation), two pointed Kripke modelsare modally equivalent in either of the three logics, if and only if they arebisimilar. As a result, the information content of such a model can besimilarly expressed in the logic of conditional belief, or the logic of degreesof 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 ofconditional and degrees of belief are incomparable, the logics of degrees ofbelief and safe belief are incomparable, while the logic of safe belief is moreexpressive than the logic of conditional belief. In view of the result onbisimulation characterisation, this is an equally surprising result. We hopeour insights may contribute to the growing community of formal epistemology andon the relation between qualitative and quantitative modelling.

Bisimulation and expressivity for conditional belief, degrees of belief, and safe belief