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

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 preimagefinite 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 B Thomas Bolander 123 Synthese 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.

Bisimulation for Conditional Modalities

“…Second, the notion of bisimulation for conditional belief offered here is modular, in the sense that it can be merged with other conditions when we consider languages with additional operators. In contrast, some results in [3] depend crucially on the existence of the knowledge operator. 10 A notion of bisimulation containing a quantification over subsets has been proposed originally in [20], adapted in [16] to epistemic lottery models and later again reshaped to work in the context of epistemic neighborhood models in [15].…”

“…The intuition behind the selection function is that f (w, X) selects the worlds in X that are 'relevant' at w. 3 For a given model M, the semantics of the language is defined recursively via an interpretation function − M : L → ℘(W ), where for the propositional part of the language the clauses are the usual ones and for conditionals we have the Stalnaker-Lewis semantics:…”

“…This observation aligns with the intentions of the advocates of relevant implications, who contemplate the possibility of p ⇒ p failing at some worlds. A different notion of bisimulation for conditional belief on multi-agent plausibility models was recently introduced in [3]. The authors prove the correspondence between bisimilarity and modal equivalence, respectively for the languages containing conditional belief and knowledge, safe belief and knowledge, degrees of belief and knowledge.…”

“…The notable exception is conditional belief. The problem of finding the right notion of bisimulation for conditional belief has been the focal point of some recent publications in the field of formal epistemology [1][2][3]13,14].…”

Dynamic Graded Epistemic Logic

“…van der Hoek (1993); Laverny (2006). Belief revision based on degrees of belief has been investigated in Aucher (2003); van Ditmarsch (2005); Andersen et al (2017).…”