The Logic of Public Announcements, Common Knowledge, and Private Suspicions

Abstract

This paper presents a logical system in which various group-level epistemic actions are incorporated into the object language. That is, we consider the standard modeling of knowledge among a set of agents by multimodal Kripke structures. One might want to consider actions that take place, such as announcements to groups privately, announcements with suspicious outsiders, etc. In our system, such actions correspond to additional modalities in the object language. That is, we do not add machinery on top of models (as in Fagin et al [4]), but we reify aspects of the machinery in the logical language.Special cases of our logic have been considered in Plaza [13], Gerbrandy [5,6], and Gerbrandy and Groeneveld [7]. The latter group of papers introduce a language in which one can faithfully represent all of the reasoning in examples such as the Muddy Children scenario. In that paper we find operators for updating worlds via announcements to groups of agents who are isolated from all others. We advance this by considering many more actions, and by using a more general semantics.Our logic contains the infinitary operators used in the standard modeling of common knowledge. We present a sound and complete logical system for the logic, and we study its expressive power.

A Qualitative Theory of Dynamic Interactive Belief Revision

“…We denote the updated state model by S ⊗ Σ, and call it the update product of the two models. The construction is similar to a point to the one in [6,5], and thus also somewhat similar to the ones in [2,23]. In fact, the set of updated states, the updated valuation and the updated indistinguishability relation are the same in these constructions.…”

“…As noticed in [28,6,5], the original model does not usually include enough states to capture all the epistemic possibilities that arise in this way. While in the previous section the models were kept unchanged during the revision, all the possibilities being already there (so that both the unconditional and the conditional beliefs referred to the same model ), we now have to allow for belief updates that change the original model.…”

“…Since in this paper we enriched our state models with doxastic plausibility relations to deal with (conditional) beliefs, it is natural to follow [5] into extending the similarity between actions and states to this setting, thus obtaining (epistemic) action plausibility models. The idea of such an extension was first developed in [2] (for a different notion of plausibility model and a different notion of update product), then generalized in [23], where many types of action plausibility models and notions of update product, that extend the so-called BMS update product from [6,5], are explored. But both these works are based on a quantitative interpretation of plausibility ordinals (as "degrees of belief"), and thus they define the various types of products using complex formulas of transfinite ordinal arithmetic, for which no intuitive justification is provided.…”

“…DEL originates in the work of Gerbrandy and Groeneveld [29,28], anticipated by Plaza in [43], and further developed by numerous authors [6,30,21,4,22,38,5,15,16] etc. In its standard incarnation, as presented e.g.…”

“…Our approach can be seen as a natural extension of the work in [5,6] on "epistemic actions", incorporating ideas from the AGM theory along the lines pioneered in [2] and [23], but using a qualitative approach based on conditional beliefs, in the line of [49,20,19,14].…”