We investigate a new logic that extends Dynamic Epistemic Logic (DEL), by combining standard epistemic modalities for (individual and distributed) propositional knowledge with operators for (conditional) non-propositional knowledge of a number (in which an agent or a group have knowledge of the value of some variable x, conditional on some additional information). We also generalize these operators, by considering formulas that express the fact that an agent or group can (conditionally) narrow down the possible values of the variable x to at most N possibilities (for some natural number N). In order to name and compare such hypothetical values, we extend the logic further with definite descriptions based on minimization operators, denoting the least of the N possible values of x (according to some fixed order) that are considered possible by the agent or group. On this static base, we consider DEL-style extensions with dynamic modalities for general 'data-exchange events' (covering private and public propositional announcements, but also secret hacking of a private database, or public sharing of one's data via open-source repositories, etc.). In such scenarios, whole 'chunks' of information may be exchanged or modified: once access to a given source is gained, all the 'data' stored at that specific location becomes available. We give complete axiomatizations for the resulting logics, and prove their decidability and co-expressivity.
We investigate a logic for asynchronous announcements wherein the sending of the messages by the environment is separated from their reception by the individual agents. Both come with different modalities. In the logical semantics, formulas are interpreted in a world of a Kripke model but given a history of prior announcements and receptions that already happened. An axiomatisation AA for such a logic has been given in prior work, for the formulas that are valid when interpreted in the Kripke model before any such announcements have taken place. This axiomatisation is a reduction system wherein one can show that every formula is equivalent to a purely epistemic formula without dynamic modalities for announcements and receptions. We propose a generalisation AA* of this axiomatisation, for the formulas that are valid when interpreted in the Kripke model given any history of prior announcements and receptions of announcements. It does not extend the axiomatisation AA, for example it is no longer valid that nobody has received any message. Unlike AA, this axiomatisation AA* is infinitary and it is not a reduction system.
Philippe Balbiani, Hans van Ditmarsch, Clara Lerouvillois
Distributed knowledge is a notion of group knowledge studied in multi-agent epistemic logic. Semantically, the distributed knowledge of a group is interpreted via an accessibility relation given by the intersection of the epistemic accessibility relations of the agents in that group. This paper investigates sequent calculi for epistemic logics of distributed knowledge based on K45, KD45, and S5. While cut elimination holds in existing sequent calculi for modal logics K45 and KD45, it fails in all the systems mentioned above. Instead, we establish the analytic cut property for all three systems by adapting Takano' s (2018) strategy, which restricts the cut formulas to the set of subformulas of the conclusion of the cut rule. As a corollary, the Craig interpolation theorem holds for all logics considered. We also show that all proof-theoretic results remain valid when the empty group is allowed for the distributed-knowledge operator, in which case the distributed knowledge for the empty group is interpreted as the global modality.
Dynamic epistemic logic represents belief change via model transformations induced by epistemic events. Its standard formulation (Baltag, Moss, Solecki, 1998) provides a natural account of belief expansion through the elimination of possibilities, but it cannot model belief contraction about factual propositions. A classic response enriches Kripke models with plausibility orderings, representing contraction as an update that promotes certain possibilities over others. We show that this approach has expressive limitations. In particular, the approach cannot model belief that violates positive introspection and contraction dynamics in response to a hedged public announcement that phi might be false. Motivated by these considerations, we introduce a mechanism for belief contraction defined directly on standard Kripke models, without any constraints on the doxastic accessibility relation. We show that it satisfies some of the standard properties of belief contraction but not others, study the conditions under which contraction may be unsuccessful, and provide a sound and complete axiomatization of the logic via reduction axioms. We also define a more general dynamic logic that is an extension of standard DEL and accommodates belief contractions due to events such as private or semi-private announcements, and provide a complete and sound axiomatization of the general logic.