Eventual and Strong Eventual Notions in Public Announcements

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 It does not extend the axiomatisation AA, for example it is no longer valid that nobody has received any message.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.

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.

In this work we propose a dynamic turn in paraconsistency. We introduce AMLFI1, the action model extension of the paraconsistent logic LFI1. A special case is PALFI1, a paraconsistent logic of public announcements. It corresponds to another, recently published, paraconsistent public announcement logic: the differences in their axiomatizations are mutually admissible. We also introduce UMLFI1, that extends AMLFI1 with factual change. Soundness and completeness are proven for all logics, and all extend the epistemic paraconsistent logics KLFI1, KB4LFI1 and S5LFI1, known from the literature. With such dynamic epistemic paraconsistent logics we can formalize obtaining and resolving provisional contradictions.