From positive PDL to its non-classical extensions

Abstract

We provide a complete binary implicational axiomatization of the positive fragment of propositional dynamic logic (PDL). The intended application of this result are completeness proofs for non-classical extensions of positive PDL. Two examples are discussed in this article, namely, a paraconsistent extension with modal De Morgan negation and a substructural extension with the residuated operators of the non-associative Lambek calculus. Informal interpretations of these two extensions are outlined.

First Degree Entailment with Group Attitudes and Information Updates

“…This feature of the model derives from the goal of formulating a general representation of information update on an inconsistency-tolerant background. The indexing set of situations, "the proposition triggering the update" [8,32], corresponds to the information content of the update. We do not assume the content of an update to correspond to a prime situation; typically the "incoming" information corresponds to a part of a prime situation.…”

“…Both of these papers contain only single-agent epistemic operators. An FDE-based group epistemic logic with universal and common knowledge is a fragment of paraconistent Propositional Dynamic Logic studied in [31,32]. Bílková et al [10] outline an extension of their substructural epistemic framework with common knowledge, but completeness is left for future research.…”