An Abstract Look at Awareness Models and Their Dynamics

TL;DR

This paper explores awareness models in modal logic, establishing their connections with other logical systems and analyzing their dynamic properties through event models, with a focus on axiomatization and property preservation.

Contribution

It introduces bridges between awareness logics and other modal systems, and provides closure theorems for dynamic updates preserving key properties.

Findings

Characterized properties of awareness sets with canonical formulas

Proved closure theorems for dynamic updates

Axiomatized dynamic logics using reduction axioms

Abstract

This work builds upon a well-established research tradition on modal logics of awareness. One of its aims is to export tools and techniques to other areas within modal logic. To this end, we illustrate a number of significant bridges with abstract argumentation, justification logics, the epistemic logic of knowing-what and deontic logic, where basic notions and definitional concepts can be expressed in terms of the awareness operator combined with the box modality. Furthermore, these conceptual links point to interesting properties of awareness sets beyond those standardly assumed in awareness logics, i.e. positive and negative introspection. We show that the properties we list are characterised by corresponding canonical formulas, so as to obtain a series of off-the-shelf axiomatisations for them. As a second focus, we investigate the general dynamics of this framework by means of event models. Of specific interest in this context is to know under which conditions, given a model that satisfies some property, the update with an event model keeps it within the intended class. This is known as the closure problem in general dynamic epistemic logics. As a main contribution, we prove a number of closure theorems providing sufficient conditions for the preservation of our properties. Again, these results enable us to axiomatize our dynamic logics by means of reduction axioms.