Simple Axioms for Local Properties

TL;DR

This paper introduces simpler axiomatizations for modal logics with local properties, allowing reasoning about properties that hold only in parts of the model rather than globally.

Contribution

It provides significantly simpler axiomatizations for modal logics with local properties compared to previous complex systems.

Findings

Simpler axiomatizations for local properties in modal logic

Applicable to a wide range of local properties

Facilitates reasoning about non-global frame properties

Abstract

Correspondence theory allows us to create sound and complete axiomatizations for modal logic on frames with certain properties. For example, if we restrict ourselves to transitive frames we should add the axiom $\square \phi \rightarrow \square\square\phi$ which, among other things, can be interpreted as positive introspection. One limitation of this technique is that the frame property and the axiom are assumed to hold globally, i.e., the relation is transitive throughout the frame, and the agent's knowledge satisfies positive introspection in every world. In a modal logic with local properties, we can reason about properties that are not global. So, for example, transitivity might hold only in certain parts of the model and, as a result, the agent's knowledge might satisfy positive introspection in some worlds but not in others. Van Ditmarsch et al. (2012) introduced sound and complete axiomatizations for modal logics with certain local properties. Unfortunately, those axiomatizations are rather complex. Here, we introduce far simpler axiomatizations for a wide range of local properties.