Topological Subset Space Models for Public Announcements

TL;DR

This paper introduces a topological subset space semantics for public announcements, emphasizing the role of local truth and topological structure, and provides a complete axiomatization of the logic.

Contribution

It reformulates public announcement semantics using topological subset spaces, enhancing the original approach and establishing a complete axiomatization.

Findings

Revised semantics improve understanding of public announcements

Topological structure is crucial for the semantics

A complete axiomatization is provided

Abstract

We reformulate a key definition given by Wang and Agotnes (2013) to provide semantics for public announcements in subset spaces. More precisely, we interpret the precondition for a public announcement of {\phi} to be the "local truth" of {\phi}, semantically rendered via an interior operator. This is closely related to the notion of {\phi} being "knowable". We argue that these revised semantics improve on the original and offer several motivating examples to this effect. A key insight that emerges is the crucial role of topological structure in this setting. Finally, we provide a simple axiomatization of the resulting logic and prove completeness.