Topological Subset Space Models for Public Announcements

We reformulate a key definition given by Wang and Agotnes [18] to provide semantics for public announcements in subset spaces. More precisely, we interpret the precondition for a public announcement of \(\varphi \) to be the “local truth” of \(\varphi \), semantically rendered via an interior operator. This is closely related to the notion of \(\varphi \) 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.