Arbitrary Announcements on Topological Subset Spaces

Abstract

Subset space semantics for public announcement logic in the spirit of the effort modality have been proposed by Wang and Ågotnes [18] and by Bjorndahl [6]. They propose to model the public announcement modality by shrinking the epistemic range with respect to which a postcondition of the announcement is evaluated, instead of by restricting the model to the set of worlds satisfying the announcement. Thus we get an “elegant, model-internal mechanism for interpreting public announcements” [6, p. 12]. In this work, we extend Bjorndahl’s logic $PAL_{int}$ of public announcement, which is modelled on topological spaces using subset space semantics and adding the interior operator, with an arbitrary announcement modality, and we provide topological subset space semantics for the corresponding arbitrary announcement logic $APAL_{int}$, and demonstrate completeness of the logic by proving that it is equal in expressivity to the logic without arbitrary announcements, employing techniques from [2, 13].