Structures of Oppositions in Public Announcement Logic

Abstract

In this paper we study dynamic epistemic logic, and in particular public announcement logic, using the tools of n-opposition theory. Dynamic epistemic logic is a contemporary development of epistemic logic, which takes into account changes of knowledge through time. It studies, for example, public announcements and the influence they have on the agents’ knowledge. n-Opposition theory is a systematic generalization of the traditional squares of oppositions. In the paper we provide introductions to the intuitions behind and the formal details of both of these disciplines. We construct a square and a hexagon of oppositions for the structural properties of public announcement. We then generalize this to opposition structures for any partially functional process, and show how this generalization can be used to support the structuralist philosophy surrounding n-opposition theory. Next, we focus on the epistemic properties of public announcement, and construct an octagon and a (three-dimensional) rhombic dodecahedron of oppositions. Many of the techniques applied in this paper were originally developed by Smessaert (Logica Univers. 3:303–332, 2009); they are thus shown to be very powerful and widely applicable. Summing up: we establish a strong connection between public announcement logic and n-opposition theory, and show that this connection has definite advantages for both of the disciplines involved.