Reasoning About Space: The Modal Way

Abstract

We investigate the topological interpretation of modal logic in modern terms, using a new notion of bisimulation. Next, we look at modal logics with interesting topological content, presenting, amongst others, a new proof of McKinsey and Tarski's theorem on completeness of S4 with respect to the real line, and a completeness proof for the logic of finite unions of convex sets of reals. We conclude with a broader picture of extended modal languages of space, for which the main logical questions are still wide open.

Structures for Epistemic Logic

“…In fact, there is a deeper connection between these two structures. We refer to [3] and [15] for more details on the connection of the spatial logic of R and the logic of the 2-fork.…”

“…As every reflexive and transitive Kripke frame corresponds to an Alexandroff space, the completeness follows. Van Benthem and his collaborators, however, gave a different, elegant and more self-contained proof of this result by introducing a topo-canonical model (a topological analogue of a canonical Kripke model) [3], [14]. We will quickly sketch the basic idea of this construction.…”

Johan van Benthem on Logic and Information Dynamics

“…Johan van Benthem's interest in spatial logics is in fact deeper and goes way beyond the needs of epistemic logic [58,59], but he has been particularly interested in the topological interpretation of knowledge, and its generalization to neighborhood semantics. In joint work with his Stanford student Sarenac [70], he investigated the properties of common knowledge in a topological setting.…”

Logic for physical space

“…Indeed, relational frames for S4 can be thought of as special topological spaces (see, e.g., Aiello et al (2003)). In fact, topological semantics for S4 is more powerful than its relational semantics because, as follows from Gerson (1975), there exist extensions of S4 which are topologically complete, but relationally incomplete.…”