Polyhedral Completeness in Intermediate and Modal Logics
Sam Adam-Day
Abstract:
This thesis explores a newly-defined polyhedral semantics for intuitionistic and modal logics. Formulas are interpreted inside the Heyting algebra of open subpolyhedra of a polyhedron, and the modal algebra of arbitrary subpolyhedra with the topological interior operator. This semantics enjoys a Tarski-style completeness result: IPC and S4.Grz are complete with respect to the class of all polyhedra. In this thesis I explore the general phenomenon of completeness with respect to some class of polyhedra.
I present a criterion for the polyhedral completeness of a logic based on Alexandrovâ€™s nerve construction. I then use this criterion to exhibit an infinite class of polyhedrally-complete logics of each finite height, as well as demonstrating the polyhedral completeness of Scottâ€™s logic SL. Taking a different approach, I provide an axiomatisation for the logic of all convex polyhedra of each dimension n.
The main conceptual contribution of this thesis is the development of a combinatorial approach to the interaction between logic and geometry via polyhedral semantics.