MoL-2019-08: Polyhedral Completeness in Intermediate and Modal Logics

MoL-2019-08: Adam-Day, Sam (2019) Polyhedral Completeness in Intermediate and Modal Logics. [Report]

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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 Alexandrov’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 Scott’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.

Item Type: Report
Report Nr: MoL-2019-08
Series Name: Master of Logic Thesis (MoL) Series
Year: 2019
Subjects: Logic
Depositing User: Dr Marco Vervoort
Date Deposited: 03 Aug 2019 21:36
Last Modified: 03 Aug 2019 21:36

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