MoL-2019-08: Adam-Day, Sam (2019) Polyhedral Completeness in Intermediate and Modal Logics. [Report]
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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 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 Mathematics |
| Depositing User: | Dr Marco Vervoort |
| Date Deposited: | 03 Aug 2019 21:36 |
| Last Modified: | 03 Aug 2019 21:36 |
| URI: | https://eprints.illc.uva.nl/id/eprint/1690 |
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