A simple propositional calculus for compact Hausdorff spaces
Guram Bezhanishvili, Nick Bezhanishvili, Thomas Santoli, Yde Venema
Abstract:
We introduce a simple propositional calculus for compact Hausdorff spaces. Our approach is based on de Vries duality. The main new connective of our calculus is that of strict implication. We define the strict implication calculus SIC as our base calculus. We show that the corresponding variety SIA of strict implication algebras is a discriminator and locally finite variety. We prove that SIC is strongly sound and complete with respect to the universal subclass RSub of SIA, where the modality associated with the strict implication only takes on the values of 0 and 1. We develop \Pi_2-rules for strict implication algebras, and show that every \Pi_2-rule defines an inductive subclass of RSub. We prove that every derivation system axiomatized by Pi_2-rules is strongly sound and complete with respect to the inductive subclass of RSub it defines. We introduce the de Vries calculus DVC and show that it is strongly sound and complete with respect to the class of compingent algebras, and then use MacNeille completions to prove that DVC is strongly sound and complete with respect to the class of de Vries algebras. We then utilize de Vries duality to introduce topological models of our calculus, and conclude that DVC is strongly sound and complete with respect to the class of compact Hausdorff spaces. We also develop strongly sound and complete calculi for zero-dimensional and connected compact Hausdorff spaces, and give a general criterion of admissibility for \Pi_2-rules. We finish the paper by comparing our approach to the existing approaches in the literature that are related to our work.