Algebraic aspects of propositional logic

Leisegang, Nicholas

Algebraic aspects of propositional logic

Thesis / Dissertation

2024

Publisher

University of Cape Town

Department

Department of Mathematics and Applied Mathematics

Faculty

Faculty of Science

Abstract

In this dissertation, we seek to examine the connection between abstract algebra and propositional logic. We start by considering the category Bool of Boolean algebras, the algebraic counterpart of classical propositional logic. We provide an algebraic definition of theories and models of classical logic and provide algebraic algorithms to determine whether a chosen formula is a theorem of a given theory of classical logic. In order to generalize this approach, we then describe varieties of universal algebra and some of their properties. Using this framework, we show in a general setting how a formal theory of propositional logic induces a variety of universal algebra in which logical connectives become algebraic operations and logical formulae are considered equal when they are logically equivalent. We then discuss algebraic varieties corresponding to various non-classical propositional logics. In particular, we consider the variety of Heyting algebras Heyt which corresponds to intuitionistic logic, and certain subvarieties of Heyt which correspond to intermediate logics. We then describe several algebraic varieties which correspond to theories of normal modal logic. Moreover, by considering free algebras and completeness in Heyt, we establish that we are unable to use the same methods used in Bool to construct algorithms to determine theorems of intuitionistic logic. Lastly, we construct an adjunction between Heyt and the category of topological Boolean algebras, and through this show that we again cannot construct similar algebraic algorithms to determine theorems in the modal logic S4.

Keywords

Bool of Boolean algebras

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