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Algebraic aspects of propositional logic
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...
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| Format: | Thesis |
| Language: | English English |
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Department of Mathematics and Applied Mathematics
2025
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| _version_ | 1867613944587223040 |
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| access_status_str | Open Access |
| author | Leisegang, Nicholas |
| author2 | Janelidze-Gray, Tamar |
| author_browse | Janelidze-Gray, Tamar Leisegang, Nicholas |
| author_facet | Janelidze-Gray, Tamar Leisegang, Nicholas |
| author_sort | Leisegang, Nicholas |
| collection | Thesis |
| description | 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. |
| format | Thesis |
| id | oai:open.uct.ac.za:11427/41004 |
| institution | University of Cape Town (South Africa) |
| language | English eng |
| last_indexed | 2026-06-10T12:44:11.915Z |
| license_str | Not specified — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from UCTD — University of Cape Town Open Access Repository |
| publishDate | 2025 |
| publishDateRange | 2025 |
| publishDateSort | 2025 |
| publisher | Department of Mathematics and Applied Mathematics |
| publisherStr | Department of Mathematics and Applied Mathematics |
| record_format | dspace |
| source_str | UCTD — University of Cape Town Open Access Repository |
| spelling | oai:open.uct.ac.za:11427/41004 Algebraic aspects of propositional logic Leisegang, Nicholas Janelidze-Gray, Tamar Janelidze, George Bool of Boolean algebras 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. 2025-02-25T07:51:45Z 2025-02-25T07:51:45Z 2024 2025-02-25T07:48:42Z Thesis / Dissertation Masters MSc http://hdl.handle.net/11427/41004 en eng application/pdf Department of Mathematics and Applied Mathematics Faculty of Science University of Cape Town |
| spellingShingle | Bool of Boolean algebras Leisegang, Nicholas Algebraic aspects of propositional logic |
| thesis_degree_str | Master's |
| title | Algebraic aspects of propositional logic |
| title_full | Algebraic aspects of propositional logic |
| title_fullStr | Algebraic aspects of propositional logic |
| title_full_unstemmed | Algebraic aspects of propositional logic |
| title_short | Algebraic aspects of propositional logic |
| title_sort | algebraic aspects of propositional logic |
| topic | Bool of Boolean algebras |
| url | http://hdl.handle.net/11427/41004 |
| work_keys_str_mv | AT leisegangnicholas algebraicaspectsofpropositionallogic |