Tokuda, Naomi (2024). Axiomatizing One-Variable Fragments of First-Order Logics. (Thesis). Universität Bern, Bern
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Abstract
First-order logics are much more expressive than their propositional counterparts, but often lack decidability. To remedy this problem while still maintaining some of the expressivity of first-order logics, we consider one-variable fragments of first-order logics. The one-variable fragment of a first-order logic consists of consequences in the logic constructed using one distinguished variable x, unary predicates, propositional operations, and the quantifiers (∀x) and (∃x). We define a semantics for a first-order logic based on a class of L-lattices, which induces a semantics for its one-variable fragment. However, finding an axiomatization of this fragment is generally not trivial. A Hilbert-style axiomatization of a first-order logic does not yield a Hilbert-style axiomatization of its one-variable fragment, since derivations of one-variable formulas may introduce new variables. Consequence in one-variable first-order logics can be translated into consequence in a class of algebras with modalities. Therefore, the challenge of finding axiomatizations for one-variable fragments may be interpreted as the challenge of finding an equational basis for the corresponding classes of algebras. Despite axiomatization results for certain fragments, a general approach has been missing. In this thesis, we take a first step towards overcoming this challenge, by proving that the class of algebras corresponding to the one-variable fragment of a first-order logic based on a variety of L-lattices that has the superamalgamation property admits an axiomatization by “S5-like” equations. This is achieved through both algebraic and proof-theoretic approaches.
| Item Type: | Thesis |
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| Dissertation Type: | Single |
| Date of Defense: | 29 August 2024 |
| Subjects: | 500 Science > 510 Mathematics |
| Institute / Center: | 08 Faculty of Science > Department of Mathematics and Statistics > Institute of Mathematics |
| Depositing User: | Hammer Igor |
| Date Deposited: | 01 Oct 2024 15:45 |
| Last Modified: | 01 Oct 2024 15:45 |
| URI: | https://boristheses.unibe.ch/id/eprint/5470 |
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