Tokuda, Naomi (2024). Axiomatizing OneVariable Fragments of FirstOrder Logics. (Thesis). Universität Bern, Bern

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Abstract
Firstorder 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 firstorder logics, we consider onevariable fragments of firstorder logics. The onevariable fragment of a firstorder 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 firstorder logic based on a class of Llattices, which induces a semantics for its onevariable fragment. However, finding an axiomatization of this fragment is generally not trivial. A Hilbertstyle axiomatization of a firstorder logic does not yield a Hilbertstyle axiomatization of its onevariable fragment, since derivations of onevariable formulas may introduce new variables. Consequence in onevariable firstorder logics can be translated into consequence in a class of algebras with modalities. Therefore, the challenge of finding axiomatizations for onevariable 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 onevariable fragment of a firstorder logic based on a variety of Llattices that has the superamalgamation property admits an axiomatization by “S5like” equations. This is achieved through both algebraic and prooftheoretic approaches.
Item Type:  Thesis 

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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