Tuyt, Olim Frits (2021). OneVariable Fragments of FirstOrder ManyValued Logics. (Thesis). Universität Bern, Bern

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Abstract
In this thesis we study onevariable fragments of firstorder logics. Such a onevariable fragment consists of those firstorder formulas that contain only unary predicates and a single variable. These fragments can be viewed from a modal perspective by replacing the universal and existential quantifier with a box and diamond modality, respectively, and the unary predicates with corresponding propositional variables. Under this correspondence, the onevariable fragment of firstorder classical logic famously corresponds to the modal logic S5. This thesis explores some such correspondences between firstorder and modal logics. Firstly, we study firstorder intuitionistic logics based on linear intuitionistic Kripke frames. We show that their onevariable fragments correspond to particular modal Gödel logics, defined over manyvalued S5Kripke frames. For a large class of these logics, we prove the validity problem to be decidable, even coNPcomplete. Secondly, we investigate the onevariable fragment of firstorder Abelian logic, i.e., the firstorder logic based on the ordered additive group of the reals. We provide two completeness results with respect to Hilbertstyle axiomatizations: one for the onevariable fragment, and one for the onevariable fragment that does not contain any lattice connectives. Both these fragments are proved to be decidable. Finally, we launch a much broader algebraic investigation into onevariable fragments. We turn to the setting of firstorder substructural logics (with the rule of exchange). Inspired by work on, among others, monadic Boolean algebras and monadic Heyting algebras, we define monadic commutative pointed residuated lattices as a first (algebraic) investigation into onevariable fragments of this large class of firstorder logics. We prove a number of properties for these newly defined algebras, including a characterization in terms of relatively complete subalgebras as well as a characterization of their congruences.
Item Type:  Thesis 

Dissertation Type:  Single 
Date of Defense:  2 July 2021 
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:  02 Aug 2021 12:37 
Last Modified:  02 Aug 2021 12:41 
URI:  https://boristheses.unibe.ch/id/eprint/2864 
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