Algebraic semantics for one-variable lattice-valued logics

Cintula, Petr; Metcalfe, George; Tokuda, Naomi (2022). Algebraic semantics for one-variable lattice-valued logics. In: Fernández-Duque, David; Palmigiano, Alessandra; Pinchinat, Sophie (eds.) Proceedings of AiML 2022. Advances in Modal Logic: Vol. 14 (pp. 237-257). College Publications

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The one-variable fragment of any first-order logic may be considered as a modal logic, where the universal and existential quantifiers are replaced by a box and diamond modality, respectively. In several cases, axiomatizations of algebraic semantics for these logics have been obtained: most notably, for the modal counterparts S5 and MIPC of the one-variable fragments of first-order classical logic and intuitionistic logic, respectively. Outside the setting of first-order intermediate logics, however, a general approach is lacking. This paper provides the basis for such an approach in the setting of first-order lattice-valued logics, where formulas are interpreted in algebraic structures with a lattice reduct. In particular, axiomatizations are obtained for modal counterparts of one-variable fragments of a broad family of these logics by generalizing a functional representation theorem of Bezhanishvili and Harding for monadic Heyting algebras. An alternative proof-theoretic proof is also provided for one-variable fragments of first-order substructural logics that have a cut-free sequent calculus and admit a certain bounded interpolation property.

Item Type:

Book Section (Book Chapter)

Division/Institute:

08 Faculty of Science > Department of Mathematics and Statistics > Institute of Mathematics

UniBE Contributor:

Cintula, Petr, Metcalfe, George, Tokuda, Naomi Maja

Subjects:

500 Science > 510 Mathematics

ISBN:

978-1-84890-413-2

Series:

Advances in Modal Logic

Publisher:

College Publications

Language:

English

Submitter:

George Metcalfe

Date Deposited:

21 Sep 2022 11:08

Last Modified:

05 Dec 2022 16:24

BORIS DOI:

10.48350/173082

URI:

https://boris.unibe.ch/id/eprint/173082

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