One-variable fragments of first-order logics

0585222 - ÚI 2025 GB eng J - Journal Article
Cintula, Petr - Metcalfe, G. - Tokuda, N.
One-variable fragments of first-order logics.
Bulletin of Symbolic Logic. Online 01 April 2024 (2024). ISSN 1079-8986. E-ISSN 1943-5894
R&D Projects: GA ČR(CZ) GA22-01137S
EU Projects: European Commission(XE) 101007627 - MOSAIC
Institutional support: RVO:67985807
Keywords : First-Order Logic * One-Variable Fragment * Modal Logic * Substructural Logic * Superamalgamation * Sequent Calculus
OECD category: Pure mathematics
Impact factor: 0.6, year: 2022
Method of publishing: Open access
https://doi.org/10.1017/bsl.2024.22

The one-variable fragment of a first-order logic may be viewed as an “S5-like” modal logic, where the universal and existential quantifiers are replaced by box and diamond modalities, respectively. Axiomatizations of these modal logics have been obtained for special cases — notably, the modal counterparts S5 and MIPC of the one-variable fragments of first-order classical logic and first-order intuitionistic logic, respectively — but a general approach, extending beyond first-order intermediate logics, has been lacking. To this end, a sufficient criterion is given in this paper for the one-variable fragment of a semantically-defined first-order logic — spanning families of intermediate, substructural, many-valued, and modal logics — to admit a certain natural axiomatization. More precisely, an axiomatization is obtained for the one-variable fragment of any first-order logic based on a variety of algebraic structures with a lattice reduct that has the
superamalgamation property, using a generalized version of a functional representation theorem for monadic Heyting algebras due to Bezhanishvili and Harding. An alternative proof-theoretic strategy for obtaining such axiomatization results is also developed for first-order substructural logics that have a cut-free sequent calculus and admit a certain interpolation property
Permanent Link: https://hdl.handle.net/11104/0352990