ONE-VARIABLE FRAGMENTS OF FIRST-ORDER LOGICS

• Published in: The bulletin of symbolic logic 2024-06, Vol.30 (2), p.253-278
• Main Authors: CINTULA, PETR, METCALFE, GEORGE, TOKUDA, NAOMI
• Format: Article
• Language: English
• Subjects: Algebra; Axioms; Calculus; Fragments; Logic; Semantics
• ISSN: 1079-8986; 1943-5894
• DOI: 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 $\mathrm {S5}$ and $\mathrm {MIPC}$ 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.