One-variable fragments of first-order logics

TL;DR

This paper provides a general criterion for axiomatizing the one-variable fragment of various first-order logics, including intermediate, substructural, many-valued, and modal logics, based on algebraic and proof-theoretic properties.

Contribution

It introduces a unified approach to axiomatize one-variable fragments of diverse first-order logics using algebraic structures with the superamalgamation property and a proof-theoretic method for certain substructural logics.

Findings

Axiomatization criterion for one-variable fragments of first-order logics.

Application to logics with algebraic structures having superamalgamation.

Alternative proof-theoretic approach for substructural logics with cut-free calculus.

Abstract

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 intuitionistic logic -- 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 natural axiomatization. More precisely, such 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, building on 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.