Algebraic semantics for one-variable lattice-valued logics

TL;DR

This paper develops an algebraic semantics framework for one-variable lattice-valued logics, generalizing existing modal logic approaches and providing axiomatizations for broad classes of these logics.

Contribution

It introduces a general approach to algebraic semantics for one-variable lattice-valued logics, extending previous modal logic results to a wider family of logics.

Findings

Axiomatizations for modal counterparts of one-variable lattice-valued logics

Generalization of a functional representation theorem for monadic Heyting algebras

Alternative proof-theoretic proof for certain substructural logics

Abstract

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.