Algebraic logic for the negation fragment of classical logic

TL;DR

This paper explores the algebraic structures underlying the negation fragment of classical logic, providing axiomatizations, model characterizations, and classification within logical hierarchies.

Contribution

It introduces a Hilbert-style axiomatization and characterizes algebraic models for the negation fragment of classical logic, advancing algebraic logic understanding.

Findings

Identified three classes of algebras associated with the negation fragment

Characterized reduced and generalized matrix models for the logic

Classified the negation fragment within Leibniz and Frege hierarchies

Abstract

The general aim of this article is to study the negation fragment of classical logic within the framework of contemporary (Abstract) Algebraic Logic. More precisely, we shall find the three classes of algebras that are canonically associated with a logic in Algebraic Logic, that is, we find the classes $\mathrm{Alg}^*$, $\mathrm{Alg}$ and the intrinsic variety of the negation fragment of classical logic. In order to achieve this, firstly we propose a Hilbert-style axiomatization for this fragment. Then, we characterize the reduced matrix models and the full generalized matrix models of this logic. Also, we classify the negation fragment in the Leibniz and Frege hierarchies.