Algebraic logic for the negation fragment of classical logic

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.