Logics of varieties, logics of semilattices and conjunction

Abstract

This article starts with a general analysis of the problem of how to associate a logic with a given variety of algebras, and shows that it has a positive solution for two of the standard procedures of performing this association, and a negative one in the third. Then the article focuses on the case of the ‘logics of semilattices’, which are defined as the logics related to the variety of semilatices via two of the standard procedures of abstract algebraic logic. We study their main properties, classify them in the Leibniz hierarchy and the Frege hierarchy (the two hierarchies of abstract algebraic logic), and study the poset they form (in particular, we find its least element and its two unique maximals, and prove it is atomless). Even if there is an infinity of such logics of semilattices, it is not known whether there are logics related to the variety of semilattices via the Leibniz reduction too; we discuss this issue and provide a partial solution to this problem. The final section studies one of the maximals of this poset, the conjunctive fragment of classical propositional logic; among other properties we give two new characterisations of this logic, one of them in terms of a property of the Leibniz operator.