Abelian logic and the logics of pointed lattice-ordered varieties
Abelian logic, the logic of abelian \(l\)-groups, was introduced by Meyer and Slaney and, independently, by Casari, in the 1980s. Since then, several papers have been devoted to the topic, which, however, has never really made its way into the mainstream of logical research. At the time when abelian logic was introduced, the paradigm of abstract algebraic logic (AAL) was not yet dominant in the investigations of the relation between logics and the corresponding classes of algebras; furthermore, it was still commonplace to characterize propositional logics mainly as collections of theorems, rather than by means of consequence relations. This was, in fact, the point of view that Meyer and Slaney adopted most of the time in their paper (unlike Casari, whose article does not have abelian logic, however, as its main focus). In this paper, the authors aim at departing from the above mentioned trends and at reconsidering abelian logic from a different perspective. They place themselves in the wider context of a study of pointed varieties of algebras with a lattice term reduct -- a class of varieties which includes several logically and algebraically significant examples (e.g., Boolean algebras, MV algebras, Boolean algebras with operators, residuated lattices and their subvarieties, algebras from quantum logic or from depth relevant logic) -- and they show that, given a rather minimal set of assumptions, each such variety gives rise in a natural way, and according to a regular pattern, to at least three interesting logics, two of which are finitely and strongly algebraisable, while the third need not even be protoalgebraic. In the case of abelian \(l\)-groups, such logics, respectively, correspond to: i) the consequence relation associated by Meyer and Slanley to abelian logic; ii) Galli et al.'s logic of equilibrium; iii) a new non-protoalgebraic but selfextensional logic of ``preservation of truth degrees''. The authors introduce first the notion of pointed \(l\)-variety w.r.t. \(e\) (\(e\)-\(l\)-variety, for short) and then three logics (\(\models^{V}_{1},\models^{V}_{2},\models^{V}_{3}\)) out of this variety, and, by different examples and lemmas, they present some relationships between this logics. Then they restrict their attention to \(e\)-\(l\)-varieties that have suitable candidates for implication or equivalence connectives. In general, the languages with which they are working need not contain anything like a fusion connective with respect to which a candidate implication should behave as a residual. Therefore, defining the implication and the equivalence on \(e\)-\(l\)-varieties, the authors try to get as close as they can to a residuation requirement given the expressive means they have at their disposal. Also, they define some other sensible connectives with implicative features which arise in the languages of \(e\)-\(l\)-varieties with implication. Next, the authors presents different conditions for a logic to be (strongly, finitely, regulary) algebraisable; they illustrates, for example, that \(\models^{V}_{1},\models^{V}_{2}\) are different algebraisable logics with the same equivalent algebraic semantics, a well-known but not too usual phenomenon in AAL. Moreover, there are investigated the deductive filters of logics from \(e\)-\(l\)-varieties and their relationships with the congruence filters of an abelian \(l\)-group. Finally, the authors set aside the general framework of logics from \(e\)-\(l\)-varieties and focus more closely on the special case of abelian logics. Whenever an abstract logic is defined semantically, a question naturally arises as to whether such a logic admits a finite axiomatisation by means of a Hilbert-style system, and the authors try to give comprehensive answers to this question with respect to abelian logics. The paper concludes by proving in detail three deduction theorems for the three logics under consideration at the beginning.
scientific article; zbMATH DE number 5508678