Abelian logic and the logics of pointed lattice-ordered varieties
type: Item
instance of: scholarly article
title: Abelian logic and the logics of pointed lattice-ordered varieties
zbMATH Open document ID: 1162.03036
author: Francesco Paoli
author: Matthew Spinks
author: Robert Veroff
published in: Logica Universalis
publication date: 2009-02-17T00:00:00Z
review text: 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…
reviewed by: Florentina Chirteş
Mathematics Subject Classification ID: 03G99
Mathematics Subject Classification ID: 03B22
Mathematics Subject Classification ID: 03C05
Mathematics Subject Classification ID: 06F20
zbMATH DE Number: 5508678
zbMATH Keywords: abelian logic
zbMATH Keywords: lattice-ordered variety
zbMATH Keywords: assertional logic
zbMATH Keywords: logics preserving degrees of truth
MaRDI profile type: Publication
full work available at URL: s11787-008-0034-2
OpenAlex ID: W2014657478
DOI: 10.1007/S11787-008-0034-2
description: scientific article; zbMATH DE number 5508678