Order algebraizable logics
type: Item
instance of: scholarly article
title: Order algebraizable logics
zbMATH Open document ID: 1263.03061
author: J. G. Raftery
published in: Annals of Pure and Applied Logic
publication date: 2013-02-19T00:00:00Z
review text: In the article under review the author provides an order-theoretic generalization of the classical notion of an algebraizable logic. I recall that a deductive system is said to be algebraizable if it is fully equivalent to the equational consequence relation of a class of algebras (with a common algebraic signature). For a given language \(\mathcal{L} = (\mathbf{\Sigma},V)\), where \(\mathbf{\Sigma}\) is an algebraic signature and \(V\) a set of variables, \(\mathbf{Tm}\), the free \(\mathbf{\Sigma}\)-algebra on \(V\), and a class \(\mathsf{K}\) of partially ordered \(\mathbf{\Sigma}\)-algebras, where a partially ordered \(\mathbf{\Sigma}\)-algebra is a pair \((\mathbf{A},\leq)\) with \(\mathbf{A}\) a \(\mathbf{\Sigma}\)-algebra and \(\leq\) a partial order on the underlying set \(A\) of \(\mathbf{A}\) (no compatibility between \(\leq\) and the operations of \(\mathbf{A}\) is assumed), the author defines the inequational consequence relation \(\models^{\leq}_{\mathsf{K}}\) of \(\mathsf{K}\) as the subset of \(\mathrm{Sub}(Tm^{2})\times Tm^{2}\), where \(Tm\) is the underlying set of \(\mathbf{Tm}\), consisting of all pairs \((\Gamma,\alpha\preccurlyeq\beta)\) in \(\mathrm{Sub}(Tm^{2})\times Tm^{2}\) such that, for every \((\mathbf{A},\leq)\) in \(\mathsf{K}\) and every homomorphism \(f\) from \(\mathbf{Tm}\) to \(\mathbf{A}\), if, for all \(\mu\preccurlyeq\nu\) in \(\Gamma\), \(f(\mu)\leq f(\nu)\), then \(f(\alpha)\leq f(\beta)\). Then, after defining for \((\mathcal{L},\mathbf{R})\), where \(\mathbf{R} = (R,\mathrm{rk})\), with \(\mathrm{rk}\) a mapping from \(R\) to \(\mathbb{N}-\{0\}\), the concept of \(\mathbf{R}\)-formula over \(\mathcal{L}\) as an expression \(r(\alpha_{0},\dots,\alpha_{n-1})\), where \(r\in R\) and \(\alpha_{0},\dots, \alpha_{n-1}\in Tm\), with \(\mathrm{rk}(r) = n\), the author defines an \(R\)-deductive system over \(\mathcal{L}\) as any consequence relation on the set of all \(\mathbf{R}\)-formulas that is substitution invariant. After thi…
reviewed by: Juan Climent Vidal
Mathematics Subject Classification ID: 03G27
zbMATH DE Number: 6137399
zbMATH Keywords: consequence relation
zbMATH Keywords: deductive system
zbMATH Keywords: order algebraizable logic
zbMATH Keywords: polarity
zbMATH Keywords: quasi-order
zbMATH Keywords: abstract algebraic logic
MaRDI profile type: Publication
full work available at URL: j.apal.2012.10.013
OpenAlex ID: W2010311295
DOI: 10.1016/J.APAL.2012.10.013
description: scientific article; zbMATH DE number 6137399