Algebraic study of two deductive systems of relevance logic
type: Item
instance of: scholarly article
title: Algebraic study of two deductive systems of relevance logic
zbMATH Open document ID: 0833.03007
author: Gonzalo Rodríguez
author: Josep Maria Font
published in: Notre Dame Journal of Formal Logic
publication date: 1996-03-17T00:00:00Z
review text: The paper studies from an algebraic point of view both the classical relevant logic \({\mathbf R}\) and a weaker version \({\mathbf W} {\mathbf R}\), which arises from \({\mathbf R}\) by dropping the Identity Law \(p \to p\). \({\mathbf W} {\mathbf R}\) is given by its consequence operation \(\vdash_{{\mathbf W} {\mathbf R}}\) defined as \[ \Gamma \vdash_{{\mathbf W} {\mathbf R}} \varphi \Leftrightarrow \exists \varphi_1, \ldots, \varphi_n \in \Gamma:\;\vdash_{\mathbf R} (\varphi_1 \wedge \ldots \wedge \varphi_n) \to \varphi. \] It is provided with a classical style Gentzen calculus. Both \({\mathbf W} {\mathbf R}\) and \({\mathbf R}\) are investigated by means of abstract logic, i.e., abstract algebras with closure operators. One of the most interesting results is that on any algebra of suitable type there is an isomorphism between the lattice of all \({\mathbf W} {\mathbf R}\)-logics on it and the lattice of all congruences of the algebra whose quotient is an \({\mathbf R}\)-algebra.
reviewed by: Max Urchs
Mathematics Subject Classification ID: 03B47
Mathematics Subject Classification ID: 03G25
Mathematics Subject Classification ID: 03B22
zbMATH DE Number: 769230
zbMATH Keywords: relevant logic
zbMATH Keywords: consequence operation
zbMATH Keywords: abstract logic
zbMATH Keywords: algebras with closure operators
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1305/ndjfl/1040511344
OpenAlex ID: W2090474572
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DOI: 10.1305/NDJFL/1040511344
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description: scientific article; zbMATH DE number 769230