CE is not a conservative extension of E
type: Item
instance of: scholarly article
title: CE is not a conservative extension of E
zbMATH Open document ID: 0958.03014
DOI: 10.1023/A:1004731401855
author: Edwin D. Mares
published in: Journal of Philosophical Logic
publication date: 2001-03-22T00:00:00Z
review text: The logic \({\mathbf {CE}}\) results from adding Boolean negation, \(\neg\), to the Anderson-Belnap logic \({\mathbf E}\) of entailment. Semantically, this is treated classically, so that \(\neg A\) is true at a point, or `possible world', iff \(A\) is not true at that point. Boolean negation contrasts with the de Morgan negation, \(\sim\), of \({\mathbf E}\) and other relevant logics, which requires a more complex semantical rule. As the title indicates, this paper demonstrates that \({\mathbf {CE}}\) is not a conservative extension of \({\mathbf E}\). The proof is based on the Routley-Meyer ternary relational semantics for relevant logics, wherein \({\mathbf {CE}}\) can be proved sound and complete, and to contain a formula in the vocabulary of \({\mathbf E}\) that is not provable in \({\mathbf E}\). The result is interesting since the addition of Boolean negation to the logic \({\mathbf R}\), which is stronger than \({\mathbf E}\), is conservative, and so is its addition to relevant logics weaker than \({\mathbf E}\).
Mathematics Subject Classification ID: 03B47
Mathematics Subject Classification ID: 03B45
zbMATH DE Number: 1491206
zbMATH Keywords: relevant logic
zbMATH Keywords: relevance logic
zbMATH Keywords: modal logic
zbMATH Keywords: Boolean negation
zbMATH Keywords: entailment
reviewed by: Louis F. Goble
MaRDI profile type: Publication
cites work: Q4297115
cites work: Gentzenizations of relevant logics without distribution. I
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cites work: Classical relevant logics. I
cites work: Classical relevant logics. II
cites work: The semantics of entailment. III
cites work: Q3701398
description: scientific article; zbMATH DE number 1491206