Perfect validity, entailment and paraconsistency
type: Item
instance of: scholarly article
title: Perfect validity, entailment and paraconsistency
zbMATH Open document ID: 0582.03011
DOI: 10.1007/BF00935749
author: Neil Tennant
published in: Studia Logica
publication date: 1984-01-01T00:00:00Z
review text: This paper treats entailment as a subrelation of classical consequence and deducibility. Working with a Gentzen set-sequent system, we define an entailment as a substitution instance of a valid sequent all of whose premisses and conclusions are necessary for its classical validity. We also define a sequent Proof as one in which there are no applications of cut or dilution. The main result is that the entailments are exactly the Provable sequents. There are several important corollaries. Every unsatisfiable set is Provably inconsistent. Every logical consequence of a satisfiable set is Provable therefrom. Thus our system is adequate for ordinary mathematical practice. Moreover, transitivity of Proof fails upon accumulation of Proofs only when the newly combined premisses are inconsistent anyway, or the conclusion is a logical truth. In either case Proofs that show this can be effectively determined from the Proofs given. Thus transitivity fails where it least matters - arguably, where it ought to fail! We show also that entailments hold by virtue of logical form insufficient either to render the premisses inconsistent or to render the conclusion logically true. The Lewis paradoxes are not Provable. Our system is distinct from Anderson and Belnap's system of first degree entailments, and Johansson's minimal logic. Although the Curry set paradox is still Provable within naive set theory, our system offers the prospect of a more sensitive paraconsistent reconstruction of mathematics. It may also find applications within the logic of knowledge and belief.
Mathematics Subject Classification ID: 03B45
Mathematics Subject Classification ID: 03B60
zbMATH DE Number: 3930972
zbMATH Keywords: paraconsistency
zbMATH Keywords: relevance
zbMATH Keywords: entailment
zbMATH Keywords: Gentzen set-sequent system
MaRDI profile type: Publication
cites work: A proof-theoretic approach to entailment
cites work: Q3941377
cites work: Proof and Paradox
cites work: First degree entailments
cites work: Q4198728
description: scientific article; zbMATH DE number 3930972