Studia Logica

, Volume 43, Issue 1–2, pp 181–200

# Perfect validity, entailment and paraconsistency

• Neil Tennant
Article

## Abstract

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.

## Keywords

Logical Consequence Computational Linguistic Mathematical Practice Logical Truth Minimal Logic
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. [1]
N. Tennant,A proof-theoretic approach to entailment,Journal of Philosophical Logic 9 (1980), pp. 185–209.Google Scholar
2. [2]
N. Tennant,Natural Logic, Edinburgh University Press, 1978.Google Scholar
3. [3]
4. [4]
T. J. Smiley,Entailment and deducibility,Proceedings of the Aristotelian Society n. s. 59 (1968–9), pp. 233–254.Google Scholar
5. [5]
A. R. Anderson andN. D. Belnap,First degree entailments,Mathematische Annalen 149 (1963), pp. 302–319.Google Scholar
6. [6]
I. Johansson,Der Minimalkalkül, ein reduzierter intuitionistischer Formalismus,Compositio Mathematica 4 (1936), pp. 119–136.Google Scholar
7. [7]
R. K. Meyer, R. Routley andJ. M. Dunn,Curry's paradox,Analysis 40 (1980), pp. 124–128.Google Scholar