The paper explains how a paraconsistent logician can appropriate all classical reasoning. This is to take consistency as a default assumption, and hence to work within those models of the theory at hand which are minimally inconsistent. The paper spells out the formal application of this strategy to one paraconsistent logic, first-order LP. (See, Ch. 5 of: G. Priest, In Contradiction, Nijhoff, 1987.) The result is a strong non-monotonic paraconsistent logic agreeing with classical logic in consistent situations. It is shown (...) that the logical closure of a theory under this logic is trivial only if its closure under LP is trivial. (shrink)

Reviewed Works:Gaisi Takeuti, Proof Theory.Georg Kreisel, Proof Theory: Some Personal Recollections.Wolfram Pohlers, Contributions of the Schutte School in Munich to Proof Theory.Stephen G. Simpson, Subsystems of $\mathbf{Z}_2$ and Reverse Mathematics.Solomon Feferman, Proof Theory: A Personal Report.

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. (shrink)