David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
Learn more about PhilPapers
Studia Logica 43 (1-2):181 - 200 (1984)
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||No keywords specified (fix it)|
|Categories||categorize this paper)|
|Through your library||Configure|
Similar books and articles
Thomas Lukasiewicz (2005). Nonmonotonic Probabilistic Reasoning Under Variable-Strength Inheritance with Overriding. Synthese 146 (1-2):153 - 169.
Jeffry L. Hirst & Carl Mummert (2010). Reverse Mathematics and Uniformity in Proofs Without Excluded Middle. Notre Dame Journal of Formal Logic 52 (2):149-162.
Heinrich Wansing (2010). The Power of Belnap: Sequent Systems for SIXTEEN ₃. [REVIEW] Journal of Philosophical Logic 39 (4):369 - 393.
Yaroslav Shramko & Heinrich Wansing (2005). Some Useful 16-Valued Logics: How a Computer Network Should Think. [REVIEW] Journal of Philosophical Logic 34 (2):121 - 153.
J. W. Degen (1999). Complete Infinitary Type Logics. Studia Logica 63 (1):85-119.
Stewart Shapiro (2002). Incompleteness and Inconsistency. Mind 111 (444):817-832.
Neil Tennant (1994). Intuitionistic Mathematics Does Not Needex Falso Quodlibet. Topoi 13 (2):127-133.
J. Michael Dunn (1980). A Sieve for Entailments. Journal of Philosophical Logic 9 (1):41 - 57.
Added to index2009-01-28
Total downloads12 ( #104,751 of 1,010,234 )
Recent downloads (6 months)3 ( #28,090 of 1,010,234 )
How can I increase my downloads?