General-Elimination Stability

• Published in: Studia logica Vol. 105; no. 2; pp. 361 - 405
• Main Authors: Jacinto, Bruno, Read, Stephen
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer 01-04-2017; Springer Netherlands
• Subjects: Computational Linguistics; Education; Logic; Mathematical Logic and Foundations; Philosophy; Grounds; Local completeness; Stability; Harmony; General-elimination rules; Gentzen
• ISSN: 0039-3215; 1572-8730
• DOI: 10.1007/s11225-016-9692-x

General-elimination harmony articulates Gentzen's idea that the eliminationrules are justified if they infer from an assertion no more than can already be inferred from the grounds for making it. Dummett described the rules as not only harmonious but stable if the E-rules allow one to infer no more and no less than the I-rules justify. Pfenning and Davies call the rules locally complete if the E-rules are strong enough to allow one to infer the original judgement. A method is given of generating harmonious general-elimination rules from a collection of I-rules. We show that the general-elimination rules satisfy Pfenning and Davies' test for local completeness, but question whether that is enough to show that they are stable. Alternative conditions for stability are considered, including equivalence between the introduction-and elimination-meanings of a connective, and recovery of the grounds for assertion, finally generalizing the notion of local completeness to capture Dummett's notion of stability satisfactorily. We show that the general-elimination rules meet the last of these conditions, and so are indeed not only harmonious but also stable.