General-Elimination Stability

Studia Logica 105 (2):361-405 (2017)

Abstract

General-elimination harmony articulates Gentzen’s idea that the elimination-rules 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.

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Author Profiles

Bruno Jacinto
University of St. Andrews
Stephen Read
University of St. Andrews

References found in this work

The Logical Basis of Metaphysics.Michael Dummett, Hilary Putnam & James Conant - 1994 - Philosophical Quarterly 44 (177):519-527.
Elements of Intuitionism.Michael Dummett - 1980 - British Journal for the Philosophy of Science 31 (3):299-301.
General-Elimination Harmony and the Meaning of the Logical Constants.Stephen Read - 2010 - Journal of Philosophical Logic 39 (5):557-576.
Natural Deduction: A Proof-Theoretical Study.Richmond Thomason - 1965 - Journal of Symbolic Logic 32 (2):255-256.
A Natural Extension of Natural Deduction.Peter Schroeder-Heister - 1984 - Journal of Symbolic Logic 49 (4):1284-1300.

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