Harmony and autonomy in classical logic

Journal of Philosophical Logic 29 (2):123-154 (2000)
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Michael Dummett and Dag Prawitz have argued that a constructivist theory of meaning depends on explicating the meaning of logical constants in terms of the theory of valid inference, imposing a constraint of harmony on acceptable connectives. They argue further that classical logic, in particular, classical negation, breaks these constraints, so that classical negation, if a cogent notion at all, has a meaning going beyond what can be exhibited in its inferential use. I argue that Dummett gives a mistaken elaboration of the notion of harmony, an idea stemming from a remark of Gerhard Gentzen's. The introduction-rules are autonomous if they are taken fully to specify the meaning of the logical constants, and the rules are harmonious if the elimination-rule draws its conclusion from just the grounds stated in the introduction-rule. The key to harmony in classical logic then lies in strengthening the theory of the conditional so that the positive logic contains the full classical theory of the conditional. This is achieved by allowing parametric formulae in the natural deduction proofs, a form of multiple-conclusion logic



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Stephen Read
University of St. Andrews

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References found in this work

The logical basis of metaphysics.Michael Dummett - 1991 - Cambridge: Harvard University Press.
Logic, semantics, metamathematics.Alfred Tarski - 1956 - Oxford,: Clarendon Press. Edited by John Corcoran & J. H. Woodger.
Natural deduction: a proof-theoretical study.Dag Prawitz - 1965 - Mineola, N.Y.: Dover Publications.
The logical syntax of language.Rudolf Carnap - 1937 - London,: K. Paul, Trench, Trubner & co.. Edited by Amethe Smeaton.
Introduction to mathematical logic.Alonzo Church - 1944 - Princeton,: Princeton University Press. Edited by C. Truesdell.

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