A Note on Harmony

Journal of Philosophical Logic 41 (3):613-628 (2012)
Abstract
In the proof-theoretic semantics approach to meaning, harmony , requiring a balance between introduction-rules (I-rules) and elimination rules (E-rules) within a meaning conferring natural-deduction proof-system, is a central notion. In this paper, we consider two notions of harmony that were proposed in the literature: 1. GE-harmony , requiring a certain form of the E-rules, given the form of the I-rules. 2. Local intrinsic harmony : imposes the existence of certain transformations of derivations, known as reduction and expansion . We propose a construction of the E-rules (in GE-form) from given I-rules, and prove that the constructed rules satisfy also local intrinsic harmony. The construction is based on a classification of I-rules, and constitute an implementation to Gentzen’s (and Pawitz’) remark, that E-rules can be “read off” I-rules
Keywords Natural-deduction  Proof-theoretic semantics  Introduction-rules  Elimination-rules  Harmony  Local soundness/completeness
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    References found in this work BETA
    H. P. Barendregt (1984). The Lambda Calculus: Its Syntax and Semantics. Sole Distributors for the U.S.A. And Canada, Elsevier Science Pub. Co..
    Gerhard Gentzen (1964). Investigations Into Logical Deduction. American Philosophical Quarterly 1 (4):288 - 306.

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    Stephen Read (2000). Harmony and Autonomy in Classical Logic. Journal of Philosophical Logic 29 (2):123-154.
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