A Note on Harmony

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.

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