The Harmony of Identity

Abstract

The standard natural deduction rules for the identity predicate have seemed to some not to be harmonious. Stephen Read has suggested an alternative introduction rule that restores harmony but presupposes second-order logic. Here it will be shown that the standard rules are in fact harmonious. To this end, natural deduction will be enriched with a theory of definitional identity. This leads to a novel conception of canonical derivation, on the basis of which the identity elimination rule can be justified in a proof-theoretical manner.

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Acknowledgements

The justification of =-elim given here is an adaptation to natural deduction of the justification of the corresponding rule in Martin-Löf’s type theory given in [7]. Correspondence with Per Martin-Löf pertaining to the latter work suggested to me the conception of canonical derivation that lies at the basis of the current work. While writing the paper I have been financially supported by grant nr. 17-18344Y from the Czech Science Foundation, GAČR.

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Correspondence to Ansten Klev.

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Klev, A. The Harmony of Identity. J Philos Logic 48, 867–884 (2019). https://doi.org/10.1007/s10992-018-09499-0

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Keywords

  • Identity
  • Natural deduction
  • Definition