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WHY SHOULD IDENTITY BE LOGICAL?

Part of: Proof theory and constructive mathematics

Published online by Cambridge University Press:  17 September 2025

CHRIS MITSCH Open the ORCID record for CHRIS MITSCH [Opens in a new window]
CHRIS MITSCH*
Affiliation:
DEPARTMENT OF COMPUTER SCIENCE BUCKNELL UNIVERSITY LEWISBURG PA 17837 USA
*
E-mail: cr.mitsch@gmail.com
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Abstract

Logical inferentialists have expected identity to be susceptible of harmonious introduction and elimination rules in natural deduction. While Read and Klev have proposed rules they argue are harmonious, Griffiths and Ahmed have criticized these rules as insufficient for harmony. These critics, moreover, suggest that no harmonious rules are forthcoming. We argue that these critics are correct: the logical inferentialist should abandon hope for harmonious rules for identity. The paper analyzes the three major uses of identity in presumed-logical languages: variable coordination, definitional substitution, and co-reference. We show that identity qua variable coordination is not logical by providing a harmonious natural-deduction system that captures this use through the quantifiers. We then argue that identity qua definitional substitution or co-reference faces a dilemma: either its rules are harmonious but they obscure its actual use in inference, or its rules are not harmonious but they make its actual use in inference plain. We conclude that the inferentialist may have harmonious rules for identity only by disrespecting its inferential use.

Keywords

harmonyidentityproof-theoretic semanticsinferentialismWittgensteinW-logicmeaning as use

MSC classification

Primary: 03F03: Proof theory, general

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Type
Research Article
Information
The Review of Symbolic Logic , First View , pp. 1 - 24
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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