Mitsch, Chris
(2023)
Why Should Identity Be Harmonious?
[Preprint]
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. I 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.
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