Categorical harmony and path induction

Review of Symbolic Logic 10 (2):301-321 (2017)

Authors
Patrick Walsh
Carnegie Mellon University
Abstract
This paper responds to recent work in the philosophy of Homotopy Type Theory by James Ladyman and Stuart Presnell. They consider one of the rules for identity, path induction, and justify it along ‘pre-mathematical’ lines. I give an alternate justification based on the philosophical framework of inferentialism. Accordingly, I construct a notion of harmony that allows the inferentialist to say when a connective or concept is meaning-bearing and this conception unifies most of the prominent conceptions of harmony through category theory. This categorical harmony is stated in terms of adjoints and says that any concept definable by iterated adjoints from general categorical operations is harmonious. Moreover, it has been shown that identity in a categorical setting is determined by an adjoint in the relevant way. Furthermore, path induction as a rule comes from this definition. Thus we arrive at an account of how path induction, as a rule of inference governing identity, can be justified on mathematically motivated grounds.
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DOI 10.1017/S1755020317000077
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References found in this work BETA

The Logical Basis of Metaphysics.Michael Dummett, Hilary Putnam & James Conant - 1991 - Philosophical Quarterly 44 (177):519-527.
Category Theory.Steve Awodey - 2007 - Studia Logica 86 (1):133-135.
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[Omnibus Review].Dag Prawitz - 1991 - Journal of Symbolic Logic 56 (3):1094-1096.

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