Categorical harmony and path induction

Review of Symbolic Logic 10 (2):301-321 (2017)
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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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Patrick Walsh
Carnegie Mellon University

Citations of this work

The Hole Argument, take n.John Dougherty - 2020 - Foundations of Physics 50 (4):330-347.
Category theory.Jean-Pierre Marquis - 2008 - Stanford Encyclopedia of Philosophy.

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References found in this work

The Runabout Inference-Ticket.A. N. Prior - 1960 - Analysis 21 (2):38-39.
The Logical Basis of Metaphysics.Michael Dummett, Hilary Putnam & James Conant - 1994 - Philosophical Quarterly 44 (177):519-527.
Tonk, Plonk and Plink.Nuel Belnap - 1962 - Analysis 22 (6):130-134.
The runabout inference ticket.Arthur Prior - 1967 - In P. F. Strawson (ed.), Philosophical logic. London,: Oxford University Press. pp. 38-9.
Natural Deduction: A Proof-Theoretical Study.Richmond Thomason - 1965 - Journal of Symbolic Logic 32 (2):255-256.

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