Skip to main content



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.

Corresponding author
Hide All
[1] Awodey S. (2009). Introduction to Categorical Logic. Available at: (accessed October 31, 2009).
[2] Awodey S. (2010). Category Theory (second edition). Oxford, UK: Clarendon Press.
[3] Belnap N. D. (1962). Tonk, plonk and plink. Analysis, 22(6), 130134.
[4] Bernays P. (1922). Über hilberts gedanken zur grundlegung der arithmetik. Jahresbericht der Deutschen Mathematiker-Vereinigung, 31, 1019.
[5] Coquand T. (2011). Equality and Dependent Type Theory. Available at:∼coquand/equality.pdf (accessed October 5, 2015).
[6] Coquand T. (2014). A Remark on Singleton Types. Available at:∼coquand/singl.pdf (accessed March 6, 2017).
[7] Dummett M. (1981). Frege: Philosophy of Language. Cambridge, MA: Harvard University Press.
[8] Dummett M. (1991). The Logical Basis of Metaphysics. Cambridge, MA: Harvard University Press.
[9] Schreiber U. (2009). ‘Philosophical’ meaning of the yoneda lemma, MathOverflow. Available at: schreiber.
[10] Jacobs B. (1999). Categorical Logic and Type Theory. Amsterdam: Elsevier.
[11] Koslow A. (2005). A Structuralist Theory of Logic. New York: Cambridge University Press.
[12] Ladyman J. & Presnell S. (2014). Does homotopy type theory provide a foundation for mathematics. The British Journal for the Philosophy of Science, forthcoming.
[13] Ladyman J. & Presnell S. (2015). Identity in homotopy type theory, part I: The justification of path induction. Philosophia Mathematica, 23(3), 386406.
[14] Lawvere F. W. (1969). Adjointness in foundations. Dialectica, 23(3–4), 281296.
[15] Lawvere F. W. (1970). Equality in hyperdoctrines and comprehension schema as an adjoint functor. Applications of Categorical Algebra, 17, 114.
[16] Martin-Löf P. (1995). Verificationism then and now. In van der Schaar M., editor. The Foundational Debate: Complexity and Constructivity in Mathematics and Physics. Vienna Circle Institute Yearbook, Vol. 3. Dordrecht: Springer Netherlands, pp. 314.
[17] Martin-Löf P. (1996). On the meanings of the logical constants and the justifications of the logical laws. Nordic Journal of Philosophical Logic, 1(1), 1160.
[18] Maruyama Y. (2016). Categorical harmony and paradoxes in proof-theoretic semantics. In Piecha T. and Schroeder-Heister P., editors. Advances in Proof-Theoretic Semantics. Trends in Logic, Vol. 43. Heidelberg: Springer, pp. 95114.
[19] Mayberry J. (1994). What is required of a foundation for mathematics? Philosophia Mathematica, 2(1), 1635.
[20] Morehouse E. (2013). An Adjunction-Theoretic Foundation for Proof Search in Intuitionistic First-Order Categorical Logic Programming, Ph.D. Thesis, Wesleyan University.
[21] Peregrin J. (2012). What is inferentialism? In Gurova L., editor. Inference, Consequence, and Meaning: Perspectives on Inferentialism. Newcastle upon Tyne, UK: Cambridge Scholars Publishing, pp. 316.
[22] Peregrin J. (2014). Inferentialism: Why Rules Matter. New York: Palgrave Macmillan.
[23] Pfenning F. & Davies R. (2001). A judgemental reconstruction of modal logic. Mathematical Structures in Computer Science, 11(4), 511540.
[24] Prawitz D. (1965). Natural Deduction: a Proof-Theoretical Study. Stockholm: Almqvist & Wiksell.
[25] Prior A. N. (1960). The runabout inference-ticket. Analysis, 21(2), 3839.
[26] Schroeder-Heister P. (2007). Generalized definitional reflection and the inversion principle. Logica Universalis, 1(2), 355376.
[27] Steinberger F. (2009). Harmony and Logical Inferentialism. Ph.D. Thesis, University of Cambridge.
[28] Steinberger F. (2009). Not so stable. Analysis, 69(4), 655661.
[29] Stevenson J. T. (1961). Roundabout the runabout inference-ticket. Analysis, 21(6), 124128.
[30] Tennant N. (1978). Natural Logic. Edinburgh: Edinburgh University Press.
[31] Tennant N. (1987). Anti-realism and Logic: Truth as Eternal. Oxford: Clarendon Press.
[32] Tennant N. (2007). Inferentialism, logicism, harmony, and a counterpoint. Essays for Crispin Wright: Logic, Language, and Mathematics, 2, 105132.
[33] Tennant N. (2010). Inferential semantics for first-order logic: Motivating rules of inference from rules of evaluation. In Smiley T. J., Lear J., and Oliver A., editors. The Force of Argument: Essays in Honor of Timothy Smiley. London: Routledge, pp. 223257.
[34] The Univalent Foundations Program (2013). Homotopy Type Theory: Univalent Foundations of Mathematics. Institute for Advanced Study. Available at:
[35] Walsh P. (2015). Justifying Path Induction: An Inferentialist Analysis of Identity Elimination in Homotopy Type Theory. Master’s Thesis, Carnegie Mellon University.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

The Review of Symbolic Logic
  • ISSN: 1755-0203
  • EISSN: 1755-0211
  • URL: /core/journals/review-of-symbolic-logic
Please enter your name
Please enter a valid email address
Who would you like to send this to? *



Full text views

Total number of HTML views: 2
Total number of PDF views: 44 *
Loading metrics...

Abstract views

Total abstract views: 379 *
Loading metrics...

* Views captured on Cambridge Core between 14th March 2017 - 20th November 2017. This data will be updated every 24 hours.