Abstract
On the basis of Martin-Löf’s meaning explanations for his type theory a detailed justification is offered of the rule of identity elimination. Brief discussions are thereafter offered of how the univalence axiom fares with respect to these meaning explanations and of some recent work on identity in type theory by Ladyman and Presnell.
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Notes
The already complicated terminological situation is not helped by the fact that in homotopy type theory, ‘set’ and ‘proposition’ are given novel meanings; see definitions 3.1.1 and 3.3.1 in (The Univalent Foundations Program, 2013).
On the origin of the conjecture, see also Prawitz (2015, p. 45).
Perhaps the first to introduce a special term for what are here called canonical objects was Husserl, who in the second part of his Philosophie der Arithmetik (Husserl 1891, pp. 295–297), speaks of the reduction of problematische Zahlen to Normalzahlen.
The system of Martin-Löf (1975) differs in this respect, since there evaluation works from within.
This introduction rule together with the naturally associated elimination- and equality rules give rise to Curry’s Paradox (Curry 1942).
That an inductively defined predicate cannot occur negatively in the premiss of its own introduction rule is in effect required by the schemes for such definitions given by Martin-Löf (1971, pp. 182–183). The requirement is also made by Dybjer (1994) for his general form of introduction rules in type theory.
This extension of the notion of canonicity can be used to argue that, for instance, \(A\wedge \lnot A\) and \(\bot\) are not equal \(\mathbf {prop}\)s. In the context \(x:A, y: \lnot A\), the term \(\langle x,y\rangle\) is a canonical element of \(A\wedge \lnot A\), but not of \(\bot\).
In the case of \(\Pi\) the higher-order presentation is needed; see Garner (2009, Thm. 4.2’).
That (UIP) is not derivable in Martin-Löf type theory with the \(\text{Id}\)-elimination rule assumed in this paper is a non-trivial meta-mathematical result first proved by Hofmann and Streicher (1998).
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Acknowledgements
I am grateful to Göran Sundholm for many discussions on the philosophical foundations of type theory and to Per Martin-Löf, who in correspondence clarified what is the canonical form of an element of an Id-set. The comments of two referees at Topoi were of great help when preparing the final version of the paper. Thanks are also due to the referees of a predecessor of this paper that was submitted to another journal. While writing the paper, the author has been supported by Grant No. 17-18344Y from the Czech Science Foundation, GAČR.
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Klev, A. The Justification of Identity Elimination in Martin-Löf’s Type Theory. Topoi 38, 577–590 (2019). https://doi.org/10.1007/s11245-017-9509-1
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DOI: https://doi.org/10.1007/s11245-017-9509-1