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Topoi

pp 1–14 | Cite as

The Justification of Identity Elimination in Martin-Löf’s Type Theory

  • Ansten Klev
Article

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.

Keywords

Justification of logical laws Type theory Identity 

Notes

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.

References

  1. Boolos G (1971) The iterative conception of set. J Philos 68:215–232CrossRefGoogle Scholar
  2. Curry HB (1942) The inconsistency of certain formal logics. J Symb Log 7:115–117CrossRefGoogle Scholar
  3. Dybjer P (1994) Inductive families. Form Asp Comput 6:440–465CrossRefGoogle Scholar
  4. Garner R (2009) On the strength of dependent products in the type theory of Martin-Löf. Ann Pure Appl Log 160:1–12CrossRefGoogle Scholar
  5. Hofmann M, Streicher T (1998) The groupoid interpretation of type theory. In: Sambin G, Smith JM (eds) Twenty-five years of constructive type theory. Oxford University Press, Oxford, pp 83–111Google Scholar
  6. Husserl E (1891) Philosophie der Arithmetik. C.E.M. Pfeffer, HalleGoogle Scholar
  7. Ladyman J, Presnell S (2015) Identity in homotopy type theory, part I: the justification of path induction. Philos Math 23:386–406CrossRefGoogle Scholar
  8. Ladyman J, Presnell S (2016) Does homotopy type theory provide a foundation for mathematics? Br J Philos Sci. doi: 10.1093/bjps/axw006 Google Scholar
  9. Martin-Löf P (1971) Hauptsatz for the intuitionistic theory of iterated inductive definitions. In Fenstad JE (ed) Proceedings of the second Scandinavian logic symposium, pp 179–216. North-Holland, AmsterdamGoogle Scholar
  10. Martin-Löf P (1975) An intuitionistic theory of types: predicative part. In: Rose HE, Shepherdson JC (eds) Logic colloquium ‘73. North-Holland, Amsterdam, pp 73–118Google Scholar
  11. Martin-Löf P (1982) Constructive mathematics and computer programming. In: Cohen JL, Łoś J et al (eds) Logic, methodology and philosophy of science VI. North-Holland, Amsterdam, pp 153–175Google Scholar
  12. Martin-Löf P (1984) Intuitionistic type theory. Bibliopolis, NaplesGoogle Scholar
  13. Nordström B, Petersson K, Smith J (1990) Programming in Martin-Löf’s type theory. Oxford University Press, OxfordGoogle Scholar
  14. Nordström B, Petersson K, Smith JM (2000) Martin-Löf’s type theory. In: Abramsky S, Gabbay D, Maibaum TSE (eds) Handbook of logic in computer science. Logic and algebraic methods, vol 5. Oxford University Press, Oxford, pp 1–37Google Scholar
  15. Prawitz D (1965) Natural deduction. Almqvist & Wiksell, StockholmGoogle Scholar
  16. Prawitz D (1971) Ideas and results in proof theory. In Fenstad JE (ed) Proceedings of the second Scandinavian logic symposium, North-Holland, Amsterdam, pp 235–307 Google Scholar
  17. Prawitz D (2015) A short scientific autobiography. In: Wansing H (ed) Dag prawitz on proofs and meaning. Outstanding contributions to logic. Springer, Dordrecht, pp 33–64Google Scholar
  18. Shoenfield J (1977) The axioms of set theory. In: Barwise J (ed) Handbook of mathematical logic. North-Holland, Amsterdam, pp 321–344CrossRefGoogle Scholar
  19. The Univalent Foundations Program (2013) Homotopy type theory: Univalent foundations of mathematics. Institute for Advanced Study, Princeton. http://homotopytypetheory.org/book
  20. Walsh P (2017) Categorical harmony and path induction. Rev Symb Log 10:301–321CrossRefGoogle Scholar
  21. Zermelo E (1930) Über Grenzzahlen und Mengenbereiche. Fundam Math 16:29–47Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  1. 1.Department of Logic, Institute of Philosophy, Czech Academy of SciencesPraha 1Czech Republic

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