Journal of Symbolic Logic

Interpreting descriptions in intensional type theory

Jesper Carlström

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Natural deduction systems with indefinite and definite descriptions (ε-terms and ℩-terms) are presented, and interpreted in Martin-Löf's intensional type theory. The interpretations are formalizations of ideas which are implicit in the literature of constructive mathematics: if we have proved that an element with a certain property exists, we speak of ‘the element such that the property holds' and refer by that phrase to the element constructed in the existence proof. In particular, we deviate from the practice of interpreting descriptions by contextual definitions.

Article information

Source
J. Symbolic Logic Volume 70, Issue 2 (2005), 488-514.

Dates
First available: 1 July 2005

Permanent link to this document
http://projecteuclid.org/euclid.jsl/1120224725

Digital Object Identifier
doi:10.2178/jsl/1120224725

Zentralblatt MATH identifier
1089.03051

Mathematical Reviews number (MathSciNet)
MR2140043

Citation

Carlström, Jesper. Interpreting descriptions in intensional type theory. Journal of Symbolic Logic 70 (2005), no. 2, 488--514. doi:10.2178/jsl/1120224725. http://projecteuclid.org/euclid.jsl/1120224725.


Export citation

References

  • M. Abadi, G. Gonthier, and B. Werner Choice in dynamic linking, Foundations of software science and computation structures, 7$^\textth$ international conference, FOSSACS 2004, Barcelona, Spain, March 29--April 2, 2004, proceedings (Igor Walukiewicz, editor), Lecture Notes in Computer Science, vol. 2987, Springer,2004, pp. 12--26.
  • P. Aczel and N. Gambino Collection principles in dependent type theory, Types for proofs and programs, international workshop, TYPES 2000, Durham, UK, December 8--12, 2000, selected papers (P. Callaghan, Z. Luo, J. McKinna, and R. Pollack, editors), Lecture Notes in Computer Science, vol. 2277, Springer,2002, pp. 1--23.
  • E. Bishop Foundations of constructive analysis, McGraw-Hill Book Co., New York,1967.
  • J. Carlström Subsets, quotients and partial functions in Martin-Löf's type theory, Types for proofs and programs, second international workshop, TYPES 2002, Berg en Dal, The Netherlands, April 24--28, 2002, selected papers (H. Geuvers and F. Wiedijk, editors), Lecture Notes in Computer Science, vol. 2646, Springer,2003, pp. 78--94.
  • G. Frege Über Sinn und Bedeutung, Zeitschrift für Philosophie und philosophische Kritik, vol. NF 100 (1892), pp. 25--50, English translation in [?].
  • A. Heyting Intuitionism: An introduction, North-Holland, Amsterdam,1956.
  • D. Leivant Existential instantiation in a system of natural deduction for intuitionistic arithmetics, Technical Report ZW 13/73, Stichtung Mathematisch Centrum, Amsterdam,1973.
  • S. Maehara The predicate calculus with $\varepsilon$-symbol, Journal of the Mathematical Society of Japan, vol. 7 (1955), pp. 323--344.
  • P. Martin-Löf Intuitionistic type theory, Bibliopolis, Naples,1984, Notes by Giovanni Sambin.
  • R. Mines, F. Richman, and W. Ruitenburg A course in constructive algebra, Springer-Verlag, New York,1988.
  • G. E. Mints Heyting predicate calculus with epsilon symbol, Journal of Soviet Mathematics, vol. 8 (1977), pp. 317--323.
  • B. Nordström, K. Petersson, and J. Smith Programming in Martin-Löf's type theory, Oxford University Press,1990, http://www.cs.chalmers.se/Cs/Research/Logic/book/.
  • D. Prawitz Natural deduction: a proof-theoretical study, Stockholm Studies in Philosophy, Almqvist & Wiksell,1965.
  • A. Ranta Type-theoretical grammar, Oxford University Press, Oxford,1994.
  • B. Russell On denoting, Mind, vol. 14 (1905), pp. 479--493.
  • G. Sambin and S. Valentini Building up a toolbox for Martin-Löf's type theory: subset theory, Twenty-five years of constructive type theory (Venice, 1995) (G. Sambin and J. Smith, editors), Oxford Logic Guides, vol. 36, Oxford University Press,1998, pp. 221--244.
  • D. Scott Identity and existence in intuitionistic logic, Applications of sheaves (M. P. Fourman, C. J. Mulvey, and D. S. Scott, editors), Lecture Notes in Mathematics, vol. 753, Springer, Berlin,1979, pp. 660--696.
  • K. Shirai Intuitionistic predicate calculus with $\varepsilon$-symbol, Annals of the Japan Association for Philosophy of Science, vol. 4 (1971), pp. 49--67.
  • S. Stenlund The logic of description and existence, Filosofiska studier, no. 18, Filosofiska föreningen och Filosofiska institutionen vid Uppsala universitet, Uppsala,1973.