Skip to main content
Log in

A Negationless Interpretation of Intuitionistic Theories. II

  • Published:
Studia Logica Aims and scope Submit manuscript

Abstract

This work is a sequel to our [16]. It is shown how Theorem 4 of [16], dealing with the translatability of HA(Heyting's arithmetic) into negationless arithmetic NA, can be extended to the case of intuitionistic arithmetic in higher types.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Beth, E. W., The Foundations of Mathematics, North-Holland, Amsterdam, 1959.

    Google Scholar 

  2. Bishop, E., Foundations of Constructive Analysis, McGraw-Hill, New York, 1967.

    Google Scholar 

  3. Franchella, M., ‘Griss' contribution to intuitionism’, in Philosophy of Mathematics. Proceedings of the 15th International Wittgenstein-Symposium, August 16–23, 1992, J. Czermak (ed.), Kirchberg am Wechsel (Austria), Hölder-Pichler-Tempsky, Vienna, 1993, pp. 119–126.

    Google Scholar 

  4. Franchella, M., ‘Brouwer and Griss on intuitionistic negation’, Modern Logic 4 (1994), 256–265.

    Google Scholar 

  5. Franchella, M., ‘Negation in the work of Griss’, in Perspectives on Negation. Essays in Honour of Johan J. de Jongh on His 80th Birthday, H. C. M. de Swart and L. J. M. Bergman (eds.), Tilburg University Press, Tilburg, 1995, pp. 29–40.

    Google Scholar 

  6. Gilmore, P. C. G., ‘The effect of Griss' criticism of the intuitionistic logic on deductive theories formalized within the intuitionistic logic II’, Indagationes Mathematicae 15 (1953), 175–186.

    Google Scholar 

  7. Goodman, N. D., and J. Myhill, ‘The formalization of Bishop's constructive mathematics’, in Toposes, Algebraic Geometry and Logic, Lecture Notes in Mathematics, vol. 274, F. W. Lawvere (ed.), Springer, Berlin, Heidelberg, New York, 1972, pp. 83–96.

    Google Scholar 

  8. Griss, G. F. C., Idealistische Filosofie, Van Loghum Slaterus, Arnhem, 1946.

    Google Scholar 

  9. Griss, G. F. C., ‘Over de negatie’, in Feestbundel aangeboden door vrienden en leerlingen aan Prof. H. J. Pos, N. V. Noord-Hollandsche Uitgevers Maatschappij, Amsterdam, 1948, pp. 96–106.

    Google Scholar 

  10. Griss, G. F. C., ‘Sur la négation (dans les mathématiques et la logique)’, Synthese 7 (1948), 71–74.

    Google Scholar 

  11. Griss, G. F. C., ‘Logique des mathématiques intuitionnistes sans négation’, Comptes Rendus de l'Académie des Sciences, Paris 227 (1948), 946–948.

    Google Scholar 

  12. Griss, G. F. C., ‘Negationless intuitionistic mathematics II’, Indagationes Mathematicae 12 (1950), 108–115.

    Google Scholar 

  13. Griss, G. F. C., ‘Logic of negationless intuitionistic mathematics’, Indagationes Mathematicae 13 (1951), 41–49.

    Google Scholar 

  14. Griss, G. F. C., ‘La mathématique intuitioniste sans negation’, Nieuw Archief voor Wiskunde 3(3) (1955), 134–142.

    Google Scholar 

  15. Krivtsov, V. N., ‘A negationless interpretation of intuitionistic axiomatic theories: higher-order arithmetic’, research report ML-1998-07, Institute for Logic, Language and Computation, University of Amsterdam, ILLC Scientific Publications, Amsterdam, 1998.

    Google Scholar 

  16. Krivtsov, V. N., ‘A negationless interpretation of intuitionistic theories I’, Studia Logica 64 (2000), xyz–XYZ.

    Google Scholar 

  17. Myhill, J., ‘Formal systems of intuitionistic analysis I’, in Logic, Methodology and Philosophy of Science, vol. 3, B. van Rootselaar and J. F. Staal (eds.), North-Holland, Amsterdam, 1968, pp. 161–178.

    Google Scholar 

  18. Myhill, J., ‘Embedding classical type theory in “intuitionistic” type theory, a correction’, in Axiomatic Set Theory, Part II, T. Jech (ed.), Providence, RI, American Mathematical Society, 1974, pp. 185–188.

    Google Scholar 

  19. Nelson, D., ‘Non-null implication’, The Journal of Symbolic Logic 31 (1966), 562-572.

    Google Scholar 

  20. Takeuti, G., Proof Theory, North-Holland, Amsterdam, 1975.

    Google Scholar 

  21. Troelstra, A. S., ‘Intuitionistic formal systems’, in Metamathematical Investigation in Intuitionistic Arithmetic and Analysis, Lecture Notes in Mathematics, vol. 344, A. S. Troelstra (ed.), Springer, Berlin, Heidelberg, New York, 1973, pp. 1–96.

    Google Scholar 

  22. Troelstra A. S., and D, van Dalen, Constructivism in Mathematics. An Introduction, vol. I, North-Holland, Amsterdam, 1988.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Krivtsov, V.N. A Negationless Interpretation of Intuitionistic Theories. II. Studia Logica 65, 155–179 (2000). https://doi.org/10.1023/A:1005207512630

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1005207512630

Navigation