Studia Logica

, Volume 107, Issue 1, pp 11–29 | Cite as

The Fundamental Problem of General Proof Theory

  • Dag PrawitzEmail author
Open Access


I see the question what it is that makes an inference valid and thereby gives a proof its epistemic power as the most fundamental problem of general proof theory. It has been surprisingly neglected in logic and philosophy of mathematics with two exceptions: Gentzen’s remarks about what justifies the rules of his system of natural deduction and proposals in the intuitionistic tradition about what a proof is. They are reviewed in the paper and I discuss to what extent they succeed in answering what a proof is. Gentzen’s ideas are shown to give rise to a new notion of valid argument. At the end of the paper I summarize and briefly discuss an approach to the problem that I have proposed earlier.


Proof theory Proof Valid inference Valid argument Gentzen’s naturaldeduction Intuitionism 


  1. 1.
    Diller, J., and A. S. Troelstra, Realizability and intuitionistic logic, Synthese 60:253–282, 1984.Google Scholar
  2. 2.
    Dummett, M., The philosophical basis of intuitionistic logic, in H. E. Rose et al. (eds.), Logic Colloquium ’73, North-Holland Publishing Company, Amsterdam, 1975, pp. 5–40.Google Scholar
  3. 3.
    Dummett, M., The Logical Basis of Metaphysics, Duckworth, London, 1991.Google Scholar
  4. 4.
    Gentzen, G., Untersuchungen über das logische Schließen, Mathematische Zeitschrift 39:176–210, 1935.Google Scholar
  5. 5.
    Hilbert, D., Axiomatisches Denken, Mathematischen Annalen 78:405–415, 1918.Google Scholar
  6. 6.
    Hilbert, D., Neubegründung der Mathematik, Erste Mitteilung, Abhandlungen aus dem mathematischen Seminar der Hamburgischen Universität 1:157–177, 1922.Google Scholar
  7. 7.
    Heyting, A., Die intuitionistische Grundlegung der Mathematik, Erkenntnis 2:106–115, 1931.Google Scholar
  8. 8.
    Heyting, A., Mathematische Grundlagenforschung, Intuitionismus, Beweistheorie, Springer, Berlin, 1934.Google Scholar
  9. 9.
    Heyting, A., Intuitionism in mathematics, in R. Klibansky (ed), Philosophy in the Mid-Century, La Nuova Italia, Florence, 1958, pp. 101–115.Google Scholar
  10. 10.
    Howard, W., The formulae-as-types notion of construction, in J. Seldin et al. (eds.), To H. B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism, Academic Press, London, 1980, pp. 479–490.Google Scholar
  11. 11.
    Kreisel, G., A survey of proof theory, Journal of Symbolic Logic 33:321–388, 1968.Google Scholar
  12. 12.
    Kreisel, G., Book reviews, The Collected Papers of Gerhard Gentzen, The Journal of Philosophy 68:238–265, 1971.Google Scholar
  13. 13.
    Kreisel, G., A survey of proof theory II, in J. Fenstad (ed.), Proceedings of the Second Scandinavian Logic Symposium, North-Holland, Amsterdam, 1971, pp. 109–170.Google Scholar
  14. 14.
    Martin-Löf, P., Intuitionistic Type Theory, Bibliopolis, Napoli, 1984.Google Scholar
  15. 15.
    Martin-Löf, P., Truth of a proposition, evidence of a judgement, validity of a proof, Synthese 73:407–420, 1987.Google Scholar
  16. 16.
    Martin-Löf, P., Truth and knowability: on the principles C and K of Michael Dummett, in H. G. Dales and G. Oliveri (eds.), Truth in Mathematics, Clarendon Press, Oxford, 1998, pp. 105–114.Google Scholar
  17. 17.
    Prawitz, D., Natural Deduction. A Proof-Theoretical Study, Almqvist & Wiksell, Stockholm, 1965. (Republished by Dover Publications, New York, 2006.)Google Scholar
  18. 18.
    Prawitz, D., Ideas and results in proof theory, in J. Fenstad (ed.), Proceedings of the Second Scandinavian Logic Symposium, North-Holland, Amsterdam, 1971, pp. 237–309.Google Scholar
  19. 19.
    Prawitz, D., Towards a foundation of general proof theory, in P. Suppes et al. (eds.), Logic, Methodology and Philosophy of Science IV, North-Holland, Amsterdam, 1973, pp. 225–250.Google Scholar
  20. 20.
    Prawitz, D., Meaning approached via proofs, Synthese 148:507–524, 2006.Google Scholar
  21. 21.
    Prawitz, D., Explaining deductive inference, in H. Wansing (ed.), Dag Prawitz on Proofs and Meaning (Outstanding Contributions to Logic vol. 7), Springer, Cham, 2015, pp. 65–100.Google Scholar
  22. 22.
    Prawitz, D., On the relation between Heyting’s and Gentzen’s approaches to meaning, in T. Piecha and P. Schroeder-Heister (eds.), Advances in Proof-Theoretic Semantics (Trends in Logic vol. 43), Springer, Cham, 2016, pp. 5–25.Google Scholar
  23. 23.
    Prawitz, D., Gentzen’s justification of inferences, in T. Piecha and P. Schroeder-Heister (eds.), General Proof Theory. Celebrating 50 Years of Dag Prawitz’s “Natural Deduction”. Proceedings of the Conference held in Tübingen, 27–29 November 2015, URI:, University of Tübingen, 2016, pp. 263–276.
  24. 24.
    Schroeder-Heister, P., Validity concepts in proof-theoretic semantics, Synthese 148:525–571, 2006.Google Scholar
  25. 25.
    Sundholm, G., Questions of Proof, Manuscrito (Campinas) 16:47–70, 1993.Google Scholar
  26. 26.
    Sundholm, G., Existence, proof and truth-making: A perspective on the intuitionistic conception of truth, Topoi 13:117–126, 1994Google Scholar
  27. 27.
    Troelstra, A. S., Aspects of constructive mathematics, in J. Barwise (ed.), Handbook of Mathematical Logic, North-Holland Publishing Company, Amsterdam, 1977, pp. 973–1052.Google Scholar
  28. 28.
    Troelstra, A. S., and D. Van Dalen, Constructivism in Mathematics, vol. 1, North Holland Publishing Company, Amsterdam, 1988.Google Scholar

Copyright information

© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Stockholm UniversityStockholmSweden

Personalised recommendations