Intuitive counterexamples for constructive fallacies

  • James Lipton
  • Michael J. O'Donnell
Invited Papers
Part of the Lecture Notes in Computer Science book series (LNCS, volume 841)


Formal countermodels may be used to justify the unprovability of formulae in the Heyting calculus (the best accepted formal system for constructive reasoning), on the grounds that unprovable formulae are not constructively valid. We argue that the intuitive impact of such countermodels becomes more transparent and convincing as we move from Kripke/Beth models based on possible worlds, to Läuchli realizability models. We introduce a new semantics for constructive reasoning, called relational realizability, which strengthens further the intuitive impact of Läuchli realizability. But, none of these model theories provides countermodels with the compelling impact of classical truth-table countermodels for classically unprovable formulae.

We outline a proof that the Heyting calculus is sound for relational realizability, and conjecture that there is a constructive choice-free proof of completeness. In this respect, relational realizability improves the metamathematical constructivity of Läuchli realizability (which uses choice in two crucial ways to prove completeness) in the same sort of way Beth semantics improves Kripke semantics.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Allan Ross Anderson and Nuel D. Belnap, Entailment: the Logic of Relevance and Necessity, volume I, Princeton University Press, Princeton, NJ, 1975.Google Scholar
  2. 2.
    E. W. Beth, The Foundations of Mathematics, A Study in the Philosophy of Science, Studies in Logic and the Foundations of Mathematics, North-Holland Publishing Company, Amsterdam, 1959.Google Scholar
  3. 3.
    H. B. Curry and R. Feys, Combinatory Logic Volume I, Studies in Logic and the Foundations of Mathematics, North-Holland Publishing Company, Amsterdam, 1958.Google Scholar
  4. 4.
    H. C. M. de Swart, “Another Intuitionistic Completeness Proof”, Journal of Symbolic Logic 41 (1976) 644–662.Google Scholar
  5. 5.
    M. A. E. Dummett, Elements of Intuitionism, Oxford University Press, 1977.Google Scholar
  6. 6.
    M. C. Fitting, Intuitionistic Logic, Model Theory, and Forcing, Studies in Logic and the Foundations of Mathematics, North-Holland Publishing Company, Amsterdam, London, 1969.Google Scholar
  7. 7.
    J.-Y. Girard, Y. Lafont, and P. Taylor, Proofs and Types, Cambridge Tracts in Theoretical Computer Science, Cambridge University Press, 1989.Google Scholar
  8. 8.
    R. Grayson, “Forcing in Intuitionistic Set Theory Without Power Set”, Journal of Symbolic Logic 48 (1983) 670–682.Google Scholar
  9. 9.
    A. Heyting, “Die Formalen Regeln der Intuitionistischen Logik”, Sitzungsberichte der Preussischen Academie der Wissenschaften, Physikalisch-Matematische Klasse (1930) 42–56.Google Scholar
  10. 10.
    W. A. Howard, “The Formulae-As-Types Notion of Construction”, in: To H. B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism (J. P. Seldin and J. R. Hindley, eds.), pp. 479–490, Academic Press, 1980.Google Scholar
  11. 11.
    S. C. Kleene, “On the Interpretation of Intuitionistic Number Theory”, The Journal of Symbolic Logic 10(4) (1945) 109–124.Google Scholar
  12. 12.
    S. C. Kleene, “Realizability”, in: Constructivity in Mathematics (A. Heyting, ed.) pp. 285–289, North-Holland Publishing Company, Amsterdam, 1959. Proceedings of the Colloquium Held in Amsterdam, August 26–31, 1957.Google Scholar
  13. 13.
    S. C. Kleene and R. E. Vesley, The Foundations of Intuitionistic Mathematics, Especially in Relation to Recursive Functions, Studies in Logic and the Foundations of Mathematics, North-Holland Publishing Company, Amsterdam, London, 1965.Google Scholar
  14. 14.
    S. A. Kripke, “Semantical Analysis of Modal Logic I: Normal Modal Propositional Calculi”, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 9 (1963) 67–96.Google Scholar
  15. 15.
    S. A. Kripke, “Semantical Analysis of Intuitionistic Logic, I”, in: Formal Systems and Recursive Functions (J. N. Crossley and M. A. E. Dummett, eds.), pp. 92–130, North-Holland Publishing Company, Amsterdam, 1965. Proceedings of the Eighth Logic Colloquium, Oxford, July 1963.Google Scholar
  16. 16.
    Stuart A. Kurtz, John C. Mitchell, and Michael J. O'Donnell, “Connecting Formal Semantics to Constructive Intuitions”, in: Constructivity in Computer Science (J. P. Myers and M. J. O'Donnell, eds.), pp. 1–21, Lecture Notes in Computer Science 613, Springer-Verlag, Berlin, 1992. Proceedings of the Summer Symposium, San Antonio, TX, June 1991.Google Scholar
  17. 17.
    H. Läuchli, “An Abstract Notion of Realizability for which Intuitionistic Predicate Calculus is Complete”, in: Intuitionism and Proof Theory (A. Kino, J. Myhill, and R. E. Vesley, eds.), pp. 277–234, Studies in Logic and the Foundations of Mathematics, North-Holland Publishing Company, Amsterdam, London, 1970. Proceedings of the Conference on Intuitionism and Proof Theory, Buffalo, New York, August 1968.Google Scholar
  18. 18.
    J. C. Mitchell, “Type Systems for Programming Languages”, in: Handbook of Theoretical Computer Science, Volume B (J. van Leeuwen, ed.), pp. 365–458, North-Holland Publishing Company, Amsterdam, 1990.Google Scholar
  19. 19.
    Andrew M. Pitts, “Poloymorphism is Set-Theoretic, Constructively”, in: Proceedings of the Conference on Category Theory and Computer Science, Edinburgh, 1987 (D. Pitt, ed.), pp. 12–39, Lecture Notes in Computer Science 283, Springer-Verlag, Berlin, 1987.Google Scholar
  20. 20.
    D. Prawitz, Natural Deduction, Almqvist & Wiksell, Stockholm, 1965.Google Scholar
  21. 21.
    John C. Reynolds, “Polymorphism is Not Set-Theoretic”, in: Semantics of Data Types, pp. 145–156, Lecture Notes in Computer Science 173, Springer-Verlag, Berlin, 1984.Google Scholar
  22. 22.
    G. F. Rose, “Propositional Calculus and Realizability”, Transactions of the American Mathematical Society 75 (1953) 1–19.Google Scholar
  23. 23.
    R. Statman, “Logical Relations and the Typed Lambda Calculus”, Information and Control 65 (1985) 85–97.Google Scholar
  24. 24.
    Sören Stenlund, Combinators, λ-terms, and Proof Theory, D. Riedel Publishing Company, Dordrecht-Holland, 1972.Google Scholar
  25. 25.
    Alfred Tarski, “Pojecie Prawdy W Jezykach Nauk Dedukcyjnch”, Prace Towarzystwa Naukowego Warzawskiego (1933). English translation in [26].Google Scholar
  26. 26.
    Alfred Tarski, Logic, Semantics, and Metamathematics, Oxford University Press, 1956.Google Scholar
  27. 27.
    A. S. Troelstra and D. van Dalen, Constructivism in Mathematics: an Introduction, Studies in Logic and the Foundations of Mathematics, North-Holland, 1988.Google Scholar
  28. 28.
    D. van Dalen, “Intuitionistic Logic”, in: Handbook of Philosophical Logic III (D. Gabbay and F. Guenther, eds.), pp. 225–339, D. Reidel, Dordrecht, 1986.Google Scholar
  29. 29.
    W. Veldman, “An Intuitionistic Completeness Theorem for Intuitionistic Predicate Logic”, Journal of Symbolic Logic 41 (1976) 159–166.Google Scholar
  30. 30.
    L. Wittgenstein, “Tractatus Logico-Philosophicus”, Annalen der Natur-philosophie (1921). English translation in [31].Google Scholar
  31. 31.
    L. Wittgenstein, Tractatus Logico-Philosophicus, Routledge and Kegan Paul, 1961.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1994

Authors and Affiliations

  • James Lipton
    • 1
  • Michael J. O'Donnell
    • 2
  1. 1.Wesleyan UniversityUSA
  2. 2.The University of ChicagoUSA

Personalised recommendations