Skip to main content

A semantics of evidence for classical arithmetic

  • Thierry Coquand (a1)

If it is difficult to give the exact significance of consistency proofs from a classical point of view, in particular the proofs of Gentzen [2, 6], and Novikoff [14], the motivations of these proofs are quite clear intuitionistically. Their significance is then less to give a mere consistency proof than to present an intuitionistic explanation of the notion of classical truth. Gentzen for instance summarizes his proof as follows [6]: “Thus propositions of actualist mathematics seem to have a certain utility, but no sense. The major part of my consistency proof, however, consists precisely in ascribing a finitist sense to actualist propositions.” From this point of view, the main part of both Gentzen's and Novikoff's arguments can be stated as establishing that modus ponens is valid w.r.t. this interpretation ascribing a “finitist sense” to classical propositions.

In this paper, we reformulate Gentzen's and Novikoff's “finitist sense” of an arithmetic proposition as a winning strategy for a game associated to it. (To see a proof as a winning strategy has been considered by Lorenzen [10] for intuitionistic logic.) In the light of concurrency theory [7], it is tempting to consider a strategy as an interactive program (which represents thus the “finitist sense” of an arithmetic proposition). We shall show that the validity of modus ponens then gets a quite natural formulation, showing that “internal chatters” between two programs end eventually.

We first present Novikoff's notion of regular formulae, that can be seen as an intuitionistic truth definition for classical infinitary propositional calculus. We use this in order to motivate the second part, which presents a game-theoretic interpretation of the notion of regular formulae, and a proof of the admissibility of modus ponens which is based on this interpretation.

Hide All
[1]Abramsky, S., Computational interpretations of linear logic, Theoretical Computer Science (1993), pp. 357, Special issue for MFPS 1990.
[2]Bernays, P., On the original Gentzen consistency proof for number theory, Intuitionism and proof theory (Kino, A., Myhill, J., and Vesley, J. E., editors), North Holland, Amsterdam, 1970, pp. 409417.
[3]Blass, A., Degrees of indeterminancy of games, Fundamental Mathematicae, vol. LXXVII (1972), pp. 151166.
[4]Church, A., Review of Novikoff's article, this Journal, vol. 11 (1946), pp. 129131.
[5]Constable, R., The semantics of evidence, Technical Report TR 85–684, Cornell University, Department of Computer Science, Ithaca, New York, 1985.
[6]Gentzen, G., The collected papers of Gerhard Gentzen, North-Holland, Amsterdam, 1969, edited by Szabo, M. E..
[7]Hoare, C.A.R., Communicating sequential processes, Prentice-Hall, 1985.
[8]Kleene, S. C., Introduction to metamathematics, North-Holland, 1952.
[9]Kreisel, G., Interpretation of non-finitist proofs, this Journal, vol. 17 (1952), pp. 5056.
[10]Lorenzen, K., Ein dialogisches konskruktivitatskriterium, Infinitistic methods, Pergamon Press, 1962, Proceedings of the Symposium on the Foundations of Mathematics, PWN, Warszawa, 1959, pp. 193200.
[11]Martin-Löf, S. C., Notes on constructive mathematics, Almqvist and Wiksell, Stockholm, 1970.
[12]Mints, G., Proof theory in the USSR, 1925-1969, this Journal, vol. 56 (1991), pp. 385424.
[13]Murthy, C., Extracting constructive content from classical proofs, Ph.D. thesis, Cornell University, 1990.
[14]Novikoff, P. S., On the consistency of certain logical calculus, Matematiceskij sbornik (Recueil-Mathématique, T.12), vol. 54 (1943), pp. 230260.
[15]Tait, W. W., Normal derivability in classical logic, The syntax and semantics of infinitary languages (Barwise, Jon, editor), Lecture Notes in Mathematics, vol. 72, Springer Verlag, 1968, pp. 204236.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Full text views

Total number of HTML views: 0
Total number of PDF views: 5 *
Loading metrics...

Abstract views

Total abstract views: 165 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 23rd September 2018. This data will be updated every 24 hours.