Intuitionistic logic considered as an extension of classical logic : some critical remarks

Legris, Javier; Molina, Jorge A.

Philosophia Scientiae, Tome 5 (2001) no. 2, pp. 27-50.

Dans cet article, nous analysons la conception de la logique intuitionniste comme une extension de la logique classique. Ce point de vue - surprenant au premier abord - a été explicitement soutenu par Jan Łukasiewicz sur la base d'une projection de la logique propositionnelle classique dans la logique propositionnelle intuitionniste, réalisée par Kurt Gödel en 1933. Au même moment, Gerhard Gentzen proposait une autre projection de l'arithmétique de Peano dans l'arithmétique de Heyting. Nous discutons ces projections en lien avec le problème de la détermination des symboles logiques qui expriment adéquatement les idiosyncrasies de la logique intuitionniste. De nombreux philosophes et logiciens ne semblent pas suffisamment conscients des difficultés soulevées par le fait de considérer la logique classique comme un sous-système de logique intuitionniste. Un résultat de cette discussion sera de faire ressortir ces difficultés. La notion de traduction logique jouera un rôle essentiel dans l'argumentation, et nous esquisserons quelques conséquences concernant la signification des constantes logiques.

In this paper we analyze the consideration of intuitionistic logic as an extension of classical logic. This - at first sight surprising - point of view has been sustained explicitly by Jan Łukasiewicz on the basis of a mapping of classical propositional logic into intuitionistic propositional logic by Kurt Gödel in 1933. Simultaneously with Gödel, Gerhard Gentzen had proposed another mapping of Peano´s arithmetic into Heyting´s arithmetic. We shall discuss these mappings in connection with the problem of determining what are the logical symbols that properly express the idiosyncracy of intuitionistic logic. Many philosophers and logicians do not seem to be sufficiently aware of the difficulties that arise when classical logic is considered as a subsystem of intuitionistic logic. As an outcome of the whole discussion these difficulties will be brought out. The notion of logical translation will play an essential role in the argumentation and some consequences related to the meaning of logical constants will be drawn.

[1] Brouwer, Luitzen Egbertus Jan 1929.- “Mathematik, Wissenschaft und Sprache”. In Monatsheft für Mathematik und Physik 36, 153-164. | JFM

[2] Brouwer, Luitzen Egbertus Jan 1949.- “Consciousness, Philosophy, and Mathematics”. In Proceedings of the 10th International Congress of Philosophy, Amsterdam 1948. Amsterdam, North-Holland, 1949, 1243-1249. Reprinted in Philosophy of Mathematics. Selected Readings, edited by Paul Benacerraf & Hilary Putnam, 2nd. ed., Cambridge et al., Cambridge University Press. 1983, 90-96.

[3] Došen, Kosta 1993.- “A Historical Introduction to Substructural Logics”. In Substructural Logics edited by Peter Schroeder-Heister and Kosta Došen, Oxford, Clarendon Press, 1-30. | MR | Zbl

[4] Dummett, Michael 1973.- “The Philosophical Basis of Intuitionistic Logic”. In Truth and Other Enigmas by Michael Dummett, Cambridge (Mass.), Harvard University Press, 1978, 215-247.

[5] Gabbay, Dov M. 1981.- Semantical Investigations in Heyting's Intuitionistic Logic. Dordrecht-Boston-London, Reidel. | MR | Zbl

[6] Gentzen, Gerhard 1933.- “On the Relation between Intuitionistic and Classical Arithmetic”. In [Szabo 1969], 53-67.

[7] Gentzen, Gerhard 1936.- “The Consistency of Elementary Number Theory”. In [Szabo 1969], 132-213.

[8] Glivenko, Valerij Ivanovic 1929.- “Sur quelques points de la logique de M. Brouwer”. Acad. Roy. Belg.Bull. Cl.Sci., Ser. 5, 15, 183-188. | JFM

[9] Gödel, Kurt 1933a.- “Zur intuitionistischen Arithmetik und Zahlentheorie”. In Ergebnisse eines mathematischen Kolloquiums, 4, 34-38. | JFM

[10] Gödel, Kurt 1933b.- “Eine Interpretation des intuitionistischen Aussagenkalküls”. In Ergebnisse eines mathematischen Kolloquiums, 4, 39-40. | Zbl

[11] Haack, Susan 1973.- Deviant Logic. Cambridge, Cambridge, University Press.

[12] Heyting, Arendt 1930.- “Die formalen Regeln der intuitionistischen Logik”. Sitzungsber. preuss. Ak. Wiss. Phys.- Math. Klasse II, 42-56. | JFM

[13] Heyting, Arendt 1956.- Intuitionism. An Introduction. Amsterdam, North-Holland. | MR | Zbl

[14] Herbrand, Jacques 1932.- “Sur la non-contradiction de l'arithmetique”. J. reine und angew. Math. 166, 1-8. | Zbl

[15] Kleene, Stephen Cole 1952.- Introduction to Metamathematics. New York-Toronto, Van Nostrand. | MR

[16] Kripke, Saul 1965.- “Semantical Analysis of Intuitionistic Logic”. In Formal Systems and Recursive Functions, ed. by J.N. Crossley and M.A.E. Dummett. Amsterdam, North-Holland, 92-130. | MR | Zbl

[17] Legris, Javier 1990.- Eine epistemische Interpretation der intuitionistischen Logik. Würzburg, Königshausen & Neumann. | Zbl

[18] Lenzen, Wolfgang 1991.- “What is (Or at Least Appears to Be) Wrong with Intuitionistic Logic?”. In Advances in Scientific Philosophy. Essays in Honour of Paul Wengartner ed. by G. Schurz & G. Dorn. Amsterdam, Rodopi, 173-186.

[19] Łukasiewicz, Jan 1951.- “On Variable Functors of Propositional Arguments” 1970], 311-324.

[20] Łukasiewicz, Jan 1952.- “On The Intuitionistic Theory of Deduction”. In [Łukasiewicz 1970], 325-340

[21] Łukasiewicz, Jan 1970.- Selected Works. ed. by L. Borkowski. Amsterdam-London, North-Holland. | MR

[22] Prawitz, Dag & P.-E. Malmnäs 1968.- “A Survey of Some Connections Between Classical, Intuitionistic and Minimal Logic”. In Contributions to Mathematical Logic. Proceedings of the Logic Colloquium, Hannover 1966 ed. by H. Arnold Schmidt, Kurt Schütte & H.J. Thiele. Amsterdam, North-Holland, 215-229. | MR | Zbl

[23] Prior, A.N 1962.- Formal Logic. Oxford, at the Clarendon Press. | MR | Zbl

[24] Quine, Willard Van Orman 1970.- Philosophy of Logic. Englewood Cliffs, N.J., Prentice-Hall.

[25] Schütte, Kurt 1968.- Vollständige Systeme modaler und intuitionistischer Logik. Berlin-Heidelberg-N.York, Springer. | MR | Zbl

[26] Sundholm, Göran 1983.- “Systems of Deduction”. In Handbook of Philosophical Logic, vol. I, ed. by Dov. Gabbay and Franz Guenther, Dordrecht, Reidel, 133-188. | Zbl

[27] Szabo, M.E. 1969.- The Collected Papers of Gerhard Gentzen. Amsterdam-London, North-Holland. | MR | Zbl

[28] Troelstra, Anne S. 1969.- Principles of Intuitionism. Heidelberg-N.York, Springer. | MR

[29] Van Dalen, Dirk 1973.- “Lectures on Intuitionism”. In Cambridge Summer School in Mathematical Logic ed. by A.R.D. Mathias and H. Rogers. Berlin-Heidelberg-New York, Springer, 1-94. | MR | Zbl

[30] Wójcicki, Rizsard 1988.- Theory of Logical Calculi. Basic Theory of Consequence Operations. Dordrecht-Boston-London, Kluwer. | MR | Zbl