Logica Universalis

, Volume 6, Issue 3–4, pp 553–586 | Cite as

In Defense of Logical Universalism: Taking Issue with Jean van Heijenoort

Article

Abstract

Van Heijenoort’s main contribution to history and philosophy of modern logic was his distinction between two basic views of logic, first, the absolutist, or universalist, view of the founding fathers, Frege, Peano, and Russell, which dominated the first, classical period of history of modern logic, and, second, the relativist, or model-theoretic, view, inherited from Boole, Schröder, and Löwenheim, which has dominated the second, contemporary period of that history. In my paper, I present the man Jean van Heijenoort (Sect. 1); then I describe his way of arguing for the second view (Sect. 2); and finally I come down in favor of the first view (Sect. 3). There, I specify the version of universalism for which I am prepared to argue (Sect. 3, introduction). Choosing ZFC to play the part of universal, logical (in a nowadays forgotten sense) system, I show, through an example, how the usual model theory can be naturally given its proper place, from the universalist point of view, in the logical framework of ZFC; I outline another, not rival but complementary, semantics for admissible extensions of ZFC in the very same logical framework; I propose a way to get universalism out of the predicaments in which universalists themselves believed it to be (Sect. 3.1). Thus, if universalists of the classical period did not, in fact, construct these semantics, it was not that their universalism forbade them, in principle, to do so. The historical defeat of universalism was not technical in character. Neither was it philosophical. Indeed, it was hardly more than the victory of technicism over the very possibility of a philosophical dispute (Sect. 3.2).

Mathematics Subject Classification

03A05 03-03 03Cxx A1A60 01A85 

Keywords

Jean van Heijenoort philosophy of logic history of logic logical universalism model theory 

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References

  1. 1.
    Ajdukiewicz, K.: Z metodologji nauk dedukcyjnych. Nakładem Polskiego Towarzystwa Filozoficznego, Lwów (1921)Google Scholar
  2. 2.
    Bynum, T.W.: Editor’s Introduction. In: [18], pp. 55–100 (1972)Google Scholar
  3. 3.
    Carroll, L.: What the Tortoise said to Achilles, Mind (n.s) 4, 278–280 (1895)Google Scholar
  4. 4.
    Carroll, L.: (Gattégno, J., Coumet, E., trans. & ed.), Logique sans peine. Hermann, Paris (1966)Google Scholar
  5. 5.
    Coumet, E.: Lewis Carroll logicien. In: [4], pp. 255–288 (1966)Google Scholar
  6. 6.
    Dedekind, R.: Was sind uns was sollen die Zahlen? Vieweg, Braunschweig (1888)Google Scholar
  7. 7.
    Enderton, H.B.: A Mathematical Introduction to Logic. Academic Press, London (1972) (2nd edn. 2001)Google Scholar
  8. 8.
    Feferman, A.B.: Politics, Logic, and Love: The Life of Jean van Heijenoort. Jones and Bartlett, Boston; A K Peters, Wellesley, MA (1993); reprinted as: From Trotsky to Gödel: The Life of Jean van Heijenoort. A K Peters, Natick, Mass (2001)Google Scholar
  9. 9.
    Feferman, A.B., Feferman, S.: Jean van Heijenoort (1912–1986). In: [56], pp. 1–7 (1987)Google Scholar
  10. 10.
    Feferman A.B., Feferman S.: Alfred Tarski: Life and Logic. Cambridge University Press, Cambridge (2004)MATHGoogle Scholar
  11. 11.
    Fraenkel, A.A.: Der Begriff “definit” und die Unabhängigkeit des Auswahlsaxioms, Sitzungsberichte der Preussischen Akademie der Wissenschaften, Physikalisch-mathematische Klasse, pp. 253–257 (1922)Google Scholar
  12. 12.
    Fraenkel A.A.: Zu den Grundlagen der Cantor-Zermeloschen Mengenlehre. Math. Ann. 86, 230–237 (1922)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Frege, G.: Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens. L. Nebert, Halle (1879); English translations in: [60], pp. 1–82 (1967) and [18], pp. 101–203 (1972)Google Scholar
  14. 14.
    Frege, G.: Über den Zweck der Begriffsschrift, Sitzungsberichte der Jenaischen Gesellschaft für Medicin und Naturwissenschaft für das Jahr 1882 16, Suppl.-Heft II, 1–10 (1882-83)Google Scholar
  15. 15.
    Frege, G.: Über Begriff und Gegenstand, Vierteljahrsschrift für wissenschaftliche Philosophie (N.F.) 16, 192–205 (1892)Google Scholar
  16. 16.
    Frege, G.: Grundgesetze der Arithmetik begriffsschriftlich abgeleitet, 2 Bde. H. Pohle, Jena (1893–1903)Google Scholar
  17. 17.
    Frege, G.: Letter to Russell dated June 22, 1902 (1902); English translation in: [60], pp. 126–128 (1967)Google Scholar
  18. 18.
    Frege G.: Conceptual Notation and Related Articles (Bynum, T.W., trans. & ed.). Clarendon Press, Oxford (1972)Google Scholar
  19. 19.
    Girard, J.-Y.: La mouche dans la bouteille. In: [56], pp. 9–12 (1987)Google Scholar
  20. 20.
    Gödel, K.: Über die Vollstänsdigkeit des Logikkalküls, Dissertation, University of Vienna (1929); reprinted, with English translation, in: [23], vol. I, pp. 60–101 (1986-2003)Google Scholar
  21. 21.
    Gödel, K.: Die Vollständigkeit der Axiome des logischen Funktionenkalküls. Monatsh. Math. Phys. 37, 349–360 (1930); English translation in: [60], pp. 582–591 (1967); reprinted in, with English translation, in: [23], vol. I, pp. 102–123 (1986-2003)Google Scholar
  22. 22.
    Gödel, K.: Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I. Monatsh. Math. Phys. 38, 173–198 (1931); English translation in: [60], pp. 596–617 (1967); reprinted with English translation in: [23], vol. I, pp. 144–195 (1986–2003)Google Scholar
  23. 23.
    Gödel K.: Collected Works, 5 vols (Feferman, S., et al., eds.). Oxford University Press, Oxford (1986–2003)Google Scholar
  24. 24.
    Goldfarb W.D.: Logic in the twenties: the nature of the quantifier. J. Symbol. Logic 44, 351–368 (1979)MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Herbrand J.: Écrits logiques (van Heijenoort, J., ed.). Presses Universitaires de France, Paris (1968)Google Scholar
  26. 26.
    Hilbert D., Ackermann W.: Grundzüge der theoretischen Logik. Springer, Berlin (1928)MATHGoogle Scholar
  27. 27.
    Hintikka, J.: On the development of the model-theoretic viewpoint in logical theory. Synthèse 77, 1–36 (1988); reprinted in: [28], pp. 104–139 (1997)Google Scholar
  28. 28.
    Hintikka, J.: Lingua Universalis vs. Calculus Ratiocinator: An Ultimate Presupposition of Twentieth-Century Philosophy. Kluwer, Dordrecht (1997)Google Scholar
  29. 29.
    Hylton P.: Russell’s substitutional theory. Synthese 45, 1–31 (1980)MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    Hylton P.: Russell, Idealism and the Emergence of Analytic Philosophy. Clarendon Press, Oxford (1990)Google Scholar
  31. 31.
    Kripke S.: Outline of a theory of truth. J. Philos. 72, 690–716 (1975)CrossRefGoogle Scholar
  32. 32.
    Löwenheim, L.: Über Möglichkeiten im Relativkalkül, Math. Ann. 76, 447–470; English translation in: [60], pp. 228–251 (1967)Google Scholar
  33. 33.
    Martin-Löf P.: Intuitionistic Type Theory. Bibliopolis, Naples (1984)MATHGoogle Scholar
  34. 34.
    Partee B.H.: Montague grammar and transformational grammar. Linguist. Inquiry 6, 203–300 (1975)Google Scholar
  35. 35.
    Peano, G.: Arithmetices principia, nova methodo exposita. Bocca, Turin (1889); English partial translation in: [60], pp. 83–97 (1967)Google Scholar
  36. 36.
    Quine, W.V.: Letter to Jean van Heijenoort dated December 4, 1974. In: [76], p. 39 (1985)Google Scholar
  37. 37.
    Rayo, A., Uziquiano, G. (eds.): Absolute Generality. Clarendon Press, Oxford (2006)Google Scholar
  38. 38.
    Rosser J.B.: Extensions of some theorems of Gödel and Church. J. Symbol. Logic 1, 87–91 (1936)MATHCrossRefGoogle Scholar
  39. 39.
    de Rouilhan, Ph.: J.v.H; (speech delivered in memory of Jean van Heijenoort at the Institut Henri Poincaré on May 14, 1986). In: [56], pp. 13–16 (1987)Google Scholar
  40. 40.
    de Rouilhan, Ph.: Frege. Les paradoxes de la représentation. Les Editions de Minuit, Paris (1988)Google Scholar
  41. 41.
    de Rouilhan, Ph.: De l’universalité de la logique. In: Bouveresse, J. (éd.) L’âge de la science. Lectures philosophiques, vol. 4. I: Philosophie de la logique et philosophie du langage, pp. 93–113. Éditions Odile Jacob, Paris (1991)Google Scholar
  42. 42.
    de Rouilhan, Ph.: Tarski et l’universalité de la logique. In: Nef, F., Vernant, D. (eds.) Le formalisme en question. Le tournant des années, vol. 30, pp. 85–102. Vrin Paris (1998)Google Scholar
  43. 43.
    de Rouilhan Ph.: La théorie des modèles et l’architecture des mathématiques. In: Gochet, P., Rouilhan, Ph. (eds.) Logique épistémique et philosophie des mathématiques, pp. 39–114. Vuibert, Paris (2007)Google Scholar
  44. 44.
    de Rouilhan, Ph.: Generalizing Frege’s paradox and putting truth to work to solve it. In: Achourioti, D., Fujimoto, K., Galinon, H., Martinez, J. (eds.) Unifying the Philosophy of Truth. Springer, Berlin (forthcoming)Google Scholar
  45. 45.
    Russell, B.: Letter to Frege dated June 16, 1902. In: [60], pp. 124–125 (1967)Google Scholar
  46. 46.
    Russell, B.: The Principles of Mathematics. G. Allen & Unwin, London (1903)Google Scholar
  47. 47.
    Skolem, Th.: Logisch-kombonatorische Untersuchungen über die Erfüllbarkeit oder Beweisbarkeit mathematischer Sätze nebst einem Theoreme über dichte Mengen, Skrifter utgit av det Norske Videnskapsakademiet i Kristiana, I. Matematisk-naturvidenskapelig klasse 4, 1–36 (1920); English translation in: [60], pp. 252–263 (1967)Google Scholar
  48. 48.
    Skolem, Th.: Einige Bemerkungen zur axiomatischen Begründung der Mengenlehre. In: Mathematikerkongressen i Helsingfors den 4–7 Juli 1922, Den femte skandinaviska mathematikerkongressen, Redogörelse, pp. 217–232. Akademiska Bokhandeln, Helsinki (1923); English translation in: [60], pp. 290–301 (1967)Google Scholar
  49. 49.
    Tarski, A.: Pojęcie prawdy w językach nauk dedukcyjnych, Prace Towarzystwa Naukowego Warszawskiego, Wydział III, Nauk matematyczno-fizycznych (Travaux de la Société des Sciences et des Lettres de Varsovie, Classe III, Sciences Mathématiques et Physiques), nr. 34 (1933)Google Scholar
  50. 50.
    Tarski, A.: Der Wahrheitsbegriff in den formalisierten Sprachen. Stud. Philos. 1, 261–405 (1936); English translation in: [54], pp. 152–278 (1956)Google Scholar
  51. 51.
    Tarski, A.: O pojciu wynikania logicznego, Przegląd Filozoficzny 39, 58–68 (1936); German translation: Über den Begriff der logischen Folgerung, Actes du Congrès International de Philosophie Scientifique. Sorbonne Paris 1935 (Actualités Scientifiques et Industrielles, vol. 388–395), vol. VII (A.S.I., vol. 394). Hermann, Paris, pp. 1–11 (1936); English translation in: [54], pp. 409–420 (1956)Google Scholar
  52. 52.
    Tarski, A.: A general method in proofs of undecidability. In: Tarski, A., with Mostowski, A., Robinson, R.M. (eds.) Undecidable Theories, pp. 1–35. North-Holland, Amsterdam (1953; reprinted: Dover Publications, Mineola, N.Y., 2010)Google Scholar
  53. 53.
    Tarski, A.: Contributions to the theory of models, Koninklijke Nederlandse Akademi van Wetenschappen, Proceedings, series A 57 (= Indagationes mathematicae 16), 572–588, 58 (= Indag. math. 17), 56–64 (1954-1955)Google Scholar
  54. 54.
    Tarski, A.: Logic, Semantics, Metamathematics: Papers from 1923 to 1938 (Woodger, J.H., trans.). Clarendon Press, Oxford (1956)Google Scholar
  55. 55.
    Tarski A., Vaught R.: Arithmetical extensions of relational systems. Composit. Math. 13, 81–102 (1957)MathSciNetGoogle Scholar
  56. 56.
    The Paris Logic Group (eds.): Logic Colloquium ’85. In: Proceedings of the Colloquium held in Orsay, France, July 1985. North-Holland, Amsterdam (1987)Google Scholar
  57. 57.
    Tichý P.: The Foundations of Frege’s Logic. Walter de Gruyter, Berlin (1988)MATHGoogle Scholar
  58. 58.
    Trotsky L., Trotsky N.: Correspondance 1933–1938 (van Heijenoort, J., trans. & ed.). Gallimard, Paris (1980)Google Scholar
  59. 59.
    van Heijenoort, J.: Friedrich Engels and mathematics (1948). In: [76], pp. 123–151 (1985)Google Scholar
  60. 60.
    van Heijenoort, J. (ed.): From Frege to Gödel: A source book in mathematical logic, 1879–1931. Harvard University Press, Cambridge (1967)Google Scholar
  61. 61.
    van Heijenoort, J.: Logic as calculus and logic as language. In: Cohn, R.S., Wartofsky, M.W. (eds.) Proceedings of the Boston Studies in the Philosophy of Science, vol. 3, pp. 440–446. D. Reidel, Dordrecht (1967); reprinted in: [76], pp. 11–16 (1985)Google Scholar
  62. 62.
    van Heijenoort, J.: Subject and predicate in Western logic. Philos. East West 24, 253–268 (1974); reprinted in: [76], pp. 17–34 (1985)Google Scholar
  63. 63.
    van Heijenoort, J.: On the number of planets. In: [76], p. 35 (1985)Google Scholar
  64. 64.
    van Heijenoort, J.: On Kripke’s puzzle. In: [76], pp. 37–38 (1985)Google Scholar
  65. 65.
    van Heijenoort, J.: Letter to Quine dated December 9, 1974. In: [76], p. 41 (1985)Google Scholar
  66. 66.
    van Heijenoort, J.: El desarrollo de la teoria de la cuantification. Universidad Nacional Autonoma de Mexico, Instituto de Investigaciones Filosoficas, Mexico City (1976)Google Scholar
  67. 67.
    van Heijenoort, J.: Set-theoretic semantics. In: Gandy, R.O., Hyland, J.M.E. (eds.) Logic Colloquium ’76. Proceedings of a Conference Held in Oxford in July 1976, pp. 183–190. North-Holland, Amsterdam (1977); reprinted in: [76], pp. 43–53 (1985)Google Scholar
  68. 68.
    van Heijenoort, J.: Sense in Frege. J. Philos. Logic 6, 93–102 (1977); reprint in: [76], pp. 55–63 (1985)Google Scholar
  69. 69.
    van Heijenoort, J.: Frege on sense identity. J. Philos. Logic 6, 103–108 (1977); reprinted in: [76], pp. 65–69 (1985)Google Scholar
  70. 70.
    van Heijenoort, J.: Introduction à à la sémantique des logiques non classiques. Collection de l’École Normale Supérieure de Jeunes Filles, Paris (1978)Google Scholar
  71. 71.
    van Heijenoort, J.: De Prinkipo à à Coyoacán: Sept ans auprès de Léon Trotsky. Les Lettres Nouvelles Maurice Nadeau and Robert Laffont, Paris (1978)Google Scholar
  72. 72.
    van Heijenoort J.: With Trotsky in Exile: from Prinkipo to Coyoacan. Harvard University Press, Cambridge (1978)Google Scholar
  73. 73.
    van Heijenoort, J.: Ostension and vagueness. In: [76], pp. 71–73 (1985)Google Scholar
  74. 74.
    van Heijenoort, J.: Absolutism and relativism in logic. In: [76], pp. 75–83 (1985)Google Scholar
  75. 75.
    van Heijenoort, J.: L’œuvre logique de Jacques Herbrand et son contexte historique. In: Stern, J. (ed.) Logic Colloquium ’81: Proceedings of the Herbrand Symposium held in Marseille, France, July 1981, pp. 111–122. North-Holland, Amsterdam (1982)Google Scholar
  76. 76.
    van Heijenoort J.: Selected Essays. Biblopolis, Naples (1985)Google Scholar
  77. 77.
    van Heijenoort, J.: Frege and vagueness. In: Haaparanta, L., Hintikka, J. (eds.) Frege synthetized: studies of the philosophical and foundational work of Gottlob Frege, pp. 31–46. D. Reidel, Dordrecht (1985); reprinted in: [76], pp. 85–97 (1985)Google Scholar
  78. 78.
    van Heijenoort, J.: Jacques Herbrand’s work in logic and its historical context. In [76], pp. 99–121 (1985); English translation, with emendations of [75].Google Scholar
  79. 79.
    van Heijenoort, J.: Système et métasystème chez Russell. In: [56], pp. 111–121 (1987)Google Scholar
  80. 80.
    Whitehead, A.N. Russell, B.: Principia Mathematica. Cambridge University Press, Cambridge (1910–1913; 2nd edn, 1925–1927)Google Scholar
  81. 81.
    Wittgenstein, L. Logisch-philosophische Abhandlung. Annalen der Naturphilosophie 14. Reinhold Berger, Leipzig (1921)Google Scholar
  82. 82.
    Zermelo, E.: Untersuchungen über die Grundlagen der Mengenlehre, I. Math. Ann. 65, 261–281 (1908); English translation in [60], pp. 199–215 (1967)Google Scholar

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Authors and Affiliations

  1. 1.ParisFrance

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