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Standard vs. Nonstandard Distinction: A Watershed in the Foundations of Mathematics

  • Jaakko Hintikka
Chapter
Part of the Synthese Library book series (SYLI, volume 251)

Abstract

In this paper, I will discuss a conceptual distinction, or a contrast, between two opposing ideas, that has played an extremely important role in the foundations of mathematics. The distinction was first formulated explicitly, though not quite generally, by Leon Henkin in 1950.1 He called it a distinction between the standard and the nonstandard interpretation of higher-order logic. I will follow his terminology, even though it may not be the most fortunate one. One reason for saying that this nomenclature is not entirely happy is that it is not clear which interpretation is the “standard” one in the sense of being a more common one historically. Another reason is that one can easily characterize more than one nonstandard interpretation of higher-order logic, even though Henkin considered only one.2 Furthermore, it turns out that the distinction (rightly understood) is not restricted to higher-order logics.’ Last but not least, it is far from clear how Henkin’s notion of standard interpretation or standard model is related to logicians’ idea of the standard model of such first-order theories as elementary arithmetic, in which usage “standard model” means simply “intended model”.

Keywords

Arbitrary Function Standard Interpretation Propositional Function Intended Model Nonstandard Interpretation 
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NOTES

  1. 1.
    Leon Henkin: 1950, Completeness in the Theory of Types, Journal of Symbolic Logic 15 81–91.Google Scholar
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  3. 2.
    See section 2.Google Scholar
  4. 3.
    Cf. here Jaakko Hintikka: 1980, Standard vs. Nonstandard Logic: Higher Order, Modal and First-Order Logics, in E. Agazzi (ed.), Modern Logic: A Survey, D. Reidel, Dordrecht, 283–296.Google Scholar
  5. 4.
    The following paragraphs as well as section 3 below follow closely the exposition in Jaakko Hintikka and Gabriel Sandu: 1992, The Skeleton in Frege’s Cupboard: The Standard versus Nonstandard Distinction, Journal of Philosophy 89 290–315.Google Scholar
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    See Über eine bisher noch nicht benützte Erweiterung des finiten Standpunktes, in Solomon Feferman et al. (eds.), Kurt Gödel: Collected Works, Vol. 2: Publications 1938–1974, Oxford University Press, New York, 1990, pp. 240–251. (Cf. also pp. 217–241.)Google Scholar
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  13. 12.
    For instance, it appears that some commentators have misinterpreted Frege because he does not assume the definability interpretation. (For Frege, functions exist objectively independently of their representability in language.) From this they have in effect mistakenly inferred that Frege accepted the standard interpretation. See here Hintikka and Sandu, op. cit.Google Scholar
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    Op. cit., p. 165 of the reprint.Google Scholar
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    For a discussion of the history of this contrast, see Gregory H. Moore: 1988, “The Emergence of First-Order Logic”, in William Aspray and Philip Kitcher (eds.), History and Philosophy of Modern Mathematics (Minnesota Studies in the Philosophy of Science, vol. II), University of Minnesota Press, Minneapolis, 1988, pp. 95–135.Google Scholar
  35. 34.
    See here Merrill B. Hintikka and Jaakko Hintikka: 1986, Investigating Wittgenstein, Basil Blackwell, Oxford, chapters 2 and 4.Google Scholar
  36. 35.
    Cf. Gregory H. Moore, op. cit., note 10 above.Google Scholar
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    Bertrand Russell: 1919, Introduction to Mathematical Philosophy, Allen and Unwin, London, chapter 12, especially p. 126.Google Scholar
  38. 37.
    Op. cit., p. 309.Google Scholar
  39. 38.
    See op. cit., note 5 above, and cf. Jaakko Hintikka: 1993, `Gödel’s Functional Interpretation in Perspective’, in M. D. Schwabl (ed.), Yearbook of the Kurt Gödel Society, Vienna, pp. 5–43.Google Scholar
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    See Hintikka, op. cit., note 3 above.Google Scholar
  41. 40.
    Michael Dummett, Elements of Intuitionism, Clarendon Press, Oxford, 1977, pp. 52–53 and 314.Google Scholar
  42. 41.
    Op. cit., note 10 above, especially pp. 64–76.Google Scholar
  43. 42.
    Op. cit., note 20 above, p. 71.Google Scholar
  44. 43.
    See here Jaakko Hintikka: 1988, `What Is the Logic of Experimental Inquiry?’, Synthese 74, 173–190.Google Scholar
  45. 44.
    For an early discussion of the nature and role of conclusiveness conditions, see Jaakko Hintikka, The Semantics of Questions and the Questions of Semantics (Acta Philosophica Fennica vol. 28, no. 4) Societas Philosophica Fennica, Helsinki, 1976, especially ch. 3. The analysis presented there is now being generalized, especially to questions whose answers are functions.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1995

Authors and Affiliations

  • Jaakko Hintikka
    • 1
  1. 1.Boston UniversityUSA

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