Metatheoretical Considerations

  • Giovanni Sommaruga
Chapter
Part of the Synthese Library book series (SYLI, volume 290)

Abstract

There is a wealth of metatheoretical or philosophical considerations behind the technicalities of constructive type theory. The purpose of this chapter is to cast at least some light on at least a few of those considerations and to draw attention to the philosophical underpinnings and intricacies of type theory. These issues have been selected for consideration: the philosophical evergreen called identity (Sect. i), one of the few main pillars of any logic, namely the notion of judgement, which is also elucidated by means of the theory of knowledge (Sect. ii), the great tradition of metalogic with its principal targets: consistency, completeness and decidability (Sect. iii), and last but not least the positioning of constructive type theory within the philosophy or foundations of mathematics (Sect. iv). Needless to write, the real bulk of metatheoretical or philosophical consideration remains still to be carried out — the following is but an initiation.

Keywords

Type Theory Correct Judgement Extensional Identity Analytic Judgement Definitional Identity 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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Notes

  1. 1.
    Martin-Löf (1972A).Google Scholar
  2. 2.
    Cf. the theorem on extensional equality of fcts., Nordström et al. (1990), pp. 84,85, stating that extensionally identical fcts. are identical in the sense of E. Google Scholar
  3. 3.
    Version (6), p. 62.Google Scholar
  4. 4.
    Martin-Löf (1972A), p. 102.Google Scholar
  5. 5.
    pp. 102,103.Google Scholar
  6. 6.
  7. 7.
    Nordström et al. (1990), p. 67.Google Scholar
  8. 8.
    What has been called an expression or mode of containment Φ is just an abbreviatory formulation of the phrase ‘a mode of containment of singular terms or sentences in singular terms or sentences’. Cf. Quine (1960), p. 144.Google Scholar
  9. 9.
    Martin-Löf (1972A), pp. 104,105.Google Scholar
  10. 10.
    Cf. the example in Nordström et al. (1990), p. 66.Google Scholar
  11. 11.
    Nordström et al. (1990), p. 19.Google Scholar
  12. 12.
    E.g. p. 61.Google Scholar
  13. 13.
  14. 14.
  15. 15.
    Thompson (1991), e.g. p. 166.Google Scholar
  16. 16.
    E.g. pp. 133, 166.Google Scholar
  17. 17.
  18. 18.
    E.g. p. 118.Google Scholar
  19. 19.
  20. 20.
  21. 21.
    Kneale and Kneale (1978), p. 55.Google Scholar
  22. 22.
  23. 23.
    pp. 160–163.Google Scholar
  24. 24.
    Broadie (1993), chapts. 8 and 7, resp.Google Scholar
  25. 25.
    Broadie (1993), pp. 100–109.Google Scholar
  26. 26.
    Kneale and Kneale (1978), p. 489.Google Scholar
  27. 27.
    Sluga (1980), pp. 83, 84.Google Scholar
  28. 28.
    Dummett (1981), p. 316.Google Scholar
  29. 29.
    Berg (1962), p. 64.Google Scholar
  30. 30.
    For the historical evolution of the terms ‘proposition’, ‘judgement’, ‘assertion’, cf. generally Martin-Löf (1983A), pp. 205–211.Google Scholar
  31. 31.
    Salvesen (1989), p. 62.Google Scholar
  32. 32.
  33. 33.
  34. 34.
  35. 35.
  36. 36.
  37. 37.
    pp. 79, 80.Google Scholar
  38. 38.
  39. 39.
    p. 71 and pp. 83, 84.Google Scholar
  40. 40.
    pp. 84, 85.Google Scholar
  41. 41.
    Sundholm (1990), pp. 60, 61, esp. p. 61.Google Scholar
  42. 42.
    For more history on this point, cf. Note 30 in Sundholm (1994).Google Scholar
  43. 43.
    Cf. Martin-Löf’s short discussion of Lovejoy’s principle of plenitude and the medieval Ab posse ad esse non valet consequentia, in Martin-Löf (1990), p. 143.Google Scholar
  44. 44.
    E.g. in Dummett (1977).Google Scholar
  45. 45.
    Prawitz (1980).Google Scholar
  46. 46.
    putnam (1981).Google Scholar
  47. 47.
    All these explanations can be found almost literally in Sundholm (1994), p. 117 and pp. 121–123; note, however, that what Sundholm calls ageneral truth-maker analysis of truth’ is here called a truth-maker theory of truth.Google Scholar
  48. 48.
    Dummett (1976) and Prawitz (1987).Google Scholar
  49. 49.
    About the need for further transcendental exploration of evidence, cf. Pietersma (1977), pp. 47–49.Google Scholar
  50. 50.
    Follesdal (1991), pp. 27, 28.Google Scholar
  51. 51.
    Patzig (1973A).Google Scholar
  52. 52.
    These explanations are based on Chisholm (1967), p. 367, and Chisholm (1982).Google Scholar
  53. 53.
    This alternative justification was suggested to me by A.S. Troelstra.Google Scholar
  54. 54.
    Cf. Chisholm (1982).Google Scholar
  55. 55.
    Cf. Boolos (1982).Google Scholar
  56. 56.
    Cf. Martin-Löf (1985).Google Scholar
  57. 57.
    Cf. Carnap (1931).Google Scholar
  58. 58.
    Stegmüller (1969), pp. 265–270.Google Scholar
  59. 59.
    The main sources for Sect. iv. are Körner (1960) and Martin-Löf’s version (8) of constructive type theory. Other auxiliary sources are Heyting (1934) and (1971), von Neumann (1931), Parsons (1967) and Prawitz (1971).Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2000

Authors and Affiliations

  • Giovanni Sommaruga
    • 1
  1. 1.Albert-Ludwigs UniversityFreiburgGermany

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