• Sjoerd D. Zwart
Part of the Synthese Library book series (SYLI, volume 307)


We encountered the distinction between verisimilitude and truthlikeness definitions in Chapter 1. Verisimilitude definitions define distance to the truth using truth-value and logical strength, and were presented in Chapter 2. Truthlikeness definitions establish distance to the truth using similarity between the possible worlds, and are the subject of the present chapter.


Atomic Proposition True Proposition Negative Content False Proposition Propositional Language 
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  1. 1.
    Interestingly, Niiniluoto and Kuipers also attended this conference.Google Scholar
  2. 2.
    Hilpinen (1976) is Hilpinen’s contribution to the Warsaw conference for Formal Methodology in the Methodology of the Empirical Sciences, June 17–21, 1974.Google Scholar
  3. 3.
    Oddie (1986).Google Scholar
  4. 4.
    Brink and Heidema (1987), and Burger and Heidema (1994).Google Scholar
  5. 5.
    Ibid p.25Google Scholar
  6. 6.
    See Chapter 5. For instance, if the characteristics Ci are p,and q,then p A q is more similar top A q then p A q; however, this similarity ordering reverses if the characteristics are p and p H q. Google Scholar
  7. 7.
    Note that both E u (P) ç E u (Q) y E u (P) E u (Q),and I u (Q) ç Iu(P) v l u (Q) I u (P) hold.Google Scholar
  8. 8.
    See also Oddie (1990). If minu(cp):=K E E u (cp): K? K’ for VK’ e E u (P), and maxu((p):= K E lu(cp): K ç K’ for VK’ E u (P) then it holds that yr Û cp if mina(yr) ç minu(cp) and maxu(W) ç maxu(cp).Google Scholar
  9. 9.
    See also Niiniluoto (1987), p. 233.Google Scholar
  10. 10.
    For the more general case see appendix.Google Scholar
  11. 11.
    See Niiniluoto (1987), p.126.Google Scholar
  12. 12.
    Niiniluoto (1987), p.311.Google Scholar
  13. 13.
    Cf. W. K. Clifford in Proceedings of the Manchester Literary and Philosophical Society (February 1877, vol. xvi).Google Scholar
  14. 14.
    If M 1 …M k are the atomic predicates of a monadic language 2N then a Q-predicate is defined by Q(x):= (±)M 1 (x) A… (±)M k (x),in which (±)M ; means that M t is affirmed or negated.Google Scholar
  15. 15.
    Niiniluoto (1987), p. 212. It is similar to what Oddie (1990) calls the averaging method.Google Scholar
  16. 16.
    0(h ;,g) = red((AiiI j E Is)) means that the value of the function A for the argument h i and g equals the function value of red for the arguments (All,…, Air…, D in ) for j E Ig. Google Scholar
  17. 17.
    Ibid. p.212Google Scholar
  18. 18.
    Ibid p.217Google Scholar
  19. 19.
    Ibid. p.214–216Google Scholar
  20. 20.
    The diagram is a copy of Fig.5 of Niiniluoto (1987), p.214.Google Scholar
  21. 21.
    See Niiniluoto (1987), p. 25 and Czekanowski (1909).Google Scholar
  22. 22.
    Sokal (1961).Google Scholar
  23. 23.
    See Niiniluoto (1987) p. 232–234.Google Scholar
  24. 24.
    Ibid. p.248.Google Scholar
  25. 25.
    Niiniluoto (1987) sections 6.7 and 6.8, respectively.Google Scholar
  26. 26.
    The numbers between parenthesis refer to the sections in Niiniluoto (1987) describing the relevant distance functions.Google Scholar
  27. 27.
    Niiniluoto (1987) section 2.2 and section 3.2; Niiniluoto (1987b), p.14.Google Scholar
  28. 28.
    Niiniluoto (1987), section 6.2, p. 207–209.Google Scholar
  29. 29.
    Perhaps the choice in favour of the Clifford measure has to do with the expected verisimilitude. We will come to this in the next chapter. Niiniluoto suggests that his Jyväskylä measure would produce nice results. This measure is a weighted, quantitative version of the Clifford measure. Niiniluoto (1987), p.313–320]. Although this measure would somewhat qualify the content character of the present example, it does not wipe out its overall content character.Google Scholar
  30. 30.
    Oddie (1987 a), p.37.Google Scholar
  31. 31.
    Tichÿ and Oddie define their truthlikeness ordering for higher order, intensional interpreted languages.Google Scholar
  32. 32.
    E.g. see Oddie (1987a), p. 30–34 or Oddie (1986), p. 60–63.Google Scholar
  33. 33.
    Oddie (1986), p.62.Google Scholar
  34. 34.
    Ibid. p. 126–127.Google Scholar
  35. 35.
    L.J. Cohen (1980).Google Scholar
  36. 36.
    ibid p.34–35.Google Scholar
  37. 37.
    Miller (1975).Google Scholar
  38. 38.
    Oddie (1987a).Google Scholar
  39. 39.
    Niiniluoto (1987), p.324. Section 9.3 gives a good analysis of difference between the Tich-Oddie measure and the Clifford measure.Google Scholar
  40. 40.
    Oddie (1990).Google Scholar
  41. 41.
    The proof can be found in the Appendix.Google Scholar
  42. 42.
    Ibid. p.132.Google Scholar
  43. 43.
    In fact, it is easy to see that for all sentences cp, yr of 2 [p,q obtains: =andO 9 w BuandH 9 Google Scholar
  44. 44.
    Kuipers (1987), van Benthem (1987) and Kuipers (1992).Google Scholar
  45. 45.
    The relation between ideal gas models and Van der Waals gas models is an example of concretization; Kuipers (1992), section 4.Google Scholar
  46. 46.
    In the latest version of definition the t must be an element of T-X instead of T. Google Scholar
  47. 47.
    Kieseppä (1995, p.184) makes related observations, although he considers the case in which there are no incomparable elements and in which the truth is complete.Google Scholar
  48. 48.
    The notation of the definition and those which are to come suggest that the relevant sets are at most denumerable. Adaptation to continuous sets does not seem to produce any substantial difficulties.Google Scholar
  49. 49.
    Kuipers (1992), p. 332–333. It equals Niiniluoto’s minsum measure Sms(X, 1 ), (1987), p. 246.Google Scholar
  50. 50.
    Ibid. p.332.Google Scholar
  51. 51.
    Ibid. section 3.7.Google Scholar
  52. 52.
    Niiniluoto (1987), p. 247.Google Scholar
  53. 53.
    Kuipers (1992), p.319.Google Scholar
  54. 54.
    Kieseppä (1995), p.180 and Kuipers (1997), p. 161.Google Scholar
  55. 55.
    Private correspondence with Kieseppä.Google Scholar
  56. 56.
    Kuipers (1992), p.333.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2001

Authors and Affiliations

  • Sjoerd D. Zwart
    • 1
  1. 1.Delft University of Technology and University of AmsterdamThe Netherlands

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