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


In Chapter 1, I introduced Popper’s comparative content definition, which is based on truth-value and logical strength, and showed how it excludes the comparison of two different false theories. After the publication of this peculiarity in 1974,1 Miller and Kuipers searched independently for another way to formalize Popper’s intuitions about verisimilitude. Their endeavours resulted in comparative content definitions (Kuipers calls his version the naive comparative definition). Miller formulates a distance function which has as its codomain the original Boolean algebra instead of the real numbers. Both definitions are almost identical to the consequence definition also hinted at in the first chapter (subsection 1.4.1). Here we examine Miller’s and Kuipers’s content proposals.


Boolean Algebra Actual World Symmetric Difference Stone Space False Proposition 
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  1. 1.
    Miller (1974) and Tichÿ (1974).Google Scholar
  2. 2.
    Miller (1978, p. 429).Google Scholar
  3. 3.
    This is in accordance with Miller (1994).Google Scholar
  4. 4.
    Ibid p. 416.Google Scholar
  5. 5.
    Miller (1978, p.423).Google Scholar
  6. 6.
    Note that Miller calls here A+B, what he calls A.B in Miller (1974a) and vice versa.Google Scholar
  7. 7.
    Miller (1978) p. 424. Further, the Brouwerian algebra is also known as the dual of the Heyting algebra. Curry calls it the subtractive lattice, and Popper the dual­intuitionistic calculus.Google Scholar
  8. 8.
    Miller (1974) p.168.Google Scholar
  9. 9.
    Ibid. p.169 note 1.Google Scholar
  10. 10.
    Miller (1978, p.427).Google Scholar
  11. 11.
    Miller (1974, p.174) second corollary of Theorem 5.Google Scholar
  12. 12.
    Ibid. Lemma on p.173.Google Scholar
  13. 13.
    Ibid.p.427.Google Scholar
  14. 14.
    Obviously, it H -’Tcp H T equals (yr A -t) y (-j, A t)(cp A ‘t) y (p A t). Consequently, [Mod(yr Ai) v Mod(--Air A t)] S [Mod(q) A --ti) v Mod(’cp A t)], which is abbreviated by: Mod(w) A Mod(t) Mod(cp) A Mod(t)].Google Scholar
  15. 15.
    See for more detailed information Bell and Slomson 1969.Google Scholar
  16. 16.
    However, the theory of densely ordered sets without end-points is complete and first order axiomatizable (see Bell and Slomson 1969 chapter 9 section 5).Google Scholar
  17. 17.
    See also Kuipers (2000).Google Scholar
  18. 18.
    The differences between the logicistic and structuralist theory representation emerge while using more sophisticated theory representations.Google Scholar
  19. 19.
    Kuipers (1992), p.300–301.Google Scholar
  20. 20.
    Ibid.p.301.Google Scholar
  21. 21.
    Ibidp.301. and Kuipers (1987) p.82.Google Scholar
  22. 22.
    Cf. Kuipers (1982), p.347.Google Scholar
  23. 23.
    Ibid.section 2.Google Scholar
  24. 24.
    Kuipers (1987a), p.82.Google Scholar
  25. 25.
    Ibid. p.83. Compare with the “potential falsifiers” in Popper (1963), p.385.Google Scholar
  26. 26.
    See for Kuipers’s version Kuipers (1982) and Kuipers (1992).Google Scholar
  27. 27.
    Kuipers (1982, p.347) gives the same list of the possible combinations of weak and strong, theoretical and descriptive truth.Google Scholar
  28. 28.
    Note that according to Popper (1963, fifth edition, p.232, and Addendum 1) the empirical content of a theory is proportional to its logical content.Google Scholar
  29. 29.
    Niiniluoto’s remarks (1987, p.381–382) seem to be inspired by the same observations.Google Scholar
  30. 30.
    See Kuipers (1982), p.353 and (1992), p.304.Google Scholar
  31. 31.
    Kuipers (1992), p.314.Google Scholar
  32. 32.
    See Miller (1978, last page) for the same assessment.Google Scholar
  33. 33.
    See for more properties of the A-definition van Benthem (1987).Google Scholar
  34. 34.
    Kuipers (1982), p.357. Although Kuipers (1992) refers to worlds on p. 301, the paper does not mention the incompleteness of the truth.Google Scholar
  35. 35.
    Niiniluoto (1987), p.381.Google Scholar
  36. 36.
    Kuipers (1982), the second section.Google Scholar
  37. 37.
    Although it cannot be denied that the descriptive theory of plate tectonics has been a big step in the right direction.Google Scholar
  38. 38.
    Niiniluoto (1987), p.381–382.Google Scholar
  39. 39.
    Actually, Niiniluoto proposes to replace first order constituentsGoogle Scholar
  40. 40.
    Cohen (1980) and (1987).Google Scholar
  41. 41.
    Hughes and Cresswell (1984), p.7.Google Scholar
  42. 42.
    At least according to Kuipers (1982).Google Scholar
  43. 43.
    Here again we may restrict theories to be elements of SProp(2s5), but we leave the details to the reader.Google Scholar
  44. 44.
    In 2s5 [p,q] with i = q(p A q) the complete falsehood 4 is the most distinct constituentGoogle 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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