# Verisimilitude

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

## Abstract

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.

## Keywords

Boolean Algebra Actual World Symmetric Difference Stone Space False Proposition
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.

## Notes

1. 1.
Miller (1974) and Tichÿ (1974).Google Scholar
2. 2.
3. 3.
This is in accordance with Miller (1994).Google Scholar
4. 4.
5. 5.
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.
9. 9.
10. 10.
11. 11.
Miller (1974, p.174) second corollary of Theorem 5.Google Scholar
12. 12.
13. 13.
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.
18. 18.
The differences between the logicistic and structuralist theory representation emerge while using more sophisticated theory representations.Google Scholar
19. 19.
20. 20.
21. 21.
Ibidp.301. and Kuipers (1987) p.82.Google Scholar
22. 22.
23. 23.
24. 24.
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.
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.
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.
39. 39.
Actually, Niiniluoto proposes to replace first order constituentsGoogle Scholar
40. 40.
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 