Refined Verisimilitude pp 74-120 | Cite as

# Truthlikeness

Chapter

## Abstract

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.

## Keywords

Atomic Proposition True Proposition Negative Content False Proposition Propositional Language
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## Notes

- 1.Interestingly, Niiniluoto and Kuipers also attended this conference.Google Scholar
- 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.Oddie (1986).Google Scholar
- 4.Brink and Heidema (1987), and Burger and Heidema (1994).Google Scholar
- 5.
*Ibid*p.25Google Scholar - 6.See Chapter 5. For instance, if the characteristics C
_{i}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.Note that both
*E*_{u}*(P)*ç*E*_{u}*(Q) y E*_{u}*(P) E*_{u}*(Q)*,and*I*_{u}*(Q)*ç I*u(P) v l*_{u}*(Q) I*_{u}*(P)*hold.Google Scholar - 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 l*u(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.See also Niiniluoto (1987), p. 233.Google Scholar
- 10.For the more general case see appendix.Google Scholar
- 11.See Niiniluoto (1987), p.126.Google Scholar
- 12.Niiniluoto (1987), p.311.Google Scholar
- 13.Cf. W. K. Clifford in
*Proceedings of the Manchester Literary and Philosophical Society*(February 1877, vol. xvi).Google Scholar - 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.Niiniluoto (1987), p. 212. It is similar to what Oddie (1990) calls the averaging method.Google Scholar
- 16.
*0(h*_{ ;},*g) =*red((A_{ii}I*j*E I_{s})) 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.
*Ibid.*p.212Google Scholar - 18.
*Ibid*p.217Google Scholar - 19.
*Ibid.*p.214–216Google Scholar - 20.The diagram is a copy of Fig.5 of Niiniluoto (1987), p.214.Google Scholar
- 21.See Niiniluoto (1987), p. 25 and Czekanowski (1909).Google Scholar
- 22.Sokal (1961).Google Scholar
- 23.See Niiniluoto (1987) p. 232–234.Google Scholar
- 24.
*Ibid.*p.248.Google Scholar - 25.Niiniluoto (1987) sections 6.7 and 6.8, respectively.Google Scholar
- 26.The numbers between parenthesis refer to the sections in Niiniluoto (1987) describing the relevant distance functions.Google Scholar
- 27.Niiniluoto (1987) section 2.2 and section 3.2; Niiniluoto (1987b), p.14.Google Scholar
- 28.Niiniluoto (1987), section 6.2, p. 207–209.Google Scholar
- 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.Oddie (1987 a), p.37.Google Scholar
- 31.Tichÿ and Oddie define their truthlikeness ordering for higher order, intensional interpreted languages.Google Scholar
- 32.E.g. see Oddie (1987a), p. 30–34 or Oddie (1986), p. 60–63.Google Scholar
- 33.Oddie (1986), p.62.Google Scholar
- 34.
*Ibid.*p. 126–127.Google Scholar - 35.L.J. Cohen (1980).Google Scholar
- 36.
*ibid*p.34–35.Google Scholar - 37.Miller (1975).Google Scholar
- 38.Oddie (1987a).Google Scholar
- 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.Oddie (1990).Google Scholar
- 41.The proof can be found in the Appendix.Google Scholar
- 42.
*Ibid.*p.132.Google Scholar - 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.Kuipers (1987), van Benthem (1987) and Kuipers (1992).Google Scholar
- 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.
- 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.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.Kuipers (1992), p. 332–333. It equals Niiniluoto’s minsum measure Sms(X, 1 ), (1987), p. 246.Google Scholar
- 50.
*Ibid.*p.332.Google Scholar - 51.
*Ibid.*section 3.7.Google Scholar - 52.Niiniluoto (1987), p. 247.Google Scholar
- 53.Kuipers (1992), p.319.Google Scholar
- 54.Kieseppä (1995), p.180 and Kuipers (1997), p. 161.Google Scholar
- 55.Private correspondence with Kieseppä.Google Scholar
- 56.Kuipers (1992), p.333.Google Scholar

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