# Statistical Explanations

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## Abstract

The purpose of this paper is to provide a systematic appraisal of the covering law and statistical relevance theories of statistical explanation advanced by Carl G. Hempel and by Wesley C. Salmon, respectively. The analysis is intended to show that the difference between these accounts is in principle analogous to the distinction between truth and confirmation, where Hempel’s analysis applies to what is taken to be the case and Salmon’s analysis applies to what is the case. Specifically, it is argued

- (a)
that statistical explanations exhibit the nomic expectability of their explanandum events, which in some cases may be strong but in other cases will not be

- (b)
that the statistical relevance criterion is more fundamental than the requirement of maximal specificity and should therefore displace it; and,

- (c)
that if statistical explanations are to be envisioned as inductive arguments at all, then only in a qualified sense since, in particular, the requirement of high inductive probability between explanans and explanandum must be abandoned.

## Keywords

Logical Probability Inductive Argument Statistical Explanation Knowledge Situation Maximal Specificity
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## Notes

- 1.Carl G. Hempel, ‘Aspects of Scientific Explanation’,
*Aspects of Scientific Explanation*, Part 3, The Free Press, New York, 1965, and Carl G. Hempel ‘Maximal Specificity and Lawlikeness in Probabilistic Explanation’,*Philosophy of Science*(June 1968).Google Scholar - 2.Wesley C. Salmon,
*Statistical Explanation and Statistical Relevance*, University of Pittsburgh Press, Pittsburgh, 1971.Google Scholar - 3.Carl G. Hempel, ‘Deductive-Nomological
*vs*.> Statistical Explanation’, in*Minnesota Studies in the Philosophy of Science*, Vol. III (ed. by H. Feigl and G. Maxwell), University of Minnesota Press, Minneapolis, 1962, p. 125.Google Scholar - 4.Hempel, ‘Maximal Specificity’, p. 117. Note that this quotation and those following have been slightly revised to apply to the examples at hand.Google Scholar
- 5.Hempel, ‘Maximal Specificity’, p. 118.Google Scholar
- 6.
*Ibid*.> (I) and (II) may be viewed as illustrating both kinds of ambiguity.Google Scholar - 7.Hempel, ‘Maximal Specificity’, p. 121.Google Scholar
- 8.
*Ibid*.> Google Scholar - 9.Hempel, ‘Maximal Specificity’, p. 131.Google Scholar
- 10.Hempel, ‘Maximal Specificity’, p. 128.Google Scholar
- 11.Hempel, ‘Maximal Specificity’, p. 131.Google Scholar
- 12.Cf. Salmon,
*op. cit*.>, pp. 42–43 and pp. 106–108.Google Scholar - 13.Carl G. Hempel and Paul Oppenheim, ‘Studies in the Logic of Explanation’,
*Aspects of Scientific Explanation*, pp. 248–249.Google Scholar - 14.Salmon,
*op. cit*.>, pp. 9–10.Google Scholar - 15.Salmon,
*op. cit*.>, p. 63.Google Scholar - 16.Salmon,
*op. cit*.>, pp. 106–108.Google Scholar - 20.On Carnap’s conception, of course, the logical probability of an hypothesis, given certain evidence, is equal to the prior probability of that hypothesis and that evidence together divided by the prior probability of that evidence. In application to explanations, therefore, the logical probability of such an explanandum, given such an explanans, is equal to the prior probability of that explanans and that explanandum together divided by the prior probability of that explanans. Since the explanans — regarded as evidence for its explanandum, regarded as an hypothesis — belongs to a particular knowledge situation
*K*, the elimination of epistemic relativization from Hempel’s account requires (a) displacement of an epistemic concept of logical probability (i.e., Carnap’s) by a non-epistemic concept (e.g., Reichenbach’s) as well as (b) substitution of an ontic requirement of statistical relevance (e.g., Salmon’s) for an epistemic requirement (i.e., Hempel’s). For an elementary introduction to Carnap’s conception, see Bryan Skyrms,*Choice and Chance*, Dickinson, Belmon, Calif., 1966, Chapter V.3;Google Scholar - 20a.for a detailed account, see Rudolf Carnap,
*Logical Foundations of Probability*(2nd ed.), University of Chicago Press, Chicago, 1962, Chapter III, and Appendix.Google Scholar - 21.Hans Reichenbach,
*The Theory of Probability*, University of California Press, Berkeley, 1949, pp. 378–382. Hempel has observed that Reichenbach himself apparently never explicitly considered the logical structure of explanations invoking statistical laws (‘Maximal Specificity’, p. 122). Under Reichenbach’s logical interpretation, however, it would be possible to maintain a modified Hempelian account along the following lines:(i) the explanandum must be an inductive consequence of its explanans;(ii) the explanans must contain at least one statistical law that is actually required for the derivation of its explanandum;(iii) the statistical laws invoked in the explanans must satisfy the requirement of reference class homogeneity; and,(iv) the sentences constituting the explanation — both the explanans and its explanandum — must be true.These conditions, moreover, may obviously be generalized to encompass those explanations invoking universal laws.Google Scholar - 21a.For a discussion of the necessity for condition (iii) even in the case of explanations invoking universal laws, see Salmon,
*op. cit*.>, pp. 33–35.Google Scholar

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© D. Reidel Publishing Company, Dordrecht-Holland 1974