^* The author is indebted to Carl G. Hempel and to Wesley C. Salmon for their critical comments on an earlier version of this paper.

¹ Hempel, Carl G., ‘Aspects of Scientific Explanation’, Aspects of Scientific Explanation, Part 3, The Free Press, New York, 1965Google Scholar, and Hempel, Carl G. ‘Maximal Specificity and Lawlikeness in Probabilistic Explanation’, Philosophy of Science (June 1968).CrossRefGoogle Scholar

² Salmon, Wesley C., Statistical Explanation and Statistical Relevance, University of Pittsburgh Press, Pittsburgh, 1971.CrossRefGoogle Scholar

³ Hempel, Carl G., ‘Deductive-Nomological vs. Statistical Explanation’, in Minnesota Studies in the Philosophy of Science, Vol. III (ed. by Feigl, H. and Maxwell, G.), 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.

⁵ Hempel, , ‘Maximal Specificity’, p. 118.Google Scholar

⁶ Ibid. (I) and (II) may be viewed as illustrating both kinds of ambiguity.

⁷ Hempel, ‘Maximal Specificity’, p. 121.

⁸ Ibid.

⁹ Hempel, ‘Maximal Specificity’, p. 131.

¹⁰ Hempel, ‘Maximal Specificity’, p. 128.

¹¹ Hempel, ‘Maximal Specificity’, p. 131.

¹² Cf. Salmon, op. cit., pp. 42-43 and pp. 106-108.

¹³ Hempel, Carl G. and Oppenheim, Paul, ‘Studies in the Logic of Explanation’, Aspects of Scientific Explanation, pp. 248-249.Google Scholar

¹⁴ Salmon, op. cit., pp. 9-10.

¹⁵ Salmon, op. cit., p. 63.

¹⁶ Salmon, op. cit., pp. 106-108.

¹⁷ It is interesting to observe, therefore, that although Hempel has implicitly differentiated between the explanation and the confirmation contexts in the process of distinguishing between the rationale for the requirement of maximal specificity (RMS*) as opposed to the rationale for the requirement of total evidence, he continues to envision r as it occurs within the explanation context as essentially the same r which occurs within the confirmation context. This suggests the possibility that Hempel's present account of statistical explanation represents a ‘transitional phase’ toward a more adequate analysis.

¹⁸ Salmon maintains that in Carnap's system of inductive logic, “there is no such thing as inductive inference in the sense required for Hempel's account of inductive-statistical explanation. In Carnap's inductive logic there are no inductive arguments consisting of premisses and conclusions, which allow you to affirm the conclusion (with some degree of probability) if you are prepared to assert the premisses. On this view, inductive logic is strongly disanalogous to deductive logic, even to the extent of proscribing inference entirely . Op. cit., pp. 8-9.

¹⁹ Hempel,‘Aspects’, p. 389.

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; for a detailed account, see Carnap, Rudolf, Logical Foundations of Probability (2nd ed.), University of Chicago Press, Chicago, 1962Google Scholar, Chapter III, and Appendix.

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:

1. (i)

   (i) the explanandum must be an inductive consequence of its explanans;

2. (ii)

   (ii) the explanans must contain at least one statistical law that is actually required for the derivation of its explanandum;

3. (iii)

   (iii) the statistical laws invoked in the explanans must satisfy the requirement of reference class homogeneity; and,

4. (iv)

   (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. 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.

²² It might be supposed that such an approach would encounter a lottery paradox; however, that this is not the case follows from the fact that it requires no rule of acceptance or of rejection (which, I believe, further reinforces the significance of distinguishing between the two concepts of inductive argument).