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How to Think about Indirect Confirmation

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Abstract

Suppose a theory T entails hypotheses H and \(H'\), neither of which entails the other. A number of authors have argued that a piece of evidence E “indirectly confirms” H when E confirms either T or \(H'\). But there has been a protracted and unsettled debate about whether indirect confirmation is a sound inference procedure. Skeptics argue that the procedure employs conditions of confirmation that jointly lead to absurdity. Proponents argue that this criticism is unfounded or that its import is exaggerated. I will argue that no side has the story quite right, and some have the story quite wrong. Indirect confirmation, as characterized above, is unsound, and a good chunk of this paper will be concerned with showing why most extant defenses of the procedure err. On the other hand, when certain probabilistic (in)dependence relations hold between T, H, and \(H'\), indirect confirmation can work, for reasons that trace back to Reichenbach’s principle of the common cause. I illustrate these matters with some contemporary and historical examples, with a particular focus on Kepler’s use of data about mars’s elliptical orbit to justify a claim about earth’s.

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Notes

  1. Newton-Smith was reflecting on the choice in 1905 between Lorentz’s ether theory and special relativity. The two theories were equally compatible with the available data, but Newton-Smith claimed the empirical success of general relativity (e.g., in explaining the precession of Mercury) provided “empirical grounds” (1980, p. 105) to prefer special relativity.

  2. For instance, Newton-Smith (1980) and Bergström (1984) also employ a form of indirect confirmation. While it’s not clear that their accounts rely on the CCC, the accounts clearly rely on the SCC.

  3. Note that X confirms Y whenever X entails Y if X and Y are contingent propositions.

  4. And here’s an analogous counterexample to the CCC: If you observe that a card is a heart, this confirms that the card is red, which confirms that the card is the Jack of Diamonds. But observing that the card is a heart doesn’t raise the probability that the card is the Jack of Diamonds.

  5. I thank John Norton for alerting me to the relevance of \(\lambda \text {CDM}\) to this issue.

  6. Quite similar ideas appear in Lehtinen (2016). I take Lehtinen (2018) to be the more definitive (because more recent) statement of his views.

  7. In fact, Skellam (1955) appears to have been the first to use “robust” in the sense that Levins later used that term. Box (1953) used “robust” to describe an inference from data that did not depend on certain assumptions of the statistical model (e.g., normality).

  8. Only the left-hand side of (3) appears in what Lehtinen actually wrote (Ibid., p. 551). Context suggests that he meant this to be an equation and that the right-hand side was supposed to be \(P(R_M)\).

  9. In fact, each model entails this prediction, but that’s irrelevant.

  10. Contrast this scenario with, say, the judgment that vestigial organs raise the probability of the truth of common ancestry, so that \(Pr(CA |\ \text {Vestigial Organs}) > Pr(CA)\). Because CA appears on both the left- and right-hand side of the inequality, one can more reasonably assess whether the inequality is true or false, though some will attest that speaking of the “prior” probability of common ancestry is ill advised (see, e.g., Sober 2008).

  11. Kotzen (2012, pp. 66-67) observes that knowing that this condition is satisfied requires one to already know something about the probabilistic relationship between X and Z. Kotzen therefore does not believe the condition is “usable” in that it does not tell us something we do not already know. This is correct, but I am here primarily concerned with the confirmational relationships between propositions, not articulating an inferential procedure that an agent might employ to reason about the propositions.

  12. Suppes (1986) showed that in fact confirmation is transitive when the weaker condition \(Pr(Z|\, X\ \& \ Y) \ge Pr(Z|\, Y)\) and \(Pr(Z|\, X\ \& \ \lnot Y) \ge Pr(Z|\, \lnot Y)\) is satisfied (see also Hesse (1970) and Roche (2012)). Atkinson and Peijnenburg (2021) have weakened the condition still further. These results are less relevant to the examples I am concerned with, however.

  13. See also Eells and Sober (1983).

  14. I’ll use CD to refer both to the theory of continental drift and to the causal process that results in \(H_1\) and \(H_2\). Context will disambiguate.

  15. That is, \(Pr(H_1 \& \, E_2 |\, CD) = Pr(H_1 |\, CD) \times Pr(E_2 |\, CD)\) and \(Pr(H_1 \& \, E_2 |\, \lnot CD) = Pr(H_1 |\, \lnot CD) \times Pr(E_2 |\, \lnot CD)\).

  16. While Dietrich and Moretti (2005) are concerned with indirect confirmation of the sort described in Laudan and Leplin (1991), those authors appear to believe a conjunctive fork is not the correct way to understand Laudan and Leplin’s example of continental drift (2005, p. 417).

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Acknowledgements

This paper benefited greatly from comments and criticism from Tony Beavers, Mark Bedau, Hyundeuk Cheon, Bryan Cwik, Avram Heller, Aki Lehtinen, Edouard Machery, John Norton, Steven Orzack, Matt Parker, Darrell Rowbottom, Elliott Sober, and various anonymous reviewers.

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  1. Department of Philosophy, Portland State University, Portland, OR, USA

    Brian McLoone

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McLoone, B. How to Think about Indirect Confirmation. Erkenn (2023). https://doi.org/10.1007/s10670-023-00713-3

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