Abstract
Among Bayesian confirmation theorists, several quantitative measures of the degree to which an evidential proposition E confirms a hypothesis H have been proposed. According to one popular recent measure, s, the degree to which E confirms H is a function of the equation P(H|E) − P(H|~E). A consequence of s is that when we have two evidential propositions, E1 and E2, such that P(H|E1) = P(H|E2), and P(H|~E1) ≠ P(H|~E2), the confirmation afforded to H by E1 does not equal the confirmation afforded to H by E2. I present several examples that demonstrate the unacceptability of this result, and conclude that we should reject s (and other measures that share this feature) as a measure of confirmation.
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Notes
For example, these are the four considered in Fitelson (2001).
Or \( \frac{{{\text{P(H}}\left| {\text{E}} \right.) - {\text{P(H)}}}}{{{\text{P(}} \sim {\text{E)}}}} \). These are equivalent provided that P(E) is not 1. See Christensen (1999, p. 450). I call measure s the alternative difference measure of confirmation because it was most prominently defended by Christensen (1999) as an alternative to the traditional difference measure d.
As evidenced, for example, by Fitelson’s exclusion of it in favor of d, r, and l when he briefly mentions ‘the three most popular Bayesian relevance measures of … confirmation’ in Fitelson (2007, p. 478).
See also Joyce (2004, pp. 144–145), in which s is again defended as one of multiple legitimate measures.
As I will note, the problem I raise for s also affects other, less popular, measures that have been proposed, but I focus on s because it is seen as more of a live option than these other measures.
(CC) is also invoked by Crupi et al. (2010, p. 79) and Crupi et al. (2007, p. 234). The former propose it as a basic desideratum on Bayesian measures of confirmation, and the latter offer the fact that their preferred measure z (a measure not mentioned above) satisfies (CC) as evidence for z. In both cases, however, the authors’ only argument for (CC) is that it is necessary for popular Bayesian analyses, such as Horwich’s (1982) solution to the ravens paradox. Steel (2003, pp. 219–220) also mentions principles equivalent to (CC), but does not argue that they are true, only that they are assumed by many Bayesians.
I should also note that Crupi et al.’s (2007) recently proposed measure z satisfies (CC), and so does not suffer from this problem either. According to \( z, c\,\left( {{\text{H}},{\text{E}}} \right) = \frac{{{\text{P(H}}\left| {\text{E}} \right. )- {\text{P(H)}}}}{{{\text{P(}} \sim {\text{H)}}}} \) if P(H|E) ≥ P(H), and \( \frac{{{\text{P(H}}\left| {\text{E}} \right. )- {\text{P(H)}}}}{\text{P(H)}} \) otherwise.
My thanks to David Christensen for suggesting this possible motivation for (ES) to me.
Cf. Joyce’s (1999, p. 206) claim, in defence of s, that one way we think about the degree to which a proposition confirms a hypothesis is by contrasting the effects of learning that proposition with the effects of learning its negation.
And holding that the confirmation of H by E can be measured by the difference between P(H|E) and P(H|~E) because this satisfies (ES) seems just as arbitrary as holding that it can be measured by the difference between P(H|E) and P(H|(~E&P)). Of course, this latter function could not be a general Bayesian measure of confirmation because it might be possible that P(H|(~E&P)) > P(H|E) > P(H) (e.g., if P entailed H and E did not), in which case E will confirm H but the measure will report that it disconfirms it. But it is interesting to note that the “measure” P(H|E) − P(H|(~E&P)) has the same property that Christensen and Joyce find so appealing in the measure P(H|E) − P(H|~E), namely, it is “invariant under learning” (Joyce 1999, p. 207)—changes in P(E) do not lead to changes in c(H,E). This property is crucial to Christensen’s (1999, pp. 451–452) use of s to solve the old-evidence problem.
My thanks to an anonymous reviewer for making me aware of the need to consider this possibility.
For example, if we let each location be equal to 1/10,000 of a sector, and set n to 200,000, then the locations will cover one-fifth of the forest. (If m/n = .003505, m will be equal to 701, and the locations will cover 701/10,000 of sector 12.).
It is not necessary that this positive relevance be entailment in either case. To see this, imagine that one (but only one) of the locations in Wm/n had been set to outside the forest. Then Wm/n would not entail W, but it would still make W (and by extension, H) highly probable, and the counter intuitiveness of Wm/n confirming H more than W12 would remain.
Also, (E1vE2), (HvP) for any P with probability <1, etc.
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Acknowledgments
This paper had its genesis in a graduate seminar on probability at Western Michigan University in Fall 2009. I am grateful to Timothy McGrew for teaching that class and helping me think through these issues. I would also like to thank David Christensen for insightful correspondence on this project, as well as Matthew Lee and an anonymous reviewer for helpful comments on earlier drafts.
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Climenhaga, N. A problem for the alternative difference measure of confirmation. Philos Stud 164, 643–651 (2013). https://doi.org/10.1007/s11098-012-9872-0
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DOI: https://doi.org/10.1007/s11098-012-9872-0