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Does the Bayesian solution to the paradox   of confirmation really support  Bayesianism?

Brian Laetz

Abstract Bayesians regard their solution to the paradox of confirmation as grounds for preferring their theory of confirmation to Hempel’s. They point out that, unlike Hempel, they can at least say that a black raven confirms “All ravens are black” more than a white shoe. However, I argue that this alleged advantage is cancelled out by the fact that Bayesians are equally committed to the view that a white shoe confirms “All non-black things are non-ravens” less than a black raven. In light of this, I reexamine the dialectic between Hempel and the Bayesians.

Keywords Paradox of confirmation . Paradox of the ravens . Bayesianism . Carl Hempel

1. Introduction

One of the major selling-points of a Bayesian theory of confirmation is its alleged ability to cope with classic problems of confirmation better than its qualitative predecessors. One purported success story in this vein is the Bayesian solution to Hempel’s paradox of confirmation (1945), which is frequently trumpeted as a significant advance over Hempel’s view. However, even granting all the assumptions of a Bayesian treatment, I argue that if the supposed advantages of the solution are genuine, they are nevertheless offset by disadvantages that do not afflict Hempel’s analysis. Thus, Bayesians have failed to explain how their solution to the paradox of confirmation provides independent grounds for preferring their theory of confirmation to Hempel’s. Of course, this is not to say that Bayesianism is not superior on other grounds—even many other grounds—but it is to say that typical accounts of Bayesian progress in confirmation theory are mistaken on this single, but noteworthy, detail.

Hempel derived the paradox of confirmation from trivial logical equivalences and two seemingly innocuous principles. For our purposes, these principles can be informally stated as:

Nicod’s Condition: A positive-instance of a hypothesis confirms it.  Equivalence Condition: Whatever confirms a hypothesis will confirm any logically equivalent hypothesis.

From these, Hempel delivered the following memorable illustration of the paradox. Consider the hypothesis “All ravens are black.” This universal generalization is logically equivalent to “All non-black things are non-ravens”—its contrapositive. By Nicod’s criterion, this latter statement is confirmed by any non-black non-raven, like a white shoe, purple shirt, yellow duck, and much, much else. But now, by the equivalence principle, it follows that such things also confirm “All ravens are black”—a puzzling result. Moreover, this is just a random, though admittedly famous, example; there are innumerable instances of the paradox. For now, this brief statement of the puzzle will suffice.

2. Hempel  vs the Bayesians  part I

Not surprisingly, Hempel’s discussion and treatment of the paradox is more nuanced than much literature would suggest. That being said, however, my argument does not hinge on this. Thus, we can sum up Hempel’s approach rather briefly. First, he accepts that white shoes are relevant to assessing whether all ravens are black; in short, he embraces the paradox. Second, he accepts that a white shoe confirms “All ravens are black” just as much as a black raven. This stems from Hempel’s qualitative approach, which, ipso facto, precludes incremental confirmation. How do these conclusions compare to those of  Bayesians?1

Although Bayesians reject Nicod’s condition (see Good 1961, 1967), they too accept the paradoxical conclusion that white shoes are relevant to determining whether all ravens are black. So, they can claim no advantage over Hempel there and, as a matter of fact, they do not. The alleged advances stem from different views regarding the degree to which white shoes and black ravens respectively confirm the hypothesis. Two basic claims are presented in this vein. First, Bayesians maintain that a white shoe provides less support for the hypothesis than a black raven, something that is precluded on Hempel’s account. Second, going further, some Bayesians maintain that a white shoe confirms “All ravens are black” to a negligible degree,  more  specifically,  nearly  0  on  some  acceptable  relevance  measure.2  In

addition, those who embrace the last point also claim the Bayesian position has greater explanatory power than Hempel’s. Specifically, it is said that since the degree of confirmation is so small in such cases, this explains why, upon first encountering the paradox, people think that white shoes are confirmationally irrelevant, for it is easy to confuse the minute degree of confirmation they provide with no confirmation at all.

Now it’s important to note that many Bayesians do not push all three points we’ve just viewed. However, let’s grant whatever assumptions are necessary for the Bayesian to establish all three.3 If so, it looks like this is the best response that a Bayesian can manage. In short, what more can a Bayesian ask for? It’s also clear that these features are not shared by Hempel’s account. However, if they are genuine advantages, I suggest that they are cancelled out by other implications of the Bayesian treatment.

3. Hempel vs the Bayesians part  II

To clearly see the problem, consider the paradox of confirmation again, but this time, let’s begin with the hypothesis “All non-black things are non-ravens.” Ordinarily, we think this would only be confirmed by things like white shoes, purple shirts, yellow ducks, and the like. However, this statement is logically equivalent to “All ravens are black,” which, by Nicod’s condition, is confirmed by a black raven. Finally, by the equivalence condition, black ravens also confirm “All non-black things are non- ravens”—a puzzling result. Let’s compare how Hempel and the Bayesians will respond. To begin, although Bayesians reject Nicod’s condition, both will accept that black ravens are relevant to assessing whether all non-black things are non-ravens. In short, both will embrace the paradox. So, neither side could claim superiority here, and, as a matter of fact, neither would. The differences between Hempel and the Bayesians will regard the degree of confirmation in such cases. On the one hand, Hempel must claim that black ravens and white shoes equally confirm “All non- black things are non-ravens.” On the other hand, we have already granted Bayesians that a black raven confirms “All ravens are black” more than a white shoe. But, by contraposition, “All ravens are black” is logically equivalent to “All non-black things are non-ravens.” And, by the equivalence condition, we find that “All non- black things are non-ravens” is confirmed more by a black raven than a white shoe. Further, we have already granted Bayesians that a white shoe confirms “All ravens are black” to a negligible degree, more specifically, nearly 0, unlike a black raven. So, by similar reasoning, we find that white shoes confirm “All non-black things are non-ravens” to a negligible degree, more specifically, nearly 0. Finally, we find that Bayesianism does not at all explain the intuition people have that white shoes do indeed confirm “All non-black things are non-ravens.” In sum, every single thing Bayesians claim as an advantage over Hempel with respect to the hypothesis “All ravens are black” is exactly reversed when considering the hypothesis “All non- black things are non-ravens.” At this point, a Bayesian might well say, “So what?” To them, the results they obtain in each case will seem intuitively superior to those

of Hempel. However, I argue that this is only the case if one is already assuming Bayesian ideas, and thus conclude that the Bayesian solution to the paradox of confirmation ultimately fails to provide an independent reason for preferring Bayesianism to Hempel’s theory of confirmation.

4. Why the Bayesian solution does not support Bayesianism

Generally stated, the original intuition the paradox violates—and which logic dictates we must abandon—is that positive-instances of a hypothesis’ contrapositive never confirm the relevant hypothesis. Hempel and the Bayesians both ultimately accept this violation—as they rightly should—but Bayesians claim to improve upon Hempel’s view, on the grounds that their specific results are more intuitive. As it happens, Bayesians expend little effort explaining why their results are supposed to be more appealing; rather, their usual strategy is just to show how they can be rigorously derived from a small set of assumptions on a Bayesian view. Understanding why,  however, is crucial to the  issue at  hand, as will become clear. I think there are two possibilities. To approach them, begin by again considering the statement, “All ravens are black,” the favored Bayesian hypothesis for illustrating the paradox. Why is the Bayesian solution supposed to be more appealing here?

One explanation exploits our pre-theoretical or prima facie intuitions. Like Hempel, Bayesians accept the paradox that white shoes can confirm “All ravens are black,” but they also claim to “soften” the paradox in two ways that Hempel does not. First, according to Bayesians, ravens are still more important to evaluating the hypothesis than non-black things. On their view, it is still fine to privilege a black raven over a white shoe here, as everyone does. On Hempel’s view, however, this becomes a rationally unfounded prejudice, since each confirms the hypothesis equally. In this regard, Bayesianism deviates less from scientific practice and our prima facie intuitions. Second, white shoes may, for all practical purposes, be ignored on the Bayesian view, since they offer so little confirmation to begin with. This, of course, is what everyone does, but this practice again seems unwarranted on Hempel’s view, just as unwarranted as it would be to ignore black ravens, since they are confirmationally equal. In this respect, the Bayesian results again seem more accommodating of our everyday practices. This sort of explanation of the superiority of the Bayesian solution clearly seems to privilege Bayesianism over Hempel’s account of confirmation, so long as we just concentrate on their respective solutions to the paradox of white shoes confirming “All ravens are black.” But, as should be clear from the preceding section, this sort of explanation is no longer available to Bayesians when considering the hypothesis, “All non-black things are non-ravens.” Again, like Hempel, Bayesians accept the paradoxical conclusion that black ravens can confirm “All non-black things are non-ravens,” but instead of softening this result, as they claim to do in the preceding case, here they instead intensify it by holding that black ravens are more important than white shoes and that white shoes can, for all practical purposes, be ignored in this instance. Thus, Bayesians fare correspondingly worse where they previously did better—at least with respect to our initial intuitions. Moreover, though we have been concentrating on particular hypotheses, it should be clear that the problem is perfectly general. If the Bayesians’

results are supposed to be more appealing than Hempel’s, on the grounds that they better accommodate our pre-theoretical instincts, what we find is that they do so regarding one formulation of a hypothesis, while Hempel comes off better with respect to other formulations. So, this sort of explanation fails to favor the Bayesian solution over Hempel’s, and thus provides no reason to choose Bayesianism over Hempel’s theory of confirmation. Excessive focus on the hypothesis, “All ravens are black,” unduly highlights the immediate appeal of the Bayesian solution, and neglects its prima facie weaknesses with respect to other cases. To be sure, the Bayesian solution to any instance of the paradox will vary with the details of each particular case—the hypothesis under consideration and the size of the classes involved. But fairly assessing the Bayesian treatment of the paradoxes of confirmation, as Hempel originally dubbed them, demands considering a represen- tative sample of them. And with respect to our initial intuitions, it should be clear that the case of “All ravens are black” is not the only hypothesis worth considering. In sum, this explanation of why the Bayesian results are more attractive then Hempel’s fails to provide a clear rationale for thinking that the Bayesian solution to the paradox favors Bayesian confirmation theory over Hempel’s. And this point is especially important to appreciate, because this appears to be the only sort of explanation that has ever been offered by Bayesians. But perhaps there is a better explanation of why the Bayesian solution is more  attractive.

A different account might appeal to our considered intuitions. Along these lines, a Bayesian might grant that their solution is ultimately not more attractive than Hempel’s with respect to our prima facie intuitions, but insist that the Bayesian results are sensitive to our deeper intuitions, ones that might not be apparent initially, but that are far more important. I suspect more than one argument could be made in this vein. On closer inspection though, I think it is very difficult to produce such an argument without presupposing some Bayesian idea and thus begging the question against Hempel. To illustrate, I’ll focus on what I think is the strongest argument that could be made along these  lines.

A Bayesian might claim that one of our deeper intuitions is that degree of confirmation depends on class size: members of smaller classes offer more confirmation than members of larger classes. Pressing further, they will say that   the Bayesian solution perfectly captures this, since there are far more non-black things than ravens in our world. Thus, the initially odd results regarding “All non- black things are non-ravens” ultimately turn out to be just as intuitively appealing as those regarding “All ravens are black.” There is a problem here, however. Specifically, it is not clear that class size seems relevant to confirmation in absence of Bayesian ideas—not every tenet of Bayesianism, mind you, but core ideas, nonetheless, which are not neutral grounds between these opposing views. For if one finds it strange to think that a black raven confirms “All non-black things are non- ravens” more than a white shoe, I doubt this feeling weakens merely by being reminded that there are more non-black things than ravens. Strictly speaking, for this to matter, one must already grant that background knowledge is confirmationally relevant. But, crucially, this is one issue of dispute between Hempel and the Bayesians. Moreover, the respective frequencies of ravens and non-black things only seem to favor the Bayesian results by implicitly relying on a connection between probability  and  confirmation  (since  probabilities,  on  typical  views,  should  be

informed by frequencies). But, again, explicating confirmation along probabilistic lines, as does the relevance criterion of confirmation, is another key proposal of Bayesianism, which is also at issue between Hempel and the Bayesians. Thus, this sort of explanation of the Bayesian solution’s superiority silently relies on tenets of Bayesianism. To then argue for Bayesianism over Hempel’s theory of confirmation on the basis of the Bayesian solution would be question-begging; it would already presuppose Hempel is wrong on key issues, and that Bayesians are right. If we must do this to favor the Bayesian treatment of the paradox, then we must already reject Hempel, in which case it is not clear that the Bayesian solution to the paradox provides an independent reason to reject Hempel’s theory. Less subtle explanations of why the Bayesian solution is better also suffer this fate. For example, merely showing how the Bayesian results can be rigorously derived from Bayesian premises fails to provide neutral grounds for someone to choose Bayesianism over Hempel’s theory of confirmation.

To better appreciate the dialectical situation now surrounding Hempel’s paradox, it helps to contrast it with the situation regarding another classic puzzle in confirmation theory. Consider the tacking paradox or problem of irrelevant conjunction, originally raised against hypothetico-deductivism (H-D).4 Here too, Bayesians present their solution as providing independent grounds for preferring Bayesianism to a qualitative rival. But, I contend, they can do so without encountering any of the sorts of problems I have raised for Bayesians who similarly present their solution to Hempel’s paradox. As is well known, according to H-D, an observation confirms some particular hypothesis if and only if it can be soundly deduced from it. In other words, theories are confirmed by their successful predictions—a plausible notion. Notoriously, however, this clean deductive framework also leads to an odd implication. To take a familiar example, Einstein’s General Theory of Relativity predicts that light will bend around large objects, like the Sun. Of course, light does bend around the Sun, which confirms Einstein’s theory, on H-D. By the same token, however, this observation can equally be deduced from Einstein’s theory conjoined to any irrelevant hypothesis, like “The moon is made of cheese.” So, hypothetico- deductivists must accept that light bending around the Sun also confirms the conjunction of Einstein’s theory and the cheese—moon hypothesis. Here, Bayesians similarly tout their response as a reason for favoring their theory of confirmation.

In brief, like hypothetico-deductivists, Bayesians also accept that when evidence confirms some hypothesis it will also confirm the conjunction of that hypothesis and any irrelevant statement. However, unlike hypothetico-deductivists, they note that within a Bayesian framework one can “soften” this consequence by at least saying that when evidence confirms a hypothesis it does so to a greater degree than it does the conjunction of that hypothesis and an irrelevant statement. On ordinary intuitions, this response straightforwardly favors Bayesianism. For, even if we cannot wholly escape the paradox, it at least seems like progress to hold that the evidence privileges the hypothesis over complex statements of which it is a constituent. Note too that, with respect to our prima facie intuitions, there are no problem-cases, like the one I have posed for the Bayesian solution to Hempel’s paradox; there are no odd instances where the evidence supports a hypothesis and

4 Fitelson (2002) provides a good summary and important recent discussion of the problem.

irrelevant conjunct more than it does the hypothesis alone. And thus, Bayesians need not struggle to offer a subtler account of why their solution is better. In short, the Bayesian solution to the tacking paradox clearly provides independent grounds for preferring Bayesianism to hypothetico-deductivism. Thus, from the standpoint of this essay, it is a mistake to place the Bayesian solution to the paradox of confirmation alongside their solution to the problem of irrelevant conjunction as another independent reason for preferring a probabilistic theory to a qualitative rival. Typical accounts of Bayesian progress in confirmation theory are mistaken in doing so.

We can sum up these results as a dilemma for Bayesians that take their solution to the paradox of confirmation as a reason to choose Bayesianism over Hempel’s theory of confirmation. Either the Bayesian results are supposed to be more attractive because they appeal to our prima facie intuitions or because they appeal to our considered intuitions. If the former, then the Bayesian results do not favor Bayesianism over Hempel’s account. To the extent that their treatment of some instances of the paradox seems better, their treatment of other instances of the paradox seems worse. If the latter, I suggest that it is difficult to find any considered intuitions that do not already presuppose Bayesian ideas which are at issue, in which case arguing that the Bayesian results favor Bayesianism over Hempel’s account is question-begging.

5. Some objections and replies

One point that a Bayesian might make, with which I agree, is simply that there are other reasons to prefer Bayesianism to Hempel’s theory of confirmation. Indeed, some of these even historically came out of discussions of the paradox, most notably, problems with Nicod’s criterion, which early Bayesians deserve credit for identifying. However, rigor demands knowing precisely what neutral grounds there are for preferring Bayesianism to alternative theories of confirmation, even very old- fashioned ones, like Hempel’s. If received wisdom is mistaken in taking the Bayesian solution to the paradox to be among them, as I have argued, this is worth noting.

One might also complain that I have misconstrued the Bayesian position on the paradox. Specifically, it might be said that Bayesians, or at least typical ones, do not take their solution to provide a reason for choosing Bayesianism over Hempel’s view. To the casual observer, this might seem rather implausible, although it should be considered, for a careful reading of much of the relevant Bayesian literature suggests that interest in the paradox long ago acquired a life far beyond any initial dispute with Hempel. Not only Hempel’s view, but seemingly all qualitative theories of confirmation have been dismissed as inadequate for quite some time, and yet interest and discussion of the paradox has continued unabated. Thus, much work on the paradox seems rather unconcerned with engaging Hempel and far more interested in simply working out the details of the Bayesian treatment, presumably, to develop the Bayesian view as comprehensively as possible. Despite all this, it still seems that one small push toward Bayesianism, early on, stemmed from a general consensus that the Bayesian conclusions regarding the paradox are less counterin- tuitive than Hempel’s. Moreover, even if much recent literature on the paradox