1. Introduction

In the course of everyday reasoning, we often explain the frequency with which some type of event occurs by appealing to the probability of that type of event. Why haven’t you ever seen a roulette ball land on the same number twice in a row? Because it is very unlikely for it to do so. Why does an evenly weighted coin land heads roughly half the time? Because the probability of it landing heads is 1/2. Why do you always hit at least one red light on your drive home from work? Because the probability of hitting at least one red light is so high.

You might think that this sort of reasoning should not be taken too seriously. You might think that using probabilities to explain the frequencies of everyday macro-physical events is a mistake or at most a useful shorthand, a placeholder of sorts for some more complicated nonprobabilistic explanation. But a similar sort of reasoning also plays a crucial role in some of our best scientific theories. And insofar as it does, it is not so easily dismissed.

I will use the name ‘nomological probability’ for the type of probability that plays the explanatory role just described.

‘Nomological’ probabilities are so named because the explanatory role that such probabilities play with respect to stable, long-run relative frequencies is reminiscent of the explanatory role that laws play with respect to associated regularities. If it is a law that events of type E never occur, that explains why events of type E never occur. If it is a law that all Fs are Gs, that explains why all Fs are Gs. Nomological probabilities, then, are probabilities that play a lawlike role.

In this paper, I argue that we ought to believe that there are such things as nomological probabilities even if the further metaphysics we are able to give of such probabilities turns out to be problematic in various ways. In particular, I will argue that we ought to believe that there are nomological probabilities even if it turns out that the only successful analysis of such probabilities involves entities that are supposed to be in some sense metaphysically weird, or even if there is no successful analysis of such entities at all, and nomological probabilities turn out to be sui generis. I then give an example of one way in which this argument should shape future work on the metaphysics of chance. In particular, I describe a challenge associated with understanding one common group of analyses of objective probability—Humean analyses—as analyses of nomological probability.

Of course, the idea that some sort of objective probability plays an important role in our best scientific theories is hardly new. But the particular explanatory role that I will identify has seen relatively little discussion in the literature,Footnote ² and its implications have largely gone unrecognized. Insofar as philosophers acknowledge that we want objective probabilities to explain frequencies, they seem to think that desideratum is just one among many that compete to influence our metaphysics of chance—one that might be trumped, for instance, by concerns about the type of entities that explanatorily powerful probabilities would be and how those entities fit into our broader metaphysics. But given the argument I present below, this is a mistake. We should think that there are nomological probabilities even if we are forced to accept that such probabilities violate various principles we had hoped our metaphysics would respect and even if such probabilities do not fit into our broader metaphysics at all. Insofar as the existence of such probabilities raises a challenge for Humean analyses of objective probability, for instance, it is not a challenge that can be postponed or dismissed on the grounds that the Humean approach’s virtues outweigh any weakness that the challenge exposes.

Two clarifications before we begin. First, as the title says, this is supposed to be a naturalist’s guide to objective probability. As I will be using the term here, to be a naturalist is to take standard scientific methodology as a good guide to successful inquiry into what the world is like. Insofar as one is a naturalist, then, one should think that the norms and constraints governing scientific theory choice should govern metaphysical theory choice as well. This does not mean, however, that one must take everything that scientists claim at face value. Nor does it mean that one should think that philosophers have nothing to add to the understanding of scientific practice. Naturalism is one thing. Scientism is another.

Second, as the title of the article says, this is supposed to be a naturalist’s guide to objective chance. I take it that it is obvious, even just from what has been said so far, that whatever else they are, nomological probabilities are objective probabilities—probabilities that are determined by physical features of the world independently of any particular epistemic position that we might occupy within that world. However much we might wish it to be otherwise, the behavior of roulette wheels and coin tosses is not explained by facts about the beliefs or evidence we happen to have. You might, of course, explain our expectations regarding how roulette wheels and the like behave by appealing to facts about the beliefs or evidence that we have, but explaining our expectations regarding some phenomena and explaining the phenomena itself are two very different things.Footnote ³

So the argument presented in the first part (secs. 2 and 3)—the argument for the existence of nomological probabilities—is also an argument for the existence of objective probabilities, or chances. (I use the terms ‘objective probability’, ‘chance’, and ‘objective chance’ interchangeably.) But nowhere in what follows do I commit myself to the view that nomological probability is the only type of objective probability. Insofar as you are a naturalist, the arguments below should convince you to believe in nomological probabilities. If, in addition, you have antecedent metaphysical commitments or intuitions that commit you to some notion of objective probability that comes apart from nomological probability, by all means feel free to accept such objective nonnomological probabilities into your metaphysics. But you must accept them in addition to nomological probabilities. Or so I will argue.

2. The Argument for Nomological Probability

Why think that there are such things as nomological probabilities? In short, because they play a crucial explanatory role in our best scientific theories. In particular, they explain certain patterns of events that would otherwise go unexplained.

Here is a paradigm case of the sort I have in mind. Consider carbon-11, a radioactive isotope of carbon that decays into boron. According to the standard interpretation of quantum mechanical phenomena—the interpretation taught in physics textbooks and in physics classrooms—the fundamental laws governing the decay of carbon-11 atoms are indeterministic. More specifically, there is no feature F such that all and only the carbon-11 atoms that have F will decay within a given time period. Nonetheless, over a long series of trials, we observe a relatively stable pattern in the decay of such atoms. And we take this relatively stable pattern in our data to be indicative of an even more widespread pattern in the sorts of events that occur in similar circumstances, namely (1):

The standard explanation for (1)—the explanation that appears in physics textbooks and that is taught in a physics classrooms—is a probabilistic explanation. It is standard, in other words, to explain (1) by appeal to (2):

One way to argue for the existence of nomological probabilities, then, would be to take this standard explanation at face value. Scientists assert that (2) explains (1). So (2) explains (1). So the probability referred to in (2) is a nomological probability.

Insofar as you find that argument convincing, great! You are already on board with the existence of nomological probability. But insofar as you are not convinced, there is another way of arguing for the existence of nomological probability, one that takes its cue not just from what scientists say about particular cases like the one described above, but from a more general feature of standard scientific practice. That argument goes like this: assuming that the standard interpretation of quantum mechanical phenomena is correct, if (2) does not explain (1), then nothing does. And (1) describes the sort of pattern such that leaving that pattern unexplained is a serious cost. In particular, the costs associated with leaving (1) unexplained are higher than the costs associated with using (2) to explain (1). So, we ought to adopt a theory according to which (2) explains (1). So, the probability referred to in (1) is a nomological probability. This is the argument that I will focus on here. I will refer to it as the argument for nomological probability.

Why think that if (2) does not explain (1), then nothing does? Because the only other other candidate explanation for (1) is:

And, as I mentioned above, according to the standard interpretation of quantum mechanical phenomena, (3) is false.

Why think that (1) describes the sort of pattern such that leaving that pattern unexplained would be a serious cost? The first step in answering this question is to see that (1) describes what I will call a robust pattern of events—a pattern that holds under a variety of temporal, spatial, and counterfactual conditions. It does not matter whether you consider a sample of carbon-11 before 1950, or between 1950 and 1975, or after 2000, or whether that sample was observed in Austria or Australia, or whether it contained a few additional atoms or a few fewer. The pattern described in (1) is supposed to hold under all of these—and many more—conditions.Footnote ⁵

Why think that it is a serious cost to leave robust patterns, like (1), unexplained? Because it is a widespread and well-established part of scientific practice to view robust patterns in this way. Scientists do not just accept that certain kinds of collisions between particles are reliably followed by certain patterns in the data collected at the Large Hadron Collider—they adopt a theory according to which those patterns are explained by short-lived and otherwise unobservable particles that result from those collisions. They do not just make note of slight periodic oscillations in the light emitted from distant stars—they adopt a theory according to which those oscillations are explained by a nearby exoplanet. None of our best scientific theories, whether at the level of fundamental physics or in the special sciences, leave robust patterns unexplained. Why not? Presumably, because to do so would be considered a serious cost.Footnote ⁶

But it is not enough to establish that (1) describes the sort of pattern such that leaving that pattern unexplained would be a serious cost. We also need to show that the costs associated with leaving (1) unexplained are higher than the costs associated with using (2) to explain (1). The costs associated with using (2) to explain (1) depend on the metaphysical account that one gives of nomological probabilities. I will say more about various candidate accounts in the second part of the paper (sec. 4). For now it will suffice to note that a metaphysical account of nomological probabilities could be costly in two very broad sorts of ways.

First, it could be metaphysically weird. Nomological probabilities could turn out to be the sorts of entities such that, even if they play a role in folk theory or ordinary language, we should strive to eliminate from our metaphysics wherever we can. What counts as a metaphysically weird entity will, of course, depend on your background metaphysical commitments. But the obvious way in which nomological probabilities could turn out to be metaphysically weird by the lights of many contemporary metaphysicians is that they could turn out to be non-Humean—they could involve primitive modal, causal, or dispositional properties. According to Humeans, at least, such properties are metaphysically suspect; insofar as we can do without them, we should.Footnote ⁷

But to be metaphysically weird is not the only way in which an entity can be costly. Entities can also be costly because they are novel—because they cannot be wholly or partly identified with or analyzed in terms of other, more familiar entities. Novel entities, most philosophers agree, are a significant cost; insofar as we can do without them, we should.

Novelty, of course, comes in degrees, and the extent to which an entity is novel may be controversial. But by way of illustrating the difference between metaphysical weirdness and metaphysical novelty, consider various potential propensity analyses of nomological probabilities, according to which nomological probabilities are some sort of disposition or tendency for a certain type of physical system to produce a certain type of result.Footnote ⁸ These accounts are non-Humean, and thus, at least by the Humeans’ lights, metaphysically weird. But they do not incur any especially significant further costs associated with being novel. Dispositions and tendencies are the sorts of things that are familiar from everyday language, from folk theorizing, and from other areas of philosophical inquiry. So even if the propensity interpretation does not provide us with a fully reductive or Humean analysis of nomological probability, it identifies probabilities with a broad, relatively familiar class of metaphysical entities. On a propensity account, then, nomological probabilities might count as weird, but they would not be especially novel.

The sort of analysis that would incur significant costs associated with novelty is an analysis on which nomological probabilities are sui generis entities, which cannot be wholly or partly identified with or analyzed in terms of other, more familiar classes of entities.Footnote ⁹ On a sui generis account of nomological probabilities, the explanatory role that nomological probabilities play would be our main and perhaps only way of grasping what such entities are. We cannot say what nomological probabilities are. We can only say what they do.

All that is by way of spelling out two broad ways in which nomological probabilities might turn out to be costly. The crucial point here is that even if nomological probabilities turn out to be costly in one or both of these ways, we still ought to think that they exist. Let’s focus first on the possibility that the only successful analysis of nomological probabilities involves entities that are supposed by some to be metaphysically weird. It is clearly a widespread and well-established part of scientific practice not to leave robust patterns—like the one described in (1)—unexplained even if the only plausible explanation for them requires introducing metaphysically weird entities. The history of scientific inquiry is replete with examples of scientists allowing into their metaphysics entities that would previously have been thought of as strange or suspect, as long as those entities play a robust explanatory role. Think of Newton positing a force that operates on heavenly and earthly bodies alike. Or Planck suggesting that light exhibits properties associated with both a particle and a wave. Or contemporary physicists positing fundamental particle after fundamental particle in order to explain their experimental results, despite the fact that their standard model is becoming increasingly unwieldy.Footnote ¹⁰

In each of these cases the need to explain a robust pattern in the phenomena trumps concerns about metaphysical weirdness. And with examples like these in mind, it is hard to imagine how the worry that nomological probabilities may not admit of, for instance, a Humean analysis, could justify the attitude that such entities are just too strange to be allowed into our metaphysics. If we use scientific practice as our guide, and if the alternative is leaving a robust pattern unexplained, it seems that we ought to swallow our metaphysical scruples and introduce whatever entities necessary—whether Humean or not.

This may sound surprising. Philosophers who think of themselves in the empiricist tradition, and who place significant constraints on the kinds of entities they allow into their metaphysics, also tend to think of themselves as being more scientifically respectable than their opponents. What the argument above shows is that, in at least some cases, by invoking the very constraints that are supposed to make them scientifically respectable, such philosophers risk running afoul of actual scientific practice. According to Lewis (Reference Lewis1994, 474), the motivation for Humeanism is the desire to, “resist philosophical arguments that there are more things in heaven and earth than physics has dreamt of.” But surely those who think that we should resist philosophical arguments that there are more things in heaven and earth than appear in our best physics should also think that we should resist philosophical arguments that there are fewer things in heaven and earth than appear in our best physics. As Maudlin (Reference Maudlin2007a, 77) puts it, “Philosophical accounts that force upon us something that physics rejects ought to be viewed with suspicion. But equally suspect are philosophical scruples that rule out what physics happily acknowledges.” And, as the above argument shows, Humeanism runs the risk of turning on just such scruples.

So the argument for the existence of nomological probabilities goes through even if the only successful analysis of nomological probabilities turns out to be non-Humean (or otherwise (supposedly) metaphysically weird). The same is true even if the only plausible account of nomological probabilities turns out to be one on which such probabilities are also novel. After all, scientists not only allow metaphysically strange entities into their theories as long as those entities explain robust patterns that would otherwise go unexplained—they also allow sui generis entities in order to play the same role.

Here is an example.Footnote ¹¹ Astronomers have observed that the rate of expansion of the universe is accelerating, and physicists have ruled out all possible explanations for this acceleration that are based on those entities to which our best scientific theories are already committed. Given that no other explanation is in the offing, physicists have, by and large, reluctantly decided to posit a new sort of entity—dark energy. What is dark energy? Physicists cannot tell us. They can only tell us what dark energy does. It causes the rate of expansion of the universe to accelerate.

Now it may be that in the future we will learn more about what dark energy is—we will be able to do more than identify its explanatory role. Indeed it may be that one of the key desiderata of future theories is that they be able to provide further insight along such lines. But the inclusion of dark energy in our best scientific theories does not appear to be conditional on that kind of future success. Scientists are, by and large, confident that dark energy exists—if it does not, there is no explanation for the rate at which the expansion of the universe is accelerating, and that is the sort of fact for which there must be some sort of explanation. If it turns out that we need to accept dark energy as a sui generis entity, so be it.

My claim is that philosophers should be willing to take the same attitude toward nomological probabilities. Given the crucial explanatory role that they play, we should be more convinced that nomological probabilities exist than of any particular story about what they are. In fact, we should be open to the idea that there is no further story about what they are. It may turn out that we cannot say what nomological probabilities are; we can only say what they do. Nomological probabilities are those objective probabilities that explain associated frequencies in the way described above.

Let’s recap. I have argued that on the standard interpretation of quantum mechanical phenomena, either (2) explains (1) or nothing does. And I have argued that the following is an important principle governing standard scientific practice.

Insofar as we are naturalists (in the sense described in the introduction), it follows that we ought to use the pattern-explanation constraint as a guide to metaphysical theory choice as well. So even if the only plausible analysis of nomological probabilities involves entities that are supposed to be in some sense metaphysically weird—even if, for instance, the only plausible analysis is non-Humean—we still ought to accept that there are nomological probabilities. Indeed, even if the only plausible account of nomological probabilities is one on which they are sui generis, we ought to accept that there are nomological probabilities.Footnote ¹²

In section 4, I will demonstrate one way in which this argument should shape future work on the metaphysics of chance. But before I do, let me briefly address three objections that might be thought to create difficulty for the argument for nomological probability as I have presented it above.

3. Three Objections

The first two objections I will consider are to do with my appeal to the standard interpretation of quantum mechanical phenomena in the argument for nomological probability. That argument relied on the premise that if (2) does not explain (1), then nothing does. The reason I gave in support of that premise was that the only other plausible explanation of (1) was (3), and according to our best scientific theories—in particular, according to the standard interpretation of quantum mechanical phenomena—(3) is false.

Consider first the fact that the standard interpretation of quantum mechanical phenomena—the one taught in physics textbooks—is not obviously one of our best scientific theories. In particular, the dynamics posited by that standard interpretation is deeply problematic. It relies on a fundamental distinction between measurement processes, which cause a certain type of evolution, called collapse, and nonmeasurement processes, which do not. Doesn’t this undermine the argument for the existence of nomological probabilities?

In fact, it does not. The argument for nomological probabilities goes through even if one eschews the standard interpretation and adopts instead some version of the Ghirardi-Rimini-Weber (GRW) interpretation, according to which there is no fundamental distinction between measurement and nonmeasurement process; instead collapses are the result of a spontaneous, indeterministic process. On a GRW interpretation, the fundamental laws are still indeterministic and (3) is still false. And versions of the GRW interpretation are clearly among the live candidates for our best interpretation of quantum mechanical phenomena.

This gives rise to a related objection, however. Although versions of the GRW interpretation are clearly among the live candidates for our best interpretation of quantum mechanical phenomena, they are not the only live candidates. Bohmian mechanics is also a live candidate, and according to Bohmian mechanics, (3) is true.Footnote ¹³

How does the argument for the existence of nomological probability fare once we acknowledge that Bohmian mechanics is also a live candidate for our best scientific theory of quantum mechanical phenomena? It still establishes a substantive conclusion—it establishes that there are nomological probabilities in the region of logical space that is currently compatible with our best scientific theories. And it still follows from the argument that our metaphysics needs to leave room for the existence of nomological probabilities—we ought not take on commitments from which it would follow that there are no such things, or on which such things would be especially difficult to countenance or understand. To do so would be to settle, by stipulation, what is otherwise an open scientific question. So the existence of Bohmian mechanics as an alternative interpretation does not undermine the importance of the argument for nomological probability.

Those were the first two objections. The third objection that I will consider is of a more general philosophical nature. So far I have said nothing whatsoever about a theory of explanation. But surely we cannot establish that (2) explains (1) without first establishing under what conditions one thing explains another.

In fact, there are good reasons to think that we can do just that. For one thing, since satisfying the pattern-explanation constraint warrants the introduction of weird (or even novel) entities in order to explain robust patterns, positing that (2) explains (1) places only minimal constraints on one’s theory of explanation. If there is no way of plausibly interpreting (2) as being both explanatorily robust and about familiar entities, one can simply reinterpret (2) as in fact being about some novel entities instead. But note that it is also plausible that the considerations above support not only the introduction of novel entities, but also the introduction of novel explanatory relations. Insofar as our favored theory of explanation gives us no reason for thinking that (2) explains (1) on any interpretation of (2), then we might simply conclude that there exists a novel sort of explanatory relation, which was previously unrecognized in our theorizing, and which holds between (2) and (1). Such a novel explanatory relation would surely be a cost.Footnote ¹⁴ But it appears to be precisely the sort of cost that is trumped by the pattern-explanation constraint. We do not know what dark energy is, so we cannot be sure that the way in which it explains the accelerating rate of expansion of the universe is compatible with any particular theory of explanation. But that does not seem to worry the scientists who posit dark energy. They are more convinced of the fact that something must explain the accelerating rate of expansion of the universe than they are of any particular story about what explanation is. I suggest that as metaphysicians we take the same attitude toward the explanatory role of nomological probability. We should be more convinced of the fact that something explains (1) than we are of any particular story about what explanation is.

4. What Are Nomological Probabilities?

In this section, I will illustrate one way in which the argument in section 2 should shape future work on the metaphysics of chance. In particular, I will present a challenge to one group of analyses of objective probability—Humean analyses—understood as analyses of nomological probability.

Humeans think that all that exists at the fundamental level is the Humean mosaic—the distribution of categorical properties across space and time.Footnote ¹⁵ Insofar as there are facts about what caused what, or what could have happened, or what was disposed to occur under various circumstances, those facts depend, in some important sense, on the facts about the mosaic.Footnote ¹⁶

Insofar as Humeans think that everything depends on the mosaic, they must think that the nomological probabilities depend on the mosaic. As a result, Humeans face an immediate worry. In broad strokes, the worry is this: Humeans think that the nomological probabilities depend on the mosaic. But they must—given the argument in section 2—also think that the nomological probabilities explain features of that mosaic. And how can it be that nomological probabilities explain features of the mosaic if the nomological probabilities themselves depend on the mosaic?

Call this the explanatory challenge. Surprisingly, although versions of this challenge are familiar from the debate over Humean analyses of laws of nature, whether and how these challenges extend to the debate over Humean analyses of chance has received relatively little attention.Footnote ¹⁷ Different versions of the explanatory challenge will target different Humean analyses of objective probability. The most straightforward Humean analysis is an actual frequency analysis, according to which the objective probability of an event of type E happening in a situation of type S is just the relative frequency with which events of type E actually occur in situations of type S.

Understood as an analysis of nomological probability, the actual frequency analysis faces a particularly serious version of the explanatory challenge. After all, according to that sort of analysis, (2) is equivalent to (1). So (2) cannot explain (1). Nothing can explain itself.

The actual frequentist may be tempted to claim that even if (2) does not explain (1) it explains parts of the pattern described in (1). For instance, it explains (4):

But this is a non sequitur. If the discussion in section 2 is correct then it does not matter whether the actual frequentist can show that (2) explains (4). Given the argument in section 2, they also need to show that (2) explains (1). Because if (2) does not explain (1) then nothing does, and (1) describes the sort of pattern that requires an explanation.

Of course, the actual frequentist might suggest that the argument in section 2 was mistaken. It is not patterns like the one described in (1) that require explanation at all. What require explanation are patterns like the one described in (4). But surely that cannot be correct. To say that (4) requires explanation while (1) does not is to say that whether a pattern requires explanation depends on whether we happen to have observed all instances of that pattern. But whether we happen to have observed every instance of some particular pattern is highly accidental. It depends on where we happen to have constructed our experiments, when we happen to have run them, whether we happen to have been paying attention on any particular occasion, and so on. And surely whether some otherwise robust pattern requires explanation cannot be so accidental.Footnote ¹⁸

That the actual frequency analysis faces this sort of explanatory challenge has been noted before (see Hájek Reference Hájek1996). What is important to observe here is the strength of the challenge. It looks like one cannot give an actual frequency analysis of nomological probability. Where does that leave those who are otherwise inclined toward such analyses? Should they give up on the existence of nomological probability? They should not. Should they attempt to engage the rest of us by listing the other virtues of their analysis or the costs of the relevant alternatives? Again, no. The argument in section 2 shows that we should think that there are nomological probabilities, pretty much regardless of the costs associated with accepting such entities into our metaphysics. Insofar as you are a committed actual frequentist, then, you cannot simply dismiss the explanatory challenge outlined above. You must meet it. Until you have done so, your position is untenable.

Of course, many philosophers already think actual frequentism is untenable. The explanatory challenge becomes more interesting, therefore, when it targets other, more prominent Humean analyses of objective probability.Footnote ¹⁹ Perhaps the most prominent such analysis is the best systems analysis or the BSA.Footnote ²⁰ The BSA is first and foremost an analysis of laws. It says that (i) the best system is the set of propositions describing the Humean mosaic that best balances simplicity and strength and (ii) the laws are those propositions that appear in the best system.

In order to extend the BSA to objective probability, one takes all propositions that describe the Humean mosaic and supplements those propositions with an uninterpreted function P(x) that assigns to each proposition a number between 0 and 1. One then considers a further theoretical virtue alongside simplicity and strength—the virtue of fit. A system is better fitting to the extent that P(x) is higher for true propositions and lower for false propositions, and the best system is the one that best balances simplicity, strength, and fit. The function P(x) is then interpreted as the objective probability function.Footnote ²¹ Insofar as we are hoping to give a best systems analysis of nomological probability, the function P(x) is interpreted as the nomological probability function.

The BSA clearly avoids the specific version of the explanatory challenge that faced the actual frequency account. On the BSA, nomological probabilities are not actual frequencies, they are magnitudes that play a certain sort of role in the best system. So the advocate of the BSA is not committed to anything explaining itself. But there is still a nearby challenge. This challenge arises because although (on the BSA) probabilities are not equivalent to frequencies, the reason why some probability function ends up being a part of the best system will, at least in paradigm cases, be because of the existence of some corresponding frequency. Insofar as there is a stable, long-run relative frequency in the mosaic, a probability function that assigns a probability close to the relative frequency will in general yield quite a bit of simplicity with little trade-off in terms of fit.

Consider, for instance, the relation between (2) and (1). Although (2) is not equivalent to (1) (according to the BSA), it is still the case that (2) is true because (1) is true—it is because the frequency described in (1) obtains that the probability function that generates (2) is a part of the simplest, strongest, and best-fitting system. And if p is true because q is true, then surely in some sense q explains p. But then it follows that (1) explains (2) and that (2) explains (1). And this seems to be a worrisome sort of explanatory circularity. How can it be that q explains p and that p explains q?Footnote ²²

It is important to note that the point generalizes beyond the particular worry about circular explanation. We already established that, according to the Humean, facts about chances must depend on the mosaic, in some sense. More specifically, given the best systems analysis, it seems natural to say that (2) depends on (1). But it is also natural to think that explanatory relations (especially explanatory relations of the type that scientists are after) track dependence relations. When we introduce (2) to explain (1), then, we are saying that (1) depends on (2). But then, according to the Humean, (2) depends on (1) and (1) depends on (2). And that seems to be a worrisome sort of symmetric dependence. How can it be that q depends on p and that p depends on q?

There are two general sorts of moves that an advocate of the best systems analysis might make in response to these challenges. First, they could try to insist that (1) does not explain (2), or that (2) does not depend, in any sense, on (1). But that seems, at the very least, difficult to square with both the Humean motivation and the details of the BSA. Second, they might try to embrace the relevant sorts of explanatory circularity or symmetric dependence and insist that it is just fine for p to explain q while q explains p and for q to depend on p while p depends on q.Footnote ²³ It is not, given the argument presented in section 2 of this paper, an option to simply claim that (2) does not explain (1), or that (1) does not, in any sense, depend on (2).

One way to make the second strategy described above (embracing circular explanation and symmetric dependence) more palatable is to follow a suggestion put forward by Loewer (Reference Loewer2012) in response to the explanatory challenge that targets advocates of the best systems analysis of laws. According to that challenge, since Humean laws depend on the mosaic—and thus on the instances of those very laws, they cannot in turn explain those instances. Loewer’s suggestion is that Humeans (i) distinguish between two types of explanation—metaphysical explanation, on the one hand, and scientific explanation, on the other, and then (ii) claim that the mosaic metaphysically explains the Humean laws, while the laws scientifically explain the mosaic. Extended to Humean analyses of chance, the suggestion would be that (1) explains (2) in the sense that it metaphysically explains (2), while (2) explains (1) in the sense that it scientifically explains (1). And that sort of circular dependence, according to Loewer, is not in fact problematic.Footnote ²⁴

Is this distinction between metaphysical and scientific explanation legitimate? Is it the sort of distinction that a Humean should be happy to accept? Does it suffice as a response to the worry about circular explanation? Can it be extended to ameliorate the worries about symmetric dependence? Perhaps. But even if the answer to all of these questions is yes, there is good reason to be suspicious that the sort of promissory note that the Humeans are issuing here can in fact be cashed. Many of us are operating with a background view on which scientific explanation—at least in paradigm cases—is some kind of causal explanation and on which metaphysical explanation tracks grounding relations, which—whatever else they are—are noncausal.Footnote ²⁵ So many of us are operating with a background view on which there is a prima facie reason to think that there is a substantive difference between scientific explanation and metaphysical explanation—one is causal, while the other is not. But it is not at all obvious that this sort of straightforward distinction maps onto the case at hand. In particular, it is not at all obvious that the Humean should think that nomological probabilities cause frequencies in any straightforward sense. Perhaps such a view is plausible on a propensity analysis, according to which probabilities just are causal dispositions or tendencies. But can a Humean tell a similar story? Should she want to?

This point is by no means decisive, but it does serve to shift the burden on to those who wish to defend something like Loewer’s move. Perhaps one could say that nomological probabilities mediate causal relations in the mosaic in some more sophisticated sense. Perhaps we can tell some alternative story of scientific explanation that makes sense of the relation between (2) and (1) and maintains a robust distinction between scientific and metaphysical explanation. But until we see that sort of account, we ought to be skeptical of Loewer’s way of accepting circles of explanation, and of any analogous attempt to respond to worries about symmetric dependence.

In any case, my goal here is not to argue that there are decisive reasons for thinking that any successful analysis of nomological probability must be non-Humean. My goal is to show that those who wish to give a Humean analysis of nomological probability have their work cut out for them and to emphasize that such work is not work that can be postponed or set aside. The argument in sections 2 and 3 of this article shows that we should think that there are nomological probabilities pretty much regardless of the costs associated with accepting such entities into our metaphysics. Insofar as you are a committed Humean, then, you cannot simply dismiss the explanatory challenge outlined here. You must meet it head on. Until you have done so, your position is untenable.

5. Conclusion

In sections 2 and 3, I argued that insofar as we are naturalists—insofar as we take standard scientific methodology as a good guide to successful inquiry into what the world is like—we ought to believe in the existence of nomological probabilities. This result holds even if it turns out that we cannot give a Humean analysis of such probabilities or if the only available analysis of such probabilities appeals to entities that are supposed to be metaphysically weird in some other sense. Indeed the result holds even if it turns out that we cannot give any robust metaphysical analysis of such probabilities at all and instead must accept them as sui generis entities. Scientists are willing to accept metaphysically strange and novel entities into their theories when the alternative is leaving a robust pattern unexplained. Metaphysicians should be too.

In the second part (sec. 4), I presented a challenge for one common group of metaphysical analyses of objective probability—Humean analyses—understood as candidate metaphysical analyses of nomological probability. Perhaps this challenge can be met, but at the very least advocates of Humean analyses of nomological probabilities have quite a bit of work ahead. And insofar as that work cannot be completed we ought to take seriously the possibility that we will need to accept an alternative analysis of nomological probabilities.

The reader will note that I have not attempted to answer the following question: insofar as a Humean analysis of nomological probability fails, what sort of non-Humean analysis should we give? A plausible initial starting point for such an analysis would, of course, be various propensity analyses. But the point I am making here is a more general one. Even if you think that various propensity analyses ultimately fail,Footnote ²⁶ you should not give up on the notion of nomological probability. Even if you think that there is no successful analysis of nomological probabilities at all, and that they are instead wholly sui generis entities, you should still think that they exist.

By way of conclusion, let me say something about the particular methodology I have employed above. It can sometimes seem like metaphysicians who are interested in concepts like probability, lawhood, time, or space—concepts that clearly arise in our best scientific theories—face a choice between taking everything that those theories say at face value or being scientifically revisionary. But as the above argument demonstrates, there is a way of doing scientifically respectable metaphysics that goes beyond the former but falls short of the latter. On this way of doing metaphysics we take standard scientific practice as a model and paradigm instance of successful inquiry into what the world is like, while leaving room for the possibility that the particular theories that scientists endorse might be problematic.

In this paper I have argued that in scientific theorizing it is standard practice to be more certain that there are things that play certain types of explanatory roles—in particular that there are things that explain certain sorts of patterns—than of any particular metaphysical constraints on what sorts of things there are in the world. The same should be true of metaphysical theorizing. And insofar as it is, our metaphysics ought to include nomological probabilities.