1. Introduction

To paraphrase Butler (Reference Butler1736), chances are the very guide to life; ordinarily, an agent’s beliefs about the objective chances of various outcomes determine her rational credences in those outcomes. Lewis (Reference Lewis1980) famously offered a codification of the relationship between chance and credence. Roughly, Lewis’s principle implies that if a rational agent (i.e., an agent whose opinions can be represented as having come from a reasonable initial credence function by conditionalizing on her total evidence) believes an outcome to have a given chance of occurring, then her credence in that outcome is equal to that chance. For example, if I believe the chance of rain tomorrow is .7, then my credence in rain tomorrow (if I am rational) is .7.

The principle holds with one exception. A reliable crystal ball might predict that there will be no rain tomorrow despite there now being a high chance that it will rain tomorrow. If I believe that the crystal ball is reliable, I should put in my lot with the ball’s prediction and not set my credence to what I believe the chance of rain to be. Reliable crystal balls provide a kind of information about future outcomes that, to use Lewis’s metaphor, does not “go by way of” the chances of those outcomes. Every such case (where an agent ought to ignore the chances when forming her credences), argued Lewis, is one in which an agent has “inadmissible” information.

Lewis named his formulation the Principal Principle (PP), since it describes perhaps the essential feature of objective chance, and argued that any adequate theory of chance must explain why we ought to form our credences in accord with the PP. Although there is debate about whether the PP best captures the relationship between chance and credence (e.g., Hall Reference Hall1994; Lewis Reference Lewis1994; Nelson Reference Nelson2009; Meacham Reference Meacham2010), many philosophers (e.g., Salmon Reference Salmon1967; Hájek Reference Hájek2007) agree with Lewis that theories of objective chance are burdened with showing how it is that chances are the very guide to life. But this has proved a difficult burden to meet. No theory of chance is widely agreed to give a satisfactory explanation of the PP (or of some appropriately similar principle), and some philosophers (e.g., Strevens Reference Strevens1999) doubt that such an explanation can be found.Footnote ¹

In what follows, I argue for an alternative approach to explaining the PP. To begin, we can distinguish between two distinct aspects of the PP that call for explanation. The first is the particular quantitative relationship it implies between a rational agent’s credences regarding the chances of various outcomes and her credences in those outcomes. If a rational agent is certain that the chance of some outcome is n (a real number between 0 and 1 inclusive), then her credence in that outcome is equal to n (so long as she has only admissible information). But why does rationality require that the values be equal?

A second aspect of the PP that calls for explanation is more basic. Never mind wondering why beliefs about chances determine rational credences in the particular way described by the PP; why do beliefs about chances determine rational credences in any way at all? There is no particular credence in rain, for example, that I am rationally obliged to have, given that I believe there are storm clouds overhead. But unlike my opinions about storms clouds, my opinions about the chance of rain (if I have any such opinions) rationally compel me to have a very particular credence that it rains, no matter what other admissible information I have. I call this second aspect of the relationship between chance and credence “Swamping”:

Swamping is at the heart of the relationship between chance and credence. Chance is aptly described as the very guide to life, precisely because a rational agent’s credence in an outcome’s chance “swamps” all of her other (admissible) evidence about that outcome.Footnote ³ Explaining Swamping, then, is a significant step toward explaining the PP. Unfortunately, theories of chance have had no better luck at explaining Swamping than they have had at explaining the PP.Footnote ⁴

Perhaps we have failed to discover an explanation of Swamping (and ultimately, the PP) because we have been looking in the wrong place. The goal of this essay is to show how Swamping might be explained without appeal to any particular metaphysics of chance. On the view I develop, Swamping holds not because of what chances are but rather because of the explanatory role they play for the rational agents who believe in them.

I proceed as follows. In section 2, I introduce a principle that articulates one aspect of the relationship between beliefs about scientific explanations of outcomes and rational expectations in those outcomes. I call this principle the Hypothetical Explanation-Credence Principle (HCP). In section 3, I give an argument from HCP to Swamping, and I identify two problems this argument faces: the admissibility problem and the generalization problem. The admissibility problem arises because HCP, like Swamping, applies only to cases in which rational agents have no inadmissible information. In section 4, I sketch a novel account of admissible information and argue that, although this account is imprecise, it plausibly solves the admissibility problem. The generalization problem arises because Swamping is an instance of HCP only if objective chances play a certain explanatory role for rational agents. In section 5, I articulate a novel account of this explanatory role that I call the “causal chain account.” Rather than argue directly for the causal chain account, I show that it solves the generalization problem. I conclude in section 6 by arguing that, although my strategy for explaining Swamping is consistent with a wide array of metaphysical theories of chance, some well-known theories are a worse fit with my strategy than are others.

2. The Hypothetical Explanation-Credence Principle

According to Swamping, beliefs about chances determine rational credences in typical cases in which a rational agent is making a prediction (i.e., cases in which a rational agent has no “inadmissible” information). If our goal is to better understand why Swamping is true, a good way to start is by exploring what other kinds of beliefs stand in a similar relationship to our rational credences in typical cases of prediction.

Suppose that a rational agent is predicting whether I will get a cold. If she has standard background beliefs about colds, she bases her expectations about my future health on factors such as whether I have recently had contact with sick colleagues, the frequency with which I disinfect my hands, and so on. She bases her expectations on these particular factors (rather than on, say, whether I drive a Volkswagen) because she believes that these factors explain why I get a cold if I do. The moral I draw from this example and others like it is that beliefs about explanations (typically) play an important role in determining rational credences about the targets of those explanations. The HCP is my attempt to articulate at least one aspect of that role.

Two points of clarification will be helpful for motivating HCP. First, it is notoriously difficult to give an analysis of “scientific explanation” that is neither too restrictive nor too broad. Thankfully, we can largely do without any particular analysis for my present purposes. I am interested in the relationship between beliefs about explanation and rational credence: independent of the truth of such beliefs. Accordingly, I assume as little as possible about the correct theory of scientific explanation in motivating HCP.Footnote ⁵

Second, our rational agent need not believe that, say, my recent contact with sick colleagues explains my cold in order for her confidence in the former to influence her confidence in the latter. The fact that P explains Q (at least on one common understanding of “explains”) guarantees that P and Q obtain, but our agent may be skeptical that I will get a cold precisely because she’s skeptical that I have had contact with sick colleagues. For her confidence that I get a cold to be constrained by her confidence that I have had contact with sick colleagues, it is sufficient that she believe that if it is true that I have had contact with sick colleagues and if it is true that I get a cold, then my contact with my colleagues explains my cold. In other words, it is enough that she believes my contact with sick colleagues is a “hypothetical explanation” of my cold:

Our example of the rational agent who is predicting whether I get a cold can now be made more precise. Our agent has various degrees of confidence in propositions of the form “X is a hypothetical explanation of P.”Footnote ⁷ She may, for example, be very confident that my recent contact with sick colleagues is a hypothetical explanation of my cold and sure that my car’s make is not a hypothetical explanation of my cold.Footnote ⁸ Furthermore, she has various degrees of confidence that X’s she takes to be (or not to be) hypothetical explanations obtain. She may be unsure whether I have had contact with sick colleagues, for example, but very confident that I own a Volkswagen.

Our agent’s expectations about whether I get a cold are sensitive to her opinions about which hypothetical states of affairs are hypothetical explanations of my cold in tandem with her opinions about which hypothetical states of affairs actually obtain. Because she is confident that my contact with sick colleagues is a hypothetical explanation of my cold, for example, her expectation about whether I get a cold may vary with her credence that I have had contact with sick colleagues. Similarly, if she is confident that I have had contact with sick colleagues, her expectation about whether I get a cold may vary with her credence that my contact with sick colleagues is a hypothetical explanation of my cold. Finally, because she is sure that my car’s make is not a hypothetical explanation of my cold, her credence that my car is a Volkswagen makes no difference to her expectation about my cold.

This example is one instance of a broad range of cases in which a rational agent’s expectations about P are sensitive to her opinions about propositions of the form “X is a hypothetical explanation of P” and “X obtains.” In such cases, her credence in P may change when she acquires new information that changes her opinions about whether X is a hypothetical explanation of P or about whether X obtains. However, I will argue that once a rational agent is certain that a particular X obtains and certain that X is a hypothetical explanation of P, new information typically makes no difference to her credence in P. The rare cases in which a rational agent does change her credence in P despite being certain that X obtains and that X is a hypothetical explanation of P are ones in which the agent has inadmissible information. This claim is the germ of a principle that I call the Hypothetical Explanation-Credence Principle:

Roughly, HCP says that if a rational agent is certain that X is a hypothetical explanation of P and she has no inadmissible information, the value of her credence in P given that X obtains is independent of the content of her admissible body of evidence.

Is HCP true? That depends on what it is for information to be “admissible.” But if “admissible” turns out to have the same meaning when used in HCP as it does when used in Swamping, then HCP implies that a rational agent’s admissible evidence about P is swamped by her beliefs about hypothetical explanations of P in much the same way that a rational agent’s admissible evidence about P is swamped by her beliefs about the chance of P.

3. From HCP to Swamping

Consider an instance of HCP in which X is that the chance of P is equal to x (a real number on the unit interval). Suppose some rational agent is certain that the chance of P is equal to x and certain that the chance of P (i.e., the fact that P has a given chance) is a hypothetical explanation of P.Footnote ⁹ Finally, assume she has only “admissible” information (whatever that might mean). Then, by HCP, her credence in P is determined by her initial credence function C (i.e., her credence function before updating on X and her admissible body of evidence) and the content of X and P. Because X is the proposition that the chance of P is equal to x, X specifies the content of P. So, in this special case of HCP, C and the content of X determines the agent’s credence in P. Holding fixed whatever C with which our agent began, it follows that her credence in P is determined by her opinions about the chance of P.Footnote ¹⁰ So, in this special case, HCP implies that an agent’s opinions about chances swamp all of her other admissible evidence.

HCP also has implications for agents who distribute their credences over (a potentially infinite) number of (mutually exclusive and collectively exhaustive) propositions about the value of the chance of P. If a rational agent is certain that each such chance proposition is a hypothetical explanation of P and she has only admissible information, then HCP and the theorem of total probability imply that her credence in P is determined by C, the content of each chance proposition, and her credence in each chance proposition. In this case, HCP (like Swamping) implies that admissible information has no impact on her credence in P except by way of changing her credence in some chance proposition.

What would it take for this for this argument to apply to every case to which Swamping applies? First, “is admissible” as used in HCP must be coextensive with “is admissible” as used in Swamping in every case in which X is that the chance of P is equal to x.Footnote ¹¹ To avoid confusion, let “admissible_PP” express the concept that “admissible” expresses as it appears in Swamping (and the PP). Let “admissible_HCP” express the concept that “admissible” expresses as used in HCP. On one hand, HCP requires a somewhat narrow reading of “admissible_HCP,” since potential counterexamples to HCP grow as admissibility_HCP is broadened.Footnote ¹² In particular, “admissible_HCP” must be narrow enough to exclude all information that is inadmissible_PP in every case in which X is that the chance of P is x. Otherwise, canonical cases of inadmissible_PP information (e.g., the predictions of a reliable crystal ball) are counterexamples to HCP. On the other hand, “admissible_HCP” must not be given too narrow a reading. Swamping is meant to apply to all typical cases of prediction, and so all information that we typically have when we make predictions is admissible_PP.Footnote ¹³ Similarly, in cases in which X is that the chance of P is x, “admissible_HCP” must be broad enough to include all information that is had in typical cases of prediction. The “admissibility problem” is to provide an account of admissible_HCP information that is narrow enough to protect HCP from spurious counterexamples but broad enough for HCP to apply to every case to which Swamping applies.Footnote ¹⁴

Second, the argument from HCP to Swamping seems to generalize only to cases in which a rational agent is certain that the chance of an outcome is a hypothetical explanation of that outcome. But, as we will see in section 5.1, there are cases in which a rational agent obeys Swamping but does not regard the chance of an outcome to be a hypothetical explanation of that outcome. The “generalization problem” is to provide an account of the explanatory role that objective chances play that allows the argument from HCP to Swamping to generalize to every case to which Swamping applies.

At this stage, some readers will doubt that there is a solution to the generalization problem, on the grounds that Swamping cannot be accounted for by the explanatory role that chances play. I consider two objections that might motivate such skepticism.Footnote ¹⁵ The first is simply that there are rational agents who do not believe that objective chances are hypothetical explanations.Footnote ¹⁶ Chances seem to play no explanatory role for such agents, and so HCP does not apply to them. One reason why a rational agent might deny that objective chances are hypothetical explanations is that she believes that, although chance outcomes have hypothetical explanations, objective chances are merely parts of those hypothetical explanations. Among such agents, distinguish those who believe that objective chances (by themselves) are not explanatory from those who believe that objective chances (by themselves) merely fail to satisfy some ideal of explanation. For HCP to apply to the latter kind of agent, “hypothetical explanation” must be understood as referring to minimally adequate, rather than ideal, explanations. I suspect that HCP remains plausible even when so interpreted, although an argument to that effect would require a delicate discussion of minimally adequate explanations. The greater difficulty is with the former sort of agent, who believes that chances are parts of hypothetical explanations but denies that they are (by themselves) even minimally adequate hypothetical explanations. I doubt that such agents have sufficient conceptual grasp of explanation and objective chance to count as believing in objective chances, but defending that claim would require a serious discussion of scientific explanation and its conceptual ties to objective chance. I aim only to show that if objective chances play a certain explanatory role for the rational agents who believe in them, then Swamping follows from HCP. That said, if I meet this goal then that is some (admittedly defeasible) evidence that chances do indeed play that explanatory role.

Another reason why a rational agent might deny that objective chances are hypothetical explanations is a commitment to the view that chance outcomes have no explanations. I find this view implausible, but it is neither irrational nor conceptually confused. The possibility of such agents motivates our second objection: it seems that Swamping is true of agents who are not certain of any hypothetical explanations.Footnote ¹⁷

Consider a rational agent who wonders whether a coin will land heads on a particular toss. Suppose she is certain that the relative frequency with which this coin actually lands heads (over the entire span of its existence) is .5. Further, suppose she doubts that this relative frequency, or anything else, is a hypothetical explanation of the coin’s landing heads on a given toss. Although it seems that HCP does not apply to such an agent (since there is nothing she believes to be a hypothetical explanation of the coin’s landing heads), her credence in heads is (plausibly) determined by her certainty that the actual relative frequency of heads is .5. Beliefs about actual relative frequencies obey the analogue to Swamping that we get by replacing “chances” with “actual relative frequencies.” It is implausible that HCP is a genuine explanation of Swamping if it is silent about such closely related principles.

Despite appearances to the contrary, HCP does apply to this case. To see why, start by imagining that we are about to draw a ball from an urn filled with an equal number of black and red balls. How confident should we be that a red ball will be drawn? Simply knowing that half the balls are black and half are red is not, I claim, enough to answer that question; instead, we need some further information about chances. For example, if the chance that a given red ball is picked is greater than is the chance that a given black ball is picked, our confidence that a red ball is picked should be greater than .5. If, instead, all of the balls have the same chance of being picked, our confidence that a black ball is selected should be .5. If we have no information about the chances, we are allowed to proceed as if all of the balls have the same chance of being picked.Footnote ¹⁸ But knowing that half the balls are black and half are red, without any further assumption, is insufficient to determine our credence that the drawn ball is red.

Observing one of a series of actual coin tosses, half of which land heads and half of which land tails, is importantly like drawing one ball from an urn that contains an equal number of red and black balls. Just as knowing that half the balls in the urn are red does not (by itself) determine our credence that the drawn ball is red, knowing that half the coin tosses land heads does not (by itself) determine our agent’s credence that the next toss lands heads. Instead, if she does not know each toss’s chance of being the one she observes, she proceeds as if each toss has the same chance of being the one she observes. If she is certain that chances are hypothetical explanations, HCP entails that her beliefs about the chances of observing each coin toss determine her credences about which coin toss she will observe. In turn, those credences combine with her certainty that half the tosses land heads to determine her credence in heads. So, HCP helps to explain why rational agents set their credences in accord with actual relative frequencies even if those agents do not believe that actual relative frequencies are hypothetical explanations.

What of a rational agent who believes that actual relative frequencies are not hypothetical explanations and that chances just are actual relative frequencies? HCP does not apply to such an agent, since she is not certain that chances are hypothetical explanations. Perhaps that result is not so bad. The moral of the analogy between the urn case and the coin case is that beliefs about actual relative frequencies do not, by themselves, determine our rational credences. It is not clear that agents for whom beliefs about chances are just more beliefs about actual relative frequencies (and who deny that actual relative frequencies are hypothetical explanations) obey Swamping.Footnote ¹⁹

Furthermore, it is independently plausible to me that Swamping and the PP do not apply to agents who believe that chance outcomes are inexplicable. For example, we know precisely how confident I should be that a fair coin lands heads, but we have no idea what my precise confidence should be that my wallet inexplicably begins to sing. I submit that part of the reason there are definitive answers to questions about how confident I should be in the outcomes of coin tosses, but no definitive answers to questions about how confident I should be that inanimate objects begin to sing, is that I am certain of hypothetical explanations of the former but regard the latter as inexplicable. For an agent who believes that chance outcomes are as inexplicable as I believe singing wallets to be, perhaps there are no answers (beyond mere Bayesian updating) to questions about how confident to be in the occurrence of chance outcomes.

4. The Admissibility Problem

According to HCP, a rational agent who is certain that X and that X is a hypothetical explanation of P does not change her credence in P when she learns new admissible_HCP information. Recall that, in order to get from HCP to Swamping, we need an account of admissible_HCP information that is narrow enough to protect HCP from spurious counterexamples but broad enough for HCP to apply to all typical cases of prediction (i.e., to all cases to which Swamping applies).

Suppose a rational agent is certain that I have had contact with sick colleagues and that having contact with sick colleagues is a hypothetical explanation of my cold. Does her expectation about my cold change if she learns new information? Unsurprisingly, that depends on the kind of information she learns.

A rational agent’s certainty that X and that X is a hypothetical explanation of P screens off a great deal of information that might otherwise make a difference to her credence in P. Learning that I attended a department meeting, for example, makes no difference to our agent’s credence about whether I get a cold since it is evidence that I get a cold only in virtue of being evidence that I have had contact with sick colleagues. Similarly, learning that cold viruses are passed through touch makes no difference to our agent’s credence that I get a cold, since she is already certain that contact with sick colleagues is a hypothetical explanation of my cold. So, if a rational agent is certain that X and that X is a hypothetical explanation of P, then the following kinds of information make no difference to her credence in P:

• (1) Information that is evidence about P only in virtue of being evidence about X.

• (2) Information that is evidence about P only in virtue of being evidence about whether X is a hypothetical explanation of P.

I have no precise account of “evidence” to offer.Footnote ²⁰ The rough idea is that information is evidence about some P for an agent exactly in case she should, given her other opinions, take that information to confirm or disconfirm P.Footnote ²¹ The sense of “evidence” I employ, then, is always relativized to the background beliefs of a particular agent. Since this understanding of evidence is central to my account of admissible_HCP information, whether information is admissible_HCP, on my view, is similarly relativized.

Notice that not all evidence about P is screened off by a rational agent’s certainty that X and that X is a hypothetical explanation of P. If, for example, our agent learns about events that occur subsequent to my catching a cold, she may change her credence that I get a cold. Furthermore, learning about events that occur before my catching a cold, such as the predictions of a reliable crystal ball, may change her credence that I get a cold. However, we do not have access to such information in typical cases of prediction, and so it can safely be regarded as inadmissible_HCP.

What happens if a rational agent learns that something other than X, say Y, might be a hypothetical explanation of P? If she is certain that there is a unique explanation of P, then her credence in P does not change. Because she is certain that X and that X is a hypothetical explanation of P, and certain that there is only one explanation of P, she must be either certain that Y is not a hypothetical explanation of P or certain that Y does not obtain. Either way, learning that some Y is a hypothetical explanation of P makes no difference to her credence in P.

However, rationality allows agents to believe that there are compatible yet distinct explanations.Footnote ²² Cases involving such agents are potential counterexamples to HCP because, to put the matter metaphorically, compatible hypothetical explanations of P might give a rational agent conflicting advice about what should be her credence in P. Suppose our agent is certain that I have had contact with sick colleagues and that I have a vitamin C deficiency. Suppose she then becomes convinced that my vitamin C deficiency, like my contact with sick colleagues, is a hypothetical explanation of my cold. Then, her updated credence that I get a cold—made in light of the new information that my vitamin C deficiency is a hypothetical explanation of my cold—may differ from her old credence that I get a cold. We cannot argue that information about a hypothetical explanation of my cold other than my contact with sick colleagues is somehow inadmissible_HCP for this agent, because information about alternative hypothetical explanations of chance outcomes is surely admissible_PP. So, we seem to have shown that HCP is false.

This potential counterexample relies on the assumption that X and Y might give a rational agent incompatible advice about what to expect with respect to P when she is certain that both X and Y are hypothetical explanations of P. In advocating HCP, I reject this assumption. It seems to me central to our conception of explanation that we are not certain of why an event occurs (if it does) unless we are also certain that we are in at least the best possible epistemic position (relative to that event) available to agents who have only admissible_HCP information (in the sense of “admissible_HCP” developed below). If that is right, then if a rational agent with only admissible_HCP information is certain that X obtains and that X is a hypothetical explanation of P, then she is not certain that Y is a hypothetical explanation of P or she has ruled out Y or learning that Y obtains would not (by her own lights) put her in a better epistemic situation with respect to forming expectations about P.

Influential theories of statistical explanation (Hempel Reference Hempel1965; Salmon Reference Salmon1971; Railton Reference Railton1978) imply a similarly high standard for certainty about why an event occurs, but it may strike some readers as implausibly demanding (or at least rational to deny). For example, given that we know that we are rarely in the best possible epistemic situation that is consistent with having admissible_HCP information, we are rarely rationally certain of some X that we are certain is a hypothetical explanation of P. I find this implication intuitive, but admittedly it rules out any theory of explanation according to which it is easy to know why events occur. At any rate, if we assume that rational agents never make incompatible predictions about P in light of (what they believe to be) compatible hypothetical explanations of P, then learning that Y obtains or that Y is a hypothetical explanation of P makes no difference to a rational agent’s credence in P once she is certain that X and that X is a hypothetical explanation of P.

There is one final wrinkle brought out by this case. In virtue of what is the information that I am vitamin C deficient admissible_HCP for our rational agent who is predicting whether I get a cold while a reliable crystal ball’s predictions about whether I get a cold are not? The answer is that our agent believes that it is at least a live option that a vitamin C deficiency is a hypothetical explanation of my cold, whereas she has ruled out the possibility that the predictions of a reliable crystal ball are a hypothetical explanation of my cold (so long as she has background beliefs like ours). Whatever other hypothetical explanations of my cold there may be, she is sure that the predictions of reliable crystal balls are not among them.

In light of these considerations, our list of information that is available in typical cases of prediction but that makes no difference to a rational agent’s credence in P once she is certain that X and that X is a hypothetical explanation of P grows to include the following conditions:

• (3) Information that is evidence about P only in virtue of being evidence about some hypothetical state of affairs Y such that it is a live option (for the agent) that Y is a hypothetical explanation of P.

• (4) Information that is evidence about P only in virtue of being evidence about whether some hypothetical state of affairs Y is a hypothetical explanation of P.

Finally, in a typical case of prediction, the vast majority of information a rational agent has is not evidence about P. Since information that is not evidence about P makes no difference to a rational agent’s credence in P, the following information makes our list:

• (5) Information that is not evidence about P.

An account according to which information is admissible_HCP (with respect to a particular rational agent’s credence function and to particular hypothetical states of affairs X and P) if and only if it satisfies at least one of 1–5 would be adequately broad and would protect HCP from most spurious counterexamples—but not all. Suppose, once again, that a rational agent is predicting whether I get a cold. She is certain that I have had contact with sick colleagues and certain that my contact with sick colleagues is a hypothetical explanation of my cold. Suppose also that it is a live option for her that vitamin C deficiencies are hypothetical explanations of colds, although she is currently not very confident that I have a vitamin C deficiency. Finally, suppose she is certain of the following disjunction: either I do not have a vitamin C deficiency or I will not get a cold. What happens to her credence that I get a cold if she learns that I hate citrus fruit? If she has normal background beliefs about the connection between citrus and vitamin C, she may increase her credence that I am vitamin C deficient. But, because she is certain that either I do not have a vitamin C deficiency or I do not get a cold, she may also decrease her credence that I get a cold. If all of her information in this case is admissible_HCP, HCP is false. Condition 3 implies that the information that I hate citrus fruit is admissible_HCP. Whether the disjunction (that either I do not have a vitamin C deficiency or I will not get a cold) is admissible_HCP depends on whether it counts as evidence about my cold, and my use of “evidence” (and “in virtue of”) in 1–5 is too imprecise to yield a definitive answer.

Nevertheless, I am optimistic that the admissibility problem can be solved with an account of “admissible_HCP” that is in the spirit of 1–5. Conditions 1–5 are not a disjointed collection of ad hoc stipulations. Information that fails to satisfy at least one of 1–5 is, roughly, information that is evidence about P but not only in virtue of being explanatorily relevant to P. The information that I hate citrus fruit is intuitively not admissible_HCP because it provides our agent with evidence about my cold via both an explanatory route (because it satisfies 3) and a nonexplanatory route (through her confidence in the disjunction). It is hard to say what it is for information to provide evidence through an “explanatory route” rather than not, or to be “explanatorily relevant” rather than not, but the intuition behind these distinctions is reasonably clear. Every example of inadmissible_PP information I know of involves evidence about a chance outcome that is not merely explanatorily relevant to that outcome, so it is plausible that this rough characterization of “admissible_HCP” is sufficiently narrow to protect HCP from spurious counterexamples.

Furthermore, this rough characterization of admissible_HCP information seems to be broad enough to allow HCP to apply in all typical cases of prediction (i.e., cases to which Swamping applies). The examples that motivate 1–5 suggest that, in typical cases of prediction, all of our information is not evidence about P or is explanatorily relevant to P or is screened off by information that is explanatorily relevant to P. I conclude that it is plausible that HCP is true on a reading of “admissible_HCP” that is coextensive with “admissible_PP” in every case in which X is that the chance of P is x.Footnote ²³

5. The Generalization Problem

6. Conclusion: Implications for a Theory of Chance

Lewis suspected that the PP is explained only if chances somehow reduce to (or, at least, supervene on) the Humean mosaic. He writes, “I think I see, dimly but well enough, how knowledge of frequencies and symmetries and best systems could constrain rational credence. I don’t begin to see, for instance, how knowledge that two universals stand in a certain special relation N* could constrain rational credence about the future instantiation of those universals” (Lewis Reference Lewis1994, 484).

Lewis’s thought, roughly speaking, is that if chances supervene on what actually happens (as far as facts not involving chances are concerned), then it is not entirely mysterious why one’s opinions about chances should constrain one’s expectations about what actually happens—including the outcomes of chance processes. But, contra Lewis, it is far from obvious that theories according to which chances supervene on the Humean mosaic can provide noncircular explanations of the PP.Footnote ²⁵

Rather than follow Lewis in arguing that the PP is explained by the metaphysics of objective chance, I have argued that one aspect of the PP (i.e., Swamping) follows from a theory of the particular explanatory role that chances play (i.e., the causal chain theory) and a principle governing the relationship between beliefs about scientific explanations and rational expectations (i.e., HCP). One prima facie advantage of this approach is that, strictly speaking, it is neutral with respect to what (if anything) chances reduce to or supervene on. That said, my explanation of Swamping might nevertheless have some implications for a theory of chance.

The argument from HCP to Swamping requires the truth of either the causal chain theory or a relevantly similar theory according to which chances are hypothetical explanations for the rational agents who believe in them. My strategy for explaining Swamping, then, requires that objective chances are the kind of thing that agents can rationally believe to be hypothetical explanations. But theories of chance according to which chances supervene on the Humean mosaic have a prima facie difficulty allowing that chances are explanations of (any part of) the Humean mosaic. The worry is that if the Humean mosaic helps to constitute the chances, then the chances are ill placed to explain the Humean mosaic.Footnote ²⁶ In contrast, the very theories that Lewis uses the PP to argue against (i.e., theories that deny that chance facts supervene on the Humean mosaic) have no similar difficulty; they allow that chances are responsible for features of the Humean mosaic but not vice versa. Of course, these are only prima facie considerations. But they suggest that the challenge of explaining Swamping can be met more easily by theories that do not purport to reduce chances to features of the Humean mosaic.

I have argued that HCP entails Swamping given the causal chain theory. That result, of course, does not establish that HCP and the truth of the causal chain theory are what explain Swamping. Swamping might instead be part of an explanation of HCP or of the truth of the causal chain theory. But the analysis of admissible_HCP information I sketched in section 4 makes essential reference to hypothetical explanation. This suggests that our rational opinions about hypothetical explanations are at the heart of Swamping (and ultimately the PP). Furthermore, it strikes me as less mysterious for beliefs about hypothetical explanations to constrain rational credences than for beliefs about chances to constrain rational credences. For these reasons, I am optimistic that the direction of explanation goes from HCP and the causal chain theory to Swamping.

If my explanation of Swamping is correct, then we have an explanation of a notoriously puzzling aspect of the PP that is prima facie consistent with a wide variety of theories of objective chance. Theories that deny that chances supervene on the Humean mosaic have no principled difficulty employing my explanation. Far from being the hopeless nonstarters that Lewis envisioned, such theories may be able to use my explanation of Swamping—and thus to ground Butler’s maxim that chances are the very guide to life.