1. Initial Assumptions and Problematic
Humean Supervenience, the Best Systems Analysis of Laws and Chance, and the Principal Principle form for Lewis an individually plausible but mutually inconsistent trio. The trouble is chance. On the one hand, if Humean Supervenience is true, objective chance is the sort of thing that supervenes on the distribution of qualities at points or point-sized objects. On the other, our best grasp of the notion of objective chance is mediated through the Principal Principle, which states that objective chance is the sort of thing that constrains rational credence. The tension is created by undermining futures—futures whose distribution of qualities completes a chance-making pattern in which present chances differ from what they actually are—a consequence of the thesis that chance supervenes in a Humean manner. These futures enjoin a contradiction in the Principal Principle, which assigns to them both a zero and nonzero chance. This problem is purportedly fixed by modifying the Principal Principle but, I will argue, turns out to be problematic in its own right, posing for the Humean a series of hard questions.
Here my aim is to set out and explore the repercussions of this problem from a Lewisian standpoint. As such, the range of possible solutions will be constrained in part by consistency with Humean Supervenience, the Best Systems Analysis of Laws and Chance, and the Principal Principle in either its new or updated form. A certain amount of stage setting, however, is first required. Before I go on to develop the problem I see for the updated Principal Principle, first let me briefly sketch Lewis's development of the principle and its subsequent modification.
1.1. Humean Supervenience, and Chance
In the introduction to his 1986 collection of papers, Lewis inaugurates the idea of Humean Supervenience as the guiding thesis for much of his work (Reference Lewis1986a). The central idea, refined in several places (Reference Lewis1994, Reference Lewis1999a, Reference Lewis1999b), is that the set of truths at worlds like ours supervenes on the distribution of qualities at points or point-sized objects. The picture, Lewis claims, is “inspired by classical physics,” and, as is made clear, Humean Supervenience is intended as a contingent claim about our world and those like it (Reference Lewis1994, 474).
But, as Lewis is quick to note, we have reason to think that classical physics is partly in error, or at least incomplete. The world, our current best physical theories seem to say, is a chancy place, and these chances—for instance, the chance of decay of a single uranium atom—are part of the objective fabric of the world. Thus, there are truths in our world that pertain to objectively chancy events, and if Humean Supervenience is true, these truths must supervene on the fundamental distribution of qualities at points or point-sized objects.
1.2. Objective Chance and the Principal Principle
Our world is an objectively chancy place. Truths about these chance events, if Lewis is correct, must then supervene in a Humean manner. Yet, we might still wonder, what exactly is the notion at play when we say that there is a chance x that this particular uranium atom will decay in a year? That is, how exactly are we to understand objective chance?
Humean Supervenience goes part of the way to answering this question, telling us what the world must be like in order for events to be objectively chancy. Lewis's Principal Principle goes the rest of the distance, linking the notion of objective chance with rational credence and thus, as Lewis argues, providing us with “the best grip we have on the concept of chance” (Reference Lewis1986c, 264).
Given a world w and time t, let C be some reasonable credence function, A some proposition in the domain of Pt,w, Ht,w the history of w up to t, and Tt,w a theory of chance for w at t.Footnote 1 The Principal Principle states:
The idea is simple. A fair coin will be tossed at noon, t 1; half an hour before, at t 0, you're asked to what degree do you believe the proposition, A, that the coin, when tossed, will land “heads”? The answer is C(A) = .5.Footnote 2 Thus, the principle is intended to capture the intuitive idea that whatever the objective chance turns out to be, it is the sort of thing that would ideally constrain rational credence.
1.3. The Bug
The “big bad bug,” as Lewis calls it, is that Humean Supervenience turns out to be incompatible with the Principal Principle. The trouble stems from undermining futures—futures whose chance-making distribution of qualities determines a set of chances different from what they actually are.
If we understand our theory of chance, T, as Lewis would—as the set of all true history to chance conditionals—then the problem is easy to appreciate. Let W be some world with an actual future @ in which P(A) = .7, let T be a theory of chance for W at t 0, and let A be some arbitrary proposition. Furthermore, let F be some proposition concerning a future chance distribution that completes a chance-making pattern in which P(A) = .5, whose chance according to T in W at t 1 is .4. Since, according to T, P(F) = .4,
Yet, F is, strictly speaking, incompatible with T, as it assigns a value to P(A) that is different from the actual value, so
Thus, the undermining future F, which we take to be a natural consequence of the thesis that chance supervenes in a Humean manner, generates a contradiction in the Principal Principle.
2. The Fix, and Why It Might Not Yet Be Enough
The fix, suggested in related papers by Lewis (Reference Lewis1994) and Hall (Reference Hall1994) is to recast the Principal Principle as the New Principle, where C(−|−) is some reasonable credence function, A is any proposition, Htw specifies the complete history of some world w up to time t, and Tw is a complete theory of chance for world w:
In this way, the contradiction enjoined by the Principal Principle when applied to undermining futures is avoided. If we let F be some undermining future as before, then both C(F|HtwTw) = 0 and P(F|Tw) = 0, since Tw by definition is inconsistent with F; so far, so good. Further, the new principle turns out to be equivalent to the old principle in situations lacking undermining futures; in these situations, P(T) = 1 and hence P(A|T) = P(A). Thus, as a technical solution, the New Principle seems to be everything we could want.
However, this technical victory comes at a price. As Lewis laments, “Our new version of the Principal Principle is better by Humean lights; but … I still find the old one more intuitive,” and indeed it is fairly easy to see why (Reference Lewis1994, 489). The old principle was intended to capture the idea that objective chance, understood as a probability measure over single-case events, is the sort of thing that should constrain rational credence. The new principle, as we've seen, replaces this notion of a single-case probability measure with the conditional probability of an event given a theory of chance. The immediate worry is that this modification renders the new principle, in Hall's (Reference Hall1994) words, “user hostile,” in that the probability of some event, given a theory of chance, turns out to be a rather difficult quantity to know how to calculate.
We might assuage these initial doubts, even in the face of undermining futures, by exploiting the fact that the new and old principles often enough are approximations of one another. In this way, the Old Principle might serve as a heuristic, or quick guide, to our understanding of chance and credence. Often enough in the sciences, we make use of such expediences, so here too we might not think they will be the cause of too much trouble.Footnote 3
Difficulties, however, arise in another way. To highlight these problems, first consider the Principal Principle in its original formulation: C(A|HtwTw) = P(A). The driving insight here is the intuition that whatever objective chance turns out to be, it acts as a constraint on rational credence, and this seems to hold whether or not a Humean analysis of chance turns out to be viable. Paradigmatically, we might think of objective chance as the sort of thing that attaches to the radioactive decay of a single uranium atom; let this event of decay be designated by the propositional variable A. The objective chance of A, then, is modeled by the probability measure P, a function from a σ-field F to [0, 1], where 
. In this way, there is a clear content that attaches to P(A) as it appears in the Principal Principle; it captures the objective chance of single-case events and is modeled by an appropriate probability measure. Thus, there should be no difficulty in thinking that P(A) captures the objective chance of A.
In the case of the New Principle, however, there are obstacles to this straightforward assimilation of P as a measure of objective chance. As before, the New Principle is intended to capture the same relation as the old. The difference, as we saw above, is the New Principle's substitution of P(A|T) for P(A); that is, C(A|HtwTw) = P(A|Tw). It is here that we should pause to consider whether P(A|T) can be interpreted as a function describing some objective probability.
First off, we might find this interpretation odd, given that P(A|T) expresses a conditional probability, that is, the probability of A, given T. Although nothing stands in our way of understanding A as before—say, as the proposition that this uranium atom will decay in the next 50 years—it seems puzzling to think that the probability of A, given some theory of chance, specifies some objective chance. The motivation for this sense of puzzlement is twofold. First, by appeal to intuition, it seems natural to think that the objective chance of a singular event takes some value x, where 
, regardless of what we think, what we know, or what any particular theory says. Indeed, if we think that A has some objective chance, then it seems natural to suppose that it is the quantity of P(A), instead of P(A|T), that is of interest. Put another way, conditionalizing on a theory of chance shouldn't have any bearing on the objective chance of an event; it is simply a different quantity.
Of course, we might agree to all of this and yet still maintain that the value of P(A|T) picks out some interesting objective probability different (at least potentially different) from P(A). This brings us to our second source of motivation and, as I will argue, cause of trouble. Per the New Principle, P(A|T) is understood as the objective chance of A, given T (or, if you prefer, A conditional on T). Conditional probabilities, standard in probability theory, are understood according to ratio formula:
Thus, our commitment to P(A|T) being a probability function that picks out the value of some objectively chancy event seems also to commit us to P(T) having some objective chance. That is, if P is an intelligible probability function, and we are committed to the standard analysis of conditional probabilities, it seems we are saddled with the consequence that 
and P(T) are perfectly intelligible probability functions. Yet, keeping in mind that the function P in the Principal Principle was intended to capture the values of objective chance, it appears that now we are required to think that our theory of chance itself, T, has some objective chance.
Here, we might wonder how this could be. In what sense could we think that a theory of chance itself has some objective chance? A natural suggestion is that the objective chance of a theory just is the chance that the theory is true. In our case, a theory of chance would be true just in case the chance-making patterns extended through time (future included) at a world are such that they determine the chance distribution that the theory claims. But, this suggestion, however natural it may be, runs into a familiar problem. If we take P(T) to express the chance that a theory of chance is true, then we come about-face on the problem of undermining futures. Since our theory specifies only that the chance-making distribution has a particular chance (things could have gone otherwise), it is self-undermining, claiming simultaneously that a particular chance distribution has some chance while also claiming that this chance distribution is incompatible with what the theory claims. Put another way: the theory assigns some chance to the theories assigning different chances than it does.
The argument is entirely analogous to the argument for undermining futures. Let T claim that the chance distribution A it predicts is such that P(A) = .7. Let 
, so P(B) = .3 according to T. If we apply the old principle to this case, in an attempt to understand the objective probability of our theory, P(T), we end in contradiction. If we apply the new principle, we end up, on the one hand, with the thought that since T specifies P(A) = .7, this should be taken as the objective probability of the theory, while, on the other, the new principle specifies that C(T|HT) = 1, because P(T|T) = 1. Note too that this problem is not solved by recourse to some other theory, T′, on which the probability of T is conditional—perhaps a theory of our theory of chance. If we are tempted by this option, I suggest we simply reread the prior two paragraphs, substituting T for A, and T′ for T .
To briefly take the toll, we found the New Principle to be problematized by its apparent requirement that we make sense of the notion that our theory of chance itself has some objective chance. Attempting to do this in terms of our best guide to objective chance—the Principal Principle—founders on the problem of undermining futures. If we wish to retain the New Principle, we seem required to provide some alternative answer to the question of how our theory of chance could itself be chancy.
There are at least two ways to go from here in answer to this question: one straight, the other evasive. The straight answer: theories of chance have objective chance. They do so because, as good Humeans, we think that the laws of nature, including the laws of chance, supervene on the distribution of qualities at point-sized objects. Thus, we could hope to find a way for Lewis's Best Systems Analysis of Laws and Chance to provide a satisfactory answer to how a theory of chance itself could be chancy.Footnote 4
Alternatively, we might try and evade the question altogether. We could do this if, following Hájek (Reference Hájek2003), we thought that the ratio analysis of conditional probabilities was flawed and that conditional probabilities were themselves the primitive notion with regard to probability theory. In this way, we could straightforwardly take P(A|T) to express the value of the measure of objective chance without thereby thinking that it commits us to figuring out what P(T) might be. I will consider each of these options in turn.
3. A Way Out?
3.1. Prospects for a Straight Answer
As conceived above, the straight answer to the question of how P (A|T) could be a measure of objective probability is one that would retain the ratio analysis of conditional probabilities. Our trouble here, then, is to make sense of the notion of our theory of chance, T, itself having some objective chance.
As a first pass, we might try and evade this concern altogether, claiming that the idea of a theory of chance having some objective chance is ill conceived. In its place, we might think that there is some more appropriate notion, perhaps epistemic in nature, which would more obviously apply to theories while also picking out some function modeled by the probability calculus. Epistemic credence here would seem the obvious choice, but it is one we should be quickly disabused of.
The worry is that any attempt to reinterpret P(T) as anything other than a measure of objective chance seems to undermine the very relation the Principal Principle was supposed to describe. If P(T) is understood as some epistemic notion, say credence, then its occurrence in the analysis of P(A|T) seems to undermine the thought that this measure, P(A|T), can rightly be conceived as a measure of objective probability. Such a substitution would, if you will, enjoin a mismatch in the units of the ratio analysis, turning the right-hand side of the New Principle into some chance-credence hybrid. Thus, whatever the function P() turns out to measure, we should expect a univocal interpretation for the antecedent and consequent of the conditional probability.
Alternatively, we might look for an answer to our problem in Lewis's conception of what the theory chance, T, should be. Since we are here accepting the ratio analysis of conditional probabilities, we thus need to account for the sense in which a theory of chance itself has some chance. Hints at how this could be done were given in section 2 above, and I think extending that discussion here will turn out instructive.
As mentioned earlier, I think the best sense to be made of the objective chance of a theory of chance is in terms of the chance that the theory is true. Again, the chance of truth for a theory of chance appears prima facie to be a coherent notion under the assumption of Humean Supervenience, where the truth about any chance ascribing proposition (or any proposition in general) supervenes on the distribution of qualities at point-sized objects. For Lewis, the theory of chance at a world is given by the conjunction of history-to-chance conditionals, and we might expect it to be comprehensive in the sense that given a set of events 
, T = 
.
As Lewis claims, our theory of chance is one that “never had any chance of being false” (Reference Lewis1986c, 84). This suggests at least two interpretations of what P(T) might be: on the one hand, P(T) = 1; on the other, P(T) = x, where 
. There are clear problems with both; we'll consider first the claim that P(T) = 1, as the difficulty here generalizes in an interesting manner.
Assume that P(T) = 1. Since 
, we know by probability theory that 
, since the probability of a conjunction is always less than or equal to the probability of each of its conjuncts. So, each history-to-chance condition that composes the theory of chance must, on our assumption, have a probability equal to unity. Here, again, we have two options, this time concerning how we are to interpret the conditionals that compose the theory of chance: they are either conditional probabilities or, more simply, measures of the probabilities of the conditional. We'll address each of these in order.
Assuming that the probability of the history-to-chance conditional, C, is given by a conditional probability, let P(C) = P(Ai|H); by our earlier result, P(Ai|H) = 1. Now, according to Lewis (Reference Lewis1986c), since the past is fixed, for any historical proposition H, P(H) = 1, which means that, according to the ratio analysis for P(Ai|H), P(Ai) = 1. This result is surely intolerable. From the assumption that our theory of chance never had any chance of being false, we've proved determinism; for any event Ai, our theory of chance ascribes to that event a probability of 1. A similar result follows from treating the probability of each of the conjuncts of our theory of chance as the probability of the conditionals 
. As before, we know that 
. As Lewis argues elsewhere (Reference Lewis1986b), the probability of a conditional is equivalent to the probability of the consequent, whenever the antecedent is possible. Since this applies both to the indicative and Stalnaker conditional,Footnote 5 and at any world, the past is certainly possible, 
, which again means that 
. Here, too, we've arrived at determinism. Thus, P(T) ≠ 1.
In light of this result, take P(T) = x, where 
. The argument pattern above generalizes perfectly to this case. Since we've assumed that the past is fixed and, hence, 
, on either the conditional probability or the probability of conditional reading, the probability of each of the conjuncts of our theory of chance is equivalent to the probability of each of their respective events. Where Ai designates some event, 
. But this cannot be right either. In trying to understand the chanciness of our theory of chance, we've doubled back on the unconditional chance of a single event—the very quantity the original Principal Principle was supposed to give us. Furthermore, it is at first strikingly peculiar that the chance that the theory of chance is true is determined by the conjunction of the set of chances the theory ascribes to single events. Indeed, as the number of events under the purview of T increases, the chance that T is true becomes infinitesimally small; so too for rival theories T 0, which differ from T only in the values assigned to a minimum of one event.Footnote 6
Thus, there appear to be no good options for an account of what P(T) might be. It might be better to drop the assumption that led to it entirely.
3.2. Another Way Out?
Given these difficulties, it might be better to give up on trying to account for the chance of our theory of chance. The assumption responsible for our problem was that the conditional probabilities present in the New Principle were to be understood according to the ratio analysis. Thus, it is worth considering whether abandoning that assumption might allow a better understanding of the Principal Principle in vindication of Lewis's project. The most obvious strategy here would be to follow Alan Hájek's (Reference Hájek2003) proposal treating conditional probabilities as primitive. In this way, the New Principle might plausibly be thought to represent the relationship between rational credence and chance without thereby committing us to giving an account of the chance of our theory of chance. Such a proposal would clearly avoid the problems I have raised above; however, it is not itself entirely free from trouble. Important questions remain regarding the treatment of primitive conditional probabilities, which here I can only sketch.
The most obvious concern, and indeed for our purposes most pressing, is whether or not P(A|T) even construed as a primitive notion can plausibly be thought to represent the sort of quantity that at least intuitively is meant to figure into the Principal Principle, namely, the objective chance of a single event. Here again, the conditional probability of some event given a theory of chance does not at least immediately appear to be the right sort of notion. The worry, in letting primitive conditional probabilities do the work of what we thought were unconditional probabilities of single-case events, is that without some independently plausible account of theory selection, we appear to introduce a reference-class problem into what we thought were the representatives of objective chance. Thus, while we think that the primitive conditional probability P(A|T) is to be a stand-in for P(A), what of the function P(A|T′), when it differs from P(A|T)? Is it the measure of probability of some other event not identical to A, or is it a measure of A under a different description? These questions do not receive a simple answer. This sort of reference-class problem is exemplified in Nelson's (Reference Nelson2009) recent attempts to understand the Principal Principle in light of Hájek's critiques of conditional probabilities. However, I should add, he makes it clear that he is not interested in vindicating Lewis's entire project and so allows himself to drop the concern of whether the principle he proposes is coherent with Lewis's account of the ontology of chance.
Regardless, a reinterpretation of the Principal Principle in light of Hájek's work seems minimally to require an account of how a primitive conditional probability of an event given a theory of chance can rightly be thought as a measure of objective chance. How exactly such an answer is to be given is something, for reasons of length, I cannot consider here. But this is not a strategy that we should be quick to foreclose on. While, as I've argued here, understanding the Principal Principle in terms of the ratio analysis is clearly problematic, aspects of Lewis's project may be vindicated on other grounds.
