For Lewis (1994: 480) “chances are what the probabilistic laws of the best system say they are” so I shall first consider what that means, for such a Humean, before reiterating Black’s (1998) objection, that Lewis should not also have defined chance as (1994: 489) “that feature of Reality that obeys the old Principle”. That Principle is Lewis’s Principal Principle (PP, a self-evident truth about chance and credence, see below), with which Lewis showed, rather heroically, the implausibility, in the case of chancy worlds such as this one appears to be, of his own favourite thesis, of Humean Supervenience (HS, according to which everything supervenes on the distribution of fundamental properties at space-time points; all combinatorially possible distributions being metaphysically possible). As you will see, he later tried to rescue HS by revising PP, but if Black is right then there is a fundamental incoherence in Lewis’s approach to physical probability.
To begin with, our credence (subjective probability) is the degree of our belief in some proposition. Were we rational, our credence function C (a function of propositions that we might entertain) would yield credences obeying the probability axioms (as it would for Mellor). We also have a concept of chance (physical probability), e.g. the 50% chance that a tritium atom will decay during the next 12.26 years. Whether or not chance exists, we have some definite beliefs about it, e.g. that it is time-dependent (what is past is no longer chancy) and world-dependent (e.g. the fairness of a coin may vary from world to world), and that:
[The] chance of decay is connected as follows to credence: if a rational believer knew that the chance of decay was 50%, then almost no matter what else he might or might not know as well, he would believe to degree 50% that decay was going to occur. Almost no matter; because if he had reliable news from the future about whether decay would occur, then of course that news would legitimately affect his credence. (Lewis 1994: 475)
That connection is Lewis’s (1986: 87) PP: C(A | X & E) = x, where (A | B) means A given B—i.e., C(A | B) = C(A & B)/C(B)—and X is the proposition that the chance (at time t) of A being true is x, and E can be any proposition that is compatible with X and which is admissible (at t) in the sense that the equation holds (e.g. facts from the future about the truth of A would be inadmissible).
According to HS, contingent truths about chance are made true by certain patterns in the underlying distribution of the fundamental properties, but because of the self-evident PP: “Whatever makes it true that the chance of decay is 50% must also, if known, make it rational to believe to degree 50% that decay will occur.” (Lewis 1994: 476) The PP is trivially true—it would only make a substantial assertion were the admissible kinds of propositions independently specified—so the question arises, what do we know about those kinds?
Clearly admissible is the evidence one has for believing X, while evidence for the truth of A obtained after t is inadmissible because it relates to the outcome directly, rather than via the chance (at t) of A. Since propositions are admissible insofar as they only affect our credence in them via their chances, admissible propositions appear to include those that yield purely historical information (Lewis 1986: 93-4) and those yielding purely hypothetical information about the way that chance depends upon history (Lewis 1986: 95-6). Lewis (1986: 97) reformulated his PP so as to make those two kinds of admissible propositions explicit, as follows.
Let Htw be a complete history of a world w up to time t. Let Tw be a complete theory of chance for w, i.e. “a full specification, for world w, of the way chances at any time depend on history up to that time” (Lewis 1986: 97). And let Ptw be the chance function at t and w (a function of chancy propositions). Because Htw & Tw is a conjunction of admissible propositions, it may replace E, so that (X & E) = (X & Htw & Tw). And since Tw says how chances depend upon history, it entails all the true history-to-chance conditionals at w. In particular, it entails (Htw ® X), whence (X & Htw & Tw) = (Htw & Tw). And since x may be replaced by Ptw(A), Lewis’s reformulated PP is C(A | Htw & Tw) = Ptw(A).
Unfortunately, either the history-to-chance conditionals entailed by Tw are contingent, or else they are necessary, which presents Lewis with the following dilemma.
If they are contingent then the reformulated PP yields a contradiction from the supposition of HS, as follows. In accordance with HS, the laws of nature are given by a best-system analysis of the patterns of nature (i.e. the laws are the theorems of the deductive system whose theorems are all true and which best combines simplicity with informativeness). Some of those patterns might yield the conditionals in question, but:
The trouble
is that whatever pattern it is in the arrangement of qualities that makes the
conditionals true will itself be something that has some chance of coming
about, and some chance of not coming about. What happens if there is some
chance of getting a pattern that would undermine that very chance? (Lewis 1986: xv)
Such a future pattern, described by some proposition F, would have a non-zero chance Ptw(F), whence, via the reformulated PP, C(F | Htw & Tw) ¹ 0. But F is known to be inconsistent with Tw, so C(F & Tw) = 0, whence C(F | Htw & Tw) = 0.
Despite that contradiction, undermining patterns (F) cannot be ruled out. Consider how, before any coins had been tossed, it was possible that they would all land heads up—more precisely, if different alternative futures did not give rise to different present chances, then the future would be irrelevant to the determination of present chances (and yet, e.g., initial outcomes are always going to exhibit a pattern with a deterministic appearance).
Now, were the conditionals entailed by Tw necessary, implausibly stringent criteria would (again) restrict the reasonable credence functions:
To illustrate: on this hypothesis, enough purely historical information would suffice to tell a reasonable believer whether the half-life of radon is 3.825 days or 3.852. What is more, enough purely historical information about any initial segment of the universe, however short, would settle the half-life! (It might even be a segment before the time when radon first appeared.) (Lewis 1986: 131)
Consequently, Lewis (1994: 474) wants to defend the plausibility of HS as a contingent truth (about this world). He therefore needs to accommodate the wealth of scientific evidence for the existence of chancy events (in this world) somehow, even though plausible Humean chances appear to contradict the self-evident PP.
Fortunately for Lewis, Thau (1994: 500) noted that, “A proposition is inadmissible if it provides direct information about what the outcome of some chance event is.” That is, admissibility admits of degrees, and it depends upon the proposition (A) in question, propositions about future events being (largely) admissible if they reveal (almost) nothing about the outcome of the event in question. For everyday propositions, Tw provides negligible information about their truth—only for such propositions as F would Tw be seriously inadmissible.
Inspired by Thau, Lewis (1994: 487) got his new PP by replacing Ptw(A) with Ptw(A | Tw). The reason is that, while Ptw(A | Tw) will almost equal Ptw(A) for most A (since Tw is a theory of chance), the new PP yields C(F | Htw & Tw) = 0 for undermining futures F, because Ptw(F | Tw) is 0 (F being inconsistent with Tw). The new PP, C(A | Htw & Tw) = Ptw(A | Tw), avoids the contradiction (mentioned above) by removing the one-sidedness of the conditionality upon Tw (note that simply removing the conditionality upon Tw would have crippled, rather than reformulated, the self-evident PP).
Unfortunately, since for Lewis (1994: 480) “chances are what the probabilistic laws of the best system say they are”, the following question arises:
Consider a subject who accepts this definition of ‘chance’, but doesn’t conform her credences to the ‘chances’ as so defined. What principle of rational credence will she be violating? (Black 1998: 380)
A Humean’s belief about chances is a belief about the global pattern of outcomes, so the question arises, how could such a belief be justified? Humeans should not expect future patterns to resemble past ones, and future patterns could hardly be known more directly, so it seems that any belief about a global pattern (rather than a past pattern) would be impossible to justify. And why should a Humean constrain their beliefs about individual cases according to their beliefs about any other cases? It is relatively clear why non-Humeans would—were the other cases of the same kind, they would be yielding evidence for each of the individual propensities—but Humeans?
[We] may find that one intrinsic quality is always, or usually, associated with another, but for Lewis this will always be a fact about the pattern, a brute fact which has no explanation beyond the presence of that pattern, and in particular no explanation arising from the character of the primitive qualities themselves. (Black 1998: 372)
A Humean should know better than to let her credence in some outcome of a future chancy event be influenced by her beliefs about some pattern in the previous outcomes of similar events.
Of course, knowing about future patterns would be useful, but in the absence of non-Humean inferences from past patterns such knowledge would be unavailable. And anyway, why one pattern rather than another (e.g. a sub-pattern) if they are merely patterns? Of course, we do make rational choices about such things, but it does seem that our reasons for making them ordinarily come from intuitions that are innately non-Humean.
Because Humeans believe that individual events are metaphysically independent of each other, they ought to have completely open minds about the outcomes of particular chancy events—they should not let their beliefs about global patterns constrain their credences about chancy events. So, since (each of) the PP says that their credences should be constrained by their beliefs about global patterns, Humeans should regard the PP as false. Black (1998: 382) considered the possible reply that it is just a brute fact about us, that we regard such constraints as rational, and observed that Lewis’s revision of the PP to make it compatible with the metaphysical thesis of HS would be at odds with such a sociological explanation.
Since Lewis did (1994: 489) think of chance as “that feature of Reality that obeys the old Principle”, his chances seem to be the ghosts of departed propensities (such as are ordinarily assumed). Maybe, were Humeans to regard the PP as false, the consequent counter-intuitiveness of Humean chances need be no more troubling, to them, than such consequences of HS as the superficiality of all of our scientific experiences (even those that inform us of, for example, a rainbow’s relative unreality compared to rain-clouds). But clearly, insofar as Humean chances go beyond the simple Frequentism and Personalism that we rejected above, they also inherit an additional implausibility, for most of us. Section 8.