Such obscurities aside, we have rejected Frequentism because it identifies an indeterministic physical probability with one of its consequences, i.e. with the empirical frequencies that, being displayed upon independent repetitions of its generating conditions, provide evidence for it. So, since reliable predictions are, similarly, consequences of knowing physical probabilities, rather than their entire meaning, Mellor’s identification of those two will now be similarly rejected. Mellor summarised his position as follows:
Propensities are characteristics of things warranting conditional expectations of the future. They mark non-arbitrary positions between complete ignorance of the future and complete knowledge of it. In this respect propensities are like other dispositional properties. […] Its fragility tells us what will happen to a glass only if it is dropped. […] We know what will happen when a fragile glass is dropped only if it stays fragile. Propensities differ from other dispositions only in yielding, subject to these conditions, not knowledge of what will happen in the future but mere reasonable expectation. (Mellor 1971: 169)
In this section, I address the following two points from that summary. Firstly Mellor (1969: 23) claims, “the point of ascribing chance distributions to single trials is to express the fact that their whole function is to warrant certain CBQs on the possible outcomes of such a trial.” CBQs are coherent betting quotients; i.e. odds that it would be coherent for a gambler to offer (in the sense that a clever opponent could not then ensure his own profit), given that, as Mellor put it (1971: 161), “(i) the gambler has to bet, (ii) he chooses the betting quotient, and then (iii) his opponent chooses the stake size and the direction of the bet.” I’ll consider that first claim below, once I’ve addressed the second point.
Secondly then, Mellor ascribes propensities, along with other dispositions, not to events but to things, on the grounds that dispositions are in general present whether or not they are being displayed. E.g., the Moon is massive whether or not it accelerates; and similarly, a coin may be said to be fair even when it is not being tossed (if a kind of toss is presupposed). And a disposition to respond in a certain way, to a certain situation, is always, given such a situation, displayed by such a response—e.g. were a net force exerted upon the Moon, it would accelerate, at a rate determined by its mass, cf. a coin’s fairness, which must be displayed on each and every toss—whence Mellor considers the display of a propensity to be its probability distribution, rather than the actual outcome. But prima facie a probability distribution, which is directly unobservable, could hardly be a display.
Popper regarded propensities as statistical tendencies that are not always displayed (cf. §4), and for good reason—if propensities generalise causation (for an indeterministic world) then we might think of propensities as partial causes of outcomes, which would naturally be only partially displayed. So aside from Mellor’s (therefore dubious) analogy between statistical tendencies and universal dispositions, why would Mellor have thought otherwise? One reason stands out—he (1971: 158) regarded the concept of partial compulsion as unintelligible. E.g., he (1971: 155) said, “Heads is in no way causally ‘due to’ the coin toss,” because (1971: 151) “Chance is not a sort of weak or intermittently successful causal link.” But I find that odd, because he had no problem with the concept of partial belief even though ordinarily, as Mellor (1971: 5) noted, belief is an all-or-nothing affair (for all that our beliefs may be vague).
If we can indeed consider partial beliefs (and we surely can at least consider them) then surely we can consider partial causes; e.g., just as a belief, that a coin-toss will either yield H or else T, can be associated with two partial beliefs, in H and in T, with measures that sum to 1, so similarly, tossing a coin causes an outcome that may be indeterminately H or T, but is not anything else (let us assume), and which might therefore be associated with two partial causes, of H and of T, with measures that sum to 1. Were the outcome H, it would clearly be natural to say that the tossing of the coin was a partial cause of H (the remainder of the cause being due to chance and/or the initial conditions); and while that would be less natural were the outcome T, it is surely not unintelligible to generalise causation in that way.
To pursue Mellor’s favoured analogy, consider the fall of a fragile glass onto a deep carpet. Whether or not it breaks depends upon exactly how it lands, and upon its internal structure, etc. Such a fall might even be relatively deterministic, but there will be a range of relatively indeterministic falls in between such extremes. For such falls, although the glass would remain fragile it might not break. Then again, its fragility might be displayed. Were we to define its fragility only in terms of falls onto harder surfaces, we would lose the intuitive connection between its fragility and its breaking (were that the outcome). It would be counter-intuitive for its fragility to display as breakage upon hard surfaces, if upon softer surfaces it would display as a probability distribution even were the outcome breakage.
We might say that upon hard surfaces fragility displays as a certainty of breaking; but does massiveness display, not as relatively small accelerations but similarly as their certainty? Anyway, to try to settle such terminological issues in advance of a well-founded theory of all the relevant physics might be premature. And similarly, to define the sort of things to which propensities should be ascribed might be premature. Popper mentioned both repeatable set-ups and particular events, and Mellor chose persisting things; but whichever it really is, we have still to say what it means to call |f|2 a ‘probability’ distribution (which is our concern here).
So I shall now argue that Mellor’s analysis of chance—to return to the first point—gives us no reason to adopt his terminology rather than Popper’s. Clearly, were prob(H) = ½ an objective fact (e.g. about a particularly chancy coin-toss) it would, were we to know it, justify our having (and advocating that others adopt) a partial belief of ½ in H (as the outcome of that coin-toss). But for Mellor, the entire meaning of prob(H) = ½ is that it would make advocating even odds in a CBQ scenario reasonable (i.e. rational, justified, warranted), which is much less plausible.
To begin with, in physics probabilities are usually real numbers, so warranted CBQs must, on Mellor’s analysis, be capable of a similar precision. That precision is to be achieved by the prospect of indefinite repetitions of the bet (recall that the bettor must bet). It is also that prospect, of indefinitely repeated bets at the same CBQ, which makes it unreasonable (in the sense that it would then be far more probable that losses would be acquired in the long run) to pick a CBQ that is not equal to the known physical probability. But to require indefinite repetitions that could be bet upon is to go some way beyond the self-evident rationality of having our partial belief in a particular outcome conform to its known objective probability.
For exotic particle-decays such repetitions might even be known to be extremely improbable, so that there would be no prospect at all of indefinite repetitions, yet presumably there would still (were there such exotic particle-decays) be physical probabilities associated with such decays (whose values might, for example, follow from our theory of particle physics). The temptation is to say that those probabilities are what would warrant certain CBQs were there repetitions, but what could that mean when the single case is in principle unrepeatable? (That infinitely many copies of the entire universe are to be considered, and bet upon? But then that the bettor could in principle be compelled to bet would hardly be self-evident.)
Anyway, surely we should not stipulate that, by definition, betting could not be an activity that affects the propensities that are being bet upon. E.g., not only might parapsychologists have already uncovered some evidence for some such interaction (the current published evidence being inconclusive), even sceptics about such things consider that to be an empirical question, not one that could be settled a priori. And were there such paranormal interactions, then some of us could be warranted in setting our CBQs to some other function of the fundamental physical probabilities than identity (and in advocating that similar people do likewise) because we would surely wish to retain the concept of fundamental physical probabilities (e.g. the |f|2) in order to develop our physical theories of such paranormal phenomena, were there such—and note that Mellor did not show that such phenomena should be regarded as impossible, he merely assumed (implicitly) that they were.
Similarly, while knowledge of a physical probability might well warrant that same degree of belief in that particular outcome, and/or the adoption (and the advocation) of such a CBQ, surely not every physical thing, the knowledge of which would justify such an adoption/advocation, is a physical probability. E.g., a psychic’s note about future frequencies would clearly not be a physical probability, even were it to exist.
Furthermore, CBQs are hardly an explication of rational partial beliefs. E.g. betting is a social activity, and so some people might make some of their bets solely in order to give a certain impression of what their beliefs are. In many societies, it would hardly be irrational to put such factors above mere profit, for a wide range of beliefs (even if the bettors were in a CBQ scenario), and to advocate that others do the same. (And it is plausible that we could not always allow for such interactions, especially if they were widespread.)
Presumably the reason why we should (given indefinitely repeated trials) adopt a CBQ equal to the physical probability (were that value known) is that we would be more likely to lose money if we did not. As Salmon (1979: 189-90) noted, Mellor’s (1971: 160) claim that “a partial belief is reasonable if the gambler can know he would break even after some repeated bets at the corresponding CBQ” was false, since the gambler might lose ever-increasing amounts at that CBQ (and of course, at any CBQ he might gain ever-increasing amounts).
And presumably the reason for that (expectation of losing money with different CBQs) is that the frequency of the outcome will probably tend towards that physical probability as the number of trials tends to infinity. And one reason for that (expected tendency) might be Bernoulli’s theorem (see §1), whereas Mellor simply assumes such laws of large numbers, and (see Salmon 1979: 192) that very unlikely events need not trouble us. Although Mellor failed to explain why it should be that the greater the known physical probability of A, the greater should be our tendency to act as if A will occur, Laplace’s urn indicates that a Realistic explanation of that fact is possible (since were there more white balls inside the urn, then that fact would clearly warrant a greater expectation of drawing one blindly).
Prima facie, then, indeterministic physical probability is a
measure of physical possibility, where we could picture such possibilities as
(not so much balls inside an urn, as) points in a unit continuum of physically
possible futures. Hacking (1965: 25) thought that “only
excessive metaphor makes outcomes of every chance set-up into samples from an
hypothetical population,” but the two-slit experiment indicates that
such a metaphor would not (at least in the case of quantum-mechanical
probabilities) go too far beyond what the physical facts themselves indicate.
Given the weaknesses of alternative interpretations (as above, and also
Lewis’s, which will be rejected next, in Section 7),
and of putative threats to single-case propensities (see §8, §9, and §10), such
a Realistic interpretation of quantum-mechanical probabilities retains a prima
facie plausibility (as the argument of my 2007 requires; see §10).