To sum up (my 2006), indeterministic physical probability appears to be, in the first place, some sort of measure of actual physical possibility, as Popper (1983: 286) put it. Further questions arise, e.g. what sort of measure (e.g. must it exist within ZFC) and what precisely are actual physical possibilities (e.g. might there be physical possibilities for some man’s actions that, for psychological reasons, no copy of him would perform), but mathematical paradox (see §10) and empirical inadequacy (e.g. the current inconclusiveness of parapsychological research) respectively indicate that answering them will not be straightforward.
And even the above, rather superficial defence of single-case propensities, means that the weakest links of my unpublished argument—for a potential infinity of natural numbers (my initial concern, see §10)—are now its other assumptions; so I shall now draw this glance at chance to a close, by briefly rejecting the objections to single-case propensities (of the “tendency” kind) compiled by Eagle (2004). Amongst his “twenty-one” arguments are some that I would run together, some that are against long-run propensities and Mellor’s propensities (which I need not criticise here), and some that I just cannot see as arguments at all. And since I’ve only just considered Humphrey’s “devastating” (Eagle 2004: 402) paradox, I shall now reject only a few of Eagle’s arguments explicitly (so if I’ve neglected anything important, do please let me know).
Note that Eagle was primarily criticising “propensity analyses of probability”, whereas of the two obvious ways to generalise (modern) physical probability—i.e. via our concept of probability and via (modern) physics—I have favoured the latter. To follow the former route might mean that classical statistical and subjective probabilities would need to be given a similar analysis (whence many of Eagle’s arguments), but in the latter case such things as fields of force and electrons would need to be treated similarly. Just as fields of force are posited in order to help physicists to explain observed accelerations (and note that failures of detail lead not so much to their rejection as to their evolution, into such things as spatio-temporal curvature and QED), and just as electrons are posited to explain chemistry and electricity (evolving into such things as leptons and strings), so Popper posited propensities in order to understand quantum-mechanical indeterminism—and similarly, propensities may well evolve far beyond their current theories, but if they do it will be for scientific rather than semantic reasons.
But regarding the former (Eagle’s) approach, why should the axioms for subjective probability (which concern the assignment of infinitely precise numbers to our imperfect beliefs) and the axioms for quantum-mechanical probability (which concern our scientific observations of objective reality) have more than a superficial connection (via the concepts that they both relate to, e.g. frequencies, or kinds and ratios, and uncertainty)? I can think of no good reason (after all, our concept of probability is akin to our concept of knowledge, of which there are a wide variety of kinds that need only have the formative possibility of a good informant in common; see Craig 1990). Consequently I reject Eagle’s (2004: 384-8) first four arguments; in particular:
Consider the case where in some possible world, classical physics is correct, and compare that to a different world which is correctly described by quantum theory. Both worlds have empirical phenomena to which probabilistic theories correctly attach. But are there fundamental properties in common between these worlds sufficient to give a propensity interpretation to the probabilities that appear in each of the respective probabilistic theories? (Eagle 2004: 386)
The probabilities of the former world might be interpreted as fictional propensities, corresponding to a fictional indeterminism (the reification of the ignorance, in that world, of precise initial conditions), but they need not be—there is no great mystery about classical statistical mechanics; the problem is to understand the probabilities of quantum mechanics, e.g. (as I suggest) via an analogy with Laplace’s (classical) urn.
Note that it need not be denied that there are classical physical probabilities, since propensities are not being postulated in order to explicate our concept of probability; and consequently we need not derive the standard axioms of probability from our concept of a propensity (as Eagle’s first argument would have us try to do), which is just as well because those axioms are in doubt anyway (see §10). And furthermore (Eagle’s fourth argument) the fact that “for quantum mechanical observables, Pr(AÈB) ¹ Pr(A) + Pr(B) – Pr(AÇB)” (2004: 387) prior to measurement need not trouble us, because the details of how quantum-mechanical wavefunctions interact lie beyond the problem of understanding what is meant by ‘probability’ in such contexts (e.g. when a hypothetical |f|2 is tested via frequencies; cf. §1).
Eagle’s (2004: 389-90) fifth argument concerned finkish propensities:
Consider a glass that if struck immediately anneals: until it is struck, it is fragile, but once it is struck it is not. Call such a disposition a finkish disposition. […] Whenever a finkish propensity is trialled, the outcome fails to happen, although there is a positive propensity for the outcome event to occur. […] A non-trivial probability for some event means that event is seriously possible; that there is some world where it occurs. But if finkish propensities are possible, then some non-trivial probabilities lose this connection with possibility, since there will be no world where it is possible for the propensity to manifest without the fink blocking it. (Eagle 2004: 389)
But could such a finkishly fragile glass be fragile until struck? Since it would not break were it struck, why would we call it ‘fragile’? Is it because it has the same microscopic structure as normally gives rise to a glass’s macroscopic fragility, together with something that causes it to anneal when struck? Cf. the “contesting dispositions” (Popper 1983: 287) of §4, e.g. a coin intrinsically biased towards heads, which is actually tossed only by someone who biases his tosses towards tails; or consider one’s physical ability to torture a child for pleasure, which manifests in no possible world because in no circumstances would one actually do that. But in the latter example, the net event (one doing that) is impossible, and hence has zero probability—there are underlying nonzero propensities, associated with the physics, but presumably such a body would act that way in some physically possible world. To postulate such underlying propensities is hardly immodest, because our bodies do appear to be composed of huge numbers of just a few sorts of particles (which behave quantum-mechanically).
Which also answers the reference class problem (cf. §5), i.e. Eagle’s 9th, 13th and 14th arguments: Eagle (2004: 395) asks how we should “classify the statistically relevant properties and gain information about other trials” if there is no generalisation (if the single case is the most basic case) but of course, we realists hypothesise about just such underlying generalities; and judging by the empirical evidence, actual physical chances arise from very few sorts of sources. After all, were there no such underlying sorts (such as realists posit) there would be no true laws of physics—not even deterministic ones! Indeed, Eagle’s 13th argument was entitled ‘Non-Humean’:
Single-case propensities are not underminable, and are thus radically insensitive to evidence. This is not the usual problem of characterising statistical inference, since in this case we have no connection whatsoever between the chances and the evidence. There are worlds where the frequency of A is p, but for any q, 0 £ q £ 1, the chance of A might be q. (Eagle 2004: 401)
There may well be epistemically possible worlds where the frequency of A is p and the chance of A is q ¹ p (especially if they are deterministic worlds, for which q is 0 or 1), within which we would not have direct access to the value of q; but scientists may use the frequency of the observed As (not usually the frequency of A, unless A was necessarily both observable and observed, which is just silly) as a guide—in conjunction with other evidence (e.g. patterns in their observations of A, observations relating to the reliability of their observations of A, and observations of theoretically related Bs)—towards an estimate of q (or of course, the revising of their posit of some such q).
Why should there be more of a problem with propensities, than there is with forces or energies (or indeed, with natural kinds, given the famous gap between things-in-themselves and our observations of them)? If we can cope with copper wires (in their coloured coverings) containing electrons (in their energy levels), then we can find plenty of evidence for quantum-mechanical probabilities being single-case propensities, because no other interpretation of ‘probability’ accounts half so well for what we actually observe. Still, those who suspect single-case propensities (and who have no problem with standard set theory) may well find a further reason for their suspicions in Section 10, which is a modern physical instantiation of Levy’s paradox (of equally likely integers).