For Popper (1983: 401) propensities generalise causation—cf. how, if we blindly pluck a ball from Laplace’s urn, we cause a ball to be plucked, whence that plucking is the partial cause of a white ball being plucked, with the appropriate measure being the internal proportion of white balls. For Popper (1983: 351) propensities are postulated because they would (were indeterminism indicated) explain empirical frequencies, much as similarly unobservable fields of forces explain observed accelerations. However, Humphreys (1985: 557) showed that “the causal nature of propensities cannot be adequately represented by standard probability theory”. The best introduction to his paradox remains its earlier appearance in print:
[We] do not seem to have propensities to match up with “inverse” probabilities. Given suitable “direct” probabilities we can, for example, use Bayes’s theorem to compute the probability of a particular cause of death. Suppose we are given a set of probabilities from which we can deduce that the probability that a certain person died as a result of being shot through the head is ¾. It would be strange, under these circumstances, to say that this corpse has a propensity (tendency?) of ¾ to have had its skull perforated by a bullet. Propensity […] seems to inherit the temporal asymmetry of the causal relation. (Salmon 1979: 213-4)
Given any particular corpse, it was either shot or not, whence the non-triviality of such inverse conditional probabilities as that ¾ must be due to our ignorance; and since Popper’s propensities are not subjective, it seems that a (single-case) propensity interpretation of (standard) physical probability will not, after all, be possible. Nonetheless, not only might we retain that possibility by dropping either of those parenthetical phrases, we can even (according to McCurdy 1996) retain it whilst keeping both of them.
Gillies (2000: 129-36) argued that, while Humphreys’ paradox is effective against single-case propensity interpretations, it is ineffective against the long-run kind. Consider Gillies’ (2000: 130-1) toy factory, which contains a machine that produces 800 blue toys per day, 1% of which are defective, and an older machine that only produces 200 red toys per day, 2% of which are defective. Each day a toy is selected at random and tested. Let ‘D’ mean that it was defective, let ‘B’ mean that it was blue (was made by the newer machine), and let ‘R’ mean that it was red (was made by the older machine). E.g. prob(D | B) = 0.01 may be interpreted straightforwardly as the propensity for the newer machine to produce a defective toy, given that the defects arise by chance.
By Bayes’ theorem, prob(B | D) = 2/3, and it is more of a problem to interpret that probability as a propensity because if a tested toy proves to be defective then it is already either blue or not. Gillies (2000: 131-2) interprets prob(B | D) = 2/3 as meaning that if a process of producing toys and selecting one for testing, and keeping (or noting the colour of) the defective toys, were repeated a large number of times, then blue toys would tend to appear with a frequency approximately equal to 2/3. But the whole point of the propensity interpretation (of indeterministic physical probability) is that the tendency for a propensity to manifest as a frequency-range derives from the fundamental tendency that is a single-case propensity (cf. §4). Otherwise we would have a problem pinning down (the above-mentioned) “large number” and “approximately equal”—e.g. surely the observed frequency might differ greatly from 2/3.
Gillies (2000: 147) is happy to use the standard statistical falsifying rules to bridge the gap between his long-run propensities and the observed frequencies, but it remains obscure how such rules might be explained, if not on the basis of single-case probabilities (via theorems like Bernoulli’s). And while Gillies (2000: 164) found it useful to add to Kolmogorov’s axioms an axiom of independent repetitions, he did not show that all physical probabilities might be reduced to the probabilities of independent trials.
And since long-run theories rely upon reference classes, they face the same sort of problem with uniquely instantiated cases that Frequentism fell foul of. As mentioned above (in §4), Gillies (2000: 124) treats all singular probabilities as subjective, but a uniquely instantiated particle, for example, would surely have a half-life (about which we might hypothesise) that was no less objective than all the other half-lives (that we have estimated).
And after all, the probability that one of Gillies’ toys, which is chosen at random, and which tests positive for defectiveness, will also be blue (i.e. 2/3) derives from the single-case propensities of the machines to produce defective toys, so it is not a priori implausible that single-case propensities provide (given physical indeterminism) the correct interpretation of at least part of standard probability theory, and that the remainder derives its meaning from that part.
On such approaches (for which a variety of governing principles are available, see below), we would retain a fundamental role for single-case propensities. And since we could expect such propensities to be temporally asymmetric (since so are the wavefunction collapses that generate the basic indeterministic physical probabilities), hence it would be natural to define them to be, invariably, propensities for a system to produce later events. In particular:
For any conditional propensity, inverses included, both the conditioning event and the conditioned event occur after the time at which the propensity function and the system are defined. Conditional and inverse conditional propensities are propensities of systems to produce one future event given that the other future event is also produced (by the system). (McCurdy 1996: 109)
So let us now move beyond the introductory versions of Humphreys’ paradox (considered above), and consider resolutions of Humphreys’ (2004) more formal arguments (from which he obtained his paradoxical contradictions). Humphreys assumed (within physical scenarios whose details need not concern us) both standard probabilities, in order to get his non-epistemic inverse conditional probabilities, and an intuitive principle about propensities (one of the three considered below) in order to get inverse conditional propensities that, by differing from the former probabilities, would contradict the assumption that such probabilities are propensities.
Note that both Salmon’s and Gillies’ propensities (in the introductory versions above) were evaluated at the time of the conditioning event. Although that was not unnatural, because the (absolute) propensity for the outcome A of a set-up S, say prop(A | S), will exist when S does (it being a measure of the indeterministic tendency of S to produce A), Humphreys evaluated his direct and inverse propensities at the same time (a time before both the conditioned and the conditioning events), because standard probabilities (whether absolute, direct or inverse) are defined relative a set of possibilities (i.e. F, in §1) which remain fixed.
Humphreys (2004: 669-71) considered the following three ways of defining inverse propensities.
(i) Zero influence, according to which the propensity (at t = 0) for A to occur (at t = 1) given B (at t = 2), say prop0(A1 | B2), always equals 0 because conditional propensities are measures of the degree of the causal influence that the conditioning event (B2) has upon the conditioned event (A1), and presumably a later event cannot causally influence an earlier one (whence this principle appears prima facie reasonable).
(ii) Fixity, according to which if it is given that B then, since B could only be given after t = 1, if B is chancy then it must also be given that A occurred or not (since the past is no longer chancy), whence prop0(A1 | B2) should be either 1 or 0. However, suppose that it is now t = 2. The original propensity (at t = 0) for A to occur (at t = 1) has not changed. So perhaps it is only a slightly deceptive notation (that usually leaves the conditioning set-up implicit) which suggests that the propensity should be updated to prop2(A1) (which would surely be a meaningless collection of symbols if, as seems likely, backward causation is impossible.) After all, there is a more appropriate way for B to be given, as McCurdy (1996) pointed out—the way of Kolmogorov’s conditional probabilities (see below).
(iii) Conditional independence, which concludes, from the impossibility of backwards causation, that the propensity (at t = 0) for A to occur (at t = 1) would be unaffected by later events, whence prop0(A1 | B2) = prop0(A1). Humphreys prefers this principle, and it does (as mentioned above) seem to make sense. E.g., if we toss a coin with prop(H) = ½ and get H, then it remains true that the physical probability of getting H when we tossed it was ½.
Humphreys (2004) derived contradictions from each of those principles, whence one resolution would be to regard such single-case propensities as interpretations of just part of standard probability theory—or equivalently, to regard the axioms for propensities as non-standard. But (as mentioned above) Humphreys’ arguments do not challenge McCurdy’s co-produced conditional propensities, whence the single-case propensity interpretation emerges unscathed.
According to the co-production interpretation, taking as given B2 means that we consider only the possible futures that contain B2 (assuming that some do), so it is certainly a well-motivated interpretation from the point-of-view that regards propensities as measures of the possible futures (analogous to the internal proportions of Laplace’s urn).
Since only those futures are considered, prop0(A1 | B2) is just that part of prop0(A1) that also gives rise to B2—i.e. prop0(A1 and B2)—divided by prop0(B2). The co-production definition of conditional propensities is therefore Kolmogorov’s measure-theoretic definition of conditional probability rewritten for propensities (at t = 0).
And since Humphreys’ paradoxical contradictions were consequences of a difference between standard conditional probabilities and conditional propensities, his paradox is avoided. Humphreys (2004: 674-5) tried to show that it was not, but in his first argument he assumed principle (iii) (whereas those principles are clearly false under co-production interpretations), while his other argument was as follows:
To represent a conditional propensity as a function of two absolute propensities, as co-production interpretations do, is to deny that the disposition inherent in the propensity can be physically affected by a conditioning factor. This is, at root, to commit oneself to the position that there are conditional probabilities but only absolute propensities. (Humphreys 2004: 675)
However, not only are co-produced conditional propensities exactly as conditional as conditional probabilities are, surely propensities should only be affected by their generating conditions, not by anything else. In conclusion we have, as yet, no reason to reject single-case propensities, as the most realistic way to interpret indeterministic physical probabilities, because at least three kinds—those with conditional values given by (i), by (iii), or by co-production—can, at least prima facie, cope with Humphreys’ paradox. In Section 9 (added in 2007) some further arguments against single-case propensities will be considered (and rejected) in less detail.