axh404 667..680 Brit. J. Phil. Sci. 55 (2004), 667–680, axh404 Some Considerations on Conditional Chances Paul Humphreys ABSTRACT Four interpretations of single-case conditional propensities are described and it is shown that for each a version of what has been called ‘Humphreys’ Paradox’ remains, despite the clarifying work of Gillies, McCurdy and Miller. This entails that propensities cannot be a satisfactory interpretation of standard probability theory. 1 Introduction 2 The basic issue 3 The formal paradox 4 Values of conditional propensities 5 Interpretations of propensities 6 McCurdy’s response 7 Miller’s response 8 Other possibilities 8.1 Temporal evolution 8.2 Renormalization 8.3 Causal influence 9 Propensities to generate frequencies 10 Conclusion 1 Introduction Donald Gillies’ illuminating survey article on propensities (Gillies [2000]) discusses a number of responses to what has been called ‘Humphreys’ Paradox’, 1 henceforth abbreviated to HP. The essence of HP is that single- case conditional propensities can lead to an inconsistency when principles such as Bayes’ Theorem are used to invert those conditional propensities. There now exists a body of literature responding to HP, much of which has helped to refine our understanding of what the formal constraints on 1 The name originates with Fetzer ([1981]). # British Society for the Philosophy of Science 2004 propensities must be. My purpose here is to assess the major lines of response to HP in a way that I hope will illuminate the nature of conditional propen- sities. My conclusion will be that none of the existing responses undermines the principal consequence of HP that conditional single-case propensities cannot be standard probabilities. This indicates that the nature of propensities cannot be properly captured by standard probability theory. It is not often remarked that at least three different versions of the paradox have been proposed. This variety of versions has arisen because the core idea behind the problem is easily conveyed informally, and discussions which present the problem in this informal way have fastened upon different aspects of the basic problem. As a result, not all of the replies to HP are replies to the same thing. The original version of the paradox (Humphreys [1985]) and the two other versions discussed here are formal paradoxes in the sense of giving rise to explicit inconsistencies. Other, more casual, formulations are paradoxes only in the sense of presenting a result which conflicts with ordinary expecta- tions. It is the formal versions that require our attention because, for those, a satisfactory solution is required. 2 The basic issue The essence of the issue can easily be conveyed. Suppose some conditional propensity exists, the propensity for D to occur conditional on C, Pr(D j C).2 For concreteness, consider the propensity of an individual to come down with influenza (event D), conditional upon his having been exposed to some other specific individual with this illness (event C). This is the kind of case to which propensities should be applicable, if single-case conditional propensities are countenanced at all, because a primary reason for introducing propensities was to deal with objective chances attached to specific situations. Standard theories of conditional probability require that when P(DjC) exists, so does the inverse conditional probability P(CjD). The value of P(CjD) can easily be calculated within these standard theories by using the calculus of elementary probability, for example via a basic form of Bayes’ Theorem: P(C j D)=P(D j C) P(C)/P(D) Yet the inverse propensity in the above example, Pr(C j D), the propensity of the individual to have been exposed to the carrier, given that he gets influenza at some later time, is not related to Pr(D j C) in any simple way, if indeed it is mathematically dependent at all. One might even doubt whether such an inverse propensity exists. As we shall see in a moment, it can easily be shown 2 Propensities are indicated by the notation ‘Pr’; probabilities by ‘P’. 668 Paul Humphreys that the inverse probability calculations based on Bayes’ Theorem and related results lead to an inconsistency when supplemented with a simple principle about conditional propensities. 3 The formal paradox The original formal argument in Humphreys ([1985]) used an example as an illustration, although the example was intended to be representative of any system generating non-extremal conditional propensities. It involves photons being emitted from a source at time t and impinging upon a half-silvered mirror. Some of those photons are transmitted through the mirror, others are reflected from it, and what a particular photon 3 does is irreducibly indeter- ministic. We have two events It0 (a particular photon being incident upon the mirror) and Tt00 (that photon being transmitted through the mirror), where Tt00 occurs later than It0. There is also a set of background conditions Bt which includes all the features that affect the propensity value at the initial time t, which is earlier than the times of both It0 and Tt00. The fact that this single-case propensity value Prt(.j.) is attributed at the time t and not at the time of the conditioning event is a matter of some importance, as we shall see. The assumptions fall into two groups. The first contains assignments of conditional propensity values: (i) Prt(Tt00 j It0Bt) ¼ p >0 (ii) 1 >Prt(It0 j Bt) ¼ q >0 (iii) Prt(Tt00 j :It0Bt) ¼ 0 where the arguments of the propensity functions are names designating spe- cific physical events. They do not pick out subsets of an outcome space as in the measure-theoretic approach. 4 The second group consists of a single principle of conditional independence. It asserts that the propensity values of earlier events do not depend upon the occurrence or non-occurrence of later events: ðCI) PrtðIt0 j Tt00 BtÞ ¼ PrtðIt0 j :Tt00 Bt ¼ PrtðIt0 j BtÞ (Note that here CI stands for conditional independence, not causal independence.) 3 For the purposes of the example, I believe it is correct to speak of a particular photon. One may think of this in terms of the rate of emission of photons being so low that the probability of there being more than one emitted during a typical transit time is negligible. 4 This is to keep the propensities oriented towards material, rather than formal, entities. By doing so, it requires significant metaphysical commitments to the existence of less than fully specific properties, such as ‘is even’, in order to maintain contact with common probability attributions, the discussion of which I shall not pursue here. Those who wish to avoid such commitments may simply think in terms of their favourite theory of events. Some Considerations on Conditional Chances 669 From these four assumptions alone, it is straightforward to show that the use of Bayes’ Theorem results in an inconsistent attribution of propensity values, because Prt(It0 j Tt00Bt) ¼ 1 when calculated from (i), (ii) and (iii) using Bayes’ Theorem, but Prt(It0 j Tt00Bt) < 1 when calculated using CI and (ii). In addition, a familiar principle of probability theory, the multiplication principle, also results in inconsistent attributions from the four assumptions together with the use of one standard feature of probability theory (the theorem on total probabilities) which does not require the use of inverse probabilities for its proof. 5 There are three issues of philosophical interest that emerge from the replies to this problem which have been formulated. The first is that they force us to consider how we attribute numerical values to conditional propensities. Are they values to be attributed on the basis of substantive theoretical considera- tions, on the basis of empirical data including experimental data, or, in certain cases, on the basis of broad a priori principles? There are interesting differ- ences in the attributions of such values amongst the published responses to the paradox, and it is revealing to isolate the principles on the basis of which these attributions are made. The second issue is the need to be quite clear about what conditional propensities are—what bears them, what are their argu- ments, and whether they represent degrees of causal influence. The third issue concerns the dynamics of propensities and how their values change over time. 4 Values of conditional propensities The three forms of the paradox discussed in the literature can be characterized in terms of the principles they use to attribute values to conditional propen- sities Prt(It0 j Tt00) when t00 is later than t0.6 (I assume here, in addition, that t is earlier than t0.) The first principle is the conditional independence principle CI, which claims that any event that is in the future of It0 leaves the propensity of It0 unchanged; i.e. Prt(It0 j Tt00) ¼ Prt(It0). This principle reflects the idea that there exists a non-zero propensity at t for It0 to occur, and this propensity value is unaffected by anything that occurs later than It0. A second principle, which we may call the zero influence principle, holds that when t00 is later than t0, Prt(It0 j Tt00) ¼ 0. That is, any event Tt00 that is in the future of It0 is such that the propensity at t for It0, conditional upon Tt00, is zero. This second view is appealing to those who consider conditional propensities to represent the degree of causal influence between the conditioning and the conditioned events. A third principle, which we can call the fixity principle, first represented 5 For details see Humphreys ([1985], p. 562). 6 To avoid quibbles, we can set aside the case where t0 and t00 are contemporaneous. Nothing of any great importance hinges on that case, except the truth status of Prt(It0 j It0) ¼ 1. From here onwards, I shall drop explicit notation about the background conditions Bt unless otherwise indicated. 670 Paul Humphreys in Milne ([1986]), claims that when t00 is later than t0, Prt(It0 j Tt00) ¼ 0 or Prt(It0 j Tt00) ¼ 1. This is because by the time the later event Tt00 has occurred, the occurrence or non-occurrence of the earlier event It0 is already fixed. 7 Each of these principles might be justified on the basis of an a priori argument, on the basis of a posteriori argument based on empirical evidence, or on a case- by-case basis using experimental manipulations. We have already seen how employment of principle CI leads to an inconsistency with the use of Bayes’ Theorem. A parallel argument using the zero influence principle results in a similar inconsistency. In the case of the fixity principle, because this is an indeterministic system, it is not determined whether the event Tt00 will occur once the event It0 has occurred. So there will be cases in which Prt(It0 j Tt00) ¼ 0 on the fixity view, again producing an inconsistency with the results of Bayes’ Theorem. Similar arguments give rise in each case to violations of the multi- plication principle, as the reader can easily check. 5 Interpretations of propensities A traditional taxonomy of propensities separates single-case accounts from long-run accounts. The former consider a propensity to be a disposition of a specific system to result in a specific outcome under specific test conditions. The latter takes a propensity to be a disposition of a specific system to produce specific values of frequencies under specific test conditions. A finer-grained account of single-case conditional propensities has emerged as a result of addressing HP. Four of these are the most important. A co-production interpretation considers the conditional propensity to be located in structural conditions present at an initial time t, with Prt(�j�) being a propensity at t to produce the events which serve as the two arguments of the conditional propensity. The positions of Miller ([1994], [2002]) and McCurdy ([1996]) fall into this category. Gillies ([2000]) has proposed a long-run version of the co-production interpretation. Miller, who first formulated the approach in his 1994 book Critical Rationalism, expressed the position in a later paper in this way: Pt(At0 j Ct00) is the propensity of the world at time t to develop into a world in which A comes to pass at time t0, given that it (the world at time t) develops into a world in which C comes to pass at time t00. (Miller [2002], p. 113) 8 7 There is a fourth position which asserts that inverse propensities are in general meaningless and hence that propensities are, at best, an incomplete interpretation of the probability calculus. I shall not discuss this position, because inverse propensities such as the propensity for an individual to have been exposed to a carrier given that he gets the ‘flu’ at some later time are clearly legitimate objects of discussion. Terms referring to them do not lack meaning, even if the associated propensities do not exist or, what is not the same, have value zero. 8 Also in Miller ([1994]), p. 189. Some Considerations on Conditional Chances 671 In McCurdy ([1996], pp. 108–9), we find this description: The events described by the background conditions are responsible for the assignment of particular probability values to the members of the (pre- viously established) event space, because these events are responsible for the production of the events in the event space [. . .] [T]he values assigned to conditional and inverse conditional propensities are intended to provide a measure of the strength of the propensity for the system to produce the two future events in the manner specified. The key feature of co-production interpretations is that the representations of the conditions present at the initial time t are not included in the algebra or F-algebra of events within the probability space, in contrast to the events that occur as a result of those initial conditions, representations of which are a part of the formal probability space. The probability measure is defined only over those latter events, and the conditional probability measure is then defined in the usual way (for P(B) > 0) as P(A j B) ¼ P(A&B)/P(B). In co-production interpretations, the conditional propensity is attributed to the system at the initial time t and it is a propensity for the system, characterized by the background conditions Bt, to produce events in the probability space that are related by the standard definition just given. The conditional propensity is not taken to be a property of the system at the time of the later conditioning event. A co-production interpretation is in sharp contrast to a temporal evolution interpretation, which takes the propensity to have an initial value at t, with the propensity then evolving temporally, usually with its value changing under the influence of subsequent events. Within temporal evolution interpretations, the time order of events is crucial: when t < t0 < t00, the propensity has an initial value at t, but this changes as later events occur between t and t0 and the system evolves. When the conditioning event D occurs earlier than the conditioned event C (i.e. t0 < t00), the conditional propensity Prt(Ct00 j Dt0) is simply a tem- poral update of the original propensity Prt(Ct00) which was evaluated at the initial time t. When the time of the conditioning event D is later than that of the conditioned event C (i.e. t0 > t00), the propensity evolves to the point where Ct00 either occurs or fails to occur at t 00, and anything temporally subsequent is irrelevant. Principle CI is thus true for the temporal evolution interpretation. Apparently similar to a temporal evolution interpretation, but in fact quite distinct, are renormalization interpretations. Here, instead of the temporal parameter t continuously evolving within real time, the conditioning event forces a jump from t to the time of the conditioning event, and the propensity value is determined at that new time. When t < t0 < t00, this produces the same results as for the temporal evolution interpretation. But when t0 is later than t00, the conditioning process ‘jumps over’ the events between t and t0, including t00, and determines the propensity value at time t0, in contrast to the temporal evolution interpretation, where the evolution of the propensity must pass 672 Paul Humphreys through events in their temporal order. The fixity view is entailed by the renormalization interpretation. This is because for conditional propensities such as Prt(Ct00 j Dt0), by the time the conditions Dt0 are in place, the event Ct00 either has already occurred or has failed to occur. One caveat: in standard probability theory, the renormalization procedure allows us to identify the conditional probability P(C j D) with an unconditional probability PD(C) restricted to a new domain of subsets of D. Because propensities are features of the world and are not properties of sets, that approach is inappropriate for propensities. The fourth position, the causal interpretation, takes the conditional pro- pensity to represent the degree of causal influence between the conditioning event and the conditioned event. An advocate of a causal interpretation, although in the context of probabilistic conditionals rather than conditional probabilities, is Fetzer ([1981]). The causal interpretation entails the zero influence view, on the usual position that there is no temporally backwards causation. Having laid out these positions, we can now consider the various responses to HP. I shall show that for each of the four interpretations of conditional propensities, at least one of the three principles CI—the fixity principle, or the zero influence principle, on the basis of which attributions of conditional propensity values are made—is true and hence that, for each of the four interpretations of propensities just described, a version of HP exists. 6 McCurdy’s response One direct response to the original version of HP is to argue that principle CI (i.e., Prt(It0 j Tt00Bt) ¼ Prt(It0 j :Tt00Bt) ¼ Prt(It0 j Bt) ¼ q) is false for the photon case and, by implication, that it fails for other situations having a similar structure. This position is taken in McCurdy ([1996]), where, as we have seen, he defends a co-production interpretation of conditional propensities. I shall respond to McCurdy’s arguments in two different ways. First, I shall argue that although a plausible case can be made for the co-production interpretation in my original photon example, a structurally similar example illustrates that this response will not work in that and other cases. Second, I shall argue that the co-production interpretation itself is seriously flawed as an interpretation of conditional propensities, at least in the sense that at best it preserves probability theory at the expense of losing the characteristic dis- positional content of conditional propensities. At the heart of McCurdy’s argument against the original HP is his claim that he can establish, without any appeal to the probability calculus, that Prt(It0 j Tt00Bt) ¼ 1, a value that is inconsistent with the assignment q < 1 given Some Considerations on Conditional Chances 673 to the conditional propensity by CI together with the original assignment (ii). Here is McCurdy’s argument: Instead of utilizing the inversion theorems to determine the value of Prt(It0 j Tt00Bt), the value can be arrived at as follows: the value of Prt(It0 j Tt00Bt) must be one since the description of the system indicates that the system is arranged in such a manner that if the system produces a photon that is transmitted at t00 then the system must also produce a photon that impinges upon the mirror at t0. Indeed it is assignment iii) [i.e. that Prt(Tt00 j :It0Bt) ¼ 0] that provides the information that the system is arranged in this manner, but it is the arrangement of the photon system itself—and not the value of Prt(Tt00 j :It0Bt)—that demands that Prt(It0 j Tt00Bt) ¼ 1. (McCurdy [1996], pp. 110–1) It should be clear from this quotation that McCurdy is not making the error of appealing to the fact that we can infer with certainty from the structure of the experimental arrangement that when a photon has been transmitted it must have been incident upon the mirror. Rather, it is the physical structure of the arrangement at t which he claims is the basis for the attribution of the value Prt(It0 j Tt00Bt) ¼ 1, and an appeal to a co-production interpretation is clearly at work here. The most direct evidence for this is his claim that: The fact remains that, although the events It0, Tt00, and :Tt00 lack common causal factors between the times t0 and t00, the events It0, Tt00, and :Tt00 share common causal factors that are effective between t and t0. Specifically, the photon transmission arrangement itself (described by Bt) provides a host of common causal factors. This fact is responsible for the failure of prin- ciple CI: if the system produces event Tt00, then it must have exhibited certain causal factors, some of which have an influence on the event It0. (McCurdy [1995], p. 116) Because of some previous remarks in his paper alluding to factors responsible for the momentum of the emitted photons, it is possible that the photon example misleadingly suggests some quasi-deterministic aspects of the funda- mentally indeterministic propensity at t, Prt(It0jBt). So it may help to consider a somewhat different example, consisting of a radioactive source of alpha par- ticles with a spherical radiation detector completely surrounding the source. The detector is shielded from all other sources of radiation but is of less than perfect reliability so that not all emitted particles are detected. The propensity for an alpha particle to be emitted in a specified time period is, I hope suncontroversially, taken to be fundamentally indeterministic. Let Prt(Et0 j Dt00)be the propensity at time t for an alpha particle to be emitted during the short time interval t09 conditional upon that alpha particle being detected at t00, where t < t0 < t00. This example is then formally identical to the 9 The use of a time interval rather than an instant introduces no essentially new considerations. 674 Paul Humphreys photon example, and it should be clear that because of the irreducibly indeterministic nature of radioactive decay, there are no common causal factors between t and the beginning of t0 (or the precise time of emission, if you prefer) on the basis of which one could truly assert that, if the system produces the event Dt00, then it must have exhibited certain causal factors between t and t0 some of which have an influence on Et0. Principle CI is, then, true, and I believe evidently true, for this case—i.e. Prt(Et0 j Dt00) ¼ Prt(Et0 j :Dt00) ¼ Prt(Et0). Prt(Et0 j Dt00)—is thus not equal to unity as the co-production interpretation requires. Given that this is so, HP still stands. My second reply to McCurdy’s argument is more general and applies to any co-production interpretation of conditional propensities. A co-production interpretation does preserve the formal structure of probabilities on the event space, but at a price. The price is that under this interpretation, the structural basis of the propensity presents the relation between the conditioning and conditioned events as a relation between probability measures rather than as a material relation between concrete events. There is no propensity relationship between the conditioning and conditioned events in the conditional probabil- ity P(A j B) on this view. It is thus not a single-case conditional propensity, strictly speaking. A major appeal of single-case propensities has always been their shift in emphasis from the outcomes of trials to the physical dispositions that produce those outcomes. 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. I now turn to the response given by David Miller in his 2002 paper to a different version of the paradox. 7 Miller’s response David Miller invented the co-production interpretation of propensities, ver- sions of which may be found in his ([1994]) and ([2002]). 10 Miller formulates the paradox in this way: Intuitively we may read the term Pt(AjC) as the propensity at time t for the occurrence A to be realized given that the occurrence C is realized. If C precedes A in time, this presents no extraordinary difficulty. But if C follows A, or is simultaneous (or even identical) with A, then it appears that there is no propensity for A to be realized given that C is realized; for 10 In x9.6 of his ([1994]), Miller constructs a novel relative frequency account of probability but its merits can be considered separately from those of propensity theories. Some Considerations on Conditional Chances 675 either A has been realized already, or it has not been realized and never will be. In a late work Popper wrote that ‘a propensity zero means no pro- pensity’ (Popper [1990], p.13). If the converse holds too, we may conclude that Pt(A j C) has the value zero unless C precedes A in time. But it is easy to construct examples in which none of Pt(AjC), Pt(A), and Pt(C) is zero. The simplest version of Bayes’s theorem, to the effect that Pt(A j C) ¼ Pt(C j A)Pt(A)/Pt(C), is thereby violated. (Miller [2002], p. 112) We see from this quotation that the version of HP that Miller is addressing is based on the zero influence principle, reinforced by an appeal to fixity (although not to the fixity principle as a basis for attributing propensity values). His response to the problem is contained in these two paragraphs: The objection involves a subtle misreading into the phrase ‘given that’ of an inappropriate temporal reference. Suppose that A is an occurrence that is realized, if it is realized at all, at time t0, and that C is an occurrence that is realized, if it is realized at all, at time t00. Talk of the propensity at time t for A to be realized (at time t0) given that C is realized (at time t00) does not mean that the realization of C at time t00 is supposed to be given at time t0. It means that the realization of C at time t00 is supposed to be given at time t. Of course, if t is earlier than t00, then this supposition is subjunctive. But provided that t is earlier than t00, there is no difficulty in principle in attributing a positive value to Pt(At0 j Ct00). Note that if t00 too is earlier than t0, and C comes to pass at t00, then there is an innocuous sense in which the occurrence of C is given at t0—by the time t0 is reached, C has been realized. This is not the sense of the phrase ‘given that’ that is central to the theory of relative probability. Only if t is earlier than t00 can Pt(At0 j Ct00) differ from Pt(At0), the absolute propensity at t for A to be realized at t0. We may set aside as uninteresting the case in which t is not earlier than t00. Now it should be obvious that to suppose at t that C comes to pass at t00 is not to suppose incoherently that every occurrence dated between t and t00 also comes to pass; it is not even to suppose at t that we are already at t00. Provided therefore that t is earlier than t00, to suppose at t that C comes to pass at t00 is not to suppose either that A comes to pass at t0 or that it does not come to pass at t0, even if t0 is earlier than t00. In consequence there is, if t0 is earlier than t00, nothing in principle that disallows Pt(At0 j Ct00) from taking any value greater than zero. Of course, the value of Pt(At0 j Ct00) will be either zero or unity unless t is earlier than t00. (Miller [2002], pp. 112–3) Miller’s response to HP as he has formulated the problem is indeed a powerful response to versions of HP that are based on the zero influence principle. It is also a response to versions that are based on the fixity principle. It is not, however, a response to the original HP, because subscribing to the principle CI does not entail attributing a zero value to the relevant conditional propensity. In Section 5 of Miller ([2002]), the co-production interpretation of conditional propensities is once again endorsed, and Miller asserts in Section 1 676 Paul Humphreys that he is ‘largely in agreement with’ the arguments contained in McCurdy’s paper. So it is perhaps taken to be implicit that the co-production interpreta- tion provides an effective response to principle CI. If so, then one can bring to bear against this view the arguments presented in Section 6 above to the effect that one has retained the structure of conditional probabilities at the expense of removing from conditional propensities what has traditionally been considered essential to them. 8 Other possibilities We can summarize the relation between the three principles and the co-production interpretation of conditional propensities as shown in the first line of Table 1. Our discussion of the remaining elements of the table can be brief, but it requires a little elaboration on the three other interpretations of propensities— temporal evolution interpretations, renormalization interpretations, and causal interpretations. The reader may not find all of these interpretations congenial but they do capture, perhaps somewhat crudely, attitudes towards conditional propensities that I have encountered in published and unpub- lished discussions of HP. 11 8.1 Temporal evolution Recall that in this view, the value of the propensity is firmly rooted in present material conditions, and that value dynamically evolves through time. In that sense, we can view the time t as a parameter that is continuously updated. Because only current and not hypothetical future situations affect the value of the propensity in this interpretation, principle CI is true. For similar reasons, the fixity principle and the zero influence principle are false. Table 1 t < t0 < t00 Principle CI Prt(Ct0 j Dt00) ¼ Prt(Ct0 j Dt00) ¼ Prt(Ct0) Fixity Prt(Ct0 j Dt00) ¼ 0 or 1 Zero Influence Prt(Ct0 j Dt00Bt) ¼ 0 Co-production True False False Temporal evolution True False False Renormalization False True False Causal False False True 11 An issue that I want to set aside here is the effect that HP has on our theory of rational degrees of belief through Lewis’s Principal Principle or something akin to it. Propensities are objective features of the world and the view that they must be constrained by subjective probabilities is not one that I find attractive. But there is no doubt that ignoring the way the world is can be financially ruinous if one is inclined to gamble. AQ: Please provide Table caption Some Considerations on Conditional Chances 677 8.2 Renormalization Here Prt(Ct0 j Dt00) takes the occurrence of the event Dt00 as updating the initial probability assignment so that the conditions Dt00 at t 00 are part of the basis of the propensity. With this view in mind, since t00 is later than t0, the fixity principle is true and in consequence the zero influence principle will not be true in general. The principle CI will be false because Prt(Ct0) will not have extremal values in indeterministic contexts. 8.3 Causal influence Within this interpretation, the conditional propensity captures the degree of causal influence of the conditioning event, D, on the main event of interest, C. Thus, assuming there is no temporally reversed causal influence, the zero influence principle is true for this interpretation. 12 Because the fixity principle also allows values of unity to be attributed to the conditional propensity, the fixity principle is false under this interpretation. Finally, since the degree of causal influence on Ct0 by Dt00 is zero, but in general Prt(Ct0) 6¼ 0, principle CI is also false for this interpretation. This concludes our case-by-case evaluation of the twelve possibilities. We now see that for each of the four interpretations of conditional propensities, there exists exactly one principle governing the attribution of conditional propensities that is true under that interpretation. In consequence, for each of the four interpretations there is exactly one version of HP which shows that conditional propensities cannot be probabilities. The view that propensities are probabilities cannot be saved by switching interpretations. 9 Propensities to generate frequencies Donald Gillies ([2000]) adopts what he terms a ‘long run propensity’ view. This is ‘one in which propensities are associated with repeatable conditions, and are regarded as propensities to produce, in a long series of repetitions of these conditions, frequencies which are approximately equal to the probabil- ities’ (p. 822). Although my own preference is for single-case propensities, which in certain stable conditions can ground long-run propensities through limit results, we need to consider Gillies’ solution to HP, which he discusses using Milne’s version and the frisbee-producing machine example of Earman and Salmon ([1992]). The latter example involves two frisbee-producing machines, one of which produces 800 a day with 1% defective frisbees, the 12 It is this interpretation that is explicitly rejected in Miller ([2002], p. 115) and in Gillies ([2000], p. 831). 678 Paul Humphreys other of which produces 200 a day with 2% defective. The problematical pro- pensity is the propensity for a defective frisbee to have been produced by the first machine. Gillies’ account of this propensity, which has the value 2/3, is The statement Pr(M j D&S) ¼ 2/3 means the following. Suppose we repeat S each day, but only note those days in which the frisbee selected is defective, then, relative to these conditions there is a propensity that if they are instantiated a large number of times, M will occur, i.e. the frisbee will have been produced by machine 1, with a frequency approximately equal to 2/3. (Gillies [2000], p. 829) There is in this statement a reference to only noting occurrences within which a frisbee is defective, which tints the solution with an unnecessarily epistemic colouring, but that is easily eliminated by simply considering the set of out- comes involving defective frisbees. With that minor adjustment, Gillies’ solution has the required objectivity, 13 but it reintroduces exactly the situation from which propensity accounts were intended to rescue us—the relativization of a relative frequency to a reference class and, within von Mises’ and some other frequency theories, the need to provide an objective criterion of ran- domness. In so doing, it loses exactly the features of propensities which proved attractive to many of us. As such, despite its ingenuity, Gillies’ solution cannot be a complete account of propensities. 10 Conclusion The features of propensities explored here force us to confront an important question. If conditional propensities cannot be correctly represented by stand- ard probability theory, what does that say about the status of probability theory? In Humphreys ([1985], Section IV), I tentatively suggested that prob- ability theory should be viewed as a contingent theory. David Miller ([2002], p. 115) suggests something rather different: ‘it is a factual matter whether propensities obey the calculus of probabilities,’ and floats the idea, derived from Popper, that propensities are generalized forces. Whatever is the truth about these matters, HP is not a mere puzzle. At the very least it tells us that standard probability theory does not have the status of a universal theory of chance phenomena with which many have endowed it. Acknowledgements I am indebted to David Miller for correspondence and conversations about these topics, and to Donald Gilles for similarly helpful correspondence. An 13 Objective in the sense of being a feature of the objects involved. Some Considerations on Conditional Chances 679 important catalyst over the years in these discussions has been Jim Fetzer. A preliminary version of this paper was presented to audiences at All Souls College, Oxford, and the Philosophy Department at the University of Konstanz. Their reactions were instrumental in improving the arguments. Corcoran Department of Philosophy 512 Cabell Hall University of Virginia Charlottesville Virginia 22904-4780 USA pwh2a@virginia.edu References Earman, J. and Salmon, W. [1992]: ‘The Confirmation of Scientific Hypotheses’ in M. H. Salmon et al. 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[1990]: A World of Propensities, Bristol: Thoemmes Antiquarian Books Ltd. 680 Paul Humphreys