Deterministic Chance Luke Glynn [Preprint of article published in British Journal for the Philosophy of Science, 61(1), March 2010, pp. 51-80 http://bjps.oxfordjournals.org/content/61/1/51.abstract. Please cite published version only.] ABSTRACT I argue that there are non-trivial objective chances (that is, objective chances other than 0 and 1) even in deterministic worlds. The argument is straightforward. I observe that there are probabilistic special scientific laws even in deterministic worlds. These laws project nontrivial probabilities for the events that they concern. And these probabilities play the chance role and so should be regarded as chances as opposed, for example, to epistemic probabilities or credences. The supposition of non-trivial deterministic chances might seem to land us in contradiction. The fundamental laws of deterministic worlds project trivial probabilities for the very same events that are assigned non-trivial probabilities by the special scientific laws. I argue that any appearance of tension is dissolved by recognition of the level-relativity of chances. There is therefore no obstacle to accepting non-trivial chance-role-playing deterministic probabilities as genuine chances. 1 Introduction 2 Schaffer's Incompatibilist Argument 2.1 Chance and Credence 2.2 Chance and Possibility Deterministic Chance 2 2.3 Chance and Laws 3 Special Scientific Laws 3.1 Probabilistic Special Scientific Laws in Deterministic Worlds 3.2 Lewis's Humean Analysis of Laws 3.3 Special Scientific Laws and the Law Role 4 Deterministic Chance 4.1 Chance and Laws Again 4.2 Chance and Credence Again 4.3 Chance and Possibility Again 5 Chance and Causation 6 Conclusion Appendix: Times, Levels and Chance Setups 1 Introduction It has often been assumed without argument that determinism is incompatible with non-trivial objective chance. Popper ([1982], p. 105), for example, simply says: [O]bjective physical probabilities are incompatible with determinism; [...] In a similar vein Lewis ([1986c], p. 118) states that: If the chance [of a fair coin landing heads] is zero or one, [...] then it cannot also be 50%. To the question how chance can be reconciled with Luke Glynn 3 determinism, or to the question how disparate chances can be reconciled with one another, my answer is: it can't be done. Some argument for such claims is needed. Deterministic physical theories such as Classical Statistical Mechanics (CSM) yield non-trivial probabilities that are used by physicists in prediction and explanation. Both the manner of their genesis and the nature of their employment make these probabilities good candidates to be considered objective chances.1 But Schaffer ([2007]) has recently argued that they are not chances. His argument is the most sophisticated and well-developed such argument currently to be found in the literature. He argues that although deterministic probabilities such as those generated by CSM may be formally eligible to count as chances–they are generated by a function from propositions, times and worlds onto the closed unit interval in accordance with the axioms of the probability calculus–they do not qualify as chances because they fail to play the chance role. Generating values that play the chance role is what distinguishes the chance function from the many other formally eligible functions that no-one would regard as the chance function (Schaffer, op. cit., p. 123). Schaffer claims that the role of chance is characterised by its connections with 'credence, possibility, futurity, intrinsicness, lawhood, and causation' (ibid.). He argues that, in a deterministic world, a chance function outputting non-trivial values would violate three of these connections: namely, those from chance to credence, possibility and lawhood (ibid., p. 132).2 By contrast, a function that outputs just trivial values in deterministic worlds plays the chance role perfectly, and so this should be regarded as the chance function (ibid. p. 127). Schaffer's conclusion is that there are therefore no non-trivial deterministic chances. His argument shall be outlined in §2. Deterministic Chance 4 In §§3-4 I shall seek to show that Schaffer is wrong: there are non-trivial deterministic probability functions that underwrite the connections from chance to credence, possibility and lawhood, and that therefore play the chance role. Important to the demonstration that this is so is the demonstration that chance is level-relative (this notion of level-relativity is explicated in §4). Recognition of the level-relativity of chances allows us to see the flaws in Schaffer's arguments.3 Not only are there non-trivial deterministic probability functions that play the chance role, some such functions play it better than any trivial deterministic probability function. In §4.1 and §4.2 I argue that, for at least some deterministic worlds, only a function that outputs non-trivial values can underwrite the connections from chance to lawhood and credence. I seek to reinforce the case for non-trivial objective chances in §5 by considering one of the remaining three connections discussed by Schaffer: that from chance to causation. Schaffer says (op cit. p. 132) he sees 'no problem' regarding the consistency of a non-trivial deterministic chance function with this connection, nor with the other two–from chance to futurity and intrinsicness. I argue that, not only is Schaffer correct in this, but that there are good reasons to think that a trivial deterministic chance function is inconsistent with the chance-causation connection. The conclusion (§6) is that there are non-trivial deterministic chances. 2 Schaffer's Incompatibilist Argument I will now review Schaffer's argument that (non-trivial) deterministic chances would violate the connections from chance to credence, possibility and lawhood and that chance is therefore incompatible with determinism. Luke Glynn 5 2.1 Chance and Credence Schaffer takes the chance-credence connection to be adequately captured by Lewis's ([1986b]) Principal Principle (PP), which can be stated as follows (ibid., p. 87). Let C be any reasonable initial credence function, let t be any time and let w be any world. Let Chtw be the chance function (which receives time and world indices because of the timeand worldrelativity of chance4), let x be any real number in the closed unit interval and let p be any proposition in the domain of the chance function. Finally, let X be the proposition that Ch!" p = x, and let E be any proposition compatible with X that is admissible at t and w. Then: C p X.E = x (PP) The idea is that if one started with a reasonable initial credence function (and updated by conditioning) and one learned that Ch!" p = x and if the rest of one's evidence E were admissible at time t (and consistent with the proposition that Ch!" p = x), then one would have credence in p equal to x. To get a handle on the content of the PP, we need to know what sort of evidence counts as 'admissible'. Lewis's characterization of admissible information is as follows: Admissible propositions are the sort of information whose impact on credence about outcomes comes entirely by way of credence about the chances of those outcomes. (ibid., p. 92) Deterministic Chance 6 Lewis suggests (ibid., pp. 92-6) that two different sorts of information are generally admissible at a time t and world w: historical information (or information about matters of particular fact at times no later than t) and information about the laws of w. This allows him to reformulate the PP in a manner that 'will prove easier to use' (ibid., p. 98), as follows. Let Htw be a proposition giving the complete history of w up to t and let Lw be a proposition giving the laws of nature that obtain at w. These propositions are both admissible at t and w and so, Lewis claims (ibid., p. 96), is their conjunction. Suppose that X is true: that is, Ch!" p = x. Then, since both Htw.Lw and X hold at w, they are compatible. The conjunction Htw.Lw can therefore be substituted for the proposition E to yield (ibid. p. 97): C p X.H!" . L! = x (PP!) Indeed, according to Lewis (ibid., p. 97, [1994], pp. 477-8), the laws of w together with the initial history of w through t entail the chances that obtain at t and w. Hence, Lw.Htw entails X. So X.Lw.Htw can be simplified to Lw.Htw. Since it is also true that Ch!" p = x, PP can be reformulated as follows (Lewis [1986b], pp. 96-7, [1994], p. 487): C p L! .H!" = Ch!" p (RPP) The idea is that the laws of w together with the initial history of w through t entail the chances obtaining at w and t and that reasonable credence for someone whose evidence includes just the laws and initial history (and who updates by conditioning on her evidence) is equal the chance of p that they entail. Luke Glynn 7 We are now in a position to consider Schaffer's argument for the incompatibility of non-trivial deterministic chances with the connection from chance to rational credence. His argument is as follows (op. cit., p. 128; cf. Hoefer [2007], pp. 558-9). Suppose that pe is the proposition that some event e occurs,5 and that w is a deterministic world. Suppose, moreover, that Ch!" p! is a non-trivial chance: that is 1 > Ch!" p! > 0. Now either e occurs in w or it does not. Suppose it does. Then, since w is deterministic, the laws of w together with the history of w up to (and including) any time t entail pe. Accordingly, for any reasonable credence function, C p! L! .H!" = 1. It follows immediately by the RPP that Ch!" p! = 1. Suppose, on the other hand, that e does not occur in w. Then the laws of w together with its history through t entail ~pe. Accordingly, for any reasonable credence function, C p! L! .H!" = 0. It follows immediately by the RPP that Ch!" p! = 0. So, by the assumption that Ch!" p! is a non-trivial deterministic chance, we have that 1 > Ch!" p! > 0. But, by the RPP, we have that Ch!" p! equals either 1 or 0 (depending on whether or not e occurs in w). Contradiction! Schaffer's conclusion: nontrivial deterministic chance assignments are incompatible with the connection from chance to rational credence captured by the PP (of which the RPP is a reformulation). 2.2 Chance and Possibility Schaffer argues that non-trivial deterministic chances would also violate the connection from chance to possibility. He claims (op cit., p. 124) that this connection is captured by what he calls the Realization Principle (RP):6 Deterministic Chance 8 (RP) If Ch!" p > 0, then there exists a world w' such that (i) w' matches w in laws; (ii) w' matches w in occurrent history up until time t; (iii) p is true at w'. Schaffer (ibid., p. 130) argues that non-trivial deterministic chances are inconsistent with the RP as follows. Suppose that w is a deterministic world and Ch!" p! is a non-trivial chance: that is 1 > Ch!" p! > 0. Now either e occurs in w or it does not. If it does then, since w is deterministic, the laws and initial history of w entail pe, so there cannot be a world that has the same laws and initial history as w at which ~pe is true. Therefore, by the RP, Ch!" ~p! = 0 from which it follows by complementation that Ch!" p! = 1. If, on the other hand, e does not occur at w, then the laws and initial history of w entail ~pe, so there cannot be a world with the same laws and initial history as w at which pe is true. It follows directly by the RP that Ch!" p! = 0. So, by the assumption that Ch!" p! is a non-trivial deterministic chance, we have that 1 > Ch!" p! > 0. But, by the RP, we have that Ch!" p! equals either 1 or 0 (depending on whether e occurs in w). Contradiction! Schaffer's conclusion: non-trivial deterministic chance assignments are incompatible with the connection between chance and possibility captured by the RP. 2.3 Chance and Laws Finally, Schaffer (ibid.) argues that non-trivial deterministic chances are incompatible with the connection from chance to lawhood. He claims (ibid., p. 126) that this connection is captured by the Lawful Magnitude Principle (LMP): Luke Glynn 9 (LMP) If Ch!" p = x, then the conjunction of H!" (a proposition giving the history of w up until time t) with the laws of w entails that Ch!" p = x. The LMP says, in effect, that chances are lawfully-projected magnitudes. Schaffer argues (ibid., p. 130) that non-trivial deterministic chances would be inconsistent with the LMP, since 'laws at deterministic worlds do not project chances.' The idea is as follows. Suppose that Ch!" p! is a non-trivial deterministic chance: that is 1 > Ch!" p! > 0. Since w is deterministic, L! .H!" entails either Ch!" p! = 1 or Ch!" p! = 0 depending on whether e occurs at w. It follows that the only chance assignments compatible with the LMP are Ch!" p! = 1 or Ch!" p! = 0. But now we have that 1 > Ch!" p! > 0 and that either Ch!" p! = 1 or Ch!" p! = 0. Contradiction! Schaffer's conclusion: non-trivial deterministic chance assignments are incompatible with the connection between chance and laws captured by the LMP. Since a (non-trivial) deterministic chance function would 'sever the connections from chance to credence, possibility, and lawhood' (ibid. p. 132), Schaffer concludes that '[t]his is hardly a viable conception of chance' (ibid.). In §3-4 I argue that Schaffer is wrong: there are non-trivial deterministic chance functions that are consistent with all three of these connections. Recognition of the levelrelativity of chance allows us to see that this is so and undermines Schaffer's apparent demonstration that these connections, when taken together with the supposition of non-trivial deterministic chances, land us in contradiction. Deterministic Chance 10 Indeed I will argue that, once the level-relativity of chance is recognised, it can be seen that only a chance function that outputs non-trivial values in deterministic worlds can underwrite the connections from chance to credence and lawhood. 3 Special Scientific Laws The main reason to think that there are non-trivial objective chances even in fundamentally deterministic worlds is that there exist probabilistic high-level or special scientific laws even in such worlds (§3.1).7 The probabilities projected by these laws should be regarded as genuine, objective chances because the laws in question are genuine, objective laws. Not only are they accommodated as such by Lewis's Humean analysis of lawhood (§3.2) but, because they play the law role, they must be similarly accommodated by any adequate account of lawhood (§3.3). 3.1 Probabilistic Special Scientific Laws in Deterministic Worlds Quantum mechanics–at least on standard, 'collapse' interpretations (e.g. Copenhagen and GRW)–indicates that the fundamental dynamics of our world are probabilistic. But it is not just fundamental physics that is probabilistic. Many of the high-level or special sciences also give probabilistic laws for events falling under their purview. As already noted, statistical mechanics gives such laws. So does Mendelian genetics. And probabilistic functional laws are encoded in the models of economists and meteorologists. Together with the initial history of the world, these special scientific laws entail non-trivial chances for the events that they concern. Luke Glynn 11 The Mendelian genetic laws of Segregation and of Independent Assortment, for example, give a chance 0.25 for a dihybrid cross between two parents heterozygous for each (binary) trait yielding a child that is homozygous for each trait. So suppose that Jim and Jill are common garden pea plants (pisum sativum) heterozygous for both pea shape and colour (both binary variables, with round and yellow the dominant alleles, and wrinkled and green the recessive alleles), and Tom is a plant produced by crossing Jim and Jill. Let t be the time of crossing, and p the proposition that Tom is homozygous for both shape and colour. Then the Mendelian laws together with the history of the world through t (which includes the fact that Jim and Jill are heterozygous for both shape and colour and the fact that they are crossed), entail a chance 0.25 for Tom's being homozygous for both pea shape and colour. In most cases, the special sciences make no presupposition about whether the fundamental dynamics of the world are deterministic or indeterministic. Where they do, as in CSM, the assumption is often one of microphysical determinism. It follows from the compatibility of the probabilistic special sciences with fundamental determinism that there exist fundamentally deterministic worlds (Newtonian or Bohmian worlds, perhaps) with probabilistic special scientific laws. A lot of work remains to be done to show that the non-trivial probabilities projected by the probabilistic special scientific laws of these deterministic worlds are genuine objective chances. The opponent of deterministic chance will presumably seek to dismiss this claim, arguing that such probabilities are merely epistemic (indeed Schaffer makes precisely this argument, op cit. pp. 136-9). The key to showing that these probabilities are objective chances is, of course, to show that they play the objective chance role. This will at least involve defusing Schaffer's arguments, reviewed in the previous section, that no non-trivial deterministic probability Deterministic Chance 12 function can play the chance role. I shall argue that there are some such functions that can and that, moreover, trivial deterministic functions cannot. It seems that the best place to start in showing that non-trivial deterministic probability functions can play the chance role will be with Schaffer's claim (considered in §2.3 above) that such functions violate the chance-law connection because the laws of deterministic worlds don't project chances. After all, it has just been argued that Schaffer is simply wrong in this claim. True, the fundamental laws of deterministic worlds don't project chances. But not all laws are fundamental. And it has just been argued that there are probabilistic high-level, special scientific laws even in fundamentally deterministic worlds. Schaffer is aware of the vulnerability of his position to this kind of objection. His reaction (ibid., pp. 130-2) is to argue that, on Lewis's Humean view of laws, there are good reasons for regarding these probabilistic special scientific 'laws' as failing to be genuine laws. Before considering his argument, it is worth getting a bit clearer on Lewis's analysis. 3.2 Lewis's Humean Analysis of Laws Lewis's analysis of laws ([1994], pp. 478, 480) is as follows. Consider all deductive systems whose theorems pertain to what happens in history, in the sense that they give either the outcomes or chances of outcomes in various situations. Exclude those systems whose theorems aren't true in what they say about outcomes. (Lewis is attempting to deliver a simultaneous analysis of laws and chances, so it is not yet required that the theorems must be true in what they say about the chances.) Also, exclude any that say what an outcome will be without also saying that the outcome never had a chance of not coming about. The remaining systems may differ in simplicity, strength (or informativeness), and fit (or the chance that they assign to the actual course of history). And there will be trade-offs Luke Glynn 13 between these virtues: simpler systems may be less strong or fit less well, and so on. According to Lewis: The best system is the system that gets the best balance of all three. [...] [T]he laws are those regularities that are theorems of the best system [...] [and] the chances are what the probabilistic laws of the best system say they are. (ibid., p. 480) If it can be shown that (at least some of) the special scientific laws are theorems of the Best System, then it will follow (pace Schaffer) that Lewis's Humean analysis accommodates these special scientific laws as genuine, objective laws. Indeed, there are good reasons to think that (at least some) special scientific laws are theorems of the Best System. In particular, it seems that a system that yields the special scientific laws as theorems will be much stronger or more informative than one that yields merely the fundamental, microphysical laws. This is because the microphysical laws tend to fall silent about the higher-level properties that the special scientific laws relate. The high-level properties in question are typically multiply realisable at the microphysical level. That is, different distributions of microphysical properties may realise the same high-level property. So, for example, two individuals a and b sharing some biological property F could be in rather different microphysical states, consisting of differing distributions of particles with differing values of mass, spin, charge, etc. If the property F were to be characterised in microphysical vocabulary, it would be in terms of a very long disjunction with each disjunct–itself a highly complex conjunction of propositions attributing microphysically simple properties to particles–corresponding to one possible microphysical realisation of F (cf. Hoefer [2007], p. 593). Deterministic Chance 14 This relationship of realisation grounds the distinction between higherand lowerlevels. Two or more sciences can be regarded as characterising distinct levels when various distributions of the properties of concern to one realise those of concern to the other. Where this is not the case, the sciences in question should be regarded as each providing partial characterisations of the same level. Each level, consisting of the distribution of a certain set of properties together with the nomic relations between those properties, constitutes a relatively closed system in the sense that adequate (non-reductive) explanations for the instantiation of any property within that set can be given in terms of the instantiation of other properties in that set together with the nomic connections between them. Where two or more sciences partially characterise the same level, each taken individually will lack explanatory closedness. Climate change, for instance, cannot be adequately explained without appeal to human industrial activity as well as geological and cosmological factors. This reflects the fact that climatology, economic sociology, geology and cosmology do not each characterise distinct levels, but rather each provide partial characterisations of a certain relatively 'high' level. The multiple realisability of higher-level properties at the microphysical level is the key to understanding how special scientific laws enhance the informativeness of a system that entails them. Suppose that the biological property F can be realised by any one of the complex microphysical states, α, β, γ, etc. And suppose that there is some other biological property, G, that can be realised by any one of the complex microphysical states αʹ′, βʹ′, γʹ′, etc. For each particular microphysical realisation of F, the microphysical laws will perhaps give some well-defined chance of the system's going on to exhibit one of the microphysical states that realises G. However, it cannot be expected that the microphysical laws alone will tell us anything about the relationship between F and G in general. Specifically, it cannot be Luke Glynn 15 expected that they will tell us the chance that G is instantiated given that F is instantiated, since they themselves do not give us a probability distribution over the states α, β, γ, etc., conditional upon F's being instantiated. Consequently, a system that entails only the microphysical laws will lack a certain amount of strength or informativeness in Lewis's sense: it will fail to say 'either what will happen or what the chances will be when situations of a certain kind arise' ([1994], p. 480). The kinds of situation include, of course, situations of kind F. This would be a heavy cost indeed where F and G correspond to important biological kinds as is the case, for example, if F is the property of being a crossing of two parents heterozygous for a given trait, and G is the property of the offspring's being homozygous for that trait. The fact that the microphysical laws fall silent here means that there is a gap to be filled by the special scientific laws. In this case the Mendelian laws fill the gap. The addition of axioms that entail these laws will augment a system's strength because the resulting system will tell us what the chance of G is in situations of kind F: namely (on the current interpretation of F and G) 0.5. Nor need the addition, to a system, of axioms required to entail the special scientific laws cost much in terms of simplicity. What we need to add to the axioms, in order to get the special scientific laws to fall out as theorems, is a function that takes macrostates as inputs and yields probability distributions over regions of microphysical phase space as outputs (this would yield the requisite probability distribution over the microphysical states α, β, γ, etc., conditional upon F's being instantiated, from which an overall chance of G given F could be derived). A function that yields a uniform probability distribution over those regions compossible with the macrostate might result in the best fit with the frequencies observed at a world like ours. If, for example, the various possible microphysical realisations of F lead to Deterministic Chance 16 microphysical realisations of G with an average frequency of 0.5 then a uniform probability distribution over microphysical realisations of F will yield a chance of G given F close to the actual frequency of Gs amongst Fs. In any case, the function could be tweaked to ensure the best fit with the observed frequencies in our world. The simple addition of such a function to the axioms would thus increase the strength of the system greatly by ensuring that the system tells us what the chances of G will be when situations of kind F arise and, in general, ensuring that it tells us what the chances will be when situations of kind X arise, where X is any microphysically disjunctive high-level kind. It consequently seems likely that such a system will come out best. The illustration just given of how the Lewisian Best System Analysis can accommodate special scientific laws is similar to Loewer's attempt to show that the Best System for a Newtonian world will entail CSM. Loewer ([2001]) considers a formulation of CSM given by Albert ([2001]). Albert's formulation consists of three postulates: (1) the Newtonian dynamical laws, (2) a uniform probability distribution over the possible points in microphysical phase space at the beginning of the universe, and (3) a statement characterising the beginning of the universe as a low-entropy state. I shall borrow the terminology of Schaffer (op cit., p. 122) in calling postulates (2) and (3) respectively the 'Statistical Postulate' and the 'Past Hypothesis'. Albert shows that from these three postulates the whole of statistical mechanics follows, as well as probabilistic versions of the principles of thermodynamics. Thus Loewer (op cit., p. 618) says: [T]his package is a putative Best System [...]. The contingent generalisations it entails are laws and the chance statements it entails give the chances. It is simple and it is enormously informative. [...] By being part Luke Glynn 17 of the Best System the probability distribution earns its status as a law and is thus able to confer lawfulness on those generalisations that it (together with the dynamical laws) entails. The Albert package is similar to the system sketched above in its combination of the fundamental dynamical laws with a statistical postulate specifying a uniform probability distribution over points in microphysical phase-space. The main differences are, first, that whilst the Albert package just gives a single probability distribution over phase-space points at the beginning of the universe, the postulate I described is an atemporal function from macrostates to probability distributions. Second, the Albert package includes the low-entropy condition.8 Schaffer ([2007], pp. 130-1) advances an argument against the view that the Best System will entail the special scientific laws. He argues that, even though the addition of some statistical postulate to a system may result in a gain in informativeness vis-a-vis one that entails the microphysical laws alone, these two systems aren't the only competitors. He suggests that we consider an alternative package of the fundamental dynamical laws together with the Precise Initial Conditions. Where the microphysical laws are deterministic, this package will have great strength, since it 'entails every single detail of the entire history of the world' (ibid., p. 131). A crucial question here concerns the simplicity of this latter system vis-a-vis one having just the fundamental dynamic laws and a statistical postulate as axioms. Hoefer ([2007], p. 560) puts the question well when he asks, regarding the Precise Initial Conditions, [D]o they increase the complexity of the system infinitely, or by just one 'proposition', or some amount in between? Lewis's explication does not Deterministic Chance 18 answer these questions, and intuition does not seem to supply a ready answer either. It is clear that different explications of the notion of simplicity may have strong implications for which system comes out as best. But, in any case, Schaffer is wrong to claim that the package of the microphysical laws together with the Precise Initial Conditions is 'maximally strong' (op cit. p. 132). The package is not maximally strong on Lewis's explication of the relevant notion of strength. For, as has already been seen, it fails to say 'either what will happen or what the chances will be when situations of a certain kind arise'. For example, it fails to tell us what will happen or what the chances are when situations of the microphysically disjunctive biological kind F arise, as opposed to situations of the realising microphysical kinds α, β, γ, etc. 3.3 Special Scientific Laws and the Law Role Following Schaffer and Loewer, I have focused on the Lewisian approach to laws, but there seems no good reason why other Humean approaches, as well as non-Humean approaches to laws, should not also be able to accommodate special scientific laws as genuine. Indeed there is good reason to think that, in order to be fully adequate, any analysis of laws must be able to accommodate special scientific laws. The reason is that these laws seem to play the law role just as well as the microphysical laws. In particular, the special scientific laws fulfil all the usual criteria for genuine lawhood by supporting counterfactuals, being confirmed by their instances and underwriting explanations and predictions. For example, it is true in virtue of the Mendelian laws that if I had conducted a dihybrid cross of two pea plants heterozygous for two binary traits, then there would have Luke Glynn 19 been a chance 0.25 of the offspring being homozygous for each trait. Moreover, if I know the Mendelian laws, then I should have confidence of degree 0.25 in the prediction that a given pea plant resulting from such a cross will be homozygous for each trait. Furthermore, if a resulting pea plant does in fact turn out to be homozygous for each trait, then a good explanation (if there is such a thing as a good covering law explanation) would be that I conducted the cross and that the Mendelian laws indicate that there was a 0.25 chance that this outcome would result. Finally, the Mendelian laws derive inductive confirmation from experimental results in just the same manner as do microphysical laws. Mendel ([1866]) himself conducted experimental crossings of 29,000 pea plants. His experiments with mixing one trait with another consistently yielded a 3:1 ratio between dominant and recessive phenotypes, whilst his experiments with mixing two traits resulted in 9:3:3:1 ratios. These experiments are rightly seen as having conferred rather good inductive confirmation upon the laws named after him.9 Someone might object to the foregoing along the following lines: 'If the microphysical laws project different chances from the Mendelian laws for the outcomes of dihybrid crosses, then the counterfactual "if I had conducted a cross, there would have been a chance 0.25" is false and, moreover, one should not (if one knew the microphysical laws) have confidence 0.25 in the outcome, nor does the 0.25 chance really explain the outcome.' The objector might instead maintain that it is the microphysical laws that determine which counterfactuals are true, that underwrite genuine explanations and that ground reasonable, well-informed, predictions. But such objections are misguided. The special scientific laws do not really compete in this way with the microphysical laws. Suppose, for example, that there is a fact of the matter about what the microphysics would have been had I crossed the pea plants and that the Deterministic Chance 20 microphysical laws consequently generate a chance 1 for the cross resulting in a child homozygous for each trait. Then it is true both that 'if I had conducted a cross, then there would have been a chance 1' and that 'if I had conducted a cross, then there would have been a chance 0.25'. Of course, these divergent chances cannot be the same chance. As I shall argue in §4, they are chances of different levels: the former is a micro-level chance, the latter a higher-level chance. In some cases only the latter counterfactual will be true. Suppose, for example, that there is no fact of the matter about what the microphysics would have been had I crossed. Then it is nevertheless true in virtue of the Mendelian laws that 'if I had conducted a cross, there would have been a chance 0.25'. Likewise, suppose that I am going to conduct a cross but it's still not settled what the microphysical details of the cross will be, or I just don't know the microphysical details. Then I should have confidence of degree 0.25 in the prediction that a given resultant pea plant will be homozygous for each trait. Indeed, as I shall argue in §4.2 below, I should have this degree of credence even if the microphysical details are settled and I know all the admissible information about the case. As regards explanation, there may be more than one good explanation of an event. If a resulting pea plant turns out to be homozygous, then a good explanation would be that I conducted the cross and that the Mendelian laws indicate that there was a 0.25 chance that this outcome would result. This is true even though a still more satisfying explanation might be that I conducted the cross and the microphysical laws and circumstances entailed a chance 1 for this outcome. The special scientific laws play the law role, and ought therefore to be accommodated as genuine laws by any adequate account of lawhood. And, in the previous Luke Glynn 21 subsection, I argued that there is good reason to think that they are accommodated by the Lewisian account. 4 Deterministic Chance 4.1 Chance and Laws Again In the previous section it was argued that there exist probabilistic special scientific laws even in fundamentally deterministic worlds. When taken together with an initial history of the deterministic world in question, these laws entail non-trivial probabilities for the events that they concern. Since these laws are genuine objective laws, the probabilities that they project are genuine objective chances. There therefore exist fundamentally deterministic worlds with chance-projecting special scientific laws. In light of this, it is clear that Schaffer's assertion (noted in §2.3 above) that the laws of deterministic worlds don't project chances is just false. So Schaffer is wrong that nontrivial deterministic chances are incompatible with the connection from chance to lawhood, captured by his LMP: (LMP) If Ch!" p = x, then the conjunction of H!" (a proposition giving the history of w up until time t) with the laws of w entails that Ch!" p = x. The compatibility of non-trivial deterministic chances with the LMP can be seen by reconsidering the earlier example of the crossing of pea plants. Deterministic Chance 22 Suppose that the world w at which the crossing occurs is a Newtonian one (the Mendelian laws are perfectly compatible with this). The history of w through t (the time of crossing)–which includes the fact that the crossed plants (Jim and Jill) are heterozygous for both shape and colour–together with the laws–which include the Mendelian laws–entail a chance 0.1875 for the proposition prg, that Tom (the resulting pea plant) has round, green peas.10 Thus, we have 1 > Ch!" p!" = 0.1875 > 0. But there is a difficulty here. Since w is a deterministic Newtonian world, the history of w through t–which includes the complete microphysical state of the world at t–together with the laws of w–which include the fundamental dynamical laws (alongside the Mendelian laws)–entails Ch!" p!" = 1 or Ch!" p!" = 0, depending on whether or not prg is actually true at w. Thus, we have that 1 > Ch!" p!" > 0 and that Ch!" p!" = 1 or 0. Contradiction! Or so it seems: so long as it is maintained that chance is a function of just three arguments, there is a tension in assigning the same proposition two divergent chances at the same world and time. But one can escape the contradiction by allowing that chance isn't after all just a function of propositions, worlds and times. Consider an analogy: for a given world and a given proposition, we may get more than one different chance. No one would regard this as a contradiction because chance is a function not just of worlds and propositions, but also of times. There is no tension between its being the case, for instance, that Ch!" p = x and Ch!!! p = y (where x ≠ y, t ≠ t'). Similarly, there is no tension between Ch!" p = x and Ch!"! p = y (where x ≠ y, w ≠ w'). Likewise, if chance is a function of some fourth argument, then there would be no difficulty in admitting that there may be two or more divergent chances for the same proposition, in the same world even at the same time. Luke Glynn 23 What might this fourth argument be? The consideration that puts pressure on us to accept these divergent chances is the existence of laws of different levels which entail them. It seems, then, that there is an important level-relativity of chance, as well as its proposition-, world-, and time-relativity. One can capture this additional dimension of relativity by introducing an additional subscript to the chance-function: 'Ch!"# p ' can be read 'the l-level chance of p at time t and world w'. The l-level chance of p at t and w is just the chance entailed for p by the l-level laws of w when taken together with the history of w through t. In the above example, the Mendelian laws of w together with the history of w through t entailed that Ch!" p!" = 0.1875. The microphysical laws of w together with the history of w through t, on the other hand, entail that Ch!" p!" = 1 or 0. But contrary to first appearances this is no contradiction, for the chances entailed by the Mendelian laws are not the same as those entailed by the microphysical laws: they are chances of different levels. Any appearance of contradiction evaporates when the appropriate indices are added to the chance functions. Let lh (h for higher) be the level with which the Mendelian laws are concerned, and let lf (f for fundamental) be the microphysical level. Then, making the levelrelativity of the chances explicit, we have Ch!"!! p!" = 0.1875 and Ch!"!!(p!") = 1 or 0. No contradiction there! In general, it might be that Ch!"!! p = x, but that Ch!"!! p = y (where x ≠ y, li ≠ lj). Divergent chances may exist for the same proposition at the same world and time because of the level-relativity of chance. Recognition of this fact is the key to the reconciliation of determinism with non-trivial chance.11 Whilst the LMP sanctions apparently inconsistent chances, relativizing chances to levels results in a statement of the chance-law connection that avoids this: Deterministic Chance 24 (LMP*) If Ch!"# p = x, then the conjunction of H!" (a proposition giving the history of w up until time t) with the l-level laws of w entails that Ch!"# p = x. In a deterministic world that has probabilistic laws of non-fundamental levels, the original LMP sanctions non-trivial chances (as we saw earlier). The revised LMP* just makes it clear that these non-trivial chances are not the same as the trivial chances that are also sanctioned. Thus the appearance of inconsistency is avoided. Either way, given the existence of probabilistic special scientific laws in deterministic worlds, it is clear that non-trivial deterministic chances are compatible with the chance-law connection. Indeed, it seems that the conceptual connection between chances and laws might be tighter than that captured by either the LMP or the LMP*. These principles state roughly that all chances are lawfully entailed. But one might think, in addition, that all lawfully entailed probabilities for the occurrence of events are chances. If there is this closer connection between chances and laws, then taking only the values projected by fundamental laws as chances is incompatible with the chance-law connection. That connection now not only sanctions but entails the existence of non-trivial chances in deterministic worlds with probabilistic special scientific laws and would be violated by a function that outputted only trivial chances in such a world. 4.2 Chance and Credence Again Recall that Schaffer apparently showed that a non-trivial chance assignment to a proposition p in a deterministic world contradicts the deliverances of the reformulated Principal Principle: Luke Glynn 25 C p L! .H!" = Ch!" p (RPP) The problem was that, if w is deterministic, then the laws and initial history will entail either p or ~p. Therefore, reasonable credence conditional upon the laws and initial history is equal to 1 or 0, from which it follows directly by the RPP that Ch!" p is correspondingly equal to 1 or 0. Note that it is the original PP rather than RPP that, according to Lewis, enjoys the 'direct intuitive support' ([1986b], p. 98). The PP was reformulated only because the reformulation 'prove[s] easier to use' (ibid.). Importantly, the demonstration of the incompatibility of non-trivial deterministic chances with the PP depends upon the validity of RPP as a reformulation of the PP. This is important because the reformulation is invalid. The original PP, it will be recalled, states that reasonable credence in p conditional upon the proposition X that Ch!" p = x and any other proposition E compatible with X that is admissible at time t and world w, is equal to x: C p X.E = x (PP) The reformulated PP was arrived at by assuming that the big conjunction of the complete set of laws of w with the history of w through t is an admissible proposition at t and w. But this big conjunction is not, in general, fully admissible.12 Recall that Lewis characterises admissible propositions as propositions whose impact on credence about outcomes comes entirely by way of credence about the chances of those outcomes. On the assumption (which Lewis makes) that there can only be one chance for a proposition in a world at a time, a chance that is entailed by the big conjunction of the Deterministic Chance 26 initial history and the laws, it might seem that this big conjunction is indeed admissible. But this assumption is incorrect: there exist laws of different levels which, taken together with initial history, entail divergent chances for the same propositions even in the same worlds at the same times. And, as shall now be seen, the existence of these divergent chances makes it false that the big conjunction is in general admissible. Consider the high-level chance, entailed by the Mendelian laws, for the proposition prg that Tom has round, green peas: Ch!"!! p!" = 0.1875. Relative to this chance, the conjunction of the history of w through t with the Mendelian laws (and the high level laws in general) is admissible. It is admissible because its impact on reasonable credence in the proposition prg comes entirely by way of its impact on credence about the value of Ch!"!! p!" . The proposition H!" . L!!! entails Ch!"!! p!" = 0.1875 and it contains no other information that is relevant to whether or not prg is true. But it cannot be inferred from this that Htw.Lw, the proposition giving the history of w through t together with the complete set of laws of w, is admissible. The problem is that this proposition may well carry information relevant to whether prg is true over and above that expressed by the proposition that Ch!"!! p!" = 0.1875. In particular, Lw includes the laws of levels other than lh and these may project divergent chances for prg. These chances constitute additional information relevant to whether prg is true.13 Indeed where w is fundamentally deterministic Htw.Lw, which includes the fundamental laws, entails prg and so clearly carries such additional information. Hence, relative to Ch!"!! p!" , Htw.Lw is simply inadmissible. And since the validity of the reformulation of the PP rested upon the assumption that Htw.Lw is in general admissible, it can be concluded that the reformulation is invalid and that Schaffer's demonstration that non-trivial deterministic chances are incompatible with the PP, a demonstration that depended upon the validity of the RPP, is unsuccessful. Luke Glynn 27 This argument that RPP is invalid as a reformulation of PP and that therefore nontrivial deterministic chances can't be shown to be inconsistent with the PP is one that has been made by Hoefer ([2007]). Regarding the alleged derivation of a contradiction from PP plus the supposition of a non-trivial deterministic chance for some proposition A, Hoefer says (op cit., p. 559): [T]his derivation is spurious; there is a violation of the correct understanding of admissibility going on here. For if Htw.Lw entails A, then it has a big (maximal) amount of information pertinent as to whether A, and not by containing information about A's objective chance! So Htw.Lw, so understood, must be held inadmissible, and the derivation of a contradiction fails. Whilst recognition of the level-relativity of chance makes it clear that the reformulation of the PP is invalid, it does not create any problem for the PP itself. The only necessary adjustment is that, once it is allowed that there may be more than one chance attaching to the same proposition at the same world and time, Lewis's supposition of uniqueness must be dropped. Thus, X must not be read as the proposition that the chance at w and t of p is equal to x, but rather as the proposition that the l chance at w and t of p is equal to x, where l is a variable ranging over levels. What information is admissible will depend upon which level is in question. Lewis's reformulation of the PP was intended to facilitate ease-of-use. Unfortunately, as has been seen, it involved the fallacious substitution of the not-generallyadmissible proposition Htw.Lw for the admissible proposition E. Can one give a user-friendly Deterministic Chance 28 reformulation of the PP without committing such a fallacy? The key to doing so is (as with the LMP) to restrict the laws so as to be of the same level as the chances, as in (RPP*): C p L!" .H!" = Ch!"# p (RPP∗) Intuitively: the initial history and l-level laws of the world entail the l-level chances and reasonable credence for someone whose evidence included all and only the initial history and l-level laws (and who learns by conditioning) is equal the chance for p that they entail.14 Where l = lf, the microphysical level, and w is a deterministic world, the chance entailed by L!" .H!" will be trivial and reasonable credence conditional upon just L!" .H!" will be correspondingly trivial. But where l is some non-fundamental level, lh, the chance entailed by L!" .H!" need not be trivial and, if it is not, reasonable credence conditional upon just L!" .H!" will be correspondingly non-trivial. It would, of course, be interesting to know what reasonable credence would be for someone who knew the complete laws and initial history, L! .H!", and consequently the chances projected by the laws of each of the different levels. The RPP*, which is the correct reformulation of the PP, does not tell us–it does not indicate which of the divergent chances for p would guide rational credence in p. It does seem plausible that where w is a fundamentally deterministic world the answer is the trivial, fundamental (level lf) chance of p, Ch!"!! p . 15 The reason for thinking this is that, in this case, L! .H!" actually entails p or ~p. Still RPP*, which is the correct reformulation of PP, does not yield this (or any other) answer. This is just a limitation of RPP*. In any case, PP is not after all incompatible with non-trivial deterministic chances. It is only the incorrect reformulation RPP that is inconsistent with such chances. Once the Luke Glynn 29 reformulation is corrected, it becomes clear that the existence of non-trivial chances in deterministic worlds with probabilistic special scientific laws is perfectly compatible with the connection from chance to rational credence. Indeed, it is possible to go on the offensive against the incompatibilist regarding chance and determinism. Once it is acknowledged that chances are level-relative, it can be seen that a chance function that outputted only trivial values in a deterministic world with probabilistic special scientific laws would not be one that outputted values that play the role of chance in guiding rational credence captured by the PP (and its valid reformulation, RPP*). For example, where lh is the level of concern to the geneticist and w is a deterministic world with Mendelian genetic laws, a chance function that assigned a level lh chance 1 or 0 at the time at which Jim and Jill are crossed to the proposition that Tom will have round, green peas would be one that assigned values that fail to guide rational credence given all of the relevant admissible information, information which includes just the initial history plus the lh laws (among which are the Mendelian laws but not the microphysical laws). Reasonable credence given just this information is 0.1875. Consequently, not only is it not the case that non-trivial deterministic chance functions are incompatible with the chance-credence connection, it is in fact the case that only such a function can underwrite this connection in deterministic worlds with probabilistic special scientific laws. In the previous section, we saw that the same was true with respect to the chance-law connection. 4.3 Chance and Possibility Again Deterministic Chance 30 It will be recalled that Schaffer argues that non-trivial deterministic chance functions are incompatible with the connection from chance to possibility captured by the RP: (RP) If Ch!" p > 0, then there exists a world w' such that (i) w' matches w in laws; (ii) w' matches w in occurrent history up until time t; (iii) p is true at w'. As was seen in §2.2 above, if w is a deterministic world and pe is the proposition that some event e occurs, then the initial history and laws of w will entail either pe or ~pe (depending on whether e occurs in w). Accordingly, a non-trivial chance assignment to pe is incompatible with the RP. Note that the formulation of the RP presupposes that the chance function yields a unique chance assignment for a given p, t and w. But I have been arguing that this supposition is erroneous: chances are level-relative and, consequently, there may be two or more divergent chances attaching to a proposition at a world and a time (these chances being chances of different levels). And note that the RP will be compatible with deterministic chance provided that the laws in question are restricted so as to be of the same level as the chance. Thus consider RP*: (RP*) If Ch!"# p > 0, then there exists a world w' such that (i) w' matches w in l-level laws; (ii) w' matches w in occurrent history up until time t; (iii) p is true at w'. Where lf is the microphysical level, and w is a deterministic world in which event e fails to occur, then L!!! .H!" will entail ~pe. There is therefore no world w' at which L!!! .H!" .p! is Luke Glynn 31 true. So, by RP*, Ch!"!! p! = 0. If, on the other hand, e does occur at w then analogous considerations show that RP* implies Ch!"!! p! = 1. But, where lh is some non-fundamental level, L!!! .H!" needn't entail either pe or ~pe, but may instead entail some non-trivial chance for pe. If so, there are some worlds at which L!!! .H!" .p! is true and some worlds at which L!!! .H!" .~p! is true. So it is consistent with RP* that 1 > Ch!"!! p! > 0. Nor is there a contradiction between its being the case that 1 > Ch!"!! p! > 0 and its being the case that Ch!"!! p! is equal either to 1 or 0. This is because Ch!"!! p! and Ch!"!! p! are not the same chance, but are chances of different levels. One might insist that high-level deterministic 'chances' aren't really chances since they merely fulfil RP* and not RP, which (it might be insisted) is the correct chancepossibility principle. In order to evaluate this objection, it is necessary to consider the nature of the platitude concerning chance and possibility, of which RP and RP* are both candidate precisifications. Let us suppose that Schaffer (ibid., p. 124) is correct when he says that the platitude is roughly that 'if there is a nonzero chance of p, this should entail that p is possible, and indeed that p is compossible with the circumstances.' Note, first of all, that non-trivial deterministic chances aren't supposed to be chances for events that are impossible in the sense that they are ruled out by the laws alone (as would be the case if, in our world, a non-trivial chance were assigned to the transmission of a superluminal signal). But it might be said that they are chances for events that, given the laws, are not compossible with the circumstances, including the microphysical initial history of the world. But, we might ask: which laws? In a fundamentally deterministic world, the events in question may be ruled out by the fundamental laws, given the circumstances. But, as has Deterministic Chance 32 been seen, they need not be ruled out by the non-fundamental laws. RP* allows that the llevel chance of an event e may be positive provided that e is not, in the circumstances, ruled out by the laws of that level. The objector might still insist that the chance-possibility connection is captured by RP, which rules out deterministic chances because the events in question are not compossible with the circumstances given all the laws, rather than RP* which allows them because the events they concern are compossible with the circumstances plus the higher-level laws. The dispute is then one over the precise content of the chance-possibility platitude. And one might wonder whether this content is rich enough to allow us to adjudicate between RP and RP* as candidates for the correct precisification of the chance-possibility connection. Indeed, these aren't the only two contenders. Schaffer (op cit. p. 124) takes the RP to be a relatively uncontroversial strengthening of Bigelow, Collins and Pargetter's Basic Chance Principle ([1993], p. 459): (BCP) If Ch!! p > 0, then there exists a world w' such that (i) Ch!!! p = Ch!" p ; (ii) w' matches w in occurrent history up until time t; (iii) p is true at w'. But there is evidently no inconsistency between BCP and non-trivial deterministic chances. Suppose that w is microphysically deterministic, and that w' is not (the BCP doesn't require w and w' to agree in laws). And suppose that both worlds are Mendelian, that Jim and Jill are crossed at time t in both worlds, and that prg is the proposition that Tom (a resulting pea plant) has round, green peas. Then Ch!"!!! p!" = Ch!"!! p!" = 0.1875. Since w is microphysically deterministic, it might be that Tom is determined to have wrinkled, yellow peas in w. It is compatible with this that Tom turns out to have round, green peas in w' (since Luke Glynn 33 w' is not microphyscally deterministic). Therefore, the BCP is compatible with the existence of a positive chance for prg in w, even though prg is incompatible with the microphysical laws plus microphysical history of w. Nevertheless, suppose BCP to be somehow ruled out as a candidate precisification of the chance-possibility connection, and take just RP and RP* as the relevant alternatives. There are the following grounds to favour RP* (which is compatible with non-trivial deterministic chances) over RP (which is not). If, as it seems one must when discussing the content of platitudes about chance, one takes seriously the layperson's talk about chances for ordinary macroscopic events–such as the 50% chance of the coin landing heads–or the scientist's talk about chances for special scientific events–such as the 35% chance of the hurricane making landfall in southern Florida–then there is pressure to conclude that chances are not the sorts of things that must (as the RP requires) indicate compossibility with the circumstances given the fundamental laws. The layperson and the meteorologist may not know, or have any opinion about, whether these events are compossible with the circumstances given the fundamental laws. Yet any doubt about this does not lead to doubt about the correctness of these chance assignments. What the layperson does know is that (as required by the RP*) heads is not ruled out by the macro-level law ascribing a 0.5 chance to coin flips. What the meteorologist knows is that (again, as required by the RP*) the south Florida landfall is not ruled out by her model, a model that encodes (at least approximations of) the probabilistic functional laws of her discipline. There is another reason to favour the RP* over the RP. Change the earlier example so that the crossing of the pea plants now occurs in a micro-indeterministic world, wi. And suppose that the Mendelian laws still hold at wi, so that Ch!!!!! ~p!" = 0.8125. Suppose, Deterministic Chance 34 moreover, that both prg and ~prg are compossible with the circumstances plus the fundamental laws of wi, so that this chance is consistent with the original RP. But suppose that ~prg is only compossible with the fundamental laws in virtue of the possibility of an extremely unlikely quantum event,16 so that the chance assigned by the fundamental laws to ~prg is exceedingly small, e.g. Ch!!!!! ~p!" = 0.1 ×10 !!,!!!,!!!. Surely it is bizarre in the extreme to maintain that it is this immensely unlikely and seemingly irrelevant possibility that grounds the positive value of Ch!!!!! ~p!" . Remove the astronomically small quantum uncertainty, and suddenly the value of Ch!!!!! ~p!" switches from 0.8125 to 0 (despite the continued existence of the 0.8125 probability projected by the Mendelian laws)! This is hardly plausible. The lh chance of 0.8125 sanctioned by the RP* clearly remains even if the fundamental dynamics of the world turn out to be deterministic. The elimination of the very slight quantum uncertainty has only the effect of changing the value of the lf chance from 0.1 ×10!!,!!!,!!! to 0, again in accordance with the RP*. So the RP* looks like a better chance-possibility principle than RP, and since the former sanctions non-trivial chances in deterministic worlds with probabilistic special scientific laws, it seems that non-trivial deterministic chances are compatible with the chance-possibility connection. 5 Chance and Causation Schaffer claims that non-trivial deterministic chances would violate the connections from chance to rational credence, possibility and lawhood. In the previous section it was shown that recognition of the level-relativity of chances makes it clear that this is not so. Indeed it Luke Glynn 35 was argued that in deterministic worlds with probabilistic special scientific laws only a nontrivial function can underwrite the chance-credence and chance-law connections. The aim of this section is to reinforce the conclusion that there are deterministic chances by considering one of the three remaining platitudes that Schaffer identifies: namely that connecting chance to causation. Schaffer says that he sees no incompatibility between the chance-causation platitude and a non-trivial deterministic chance function (he says the same about the final two–chance-futurity and chance-intrinsicness–platitudes). But I shall argue that there is an incompatibility with trivial deterministic chance functions. The argument is independent of considerations concerning the level-relativity of chances. Consequently, in what follows, the level index to the chance function shall be suppressed for simplicity. The argument is also ancillary to the main case for deterministic chance made in §§3-4 above, which goes through even if one rejects the claim that there is a platitudinious connection from chance to causation (or if one rejects the admittedly contentious precisification of that platitude that I suggest below). Schaffer (op. cit., p. 126) claims that the platitude about the connection from chance to causation is as follows: [C]hances should live within the causal transitions they impact. That is, if a given chance is to explain the transition from cause to effect, that chance must concern some event targeted within the time interval from when the cause occurs, to when the effect occurs. Otherwise that chance cannot impact the transition from cause into effect–it would be left outside of the action. Deterministic Chance 36 He states (ibid.) that this platitude may be codified as the Causal Transition Constraint (CTC): (CTC): If Chtw(pe) plays a role in the causal relation between c and d, then te∈[tc, td].17 Schaffer (ibid. p. 132) says that he sees 'no problem' about non-trivial deterministic chances fitting the CTC.18 I am sceptical about the status of this 'platitude'. The phrase 'chances should live within the causal transitions they impact' seems very obscure, as does the notion of a chance's being able to 'explain the transition from cause to effect'. Even the notion of a chance's 'playing a role in a causal relation' which appears in the alleged precisification of the chance-causation platitude, CTC, is very opaque. In so far as I can make sense of these statements, they just seem false (and can therefore hardly be platitudinous). Consider the following example.19 Suppose that Napoleon is on his way to Waterloo. I am a general in the Prussian army and set an ambush. My men attack Napoleon's troops, inflicting heavy losses (e). Napoleon reaches Waterloo with his army severely depleted and unable to withstand Wellington's charge (c). Wellington is victorious (d). Let Ch!!!! !@(p!) be the chance, just before I attacked, of my infliciting heavy losses on Napoleon. It seems that this chance helps explain why I did in fact inflict heavy losses upon Napoleon (e). This, in turn, helps explain why Wellington's charge (c) resulted in his victory (d). There does not seem to be any failure of transitivity here, and so (insofar as I can make sense of these notions) it seems that the chance of e helps explain (and 'impacts' Luke Glynn 37 upon) the causal transition from c to d. Nevertheless, te ∉ [tc, td]. Thus, insofar as the CTC is meaningful, it seems to be false. In any case, it strikes me as implausible to hold that the claim 'chances [...] live within the causal transitions they impact' is the obvious candidate for the platitude connecting causation and chance. I think many more people have thought the following to be a platitude: 'causes (tend to) raise the chance of their effects'.20 A natural way to be precise about the relevant chance-raising relation is to cash it out in terms of an inequality between two conditional chances: Ch!!!! ! p! p! > Ch!!!! ! p! ~p! (1) This inequality says that, just before c occurred, the chance of e conditional upon the occurrence of c was greater than the chance of e conditional upon the non-occurrence of c. The result is the following Chance-Causation Constraint (CCC): (CCC) If c is a cause of e then (ceteris paribus) c raises the chance of e in the sense that (1) obtains.21 Since chances obey the probability calculus (see Lewis [1986b], esp. p. 98) we have, by the axiom of conditional probability: Ch!" p q = Ch!"(p. q) Ch!"(q) , Ch!" q > 0 (2) Where Ch!" q = 0, the axiom of conditional probability leaves Ch!" p q undefined. Deterministic Chance 38 Suppose that w is deterministic, and c is a cause of e. Then, by CCC, it is the case (ceteris paribus) that c raises the chance of e in the sense that (1) holds. But (1) will hold only if both terms are well-defined. By (2) (plus the Complementation Theorem) both terms will be well-defined only if 1 > Ch!!!! ! p! > 0. But, by the assumption that there are only trivial chances in deterministic worlds, it follows that Ch!!!! ! p! is equal to 1 or 0. Contradiction! Conclusion: a chance function that outputs only trivial chances in deterministic worlds is incompatible with the connection between chance and causation captured by the CCC.22 Note that, although CCC says only that causes raise the chance of their effects ceteris paribus, a chance function that outputted only trivial values in deterministic worlds would mean that causes never raise the chance of their effects in those worlds. By CCC, this is unacceptable. 6 Conclusion The probabilistic special scientific laws of deterministic worlds, when taken together with the initial histories of those worlds, entail non-trivial probabilities for the events that they concern. In addition to being lawfully projected, these non-trivial probabilities guide rational credence and indicate compossibility with the special scientific laws plus circumstances. They therefore play the chance role and should be considered chances. Indeed, not only are there non-trivial deterministic probability functions (deterministic probability functions that output at least some non-trivial values) that play the chance role, there are such functions that play it better than any trivial deterministic probability function. In deterministic worlds with probabilistic special scientific laws, only a non-trivial probability function can underwrite the connections from chance to lawhood and Luke Glynn 39 rational credence (as was argued in §§4.1&4.2). And if the argument of §5 is found compelling there is reason to think that only a non-trivial function can underwrite the connection from chance to causation. The conclusion that there exist non-trivial deterministic chances is not in tension with the claim that there also exist trivial chances (trivial probabilities that also play the chance role) for the very same propositions at the very same times and worlds (outputted by the very same chance function). Any appearance of tension dissolves once it is recognized that chance is a function of four arguments: a proposition, a time, a world, and a level. The illusion of tension between trivial and non-trivial deterministic chances has been the reason for opposition to the latter. This is in evidence in the passage quoted from Lewis in the introduction to this paper. It is also in evidence in Schaffer's arguments (reviewed in §2) that sought to derive a contradiction from the supposition of non-trivial deterministic chances. Chance can be reconciled with determinism, and disparate chances can be reconciled with one another. The reconciliation is effected by the level-relativity of chance. Lewis is correct in observing (in the passage quoted in the introduction) that it is false that the chance of a fair coin landing heads can be both zero or one and also 50%. But it is only false because the supposition of uniqueness is false. It can be the case both that a coin has a chance zero or one and a chance 50%: the chances being chances of different levels. Once the illusion of tension between disparate chances is dispelled through recognition of chance's level-relativity, there remains no obstacle to accepting that the non-trivial, chance-roleplaying probabilities projected by the special scientific laws of deterministic worlds are indeed deterministic chances. Appendix: Times, Levels and Chance Setups Deterministic Chance 40 Hájek ([2003a], [2003b], [2007]) has convincingly argued that chances must be relativised to chance setups. This seems to be implicitly accepted by both Lewis and Schaffer. Both relativize chances to times, but as Hoefer observes ([2007], pp. 564-5):23 For Lewis, a non-trivial time-indexed objective probability Chtw(p) is, in effect, the chance of p occurring given the instantiation of a big setup: the entire history of the world up to time t.24 This suggests that the following analysis might be adopted:25 Ch!" p =!"# Ch! p|H!" Chances, on this analysis, are not fundamentally time-relative. Rather, the chance of p at t and w is just the chance for p that results from conditioning upon the history of w through t. This analysis has its merits: if p concerns what is past by t then Htw entails p and Ch! p|H!" = 1; moreover, since the big chance setup Htw changes as the value of t changes, so will the chance distribution that results from conditioning upon it. In short, conditioning the chance function on history through time t reproduces the properties of a time-indexed chance function. I argue in the paper that chances are level-relative. In §3.2, it was suggested that the relationship of realisation grounds the distinction between higherand lower-levels: two sciences can be regarded as (at least partially) characterising distinct levels when various distributions of the properties of concern to one realise those of concern to the other. Luke Glynn 41 With this in mind, define the history of world w at level l up until time t as the distribution of l-level properties throughout w up until t. The proposition giving this distribution can be written 'Htwl'. I think that the following analysis might then be adopted: Ch!"# p =!"# Ch! p|H!"# Chances, on this analysis, are not fundamentally level-relative. The l-level chance of p at t and w is just the chance for p that results by conditioning upon the l-level history of w through t.26 This analysis has its merits, foremost among which is that if the l-level history together with the laws of w don't entail p, then conditioning upon Htwl won't trivialise the chance of p. Conditioning the chance function on l-level initial history therefore reproduces this important property of a level-indexed chance function. Acknowledgements For helpful discussion and comments on earlier versions of this paper, I'd like to thank Frank Arntzenius, Roman Frigg, Marion Ledwig, Miklós Rédei, Jonathan Schaffer, Mauricio Suarez, Alistair Wilson and two anonymous referees of this journal. I would also like to thank audiences at several presentations of this paper in 2008, including at the British Society for the Philosophy of Science Annual Conference, the Joint Session of the Aristotelian Society and the Mind Association, the LSE Philosophy of Probability Graduate Conference, the Ockham Society at Oxford and the Philosophy of Probability Research Seminar at King's College, London. I'd like especially to thank Antony Eagle for detailed comments on several earlier drafts of this paper. Funding for this research was provided by an Arts and Humanities Research Council Doctoral Award. Deterministic Chance 42 St. Hugh's College Oxford, OX2 6LE luke.glynn@philosophy.ox.ac.uk 1 This point is emphasized by Loewer ([2001], see esp. p. 610). 2 My concern in this paper is with what Schaffer (op cit., p. 120) calls posterior deterministic chances: chances concerning events occurring after the first moment of the universe. Schaffer's case against initial deterministic chances–concerning events occurring at the first moment of a deterministic universe–is somewhat different, and I won't defend such chances here. 3 In a footnote, Schaffer (op. cit., p. 115n.) states that he is assuming chance to be a function of only three arguments: propositions, worlds and times. He says (ibid.) that 'whether determinism is compatible with a "chance" function that is relativized to [...] further inputs should be considered a separate question not addressed in the main text.' Perhaps Schaffer is correct that, on the assumption that the chance function has only three arguments, it follows that there are no deterministic chances. But what I am concerned with is whether there are deterministic chances, not whether there are deterministic chances on this assumption. Consequently, I regard evidence that there is a fourth argument (such as a level) as relevant to my topic. Such evidence will be presented in the course of this paper. Schaffer suggests that he has some justification for making the three-argument assumption. He says (ibid.) that in his paper the assumption 'will be defended indirectly [...] insofar as the role of chance [...] will prove explicable in ways that require no further inputs.' But I shall argue below that Schaffer's explication of the role of chance is flawed, and that the flaws can only be remedied by explicitly relativizing chance to a fourth input. Luke Glynn 43 I am far from being the first to argue that chance is relative to more than just propositions, times, and worlds. Hájek ([2003a], [2003b], [2007]), for example, has argued vigorously (and in my view convincingly) that chance must be relativised to a chance setup or reference class. In the appendix, I discuss the relation between my claim that chance is level-relative, and Hájek's that chance is setup-relative. 4 Lewis (ibid., pp. 91-2) advances arguments for this timeand world-relativity. The issue of whether chance is fundamentally time-relative shall be taken up in the appendix. 5 Schaffer (ibid., p. 121n.) explicitly restricts his attention to deterministic chances pertaining to propositions concerning individual, momentary event occurrences. I shall follow him in this. 6 The RP is a strengthened version of Bigelow, Collins, and Pargetter's Basic Chance Principle ([1993], p. 459). 7 Or at least in some such worlds. There might be deterministic worlds lacking probabilistic special scientific laws. I do not argue that there are non-trivial chances in these worlds. My thesis is merely that there are some deterministic worlds with non-trivial chances. In particular, there are non-trivial chances in those deterministic worlds that ours might have turned out to be (if it had turned out to be Newtonian) or might still turn out to be (if it turns out to be Bohmian): worlds that are deterministic at the microphysical level but at which there are probabilistic special scientific laws. 8 This condition creates a difficulty for the view that the Albert package is the Best System for some not-too-distant Newtonian world. The problem is that, in an earlier paper, Lewis restricts candidates for best systemhood to those systems whose axioms refer only to perfectly natural properties (Lewis [1983], pp. 368-9). Lewis holds that the perfectly natural properties are those to be discovered by fundamental physics (ibid., pp. 365, 368). And, as Schaffer (ibid., p. 130) points out: Deterministic Chance 44 The Albert package contains predicates such as 'low entropy' that refer to properties that are not perfectly natural–in microphysical vocabulary, that property is infinitely disjunctive. Hence the Albert package is not even in the running for the Lewis laws. It is ineligible from the start. I think that Lewis's restriction is unduly severe (he does not mention it in his final statement of his Best System Analysis [1994]). His ([1983], p. 367) justification for it was that the simplicity of a system is relative to the vocabulary in which it is expressed and that, by employing a very unnatural predicate, we might make a strong system very (syntactically) simple indeed. The concern was that, if just any predicates are allowed, it is difficult to see how simplicity is a virtue of a system, and how it can put any constraints upon a system's acceptability. But it does not follow that a restriction to perfectly natural predicates is required. Naturalness, as Lewis recognises (ibid. p. 368), admits of degrees. And the fact that, as Lewis also recognises (ibid.), our language 'has words mostly for not-too-unnatural properties' suggests that we are fairly adept at distinguishing reasonably natural properties from unnatural ones. Simplicity will not be a vacuous criterion so long as we require that the predicates employed in the axioms of the candidate systems refer to reasonably natural properties. Requiring reasonable naturalness may introduce a certain amount of indeterminacy or subjectivity into the Best System Analysis but, as Lewis ([1994], p. 479) is aware, there is already indeterminacy and subjectivity in the analysis as a result of its employment of the notions of simplicity, strength and balance. Relaxing a little the requirement of perfect naturalness of the predicates employed in the axioms is perhaps reasonable if the consequence of maintaining it in its strictest form is Luke Glynn 45 the exclusion of certain systems, such as the Albert package, which yield great gains in strength. 9 If a Humean analysis of laws is correct, one might wonder whether laws are genuinely confirmable by their instances and, indeed, whether they can really support counterfactuals and underwrite explanations and predictions. But there is no special problem for special scientific laws here. The problem, if there is one, is with Humeanism and not with special scientific laws. 10 In the case of pea crosses, genotypes are reliably reflected in phenotypes. 11 More generally, it is the key to the reconciliation of divergent chances projected by the laws of different levels. In the special case of interest to us, where the world in question is micro-deterministic, it allows a reconciliation of the trivial chances projected by the microphysical laws with non-trivial chances projected by higher-level laws. Note that the reconciliation thus effected between chance and determinism is a relatively weak form of reconciliation: specifically, it is not claimed that determinism at a level l can be reconciled with non-trivial objective l-level chances. This point has as a corollary the observation (footnote 7 above) that there may not exist non-trivial objective chances in thoroughly deterministic worlds: that is, worlds in which the laws of all levels are deterministic. 12 Lewis himself ([1994]) recognises that if a Humean view of laws is correct the laws are not fully admissible. He consequently acknowledges that the RPP is not a strictly valid reformulation of the PP. However, Lewis's reason for taking the laws to be strictly inadmissible centres around the problem of undermining futures and is somewhat different from that which is of concern here. It shall be set aside in what follows. 13 This is analogous to the manner in which the chances of prg at times later than t constitute inadmissible additional information relevant to whether prg is true. Deterministic Chance 46 14 Even this formulation is problematic if a Humean view of laws is correct (and not just because of the problem of undermining futures). On a Humean view the laws of all levels supervene upon history. Consequently, for some level l' (≠ l), the initial history Htw may constrain what the l' laws are. If so, someone who knew Htw would have information about what the l' laws and therefore the l' chances are, and would thereby have information about whether p is true that is not just information about the value of Ch!"# p . The proposition Htw would therefore not be fully admissible relative to Ch!"# p , and so even (RPP*) would not be a valid reformulation of (PP). The solution to this problem of inadmissibility might be to restrict the historical proposition to be a proposition (Htwl) giving information only about the history of w at level l. Such propositions are discussed further in the appendix. 15 Hoefer ([2007], pp592-3) argues that in fundamentally indeterministic worlds reasonable credence needn't track the fundamental level chances in cases of conflict. 16 This might be Railton's ([1978], p. 224) fantastic event in which 'all the naturally unstable nuclides on earth [...] commenced spontaneous nuclear fission in rapid succession'. 17 tc, td, and te here denote the times at which the subscripted events occur. 18 At least non-trivial deterministic chances of the posterior sort with which this paper is concerned (see footnote 2 above). 19 The example is due to John Hawthorne, who gave it at a seminar in Oxford on Schaffer's paper. I have embellished it somewhat. 20 This has been a principal motivation for those (such as Good [1961a], [1961b], [1962]; Reichenbach [1971]; Suppes [1970]; Lewis [1986d]; Menzies [1989]; Eells [1991]; and Kvart [2004]) who have attempted to develop probabilistic analyses of causation. Even those who don't seek a probabilistic analysis tend to agree. For instance, Mellor ([1995], esp. p. 67) Luke Glynn 47 takes the tendency for causes to raise the chance of their effects to follow from the 'connotations of causation'. 21 The level-relativity of chance makes for further subtleties in the precisification of the chance-causation connection. In particular there is the question of which level's chance of e must (ceteris paribus) be raised by c if c is to be a cause of e. Such subtleties are rather similar to those arising from the time-relativity of causation (inequality (1) incorporates the implicit assumption of one possible–and somewhat vague–answer to the question of which time's chances are relevant to whether c is a cause of e). 22 Caveat: Hájek ([2003a], [2003b], [2007]) has–in my view rather convincingly–challenged the validity of the axiom of conditional probability. Since the above demonstration of incompatibility between the CCC and a trivial deterministic chance function depends upon that axiom, it will (at least in its present form) not be found entirely convincing by those who endorse Hájek's arguments. 23 I have modified Hoefer's notation to render it consistent with my own. 24 This observation applies just as well to Schaffer as it does to Lewis. 25 Eagle ([unpublished]) gives such an analysis. 26 Hoefer (op cit., esp. pp. 562-5) too allows for the existence of well-defined chances conditional upon chance setups rather smaller than the complete initial history of the world. He explores some of the consequences of this, but I shall not enter into a discussion of these here. References Albert, D. [2001]: Time and Chance, Cambridge, MA: Harvard University Press. Deterministic Chance 48 Bigelow, J., Collins, J. and Pargetter, R. [1993]: 'The Big Bad Bug: What are the Humean's Chances?', British Journal for the Philosophy of Science, 44, pp. 443–62. Eagle, A. [unpublished]: 'Chance, Abilities, and Habituals'. Eells, E. [1991]: Probabilistic Causality, Cambridge: Cambridge University Press. Good, I. J. [1961a]: 'A Causal Calculus (I)', British Journal for the Philosophy of Science, 11, pp. 305-18. Good, I. J. [1961b]: 'A Causal Calculus (II)', British Journal for the Philosophy of Science, 12, pp. 43-51. Good, I.J. [1962]: 'Errata and Corrigenda', British Journal for the Philosophy of Science, 13, p. 88. Hájek, A. [2003a]: 'Conditional Probability Is the Very Guide of Life', in H. Kyburg Jr. and M. Thalos (eds), Probability is the Very Guide of Life: The Philosophical Uses of Chance, Chicago: Open Court, pp. 183-203. Hájek, A. [2003b] 'What Conditional Probability Could Not Be', Synthese, 137, pp. 273-323. Hájek, A. [2007] 'The Reference Class Problem is Your Problem Too', Synthese, 156, pp. 563-85. Luke Glynn 49 Hoefer, C. [2007]: 'The Third Way on Objective Probability: A Sceptic's Guide to Objective Chance', Mind, 116, pp. 549-96. Kvart, I. [2004]: 'Causation: Probabilistic and Counterfactual Analyses' in J. Collins, N. Hall, and L.A. Paul (eds), Causation and Counterfactuals, Cambridge, MA: MIT Press, pp. 359-86. Lewis, D. [1983]: 'New Work for a Theory of Universals', Australian Journal of Philosophy, 61, pp. 343-77. Lewis, D. [1986a]: Philosophical Papers, Vol. II, Oxford: Oxford University Press. Lewis, D. [1986b]: 'A Subjectivist's Guide to Objective Chance' in Lewis ([1986a]), pp. 83113. Lewis, D. [1986c]: 'Postscripts to "A Subjectivist's Guide to Objective Chance"' in Lewis ([1986a]), pp. 114-32. Lewis, D. [1986d]: 'Postscripts to "Causation"' in Lewis ([1986a]), pp. 172-213. Lewis, D. [1994]: 'Humean Supervenience Debugged', Mind, 103, pp. 473-90. Loewer, B. [2001]: 'Determinism and Chance', Studies in History and Philosophy of Modern Physics, 32, pp. 609-20. Deterministic Chance 50 Mellor, D.H. [1995]: The Facts of Causation, London: Routledge. Mendel, G. [1866]: 'Versuche über Pflanzen-Hybriden', Verhandlungen des Naturforschenden Vereins in Brünn, 4, pp. 3-47. Published in English [1901] as 'Experiments on Plant Hybridization', Journal of the Royal Horticultural Society, 26, pp. 1-32. Menzies, P. [1989]: 'Probabilistic Causation and Causal Processes: A Critique of Lewis', Philosophy of Science, 56, pp. 642-63. Popper, K. [1982]: Quantum Theory and the Schism in Physics, London: Hutchinson. Railton, P. [1978]: 'A Deductive-Nomological Model of Probabilistic Explanation', Philosophy of Science, 45, pp. 206-26. Reichenbach, H. [1971]: The Direction of Time, M. Reichenbach (ed.), Berkeley and Los Angeles, CA: University of California Press. First published in 1956. Schaffer, J. [2007]: 'Deterministic Chance?', British Journal for the Philosophy of Science, 58, pp. 113-40. Suppes, P. [1970]: A Probabilistic Theory of Causality, Acta Philosophica Fennica, 24, Amsterdam: North-Holland.