Backwards Causation and the Chancy Past

Backwards Causation and the Chancy Past Abstract I argue that the past can be objectively chancy in cases of backwards causation, and defend a view of chance that allows for this. Using a case, I argue against the popular temporal view of chance, according to which (i) chances are defined relative to times, and (ii) all chancy events must lie in the future. I then state and defend the causal view of chance, according to which (a) chances are defined relative to causal histories, and (b) all chancy events must lie causally downstream. The causal view replicates the intuitively correct results of the temporal view in cases of ordinary forwards causation, while correctly handling cases of backwards causation. I conclude that objective chance is more closely related to the direction of causation than it is to the direction of time. Objective, physical probability (or ‘chance’) is more closely related to causation, and less closely related to time, than is commonly supposed. The usual view of chance, which I’ll call the temporal view, says that chance and time are connected in two ways: chances are defined at times, and all chancy events (those with chances in between 0 and 1) must lie in the future. While this view is initially appealing, I’ll argue that both of its central claims should be rejected. I’ll give a case of so-called ‘backwards’ causation in which a past event inherits the chanciness of its future cause. When applied to this case, the temporal view breaks down. In its place, I’ll defend a causal view of chance. On this view, chances are defined relative to causal histories (not times), and all chanciness must lie causally downstream (not in the future). I’ll argue that the causal view affords all the benefits of its temporal cousin, without the drawbacks: it’s motivated by the very same considerations, and indeed replicates the results of the temporal view in cases of ordinary ‘forwards’ causation; but it correctly handles cases of backwards causation, where the temporal view breaks down. I’ll conclude that we ought to accept the causal view, and with it the primacy of the chance–causation relationship. In §1, I’ll state and motivate the temporal view. In §2, I’ll argue that the view faces serious objections. In §3, I’ll set out the causal view, and show that it overcomes the problems of the temporal view while retaining its strengths. In §4, I’ll consider the consequences for chance. 1. The temporal view of chance Chances are objective, physical probabilities. Their objectivity distinguishes them from subjective probabilities (for example, credences): if a particular coin has a chance of of landing heads when tossed on a particular occasion, then this is an objective fact, not a mere matter of opinion.1 The physicality of chances distinguishes them from other kinds of objective probabilities (for example, logical and evidential probabilities): chances reside within physical systems, such as atoms, coins and galaxies, and are governed by physical laws. A system’s chance properties are thus among its physical properties, akin to its mass, shape and electric charge. Chance is commonly taken to be closely connected to time. Here’s an influential example from Lewis: Suppose you enter a labyrinth at 11:00 a.m., planning to choose your turn whenever you come to a branch point by tossing a coin. When you enter at 11:00, you may have a 42% chance of reaching the center by noon. But in the first half hour you may stray into a region from which it is hard to reach the center, so that by 11:30 your chance of reaching the center by noon has fallen to 26%. But then you turn lucky; by 11:45 you are not far from the center and your chance of reaching it by noon is 78%. At 11:49 you reach the center; then and forevermore your chance of reaching it by noon is 100%. (Lewis, 1980, p. 91) There are two attractive thoughts here: that chances are time-dependent (in the sense that they can vary with time); and that at any given time, all the chancy events (those with chances in between 0 and 1) must lie in the future. The first thought can be formulated as follows:2 Time-Dependence Principle: The chance function ch is a function of three variables: a proposition A, a time t, and a possible world w. This says that time is one of the inputs to the chance function. Canonical chance propositions thus take the form , where . Lewis’s labyrinth case motivates the time-dependence of chance. Let centre be the proposition that you reach the centre by noon. There’s clearly some variation in the chance of centre within the world described: when we consider your respective situations at 11:00 and 11:30, for example, it’s clear that your chance of reaching the centre by noon is higher in the former situation than in the latter. Chances thus exhibit intra-world variation: relative to different situations within a given world, the very same possibility can have different chances. Chances are situation-dependent. In Lewis’s example, it’s natural to capture situation-dependence as time-dependence: a salient difference between the situations relative to which chances vary, and a convenient way of indexing them, is given by the times at which they occur.3 This leads naturally to the Time-Dependence Principle. With the Time-Dependence Principle in place, we can introduce some useful formalism. By holding fixed two of the chance function’s input parameters (the possible world and the time), we induce a real-valued, one-place function over propositions. For each possible world w and time t, let be the set of all propositions A such that the chance of A at t in w is well-defined. Define a one-place function such that for all . Call the (restricted) chance function at t in w. This function gives us the chances of various propositions at t in w. For example, if w is the labyrinth world, we have . (I’ll sometimes suppress the world parameter, writing simply cht.) I’ll assume that each (restricted) chance function satisfies the standard probability axioms. The second intuitively attractive thought brought out by Lewis’s example is that all chanciness must lie in the future. While your chance of reaching the labyrinth’s centre by noon may be intermediate before noon, it must go to either 0 or 1 afterwards. Where e is an event, let be the time of occurrence of e, that is, the time at which e either occurs or fails to do so. Let be the proposition that e occurs (at ). Then our second thought can be spelled out as follows:4 Future Principle: If , then . This says: if at t an event e has an intermediate chance of occurring, then e’s time of occurrence must lie in the future at t. All chanciness lies in the future, at all times and in all possible worlds: past chanciness is impossible. The Future Principle is intuitively compelling in run-of-the-mill cases like Lewis’s labyrinth: in such cases, the past is intuitively ‘fixed’, which goes naturally with extremal chance.5 Define the temporal view of chance as the conjunction of the Time-Dependence Principle and the Future Principle. According to the temporal view, chances vary with time, and past events must have extremal chance. In the context of everyday examples like Lewis’s labyrinth, the temporal view is plausible. For this reason, it’s the dominant view of chance in the literature. Let me now set out a more general formal framework for chance, of which the temporal view is a special case. (This will prove useful when we come to consider another special case.) This framework defines chances relative to background propositions.6 Take the chance function ch to be a real-valued function of three variables: two propositions A and B, and a possible world w. Canonical chance propositions thereby take the form , to be read as: the chance of A relative to B in w is x. The idea is that B describes a situation that arises in w and generates chances (perhaps a particular coin toss) and A describes a possible outcome of that situation (perhaps the coin’s landing heads) to which a chance value x is assigned. Call A an outcome proposition and B a background proposition. If B is a background proposition in w, then we have a tailored set of outcome propositions , to which chances are assigned. This set contains all the propositions to which a well-defined chance is assigned in w, relative to B. For example, if B describes a coin toss in w, then might include heads and tails. For each chance-generating combination of a background proposition B and a world w, define a (restricted) chance function such that for all . The restricted chance function assigns a chance value to each of the propositions in , that is, to each possible outcome corresponding to the background proposition described by B in w. For example, where B describes a coin toss in w, we might have . (I’ll sometimes suppress the world parameter, writing .) I’ll assume that each chance function satisfies the standard probability axioms. I’ll make two assumptions about background propositions. First, they are true in those worlds in which they generate chances: if B is a background proposition in w, then B is true in w.7 If a particular coin toss fails to take place in w, then in w there are no chances associated with that toss. Second, each background proposition is non-chancy, relative to itself: if B is a background proposition in w, then . Relative to a given coin toss, while the coin’s landing might be chancy, its tossing cannot be. (The tossing may be chancy relative to another background proposition.) It follows that no proposition entailed by B is chancy relative to B.8 We can restate the temporal view of chance in proposition-dependent terms. For any world w and time t, define the temporal history of w up to t, written , as the strongest proposition entirely about the past-and-present at t that’s true in w.9 Thus describes in maximal detail everything that happens in w up to and including t, without saying anything about what happens after t. Call T a temporal history of w just in case T is the temporal history of w up to some time t. We can reproduce the temporal view by taking our background propositions to be temporal histories. Defining chances relative to temporal histories is formally equivalent to defining them relative to times, in the sense that there is a bijective correspondence between pairs, where T is a temporal history of w, and pairs, where t is a time in w. We thus have a propositional surrogate for the Time-Dependence Principle. We also get the non-chanciness of the past: if , then either entails or else it entails , and so . This gives a proposition-dependent surrogate for the Future Principle, with temporal histories in place of their corresponding times. The proposition-dependent framework subsumes the temporal view. The temporal view of chance is plausible and widely endorsed. I will now argue that it should be rejected. 2. Problems for the temporal view My argument against the temporal view relies on a case of ‘backwards’ causation: a case in which a later event causes an earlier one. I will assume that such cases are conceptually possible.10 I’ll describe the case (§2.1), and then show how it makes trouble for both the Future Principle (§2.2) and the Time-Dependence Principle (§2.3). 2.1 Retrograde’s receiver Professor Retrograde has invented an ingenious device capable of backwards-in-time signalling. The device consists of two components: the sender, which sends out a signal when activated, and the receiver, which receives that signal at a designated earlier time, whereupon it beeps. Testing has shown this device to be supremely reliable. Retrograde performs the following experiment. At 9:00, she meets with her assistant. She then proceeds to her sending laboratory, which houses the sender, while the assistant travels across town to the receiving laboratory, where the receiver is located. By 11:00, Retrograde has set up an experiment: two exactly similar coins will be tossed in precisely the same way at 11:00, and land at 11:01. Each toss is carefully shielded from all outside influences. The first coin is coupled to the sender, so that a backwards-in-time signal will be sent iff the coin lands heads. This signal, if sent, will make the receiver beep at 10:00, where the assistant is waiting. If the first coin lands tails, no signal will be sent. Retrograde’s device is so reliable and her preparations so meticulous that if the first coin lands heads, the receiver is nomologically guaranteed to beep: the situation at 11:00 together with the laws of nature entail that the receiver beeps at 10:00 iff the first coin lands heads at 11:01. The second coin is a control: it will have no interesting effects regardless of how it lands. As it happens, both coins land heads at 11:01. Retrograde observes this and records it. The first coin activates the sender, and the resulting signal is received by the receiver at 10:00, whereupon the receiver beeps. The assistant records this. At 12:00, the two meet to compare notes. Let the proposition meeting1 describe the 9:00 meeting. Let beep say that the receiver beeps at 10:00. Let toss1 and toss2 say that the first and second coins (respectively) are tossed in the particular way that they are at 11:00; and let heads1 and heads2 say that they (respectively) land heads at 11:01. Let meeting2 describe the 12:00 meeting. For simplicity, suppose that the two coin tosses are the only chancy processes in the scenario. I’ll argue that this case makes trouble for the temporal view of chance. The trouble arrives in two waves. First, if we assume the Time-Dependence Principle, then the case is plausibly a counterexample to the Future Principle. Furthermore, rejecting the Future Principle does not help: problems persist if we retain the Time-Dependence Principle. 2.2 The chancy past Suppose the Time-Dependence Principle, and that heads1, heads2 and beep have well-defined chances at 11:00. Consider the following argument: The Chancy Past Argument (4) The argument is valid and its conclusion contradicts the Future Principle. I’ll now argue that if we accept the temporal view, then denying any of the premisses is problematic. 2.2 Premiss 1 Premiss 1 says that at 11:00, the second coin has a chance of of landing heads. The temporalist ought to accept this. Retrograde’s second coin is just an ordinary chancy coin, tossed in an ordinary way. The relevant physical process involves no backwards causation: the second coin’s behaviour has no future causes and no past effects. So, even if we were worried that an event’s having past effects might have an impact on its chances, that’s of no consequence here. If we accept the temporal view, then we must accept premiss 1. Here’s an objection. Perhaps backwards causation and chanciness are incompatible, in the strong sense that no single world can accommodate both. If this is correct, then the second coin’s behaviour cannot be chancy, despite its not being personally involved in backwards causation: the mere presence of backwards causation somewhere in the world is enough to rule out chanciness throughout the world. If backwards causation and chanciness are incompatible in this sense, then premiss 1 is false. This objection fails. There’s no good reason to endorse this strong incompatibility claim. Granted, there might be specific reasons for worrying about chanciness in certain scenarios involving backwards causation, for example, cases involving causal loops.11 But that’s not relevant here: we can suppose that there are no causal loops in the scenario. Rather than placing a blanket ban on chanciness in worlds featuring backwards causation, the temporalist ought to accept premiss 1 and insist on an account of chance that accommodates it. 2.2 Premiss 2 Premiss 2 says that heads1 and heads2 have the same chance at 11:00. If we accept the temporal view, then this is difficult to deny. Retrograde’s coin tosses are exactly alike, are shielded from all external influences, and are governed by the same physical laws. Surely they have the same chance of landing heads. The underlying thought here is that chances are ‘stable’: intrinsic duplicate physical processes that are isolated from external causal influences must give rise to the same chances. This intuitively compelling idea plays an important role in the chance literature. For example, [C]hance values should remain constant across intrinsically duplicate trials. The intuitive rationale for this is that if you repeat an experiment, the chances should stay the same. For instance, if the chance that the first coin toss lands heads is , and the second coin toss represents an intrinsically duplicate trial (exactly the same sort of coin, tossing, and environment), then the chance that the second coin toss lands heads should also be . (Schaffer, 2007, p. 125) And [C]hances should be sensitive to the basic symmetries of space and time—so that if, for example, two processes going on in different regions of spacetime are exactly alike, your recipe assigns to their outcomes the same single-case chances. (Arntzenius and Hall, 2003, p. 178) The stability of chance pushes the temporalist to accept premiss 2. Note that we must spell out the stability intuition carefully. Suppose I carry out duplicate die rolls at midnight. The stability intuition says that at midnight the dice have the same chance of landing six. We mustn’t demand that at one minute past midnight they have the same chance of landing six: if one lands six but the other doesn’t, the post-midnight chances of six are 1 and 0, respectively. We should demand stability only relative to background propositions describing the rolling of the dice, but not the outcomes of the rolls. We can spell this out as follows. Suppose that in world w we have two experimental trials, E and . (These might be coin tosses, die rolls, etc.) Suppose: (a) these trials have intrinsic duplicate set-ups; and (b) each trial is shielded from all external influences, so that no outside influence can affect their outcomes. Let A and describe (respectively) intrinsic duplicate possible outcomes of the trials, and let B and be background propositions in w describing (respectively) the set-ups of E and in w, but not their outcomes in w.12 Then by the stability intuition, we have . When combined with the temporal view, this gives the desired results in the dice case. Our trials have duplicate set-ups, and we can suppose they are shielded from outside influences. If T is the temporal history up to midnight, then T describes both rolls but neither outcome. By stability, the dice have the same chances of landing six, relative to T. We do not demand that the rolls have the same chances at 12:01, since the temporal history up to 12:01 does describe the trials’ outcomes. In the Retrograde case, the stability intuition combined with the temporal view entails premiss 2. Retrograde’s trials (the coin tosses) have intrinsic duplicate set-ups, and are shielded from external causal influences. If T is the temporal history of Retrograde’s world up to 11:00, then T describes both toss1 and toss2, but neither heads1 nor heads2. Stability demands that the trials have the same chances at 11:00, and presses the temporalist to accept premiss 2. Consider the following objection. Despite the general appeal of the stability intuition, the Retrograde case should be seen as an exception to it. Although Retrograde’s tosses are shielded intrinsic duplicates, there’s a difference between them that drives their 11:00 chances apart: the first toss has an effect that occurs before 11:00, namely, beep. The occurrence of this effect, together with the details of Retrograde’s experiment, guarantees that heads1 is true. By contrast, the second toss’s outcome has no earlier effect that guarantees its truth. Therefore, the objection goes, the temporalist can happily accept that and . Call this the instability objection. The objection might be buttressed by either of the following analyses of chance. The first is discussed (but not endorsed) by Handfield (2012, p. 27): Availability Analysis: The chance of A at t in w is the credence assigned to A by an ideally rational agent in possession of all the evidence that is available at t in w. The idea is that the chances are the best credences to adopt, given all the available evidence. This analysis stresses the evidential connections between a trial and its effects. It thus leads to unstable chances: a trial’s effects can carry evidence about its outcome, thereby making more evidence available, and so disturbing the trial’s chances. This in turn motivates rejecting premiss 2, as follows. At 11:00, the available evidence plausibly includes the details of Retrograde’s experimental set-up and the track record of her device, along with (crucially) beep. Presumably, an ideally rational agent in possession of this evidence would assign a high credence (perhaps 1) to heads1, while assigning credence to heads2. Therefore, on the Availability Analysis, we plausibly have , contrary to premiss 2. The second analysis is as follows:13 Nomological Analysis: The chance of A at t in w is the probability assigned to A by the physical laws given the past and present of w as of t. To calculate the chances at any given time, we take all the events that lie in the past or present at that time, feed them into the physical laws, and see what probabilities result. This stresses the nomological connections between a trial and its effects: we get unstable chances if a trial’s effects can make a difference to the nomological probabilities of its outcomes. To calculate Retrograde’s chances at 11:00, we take the laws, feed in everything that happens prior to 11:00, and see what probabilities we get for heads1 and heads2. We thus feed in beep, along with the details of the experimental set-up at 11:00. Because of the nomological connection between beep and heads1, the laws will arguably assign heads1 a high probability (perhaps 1), while assigning heads2 probability . We arguably have , contrary to premiss 2. I don’t think the instability objection works, because the kind of instability it involves is particularly implausible. The objection turns on the claim that duplicate trials can have different chances in virtue of having effects that occur at different times. Call this effect-based instability. Both the above analyses allow this: that’s why they leave room for the temporalist to deny premiss 2. But it’s not plausible that chance exhibits effect-based instability. To see this, contrast effect-based instability with two perfectly acceptable kinds of instability. First, consider set-up-based instability: there is nothing strange about a trial’s chances being sensitive to the intrinsic nature of its set-up. (The chance of a radioactive atom decaying in the next minute is uncontroversially sensitive to the details of its current quantum state.) Second, consider cause-based instability: there is nothing strange about a trial’s chances being sensitive to external causal influences on the trial. (Intrinsic duplicate atoms can unproblematically have different decay chances, if one but not the other will shortly be bombarded with high-energy particles.) But effect-based instability is much less intuitive: it means that two trials with the same intrinsic features, which are subject to the same external causal inputs, might nevertheless have different chances simply in virtue of the temporal locations of their effects. The objection proposes that a physical system’s chances depend not only on its intrinsic properties and the causal inputs to the system, but also on the causal outputs of the system. I find this implausible. It means that one can manipulate the chances associated with an experimental trial without exerting any causal influence whatsoever on that trial. On the Availability Analysis, one can manipulate a trial’s chances merely by controlling the flow of information, that is, simply by making more (or less) information available. On the Nomological Analysis, one can manipulate a trial’s chances merely by arranging for its outcomes to have certain effects at certain times. But it’s not plausible that chances behave in this way. The point can be pressed by noting that chances are physical probabilities.14 This is what distinguishes chances from other kinds of objective probabilities, such as logical and evidential probabilities. Given this, we might have hoped to assimilate a physical system’s chance properties to its archetypal physical properties: its temperature, length, mass, and so on. Such properties do not exhibit effect-based instability. You can’t manipulate a physical system’s temperature (etc.) merely by making more information available, or by arranging for its temperature to have certain effects. You can find out about its temperature in these ways, but you cannot influence it. To influence it, you must provide a causal input to the system. But according to the instability objection and the analyses that support it, you can manipulate a system’s chance properties merely by arranging for the system’s behaviour to have certain effects. This makes chance properties quite unlike archetypal physical properties. For this reason, the temporalist faces pressure to reject the Availability Analysis and the Nomological Analysis, and to accept premiss 2 and the stability of chance. Having said this, I concede that accepting unstable chances in the Retrograde case is an intelligible option. One could coherently insist that there are two kinds of physical features: those that exhibit effect-based instability (like chances) and those that don’t (like temperature, mass, length, and so on). One could then try to incorporate the unstable properties into physical theory, alongside the stable ones. For what it’s worth, I find this inelegant. Better, I think, to seek a view of chance on which chance properties are as stable, and as much like archetypal physical properties, as is feasible. But I accept that some (for example, those who emphasize chance’s epistemic role) may not value stability so highly, and may be less concerned with assimilation.15 However, let me make two final points. First, as the above quotations attest, some do value stability. This motivates the search for a view of chance that maintains stability in the Retrograde case. Second, the importance of stability as a theoretical desideratum is surely a matter of degree: everyone should agree that it carries some weight, even if only as a tiebreaker, while admitting that other desiderata are also important. In §3, I’ll argue that the causal view of chance does just as well as the temporal view with respect to other important desiderata, and better with respect to stability. This should be of interest to those who place any weight on stability. Temporalists who find the stability of chance intuitively attractive, including those who would assimilate chance properties to other archetypal physical properties, should be reluctant to reject premiss 2. 2.2 Premiss 3 Premiss 3 says that at 11:00, the chance of Retrograde’s first coin landing heads is the same as the chance of the receiver beeping. This is difficult for the temporalist to deny. Due to Retrograde’s meticulous preparations and the supreme reliability of her device, the situation at 11:00 together with the laws of nature guarantee that the receiver beeps iff the first coin lands heads. Therefore, beep and heads1 have the same chance. We can formulate this reasoning as follows. Let state describe in maximal detail the state of Retrograde’s world at 11:00. Let laws describe its laws. We have: Sub-argument for premiss 3 (state & laws) entails (heads1 iff beep)  3. The sub-argument is valid and its conclusion is premiss 3.16 I’ll now argue that for the temporalist, denying any of (i), (ii) or (iii) is problematic. Denying (i) means rejecting the possibility of the Retrograde scenario, since (i) follows from the foolproof reliability of Retrograde’s device. Denying (ii) is off the table, since it follows from the temporal view. At first sight, (iii) seems very plausible: how could the laws fail to have maximal chance? Surely they don’t deserve to be called ‘the laws’ if they have some chance of being violated. I find this persuasive, and temporalists who agree will be inclined to accept (iii). But not everyone accepts that the laws have maximal chance. It’s denied by some Humeans about chance, for example, who take the laws to supervene on patterns of actual events (Lewis, 1980, 1994): since patterns of actual events can be chancy, they say, the laws can be too. This offers hope to the temporalist: perhaps she can reject (iii), and so premiss 3, while accepting (i) and (ii), along with premisses 1 and 2 and the Future Principle. This won’t work. By (i), entails . So at 11:00, we have . By the Future Principle, , and so the inequality reduces to . By premisses 1 and 2, we have . Thus . On the current proposal, the temporalist must say that at 11:00, the chance of the laws being true is at most . Not only do the laws have some chance of being violated, they have a fifty-fifty chance of being violated: the truth of the laws depends quite literally on the toss of a coin.17 It gets worse. At 11:00, the truth or falsity of will be conclusively settled in the next minute. So on the current proposal, at 11:00 there’s a fifty-fifty chance that by 11:01, will be conclusively settled to be true: the first two conjuncts already have chance 1, while the third has chance and will be settled by 11:01. But is incompatible with laws. So if is conclusively settled to be true, then laws will be conclusively settled to be false. At 11:00, the laws have a fifty-fifty chance of being violated in the next minute. It gets even worse. We can construct situations in which the laws have an arbitrarily high chance of being violated arbitrarily soon. Suppose Retrograde’s experiment takes not a minute, but a second, and that her first coin is biased towards tails, so that the chance of heads at 11:00 is one in a million. Suppose that the coin in fact lands heads. Then at 11:00, the coin has only a one-in-a-million chance of landing heads; and yet its failure to do so would violate the laws. At 11:00 there’s an extremely high chance that the laws will be violated in the next second. Further refinements of the example drive the chance of a violation arbitrarily close to 1, and the time-frame for this violation arbitrarily close to zero. This is bad. Whatever else we say about the laws, we don’t want to say that they sometimes have an arbitrarily high chance of being violated arbitrarily soon. Even the temporalist who holds that the laws have non-maximal chance should be loath to reject premiss 3. This spells trouble for the temporal view. If one accepts the view, then rejecting any of premisses 1, 2 or 3 leads to trouble. But accepting them all means rejecting the Future Principle. The Chancy Past Argument undermines the temporal view. 2.3 Against time-dependence I’ll now argue that the Retrograde case makes trouble for the Time-Dependence Principle. Suppose that the temporalist responds to the Chancy Past Argument by accepting its conclusion, thereby abandoning the Future Principle. Can she nevertheless salvage the Time-Dependence Principle, along with some suitably weakened version of the Future Principle? Recall that the original motivation for the Future Principle was the intuitive fixity of the past in cases like Lewis’s labyrinth. But the Future Principle goes beyond this, stating that the past is non-chancy in all possible cases. Plausibly, in cases where past events remain open to future influence, the claim that they are ‘fixed’ can be questioned. So perhaps the temporalist can retreat to a weaker claim: that the past must be non-chancy except in cases of backwards causation. This would maintain the intuitive non-chanciness of the past in ordinary cases like Lewis’s labyrinth while permitting past chanciness in cases like Retrograde’s. While this position is internally consistent, it’s difficult to motivate. The methodology that supports time-dependence in ordinary cases tells against time-dependence in the Retrograde case. Why does the temporalist accept time-dependence? Because intuitively, the correct chance judgements for a well-informed agent to make vary with that agent’s situation, and particularly with the time at which the agent is located. Suppose that Retrograde’s less adventurous colleague, Professor Prograde, observes while a fair coin is tossed and lands heads. According to the temporalist, when Prograde is situated before the toss, she would be intuitively correct to judge that the toss’s outcome is ‘open’ and that heads has chance ; but when situated after the toss, she would be correct to judge that the toss’s outcome is ‘fixed’ and that heads has chance 1. Which features of Prograde’s post-toss situation make this the case? This is a difficult question, but we can give some rough answers: she has observed the coin landing heads; she has memories and records of it; there is now nothing she can do about it; and so on.18 In any case, the intuition is a compelling one: well-informed agents located in different situations can correctly make different chance judgements about the very same proposition. The temporalist takes this to show that the chances themselves vary across situations, and in particular, across times. This reasoning, which we might call the methodology of situated intuitions, is what motivates the time-dependence of chance. To emphasize the importance of this methodology to the temporalist, consider the invariant view of chance, according to which chances exhibit no intra-world variation: the chance of a fair coin landing heads is just , simpliciter.19 Why prefer the temporal view to invariantism? Because of the methodology of situated intuitions. The invariantist must abandon the intuition that post-toss Prograde would be correct to judge that heads has chance 1, along with the related intuitions concerning fixity. By contrast, the temporal view vindicates these intuitions. On this basis, the temporalist takes the methodology of situated intuitions to favour her view over invariantism. Note that the invariantist has a possible reply here.20 Sure, she might say, Prograde’s post-toss credence in heads ought to be 1, but that’s only because she has inadmissible evidence; the chance of heads is nevertheless , as invariantism says. The temporalist objects to this, on the grounds that it fails to respect her intuitions about the objective chances: when we consider Prograde’s post-toss situation, she claims, we intuit that Prograde would rightly judge that the toss’s outcome is objectively fixed, and that the objective chance of heads is 1. The methodology of situated intuitions urges us to take these intuitions seriously as intuitions about the objective facts, thereby adopting a view of chance that accommodates them. Invariantism fails to do this, instead explaining the intuitions away as mere credential artefacts. Thus the temporalist rejects invariantism. Now, this is not a knock-down argument against invariantism: one could instead either reject the temporalist’s intuitions or reject the methodology of situated intuitions. But the temporalist maintains that the better path is to accept the situation-dependence of chance. The methodology of situated intuitions is thus important in motivating the temporal view. The problem for the temporalist is that the methodology of situated intuitions cuts against her view in the Retrograde case. Consider the chance of beep relative to, first, the assistant’s situation at 11:00, and second, Retrograde’s situation at 11:00. By the methodology of situated intuitions, we discern these chances by considering what the intuitively correct chance judgements would be, as made by well-informed agents located in these this situations. I claim that intuitively, given the assistant’s situation at 11:00, he would rightly judge beep to have chance 1. Why? For the very same reasons that post-toss Prograde would rightly judge heads to have chance 1: he has just heard the receiver beep; he remembers it and has records of it; there is now nothing he can do to prevent it; and so on. By our usual standards for judging fixity/openness and discerning chances, the occurrence of the beep is ‘fixed’, and beep has chance 1. Retrograde’s situation at 11:00 is different: she would rightly judge the chance of beep to be . Why? For the same reasons that pre-toss Prograde would rightly judge the chance of heads to be : the usual telltale signs of fixity are absent. So beep has chance 1 relative to the assistant’s 11:00 situation, and chance relative to Retrograde’s 11:00 situation. But the temporal view cannot accommodate this, since these situations occur at the same time. The methodology of situated intuitions thus cuts against the temporal view. The temporalist has a possible reply here.21 Sure, she might say, the assistant’s 11:00 credence in beep ought to be 1, but only because he has inadmissible evidence; the 11:00 chance of beep is nevertheless . But this reply is hard to motivate. It parallels the invariantist’s reply to the temporalist in the Prograde case: each insists that the chances are invariant across certain situations, contrary to the methodology of situated intuitions, and seeks to explain away our fixity intuitions in terms of credences. I’m inclined to resist the temporalist’s reply here, on the same grounds that the temporalist resists the invariantist’s reply. The reasons for claiming that beep is fixed relative to the assistant’s situation are just the standard ones that the temporalist herself deploys in the Prograde case. If we claim that beep is intuitively open here, explaining away its apparent fixity as merely credential, then there is no principled reason not to treat the Prograde case similarly. But this means accepting invariantism. Instead, I think, we should follow the fixity intuitions where they lead, and reject the Time-Dependence Principle. Admittedly, this is not a knock-down argument. One can consistently maintain time-dependence, either by rejecting my intuitions about situated chance judgements or by rejecting the methodology of situated intuitions. But the former response is unsatisfying, because the assistant’s post-beep situation is so similar to Prograde’s post-toss situation: surely they should be treated alike. And the latter response is troubling, because the methodology of situated intuitions is our best means of adjudicating between invariantism, the temporal view, and other views. What would we do without it? (Note that the temporalist should have a special aversion to this response, given the importance of this methodology in recommending her view over invariantism.) All things considered, the best way forward is to reject time-dependence. We should therefore seek a new view of chance: one that defines chances relative to a new kind of background proposition. These background propositions must be capable of reflecting the intuitive intra-world variation of chance in the Retrograde case. Specifically, we will need two background propositions, B and , that correspond (respectively) to Retrograde’s and the assistant’s situations at 11:00, such that the chance of beep is relative to B and 1 relative to . I’ll now give such a view. 3. A causal view of chance I’ll now present an alternative view of chance and argue that it solves the problems of the previous section. The key move is to connect chance less closely to the arrow of time—that is, the past–future distinction—and more closely to the arrow of causation—the cause–effect distinction.22 3.1 Statement of the view We begin with some definitions and notation. For any world w, define the causal network of w as the ordered pair , where is the set of events that occur in w, and is the direct causal influence relation in w, which I assume to be a binary relation on . Where , we write just in case exerts a direct causal influence on in w. Here we say that is a cause ofin w and is an effect ofin w. (We can often omit the subscript, writing simply →.) A single event can have multiple causes and multiple effects. A cause need not necessitate its effects; for example, the tossing of a coin is a cause of its landing heads. Where we have and , we write , and say that the three events form a causal chain. The convention extends to longer chains in the obvious way. We can represent a causal network using a causal network diagram, with nodes representing events and directed edges representing the direct influence relation. (See fig. 1.) Figure 1: View largeDownload slide A causal network Figure 1: View largeDownload slide A causal network For each world w, let be the ancestral of . Thus we have iff in w there is a causal chain leading from e to . Here say that e is a (causal) ancestor ofin w, and that is a (causal) descendant of e in w. We can extend the causal ancestor relation from individual events to sets of events. Where and , say that e is an ancestor of E in w, written , iff for some ; and say that e is a descendant of E in w, written , iff for some . Say that E is connected to e in w just in case either or or . Thus E is connected to e just in case E contains either e itself or some ancestor or descendant of e. Say that E is globally connected in w if E is connected to every event in w. A globally connected set of events is like an unavoidable roadblock in a town of one-way streets: just as one can’t drive through town without encountering the roadblock, one can’t trace a maximal path through the causal network without including a member of the globally connected set. (In fig. 1, is globally connected.) We now define causal histories. Let E be a globally connected set of events in w. The causal history of E in w, written , is the strongest proposition entirely about E and its ancestors that is true in w. The causal history of E in w gives the full causal story of E and its ancestors in w, and nothing else. Thus says three things at once. It says that a particular set of events occurs: the set containing all the members of E, together with all their ancestors in w. It says that these events are causally related to one another in a particular way: the way they’re related in w. And it says that none of these events has any additional causes, aside from those they have in w. Thus the causal history of E in w provides a complete description of the ancestors of E in w, and all their in-house causal connections, and says that none of the events it describes has any additional causes. (Note: does not rule out the possibility that some of these events have additional effects.) Say that a proposition C is a causal history of w if C is the causal history of some globally connected set of events E in w. Causal histories are the causal analogues of temporal histories. Note a consequence of this definition. A causal history of a world tells the causal story of every causal chain in that world ‘from the beginning’ (causally speaking). Call an event e spontaneous in w if e occurs in w and has no ancestors in w. Then each causal history of w describes all the spontaneous events in w. In worlds with no backwards causation, causal histories subsume temporal histories. To see this, let w be a world with no backwards causation, t be a time in w, and T be the temporal history of w up to t. Let E be the set of events occurring at t in w. Suppose that E is globally connected in w.23 Then is a causal history that describes the very same events as T. For since E is globally connected in w and all causation in w is forwards, for each e in w we have just in case . In w, the events that occur before t are just the ancestors of the events occurring at t. Since T describes the former events and describes the latter, the two propositions describe the very same events.24 (See fig. 2, taking t = t1.) In worlds without backwards causation, for every temporal history there is a causal history describing the very same events. Figure 2: View largeDownload slide A temporal history T and a causal history C Figure 2: View largeDownload slide A temporal history T and a causal history C In worlds with backwards causation, temporal and causal histories can diverge markedly. Let e occur at t in w. If w contains backwards causation, then e may have future ancestors or past descendants. Let T be the temporal history of w up to t, and let C be the causal history of the events occurring at t in w. Then any future ancestors of e are described by C but not by T; and any past descendants of e that are not also ancestors of e are described by T but not by C. Thus T and C may describe different sets of events. (See fig. 3, taking t = t1.) Backwards causation drives temporal and causal histories apart. Figure 3: View largeDownload slide A temporal history T and a causal history C Figure 3: View largeDownload slide A temporal history T and a causal history C Finally, say that an event e lies upstream of a causal history C if either C entails or C entails . The events lying upstream of a causal history are those whose occurrence or non-occurrence is decisively settled by that causal history. Say that an event lies downstream of C if it does not lie upstream of C. The events lying downstream of a causal history are those whose occurrence is left open by that causal history. The causal view of chance is the conjunction of two principles. First: Causal-History-Dependence Principle: The chance function ch is a real-valued function of three variables: a proposition A, a possible world w, and a causal history C of w. Chances are defined relative to causal histories. The situations that generate chances are comprehensive causal set-ups. Canonical chance propositions take the form , to be read as: the chance of A relative to C in w is x. We can develop our formalism in the usual way. For each causal history C of w, we have a tailored set of outcome propositions to which chances are assigned, relative to C in w. We then define the (restricted) chance function such that for all . I assume that each restricted chance function satisfies the standard probability axioms. The second component of the causal view is: Downstream Principle: If , then e lies downstream of C. All chanciness relative to a causal history must lie downstream of that causal history: what’s upstream cannot be chancy. If a causal history C of w entails the occurrence (or non-occurrence) of a given event, then that event has chance 1 (or 0) relative to C in w. Chance thus treats all causal inputs as given, by assigning them chance 1, while allowing causal outputs to have intermediate chance, regardless of their times of occurrence. The Downstream Principle follows from our framework for proposition-dependent chance, together with the choice of causal histories as background propositions. If e lies upstream of C, then either C entails or C entails , and so . Taking our background propositions to be causal histories delivers the Downstream Principle for free. Note: since all spontaneous events in w are described by every causal history of w, the Downstream Principle says that no spontaneous event is chancy relative to any causal history of w. Intuitively, it’s not clear how a theory of chance ought to treat spontaneous events, but I think this is one sensible treatment, which is in keeping with the spirit of the causal view. We assign chances to events on the basis of their ancestors, and if an event has no ancestors, it cannot be chancy relative to them. 3.2 Forwards causation The causal view inherits the strengths of the temporal view: its treatment of forwards causation. The temporal view explicates the situation-dependence of chance by saying that chances vary with time (or temporal histories). On the causal view, we also have situation-dependence, but the situations relative to which chances vary are represented by causal histories. In worlds where all causation is forwards, causal histories subsume temporal histories, and so causal-history-dependence subsumes temporal-history-dependence. For example, in Lewis’s labyrinth, let Ct be the causal history describing the events occurring at time t, and all their causal ancestors. The causal view defines chances relative to each Ct: the chance of centre is 42% relative to , 26% relative to , and so on. The causal view thus mimics the structure of the temporal view in cases without backwards causation. In these cases, we also get the fixity of the past: the Downstream Principle ensures that past events have chance 1, since in the absence of backwards causation, all past events lie causally upstream.25 In Lewis’s labyrinth: the events lying upstream of Ct are just those occurring before t, and these must have chance 1 relative to Ct; thus where 11:49 (the time at which you reach the centre), the chance of centre relative to Ct is guaranteed to be 1. The causal view thereby reproduces the intuitively correct results of the temporal view in cases of forwards causation. 3.3 Backwards causation The causal view permits past chanciness: it allows that an event e occurring earlier than t can be chancy, relative to a causal history Ct that describes all the events occurring at t, along with their ancestors. (Note: strictly speaking, the causalist cannot understand past chanciness as the chanciness at t of a pre-t event, since she denies that chances are defined at times.) Past chanciness requires backwards causation: if e is chancy relative to Ct, then e must lie downstream of Ct; and if e occurs before t, this requires backwards causation. The causal view permits past chanciness in cases of backwards causation. I’ll now apply the causal view to the Retrograde case and defend the results. We can suppose that heads1, heads2, and beep have well-defined chances relative to each of the causal histories Ci listed in fig. 4. Translating the Chancy Past Argument into causal terms yields: The Causal Chancy Past Argument (IV) This argument is valid for all causal histories C. Does it make trouble for the causal view? I’ll show that it does not, by showing that there is no causal history for which the causalist is committed to the truth of all three premisses and the falsity of the conclusion. Figure 4: View largeDownload slide The causal structure of the Retrograde case Figure 4: View largeDownload slide The causal structure of the Retrograde case Consider premiss I. By the Downstream Principle, wherever C describes heads2. So the causalist must reject premiss I for C5, C6, C11 and C12. For all the other causal histories, the causalist ought to accept premiss I, since the coin tosses are the only chancy processes in the scenario. (See column I of table 1.) Consider premiss II. The Downstream Principle requires that heads1 and heads2 have the same chance (namely, 1) for C11 and C12. The stability of chance presses the causalist to accept premiss II for C4, since C4 describes both tosses but neither landing. Since the coin tosses are the only chancy processes in the scenario, the causalist should also accept premiss II for C1, C2 and C3. The causalist is plausibly committed to premiss III for each of our causal histories. First, the experimental set-up guarantees (heads1 iff beep), and so by reasoning analogous to that in §2.2, heads1 and beep must have the same chance relative to any causal history describing toss1 but not heads1 (that is, C2, C4 or C6). Furthermore, relative to any causal history not describing toss1 (that is, C1, C3 or C5), the first coin is guaranteed to be tossed, since the tosses are the only chancy processes in the scenario; so heads1 and beep have the same chance relative to each of these. Finally, the Downstream Principle ensures that relative to any causal history describing both toss1 and heads1 (that is, C7, C8, C9, C10, C11 and C12), both heads1 and toss1 have chance 1. When is the causalist committed to the falsity of the conclusion, IV? By the Downstream Principle, the chance of beep must be 1 relative to any causal history describing beep. So the causalist must deny IV for C9, C10, C11 and C12. In the discussion of premiss III, I also assumed that the causalist is committed to beep having chance 1 at C7 and C8. (See column IV in table 1.) Table 1 Causalist constraints   I  II  III  IV  C1  T  T  T    C2  T  T  T    C3  T  T  T    C4  T  T  T    C5      T    C6      T    C7  T    T  F  C8  T    T  F  C9  T    T  F  C10  T    T  F  C11    T  T  F  C12    T  T  F    I  II  III  IV  C1  T  T  T    C2  T  T  T    C3  T  T  T    C4  T  T  T    C5      T    C6      T    C7  T    T  F  C8  T    T  F  C9  T    T  F  C10  T    T  F  C11    T  T  F  C12    T  T  F  Table 1 shows that the Causal Chancy Past Argument poses no problem to the causal view. For no causal history Ci is the causalist committed to accepting all three premisses and denying the conclusion. The only causal histories for which she must accept all three premisses are and C4; and for each of these she can happily accept the conclusion too. (Intuitively, C4 is the closest approximation to the temporal history up to 11:00: it describes both coins being tossed at 11:00, but no later events. Crucially though, it doesn’t describe beep, and so beep can be chancy, and the causalist can accept IV.) Table 2 gives a consistent assignment of chances. In §2.3, I argued that the temporalist cannot respect the intuitive intra-world variation of chance in the Retrograde case: she must either say that beep is chancy at 11:00 or that it is not, but neither answer is acceptable. The causal view does better. Given Retrograde’s situation at 11:00, she is intuitively right to judge beep to be chancy. On the causal view, this is reflected by the chanciness of beep relative to a particular kind of causal history: one describing Retrograde’s surroundings at 11:00 (including both tosses) and their causal ancestors, but describing neither heads-landing (for example, C4). Thus we have . Given the assistant’s situation at 11:00, he is intuitively right to judge beep to be non-chancy. On the causal view, this is reflected by the non-chanciness of beep relative to a causal history describing the assistant’s surroundings at 11:00 and their causal ancestors, including the receiver’s beep (for example, C9). Thus we have . The causal view does justice to our intuitive chance judgements in this case. Table 2 Causalist chances   beep  heads1  heads2  C1  ½  ½  ½  C2  ½  ½  ½  C3  ½  ½  ½  C4  ½  ½  ½  C5  ½  ½  1  C6  ½  ½  1  C7  1  1  ½  C8  1  1  ½  C9  1  1  ½  C10  1  1  ½  C11  1  1  1  C12  1  1  1    beep  heads1  heads2  C1  ½  ½  ½  C2  ½  ½  ½  C3  ½  ½  ½  C4  ½  ½  ½  C5  ½  ½  1  C6  ½  ½  1  C7  1  1  ½  C8  1  1  ½  C9  1  1  ½  C10  1  1  ½  C11  1  1  1  C12  1  1  1  Note that there is nothing problematically ‘subjective’ about this. The causalist assigns different chances to beep relative to different causal histories, just as the temporalist assigns different chances to beep relative to different temporal histories. In both cases, the chance assignments are fully objective: well-informed observers will agree about which propositions have which chances relative to which background propositions. The difference is that causal histories better reflect the structure of the Retrograde case, allowing the chance assignments given by the causal view to align more closely with our intuitive judgements. The temporal view fails because it requires beep to have a single chance at 11:00, contrary to our intuitive judgements. The causal view solves this problem by defining chances relative to causal histories: there is an appropriate causal history relative to which beep is chancy, and one relative to which it is not. The causal view thus inherits the strengths of the temporal view without its weaknesses. We ought to reject the temporal view in favour of the causal view. 4. Consequences of the causal view On the causal view of chance, the ‘arrow’ of chance is connected more closely to the arrow of causation than it is to the arrow of time. All chanciness must lie causally downstream, and if parts of the past lie causally downstream, they can be chancy. (Note: the causal view does not say that chance is temporally symmetric; it says that chance inherits its temporal asymmetry from causation.) How does this square with the intuitive ‘fixity’ of the past? Here’s what I think is happening. The key fixity intuition is a causal one: that the causal history is fixed. Doubtless we also have a temporal fixity intuition, that the past is fixed; but this ultimately rests on the causal intuition, together with our ordinary assumption that the future lies causally downstream. With this assumption in place, we can embrace both intuitions. But taking backwards causation seriously means choosing between them, and the Retrograde case shows that the causal intuition wins out. This explains why we should accept the causal view, and also why the temporal view seemed attractive until we took backwards causation seriously. Are scenarios involving backwards causation too bizarre to be worth worrying about? No. First, backwards causation is taken seriously by some live physical theories.26 Second, even if there is no actual backwards causation, we should prefer a view of chance that accommodates all possible cases to one that doesn’t. Third, if chance and causation are connected as the causal view suggests, then this is the case in our world too, regardless of whether or not there is actual backwards causation. When evaluating views of chance, we should take backwards causation seriously. Consider the consequences of the causal view for chance in causal loops. Call a set of events in w a causal loop in w if its members form a causal chain in w that begins and ends with the same event. Causal histories describe causal loops in an all-or-nothing fashion: if L is a causal loop in w, then any causal history C of w that entails the occurrence of some member of L entails the occurrence of all members of L. By the Downstream Principle, there can thus be no non-trivial chances within a causal loop: no causal history describing part of a loop can confer intermediate chance on another part of that loop. Isn’t this counterintuitive? And doesn’t it count against the causal view? No. First, causal loops are odd things, and we should regard our intuitions about them with caution. Second, the causal view gives one principled account of chances in causal loops, for the following reason. The chances associated with a physical process ought to ‘factor in’ all the causal inputs to that process: causally relevant events should not be ignored. Where a process and its outcome are parts of the same causal loop, the outcome is among the causal inputs to that very process, so the outcome must itself be taken for granted by the chances associated with the process. Thus the outcome can’t be chancy, relative to the process. The causal view delivers this. However, a further objection lurks. Even if the causal view’s treatment of causal loops is defensible in itself, it has consequences for the stability of chance (§2).27 The causal view entails a limited amount of instability: duplicate trials can have different chances, if one but not the other is embedded in a causal loop. Suppose Tim travels back in time and attempts to kill his grandfather before he had children: Tim rolls a bomb towards grandfather that will explode iff a radium atom inside the bomb decays.28 Nearby, Tom (not a time traveller) attempts to kill grandfather’s partner by rolling an intrinsic duplicate bomb. On the causal view, Tim’s bomb has zero chance of exploding, relative to any causal history describing its rolling: any such causal history must describe the bomb’s failure to explode, since this is a causal ancestor of the rolling. So Tim’s trial is non-chancy. Tom’s trial is chancy: his bomb has a positive chance of exploding, relative to its being rolled. The causal view thus leads to unstable chances in causal loop cases. As I’ve said, I think this is a defensible result. But even granting this, the temporalist might object. In §2, I used stability to push the temporalist to accept premiss 2: this was part of my argument for moving from the temporal view to the causal view. But now it turns out that even the causalist is committed to unstable chances in causal loops. So why not accept them in the Retrograde case? One might then respond to the Chancy Past Argument by denying premiss 2, thus maintaining the temporal view. This objection fails. Accepting unstable chances in the Retrograde case is more troubling than accepting them in the Tim/Tom case. The occurrence of the set-up of Tim’s trial (Tim’s rolling his bomb) depends on the trial’s outcome turning out in a particular way (the bomb’s not exploding): the bomb’s exploding is incompatible with Tim’s having rolled it, since grandfather’s survival is a necessary condition for the rolling. Given that the set-up occurs, consistency requires that this outcome occurs; and this is why the outcome has chance 1 relative to any background proposition describing the set-up. Because Tim’s trial is embedded in a causal loop, its non-chanciness falls out as a matter of logic. Tom’s trial is different: here the set-up does not depend on the outcome turning out in any particular way: Tom’s bomb can consistently be rolled whether it ultimately explodes or not. So Tom’s trial can consistently be chancy. This explains why causal loop cases yield unstable chances: consistency demands it. In this respect, both of Retrograde’s trials are similar to Tom’s trial. Neither of Retrograde’s trials is such that the set-up (the coin’s being tossed) depends on the outcome turning out the way that it in fact does (the coin landing heads). Each coin can consistently be tossed whether it lands heads or not: consistency does not force instability. Therefore, accepting unstable chances in causal loops should not tempt us to accept them in the Retrograde case. Finally, let’s briefly consider the chance–credence relationship. The Principal Principle (Lewis, 1980, p. 87) can be reformulated in causal terms as follows: Causal Principal Principle: Let cr be a rational initial credence function, A be a proposition, C be a causal history, x be a real number in the unit interval, and X be the proposition that the chance of A relative to C is x. Let E be a proposition that is compatible with X and admissible relative to C. Then . Of course, this must be supplemented by an account of admissibility. I think there are good prospects of success here. Lewis claims that past evidence is ‘as a rule’ admissible, making an exception for cases of backwards causation. The causalist might naturally say that evidence lying upstream of a causal history C is admissible relative to C, thereby allowing the problematic evidence (which lies downstream) to count as inadmissible, as it should.29 Footnotes 1 I’ll use coin tosses as my main examples of chancy processes. Readers who prefer other examples can freely substitute them throughout. 2 This principle is widely accepted (Lewis, 1980; Bigelow et al., 1993; Hall, 1994, 2004; Schaffer, 2007; Handfield, 2012), though not universally so (Hoefer, 2007). I’ll focus on the time parameter, taking the other two (the proposition and possible world) for granted in what follows. 3 For Lewis, although times provide a convenient way of indexing chances, it is in fact collections of historical facts, and not times themselves, on which chances depend. (Thanks to an anonymous reviewer for stressing this point.) I will shortly make this dependence explicit by indexing chances to what I will call ‘temporal histories’. 4 See Lewis (1980), Bigelow et al. (1993), Schaffer (2007), and Hall (1994, 2004). 5 See Lewis (1980, p. 93), Bigelow et al. (1993, p. 454), and Schaffer (2007, p. 125). 6 Here I follow Meacham (2005, 2010), Nelson (2009), and Briggs (2010, 2015). The idea can be traced back to Popper (1957, 1959). 7 See Nelson (2009, pp. 175–6) for a contrary view. This distinguishes the chances relative to B from the chances conditional on B, which can be well-defined in worlds where B is false. 8 Proof: If B is a background proposition in w and B entails A, then since is a probability function and the background proposition is non-chancy, we have , so . 9 Here I follow Lewis’s use of ‘entirely about’ (1980, p. 93). 10 The thought that backwards causation makes trouble for the Future Principle is not new. See Hall (1994, p. 514), Edgington (1997, p. 431), Eagle (2011, p. 291; 2014), Handfield (2012, pp. 9–10), and Suárez (2013, pp. 79–80). My aim is to develop this thought into an explicit objection to the temporal view, and give an alternative view that overcomes it. 11 Mellor (1981; 1995). See also Edgington (1997, §8), Berkovitz (2001), Dowe (2001), and §4 below. 12 Those who distinguish ‘high-level’ and ‘low-level’ chances should also require that A and describe the respective outcomes at the same level, and B and describe the respective set-ups at the same level. Thanks to the editors for pointing this out. My examples satisfy this condition. 13 Thanks to an anonymous reviewer for suggesting this. 14 Handfield (2012, p. 28) notes the failure of the Availability Analysis to capture the physicality of chance. 15 Thanks to an anonymous reviewer for pressing this point. 16 Proof of validity: From (i) we have . From (ii) and (iii) we have . Combining these gives . Thus . Since chances are probabilities, . Similarly, . Therefore . 17 Hall (1994, §6) discusses a similar issue in the context of undermining futures. 18 See Torre (2011) for an examination of these issues. 19 See Hoefer (2007, §3.2). 20 See Hoefer (2007, p. 555). 21 Thanks to an anonymous reviewer for pressing this reply. 22 Here I develop a suggestion raised by Joyce (2007) and taken up by Eagle (2011, 2014). Joyce’s view stresses the importance of the chance–causation link, but agrees with the temporal view that chance functions are time-indexed. The view presented here takes chances as defined relative to causal histories. Eagle’s proposal combines time-dependence with an emphasis on causal information: he says that ‘chance is informed, at a time, about the causes of outcomes that have occurred by that time’ (2014, p. 138), and that chances are ‘indexed to certain bodies of information … [which contain] information about causes’ (2014, p. 153). 23 Here I assume that w is Markovian connected: for every event e that occurs in w and every time t in w, either e occurs at t in w, or else e has either an ancestor or a descendant that occurs at t in w. (Thanks to the editors for clarifying this.) Dropping this assumption complicates the relationship between temporal and causal histories: w may contain spontaneous post-t events, which are not described by the temporal history of w up to t, but are described by every causal history of w; so there is no causal history of w that describes the same events as this temporal history. For simplicity I’ll assume Markovian connectedness, noting where necessary the consequences of dropping it. (See especially footnote 25.) 24 While T and describe the same events, they say different things about what doesn’t happen: while T rules out additional pre-t events, rules out additional ancestors of . 25 Dropping the assumption of Markovian connectedness (see footnote 23 above) sees the Downstream Principle go slightly beyond the Future Principle here. Suppose that w is not Markovian connected, and contains a spontaneous event e occurring after t. Then the Future Principle allows to be chancy at t, while the Downstream Principle entails that has chance 1 relative to each causal history of w. In my opinion, this difference does not tell in favour of either view, since it’s unclear how spontaneous events ought to be treated: on the one hand, it seems reasonable enough to say that e has a well-defined, non-trivial chance relative to the events that temporally precede it but do not cause it, as the Future Principle allows; but on the other, it also seems reasonable to say that since e has no causes, it has no well-defined chance (except when e forms part of the causal history, when its chance is 1), as the Downstream Principle says. 26 See Costa de Beauregard (1976), Cramer (1986), Price (1984, 2008), and Miller (1996). 27 Thanks to an anonymous reviewer for clarifying this point. 28 This is a probabilistic version of a case of Lewis’s (1976). 29 I am grateful to Hilary Greaves, Alan Hájek, Robyn Kath, Leon Leontyev, Wolfgang Schwarz, two anonymous reviewers, and the editors of Mind for useful discussion and feedback. 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Google Scholar CrossRef Search ADS   Cramer John G. 1986: ‘The Transactional Interpretation of Quantum Mechanics’. Reviews of Modern Physics , 58, pp. 647– 87. Google Scholar CrossRef Search ADS   Dowe Phil 2001: ‘Causal Loops and the Independence of Causal Facts’. Philosophy of Science , 68( 3), pp. S89– S97. Google Scholar CrossRef Search ADS   Eagle Antony 2011: ‘Deterministic Chance’. Noûs , 45( 2), pp. 269– 99. Google Scholar CrossRef Search ADS   Eagle Antony 2014: ‘Is the Past a Matter of Chance?’ In Wilson 2014, pp. 57– 92. Edgington Dorothy 1997: ‘Mellor on Chance and Causation’. British Journal for the Philosophy of Science , 48( 3), pp. 411– 33. Google Scholar CrossRef Search ADS   Hall Ned 1994: ‘Correcting the Guide to Objective Chance’. Mind , 103, pp. 505– 18. Google Scholar CrossRef Search ADS   Hall Ned 2004: ‘Two Mistakes about Credence and Chance’. Australasian Journal of Philosophy , 82( 1), pp. 93– 111. Google Scholar CrossRef Search ADS   Handfield Toby 2012: A Philosophical Guide to Chance . Cambridge: Cambridge University Press. Google Scholar CrossRef Search ADS   Hoefer Carl 2007: ‘The Third Way on Objective Probability: A Sceptic’s Guide to Objective Chance’. Mind , 116, pp. 549– 96. Google Scholar CrossRef Search ADS   Jeffrey Richard C. (ed.) 1980: Studies in Inductive Logic and Probability, Volume II . Berkeley and Los Angeles: University of California Press. Joyce James M. 2007: ‘Epistemic Deference: The Case of Chance’. Proceedings of the Aristotelian Society , 107, pp. 187– 206. Google Scholar CrossRef Search ADS   Körner S. (ed.) 1957: Observation and Interpretation: A Symposium of Philosophers and Physicists . London: Butterworth. Lewis David 1976: ‘The Paradoxes of Time Travel’. American Philosophical Quarterly , 13( 2), pp. 145– 52. Lewis David 1980: ‘A Subjectivist’s Guide to Objective Chance’. In Jeffrey 1980, pp. 183–200. Reprinted with added postscripts in Lewis, Philosophical Papers, Volume II, pp. 83–132. Oxford: Oxford University Press, 1986. Page references are to the reprint. Lewis David 1994: ‘Humean Supervenience Debugged’. Mind , 103, pp. 473– 90. Google Scholar CrossRef Search ADS   Loewer Barry, Schaffer Jonathan (eds.) 2015: A Companion to David Lewis . Oxford and Malden, MA: Wiley-Blackwell. Google Scholar CrossRef Search ADS   Meacham C. J. G. 2005: ‘Three Proposals Regarding a Theory of Chance’. Philosophical Perspectives , 19( 1), pp. 281– 307. Google Scholar CrossRef Search ADS   Meacham C. J. G. 2010: ‘Two Mistakes Regarding the Principal Principle’. British Journal for the Philosophy of Science , 61( 2), pp. 407– 31. Google Scholar CrossRef Search ADS   Mellor D. H. 1981: Real Time . Cambridge: Cambridge University Press. Mellor D. H. 1995: The Facts of Causation . London: Routledge. Google Scholar CrossRef Search ADS   Miller D. J. 1996: ‘Realism and Time Symmetry in Quantum Mechanics’. Physics Letters A , 222, pp. 31– 6. Google Scholar CrossRef Search ADS   Nelson Kevin 2009: ‘On Background: Using Two-Argument Chance’. Synthese , 166( 1), pp. 165– 86. Google Scholar CrossRef Search ADS   Popper Karl R. 1957: ‘The Propensity Interpretation of the Calculus of Probability and the Quantum Theory’. In Körner 1957, pp. 65– 70. Popper Karl R. 1959: ‘The Propensity Interpretation of Probability’. British Journal for the Philosophy of Science , 10, pp. 25– 42. Google Scholar CrossRef Search ADS   Price Huw 1984: ‘The Philosophy and Physics of Affecting the Past’. Synthese , 61, pp. 299– 324. Google Scholar CrossRef Search ADS   Price Huw 2008: ‘Toy Models for Retrocausality’. Studies in History and Philosophy of Modern Physics , 39, pp 752– 76. Google Scholar CrossRef Search ADS   Schaffer Jonathan 2007: ‘Deterministic chance?’ British Journal for the Philosophy of Science , 58( 2), pp. 113– 40. Google Scholar CrossRef Search ADS   Suárez Mauricio 2013: ‘Propensities and Pragmatism’. Journal of Philosophy , 110( 2), pp. 61– 92. Google Scholar CrossRef Search ADS   Torre Stephan 2011: ‘The Open Future’. Philosophy Compass , 6( 5), pp. 360– 73. Google Scholar CrossRef Search ADS   Wilson Alastair (ed.) 2014: Chance and Temporal Asymmetry . Oxford: Oxford University Press. Google Scholar CrossRef Search ADS   © Cusbert 2017 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mind Oxford University Press

Backwards Causation and the Chancy Past

Mind , Volume 127 (505) – Jan 1, 2018

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Abstract

Abstract I argue that the past can be objectively chancy in cases of backwards causation, and defend a view of chance that allows for this. Using a case, I argue against the popular temporal view of chance, according to which (i) chances are defined relative to times, and (ii) all chancy events must lie in the future. I then state and defend the causal view of chance, according to which (a) chances are defined relative to causal histories, and (b) all chancy events must lie causally downstream. The causal view replicates the intuitively correct results of the temporal view in cases of ordinary forwards causation, while correctly handling cases of backwards causation. I conclude that objective chance is more closely related to the direction of causation than it is to the direction of time. Objective, physical probability (or ‘chance’) is more closely related to causation, and less closely related to time, than is commonly supposed. The usual view of chance, which I’ll call the temporal view, says that chance and time are connected in two ways: chances are defined at times, and all chancy events (those with chances in between 0 and 1) must lie in the future. While this view is initially appealing, I’ll argue that both of its central claims should be rejected. I’ll give a case of so-called ‘backwards’ causation in which a past event inherits the chanciness of its future cause. When applied to this case, the temporal view breaks down. In its place, I’ll defend a causal view of chance. On this view, chances are defined relative to causal histories (not times), and all chanciness must lie causally downstream (not in the future). I’ll argue that the causal view affords all the benefits of its temporal cousin, without the drawbacks: it’s motivated by the very same considerations, and indeed replicates the results of the temporal view in cases of ordinary ‘forwards’ causation; but it correctly handles cases of backwards causation, where the temporal view breaks down. I’ll conclude that we ought to accept the causal view, and with it the primacy of the chance–causation relationship. In §1, I’ll state and motivate the temporal view. In §2, I’ll argue that the view faces serious objections. In §3, I’ll set out the causal view, and show that it overcomes the problems of the temporal view while retaining its strengths. In §4, I’ll consider the consequences for chance. 1. The temporal view of chance Chances are objective, physical probabilities. Their objectivity distinguishes them from subjective probabilities (for example, credences): if a particular coin has a chance of of landing heads when tossed on a particular occasion, then this is an objective fact, not a mere matter of opinion.1 The physicality of chances distinguishes them from other kinds of objective probabilities (for example, logical and evidential probabilities): chances reside within physical systems, such as atoms, coins and galaxies, and are governed by physical laws. A system’s chance properties are thus among its physical properties, akin to its mass, shape and electric charge. Chance is commonly taken to be closely connected to time. Here’s an influential example from Lewis: Suppose you enter a labyrinth at 11:00 a.m., planning to choose your turn whenever you come to a branch point by tossing a coin. When you enter at 11:00, you may have a 42% chance of reaching the center by noon. But in the first half hour you may stray into a region from which it is hard to reach the center, so that by 11:30 your chance of reaching the center by noon has fallen to 26%. But then you turn lucky; by 11:45 you are not far from the center and your chance of reaching it by noon is 78%. At 11:49 you reach the center; then and forevermore your chance of reaching it by noon is 100%. (Lewis, 1980, p. 91) There are two attractive thoughts here: that chances are time-dependent (in the sense that they can vary with time); and that at any given time, all the chancy events (those with chances in between 0 and 1) must lie in the future. The first thought can be formulated as follows:2 Time-Dependence Principle: The chance function ch is a function of three variables: a proposition A, a time t, and a possible world w. This says that time is one of the inputs to the chance function. Canonical chance propositions thus take the form , where . Lewis’s labyrinth case motivates the time-dependence of chance. Let centre be the proposition that you reach the centre by noon. There’s clearly some variation in the chance of centre within the world described: when we consider your respective situations at 11:00 and 11:30, for example, it’s clear that your chance of reaching the centre by noon is higher in the former situation than in the latter. Chances thus exhibit intra-world variation: relative to different situations within a given world, the very same possibility can have different chances. Chances are situation-dependent. In Lewis’s example, it’s natural to capture situation-dependence as time-dependence: a salient difference between the situations relative to which chances vary, and a convenient way of indexing them, is given by the times at which they occur.3 This leads naturally to the Time-Dependence Principle. With the Time-Dependence Principle in place, we can introduce some useful formalism. By holding fixed two of the chance function’s input parameters (the possible world and the time), we induce a real-valued, one-place function over propositions. For each possible world w and time t, let be the set of all propositions A such that the chance of A at t in w is well-defined. Define a one-place function such that for all . Call the (restricted) chance function at t in w. This function gives us the chances of various propositions at t in w. For example, if w is the labyrinth world, we have . (I’ll sometimes suppress the world parameter, writing simply cht.) I’ll assume that each (restricted) chance function satisfies the standard probability axioms. The second intuitively attractive thought brought out by Lewis’s example is that all chanciness must lie in the future. While your chance of reaching the labyrinth’s centre by noon may be intermediate before noon, it must go to either 0 or 1 afterwards. Where e is an event, let be the time of occurrence of e, that is, the time at which e either occurs or fails to do so. Let be the proposition that e occurs (at ). Then our second thought can be spelled out as follows:4 Future Principle: If , then . This says: if at t an event e has an intermediate chance of occurring, then e’s time of occurrence must lie in the future at t. All chanciness lies in the future, at all times and in all possible worlds: past chanciness is impossible. The Future Principle is intuitively compelling in run-of-the-mill cases like Lewis’s labyrinth: in such cases, the past is intuitively ‘fixed’, which goes naturally with extremal chance.5 Define the temporal view of chance as the conjunction of the Time-Dependence Principle and the Future Principle. According to the temporal view, chances vary with time, and past events must have extremal chance. In the context of everyday examples like Lewis’s labyrinth, the temporal view is plausible. For this reason, it’s the dominant view of chance in the literature. Let me now set out a more general formal framework for chance, of which the temporal view is a special case. (This will prove useful when we come to consider another special case.) This framework defines chances relative to background propositions.6 Take the chance function ch to be a real-valued function of three variables: two propositions A and B, and a possible world w. Canonical chance propositions thereby take the form , to be read as: the chance of A relative to B in w is x. The idea is that B describes a situation that arises in w and generates chances (perhaps a particular coin toss) and A describes a possible outcome of that situation (perhaps the coin’s landing heads) to which a chance value x is assigned. Call A an outcome proposition and B a background proposition. If B is a background proposition in w, then we have a tailored set of outcome propositions , to which chances are assigned. This set contains all the propositions to which a well-defined chance is assigned in w, relative to B. For example, if B describes a coin toss in w, then might include heads and tails. For each chance-generating combination of a background proposition B and a world w, define a (restricted) chance function such that for all . The restricted chance function assigns a chance value to each of the propositions in , that is, to each possible outcome corresponding to the background proposition described by B in w. For example, where B describes a coin toss in w, we might have . (I’ll sometimes suppress the world parameter, writing .) I’ll assume that each chance function satisfies the standard probability axioms. I’ll make two assumptions about background propositions. First, they are true in those worlds in which they generate chances: if B is a background proposition in w, then B is true in w.7 If a particular coin toss fails to take place in w, then in w there are no chances associated with that toss. Second, each background proposition is non-chancy, relative to itself: if B is a background proposition in w, then . Relative to a given coin toss, while the coin’s landing might be chancy, its tossing cannot be. (The tossing may be chancy relative to another background proposition.) It follows that no proposition entailed by B is chancy relative to B.8 We can restate the temporal view of chance in proposition-dependent terms. For any world w and time t, define the temporal history of w up to t, written , as the strongest proposition entirely about the past-and-present at t that’s true in w.9 Thus describes in maximal detail everything that happens in w up to and including t, without saying anything about what happens after t. Call T a temporal history of w just in case T is the temporal history of w up to some time t. We can reproduce the temporal view by taking our background propositions to be temporal histories. Defining chances relative to temporal histories is formally equivalent to defining them relative to times, in the sense that there is a bijective correspondence between pairs, where T is a temporal history of w, and pairs, where t is a time in w. We thus have a propositional surrogate for the Time-Dependence Principle. We also get the non-chanciness of the past: if , then either entails or else it entails , and so . This gives a proposition-dependent surrogate for the Future Principle, with temporal histories in place of their corresponding times. The proposition-dependent framework subsumes the temporal view. The temporal view of chance is plausible and widely endorsed. I will now argue that it should be rejected. 2. Problems for the temporal view My argument against the temporal view relies on a case of ‘backwards’ causation: a case in which a later event causes an earlier one. I will assume that such cases are conceptually possible.10 I’ll describe the case (§2.1), and then show how it makes trouble for both the Future Principle (§2.2) and the Time-Dependence Principle (§2.3). 2.1 Retrograde’s receiver Professor Retrograde has invented an ingenious device capable of backwards-in-time signalling. The device consists of two components: the sender, which sends out a signal when activated, and the receiver, which receives that signal at a designated earlier time, whereupon it beeps. Testing has shown this device to be supremely reliable. Retrograde performs the following experiment. At 9:00, she meets with her assistant. She then proceeds to her sending laboratory, which houses the sender, while the assistant travels across town to the receiving laboratory, where the receiver is located. By 11:00, Retrograde has set up an experiment: two exactly similar coins will be tossed in precisely the same way at 11:00, and land at 11:01. Each toss is carefully shielded from all outside influences. The first coin is coupled to the sender, so that a backwards-in-time signal will be sent iff the coin lands heads. This signal, if sent, will make the receiver beep at 10:00, where the assistant is waiting. If the first coin lands tails, no signal will be sent. Retrograde’s device is so reliable and her preparations so meticulous that if the first coin lands heads, the receiver is nomologically guaranteed to beep: the situation at 11:00 together with the laws of nature entail that the receiver beeps at 10:00 iff the first coin lands heads at 11:01. The second coin is a control: it will have no interesting effects regardless of how it lands. As it happens, both coins land heads at 11:01. Retrograde observes this and records it. The first coin activates the sender, and the resulting signal is received by the receiver at 10:00, whereupon the receiver beeps. The assistant records this. At 12:00, the two meet to compare notes. Let the proposition meeting1 describe the 9:00 meeting. Let beep say that the receiver beeps at 10:00. Let toss1 and toss2 say that the first and second coins (respectively) are tossed in the particular way that they are at 11:00; and let heads1 and heads2 say that they (respectively) land heads at 11:01. Let meeting2 describe the 12:00 meeting. For simplicity, suppose that the two coin tosses are the only chancy processes in the scenario. I’ll argue that this case makes trouble for the temporal view of chance. The trouble arrives in two waves. First, if we assume the Time-Dependence Principle, then the case is plausibly a counterexample to the Future Principle. Furthermore, rejecting the Future Principle does not help: problems persist if we retain the Time-Dependence Principle. 2.2 The chancy past Suppose the Time-Dependence Principle, and that heads1, heads2 and beep have well-defined chances at 11:00. Consider the following argument: The Chancy Past Argument (4) The argument is valid and its conclusion contradicts the Future Principle. I’ll now argue that if we accept the temporal view, then denying any of the premisses is problematic. 2.2 Premiss 1 Premiss 1 says that at 11:00, the second coin has a chance of of landing heads. The temporalist ought to accept this. Retrograde’s second coin is just an ordinary chancy coin, tossed in an ordinary way. The relevant physical process involves no backwards causation: the second coin’s behaviour has no future causes and no past effects. So, even if we were worried that an event’s having past effects might have an impact on its chances, that’s of no consequence here. If we accept the temporal view, then we must accept premiss 1. Here’s an objection. Perhaps backwards causation and chanciness are incompatible, in the strong sense that no single world can accommodate both. If this is correct, then the second coin’s behaviour cannot be chancy, despite its not being personally involved in backwards causation: the mere presence of backwards causation somewhere in the world is enough to rule out chanciness throughout the world. If backwards causation and chanciness are incompatible in this sense, then premiss 1 is false. This objection fails. There’s no good reason to endorse this strong incompatibility claim. Granted, there might be specific reasons for worrying about chanciness in certain scenarios involving backwards causation, for example, cases involving causal loops.11 But that’s not relevant here: we can suppose that there are no causal loops in the scenario. Rather than placing a blanket ban on chanciness in worlds featuring backwards causation, the temporalist ought to accept premiss 1 and insist on an account of chance that accommodates it. 2.2 Premiss 2 Premiss 2 says that heads1 and heads2 have the same chance at 11:00. If we accept the temporal view, then this is difficult to deny. Retrograde’s coin tosses are exactly alike, are shielded from all external influences, and are governed by the same physical laws. Surely they have the same chance of landing heads. The underlying thought here is that chances are ‘stable’: intrinsic duplicate physical processes that are isolated from external causal influences must give rise to the same chances. This intuitively compelling idea plays an important role in the chance literature. For example, [C]hance values should remain constant across intrinsically duplicate trials. The intuitive rationale for this is that if you repeat an experiment, the chances should stay the same. For instance, if the chance that the first coin toss lands heads is , and the second coin toss represents an intrinsically duplicate trial (exactly the same sort of coin, tossing, and environment), then the chance that the second coin toss lands heads should also be . (Schaffer, 2007, p. 125) And [C]hances should be sensitive to the basic symmetries of space and time—so that if, for example, two processes going on in different regions of spacetime are exactly alike, your recipe assigns to their outcomes the same single-case chances. (Arntzenius and Hall, 2003, p. 178) The stability of chance pushes the temporalist to accept premiss 2. Note that we must spell out the stability intuition carefully. Suppose I carry out duplicate die rolls at midnight. The stability intuition says that at midnight the dice have the same chance of landing six. We mustn’t demand that at one minute past midnight they have the same chance of landing six: if one lands six but the other doesn’t, the post-midnight chances of six are 1 and 0, respectively. We should demand stability only relative to background propositions describing the rolling of the dice, but not the outcomes of the rolls. We can spell this out as follows. Suppose that in world w we have two experimental trials, E and . (These might be coin tosses, die rolls, etc.) Suppose: (a) these trials have intrinsic duplicate set-ups; and (b) each trial is shielded from all external influences, so that no outside influence can affect their outcomes. Let A and describe (respectively) intrinsic duplicate possible outcomes of the trials, and let B and be background propositions in w describing (respectively) the set-ups of E and in w, but not their outcomes in w.12 Then by the stability intuition, we have . When combined with the temporal view, this gives the desired results in the dice case. Our trials have duplicate set-ups, and we can suppose they are shielded from outside influences. If T is the temporal history up to midnight, then T describes both rolls but neither outcome. By stability, the dice have the same chances of landing six, relative to T. We do not demand that the rolls have the same chances at 12:01, since the temporal history up to 12:01 does describe the trials’ outcomes. In the Retrograde case, the stability intuition combined with the temporal view entails premiss 2. Retrograde’s trials (the coin tosses) have intrinsic duplicate set-ups, and are shielded from external causal influences. If T is the temporal history of Retrograde’s world up to 11:00, then T describes both toss1 and toss2, but neither heads1 nor heads2. Stability demands that the trials have the same chances at 11:00, and presses the temporalist to accept premiss 2. Consider the following objection. Despite the general appeal of the stability intuition, the Retrograde case should be seen as an exception to it. Although Retrograde’s tosses are shielded intrinsic duplicates, there’s a difference between them that drives their 11:00 chances apart: the first toss has an effect that occurs before 11:00, namely, beep. The occurrence of this effect, together with the details of Retrograde’s experiment, guarantees that heads1 is true. By contrast, the second toss’s outcome has no earlier effect that guarantees its truth. Therefore, the objection goes, the temporalist can happily accept that and . Call this the instability objection. The objection might be buttressed by either of the following analyses of chance. The first is discussed (but not endorsed) by Handfield (2012, p. 27): Availability Analysis: The chance of A at t in w is the credence assigned to A by an ideally rational agent in possession of all the evidence that is available at t in w. The idea is that the chances are the best credences to adopt, given all the available evidence. This analysis stresses the evidential connections between a trial and its effects. It thus leads to unstable chances: a trial’s effects can carry evidence about its outcome, thereby making more evidence available, and so disturbing the trial’s chances. This in turn motivates rejecting premiss 2, as follows. At 11:00, the available evidence plausibly includes the details of Retrograde’s experimental set-up and the track record of her device, along with (crucially) beep. Presumably, an ideally rational agent in possession of this evidence would assign a high credence (perhaps 1) to heads1, while assigning credence to heads2. Therefore, on the Availability Analysis, we plausibly have , contrary to premiss 2. The second analysis is as follows:13 Nomological Analysis: The chance of A at t in w is the probability assigned to A by the physical laws given the past and present of w as of t. To calculate the chances at any given time, we take all the events that lie in the past or present at that time, feed them into the physical laws, and see what probabilities result. This stresses the nomological connections between a trial and its effects: we get unstable chances if a trial’s effects can make a difference to the nomological probabilities of its outcomes. To calculate Retrograde’s chances at 11:00, we take the laws, feed in everything that happens prior to 11:00, and see what probabilities we get for heads1 and heads2. We thus feed in beep, along with the details of the experimental set-up at 11:00. Because of the nomological connection between beep and heads1, the laws will arguably assign heads1 a high probability (perhaps 1), while assigning heads2 probability . We arguably have , contrary to premiss 2. I don’t think the instability objection works, because the kind of instability it involves is particularly implausible. The objection turns on the claim that duplicate trials can have different chances in virtue of having effects that occur at different times. Call this effect-based instability. Both the above analyses allow this: that’s why they leave room for the temporalist to deny premiss 2. But it’s not plausible that chance exhibits effect-based instability. To see this, contrast effect-based instability with two perfectly acceptable kinds of instability. First, consider set-up-based instability: there is nothing strange about a trial’s chances being sensitive to the intrinsic nature of its set-up. (The chance of a radioactive atom decaying in the next minute is uncontroversially sensitive to the details of its current quantum state.) Second, consider cause-based instability: there is nothing strange about a trial’s chances being sensitive to external causal influences on the trial. (Intrinsic duplicate atoms can unproblematically have different decay chances, if one but not the other will shortly be bombarded with high-energy particles.) But effect-based instability is much less intuitive: it means that two trials with the same intrinsic features, which are subject to the same external causal inputs, might nevertheless have different chances simply in virtue of the temporal locations of their effects. The objection proposes that a physical system’s chances depend not only on its intrinsic properties and the causal inputs to the system, but also on the causal outputs of the system. I find this implausible. It means that one can manipulate the chances associated with an experimental trial without exerting any causal influence whatsoever on that trial. On the Availability Analysis, one can manipulate a trial’s chances merely by controlling the flow of information, that is, simply by making more (or less) information available. On the Nomological Analysis, one can manipulate a trial’s chances merely by arranging for its outcomes to have certain effects at certain times. But it’s not plausible that chances behave in this way. The point can be pressed by noting that chances are physical probabilities.14 This is what distinguishes chances from other kinds of objective probabilities, such as logical and evidential probabilities. Given this, we might have hoped to assimilate a physical system’s chance properties to its archetypal physical properties: its temperature, length, mass, and so on. Such properties do not exhibit effect-based instability. You can’t manipulate a physical system’s temperature (etc.) merely by making more information available, or by arranging for its temperature to have certain effects. You can find out about its temperature in these ways, but you cannot influence it. To influence it, you must provide a causal input to the system. But according to the instability objection and the analyses that support it, you can manipulate a system’s chance properties merely by arranging for the system’s behaviour to have certain effects. This makes chance properties quite unlike archetypal physical properties. For this reason, the temporalist faces pressure to reject the Availability Analysis and the Nomological Analysis, and to accept premiss 2 and the stability of chance. Having said this, I concede that accepting unstable chances in the Retrograde case is an intelligible option. One could coherently insist that there are two kinds of physical features: those that exhibit effect-based instability (like chances) and those that don’t (like temperature, mass, length, and so on). One could then try to incorporate the unstable properties into physical theory, alongside the stable ones. For what it’s worth, I find this inelegant. Better, I think, to seek a view of chance on which chance properties are as stable, and as much like archetypal physical properties, as is feasible. But I accept that some (for example, those who emphasize chance’s epistemic role) may not value stability so highly, and may be less concerned with assimilation.15 However, let me make two final points. First, as the above quotations attest, some do value stability. This motivates the search for a view of chance that maintains stability in the Retrograde case. Second, the importance of stability as a theoretical desideratum is surely a matter of degree: everyone should agree that it carries some weight, even if only as a tiebreaker, while admitting that other desiderata are also important. In §3, I’ll argue that the causal view of chance does just as well as the temporal view with respect to other important desiderata, and better with respect to stability. This should be of interest to those who place any weight on stability. Temporalists who find the stability of chance intuitively attractive, including those who would assimilate chance properties to other archetypal physical properties, should be reluctant to reject premiss 2. 2.2 Premiss 3 Premiss 3 says that at 11:00, the chance of Retrograde’s first coin landing heads is the same as the chance of the receiver beeping. This is difficult for the temporalist to deny. Due to Retrograde’s meticulous preparations and the supreme reliability of her device, the situation at 11:00 together with the laws of nature guarantee that the receiver beeps iff the first coin lands heads. Therefore, beep and heads1 have the same chance. We can formulate this reasoning as follows. Let state describe in maximal detail the state of Retrograde’s world at 11:00. Let laws describe its laws. We have: Sub-argument for premiss 3 (state & laws) entails (heads1 iff beep)  3. The sub-argument is valid and its conclusion is premiss 3.16 I’ll now argue that for the temporalist, denying any of (i), (ii) or (iii) is problematic. Denying (i) means rejecting the possibility of the Retrograde scenario, since (i) follows from the foolproof reliability of Retrograde’s device. Denying (ii) is off the table, since it follows from the temporal view. At first sight, (iii) seems very plausible: how could the laws fail to have maximal chance? Surely they don’t deserve to be called ‘the laws’ if they have some chance of being violated. I find this persuasive, and temporalists who agree will be inclined to accept (iii). But not everyone accepts that the laws have maximal chance. It’s denied by some Humeans about chance, for example, who take the laws to supervene on patterns of actual events (Lewis, 1980, 1994): since patterns of actual events can be chancy, they say, the laws can be too. This offers hope to the temporalist: perhaps she can reject (iii), and so premiss 3, while accepting (i) and (ii), along with premisses 1 and 2 and the Future Principle. This won’t work. By (i), entails . So at 11:00, we have . By the Future Principle, , and so the inequality reduces to . By premisses 1 and 2, we have . Thus . On the current proposal, the temporalist must say that at 11:00, the chance of the laws being true is at most . Not only do the laws have some chance of being violated, they have a fifty-fifty chance of being violated: the truth of the laws depends quite literally on the toss of a coin.17 It gets worse. At 11:00, the truth or falsity of will be conclusively settled in the next minute. So on the current proposal, at 11:00 there’s a fifty-fifty chance that by 11:01, will be conclusively settled to be true: the first two conjuncts already have chance 1, while the third has chance and will be settled by 11:01. But is incompatible with laws. So if is conclusively settled to be true, then laws will be conclusively settled to be false. At 11:00, the laws have a fifty-fifty chance of being violated in the next minute. It gets even worse. We can construct situations in which the laws have an arbitrarily high chance of being violated arbitrarily soon. Suppose Retrograde’s experiment takes not a minute, but a second, and that her first coin is biased towards tails, so that the chance of heads at 11:00 is one in a million. Suppose that the coin in fact lands heads. Then at 11:00, the coin has only a one-in-a-million chance of landing heads; and yet its failure to do so would violate the laws. At 11:00 there’s an extremely high chance that the laws will be violated in the next second. Further refinements of the example drive the chance of a violation arbitrarily close to 1, and the time-frame for this violation arbitrarily close to zero. This is bad. Whatever else we say about the laws, we don’t want to say that they sometimes have an arbitrarily high chance of being violated arbitrarily soon. Even the temporalist who holds that the laws have non-maximal chance should be loath to reject premiss 3. This spells trouble for the temporal view. If one accepts the view, then rejecting any of premisses 1, 2 or 3 leads to trouble. But accepting them all means rejecting the Future Principle. The Chancy Past Argument undermines the temporal view. 2.3 Against time-dependence I’ll now argue that the Retrograde case makes trouble for the Time-Dependence Principle. Suppose that the temporalist responds to the Chancy Past Argument by accepting its conclusion, thereby abandoning the Future Principle. Can she nevertheless salvage the Time-Dependence Principle, along with some suitably weakened version of the Future Principle? Recall that the original motivation for the Future Principle was the intuitive fixity of the past in cases like Lewis’s labyrinth. But the Future Principle goes beyond this, stating that the past is non-chancy in all possible cases. Plausibly, in cases where past events remain open to future influence, the claim that they are ‘fixed’ can be questioned. So perhaps the temporalist can retreat to a weaker claim: that the past must be non-chancy except in cases of backwards causation. This would maintain the intuitive non-chanciness of the past in ordinary cases like Lewis’s labyrinth while permitting past chanciness in cases like Retrograde’s. While this position is internally consistent, it’s difficult to motivate. The methodology that supports time-dependence in ordinary cases tells against time-dependence in the Retrograde case. Why does the temporalist accept time-dependence? Because intuitively, the correct chance judgements for a well-informed agent to make vary with that agent’s situation, and particularly with the time at which the agent is located. Suppose that Retrograde’s less adventurous colleague, Professor Prograde, observes while a fair coin is tossed and lands heads. According to the temporalist, when Prograde is situated before the toss, she would be intuitively correct to judge that the toss’s outcome is ‘open’ and that heads has chance ; but when situated after the toss, she would be correct to judge that the toss’s outcome is ‘fixed’ and that heads has chance 1. Which features of Prograde’s post-toss situation make this the case? This is a difficult question, but we can give some rough answers: she has observed the coin landing heads; she has memories and records of it; there is now nothing she can do about it; and so on.18 In any case, the intuition is a compelling one: well-informed agents located in different situations can correctly make different chance judgements about the very same proposition. The temporalist takes this to show that the chances themselves vary across situations, and in particular, across times. This reasoning, which we might call the methodology of situated intuitions, is what motivates the time-dependence of chance. To emphasize the importance of this methodology to the temporalist, consider the invariant view of chance, according to which chances exhibit no intra-world variation: the chance of a fair coin landing heads is just , simpliciter.19 Why prefer the temporal view to invariantism? Because of the methodology of situated intuitions. The invariantist must abandon the intuition that post-toss Prograde would be correct to judge that heads has chance 1, along with the related intuitions concerning fixity. By contrast, the temporal view vindicates these intuitions. On this basis, the temporalist takes the methodology of situated intuitions to favour her view over invariantism. Note that the invariantist has a possible reply here.20 Sure, she might say, Prograde’s post-toss credence in heads ought to be 1, but that’s only because she has inadmissible evidence; the chance of heads is nevertheless , as invariantism says. The temporalist objects to this, on the grounds that it fails to respect her intuitions about the objective chances: when we consider Prograde’s post-toss situation, she claims, we intuit that Prograde would rightly judge that the toss’s outcome is objectively fixed, and that the objective chance of heads is 1. The methodology of situated intuitions urges us to take these intuitions seriously as intuitions about the objective facts, thereby adopting a view of chance that accommodates them. Invariantism fails to do this, instead explaining the intuitions away as mere credential artefacts. Thus the temporalist rejects invariantism. Now, this is not a knock-down argument against invariantism: one could instead either reject the temporalist’s intuitions or reject the methodology of situated intuitions. But the temporalist maintains that the better path is to accept the situation-dependence of chance. The methodology of situated intuitions is thus important in motivating the temporal view. The problem for the temporalist is that the methodology of situated intuitions cuts against her view in the Retrograde case. Consider the chance of beep relative to, first, the assistant’s situation at 11:00, and second, Retrograde’s situation at 11:00. By the methodology of situated intuitions, we discern these chances by considering what the intuitively correct chance judgements would be, as made by well-informed agents located in these this situations. I claim that intuitively, given the assistant’s situation at 11:00, he would rightly judge beep to have chance 1. Why? For the very same reasons that post-toss Prograde would rightly judge heads to have chance 1: he has just heard the receiver beep; he remembers it and has records of it; there is now nothing he can do to prevent it; and so on. By our usual standards for judging fixity/openness and discerning chances, the occurrence of the beep is ‘fixed’, and beep has chance 1. Retrograde’s situation at 11:00 is different: she would rightly judge the chance of beep to be . Why? For the same reasons that pre-toss Prograde would rightly judge the chance of heads to be : the usual telltale signs of fixity are absent. So beep has chance 1 relative to the assistant’s 11:00 situation, and chance relative to Retrograde’s 11:00 situation. But the temporal view cannot accommodate this, since these situations occur at the same time. The methodology of situated intuitions thus cuts against the temporal view. The temporalist has a possible reply here.21 Sure, she might say, the assistant’s 11:00 credence in beep ought to be 1, but only because he has inadmissible evidence; the 11:00 chance of beep is nevertheless . But this reply is hard to motivate. It parallels the invariantist’s reply to the temporalist in the Prograde case: each insists that the chances are invariant across certain situations, contrary to the methodology of situated intuitions, and seeks to explain away our fixity intuitions in terms of credences. I’m inclined to resist the temporalist’s reply here, on the same grounds that the temporalist resists the invariantist’s reply. The reasons for claiming that beep is fixed relative to the assistant’s situation are just the standard ones that the temporalist herself deploys in the Prograde case. If we claim that beep is intuitively open here, explaining away its apparent fixity as merely credential, then there is no principled reason not to treat the Prograde case similarly. But this means accepting invariantism. Instead, I think, we should follow the fixity intuitions where they lead, and reject the Time-Dependence Principle. Admittedly, this is not a knock-down argument. One can consistently maintain time-dependence, either by rejecting my intuitions about situated chance judgements or by rejecting the methodology of situated intuitions. But the former response is unsatisfying, because the assistant’s post-beep situation is so similar to Prograde’s post-toss situation: surely they should be treated alike. And the latter response is troubling, because the methodology of situated intuitions is our best means of adjudicating between invariantism, the temporal view, and other views. What would we do without it? (Note that the temporalist should have a special aversion to this response, given the importance of this methodology in recommending her view over invariantism.) All things considered, the best way forward is to reject time-dependence. We should therefore seek a new view of chance: one that defines chances relative to a new kind of background proposition. These background propositions must be capable of reflecting the intuitive intra-world variation of chance in the Retrograde case. Specifically, we will need two background propositions, B and , that correspond (respectively) to Retrograde’s and the assistant’s situations at 11:00, such that the chance of beep is relative to B and 1 relative to . I’ll now give such a view. 3. A causal view of chance I’ll now present an alternative view of chance and argue that it solves the problems of the previous section. The key move is to connect chance less closely to the arrow of time—that is, the past–future distinction—and more closely to the arrow of causation—the cause–effect distinction.22 3.1 Statement of the view We begin with some definitions and notation. For any world w, define the causal network of w as the ordered pair , where is the set of events that occur in w, and is the direct causal influence relation in w, which I assume to be a binary relation on . Where , we write just in case exerts a direct causal influence on in w. Here we say that is a cause ofin w and is an effect ofin w. (We can often omit the subscript, writing simply →.) A single event can have multiple causes and multiple effects. A cause need not necessitate its effects; for example, the tossing of a coin is a cause of its landing heads. Where we have and , we write , and say that the three events form a causal chain. The convention extends to longer chains in the obvious way. We can represent a causal network using a causal network diagram, with nodes representing events and directed edges representing the direct influence relation. (See fig. 1.) Figure 1: View largeDownload slide A causal network Figure 1: View largeDownload slide A causal network For each world w, let be the ancestral of . Thus we have iff in w there is a causal chain leading from e to . Here say that e is a (causal) ancestor ofin w, and that is a (causal) descendant of e in w. We can extend the causal ancestor relation from individual events to sets of events. Where and , say that e is an ancestor of E in w, written , iff for some ; and say that e is a descendant of E in w, written , iff for some . Say that E is connected to e in w just in case either or or . Thus E is connected to e just in case E contains either e itself or some ancestor or descendant of e. Say that E is globally connected in w if E is connected to every event in w. A globally connected set of events is like an unavoidable roadblock in a town of one-way streets: just as one can’t drive through town without encountering the roadblock, one can’t trace a maximal path through the causal network without including a member of the globally connected set. (In fig. 1, is globally connected.) We now define causal histories. Let E be a globally connected set of events in w. The causal history of E in w, written , is the strongest proposition entirely about E and its ancestors that is true in w. The causal history of E in w gives the full causal story of E and its ancestors in w, and nothing else. Thus says three things at once. It says that a particular set of events occurs: the set containing all the members of E, together with all their ancestors in w. It says that these events are causally related to one another in a particular way: the way they’re related in w. And it says that none of these events has any additional causes, aside from those they have in w. Thus the causal history of E in w provides a complete description of the ancestors of E in w, and all their in-house causal connections, and says that none of the events it describes has any additional causes. (Note: does not rule out the possibility that some of these events have additional effects.) Say that a proposition C is a causal history of w if C is the causal history of some globally connected set of events E in w. Causal histories are the causal analogues of temporal histories. Note a consequence of this definition. A causal history of a world tells the causal story of every causal chain in that world ‘from the beginning’ (causally speaking). Call an event e spontaneous in w if e occurs in w and has no ancestors in w. Then each causal history of w describes all the spontaneous events in w. In worlds with no backwards causation, causal histories subsume temporal histories. To see this, let w be a world with no backwards causation, t be a time in w, and T be the temporal history of w up to t. Let E be the set of events occurring at t in w. Suppose that E is globally connected in w.23 Then is a causal history that describes the very same events as T. For since E is globally connected in w and all causation in w is forwards, for each e in w we have just in case . In w, the events that occur before t are just the ancestors of the events occurring at t. Since T describes the former events and describes the latter, the two propositions describe the very same events.24 (See fig. 2, taking t = t1.) In worlds without backwards causation, for every temporal history there is a causal history describing the very same events. Figure 2: View largeDownload slide A temporal history T and a causal history C Figure 2: View largeDownload slide A temporal history T and a causal history C In worlds with backwards causation, temporal and causal histories can diverge markedly. Let e occur at t in w. If w contains backwards causation, then e may have future ancestors or past descendants. Let T be the temporal history of w up to t, and let C be the causal history of the events occurring at t in w. Then any future ancestors of e are described by C but not by T; and any past descendants of e that are not also ancestors of e are described by T but not by C. Thus T and C may describe different sets of events. (See fig. 3, taking t = t1.) Backwards causation drives temporal and causal histories apart. Figure 3: View largeDownload slide A temporal history T and a causal history C Figure 3: View largeDownload slide A temporal history T and a causal history C Finally, say that an event e lies upstream of a causal history C if either C entails or C entails . The events lying upstream of a causal history are those whose occurrence or non-occurrence is decisively settled by that causal history. Say that an event lies downstream of C if it does not lie upstream of C. The events lying downstream of a causal history are those whose occurrence is left open by that causal history. The causal view of chance is the conjunction of two principles. First: Causal-History-Dependence Principle: The chance function ch is a real-valued function of three variables: a proposition A, a possible world w, and a causal history C of w. Chances are defined relative to causal histories. The situations that generate chances are comprehensive causal set-ups. Canonical chance propositions take the form , to be read as: the chance of A relative to C in w is x. We can develop our formalism in the usual way. For each causal history C of w, we have a tailored set of outcome propositions to which chances are assigned, relative to C in w. We then define the (restricted) chance function such that for all . I assume that each restricted chance function satisfies the standard probability axioms. The second component of the causal view is: Downstream Principle: If , then e lies downstream of C. All chanciness relative to a causal history must lie downstream of that causal history: what’s upstream cannot be chancy. If a causal history C of w entails the occurrence (or non-occurrence) of a given event, then that event has chance 1 (or 0) relative to C in w. Chance thus treats all causal inputs as given, by assigning them chance 1, while allowing causal outputs to have intermediate chance, regardless of their times of occurrence. The Downstream Principle follows from our framework for proposition-dependent chance, together with the choice of causal histories as background propositions. If e lies upstream of C, then either C entails or C entails , and so . Taking our background propositions to be causal histories delivers the Downstream Principle for free. Note: since all spontaneous events in w are described by every causal history of w, the Downstream Principle says that no spontaneous event is chancy relative to any causal history of w. Intuitively, it’s not clear how a theory of chance ought to treat spontaneous events, but I think this is one sensible treatment, which is in keeping with the spirit of the causal view. We assign chances to events on the basis of their ancestors, and if an event has no ancestors, it cannot be chancy relative to them. 3.2 Forwards causation The causal view inherits the strengths of the temporal view: its treatment of forwards causation. The temporal view explicates the situation-dependence of chance by saying that chances vary with time (or temporal histories). On the causal view, we also have situation-dependence, but the situations relative to which chances vary are represented by causal histories. In worlds where all causation is forwards, causal histories subsume temporal histories, and so causal-history-dependence subsumes temporal-history-dependence. For example, in Lewis’s labyrinth, let Ct be the causal history describing the events occurring at time t, and all their causal ancestors. The causal view defines chances relative to each Ct: the chance of centre is 42% relative to , 26% relative to , and so on. The causal view thus mimics the structure of the temporal view in cases without backwards causation. In these cases, we also get the fixity of the past: the Downstream Principle ensures that past events have chance 1, since in the absence of backwards causation, all past events lie causally upstream.25 In Lewis’s labyrinth: the events lying upstream of Ct are just those occurring before t, and these must have chance 1 relative to Ct; thus where 11:49 (the time at which you reach the centre), the chance of centre relative to Ct is guaranteed to be 1. The causal view thereby reproduces the intuitively correct results of the temporal view in cases of forwards causation. 3.3 Backwards causation The causal view permits past chanciness: it allows that an event e occurring earlier than t can be chancy, relative to a causal history Ct that describes all the events occurring at t, along with their ancestors. (Note: strictly speaking, the causalist cannot understand past chanciness as the chanciness at t of a pre-t event, since she denies that chances are defined at times.) Past chanciness requires backwards causation: if e is chancy relative to Ct, then e must lie downstream of Ct; and if e occurs before t, this requires backwards causation. The causal view permits past chanciness in cases of backwards causation. I’ll now apply the causal view to the Retrograde case and defend the results. We can suppose that heads1, heads2, and beep have well-defined chances relative to each of the causal histories Ci listed in fig. 4. Translating the Chancy Past Argument into causal terms yields: The Causal Chancy Past Argument (IV) This argument is valid for all causal histories C. Does it make trouble for the causal view? I’ll show that it does not, by showing that there is no causal history for which the causalist is committed to the truth of all three premisses and the falsity of the conclusion. Figure 4: View largeDownload slide The causal structure of the Retrograde case Figure 4: View largeDownload slide The causal structure of the Retrograde case Consider premiss I. By the Downstream Principle, wherever C describes heads2. So the causalist must reject premiss I for C5, C6, C11 and C12. For all the other causal histories, the causalist ought to accept premiss I, since the coin tosses are the only chancy processes in the scenario. (See column I of table 1.) Consider premiss II. The Downstream Principle requires that heads1 and heads2 have the same chance (namely, 1) for C11 and C12. The stability of chance presses the causalist to accept premiss II for C4, since C4 describes both tosses but neither landing. Since the coin tosses are the only chancy processes in the scenario, the causalist should also accept premiss II for C1, C2 and C3. The causalist is plausibly committed to premiss III for each of our causal histories. First, the experimental set-up guarantees (heads1 iff beep), and so by reasoning analogous to that in §2.2, heads1 and beep must have the same chance relative to any causal history describing toss1 but not heads1 (that is, C2, C4 or C6). Furthermore, relative to any causal history not describing toss1 (that is, C1, C3 or C5), the first coin is guaranteed to be tossed, since the tosses are the only chancy processes in the scenario; so heads1 and beep have the same chance relative to each of these. Finally, the Downstream Principle ensures that relative to any causal history describing both toss1 and heads1 (that is, C7, C8, C9, C10, C11 and C12), both heads1 and toss1 have chance 1. When is the causalist committed to the falsity of the conclusion, IV? By the Downstream Principle, the chance of beep must be 1 relative to any causal history describing beep. So the causalist must deny IV for C9, C10, C11 and C12. In the discussion of premiss III, I also assumed that the causalist is committed to beep having chance 1 at C7 and C8. (See column IV in table 1.) Table 1 Causalist constraints   I  II  III  IV  C1  T  T  T    C2  T  T  T    C3  T  T  T    C4  T  T  T    C5      T    C6      T    C7  T    T  F  C8  T    T  F  C9  T    T  F  C10  T    T  F  C11    T  T  F  C12    T  T  F    I  II  III  IV  C1  T  T  T    C2  T  T  T    C3  T  T  T    C4  T  T  T    C5      T    C6      T    C7  T    T  F  C8  T    T  F  C9  T    T  F  C10  T    T  F  C11    T  T  F  C12    T  T  F  Table 1 shows that the Causal Chancy Past Argument poses no problem to the causal view. For no causal history Ci is the causalist committed to accepting all three premisses and denying the conclusion. The only causal histories for which she must accept all three premisses are and C4; and for each of these she can happily accept the conclusion too. (Intuitively, C4 is the closest approximation to the temporal history up to 11:00: it describes both coins being tossed at 11:00, but no later events. Crucially though, it doesn’t describe beep, and so beep can be chancy, and the causalist can accept IV.) Table 2 gives a consistent assignment of chances. In §2.3, I argued that the temporalist cannot respect the intuitive intra-world variation of chance in the Retrograde case: she must either say that beep is chancy at 11:00 or that it is not, but neither answer is acceptable. The causal view does better. Given Retrograde’s situation at 11:00, she is intuitively right to judge beep to be chancy. On the causal view, this is reflected by the chanciness of beep relative to a particular kind of causal history: one describing Retrograde’s surroundings at 11:00 (including both tosses) and their causal ancestors, but describing neither heads-landing (for example, C4). Thus we have . Given the assistant’s situation at 11:00, he is intuitively right to judge beep to be non-chancy. On the causal view, this is reflected by the non-chanciness of beep relative to a causal history describing the assistant’s surroundings at 11:00 and their causal ancestors, including the receiver’s beep (for example, C9). Thus we have . The causal view does justice to our intuitive chance judgements in this case. Table 2 Causalist chances   beep  heads1  heads2  C1  ½  ½  ½  C2  ½  ½  ½  C3  ½  ½  ½  C4  ½  ½  ½  C5  ½  ½  1  C6  ½  ½  1  C7  1  1  ½  C8  1  1  ½  C9  1  1  ½  C10  1  1  ½  C11  1  1  1  C12  1  1  1    beep  heads1  heads2  C1  ½  ½  ½  C2  ½  ½  ½  C3  ½  ½  ½  C4  ½  ½  ½  C5  ½  ½  1  C6  ½  ½  1  C7  1  1  ½  C8  1  1  ½  C9  1  1  ½  C10  1  1  ½  C11  1  1  1  C12  1  1  1  Note that there is nothing problematically ‘subjective’ about this. The causalist assigns different chances to beep relative to different causal histories, just as the temporalist assigns different chances to beep relative to different temporal histories. In both cases, the chance assignments are fully objective: well-informed observers will agree about which propositions have which chances relative to which background propositions. The difference is that causal histories better reflect the structure of the Retrograde case, allowing the chance assignments given by the causal view to align more closely with our intuitive judgements. The temporal view fails because it requires beep to have a single chance at 11:00, contrary to our intuitive judgements. The causal view solves this problem by defining chances relative to causal histories: there is an appropriate causal history relative to which beep is chancy, and one relative to which it is not. The causal view thus inherits the strengths of the temporal view without its weaknesses. We ought to reject the temporal view in favour of the causal view. 4. Consequences of the causal view On the causal view of chance, the ‘arrow’ of chance is connected more closely to the arrow of causation than it is to the arrow of time. All chanciness must lie causally downstream, and if parts of the past lie causally downstream, they can be chancy. (Note: the causal view does not say that chance is temporally symmetric; it says that chance inherits its temporal asymmetry from causation.) How does this square with the intuitive ‘fixity’ of the past? Here’s what I think is happening. The key fixity intuition is a causal one: that the causal history is fixed. Doubtless we also have a temporal fixity intuition, that the past is fixed; but this ultimately rests on the causal intuition, together with our ordinary assumption that the future lies causally downstream. With this assumption in place, we can embrace both intuitions. But taking backwards causation seriously means choosing between them, and the Retrograde case shows that the causal intuition wins out. This explains why we should accept the causal view, and also why the temporal view seemed attractive until we took backwards causation seriously. Are scenarios involving backwards causation too bizarre to be worth worrying about? No. First, backwards causation is taken seriously by some live physical theories.26 Second, even if there is no actual backwards causation, we should prefer a view of chance that accommodates all possible cases to one that doesn’t. Third, if chance and causation are connected as the causal view suggests, then this is the case in our world too, regardless of whether or not there is actual backwards causation. When evaluating views of chance, we should take backwards causation seriously. Consider the consequences of the causal view for chance in causal loops. Call a set of events in w a causal loop in w if its members form a causal chain in w that begins and ends with the same event. Causal histories describe causal loops in an all-or-nothing fashion: if L is a causal loop in w, then any causal history C of w that entails the occurrence of some member of L entails the occurrence of all members of L. By the Downstream Principle, there can thus be no non-trivial chances within a causal loop: no causal history describing part of a loop can confer intermediate chance on another part of that loop. Isn’t this counterintuitive? And doesn’t it count against the causal view? No. First, causal loops are odd things, and we should regard our intuitions about them with caution. Second, the causal view gives one principled account of chances in causal loops, for the following reason. The chances associated with a physical process ought to ‘factor in’ all the causal inputs to that process: causally relevant events should not be ignored. Where a process and its outcome are parts of the same causal loop, the outcome is among the causal inputs to that very process, so the outcome must itself be taken for granted by the chances associated with the process. Thus the outcome can’t be chancy, relative to the process. The causal view delivers this. However, a further objection lurks. Even if the causal view’s treatment of causal loops is defensible in itself, it has consequences for the stability of chance (§2).27 The causal view entails a limited amount of instability: duplicate trials can have different chances, if one but not the other is embedded in a causal loop. Suppose Tim travels back in time and attempts to kill his grandfather before he had children: Tim rolls a bomb towards grandfather that will explode iff a radium atom inside the bomb decays.28 Nearby, Tom (not a time traveller) attempts to kill grandfather’s partner by rolling an intrinsic duplicate bomb. On the causal view, Tim’s bomb has zero chance of exploding, relative to any causal history describing its rolling: any such causal history must describe the bomb’s failure to explode, since this is a causal ancestor of the rolling. So Tim’s trial is non-chancy. Tom’s trial is chancy: his bomb has a positive chance of exploding, relative to its being rolled. The causal view thus leads to unstable chances in causal loop cases. As I’ve said, I think this is a defensible result. But even granting this, the temporalist might object. In §2, I used stability to push the temporalist to accept premiss 2: this was part of my argument for moving from the temporal view to the causal view. But now it turns out that even the causalist is committed to unstable chances in causal loops. So why not accept them in the Retrograde case? One might then respond to the Chancy Past Argument by denying premiss 2, thus maintaining the temporal view. This objection fails. Accepting unstable chances in the Retrograde case is more troubling than accepting them in the Tim/Tom case. The occurrence of the set-up of Tim’s trial (Tim’s rolling his bomb) depends on the trial’s outcome turning out in a particular way (the bomb’s not exploding): the bomb’s exploding is incompatible with Tim’s having rolled it, since grandfather’s survival is a necessary condition for the rolling. Given that the set-up occurs, consistency requires that this outcome occurs; and this is why the outcome has chance 1 relative to any background proposition describing the set-up. Because Tim’s trial is embedded in a causal loop, its non-chanciness falls out as a matter of logic. Tom’s trial is different: here the set-up does not depend on the outcome turning out in any particular way: Tom’s bomb can consistently be rolled whether it ultimately explodes or not. So Tom’s trial can consistently be chancy. This explains why causal loop cases yield unstable chances: consistency demands it. In this respect, both of Retrograde’s trials are similar to Tom’s trial. Neither of Retrograde’s trials is such that the set-up (the coin’s being tossed) depends on the outcome turning out the way that it in fact does (the coin landing heads). Each coin can consistently be tossed whether it lands heads or not: consistency does not force instability. Therefore, accepting unstable chances in causal loops should not tempt us to accept them in the Retrograde case. Finally, let’s briefly consider the chance–credence relationship. The Principal Principle (Lewis, 1980, p. 87) can be reformulated in causal terms as follows: Causal Principal Principle: Let cr be a rational initial credence function, A be a proposition, C be a causal history, x be a real number in the unit interval, and X be the proposition that the chance of A relative to C is x. Let E be a proposition that is compatible with X and admissible relative to C. Then . Of course, this must be supplemented by an account of admissibility. I think there are good prospects of success here. Lewis claims that past evidence is ‘as a rule’ admissible, making an exception for cases of backwards causation. The causalist might naturally say that evidence lying upstream of a causal history C is admissible relative to C, thereby allowing the problematic evidence (which lies downstream) to count as inadmissible, as it should.29 Footnotes 1 I’ll use coin tosses as my main examples of chancy processes. Readers who prefer other examples can freely substitute them throughout. 2 This principle is widely accepted (Lewis, 1980; Bigelow et al., 1993; Hall, 1994, 2004; Schaffer, 2007; Handfield, 2012), though not universally so (Hoefer, 2007). I’ll focus on the time parameter, taking the other two (the proposition and possible world) for granted in what follows. 3 For Lewis, although times provide a convenient way of indexing chances, it is in fact collections of historical facts, and not times themselves, on which chances depend. (Thanks to an anonymous reviewer for stressing this point.) I will shortly make this dependence explicit by indexing chances to what I will call ‘temporal histories’. 4 See Lewis (1980), Bigelow et al. (1993), Schaffer (2007), and Hall (1994, 2004). 5 See Lewis (1980, p. 93), Bigelow et al. (1993, p. 454), and Schaffer (2007, p. 125). 6 Here I follow Meacham (2005, 2010), Nelson (2009), and Briggs (2010, 2015). The idea can be traced back to Popper (1957, 1959). 7 See Nelson (2009, pp. 175–6) for a contrary view. This distinguishes the chances relative to B from the chances conditional on B, which can be well-defined in worlds where B is false. 8 Proof: If B is a background proposition in w and B entails A, then since is a probability function and the background proposition is non-chancy, we have , so . 9 Here I follow Lewis’s use of ‘entirely about’ (1980, p. 93). 10 The thought that backwards causation makes trouble for the Future Principle is not new. See Hall (1994, p. 514), Edgington (1997, p. 431), Eagle (2011, p. 291; 2014), Handfield (2012, pp. 9–10), and Suárez (2013, pp. 79–80). My aim is to develop this thought into an explicit objection to the temporal view, and give an alternative view that overcomes it. 11 Mellor (1981; 1995). See also Edgington (1997, §8), Berkovitz (2001), Dowe (2001), and §4 below. 12 Those who distinguish ‘high-level’ and ‘low-level’ chances should also require that A and describe the respective outcomes at the same level, and B and describe the respective set-ups at the same level. Thanks to the editors for pointing this out. My examples satisfy this condition. 13 Thanks to an anonymous reviewer for suggesting this. 14 Handfield (2012, p. 28) notes the failure of the Availability Analysis to capture the physicality of chance. 15 Thanks to an anonymous reviewer for pressing this point. 16 Proof of validity: From (i) we have . From (ii) and (iii) we have . Combining these gives . Thus . Since chances are probabilities, . Similarly, . Therefore . 17 Hall (1994, §6) discusses a similar issue in the context of undermining futures. 18 See Torre (2011) for an examination of these issues. 19 See Hoefer (2007, §3.2). 20 See Hoefer (2007, p. 555). 21 Thanks to an anonymous reviewer for pressing this reply. 22 Here I develop a suggestion raised by Joyce (2007) and taken up by Eagle (2011, 2014). Joyce’s view stresses the importance of the chance–causation link, but agrees with the temporal view that chance functions are time-indexed. The view presented here takes chances as defined relative to causal histories. Eagle’s proposal combines time-dependence with an emphasis on causal information: he says that ‘chance is informed, at a time, about the causes of outcomes that have occurred by that time’ (2014, p. 138), and that chances are ‘indexed to certain bodies of information … [which contain] information about causes’ (2014, p. 153). 23 Here I assume that w is Markovian connected: for every event e that occurs in w and every time t in w, either e occurs at t in w, or else e has either an ancestor or a descendant that occurs at t in w. (Thanks to the editors for clarifying this.) Dropping this assumption complicates the relationship between temporal and causal histories: w may contain spontaneous post-t events, which are not described by the temporal history of w up to t, but are described by every causal history of w; so there is no causal history of w that describes the same events as this temporal history. For simplicity I’ll assume Markovian connectedness, noting where necessary the consequences of dropping it. (See especially footnote 25.) 24 While T and describe the same events, they say different things about what doesn’t happen: while T rules out additional pre-t events, rules out additional ancestors of . 25 Dropping the assumption of Markovian connectedness (see footnote 23 above) sees the Downstream Principle go slightly beyond the Future Principle here. Suppose that w is not Markovian connected, and contains a spontaneous event e occurring after t. Then the Future Principle allows to be chancy at t, while the Downstream Principle entails that has chance 1 relative to each causal history of w. In my opinion, this difference does not tell in favour of either view, since it’s unclear how spontaneous events ought to be treated: on the one hand, it seems reasonable enough to say that e has a well-defined, non-trivial chance relative to the events that temporally precede it but do not cause it, as the Future Principle allows; but on the other, it also seems reasonable to say that since e has no causes, it has no well-defined chance (except when e forms part of the causal history, when its chance is 1), as the Downstream Principle says. 26 See Costa de Beauregard (1976), Cramer (1986), Price (1984, 2008), and Miller (1996). 27 Thanks to an anonymous reviewer for clarifying this point. 28 This is a probabilistic version of a case of Lewis’s (1976). 29 I am grateful to Hilary Greaves, Alan Hájek, Robyn Kath, Leon Leontyev, Wolfgang Schwarz, two anonymous reviewers, and the editors of Mind for useful discussion and feedback. 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MindOxford University Press

Published: Jan 1, 2018

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