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Lewis on Backward Causation

Ryan Wasserman Western Washington University

David Lewis famously defends a counterfactual theory of causation and a non-causal, similarity-based theory of counterfactuals.1 He also famously defends the possibility of backward causation.2 I will argue that this combination of views is untenable—given the possibility of backward causation, one ought to reject both Lewis’s theory of causation and his account of counterfactuals.3 If correct, this argument would be significant for a number of different reasons. First, Lewis routinely appealed to the possibility of backward causation in arguing against rival theories of cau-sation and counterfactuals.4 If I am correct, these kinds of arguments lose all their force. Second, many people besides Lewis have found each of these views independently plausible. If I am correct, those people will face a difficult decision—one can accept Lewis’s theories or one can ac-cept the possibility of backward causation, but one cannot accept both. Finally, I will argue that considerations of backward causation bring out a surprising asymmetry in Lewis’s overall position. If I am correct, this asymmetry constitutes a serious problem for Lewis’s theories, whatever one’s position on backward causation.

1. Causation and Counterfactuals

* The published version of this paper appears in Thought 4: 141-150.

1 See, respectively, Lewis (1973a) and Lewis (1973b). 2 See, Lewis (1973a: 566) and Lewis (1976). 3 Tooley (2002) offers a different argument for a similar conclusion. For discussion, see

Noordhof (2003). 4 See, for example, Lewis (1986: 140-1) and Lewis (1973a: 566).

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Lewis (1973a) famously defends a three-step approach to causation.5 First, he analyzes causal dependence in terms of counterfactual dependence:

e depends causally on c only if, and because, (i) e and c are dis-tinct events and (ii) if c had not occurred, e would not have oc-curred.6

Second, he uses causal dependence to analyze causal chains:

e1, e2, e3… en is a causal chain only if, and because, e1 depends causally on e2 which depends causally on e3 which… depends causally on en.

Finally, he uses causal chains to define causation itself:

For any events c and e, c is a cause of e only if, and because, (i) c and e are distinct events and (ii) there is a causal chain leading from c to e.7

Lewis combines this theory of causation with a non-causal theory of counterfactuals. First, Lewis analyzes counterfactual truth in terms of world closeness:

A > C is true at world w only if, and because, the closest A-world to w is also a C-world.

(Where ‘>’ is the counterfactual connective, an A-world is a world at which the antecedent is true, and a C-world is a world at which the con-sequent is true.8)

5 Lewis (2000) modifies this analysis in various respects. Since these modifications are

irrelevant to the present topic, we will ignore them in what follows. 6 See Lewis (1973a: 562-3). Lewis states his theory as a biconditional (“e depends caus-

ally on c iff…”), but a biconditional does not constitute a theory. Biconditionals only report patterns, where as theories should explain patterns. For more on this point, see Wasserman (forthcoming).

7 Condition (i) is included to prevent every event from being its own cause. (This con-dition is required since every event is such that it would not occur, if it did not occur.)

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Second, Lewis defines world closeness in terms of comparative simi-larity:

w1 is closer to w2 than w3 only if, and because, w1 is more similar overall to w2 than w3.

Finally, Lewis defines comparative similarity in terms of how well worlds match with respect to their laws and particular matters of fact. More carefully, Lewis claims that the standard similarity metric for evaluating counterfactuals is given by the following set of rules:

It is of first importance to avoid big, widespread, diverse viola-tions of law.

It is of second importance to maximize the spatio-temporal re-gion throughout which perfect match of particular fact pre-vails.

It is of third importance to avoid even small, localized simple violations of law.

It is of little or no importance to secure approximate similarity of particular fact, even in matters that concern us greatly. (1986: 47-8)

To illustrate these rules, consider the following example.9 Suppose

that, during his term in office, Nixon actually had a launch button that would drop nuclear bombs on the USSR. Suppose further that, at some particular time, t, Nixon actually held his finger on the button. Suppose finally that all of the old fears about nuclear stockpiling were correct—if Nixon had pushed the button, the bombs would have been dropped, in which case the retaliatory missiles would have been launched, in which

8 This formulation simplifies Lewis’s account in two familiar ways. First, we assume

that there is at least one closest A-world and, second, we assume that there is at most one closest A-world. Dropping these assumptions, Lewis’s official formulation says that A > C is true at w only if, and because, there is some world, w*, such that (i) w* is an A-world, (ii) w* is a C-world, and (iii) every other A-world that is at least as close as w* to w is also a C-world.

9 The example is Kit Fine’s (1975: 452). For further discussion of this kind of case, see Wasserman (2006).

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case there would have been a nuclear holocaust. In other words, let’s suppose the following is true:

(1) If Nixon had pressed the button at t, there would have been a nuclear holocaust.

Now, Lewis assumes (at least initially, for the sake of simplicity) that our world is deterministic. He therefore assumes that every button-pressing world will differ from the actual world (w@) with respect to its past or laws (or both). There are many such worlds, but Lewis (1986: 43-8) focuses on four in particular:

w1 is a duplicate of w@ until shortly before t, at which time the laws of w@ are violated in a small way—a few extra neurons fire in Nixon’s brain and, as a result, he presses the button. The sig-nal races from the button, the bombers drop their bombs, and nuclear holocaust ensues. w1 is then very different from w@, at least insofar as the surface of the earth is concerned. w2 is a world in which there are no violations of the actual laws, and where Nixon presses the button at t. Given this difference, and assuming the truth of determinism, it follows that w2 and w@ have entirely different histories up to t.

w3 is a duplicate of w@ until shortly before t, at which point a few extra neurons fire and Nixon presses the button. Shortly thereaf-ter, another small miracle10 takes place—the fateful signal disap-

10 Terminological note: Lewis (1986: 44-5) says that an event in world w is a miracle

relative to world w* just in case its occurrence, in its circumstances, would violate the laws of w*. So, when we say that a miracle takes place in w3, we do not mean that w3 contains an event that violates the laws of w3. Rather, we mean that it contains an event whose occurrence would violate the laws of w@ (the world at which the counterfactual is being evaluated). More terminology: Lewis (1986: 55-6) says that a small miracle is, roughly, a miracle that takes place in a small region and that a big miracle is a collection of many small mira-cles. Given this terminology, we can simplify Lewis’s rules as follows: Minimize big miracles. Maximize exact match.

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pears on its way to the bombers, so that nuclear holocaust is averted. However, w3 is no longer exactly like w@, since it con-tains various traces of the button-pressing (finger prints, sound waves, etc.) w4 is a duplicate of w@ until shortly before t, at which point a few extra neurons fire and Nixon presses the button. Shortly thereaf-ter, a large miracle takes place—not only does the fateful signal disappear on its way to the bombers (so that nuclear holocaust is averted), but so do all traces of the button-pressing (fingerprints vanish, sound waves disappear, etc.). In this way, perfect match to w@ is restored.

Initially, one might think that w4 is most similar, overall, to w@. Af-ter all, those two worlds are exact duplicates, with the exception of a few moments before and after t. But Lewis suggests this would be a mistake. w4 does do better than all of the other competitors with respect to the second rule mentioned above (maximize perfect match), but it only does so at the cost of including many small miracles. Those small miracles add up to a big miracle, which violates the very first rule. On the other end of the spectrum, w2 scores better than all of the other competitors with respect to Lewis’s third rule (minimize small miracles), but it only does so at the cost of completely ignoring the second (maximize perfect match). That leaves w1 and w3. Those worlds are tied with respect to the first two rules, since neither includes a large miracle, and both match w@ perfectly up until shortly before t. However, the first world scores better on the third rule since it includes one small miracle rather than two. The upshot, Lewis says, is that w1 turns out to be the closest button-pressing world to w@. Since that world is also a nuclear-holocaust world, Lewis concludes that (1) is true—If Nixon had pressed the button, there would have been a nuclear holocaust.

2. An Argument Against Backward Causation Minimize small miracles. Maximize approximate match.

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Suppose that a single electron persists from t1 to t3, at which point it finds its way into an experimental time machine. At that moment, a sci-entist presses a button and the electron is sent back, discontinuously, to t2, where it goes on to live a long and happy life.11 This possibility is de-picted in Figure 1, where the numbered circles indicate the various stag-es12 of the electron, the solid lines indicate continuous causal chains, and the dotted line indicates direct causal dependence:

Let’s suppose that this world is deterministic, that no other electrons appear at t2, and that no other scientists are prepared to send an electron back to that particular place-time (in other words, there are no preempt-ed backups). In that case, the following counterfactual seems obviously true:

(2) If the scientist hadn’t pressed the button at t3, an electron wouldn’t have appeared at t2.

11 I focus on discontinuous time travel for the sake of simplicity. The same point can

be made in the case of continuous time travel. 12 Following Lewis (1983), I will assume that objects persist through time by having

different temporal parts or “stages” at different times.

1

Button Pressed

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6 5 4

2

Electron Appears

t1 t2 t3 t4

Figure 1: w1

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For Lewis, this means that the closest no-pressing world must also be a no-appearance world.13

But now consider the worlds depicted in Figures 2 and 3:

13 More carefully, there must be a no-pressing-and-no-appearance world that is closer to w1 than any no-pressing-and-appearance world.

1

Button not Pressed

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4 2

No electron Appears

t1 t2 t3 t4

Figure 2: w2

1

Button not Pressed

31

6 5 4

2

Electron Appears

t1 t2 t3 t4

Figure 3: w3

4

* * *

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In the first world—w2—events unfold exactly as we would expect. Just before t3, there is a small miracle—an extra neuron fires in the sci-entist’s brain—and, as a result, she decides to not press the button. The original electron then goes on about its business, unaffected by the time machine. The same things are true at w3. In that world, a small miracle occurs just before t3, the button is not pressed, and the particle does not disap-pear. However, w3 also includes a second small miracle at t2, when a new electron appears out of nowhere. Let’s suppose that the first stage of this electron—4*—perfectly matches stage 4 from w1 in every respect, includ-ing its initial location and velocity. In that case, the t2-miracle in w3 will preserve perfect match with w1 until shortly before t3 (when the second small miracle occurs). But in that case, w3 will turn out to be closer to w1. Of course, w2 does better than w3 with respect to Lewis’s third rule, since w3 contains two small miracles rather than one (an electron miraculously appears at t2 and an extra neuron miraculously fires before t3). However, w3 does better than w2 when it comes to Lewis’s second rule, since it does a better job of preserving exact match. To make this point vivid, imagine that t2 and t3 are separated by a trillion billion years. In that case, w3 scores far better than w2 when it comes to preserving exact match while only being a little worse off when it comes to avoiding small miracles. So, given Lewis’s metric, w3 will turn out to be comparatively closer to w1. Since w3 is an electron-appearance world, it follows that (2) is false. What’s true is that an electron would have appeared out of nowhere at t2 whether or not the scientist had pressed the button. This is obviously the wrong the result.14 Note that this line of reasoning does not apply in the case of ordinary future-directed counterfactuals. In the case of Nixon and the bomb, for example, one can only purchase perfect match by removing all traces of

14 One might grant that an electron would have appeared out of nowhere whether or not the scientist had pressed the button, while also insisting that this particular electron would not have. In response, I would say that this admission is problem enough—my case constitutes a counterexample to Lewis’s theory, even if his theory gives the right results in other cases. Moreover, I would suggest that this kind of response seems circular—after all, the identity of the electron depends on the causal facts and the causal facts, for Lewis, depend on the counterfactuals.

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the relevant button-pressing; as Lewis points out, this would require a large-scale miracle. In the case of time travel, the traces of the button-pressing are irrelevant, since those traces do not disrupt perfect match in the past. This is the fundamental problem for Lewis—in the case back-ward counterfactuals, perfect match can be purchased without a miracu-lous cover up. That is why we get the wrong results on (2). And it gets worse. Given the falsity of (2), the electron-appearance does not depend on the pressing of the button. Moreover, there are no causal chains in this case, so the button-pressing does not count as a cause of the appearance. Finally, since this point does not seem to de-pend on any specific features of the case, we can generalize on our con-clusion: If Lewis’s theories of causation and counterfactuals are correct, then backward causation is impossible.

3. Objections and Replies The foregoing argument raises several questions. First, one might wonder whether the sudden appearance of an electron in w3 is really a small mir-acle from the perspective of w1. To see why this is the case, we must con-sider Lewis’s description of big and small miracles:

In whatever way events can be spread out or localized, unlawful events can be spread out or localized. In whatever way several events can be alike or varied, several unlawful events can be alike or varied. In whatever way we can distinguish one simple event from many simple events, or from one complex event consisting of many simple parts, we can in particular distinguish one simple unlawful event from many, or from one complex event consisting of many simple unlawful parts. A big miracle consists of many little miracles together, preferably not all alike. (1986: 56)

For Lewis, the size of a miracle is determined by the size of the region in which it occurs, and the variety of unlawful events it includes. But the instantaneous appearance of an electron would not take up much space, and it would not have many (if any) parts. So, on Lewis’s account, the

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sudden appearance of an electron would clearly count as a small miracle from the perspective of w1. Second, one might worry about Lewis’s general appeal to miracles, since one might reject the underlying assumption of determinism. If the world is really indeterministic in the way that contemporary physics sug-gests, then the sudden appearance of an electron would not violate any laws, so it would not be a miracle. This point may be correct, but it does not provide any help. In the indeterministic case, Lewis drops all talk of small miracles and replaces talk of big miracles with talk of “quasi-miracles” (where a quasi-miracle is just like a big miracle except that its constituent events are improbable, rather than unlawful). Clearly, the appearance of the electron in w3 would not count as a quasi-miracle from the perspective of w1 since it would be neither big (spatially speaking) nor varied (with respect to its parts). In other words, that event would be completely un-miraculous. But, in that case, w3 would come out even closer to w1, since it would preserve perfect match without any additional miracles. A third worry about miracles raises a more serious concern. Consider again Figure 1 and imagine that the numbered circles in that diagram represent person-stages, rather than electron-stages. In other words, sup-pose that a person travels (discontinuously) back from t3 to t2 as the re-sult of the scientist pressing the button. Now consider the following counterfactual:

(3) If the scientist hadn’t pressed the button at t3, a person wouldn’t have appeared at t2.

This counterfactual seems clearly correct. Somewhat surprisingly, Lewis’s account seems to agree. For imagine a world (analogous to w3) in which a person appears, out of nowhere at t2. That world would maintain perfect match with the reimagined w1 up until just before t3, but it would only do so by including a large-scale miracle. After all, the sudden appearance of a human body would require the simultaneous appearance of some 2.3×1028 electrons, along with all the corresponding protons and neu-trons.15 Moreover, all of these sub-atomic particles would have to arrive

15 Here, I assume that a person is identical to his or her body. Those who reject this assumption are free to run the argument with another (suitably large) material object.

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perfectly arranged so as to compose a living, breathing human person. This seems to be a large-scale miracle, so this world seems to violate Lewis’s first rule. Hence, the existence of such a world would not cast any doubt on (3). The upshot is that Lewis can apparently allow for re-verse-counterfactual dependence—and hence backward causation—provided that the relevant events are sufficiently large and varied.16 This result might be thought to undermine the general argument against backward causation. But to my mind it only serves to underscore the oddity of Lewis’s view. Once again, this view says that a small event, all by itself, cannot depend on a future event. However, a large event can. This is an implausible asymmetry. Moreover, this asymmetry is not the product of contingent laws or technological limitations; rather, it is grounded in the very nature of counterfactuals themselves, and so holds of metaphysical necessity. This conclusion is difficult to accept. Here is one way of emphasizing the point. It is generally agreed that persistence through time requires causal dependence—in order for some person-stages to compose a person, for example, the existence and prop-erties of each stage must depend, in the right sort of way, on those that came before. Since Lewis can allow for the existence and properties of a person-stage to depend on future-stages, he can allow for reversals of per-sonal identity. In other words, Lewis can allow for people to travel back and forth in time. Unfortunately, he cannot say the same thing about electrons, at least if those electrons are taken one at a time. As noted above, Lewis cannot allow for counterfactual dependence between elec-tron-stages in a single-electron scenario. So, he cannot allow for causal dependence. So, he cannot allow for an electron to travel back in time, all by itself. This last point is important. Since Lewis allows for persons to travel back in time, and since persons are composed in part by electrons, he can allow for electrons to travel back in time en masse. Consider again the case of a person traveling discontinuously into the past. The closest world in which the scientist does not press the button is one in which nothing appears in the past.17 In particular, worlds in which a sin-

16 The qualifications in this paragraph are due to the vagueness of “sufficiently large

and varied”. Obviously, the constituent events involved in the appearance of a human being would be larger and more varied that those involved in the appearance of a single electron. But whether or not these events would be enough for a large miracle is less obvious.

17 Let’s assume, once again, that there is a unique closest world in this case.

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gle electron appears out of nowhere are not among the closest worlds since those worlds include a small miracle without preserving perfect match. (In this case, the preservation of perfect match would require all of the particles to show up together, and to be arranged in the right sort of way). Since there are no sudden appearances of electrons in those worlds, we can allow that the actual electron-stages depend (in the right sort of way) on future electron-stages. In other words, Lewis can allow for electrons to travel back in time, provided that there are enough elec-trons travelling back together. Once again, this example helps to bring out an implausible asym-metry in Lewis’s position. Whether or not a particular object can travel back in time should depend on what that object is like, in and of itself. It should also depend on more general facts, like those involving the laws of nature, the structure of space-time, and the availability of time machines. But it should not depend on whether or not there are other objects being sent back at the same time. To think otherwise is to impose an implausi-ble extrinsic constraint on time travel. Since Lewis’s views impose exactly this constraint, those views should be rejected.

4. Conclusion I have argued that Lewis’s theories of causation and counterfactuals rule out the simplest cases of backward causation. Moreover, I have argued that his theories can allow for “complex” cases of backward causation, but that this creates an implausible asymmetry in his overall position. One can reasonably accept both kinds of causation and one can reasonably reject both, but one cannot reasonably accept one while rejecting the other. Since Lewis’s theories require us to do exactly this, one of those theories will have to go. Those who wish to allow for backward causation will have to start with Lewis’s theory of counterfactuals. After all, that theory delivers the wrong results in the case of the time-travelling electron, whether or not we accept Lewis’s theory of causation. Fortunately, we can preserve much of what is right about Lewis’s theory by revising his account of world-similarity. There are various ways in which this might be done, but one particularly elegant proposal—due to Jonathan Schaffer (2004)—

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is to build a requirement for causal independence into Lewis’s second and forth rules:

It is of first importance to avoid big, widespread, diverse viola-tions of law.

It is of second importance to maximize the spatio-temporal re-gion throughout which perfect match of particular fact pre-vails provided that what goes on in that region is causally causally independent of whether or not the antecedent ob-tains.

It is of third importance to avoid even small, localized simple violations of law.

It is of fourth importance to secure approximate similarity of particular fact, provided that those facts are causally inde-pendent of whether or not the antecedent obtains.

There are various reasons to think that this kind of account is an improvement over Lewis’s own,18 but the most important point for our purposes is that this account delivers the right results in the case of the time-travelling electron. Consider again the random appearance of an electron in w3. That appearance helps to maximize the region throughout which perfect match prevails. However, what goes on in that region is not causally independent of whether or not the scientist presses the but-ton, since the button-pressing (in w1) is what causes the appearance of the electron. The upshot is that w3 does no better than w2 when it comes to the modified version of the second rule. Moreover, w3 still does worse than w2 when it comes to Lewis’s third rule. So, that world will count as comparatively distant to w1. Hence, this account delivers the right results on statement (2)—if the scientist hadn’t pressed the button, an electron wouldn’t have appeared. Of course, this modification of Lewis’s account comes at a cost, since one cannot analyze counterfactuals in terms of causation while also ana-lyzing causation in terms of counterfactuals. In other words, the revised theory of counterfactuals will also require us to give up on the counter-factual theory of causation. This may be a cost but, if the arguments of

18 See, for example, Kvart (1986), Schaffer (2004), Hiddleston (2005), and Noordhof (2003).

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this paper are correct, this is the cost of a coherent approach to the topic of backward causation.19

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19 For comments on earlier versions of this paper, I thank Hud Hudson, Neal Tognaz-

zini, Dennis Whitcomb, and two anonymous referees.