Analysis 59.3, July 1999, pp. 165–73. © Philip Percival

A note on Lewis on counterfactual dependence in a chancy world

Philip Percival

1. In a Postscript,1 David Lewis tries to extend results obtained in his(1979) from the deterministic case to the indeterministic one. In particular,he claims that under the supposition that the actual world is indeterminis-tic, the truth of the counterfactual ‘If Nixon had pressed the button, therewould have been a nuclear holocaust’ is reconciled with his truth condi-tions for counterfactual conditionals by a certain refinement of his earliertreatment. Sections 2 to 5 explain why his claim is mistaken. Section 6remarks why this matters.

2. Let us adopt Lewis’s simplifying assumptions. The supposition thatNixon presses the button is to refer implicitly to a time t, and we are toconcern ourselves with a world w0 in which, though Nixon does not pressthe button at t and there is never any nuclear holocaust, ‘our worst fanta-sies about the button [are realized]: there is such a button, it is connectedto a fully automatic command and control system, the wired-in war planconsists of one big salvo, everything is in faultless working order, there isno way for anyone to stop the attack, and so on’ (44). Moreover, althoughw0 is thereby distinguished from the actual world, it is to be assumed verysimilar otherwise in matters of fact and principle: in particular, it is to begoverned by whatever metaphysical principles the actual world is governedby. On these assumptions, Lewis concedes that on what he calls the ‘stand-ard’ resolution of what he takes to be the inherent vagueness of counter-factuals, the counterfactual ‘if Nixon had pressed the button, there wouldhave been a nuclear holocaust’ is true at w0 (44).

On Lewis’s account, A h→ C is true at a possible world w iff C is true atall the A-worlds most similar to w, the vagueness he discerns in counter-factuals stemming from the vagueness of the similarity relation betweenpossible worlds. But consider the following possible worlds to which hedirects our attention (44–46):2

w0 (Surrogate actual world): (Surrogate) actual history until Nixondoesn’t press the button at t. There is no nuclear holocaust.(Surrogate) actual future

1 Lewis 1986: 52–66. All page references are to this volume.2 I have found it convenient to alter Lewis’s numbering. My worlds w2 and w3 are

Lewis’s worlds w3 and w4. (His world w2 need not concern us.)

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w1 (Holocaust world): History of w0 prior to just before Nixonpresses the button at t. There is a nuclear holocaust. Future verydifferent from future of w0

w2 (Approximate convergence): History of w0 prior to just beforeNixon presses the button at t. There is no nuclear holocaust.Future approximately matches that of w0

w3 (Perfect convergence): History of w0 prior to just before Nixonpresses the button at t. There is no nuclear holocaust. Future soonexactly matches that of w0

The difficulty of finding a resolution of the vagueness of the similarity rela-tion between possible worlds under which the counterfactual ‘If Nixon hadpressed the button, there would have been a nuclear holocaust’ comes outtrue at w0 on Lewis’s account is apparent. This counterfactual only comesout true at w0 on Lewis’s account if w1 is more similar to w0 than are w2or w3. Yet, with respect to matters of particular fact, the worlds w1–3 areidentical until Nixon presses the button at t, and after t w1 is less similarto w0 than are w2 and w3. (The point of distinguishing w2’s approximatematching of w0 in matters of particular fact concerning times subsequentto t from w3’s eventual exact matching of w0 in matters of particular factwill soon become apparent.)

3. Lewis’s solution in the deterministic case. Lewis’s response to the prob-lem turns on the idea that w-miracles in a world w* – events of w* thatviolate the laws of another world w – come in different sizes, the size of amiracle depending on something like the number and spatio-temporaldimensions of the law-violating event(s). For he maintains that under thesupposition of determinism, w1 is more similar to w0 than are w2 or w3relative to a resolution of the vagueness of the similarity relation betweenworlds in which the following respects of similarity are ranked thus (47–48):

The Ranking

First Importance: Avoidance of big miracles.

Second importance: Maximization of exact match in matters ofparticular fact.

Third importance: Avoidance of small miracles.

Fourth importance: Maximization of approximate match in mattersof particular fact.

To be sure, the worlds w0–3 have not been specified in sufficient detail forthe proposition that w1 is more similar to w0 than are w2 or w3 to be

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derived from The Ranking under determinism without further premisses.But Lewis thinks that premisses which facilitate the derivation are readilyavailable. Foremost among them is a claim he calls ‘Asymmetry of Mira-cles’. Say that two worlds diverge at t if they are alike in matters ofparticular fact at t but different at t+ δ, and that they converge at t if theyare different in matters of particular fact at t but alike at t+ δ . Then, if theactual world, and hence w0, is governed by deterministic laws, Lewis holds(49):

Asymmetry of Miracles Divergence from a world such as w0 is easierthan perfect convergence to it. Either takes a miracle, since w0 is deter-ministic, but convergence takes very much more of a miracle.

Lewis now argues as follows. Suppose the actual world, and hence w0, isdeterministic and embodies Asymmetry of Miracles. In that case, theworlds w1–3 all involve a w0-miracle whereby Nixon presses the button att, since they share the (deterministic) history of w0 until just before t andNixon does not press the button at t in w0. This miracle is merely smallbecause divergent. However, whereas w1 need involve no further w0-miracles – it is in accordance with the laws of the actual world, and henceof w0, that a nuclear holocaust should follow upon Nixon’s pressing thebutton in w1 – the worlds w2 and w3 cannot avoid additional w0-miracles.A further w0-miracle must occur at both w2 and w3 when Nixon’s pressingof the button does not result in a nuclear holocaust, and whereas w2 needinvolve no w0-miracles thereafter – its subsequent approximate matchingof w0 in matters of particular fact after nuclear holocaust is (somehow)avoided is consistent with the laws of the actual world, and hence withthose of w0 – a still further w0-miracle is required at w3 if w3 is to achieveexact match with w0 shortly after t. By Asymmetry of Miracles, these addi-tional miracles are, respectively, small (because a divergence miracle) andlarge (because a convergence miracle). Accordingly, relative to The Rank-ing, the gains w2 and w3 make over w1 in the similarity stakes areoutweighed by their costs: w2’s approximate match with w0 after t (rankedfourth) competes with w1’s avoidance of even small miracles after that time(ranked third), whereas w3’s exact match with w0 shortly after t (rankedsecond) competes with w1’s avoidance of big miracles (ranked first).

Under determinism, then, the problem is solved. Resolving the vague-ness of the similarity relation between possible worlds in accordance withThe Ranking renders w1 more similar to w0 than are w2 or w3. So it resultsin the truth at w0 of the counterfactual ‘If Nixon had pressed the button,there would have been a nuclear holocaust’ on Lewis’s account of thiscounterfactual’s truth condition.

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4. Lewis’s treatment of the indeterministic case. Indeterminism scotchesthis derivation, since the description of the worlds w1–3 no longer guaran-tees that there are w0-miracles at them. Although these worlds exactlymatch w0 in matters of particular fact at some times, and do not exactlymatch it at other times, unless w0 is deterministic the switch from matchingto not matching (or vice versa) might simply reflect a chance process whichhas one outcome in these worlds, another in w0. This is not to say that theworlds w1–3 needn’t involve w0-miracles on the supposition w0 is indeter-ministic. Whether or not they have to do so depends on which parts of w0are indeterministic, which of its laws are indeterministic etc. And since w0is a surrogate for the actual world, this much turns on contingent claimsabout how things are. However, Lewis hypothesizes that if the actualworld, and hence w0, is indeterministic along the lines he takes quantummechanics to suggest, chance is so pervasive that ‘[the world] can containfew if any deterministic enclaves’ (59),3 and both the divergences from w0’shistory which occur in the worlds w1–3 just before Nixon presses thebutton, and the approximate and perfect convergences to w0’s futurewhich occur, respectively, in w2 and w3 sometime afterwards, can beachieved without violating the laws of w0 (59–60).

Lewis observes that this latter supposition – that perfect convergence tow0’s future can be achieved without any w0-miracle – makes the difficultyat issue harder (60). In particular, under determinism The Ranking onlyresulted in w1 being more similar to w0 than is w3 because Asymmetry ofMiracles guaranteed that w3’s perfect convergence to w0’s future incurredthe prohibitive cost of a big w0-miracle. Clearly, then, unless w3’s perfectconvergence to the future of w0 still incurs some cost under indeterminism,the game is up: relative to The Ranking, w3 will be more similar to w0 thanis w1, and on Lewis’s account the counterfactual ‘if Nixon had pressed thebutton, there would have been a nuclear holocaust’ will be false at w0.(Indeed, the counterfactual ‘if Nixon had pressed the button, there wouldnot have been a holocaust’ will be true there.) However, if w3’s perfectconvergence to w0 can be procured in accordance with the laws of w0,what could be the cost of achieving it? Lewis’s answer exploits a conceptof ‘quasi-miraculousness’ (with respect to a world), a ‘quasi-miracle’ (withrespect to our world) being an event which is not merely extremelyimprobable, but one which is ‘such a remarkable coincidence that it would

3 Lewis’s thesis that if (quantum) indeterminism is true at the micro-level, chance seepsup to the macro-level so as to be all-pervasive, is expressed at several points amongstthe newly published material in his (1986). See his ‘Causal Explanation’ (especiallyp. 230), Postscript B of ‘A Subjectivist Guide to Objective Chance’ (especially p. 119),and Postscript B to ‘Causation’ (especially p. 175).

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be quite unlike the goings-on we take to be typical of our world’ (60). Hethen makes two conjectures. Firstly, quasi-miracles count negatively in thesimilarity stakes. Secondly, under indeterminism Asymmetry of Miraclesgoes over into another asymmetry, namely (61):

Asymmetry of Quasi-Miracles Convergence to an indeterministicworld of the sort that ours might be takes a quasi-miracle; divergencefrom such a world does not.

On these two assumptions, although w3’s perfect convergence to w0needn’t be w0-miraculous if w0 is indeterministic, the actual world, andhence w0, is nevertheless such that perfect convergence to it is costly. Ofcourse, merely supposing the avoidance of quasi-miracles is of someimportance in the similarity stakes does not get Lewis what he wants. If w1is to be more similar to w0 than is w3 under indeterminism, the avoidanceof quasi-miracles must be of an importance akin to the avoidance of bigmiracles under determinism: quasi-miracles must ‘make enough of adissimilarity to outweigh perfect match throughout the future’ (61). Butprovided the Ranking is refined so as to hold the avoidance of quasi-mira-cles more important than achieving exact match in matters of particularfact, the relative performance of w1 and w3 in the similarity stakes underindeterminism mirrors their relative performance under determinism.Relative to this refined Ranking, even in the indeterministic case w3 comesout less similar to w0 than is w1, since its (eventual) perfect match with w0is ranked below w1’s avoidance of quasi-miracles. It is this fact whichprompts Lewis to draw the conclusion that if the vagueness of the similar-ity relation between possible worlds is resolved in accordance with TheRefined Ranking, ‘the case turns out as it should: the closest worlds whereNixon presses the button are worlds where a holocaust ensues’ (60).

5. Lewis’s conclusion does not follow. He has just been seen to haveargued that on the assumption that the actual world, and hence w0, is inde-terministic, w1 is more similar to w0 than is w3 relative to The RefinedRanking. But to obtain the truth of the counterfactual ‘If Nixon hadpressed the button, there would have been a nuclear holocaust’ Lewisneeds to establish a stronger conclusion – namely, that w1 is more similarto w0 than is either w3 or w2. But on his assumptions, w1 is not more simi-lar to w0 than is w2 relative to The Refined Ranking under indeterminism.

To see that it is not, return to Lewis’s strategy under the supposition ofdeterminism. Lewis argues that by The Ranking, w1 is more similar to w0than is w2, because

(iD) w2 can only buy its approximate match in matters of particularfact to w0’s future at the prohibitive cost of the additional diver-

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gence w0-miracle by which at w2 there is no holocaust eventhough Nixon presses the button.

(iD) identifies the cost determinism imposes on approximate convergenceto w0: a w0-miracle which, given Asymmetry of Miracles, is small.However, under the supposition of indeterminism Lewis thinks this costvanishes. For while he maintains that events that are big w0-miracles underdeterminism go over into something costly – w0-quasi-miracles – he holdsthat events that are small w0-miracles under determinism do not go overinto anything costly. He says (61):

I do not speak of small quasi-miracles; what corresponds to a smallmiracle in the deterministic case is a perfectly commonplace chanceoccurrence.

On this account of the relationship between miracles and quasi-miracles,in the indeterministic case (iD) becomes:

(iI) w2 can buy its approximate match in matters of particular fact tow0’s future at no cost.

This crucial fact is simply not factored into Lewis’s assessment of theperformances, under indeterminism, of w1 and w2 relative to The RefinedRanking. Before devoting most of his discussion of the indeterministic caseto the world w3, Lewis dismisses the claims of w2 in writing (59-60):

The thing to say about approximate convergence [i.e. in w2] remainsthe same. Even if approximate convergence is cheap – and even if it ischeaper still when it can be had without even a little miracle – still wecan say that it counts for little or nothing, so it is not so that if Nixonhad pressed the button there would have been approximate conver-gence to [w0], and no holocaust.

This passage contains two oversights. Firstly, the conclusion Lewis draws is not insensitive to whether approx-

imate convergence counts ‘for little or nothing’. If approximateconvergence counts for nothing, Lewis rightly concludes that (by TheRefined Ranking) ‘it is not so that if Nixon had pressed the button therewould have been approximate convergence to [w0], and no holocaust’. Butif approximate convergence counts for anything, this conclusion is incor-rect, since in that case, relative to The Refined Ranking something of somevalue is bought for nothing (cf. iI). Hence, relative to that ranking w1 is lesssimilar to w0 than is w2, and it is so that if Nixon had pressed the button,there would have been approximate convergence to w0, and no holocaust.

Secondly, even if w2’s approximate convergence to w0 counts for noth-ing, and Lewis’s conclusion that at w0 ‘it is not so that if Nixon had pressed

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the button there would have been approximate convergence to [w0], andno holocaust’ is granted, the further conclusion that he draws implicitly –that w1 is more similar to w0 than is w2 – does not follow:4 to grant thatw2 is not more similar to w0 than is w1 is not to grant that w1 is more simi-lar to w0 than is w2. Moreover, it is clear that this further conclusioncannot be obtained on Lewis’s assumptions. In allowing that in the inde-terministic case w2 can procure its approximate convergence to w0 withoutthe expense of either the w0-miracles or w0-quasi-miracles that w1 avoids,Lewis allows that relative to The Refined Ranking, this much can beprocured at no cost (cf. iI). But to say that in comparison to w1, w2 gainsnothing at no cost is to say that it is as similar to w0 as is w1. This is to saythat it is not the case that w1 is more similar to w0 than is w2. Hence, it isto say that the counterfactual ‘If Nixon had pressed the button, therewould have been a nuclear holocaust’ does not come out true at w0 onLewis’s account of its truth condition.

6. Pointing out these two oversights would be of little consequence ifLewis might easily rectify them (Option 1) or simply ignore them (Option2). But this isn’t so.

Option 1 There is little in his discussion which suggests how, under thesupposition of indeterminism, Lewis can render w1 more similar to w0than is w2, as the truth of the counterfactual ‘If Nixon had pressed thebutton, there would have been a holocaust’ requires on his account. In thelight of his view that if indeterminism is true chance is all-pervasive, theworlds w2 and w1 are relevantly asymmetrical only in that a chance process– what happens after t as a consequence of Nixon’s pressing the button –has an (extremely) improbable outcome in the former (no nuclear holo-caust) and a (very) probable outcome in the latter (nuclear holocaust). Theonly way, albeit a straightforward one, that I can see of turning this asym-metry into the requisite asymmetry in the similarity stakes is to deem theavoidance of very improbable events of some importance while insistingthat approximate match in matters of fact is of no importance. However,adding these weightings to The Refined Ranking would result in an entail-ment from A h→ [p(¬C) ≤ r] to A h→ C when r is very small, and thisconflicts with Lewis’s deepest intuitions about counterfactual dependence.For the refinement of his account of causation which he offers in Postscript

4 This conclusion is implicit at this point, because, in switching his discussion to worldw3 immediately afterwards, Lewis says nothing further concerning w2 until heconcludes, on the following page (60), that ‘the case turns out as it should: the closestworlds where Nixon presses the button are worlds where a holocaust ensues’.

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B of his (1973) is embarked upon precisely because of his conviction thatin cases in which C would have had some chance of being false if the falseproposition A had been true, A h→ C fails unless the falsity of C wouldhave been quasi-miraculous.5

Option 2 In the light of these apparently contrary intuitions, Lewis’s insist-ence in Postscript D to his (1979) that on a satisfactory analysis thecounterfactual ‘If Nixon had pressed the button, there would have been anuclear holocaust’ ought to come out true is surprising. Perhaps he simplyconfused this constraint with the weaker requirement that if Nixon hadpressed the button, perfect convergence would not have occurred, orperhaps he thought somehow that counterfactuals of this ilk could betreated as a special case. Either way, in the light of the oversights I haveidentified, perhaps he might think simply to abandon the counterfactual ‘IfNixon had pressed the button, there would have been a nuclear holocaust’in the indeterministic case. So doing would be costly, however, since itwould mean that the central claim of his (1973), originally made underdeterminism, would not survive indeterminism. This is the claim that ‘themysterious asymmetry between open future and fixed past is nothing elsethan the asymmetry of counterfactuals’ (38). Lewis characterizes this latterasymmetry as follows (35):

Asymmetry of Counterfactual Dependence Consider those counter-factuals [A h→ C] in which the supposition A is indeed false, and inwhich A and C are entirely about the states of affairs at two times tA

and tC respectively. Many such counterfactuals are true in which Calso is false, and in which tC is later than tA. These are counterfactualsthat say how the way things are later depends on the way things wereearlier. But if tC is earlier than tA, then such counterfactuals are true ifand only if C is true. These are the counterfactuals that tell us how theway things are earlier does not depend on the way things will be later.

If, as Lewis thinks, the actual world’s indeterminism would render chanc-iness all-pervasive, chanciness will infect the consequents of thosecounterfactuals A h→ C we are accustomed to assert in which the time ofthe state of affairs C concerns is later than the time of the state of affairswhich A concerns. Hence, these counterfactuals will not survive indeter-minism either if the counterfactual ‘If Nixon pressed the button, therewould have been a nuclear holocaust’ is deemed false on the grounds that

5 Compare Lewis’s remark that ‘if [a cause] c had not occurred [and an effect] e wouldstill have had some chance y of occurring, but only a very slight chance …. We cannotquite say that without the cause, the effect would not have occurred’ (176, myitalics).

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there would have been some chance of there not being a nuclear holocausteven if Nixon had pressed the button. But if these counterfactuals don’tsurvive it, nor does Asymmetry of Counterfactual Dependence.6

University of Glasgow, G12 8QQ, UKP.Percival@philosophy.arts.gla.ac.uk

References

Lewis, D. 1973. Causation. Journal of Philosophy 70: 556–67. Reprinted as chapter 21of his 1986.

Lewis, D. 1979. Counterfactual dependence and time’s arrow. Noûs 13: 455–76.Reprinted as chapter 17 of his 1986.

Lewis, D. 1986. Philosophical Papers: Volume II . New York: Oxford University Press.

6 I am grateful to Jim Edwards for helpful comments