This is a preprint of an article accepted for publication in Nous © 2003 Blackwell Publishing

David Lewis’s Counterfactual Arrow of Time

Alexander R. Pruss

July 10, 2002

I.  Introduction

             David Lewis (1979) has argued that according to his possible worlds analysis of counterfactuals, “backtracking” counterfactuals of the form “If event A were to happen at t_A, then event B would happen at t_B” where t_B precedes t_A, are usually false if B does not actually happen at t_B.  On the other hand, there are plenty of such counterfactuals true with t_A preceding t_B;  for instance it is true that were I to drop the glass now, it would hit the ground at some point in the future, even if in fact it does not do so.  Assuming some contingent facts about the arrangement and laws of our universe, this time-reversal asymmetry, Lewis claims, follows from a possible worlds analysis of counterfactuals despite the fact that this analysis of counterfactuals is entirely time symmetric.  Lewis argues, further, that this asymmetry gives meaning to the common notion of the future as “open” and the past as “closed”, even if determinism both of the future by the past and of the past by the future is true, which for the purposes of the analysis he assumes it to be and in which assumption I will follow him in this paper.

             Much of the argument of Lewis’s (1979) paper is a reply to an objection that had been raised by Fine and others against his analysis of counterfactuals.  I shall argue that Lewis’s reply succeeds in some interesting special cases but fails in others to demonstrate the asymmetry he seeks.  But even more seriously, I shall show that the asymmetry Lewis finds, if there actually is one to be found there, is grounded in the fact that there is a time-reversal asymmetric preselection in the kinds of events that figure as antecedents of ordinary language counterfactuals.  We do not in practice ask: “What would happen if p held?” for every proposition p, but only for some.   I argue that this preselection of some antecedents of counterfactuals but not others in everyday counterfactuals is based in part on the common-sensical notion that generally it is past events that are the causes of future ones.  More precisely, the kinds of everyday counterfactuals Lewis is most interested in are generally preselected in such a way that there are localized miracles or interventions in the past of the events in the antecedents that could bring their antecedents about.  Hence, the asymmetry that Lewis finds through his analysis is parasitic on people’s time-reversal asymmetric intuitions and Lewis’s analysis fails to give independent objective grounding for the counterfactual arrow of time.

             Jonathan Bennett (1984), on the other hand, has also argued against Lewis, and gave an alternate account of counterfactuals which implies that there is a perfect symmetry between backtracking and forward-tracking counterfactuals.  However, Bennett’s account of counterfactuals will be criticized at the end of Section II, below.

             There are two general kinds of time-reversal asymmetries.  On the one hand, there are intuitive asymmetries, mainly rooted in some idea of the future being open and the past being closed.  A natural way to express this asymmetry is in terms of the asymmetry Lewis identifies in counterfactuals.  A closely related asymmetry is that we take causal effects to propagate from past to future.  If one sees a close connection between causality and counterfactuals, then one is apt to see this asymmetry as also closely tied to the asymmetry in counterfactuals.  On the other hand, we have scientific asymmetries, the most interesting of which perhaps is the one embodied in the Second Law of Thermodynamics: entropy tends to increase. 

             A natural project, then, is to try to relate the intuitive and the scientific asymmetries.  It does not appear possible to ground the entropic asymmetry in the intuitive asymmetries, given that our best understanding of the entropic asymmetry is that it is simply grounded in asymmetric boundary conditions for our universe, namely our universe’s initial state being a low-entropy one, rather than in terms of an asymmetry in the basic physical laws or much less in the metaphysics of time.  Indeed, the basic physical laws may well be almost completely time-reversal symmetric, a notion that in a deterministic system we can explain by saying that if the laws allow a transition from state S₁ to state S₂ between time t and t+Dt, then they also allow a transition from S₂^† to S₁^†, where S_i^† is the time-reversal of state S_i—the positions of all particles involved in the state are kept the same, but the direction of the velocities is reversed, and any similar minor adjustments are made.[1]  Because of this, even if there is no metaphysical asymmetry in the nature of time or of causality, nonetheless the boundary conditions for our universe, namely the low-entropy initial state, are sufficient to ensure the entropic asymmetry.  Unless one can argue that the intuitive asymmetries are responsible for the low-entropy initial state, which seems unlikely, the prospects for grounding the scientific asymmetries in the intuitive ones are dim.

             But maybe the intuitive asymmetries can be grounded in the scientific ones?  For instance, Reichenbach has connected the intuition of the future being open with the fact that we have much better knowledge of the past, and has attempted to connect this with the increase in entropy.  Thus, traces of the past, such as a footprint on a beach, are areas of lower entropy, and hence are surprising unless we posit an interaction between the region of the trace and some external cause, such as a foot.  Unfortunately, as John Earman has noted, we also take local areas of high entropy as traces of the past sometimes (Sklar, 1993, pp. 397-398).  The other major attempt to ground the intuitive asymmetries in the scientific ones is Lewis’s.  Lewis grounds an asymmetry in counterfactuals in certain facts about our world, namely in the fact that a given event tends to have an isolated cause but many spread-out effects.  Since the latter fact, as Sklar (1993, p. 401-404) notes, may be thought to be grounded in an entropic asymmetry, what Lewis’s account gives us is a grounding for intuitive asymmetries in terms of the entropic asymmetry.

             Very recently, Douglas Kutach (forthcoming) has given a non-Lewisian entropy-based account of counterfactuals.  However, the relevant part of the account proceeds by restricting consideration only to those classes of possible worlds which start out in a low-entropy state.  Doing this may well yield counterfactuals that have the right kind of asymmetry, but as Kutach himself notes, does not ground this asymmetry.  For instance, our belief that when evaluating an everyday counterfactual we need only consider those worlds that start off in a low-entropy state is surely based on our knowledge that (a) our world as a matter of fact started off in a low-entropy state and (b) that everyday counterfactuals do not backtrack in such a way as to make true any counterfactual of the form: “Were A to occur, the initial state would have been a high-entropy one”, where A is a counterfactual event after the time of the initial state.  Kutach’s approach thus in effect fixes an aspect of the initial state, namely its having low entropy, across all possible worlds relevant to the discussion, and it is not surprising that counterfactuals evaluated in this way end up being asymmetric, since an asymmetry was built into their definition by fixing an aspect of the initial state.[2]

             A different counterexample to Lewis’s account of counterfactuals and the arrow of time was recently given by Adam Elga (forthcoming) in a careful thermodynamic analysis of the time-reverse of the process of cooking an egg.  Elga’s counterexample, while correct, involves the special case of a counterfactual whose antecedent is not an event but the non-occurrence of an event.  This counterexample is discussed further in Section III.

II.  Lewis’s argument

             Lewis’s analysis of counterfactuals lives in the framework of possible worlds, though it will be irrelevant for it what ontology we choose for possible worlds.  Then:

A counterfactual “If it were that A, then it would be that C” is (non-vacuously) true if and only if some (accessible) world where both A and C are true is more similar to our actual world, overall, than is any world where A is true but C false.  (Lewis, 1979, p. 465)

This definition gives rise to the following objection stated by Kit Fine (and also given by a number of other people). 

The counterfactual “If Nixon had pressed the button there would have been a nuclear holocaust” is true or can be imagined to be so.  Now suppose that there never will be a nuclear holocaust.  Then that counterfactual is, on Lewis’s analysis, very likely false.  For given any world in which the antecedent and consequent are both true it will be easy to imagine a closer world in which the antecedent is true and the consequent false.  For we need only imagine a change that prevents the holocaust but that does not require such a great divergence from reality.  (Fine, 1975, p. 452)

             In order to get out of Fine’s objection and make his own notion of counterfactuals determinate, Lewis proposes a measure of similarity of worlds that has four factors ranked as follows:

(1) It is of the first importance to avoid big, widespread, diverse violations of [physical] law.

(2) It is of the second importance to maximize the spatio-temporal region throughout which perfect match of particular fact prevails.

(3) It is of the third importance to avoid even small, localized, simple violations of law.

(4) It is of little or no importance to secure approximate similarity of particular fact, even in matters that concern us greatly. (Lewis, 1979, p. 472)

These factors are rigged to make sure that Lewis gets the right answer to Fine’s objection.  One might object to these four factors and/or to their order.  For instance, the world which is exactly like ours for all times in our future but whose past is radically different from the past of the actual world is surely further from our world than is a world which is the same as ours for all time except that the background radiation is everywhere and everywhen about 10^{–1,000,000,000} percent higher.  However, whatever we think of Lewis’s four factors and/or their order, it would be an impressive feat if Lewis could get a time-reversal asymmetry in counterfactuals out of them, as all four factors are certainly time-reversal symmetric.  So let us grant Lewis his measure of closeness of worlds.

             Lewis then evaluates Fine’s counterfactual that if Nixon had pressed the button, the world would have been blown-up.  There are, he says, four different kinds of possible worlds where Nixon pressed the button at t.

             1. In worlds of the kind[3] of w₁, shortly before time t, everything matches what happens in the actual world w₀, but then the worlds begin to diverge:

The deterministic laws of w₀ are violated at w₁ in some simple, localized, inconspicuous way.  A tiny miracle takes place.  Perhaps a few extra neurons fire in some corner of Nixon’s brain. (Lewis, 1979, p. 468)

And so Nixon presses the button.  No further divergences from law happen.  The nuclear holocaust follows.  Note that the miracle happens shortly before time t because if it were to happen exactly at t, the miracle required might be large—after all, Nixon might not even actually be close to the button then so he would have to be instantaneously transported there.  Letting the miracle happen a little before t reduces the size of the needed miracle.

             2. In worlds of the kind of w₂, on the other hand, physical laws are never violated.  Nixon presses the button.  However, both the past and the future are different, because of the assumption of bi-directional determinism.  At no time is w₂ the same as w₀. 

             3. In worlds of the kind of w₃, two small miracles, i.e., violations of the natural laws of the actual world, happen.  First the same kind of miracle as in w₁ happens.  But then a second miracle prevents the nuclear holocaust from stopping (maybe the wire miraculously breaks before the signal passes through it).  However, the world is already very different.  Nixon will write different memoirs, the wire has heated up, the very movement of the finger has changed the gravitational gradient in China, etc.  Indeed, it is plausible that nowhere in the whole of the future light cone with apex at the first miracle will the universe be exactly the same. 

             4. And, finally, the fourth kind of world is exemplified by w₄ which is like w₃.  There, two miracles happen as in w₃.  The first is the same, but the second is a lot more impressive than it was in w₃.  Not only does the nuclear holocaust not happen, but all the traces of the button press are removed, and so after the second miracle, w₄ looks just like w₀. 

             Now, which of these kinds of worlds is closest to ours?  To escape Fine’s objection, Lewis must argue that it is w₁, because it is only in w₁ that both a nuclear holocaust happens and there is no long-term backtracking as in w₂.  Now, w₁ is definitely closer to our world than w₃ by Lewis’s criteria, because the only advantage of w₃ is that it lacks a nuclear holocaust in its future and hence there is more approximate future agreement between w₃ and w₀ than there is between w₁ and w₀.  But this agreement is merely approximate in the future light cone with apex at the event that caused the pressing of the button given how physical effects propagate, whereas w₃ has an extra, small miracle.  Avoiding small miracles is Lewis’s third most important similarity factor.  Therefore, w₃ is better in terms of the fourth most important factor, and w₁ in terms of the third important factor, and so w₁ is to be preferred to w₃ as a candidate for a closest world. 

             What of w₄?  It is true that w₄ matches w₀ in a very large spatio-temporal region: all of the future of the second miracle and all of the past of the first.  This is Lewis’s second factor.  However, w₄ must have a rather large miracle.  The gravitational gradient has to be corrected throughout a large region of space.  The particles shifted around (admittedly by a tiny distance) in China by the change in gravitational gradient caused by Nixon’s hand-movement towards the button have to be shifted back.  Nixon’s apparent memories have to be altered.  The wire has to be cooled.  The vibrations from the click of the button have to be stopped from propagating.  This would violate the first criterion for closeness of worlds in a way that w₁ does not.  Hence, w₁ is closer than w₄. 

             That leaves w₂ and w₁.  But w₂, plausibly, is just about nowhere in space-time identical with our world.  And Lewis argued that in the far future and the distant past it is probably not very similar even approximately speaking, though this is open to dispute.  But in any case, the lack of exact match anywhere in w₂ means that w₂ violates the second criterion of closeness, whereas w₁ only violated the third by having a small miracle. 

             Hence, indeed, w₁ is the closest of the worlds (or, more precisely, types of worlds) in which the button is pressed—at least if no other worlds are candidates which for the nonce I shall grant Lewis (though in Section V, below, I shall argue that Lewis has passed over a crucial candidate).  And since the nuclear annihilation of humankind does happen in w₁, it follows that Fine’s counterfactual “If Nixon had pressed the button, there would have been a nuclear holocaust” is indeed true on Lewis’s account.

             Moreover, the analysis does display a past-future asymmetry.  For, given that the closest world is w₁, it follows that counterfactuals of the form “If Nixon had pressed the button then C would happen at t_C”, where in the actual world C does not happen at t_C, can only be true if t_C is after to the time of the “small miracle” in w₁, which time is slightly before pressing the button.  So, as Lewis (1979, p. 458) admits, there may be a modest amount of backtracking in the counterfactual—but only back to the time of the miracle.

             According to Lewis, what grounds the above analysis is the fact that an event like the pressing of a button has a lot of disparate effects but a fairly localized cause.  It is this temporal disanalogy that, on Lewis’s account, grounds the counterfactual arrow of time that in turn gives meaning to our intuitions about the openness of the future and the closedness of the past.  In Section V, however, we shall see that this analysis is hopelessly flawed because of the failure to consider a fifth class of worlds.  However first we will consider some less telling counterexamples in Section III, but not before looking at Bennett’s (1984) comments on Lewis.

Bennett (1984) proposes the following account of counterfactuals.  If p is a proposition solely concerning what happens at time t, then were p to hold, q would hold is true if and only if q holds at that world which is t-closest to the actual world from among all worlds at which p holds and at which the laws of nature of our world hold.  The t-closest world is the one which is closest when we restrict our measures of closeness to the time-slice at t.  In the bidirectionally deterministic case, this boils down to this: we choose the closest time-slice to the actual world’s time-slice at t but at which p holds, and then generate the rest of the world by evolving nomically forward from the time-slice to fill out what happens after t and backward from the time-slice to fill out what happened earlier.

If Bennett is right, then there is no counterfactual asymmetry.  However, in fact, Bennett’s account fails, even in the deterministic case.[4]  To use an example inspired by Elga (forthcoming), suppose that actually I drop an egg which then falls, breaks open and is lying in pieces on the ground, while heat and sound propagate outward from the impact site.  Let t be a time after the impact.  Imagine now the above scenario run backwards.  We have a bunch of calcium carbonate flakes, i.e., pieces of eggshell, on the ground and some viscous liquid.  Heat and sound propagate inward towards this and are focused so perfectly and aligned so well with the calcium carbonate flakes that they ensure that these flakes and the liquid reassemble into an egg, which then bounces off the ground and into my hand. 

Suppose that at t the largest flake of the egg has a center at position x.[5]  Consider now the absurd counterfactual: Were the center of largest calcium carbonate flake a quarter of an inch away from x, the flake would not have come from an egg.[6]  This counterfactual is indeed absurd, and plausibly can be made to come out false on Lewis’s view.  Simply imagine a world w₁ in which a small miracle made the egg spin slightly differently as it was falling, before which time w₁ matched the actual world exactly, and after which time the laws of nature took over again.  This kind of world may well come out to be the closest one to the actual on the Lewisian analysis. 

But w₁ will not be the t-closest world.  For if the egg had spun differently while falling, other shards would also have fallen in different positions.  But the t-closest world w₂ will surely be one where only the largest flake has a position different from that which it has in the actual world, since only closeness at t matters.[7]  Now, the past of w₂ is formed by evolving backwards from the state of w₂ at t.  Imagine this process running backwards.  We start with the flakes on the ground, and heat and sound waves moving inward in their direction, as in the time-reverse of the actual world.  But now exactly one flake is disturbed, and it is clear that therefore the amazing reassembly process will fail: the flakes will fail to reassemble into an egg, and in particular the largest flake will not become part of an egg.  Since this is what is true of the time reverse of w₂, it is true in w₂ that there was no intact egg before t which broke into the flakes, and so in w₂ the largest flake does not come from an egg.  Hence, the absurd counterfactual comes out true on Bennett’s view, because restricting closeness measures to features occurrent at t and insisting on the laws holding does not allow one to get a closer world by inserting a miracle before t.

Bennett in addition to an alternative to Lewis’s account also offers a criticism.  Recall that because the miracles are inserted at some time prior to the counterfactual event on Lewis’s view, there will be some backtracking counterfactuals.  Moreover, they may go some distance back.  Presumably at the closest world at which Al Gore is president in November, 2001, he was also president on Inauguration Day, 2001, and hence it is actually true that were Gore president in November, 2001, he would have been president on January 20, 2001.  But our intuitions about the past being closed and the future being open do not have the past only partially closed.  Thus, Lewis’s counterfactual account fails to explain all of the content of our intuition about the difference between past and future. 

However, this criticism was made in a dialectical setting where Bennett was refuting the argument that Lewis’s account of counterfactuals was superior to Bennett’s on the grounds that Lewis’s account explains the arrow of time, and Bennett was claiming that a merely partial success here would not outweigh the supposed advantages of Bennett’s account.  But if the analysis of the egg-drop case is correct, Bennett’s account of counterfactuals simply fails, and a partial explanation of our intuitions about the differences between past and future is nothing to scoff at.  At least, Lewis’s view supports our intuition that the more distant past is counterfactually fixed, and the view is an improvement over other entropic accounts.  Moreover, the way the backtracking is introduced fits well with our intuitions of the difference between past and future.  When we imagine a counterfactual world where A occurs, we may well, as the interventionists (whose views will be discussed further in Section VI) would have it, imagine it as a world in which some somewhat earlier event causes A, and the asymmetry will then be centered not at the time of the occurrence of A but at the time of the cause.  This story as a whole fits well with our time-reversal asymmetric intuitions.

III.  The cable cutters

             Now consider another case.

(A)   Suppose that in our world, Nixon had in fact pressed the button connected to the doomsday device.  But unbeknownst to anybody but the Soviets, over many years the Soviets had carefully located the communications cable between the White House and the doomsday device, and along the cable they buried a thousand independent reliable cable-cutting devices (or just one device which is 1000-fold redundant) that monitor the cable for the distinctive start of a signal to activate the doomsday device and cut it off before the signal can be completed.  The doomsday device, however, requires a complete signal to activate.  Moreover, Nixon’s decision to press the button was a highly conflicted one such that there was a single neuron which happened to fire, and had it not fired the decision would not have been made.  Now, consider the counterfactual, evaluated with respect to a world in which Nixon presses the button: “Were none of the Soviet cable-cutting devices to have activated, a nuclear holocaust would have ensued.”

The counterfactual is one whose truth surely cannot be denied.  But we shall see that Lewis’s analysis assigns it the wrong truth-value.

             Following Lewis’s analysis of Fine’s case, let w₀´ be the world that (A) describes as “our world” (i.e., a world with a nuclear holocaust), and let w₁´ be the world where Nixon presses the button but a miracle prevents the one thousand Soviet devices from cutting the cable.

             But now consider an alternate world.  This is the world in which a small miracle happens in Nixon’s brain.  This small miracle prevents from firing that one neuron whose firing was necessary for the pressing of the button.  In this world, call it w₅´, the urge to press the button might come over Nixon, but even if it does, it quickly passes.  It was never a very decisive urge anyway.  And not surprisingly, in w₅´, none of the Soviet devices activate, because they have no reason to do so.

             But surely w₅´ is closer to w₀´ than w₁´ is.  For, w₁´ involves a much greater miracle than w₅´ does.  Indeed, w₁´ involves a thousand different miracles, one to prevent each of the Soviet devices from going off.  It is true that w₁´ matches w₀´ exactly spatio-temporally for a very short while longer than w₅´ does, namely for the amount of time between Nixon’s button press and the activation of the Soviet cable cutters.  But we saw in Section II that Lewis is willing to allow worlds to mismatch for a short while in order to ensure that the miracle that needs to happen be a much smaller one.  Given how much smaller the miracle would be in Nixon’s brain than in the failure of the cable-cutters, w₅´ will be closer to w₀´ than w₁´ will be.  (If we are not satisfied with the difference in the size of miracles, we can easily increase the size of a miracle needed to transmit the signal just by boosting the number of cable-cutting machines and making each machine multiply redundant.)

             But then Lewis’s counterfactual analysis implies that the counterfactual “Were none of the Soviet devices to have gone off, a nuclear holocaust would have ensued” is false, since its consequent is false in w₅´.

             Note, too, how our counterfactual absurdly backtracks to Nixon’s decision.  For the above analysis implies absurdly that in case (A) the counterfactual “Were the Soviet devices not to have gone off at t₀ (which we suppose is the time when they in fact went off), Nixon would not have pressed the button” is true.  But there is still a temporal asymmetry here.  For, although the counterfactual backtracks to Nixon’s decision, which is surely in this case too far (it should backtrack at most to the detection of the signal by the Soviet cable cutting machines), it does not backtrack very far back, whereas counterfactuals with the same antecedent forwardtrack arbitrarily far.

             The objection has been made[8] that the cable-cutting counterexample occurs in a counterfactual world, and Lewis only claims his analysis works in the actual world.  However, the counterexample occurs in a world that might well have been ours, and even if this precise case did not occur in our world, a relevantly similar one might have.

             It might more seriously be objected that “none of the Soviet devices going off” is an illicit event description.  It doesn’t, after all, describe an event, but the complement of the disjunctive event of at least one of them going off.  And if we add some positive content to the event, as in: “The doomsday signal having been sent but none of the Soviet devices going off”, then the paradox disappears.  But given that Lewis intends his account of counterfactuals to provide a counterfactual relation between propositions, the objection is one that he cannot make.  And the objection does point out something crucial:  Whether Lewis’s analysis works depends on what exactly one chooses as the antecedent of the conditionals.  This observation will be further investigated throughout the rest of this paper.

             However, although Lewis did intend his account to cover propositions, there may be good reason to restrict it to events, or at least to somewhat more definite positive propositions, though as it turns out, doing this would make Lewis’s counterfactual reductive account of causation fail.[9]  Someone who did thus restrict Lewis’s account of counterfactuals might argue that in the case of a counterfactual with a negative antecedent, the context specifies some particular way in which the antecedent is supposed to happen, and perhaps in cases of some negative propositions we will not be able to describe this in a context-free manner.  It might even be that in the case at hand, when we formulated the original counterfactual in (A) we meant to specify that we are talking of a case where the cable-cutters fail to activate despite the signal being sent, and we have no intuition about the truth-value of the counterfactual apart from this specification.

             One could make exactly the same answer to Elga’s (forthcoming) counterexample to Lewis.  Elga considers the scenario where “[a]t 8:00, Gretta cracked open an egg onto a hot frying pan”, and considers counterfactuals such as: “If Gretta hadn’t cracked the egg onto the pan, then at 8:05 there wouldn’t have been a cooked egg on the pan.”  Elga shows by thermodynamic considerations that there is a possible world extremely close to ours at which Gretta does not crack the egg onto the pan but where there is a cooked egg on the pan at 8:05.  As it turns out, the kind of world Elga constructs to demonstrate this conclusion is a world is one where in fact for all of its existence, the egg has already been cracked.[10]  But when we formulate the counterfactual, it could be argued that we do not intend this way of making the negative claim true.  We do not intend to talk about a case where Gretta doesn’t crack the egg because the egg already is in a cracked state, but of a case where the intact egg is there, perhaps in the fridge or in her hand, but she does not crack it onto the pan. 

             The negative proposition that Gretta doesn’t crack the egg can be made true in many ways, such as by Gretta’s never having existed, by her not having hands, by the egg’s already being cracked, or by the pan’s not existing or its not being there.  And there seem to be too many such ways for us to have a confidence that a meaningful counterfactual can be uttered without further disambiguation.  Perhaps, then, we need to restrict our attention to cases where a positive event is given in the antecedent of a counterfactual.  Note that that was what we had in the original case of Fine.

IV.  The neuron

             Let us return to Fine’s case.  In Lewis’s description of the world w₁ where Nixon presses the button and a nuclear holocaust ensues, one goes a short amount of time back before the button press and posits a miracle that made some neurons fire in Nixon’s brain.  This miracle made for the difference between w₁ and the actual world.  Suppose, as seems possible, that the matter in this case depends on only one neuron.  Let N be the event of this neuron firing at the time it fires in w₁.  Surely a criterion of adequacy on an account of counterfactuals like Lewis’s is that one be able to say: “Were N to happen, then the button would be pressed, and the nuclear holocaust would occur.”  After all, if the neuron is responsible for pressing the button, then the button would be pressed if the neuron were to fire.  And our intuition that should the button be pressed the nuclear holocaust would occur must be saved.

             But consider this in terms of Lewis’s counterfactual analysis.  We have the actual world w₀ and we have the world w₁ which was the closest world for Fine’s counterfactual.  World w₁ matches the actual world up to around the time t of the neuron’s firing, and then a miracle occurs which makes the neuron fire, even though the physically necessary conditions for its firing are not met.  And henceforth, the laws of physics hold.  Now, consider world w₁* (see Figure 1).  This world matches ours until the end of time and starting from a time very shortly after the neuron’s firing in w₁, at which point a small miracle occurs that prevents the neuron’s firing from having those consequences which in our world would be physically necessitated.  And then prior to the miracle, the universe satisfies the usual deterministic laws of nature (one can calculate backwards what the universe should be like prior to the miracle given its posited state at the beginning of the miracle).  Alternately, we can describe w₁* by taking w₀, reversing time, inserting a miracle at an appropriate time which miracle ensures that a time-reverse of N occurs shortly after the miracle, then letting things evolve forwards deterministically, and finally taking the time-reverse of the whole universe.  This re-description shows that in fact w₁* is formed from w₀ in pretty much the same way as w₁ was, except with time “thought of” as running backwards.  Which now is closer to w₀:  is it w₁ or is it w₁*? 

If it is not the case that w₁ is closer (i.e., if w₁* is equidistant or closer), then the counterfactual “Were N to happen, the world would blow up” will be false by Lewis’s criterion, since the consequent is false in w₁*.  It may intuitively seem that w₁ needs to be closer to w₀.  After all, surely, it has a smaller miracle in it.  For whereas in w₁ the miraculous event is merely the firing of one neuron, in w₁* there is a miraculous prevention of the multitude of the natural consequences of the neuron’s firing.  However, this is a misunderstanding of what the miracle in w₁ is.  It is not the firing of the neuron per se that is miraculous.  Rather, what is miraculous is that the neuron fires without the natural causal prerequisites obtaining.  The miracle is in the severing of the lawful connection between N and the multitude of prior events (perhaps such events as charge being built up in the neuron, the neuron being properly oriented in space vis-à-vis other neurons with appropriate charges, etc.) each of which would be needed in the ordinary course of physics in order that N might happen.  The miracle in w₁* is the miraculous severing of the lawful connection between N and the multitude of events posterior to it in the ordinary course of physical law.  In both cases, the miracle severs the connection between the single event N and a cluster of other events nomically connected to it.

Since one does sometimes talk of a miracle as consisting in an “event” like a neuron firing or a leper having become well, rather than in the severing of the nomic connections between this event and other ones that lawfully should be connected with it, whereas it is the violation of law that is relevant to Lewis’s account, perhaps one should eschew the term “miracle” in this case lest one fall into the mistake of thinking that severing connections with necessary causes is less of a miracle than severing connections with necessary effects.  However, in this paper I adhere to Lewis’s terminology, keeping constantly in mind that the claim that a “miracle” has occurred is supposed to draw one’s attention to the fact of a failure of the physical laws (of our world) as opposed to the event eventuated by this failure.  There is a difficult problem with how exactly the size of a miracle, the amount of severance of nomic connections, is to be measured in a time-reversal asymmetric way that should be further explored.[11]

“But it is clear that in w₁ it was just N that miraculously occurred, whereas in w₁*, many events miraculously occurred.”  This criticism simply rejects my analysis of the miracles as constituted by the severing of the nomic connections.  To someone who insists on such a rejection the answer is that she is simply building an arrow of time into the very notion of “the size of a miracle.”  For then in order to decide whether a given miracle is to count as the prevention of a number of effects (which on this objection counts as a greater miracle) or as the occurrence of a single event without its physically required cluster of causes one must have a distinction between cause and effect, which distinction at least in this case will surely be just another version of “the arrow of time”.  It is no surprise that if one reckons the sizes of miracles in a temporally asymmetric way, one will get a temporal asymmetry in counterfactuals out of a Lewisian analysis, but we must not attribute such reckoning to Lewis himself.

“There are many more almost immediate effects of the neuron firing than the number of causes in the relevant minimal jointly sufficient set of causes for it.”  This may be true in some cases.  However, it is not completely clear that it is true in general.  After all, quite a number of causes are required for the neuron to fire, causes quite possibly lacking in w₀.  Nixon might well have been thinking of completely different matters.  Prior to that neuron firing, maybe quite a number of changes in his brain would have been needed were the neuron to have fired in accordance with the laws of nature.  Wholly different neural pathways might have been needed in fact, if, say, Nixon’s character was dead-set against activating the doomsday device.  The firing of the neuron is miraculously severed in w₁* from all these physical preconditions.  Why should this be a smaller miracle than that of keeping the neuron, while firing, from generating the gravitational gradient propagation, electrical fields, etc., that it would normally produce?

At the very least, this shows that Lewis’s analysis would need to get into the nitty-gritty of the underlying physics for the case of counterfactuals based on events like N, since in the case of such events it is by no means obvious that his asymmetry between past and future obtains.  And it seems plausible that we could come up against cases of actually true counterfactuals in which, say, the firing of the neuron would involve so much of a departure from laws of physics because so many of the preconditions for the firing had not in fact obtained in the actual world that having the neuron fire while isolating it miraculously from the natural consequences of this firing would have been preferable for the purposes constructing a world closer to ours.  But then for such firings the asymmetry would fail or go the wrong way, and moreover Lewis’s analysis of counterfactuals would also fail—since his counterfactuals would then fail to display the temporal asymmetry that “real” counterfactuals (i.e., those in terms of which we normally talk) do.

V.  The general case

             We have seen in the previous two sections that there are fairly natural occurrences which, if we allow them to figure in the antecedents of counterfactuals, (a) may not allow the Lewisian analysis to give a correct evaluation of the truth values of these counterfactuals, and (b) may not disallow backtracking in the way Lewis would like to.  Nonetheless, one might hold that such antecedents are relatively rare, and hence that Lewis has identified a genuine asymmetry in time, albeit he has overstated his case: It is true, perhaps, for most antecedents of counterfactuals that there is an objective asymmetry between forwardtracking and backtracking, just as it is true for most systems that entropy increases even if it is not true for all of them.  We shall see that even this more cautious claim is not correct, but that the asymmetry in everyday counterfactuals for which Lewis’s account works is due to the fact that a time-reversal asymmetric anthropocentric interest is involved in picking out those everyday counterfactuals we tend to be interested in.

             The task now is to see for which events Lewis’s analysis has the most chance of working, by examining the general structure of Lewis’s argument in the case of the original counterfactual that Fine had proposed.  Generalizing, we have an event A (in Fine’s case, it is Nixon pressing the button) which does not occur in the actual world.  We want to analyze a counterfactual of the form “Were A to happen, then C would happen.”  For Lewis’s argument to work, the world closest to the actual world w₀ but in which A happens must be a world w₁ of the following kind.  Up to a time t fairly shortly before A, w₁ coincides with w₀.  Then a small miracle occurs, no greater than is needed to make A happen, and after this miracle the laws of physics valid in w₀ resume holding.

             But one could also form a different world w₁*.  This world would have the time-reverse of the structure that w₁ would have were w­₁ defined starting with the time-reverse of w₀ (and with respect to the time-reverse of A).  More precisely, in w₁*, from a time t* fairly shortly after A and until the end of time, w­_­1­­* coincides with w_0­.  Around t*, a miracle occurs, of the minimal size needed for A to occur, and prior to this miracle the laws of physics valid in w₀ hold sway.  More precisely, I shall say that a physical state of affairs A at time t₀ forwardtracks (respectively, backtracks) to a state of affairs B at a time t₁ after t₀ (respectively, a time t_–1 before t₀) providing the deterministic laws of nature are such that it is nomically impossible that A obtain at t₀ but B fail to obtain at t₁ (respectively, t_–1).  The bidirectional determinism assumed by Lewis and this paper then entails that the physical state of the universe as a whole at any given time forwardtracks to give the physical state of the universe at all future times and backtracks to give it at all past times.  This will be so, e.g., in a Newtonian universe.  Thus, w₁* prior to t* is formed by backtracking from the state of the universe at the beginning of the miracle, given our bidirectional determinism. 

It is very unfortunate that in Lewis’s panoply of possible worlds in which Nixon presses the button there is no mention of w₁*.  For in order that Lewis’s time asymmetry might be shown to work in the case of counterfactuals with antecedent A, and indeed in order for his analysis of counterfactuals with antecedent A to match the common use of counterfactuals which common use does not allow much or any backtracking, it must be established that some world of w₁-type is closer to w₀ than any world of w₁*-type.  For, if a world of w₁*-type is closer than any world of w­­­₁-type, then counterfactuals with antecedent A will backtrack since w₁*-type worlds disagree with w₀ in the past.  And if there is a tie between worlds of type w₁ and those of type w₁*, then counterfactuals with antecedent A will all be false if worlds of type w₁ and those of type w₁* disagree on the truth value of the counterfactuals’ consequents—as they will in most of the interesting cases.  Neglecting to consider w₁*-type worlds, as Lewis apparently did, is inexcusable since surely it is standard operating procedure in the arrow-of-time business to check what a given arrow would yield if its construction were applied to the time-reversed universe.

             Now what must be the case if w₁-type worlds are to be closer to w₀ than w₁*-type worlds?  First, note that if the universe has a finite past but an infinite future, then providing the miracle involved in the w₁*-type worlds is not much greater than that in the w₁-type worlds, surely w₁* will be the better match by Lewis’s criteria.  For, w₁* will exactly match w₀ throughout an infinite future time span, whereas w₁ will exactly match w₀ in only a finite time span.  Secondly, even if the universe has but a finite future, w₁* might be a better match if that future is expected to be significantly longer than the past, as indeed it is on scientific grounds expected to be since the universe is still expanding and presumably the collapse time would be comparable to the expansion time.  (It is, by the way, surely itself a reductio of Lewis’s account to think that such cosmological questions are even relevant to everyday counterfactuals.)

             So we have an initial consideration in favor of w₁* being closer to w₀.  However, this consideration may be trumped if significantly greater miracles are required in w₁* than in w₁.  In the case of the Nixon example, the miracle involved in w₁ can be taken to be quite small.  Nixon presumably was on the scene, and the manipulation of a bunch of neurons would make it be the case that he had pressed the button.  I.e., in this case if we simply go back in time a little prior to the time when we “want” A to happen, we can find a favorable world-situation in w₀ in which a small miracle will bring it about that shortly afterwards A will happen.  And it may be plausible that if we look at world-situations in w₀ after the time when we want A to happen, we will not find anywhere where a comparable small miracle could be placed from which one could backtrack (using backwards determinism) to get A.[12]

             Why is this plausible?  The plausibility, I take it, comes from the fact that we have a hard time imagining what the miracle which backtracks to Nixon pressing the button would look like.  Given that we have not managed to describe such a miracle, we have a (highly defeasible) reason to suppose that there is no small miracle that can do it, though if in the final analysis turns out that a sufficiently small miracle in the future of the button-press can do it, then Lewis’s account of this case will be falsified.  On the other hand, we have a fairly clear picture of what miracle forwardtracks to Nixon pressing the button—namely, Nixon’s neurons firing in a way that gets him to do it.  However, this availability of a situation in which some small miracle a very short time before A forwardtracks to A is a special feature of the particular counterfactual Lewis was dealing with.  For consider yet another modification of the original situation:

(B)   The button is located in a sealed container which it is beyond the technical power of us human beings to open or to manipulate the insides of.  As a matter of fact, the button is never pressed.  The counterfactual is now: “If the button had been in a depressed state, then a nuclear holocaust would have happened.”

             In case (B), we have a harder time picking out what “small” miracle prior to the depressing of the button would let one forwardtrack to a depression of the button.  Perhaps the air density over the button would miraculously increase and thereby depress the button.  This is not so small a miracle, of course: a lot of air molecules would have to be moved.  But most importantly, it is by no means obvious that a miracle located after the pressing of the button that backtracks to the depression of the button couldn’t be equally small.  In fact, it could even be a similar sort of miracle: just imagine time running backwards starting with the actual future of the world, and add a miraculous increase in the density of the air over the button.  So, case (B) is yet another case for which, quite possibly, Lewis’s analysis would fail, because the miracles involved in w₁* and w₁ could be of the same, or very similar, magnitude—and so w₁* would be the better match for w₀ given that it matches it throughout a future that is significantly longer than the past that w₁ matches w₀ in (since as far as we know, the future of the universe is longer, perhaps infinitely so, than the past).

             We are now in a position to see the difference between (B) and Fine’s example.  In Fine’s example there is an intuitive canonical miracle prior to event A (or more precisely a class of canonical miracles all of which are sufficiently similar to lead to the same results vis-à-vis the consequents of the counterfactuals we are interested in).  When asked to imagine the scenario of Nixon (contrary to fact) pressing the button, we imagine him thinking differently from the way he actually did, and the miraculous inducing of this act of him thinking in one of the ways that would lead to the button being pressed is the small miracle we are after.  We know roughly where and how to locate a miracle in the past of A: namely, by a modification of the mind of Nixon at the right time.  But when asked to imagine the scenario of the button in (B) becoming depressed, our intuition does not provide us with any such easy answer.  Maybe the right miracle is an increase in air density; or maybe, if the button and the console are made of metal, a spontaneous magnetization of the button, etc., etc.  There is no canonical location for the miracle in our world, and different ways of locating a miracle will lead to different answers to counterfactual questions.

             However, counterfactuals of Fine’s kind are arguably much more common in our daily reasoning than counterfactuals of form (B).  We often use such counterfactuals, and more generally such subjunctive conditionals, in our practical reasoning:

        Were I to choose A, C would result; but were I to choose B, D would result; I prefer D to C, so I choose A.

And in all counterfactuals that deal with practical reasoning, there is a clear place to put the “small miracle”—in the mind of the agent.  The counterfactuals and subjunctive conditionals in fact already come attached to a story about where the divergence in worlds can easily start. 

             It is true that we do sometimes ask hypothetical questions of merely theoretical interest like: “What would have happened had that button become depressed in case (B)?”  Of course, we ask such theoretical counterfactual questions more rarely than we ask the practical ones.  But even when we do ask such theoretically-oriented questions, we often have in mind some story as to how the antecedent of the counterfactual “could have come about”.  In cases where the antecedent is a human action, the story typically involves the person’s having made a different act of will.  But even in other cases, we tend to have a story in our minds as to how the antecedent “could have come about”.  Otherwise, we would (a) be liable to think the counterfactual question to be ill-defined, and (b) have little reason to ask the question in the first place.  For, leaving aside per impossibile counterfactuals which Lewis’s theory does not handle[13] and each of which may have to be treated sui generis[14], to ask the question in everyday life, we have to think of the antecedent in the conditional as having been a real possibility.  And thinking of something as a real possibility tends to involve having some story about how it “could come about”.[15]  Moreover, without such a story the relevant truth-value of the counterfactual might well be impossible to determine.  In the case of (B), if our story about how the button is most easily pressed involves a massive earthquake that shakes up the box, it might not follow from the pressing that the world blows up—for the earthquake could also destroy the wire connecting the box to the doomsday machine.  This is not a verificationist objection: the point is simply that in practice we have no difficulty determining the relevant truth value of the counterfactual, and hence there is something wrong with an account which either makes this truth value undefined or inaccessible to us.

             If we ask counterfactual questions where our listener has no story in mind as to how the antecedent could have come about, we are liable to meet with puzzlement.  “Were Queen Victoria to be alive today, what would she be doing?”  The joke answer is: “Scratching at the inside of her coffin.”[16]  But actually, the meaning of the question is not clear unless it is specified what miracle story about Queen Victoria’s survival is the relevant one for consideration.  E.g., is it that she became rejuvenated in the coffin, or is it that she just never died but lived an extra long time?  The answer will be very different in both stories.  An intelligent questioner knows this and that is why she is unlikely to ask seriously a counterfactual question without specifying some process by which the antecedent could come about, unless of course the context clearly singles out a process in virtue of that process being particularly simple, canonical or “obvious”—as in the case of the process of the button being pressed by Nixon through the firing of the neurons in his brain.

             There are two kinds of intelligent counterfactual questions that might come up in everyday discourse.  The first explicitly specifies in the antecedent what process for making the antecedent true is meant (“Were Queen Victoria to be alive today as a result of her being rejuvenated in the grave…”).  The second will be one where there is an “obvious” simplest process, which simplest process is presumably the one that requires the least difference from this world (“Were Nixon to have pressed the button…”).  Actually, the two kinds of counterfactuals are not disjoint.  The process in cases of the first kind will almost always be specified incompletely, with the gaps being left to be filled in some “obvious” simplest possible way (e.g., the supposition that Queen Victoria rejuvenated in the grave ten years ago and miraculously sustained there so that she would by now be tired of clawing at the inside of her coffin is not the simplest hypothesis as it involves two miracles).  And so most if not all cases of the first kind will in fact be cases of the second kind, with the partial process description being a part of the antecedent but still not sufficient to fix the process entirely.  Moreover, both kinds of counterfactuals have as a common feature the fact that the process for producing the antecedent, whether implicit or explicitly stated, is one that is in the past of or contemporaneous with the event that is the primary concern of the antecedent.  If from looking at the antecedent one cannot get an obvious process that could have effected the antecedent, then the intelligent questioner, in every-day cases, is unlikely to ask the counterfactual, and in every-day cases when we look for processes for effecting events we never look for them in these events’ future.

             It is has long been known that the problem with counterfactuals is in determining which features of the actual world are to be retained the counterfactual world.  In the case of our ordinary thinking about counterfactuals, it is natural to locate, with some vagueness, the first event in respect of which the counterfactual world is supposed to diverge from the actual world, and then to consider how the divergence causally propagates as a result of this event.  While it is not easy to make this intuition precise and to give an account of how we pick out the initial divergent event[17], it is plausible that this is indeed how we think about those counterfactuals that we meet most commonly in daily life.  But if this is so, then the crux of the problem of specifying what to retain in the counterfactual world is intuitively that of specifying where the divergence is to take place, i.e., of specifying how the antecedent was to have been produced, since the antecedent is supposed to be a consequence of that first divergence.

             If I am right, then those counterfactuals about which we are most likely to ask in everyday life are such that there is an “obvious” process in the past of or contemporaneous with the occurrences in the antecedent which process would bring about the antecedent A.  The “obviousness” of the process means that the process cannot deviate too wildly from physical law, since where a divergence is too great, we are puzzled as to which miraculous process is meant as in the case of the Queen Victoria counterfactual.  Hence, a Lewisian miracle bringing about this process would not be too big, and so w₁ will be close to w₀.  But except in special cases, there is no equally “obvious” temporally retrograde process starting in the future of A and backtracking to A.  Hence, it is unlikely that there be a small future miracle that backtracks to A.

             I will now give a plausibility argument for this last claim.  Consider the class E of all non-actual events that can figure as possible antecedents of counterfactuals (for simplicity, I now allow negative states of affairs to count as events).  We will subdivide E into four subclasses.  For each event A in E we can form the worlds (or classes of worlds) w₁ and w₁* following the recipes near the beginning of this section.  The closeness between w₀ and w₁ corresponds to the size of the miracle in w­₁, and a corresponding thing can be said about the closeness between w₀ and w₁*.  Fix some standard of being “quite close” between worlds.  Let E₁ be the class of all events A in E such that neither of the worlds w₁ nor w₁* constructed starting with A is “quite close” to w₀.  Let E₂ be the class of all events A in E such that w₁ is “quite close” to w₀ but w₁* is not.  Let E₃ be the class of all events A in E such that w₁* is “quite close” to w₀ but w₁ is not.  And finally let E₄ be the class of all events A in E such that the corresponding worlds w₁ and w₁* are both “quite close” to w₀. 

I now claim that there is an intuitive sense in which E₁ is a much larger class than E₂, E₃ and E₄ taken all together, so that a non-actual event chosen “at random”, i.e., an event otherwise unspecified, is most likely going to be in E₁.  For, given a randomly chosen non-actual event A it is unlikely that the event should differ in exactly such a way from the events of the actual world that only a “quite small” miracle in the past of A and also a “quite small” miracle in the future of A would suffice to change the actual world into one where A occurs. It would seem likely that a randomly chosen event would require quite a lot of modification of the actual world in order to engineer the event’s production.  The event A when chosen “at random” is likely after all to represent a very different state from the one the actual universe has.[18]  It would thus take careful preselection of events to make sure they fall into one of E₂, E₃ and E₄.  Moreover, and for a similar reason, E₄ is intuitively much smaller than E₂ (i.e., it is much less probable that a random event falls in it than in E₂).  For, being a member of E₂ È E₄ requires merely that there be a possible small miracle in the recent past which could bring about the event, whereas being a member of E₄ requires additionally that there be a possible small miracle in the near future which could backtrack deterministically to the event.  If it takes careful preselection to make sure that at least one of a past or a future miracle can yield the event, it intuitively takes even more careful preselection to make sure that both can.  Hence, E₄ is much smaller than E₂ È E₄, and thus is still significantly smaller than E₂.

Now, if we preselect our counterfactuals to have as antecedents those events A for which there exists a “quite small” miracle in the past of A which could bring it about that A occurs, our preselection ensures we only consider the small minority of events that fall in the union of E₂ and E₄.  Since E₄ is intuitively significantly smaller than E₂, it follows that it is likely that any given event thus preselected in fact falls in E₂.  And for events from E₂, Lewis’s counterfactual time-reversal asymmetry holds.  Hence, for most events preselected the way common sense preselects them (namely those which could be brought about by a miracle in their past, since common sense does not consider the possibility of the availability of processes taking place in the future of an event for the production of that event), Lewis’s asymmetry holds.

             This however is not an objective temporal asymmetry since it depends on the rigging or preselection of cases.  This preselection is based on the ordinary intuition that to engineer an event, one must make a change in the event’s past, and on the fact that we tend to ask counterfactual questions only when we can imagine a “not too wild” process, i.e., a “quite small miracle”, that could have brought about or engineered the event. 

             Without such preselection, Lewis’s account fails because it is by no means guaranteed that w₁ is a better match for w₀ than w₁* is.  Given an event A without the described preselection, it seems prima facie just as likely that a small miracle in A’s future (or, more precisely, in the future of the point in time where one would like A to occur) would backtrack to A as it is that a small miracle in A’s past would forwardtrack to A.  So there is no temporal asymmetry, or if there is one, then for events happening in our time it points the wrong way because the fact that the future of the universe is longer than the current past gives one a strong consideration in favor of w₁* as a closer match to w₁, as already discussed.  We see this in case (B), and we can also see it in the case of event N discussed in Section IV.

             We also see a partial lack of such preselection, though in a somewhat different way, in the example discussed in case (A).  There, given how great a miracle would be required to override the Soviet cable cutting machines, the Lewisian will initially be at a bit of a loss as to where the miracle should be placed—in the machines or in Nixon’s mind—and, it has been argued, will implausibly have to end up having to countenance the counterintuitive backtracking to Nixon’s own decision.  Had the counterfactual in (A) been posed, however, in a more natural way as

(C)   Had none of the Soviet devices gone off after Nixon’s pressing of the button, then a nuclear holocaust would have ensued

there would have been no problem with a Lewisian analysis.  There would then have been an obvious and canonical place where the miracle should be located—in the devices, prior to the time when in the actual world they cut the cable.  And then the counterfactual would not exhibit the extended backtracking that case (A) gave.  However, counterfactual (C) is preselected in the way I described above;  it is formulated in such a way as to determine that the process that would make the antecedent true is to involve a malfunction in the Soviet devices. 

             The preselection explanation of why Lewis’s theory usually works has at least a superficial similarity to manipulability or interventionist accounts of causation.  As Woodward (2001, Section 10) has noted, the Lewisian small miracle account of counterfactuals is indeed strikingly similar to the manipulationist idea that Y causally depends on X provided that a surgical intervention that changes X would lead to Y being changed.  Seeing Lewis’s possible worlds account of counterfactuals view as applicable precisely when one can specify a localized miraculous intervention temporally prior to the event in the antecedent, an intervention that brings about this event will increase the plausibility of the interventionist’s insistence that considerations of prior interventions are relevant to causal and counterfactual phenomena, though the exact relationship between the two views remains to be explored further.  Unfortunately, it is not clear how well the interventionist account can be made to work in a deterministic setting, given that the connection between Y and the X changed by the intervention is supposed to be nomic.[19]  As in the criticism of Bennett in Section II and the criticisms of Lewis throughout, in general in evaluating counterfactuals in a deterministic setting, the availability of a time-reversal-symmetric method of determining in which spatio-temporal regions nomicity is to be preserved in the counterfactual world is unlikely.  One could always specify this in a time-reversal asymmetric manner, ensuring that the nomic connections between events in the future of the intervention are preserved, but then of course one could not use interventionism to ground an objective arrow of time.

             Finally, note that the preselection story may not apply in the same way to all counterfactuals we make use of.  One referee suggested the counterfactual: “If Jupiter did not exist, the Earth would be bombarded by many more asteroids than it is.”  To evaluate this counterfactual, it is claimed, we do not need a story about how Jupiter was to have been removed from existence.  As it happens, the example is unfortunate.  For in fact, leading theories of asteroid formation involve Jupiter’s gravitational field, or at least the gravitational field of Jupiter at a late stage in its development.[20]  If these theories are correct, then if Jupiter never existed, asteroids would not have existed either, and the referee’s counterfactual comes out false if the process because of which Jupiter was not to have existed was to have taken place before the time when Jupiter’s gravitational field was responsible for the existence of asteroids, unless this process were to have involved blowing up Jupiter in such a way that the pieces would themselves become asteroids.  On the other hand, if the process by which Jupiter was to be made not to exist happened some time later, then the counterfactual will be true if Jupiter’s gravitational field protects the earth from asteroid impact. 

             Of course, other examples can be manufactured.  It might have turned out that Jupiter was irrelevant to the formation of the asteroids.  In that case, a precise story of how the antecedent of the counterfactual was to be brought about would not have been needed.  But this would only be so because the truth of the consequent happened to be the same for all possible such stories, and so each of the counterfactuals obtained by precisifying the antecedent would have had the same truth value.  Nonetheless, there would in general be other counterfactuals with the same antecedent whose truth value would depend on how Jupiter was to have been removed from consideration. 

             A different kind of astronomical example is given by another referee.  We can derive Kepler’s Laws from Newtonian universal gravitation by asking: “How would the planets move if the only forces acting in the solar system were gravitational forces between the planet and the sun (i.e., with no interplanetary forces)?”  But we have no story about how the antecedent could have come about.  However, we need further details.  Did the interplanetary forces suddenly cease to have effect?  Or were there never any?  In the latter case, the planets would not be moving in any way, because there would not be any planets, since planets in fact were formed by gravitational accretion.  In the former case, while we do not have a story about how the antecedent was wrought, we do have a description of the process leading up to it: At some time t₀, gravity ceased to hold except between each planet and the sun.  But probably neither scenario is what the astronomer considering this counterfactual is looking at.  Rather the astronomer is more likely to have in mind a universe where the planets have always been moving in accordance with a law according to which the only force is gravity between a planet and the sun. 

             But in any case, the antecedent of the counterfactual in this example is contextually specified to such a degree that the antecedent entails that Kepler’s laws hold.  In the case of a counterfactual which is actually an entailment, where it is logically necessary that the consequent hold if the antecedent does, it is not surprising that we do not need to specify the process by which the antecedent came about.  Lewis’s theory works for such a counterfactual, because the consequent holds at each world at which antecedent holds.  But what we are dealing with is a different kind of counterfactual from that in the kinds of everyday counterfactuals Lewis was interested in for the production of the arrow of time.  The antecedent in this example is not a temporally constrained event, for instance, or even the non-occurrence of such an event.

             And in any case there is no reason for me to have to claim that my analysis works for all counterfactuals.  Remember that we are examining whether there is some objective time-reversal asymmetry present in the majority of counterfactuals that would let one at least derive an objective arrow of time from Lewis’s account.  I grant to Lewis, for reasons discussed above, that his analysis may well apply to the majority of everyday cases, but argue that because of the anthropocentric preselection involved in the “everyday” this does not yield an objective arrow of time.

VI.  Yet another counterexample

             It might be argued that examples (A) and (B) are much too contrived because they are set in worlds significantly different from ours, and that (A) as well as the work of Elga (forthcoming) are flawed through having a negative antecedent.  However, for a more easily imaginable case, one that could easily be physically constructed, consider the following more natural example.  It is highly likely that relevantly similar cases have in fact occurred.

(D)   In the actual world, for several months prior to t₀ a one kilogram steel ball has been hanging from a strong plastic cable at a height of one meter above a highly elastic rubber pad.  At t₀, the cable is severed in a highly overdetermined way (e.g., several oxyacetylene torches are turned on it), and the ball falls.  Moreover, things are set up so that it would have been very difficult for the cable to be cut prior to t₀ (e.g., the only cutting/burning implements were not available prior to t₀) so that a large prior miracle would have been required for its cutting.  The pad is so elastic that the ball will rebound to a height of 0.9 meters.  For simplicity of calculation, suppose all this is done in a vacuum.

Assuming an acceleration of gravity of 9.8 meters / second², the ball will take about 0.45 seconds to fall.  Now, let t₁ be a certain specific time which I will specify more precisely later, but about which I will now only say that it is a time between t₀ and the ball’s impact on the pad.  Let A be the (temporally extended) event of the ball being suspended at a height of 0.8 meters at t₁ and of not having been above height 0.8 meters at any time in the two seconds preceding t₁. 

             Observe that whatever the time t₁ between t₀ and the impact is, event A is a rather difficult one to produce by a miracle in its past.  One way to produce the event would have been to cut the cable prior to t₀ and then to arrest the ball at height 0.8 meters and suspend it miraculously there for two seconds.  However, it was assumed that it was hard to produce a miraculous severing of the cable prior to t₀.  Alternately, one might backtrack several months, tie the ball at a lower height of 0.8 meters, and modify the whole set-up so the cable is cut at t₁ instead of at t₀.  However, that would involve a miracle quite long before t₀ and might require a large one if the whole set-up was overdetermined.

             On the other hand, if t₁ is chosen appropriately, it is easy to find a small miracle in the future of A that backtracks to A.  Let t₂ be the time of maximum impact of the ball on the pad in the actual world, that is the time at which the pad is maximally compressed.  Since the ball will rebound to 90% of its original height, the compressed pad at t₂ stores about 90% of the gravitational potential energy the steel ball (assuming the steel ball does not itself become compressed) had at its release.  The rest of the energy went into such things as the vibrations of the laboratory floor and heat.  Consider the time-reverse, w₀^†, of the actual world w₀.  In w₀^†, we have heat energy and vibrations of the laboratory floor which together with energy stored in a compressed pad conspire to propel a steel ball that at t₂^† is lying on the compressed pad up to a height of one meter at a later time t₀^†.  Suppose that in w₀^† we added a small localized miracle prior to t₂^† that decreased the energy stored in the compression of the rubber pad by 22%, or redirected this amount of energy away from the steel ball (e.g., by directing it into the ground).  Call the modified world w₁*^†;  this world matches w₀^† in its whole past up to shortly before t₂^†.  Then, the heat energy and vibrations of the laboratory floor which contain 10% of the energy needed to lift the steel ball to height one meter are still going to transmit energy to the ball, but the pad will now only transmit (1–0.22)(90%) » 70% of the energy the steel ball would need to be raised to that height.  Hence, the ball in w₁*^† will be raised merely to a height of 70%+10% of one meter, namely to a height of 0.8 meters.  It will achieve this height at some time t₁^† shortly prior to t₀^†, and it will then presumably start falling after t₁^† so that it will never be at a height other than 0.8 meters.

             Now, let w₁* be the time reverse of w₁*^†.  In this world, at a time t₁ shortly after t₀ the ball will have zero velocity and be at height 0.8 meters.  Note that this sentence can be used to precisely define what t₁ is;  for other values of t₁, we would have to miraculously change energies by different amounts.  The ball will then fall on the rubber pad.  Moreover, in the two (or even more!) seconds prior to t₁, the ball will not have been above height 0.8 meters.  Hence, A occurs in w₁*.  There is a small miracle after t₂ in w₁* which backtracks to the occurrence of A, and after that miracle w₁* matches the actual world exactly.  The miracle in question is a small one: it is just the decrease of energy in a compressed rubber pad (note how the compression of the rubber pad makes the miracle even more localized).  So, A can be produced by a small backtracking miracle but it does not appear likely it can be produced by a small forward-tracking miracle.  Therefore, Lewis’s framework for counterfactuals will produce many non-trivial true backtracking counterfactuals with antecedent A, since the past of w₁* presumably is never exactly like the past of w₁.  This is not only absurd, but exhibits the opposite of the arrow of time Lewis tried to exhibit.

             It might be objected that there were some idealizations in this account.  For instance, I assume the experiment was done in a vacuum and that even if we decreased the energy of the rubber pad in the time-reverse world w₀^†, still the energy in the vibrations of the floor and in the heat would go to propel the steel ball.  These assumptions, however, are not essential to the example.  Suppose that these idealizations are false.  Nonetheless, miraculously decreasing the energy of the rubber pad in w₀^† by 22% around t₂^† and then forward-tracking deterministically will have to (by energy conservation) produce a world where the steel ball will achieve a maximum height lower than one meter at a time t₁^† in the future of t₂^†.  Let the height that will be achieved be h.  Perhaps h is not exactly 0.8 meters.  But whatever it is, we can then define the event A as being the event of the ball being suspended at a height h at t₁ and of not having been above height h meters at any time in the two seconds preceding t₁, where t₁ is the time corresponding to t₁^† in the non-time-reversed world, and the analysis continues to go through.  All that matters is that in the time-reversed world if we decrease the energy in the compressed pad, which is a small localized miracle, we will ensure the ball will achieve a lower height.

             Observe that the counterfactuals with A in the antecedent would not backtrack if we specified how A was to have been brought about, as suggested in Section VI.  For instance if we suppose that A was in fact brought about by the cable’s having originally been 0.2 meters longer than in the actual world and by the ball’s having been attached to the cable (and thus having been 0.2 meters lower), and build this into the antecedent of the counterfactual, the backtracking counterfactuals become less plausible, since we would need to come up with a backtracking miracle that, in the time-reverse of the counterfactual world, lengthens the cable and ensures that the ball successfully gets attached to it.  And that would indeed probably require a large miracle.

VII.  Conclusions

             The most serious defect that Lewis’s analysis of Fine’s example suffers from is its complete neglect of consideration of worlds of the form w₁* which match ours in respect of the future shortly after the event of Nixon’s pressing of the button, but which include a miracle shortly after that pressing from which one can backtrack to the pressing itself.  While this neglect might not affect Lewis’s analysis of the particular case Fine challenged him with, it does affect the analysis of other cases. 

             Lewis (1979, pp. 473–475) thought that the temporal asymmetry that he had found was based on the contingent fact that overdetermination of the past by the future was more common than overdetermination of the future by the past.  He proceeded by comparing world w₁ with world w₄ which included two miracles, one of which was designed to erase the future effects of the first, and noting that this second miracle would have to be comparatively very large.  Price (1997, p. 151) has suggested that at the microscopic level the asymmetry may disappear.  In any case, the Lewisian analysis of the temporal asymmetry in ordinary counterfactuals like “Had Nixon pressed the button, the world would have been destroyed” where we do expect the consequent to be in the antecedent’s future cannot be sufficient.

             For, the most relevant counterfactual worlds to compare to the actual world in a Lewisian setting are not w₁ and w₄, but w₁ and w₁*, and so what one requires to get the temporal asymmetry is a condition to the effect that a lesser miracle is needed if we put the miracle in the past of the event described in the antecedent of the counterfactual, from which miracle we can deterministically forwardtrack to the antecedent’s event, than if we put the miracle in the future of the antecedent’s event and backtrack. 

             Lewis could bring in here an overdetermination asymmetry according to which a single actual event is overdetermined by a number of future events but not by a number of past events.

Whatever goes on leaves widespread and varied traces at future times.  Most of these traces are so minute or so dispersed or so complicated that no human detective could ever read them;  but no matter, so long as they exist.  It is plausible that very many simultaneous disjoint combinations of traces of any present fact are determinants thereof;  there is no lawful way for the combination to have come about in the absence of the fact.  (Lewis, 1986, p. 50.)

As an example, Lewis alludes to the phenomenon of the spreading of spherical waves spreading from a point: “Countless tiny samples of the wave each determine what happens at the space-time point where the wave is emitted or absorbed” (ibid.)  If Lewis were right, then a miracle that kept the future constant would have to cut each of the nomic connections between a past event and its future overdetermining events, and that would involve a large-scale miracle.  However, Lewis must be wrong, since we have already seen in case (B) that a single small miracle in the future of the button-press can backtrack to a button-press.

             But one can do better than just saying “Lewis must be wrong”.  In fact, one can show that the overdetermination does not occur even in the paradigmatic case of the spreading of a spherical wave.  Suppose that some event happens at t₀ in the actual world such that an amount E of energy is released and starts to spread as a spherical wave, and for simplicity suppose that in the region in question this is the only relevant source of energy.  Lewis’s claim is that the release of the energy is overdetermined by various disjoint samples of the spherical wave at some time t₁ after t₀.  But this is false due to energy considerations.  For consider the allegedly overdetermining disjoint parts S₁,S₂,…,S_n of the occurrence of the spherical wave at t₁.  The S_i are events occurring in disjoint areas of space-time, and their energy comes from the originating event at t₀.  According to Lewis, none of these samples can be present without the originating event.  Now, consider the system as a whole at t₁.  By conservation of energy, the total energy of the system is E.  Each of the S_i carries some non-zero portion E_i of the energy released by the originating event.

             If Lewis is right about his overdetermination claim, then it is nomically impossible that, say, S₁ occur at t₁ without the originating event occurring at t₀, and in particular without amount E of energy being released at t₀.  However, consider a world whose state at t₁ is just like the state of the actual world, except that S_n is replaced by an event T_n of strictly smaller energy.  Then by energy conservation, what happens at t₁ cannot backtrack nomically to the release of amount E of energy at t₀, since we have ensured that the amount of energy at t₁ is less than E, even though all but one of the allegedly overdetermining events S₁,S₂,…,S_n occur.

             We can be more explicit about the construction of the world in question.  Take the time-reverse of the actual world, letting t₀^† and t₁^† be the analogues in the time-reversed world of times t₀ and t₁, respectively, so that t₁^† < t₀^†.  At t₁^† we have a number of events S₁^†,S₂^†,…,S_n^† which are the time-reverses of S₁,S₂,…,S_n.  At t₀^† an amount E of energy is absorbed.  Now suppose that by a miracle occurring shortly before t₁^† we replace S_n^† by an event T_n^† of lower energy, and then we evolve the system until t₀^†.  Since all the energy (or at least all of the relevant form of energy) in the events S_i comes from the originating event at t₀, likewise all the energy in the events S_i^† is absorbed in the event at t₀^†.  If we replace S_n^† by a lower energy event, then the amount of energy available to be absorbed at t₀^† will be lower.  Hence, if we replace S_n^† by T_n^†, then at t₀^† there will be an amount E₁ < E of energy absorbed.  Now take the world, w^†, where this replacement happens, and reverse time once again to get a new world w.  Then, in w, at t₀ there will be a release only of an amount E₁ of energy, and events S₁,S₂,…,S_n–1 will occur but S_­n will be replaced by a lower energy event T_n.  Moreover, all the right nomic connections hold in the interval between times t₀ and t₁ since in w^† all the right nomic connections held between times t₁^† and t₀^†.  Therefore, it is not the case that the occurrence of S₁ nomically necessitates the occurrence of a release of amount E of energy at t₀.  Indeed, not even the occurrence of all of the events S₁, S₂ through S_n–1 necessitates that, simply because these events do not carry enough energy to necessitate this.

             We have argued that in fact the asymmetry in Lewis’s analysis of everyday contingent counterfactuals, if and insofar as there is any asymmetry, comes from the fact that everyday contingent counterfactuals are usually preselected for having antecedents which can be effected by easily imaginable processes acting in the antecedent’s past.  But this does not reveal an objective asymmetry that would be independent of people’s intuitions that it is past processes that cause future events rather than future processes that cause past events.  For those very intuitions are the psychological ground of the preselection of everyday counterfactuals, which preselection yields the asymmetry in the counterfactuals if we analyze the counterfactuals in a Lewisian way.  If Lewis’s argument could be taken to be an analysis of our everyday intuitions, this would not be so bad.  However, it cannot be thought of as such an analysis, because it crucially depends on the fact that any localized event normally has many, often tiny, effects throughout a large spatio-temporal region (e.g., Nixon’s button press, even if no nuclear holocaust happens, affects the gravitational field in China), whereas our everyday intuitions about counterfactuals do not depend on this.[21]

             And once we depart from the realm of most everyday counterfactuals, we can find a number of cases where Lewis’s analysis breaks down.  If the antecedent of the counterfactual is something like a neuron firing (the case in Section IV), or a button depressing in a locked box, the Lewisian analysis fails to demonstrate a past-future asymmetry grounding our view of the future as open and of the past as closed.  Moreover, as the work of Elga (forthcoming) and Section VI shows, one can construct fairly ordinary counterfactuals where Lewis’s analysis breaks down, too.  It may be, however, that in some sense for most everyday counterfactuals the Lewisian analysis holds, but the time-reversal asymmetric anthropocentric interests in preselecting which counterfactuals we use on an everyday basis preclude this from being an objective asymmetry.

             None of this denies (a) that there may well be an objective basis for asymmetry in all counterfactuals, even though Lewis’s analysis has failed to give a proper objective grounding to it except in some preselected cases, (b) that this asymmetry may well be responsible for our view of the future as open and the past as closed, and (c) that there may be an asymmetry in causal overdetermination, even though Lewis’s analysis in the end fails to connect this up with the asymmetry in (a).  The asymmetry in counterfactuals remains unexplained—unless one can reduce it to the asymmetry in causation where past events are causes of future events but not the other way around.  If one is satisfied with taking the asymmetry in causation as basic—as opposed to wanting to make sense of it in terms of the counterfactual asymmetry of time—then perhaps the fact of there being such preselection rules for antecedents of counterfactuals as ordinary usage puts in place is adequately grounded in the objective reality of world, since as we have seen it is grounded in our intuition about the causal asymmetry.  But like the interventionist, a person satisfied with this would be running a project that is the opposite of Lewis’s: causal relations would be seen as logically prior to counterfactual ones.[22]

Notes

Bibliography

Bennett, Jonathan (1984), “Counterfactuals and temporal direction”, Philosophical Review, 93, 57–91.

Edginton, Dorothy (1995), “On conditionals”, Mind, 104, 235-329.

Elga, Adam (forthcoming) “Statistical mechanics and the asymmetry of counterfactual dependence”, Philosophy of Science (supp. vol., PSA 2000).

Encyclopedia Britannica (2001), online at: http://search.eb.com.

Fine, Kit (1975), Review of David Lewis, Counterfactuals (Oxford: Blackwell, 1973), Mind, 84, 451–458.

Hausman, Daniel M. (1986), “Causation and experimentation”, American Philosophical Quarterly, 23, 143–154.

Kutach, Douglas (forthcoming), “The entropy theory of counterfactuals”, Philosophy of Science.

Lewis, David (1979), “Counterfactual dependence and time’s arrow”, Nous, 13, 455–476.

Lewis, David (1986), On the Plurality of Worlds, Oxford: Blackwell.

Mackie, Penelope (1998), “Identity, time, and necessity”, Proceedings of the Aristotelian Society, 98, 59-78.

McCall, Storrs (1984), “Counterfactuals based on real possible worlds”, Noûs, 18, 463-478.

Place, Ullin T (1997), “‘De re’ modality without possible worlds”, Acta Analytica, 129-143.

Sklar, Lawrence (1993), Physics and Chance: Philosophical Issues in the Foundations of Mechanics, New York / Cambridge: Cambridge University Press.

Price, Huw (1997), Time’s Arrow and Archimedes’ Point: New Directions for the Physics of Time, New York / Oxford: Oxford University Press.

Pruss, Alexander R. (2002), “The actual and the possible”, in: Richard M. Gale (ed.), Blackwell Guide to Metaphysics, Oxford: Blackwell, pp. 317–333.

Rescher, Nicholas and Brandom, Robert (1980), The Logic of Inconsistency, Oxford: Blackwell.

Woodward, James (2001), “Causation and manipulability”, Stanford Encyclopedia of Philosophy, online at http://plato.stanford.edu.