A lot of people have mixed feelings about the counterfactual theory of causation.  Yes, it is better than any other theory yet devised. But beating the competition is not really what we want from a theory.  What we want is that the theory should give necessary and sufficient conditions for the phenomenon it's a theory of. No one has found a version of the counterfactual theory that does either of these things.

No existing version is necessary, because when a cause has "understudies" -- events that would have caused the effect in the real cause's absence -- there is causation without counterfactual dependence. This is the problem of preemption.  No existing version is sufficient, because existing versions have it that c causes e if any of its effects cause e.  And there is plenty of evidence that c needn't cause e in such circumstances.  This is the problem of transitivity.

The claim here will be that both problems have the same solution.  Take first preemption. I say that something very like counterfactual dependence is necessary for causation.  Yes, e would still have occurred even if c had not.  But that's because circumstances would have been different if c had not occurred; back-up mechanisms that in reality misfired would have stepped into the breach.  It remains that "given how things actually went down,"  e would not have occurred if c had not.  That things would have gone down differently, if c had not occurred, obscures this fact but cannot change it.  There's a question of course of what has to be held fixed between the counterfactual situation and the actual one; more on that later.  Calling dependence with the right things held fixed de facto dependence, I say that that de facto (df) dependence is required for causation, even if counterfactual (cf) dependence is not

What about the second problem, regarding transitivity? The only reason it arises is that Lewis, in the course of a quite different (and ultimately unsuccessful) attack on the preemption problem, switched his analysans from dependence to dependence*. Ancestral relations are always transitive; so causation defined as dependence* comes out transitive too. If I am right that preemption is to be addressed not by ancestralizing dependence but by qualifying it --  by switching to dependence-certain-things-held-fixed  -- then there is no longer any expectation of transitivity, and so the problem goes away.

Some of these ideas go back to my dissertation Things (Berkeley, 1986).  Mixed in with a lot of other, well, things, was the thought that you could deal with preemption worries by defining c as a cause of e iff, letting X stand for relevant aspects of the obtaining circumstances,

II. Failures of Necessity

The counterfactual theory comes in many shapes and sizes. But in simplest form -- the form in which Hume stated it -- it says that c is a cause of e iff c and e both occur, and the occurrence of e counterfactually depends on that of c:
 

(CF1) O(c) & O(e) & (¬O(c) => ¬O(e))

Unfortunately, and this was appreciated from the start, effects don't always depend on their causes. Sometimes the (actual) cause preempts a would-be cause that in the actual cause's absence would have led to the effect anyway. So counterfactual dependence is not necessary for c to count as a cause of e. This is the problem of causal preemption.

Example: A couple of characters Hit and Miss both roll bowling balls down the lane. Hit's heavier ball deflects Miss's ball en route to the pin. Hit's throw (h) caused the pin to fall (e). But there is no dependence since if h had not occurred, e would still have occurred due to a chain of events initiated by Miss's roll (m).

This is where Lewis makes his best-known contribution. Notice something about the chain of events initiated by Miss's throw, he says. It was "cut off" before it had a chance to reach the pin. The pin's falling over may not depend on the elements of Hit's chain occurring before the cut-off point, but it does depend on those occurring after it  -- for past that point, Miss's chain is dead and buried. But now, these after-the-cut-off events depend in turn on Hit's throw. So the effect depends on something that depends on Hit's throw. It follows that if we understand causation as direct dependence or indirect dependence (via intermediaries), the counterexample no longer works. This leads Lewis to define causation as dependence*, the ancestral of direct dependence. Dependence* is the relation that c bears to e iff
 

(CF2) O(c)&O(d₁) &O(d₂)&..... &O(d_n)&O(e)

III. Failures of Sufficiency

If causation were really dependence* (the ancestral of dependence), then like dependence* it would be transitive. Whenever c caused d and d caused e, it would also be the case that c caused e. Counterexamples to the transitivity thesis have been accumulating for a while. Here is one adapted from Laurie Paul:

A second example is due to Hartry Field. Someone puts a bomb under Suzyís chair; later, Suzy notices the bomb and flees the room; later still, Suzy has a medical checkup (it was already arranged) and is pronounced healthy. The introduction of the bomb played a role in Suzyís fleeing, and her fleeing was a factor in her subsequent good health; but her good health was not in any way due to the bombís being put under her chair.

A case finally of Ned Hallís. Cultists half-way around the world are instructed to assassinate you; thatís the first event. Security officers notice the cultists at the the airport and arrest them; thatís the second event. You spend a quiet evening at home; thatís the third event. The third event counterfactually depends on the second, and the second on the first. But it is hard to believe that your quiet evening at home is in any way causally beholden to the assassination orders.
 

IV. De Facto Dependence

It is an unfortunate accident that Lewis's move to dependence* works as well as it does. I say "unfortunate" because it encourages the thought that dependence* is sufficient for causation even if not quite necessary. I say "accident" because, according to me, dependence* just happens to correlate (in the cases where it helps) with the real reason why direct counterfactual dependence fails to be necessary for causation. Effects do not have to be dependent* on their causes. Rather they have to be de facto dependent on their causes.

A second stab at explaining what I mean by de facto dependence; there'll be more below.   When someone says that c causes e, there's a distinction in play between two aspects of the circumstances in which c and e take place. On the one hand there is
 

the route along which c transmits its influence towards e

On the other hand we have
 

the infrastructure (everything else).

To say that e de facto depends on c is to say that (although it does not depend simpliciter), the infrastructure held fixed, e depends on c. That is, letting the infrastructure be INF, c causes e iff
 

(CF3) O(c) & O(e) & (¬O(c) & INF => ¬O(e))

The real reason preemptive causes make trouble for the direct dependence theory has to do not with the directness but with the envisaged sort of dependence. Switch from cf dependence to df dependence, I'll be arguing, and dependence is restored to its rightful place as the main element in causation. (Side remark: The ancestral maneuver owes its limited success to the way that it "smuggles in" information about the infrastructure: essentially, the information that the preempted cause can't have caused e because the causal chain it extends towards e peters out too soon.)
 
 

V. Preemption

Is it really true, though, that we can restore the "missing" dependence of effect on preemptive cause by allowing the dependence to be de facto? A lot depends on how exactly the infrastructure is chosen.  Speaking of the Hit/Miss example discussed above, Things said the following:

Imagine our late preemption scenario filled out so that the manager doesn't intercept Miss's ball until after it hits the (already toppled) pin. Since in this case Miss's ball does get close to the pin, something else has to go into the infrastructure. How about the fact that Miss's ball never gets close to the pin when it is in an upright position, ie., when the pin is in a condition to be toppled?  (Once again it would be easy to state this positively.)  Holding fixed the fact that Miss's ball never approaches the pin at any relevant time, it remains the case that without Hit's throw there would have been nothing to knock the pin over.
 

VI. Infrastructure

At this point a skeptical question arises.  What gives me the right to include D = the no-show status of Miss's ball in the infrastructure, as opposed to the route? Conversely, what justifies me in omitting the fact D' that Hitís ball does show up? This is important, because if we donít include D in INF, then the pin's toppling ceases to depend (modulo INF) on Hitís throw. It also ceases to depend (modulo INF) on Hit's throw if we do include D' in INF. All that matters to the effect is that Hit's ball does indeed arrive; whether it arrives because Hit threw it makes no difference whatever.

So again: what justifies me in including certain things in INF and excluding others? I'll interpret this as a request for guidelines as to the construction of infrastructures.

The first guideline that comes to mind is that we don't want to include so much in the infrastructure that e loses all its dependence (modulo INF) on earlier events. There should, in other words, be at least some candidate causes left standing. The problem with an INF including D' is that given the arrival (with sufficient velocity etc.) of Hit's ball, all of our candidate causes are disqualified; whatever Hit and Miss had neglected to do, the pin was going to fall down.

Say that infrastructure-candidate INF screens c off from e iff ¬O(c) & INF => O(e), that is, given INF, e would still have occurred even had c not occurred. Our intention is that "spurious" causes should be screened off by the infrastructure while "genuine" ones are not screened off. For this to work, though, the screen should not be so opaque that nothing makes it through.  Hence our  first guideline:

Permeability: INF should not screen all candidate causes off from the effect; some  at least should be left standing.

What favors  Hit's throw over Miss's that it is easy to think of an INF that screens off Miss's throw but not Hit's -- the fact that Miss's ball never arrives at any relevant time will do nicely -- but hard to think of one that screens off Hit's throw without also screening off Miss's. Circumstances screening off Hit's throw will typically include the causal intermediaries by which Hit's throw brings about the effect. But given those intermediaries, Miss's throw is screened off as well.  Circumstance screening off Hit's throw tend to be impermeable.

I said it was hard to think of an INF that screens off Miss alone;  I didn't say it was impossible.  One strategy would be to limit INF to the fact that either Miss's throw occurred and the pin fell, or Miss's throw did not occur and the pin did not fall. (This is true because its first disjunct is true.) As a trivial matter of logic, the pin's falling does depend modulo this INF on Miss's throw. And it doesn't depend modulo INF on Hit's throw; had it been that Hit didn't throw and INF, Miss would have thrown just the same, which given INF means that the pin would have fallen.

What should we say about this? It's part of the circumstances that O(e) <-> O(m). (Just because e and m both occur.) And clearly, [¬O(m) & (O(e) <->O(m))] => ¬O(e). Notice though that in this case the consequent is not only counterfactually entailed by the antecedent, it's strictly entailed by it. It's strictly impossible for O(e) to be true when O(e) <-> O(m) is true and O(m) is false. So our second guideline is
 

Separation: INF together with O(c) (¬O(c)) should not necessitate O(e) (¬O(e)).

It speaks well for the candidacy of Hit's throw as against Miss's that the obvious way of arranging for an INF that screens off only the former gives us an INF that together with Miss's throw necessitates that the effect occurs.

I said that the obvious way of arranging for an INF favoring Miss's throw gives us one violating Separation. I didn't say it was the only way. We could switch to a biconditional linking the preempted cause not with the effect itself, but an event immediately prior to the event on which it counterfactually depends. Think of  d' = Hit's ball arriving at the pin. If the (true!) biconditional O(d') <-> O(m) were to find its way into INF, then the effect would depend INFly on Miss's throw. How to avoid this result?

The obvious strategy is to say that INF should not be too unnatural, or at least not greatly less natural than other candidates for the role of infrastructure. It's true that the problem in this case seems to be due to a particular kind of unnaturalness, one that we could try to target independently. The biconditional O(d') <-> O(m) is the disjunction of O(d')&O(m) with ¬O(d')&¬O(m). But it seems likely that other forms of unnaturalness will create problems of their own;  so to be on the safe side let's add the guideline
 

Naturalness: Make INF as natural as possible.

It speaks well for the candidacy of Hit's throw as against Miss's that an INF favoring Miss's throw would (if it got by Permeability and Separation) have to be disjunctive in nature or in some other way unnatural.  Other guidelines might be suggested as well, but these three are enough to give the flavor.  The right sort of INF shouldn't stop the candidates in their tracks; nor should it give any of them too much of a boost;  and the less artificial, the better.
 
 

VII. Transitivity

Causation is generally transitive: if c causes d, andd causes e, then c causes e. But if the accumulating counterexamples are to be believed, it is not always transitive. This creates a problem for any theory that identifies causation with an ancestral -- for ancestrals are transitive across the board.

How does de facto theory fare in this regard? De facto dependence is not "automatically" transitive,  because unlike dependence* it is not an ancestral. And 3there are at least two reasons to think that transitivity will sometimes fail for it. One is the logical fact that X => Y and Y=>Z do not entail X =>Z. Here is Lewis, using an example from Stalnaker:

If Hoover had been born Russian, he would have been a communist.
If Hoover had been a communist, he would have been a traitor
If Hoover had been born Russian, he would have been a traitor.

In general, transitivity fails in the [following] situationÖThe antecedent of the first premise must be more far-fetched than the antecedent of the second, which is the consequent of the first. Then the closest worlds where the first antecedent holds are different from --- and may differ in character from ? the closest worlds where second antecedent holdsÖ.A Communist Hoover is nowhere to be found at worlds near ours, but a Russian-born Hoover is still more remote. (C, 33)

The second reason is that the INFs change as we move from one causal claim to another. Let it be that transitivity holds as between ¬O(c), ¬O(d), and ¬O(e). It's a further question whether the truth of ¬O(c) & INF_cd => ¬O(d) and ¬O(d) & INF_de => ¬O(e) makes for the truth ¬O(c) & INF_ce => ¬O(e). Let it be that the nearest ¬O(c)-world w is the nearest ¬O(d)-world v. The INFs wouldn't be giving the help they do unless the nearest ¬O(c)&INF_de-world w' was sometimes different from w, and the nearest ¬O(d)&INF_de-world v' was sometimes different from v. If the fact that w = v gives us reason to expect that w' should be v', I don't know what it is.

Of course, that the INF theory doesn't guarantee transitivity doesn't show that it makes transitivity fail in the right sorts of cases. So let's look at some examples, starting with  e= the pin's toppling vis-à-vis m = Miss's throw. Can we find a suitable INF such that e INF-depends on m -- such that were INF to obtain without Miss's throw, the pin wouldn't have fallen over? I can't think of one. Or consider some of other examples.  Can we find a plausible INF such that the publication of Suzy's paper INF-depends on her skiing accident -- such that had INF obtained without the accident, Suzy's paper would not have been published? None come to mind. Can we find a plausible INF such that my quiet evening at home INF-depends on the assassination orders?  Nope. Can we find one such that Suzy's subsequent good health INF-depends on the bomb's being put under her chair?  I don't think so.
 
 

VIII.  Candidate Causes

I said that we couldn't find a plausible INF such that Suzy's subsequent good health INF-depends on the bomb's being put under her chair.  But you don't have to agree with me!  What about INF = Suzy is in mortal danger as long as she remains in the room?  Given the danger, one might say, it's a good thing the bomb was there, since it was noticing the bomb that led her to flee.  This feel convoluted, to be sure --  one hardly wants to thank the bomb for alerting her to the danger, for the danger is the bomb.  But it might not be so easy to block convoluted conditions like this formally.  I want to argue that this problem doesn't really arise.

A good way to think about INFs is that they are brought in to help us select from between candidate causes  -- to help us separate the genuine such causes from the spurious or merely apparent ones.  This means that the INF machinery shouldn't even come into play unless c is a candidate cause. But what is a candidate cause?  Candidate causes are of two types: they are real causes on which the effect does not depend, or they are would-be causes responsible for the lack of dependence. What these events have in common is their membership in a set of events such that if none of the events had occurred, the effect would not have occurred. Say that e depends on a set of events like that.  Then a candidate cause c of e is an event belong to a set K on which e depends. And of course it should belong non-trivially;  e would cease to depend on K if the given event were removed from it.
 

Now that we know what a candidate cause is, we can ask the following question. What would it take for the placing of the bomb under Suzy's chair to be a candidate cause of the finding of good health?    One would need another event c' such that if the bomb had not been put under Suzy's chair and c' had not occurred either, Suzy would not have been in good health; but Suzy's subsequent good health did not counterfactually depend on c' taken alone.

I don't see that there is an event like that.  If I am right, then the placing of the bomb under Suzy's chair isn't even a candidate for causing the finding of good health.   Compare this to an ordinary preemption case.  The pin's toppling may not depend on Hit's throw taken alone, but it does depend on Hit's throw together with Miss's: if neither throw had occurred, the pin would not have fallen over. So Hit's throw is a candidate cause, and we're now entitled to wheel in infrastructure to help us decide whether its candidacy is sucessful. Here goes then:
 

(CF4)

e depends on the Ks  iff (i) had none of the Ks  occurred, e would not have occurred; and (ii) no proper subset of the Ks has this property.

Suppose that e depends on the Ks and that INF is a true proposition about the circumstances in which these events occur;  INF favors c (over the other Ks) iff  e INF-depends on c and does not INF-depend on any other K.

e de facto depends on c  iff for some choice of K, INFs favoring c are less unnatural (and ....)*  than INFs (if any) favoring other Ks.

c is a cause of e iff e  de facto depends on c.

Suppose that e counterfactually depends on c.  Then it depends on the Cs, the Cs being a group of events of which c is the only one.  Any INF that favors a C, then, has got to favor c.  It follows that e de facto depends on c and so has c as a cause.  So the present theory agrees with existing counterfactual theories that counterfactual dependence is sufficient for causation. (More or less sufficient, anyway -- there are other conditions that have to be met but they are comparatively minor.) The differences are all on the side of how to deal with causation in the absence of counterfactual dependence.
 
 

IX.  Trumping Preemption

Now let's look at a particularly troubling form of preemption discussed by Jonathan Schaffer.  Most responses to the preemption problem focus on events intermediate between c and e; they attempt to exploit the fact that in all the usual cases, for the backupcause c' to have caused e in the absence of the actual cause c, the pattern of those intermediate events would have to have been in some way different from what it was actually.  What makes trumping preemption special is that there are no relevant intermediate events to make play with.  Here is Schaffer's example:

Imagine that it is a law of magic that the first spell cast on a given day [matches] the enchantment that midnight.  Suppose that at noon Merlin casts a spell (the first that day) to turn the prince into a frog, that at 6.00pm Morgana casts a spell (the only other that day) to turn the prince into a frog, and that at midnight the prince becomes a frog.  Clearly, Merlin's spell...is a cause of the prince's becoming a frog and Morgana's is not, because the laws say that the first spells are the consequential ones.  Nevertheless, there is no counterfactual dependence of the prince's becoming a frog on Merlin's spell, becausae Morgana's spell is a dependency-breaking backup.  Further, there is neither a failure of intermediary events along the Morgana process (we may dramatize this by stipulating that spells work directly, withot any intermediaries), nor any would-be difference in time or manner of the effect absent Merlin's spell...thus nothing remains by which extent [counterfactual accounts of causation] might distinguish Merlin's spell from Morgana's in causal status (TP, 165).

One idea would be to let INF be the fact that Merlin commences spell-casting at 11.59, while Morgana commences at 5.59.  (Commencing is uttering the first word of a multi-word incantation; the spell is not cast until the incantation is out in its entirety.)   But such an INF doesn't favor Merlin; instead it screens both candidate causes off from the effect.  Had it been that Merlin in INFy circumstances had failed to cast his spell, Morgana would still have cast hers, and the effect would still have occurred. And likewise if Morgana in INFy circumstances had not cast her spell.

Another possible choice of INF is the fact that nothing Morgana does causes the prince to turn into a frog.  This does seem to favor Merlin's spell.  Had Merlin held up in circumstances where nothing Morgana does turns the prince into a frog, the prince would not have turned into a frog.  Had Morgana held up in circumstances where nothing she does etc., the prince would still have turned into a frog due to the spell cast by Merlin.  But although this INF technically works, a philosophical account of X should aim if possible to predict the observed pattern of Xness without assuming anything about that pattern.   And in treating it as a fact that nothing Morgana does causes the transformation, we are assuming something about the pattern of causation.

A third and final idea is to let INF be the fact that the prince's fate is counterfactually insensitive to the content of Morgana's spell (if any); the same thing would have happened whatever Morgana had said.  This again strikes me as favoring Merlin's spell.  Had Morgana held back, with the prince's fate insensitive to the content of her spell, the prince would still have turned into a frog due to Merlin's spell.  But what would have happened had Merlin held back, with the prince's fate insensitive to the content of Morgana's spell?  Would the prince still have suffered the very fate decreed by Morgana?  Suppose for the sake of argument that he would have.  Given that the prince's fate would have been the same whatever Morgana had said, the frog transformation would still have occurred even if Morgana had said "let him be a chicken," indeed even if she had said nothing at all.  This is very hard to believe.  It is very implausible that if Merlin had cast no spell at all, and Morgana had said "let him be a chicken" (or nothing at all), the prince would have turned into a frog.

So we are forced to reject the claim that the prince would still have turned into a frog, had Merlin said nothing, and had Morgana said, with the prince's fate counterfactually insensitive to her choice of words,  "let him turn into a frog."  Which means that the given INF does indeed favor Merlin's spell over Morgana's.  And since the given INF makes no mention of causation -- it's formulated in counterfactual terms -- there's no danger anymore that we are using causal facts to predict causal facts.  Of course, the possibility remains that a more natural and more comprehensive INF can be found that favors Morgana's spell over Merlin's. But enough has been said to show that the de facto dependence account is not at an absolute loss in this area.
 

X. Overdetermination

Overdetermination occurs when an effect e depends on the Ks but no particular K can lay claim to being more of a cause than its co-dependees. E.g., two bowling balls hit the pin at the same time, either of which would have been enough to make it topple.   Lewis has said he has no intuitions about this sort of case and is happy to leave them as spoils to the victor.  His own account treats overdeterminers as non-causes, except in very particular circumstances which let's not go into here.  This has the result that the when the pin falls over, neither Hit's throw nor Knock's is a cause.   Neither is the conjunction of their throws a cause on his account, since the toppling could too easily have occurred without it; neither is the disjunction of their throws a cause, because we are hard to put to regard the disjunction as a bona fide event.  All of this goes somewhat against the grain, since it seems to leave an event which was clearly caused -- the pin's toppling was far from a miracle -- without anything to count as its cause, or even as its causes.

If Lewis is right that we are genuinely undecided about the causal status of overdeterminers, then rather than allowing the decision to be made arbitrarily, we might hope for an account of causation that shows a similar indecision.  Take for instance the de facto dependence account. As written it says that "INFs favoring c are less unnatural than INFs favoring c'."  This could be taken to mean that some INF favoring c is less unnatural than any INF favoring c'.  On that reading Hit's throw will not emerge as a cause, since any INF favoring it will have an every-bit-as-natural counterpart favoring Knock's.

But the phrase could also be taken weakly, to mean that some INF favoring c is at least as natural as any INF favoring c'.  On this reading it's enough to make Hit's throw a cause if an INF can be found favoring it  such that Knock's throw is not favored by any more natural INFs.  It seems to me that this is possible: if Hit's throw had not occurred and the pin would have done the same thing, falling-wise, regardless of whether Knock had thrown, then, I claim, the pin would not have fallen.  To think otherwise is to think that the pin would still have fallen, even if neither Hit nor Knock had thrown their balls. And that seems just wrong. So our account read the second way predicts that Hit's throw was a cause of the pin's falling, and Knock's throw was too, since each throw occurs under conditions given which the effect would not have occurred without it.
 

XI.  Redundant Prevention

Preventing an event is something like causing the negative event consisting of its absence.  So it will not take us too far off topic to look at tricky examples concerning prevention.  Here is one due to Michael McDermott:
 

Suppose that I reach out and catch a passing cricket ball.  The next thing along in the ball's direction of motion was a solid brick wall.  Beyond that was a window.  Did my action prevent the ball hitting the window?   (Did it cause the ball to not hit the window?)  Nearly everyone's initial intuition is, 'No, becaue it wouldn't have hit the window [anyway]'  To this I say, 'If the wall had not been there, and I had not acted, the ball would have hit the window.  So between us -- me and the wall -- we prevented the ball hitting the window.  Which one of us prevented the ball hitting the window -- me or the wall (or both together)?'  And nearly everyone then retracts his initial intuition and says, 'Well, it must have been your action that did it -- the wall clearly contributed nothing'  ("Redundant Causation," 525)

An INF of the first kind might be: the wall and the ball never come into contact.   But it's not clear that the non-breaking does INF-depend on the reaching out for this choice of INF.  After all, it was a big wall.  The change in trajectory required to clear it, or stop short of it, or otherwise miss it,  would not likely have left the the ball on a course leading to the window.  A better choice of INF would be this: the ball continues on its parabolic trajectory towards the window unless and until a hand intervenes. Now the non-breaking does INF-depend on the reaching out.  For suppose the reaching out had not occurred, with the ball continuing on its parabolic window-directed path absent the intervention of a hand.  Then the ball would have hit the window.  The non-breaking does not however INF-depend on the presence of the wall.  Had the wall not been there, the reaching out would still have occurred, and the the ball's window-directed motion would have been cut short.

Now we have to hunt around for an INF that favors the presence of the wall over the catch.    How about this: the window would have  wound up in exactly the same condition whatever action the player by the wall had taken.  To check whether the non-breaking INF-depends on the wall, suppose that the wall hadn't been there and that the behavior of the window had been counterfactually insensitive to the player's actions.  Then, I suppose, the ball's inertia would have called the shots and the window would have wound up smashed.  So there is INF-dependence on the presence of the wall.   Is there INF-dependence on the reaching out?  No, for even if the reaching out had not occurred the wall would still have been there to stop the ball.

The question then becomes: which condition is more natural, that the ball stays on its path unless and until meeting up with a hand, or that the window's fate had been counterfactually insensitive to the player's actions?   Some might say the first condition, since it is categorical rather than counterfactual.  Some might say that neither condition is more natural than the other.  I doubt that anyone will find the second, counterfactual insensitivity, condition more natural.  Let's assume then that there is something to be said for the view that INFs favoring the reaching out are more natural than INFs favoring the wall, and something to be said for the view that neither sort of INF beats out the other.  Then according to our theory, there is something to be said for the view that it was the player's action that prevented the window from breaking, and something to be said for the view that this was a case of symmetric overdetermination.  If we take the second option, then we can choose (as before)  between saying that both of the two events prevented the breaking, or that neither of them did.
 

XII.  Redundant Enabling

Now let's consider a case much like the last one except that the effect is a positive occurrence rather than a negative one, ie., a failure.   A trolley is bearing down on a bunch of just-laid eggs.  A boulder left on the track influences the sequence of events in two ways.  One is that the trolley's wheels are thickly encrusted with crushed rock when they hit the eggs. The second thing the boulder does is to make for a tremendous crunching noise when the train rolls over it; this crunching noise would have alerted a snoozing elephant to the trolley's approach,  if the elephant hadn't already wandered off the track of its own accord.

Is the boulder's being left on the track a cause of the eggs' breaking?   Is the elephant's wandering off the track of its own accord a cause?  I'm again undecided, as I imagine you are too.  Part of me wants to answer "yes" and "yes," or perhaps "no" and "no."   This is the part that smells (symmetric) overdetermination and is unsure whether (symmetric) overdeterminers are causes (as discussed in section X).  Another part of me wants to answer "yes" and "no."  This is the part that smells asymmetric overdetermination with the boulder coming out ahead; given the boulder, the elephant would have been gone in any case.

Assuming you share my ambivalence, what does the theory tell us?   First we should check that the effect e depends on b = the boulder's being left on the track together with w = the elephant's wandering off of its own accord. Had neither of these two events occurred, the trolley would have hit the elephant and derailed.  Clearly too the effect doesn't depend on either b or w taken alone.   The eggs still would have broken absent just the boulder, and they still would have broken absent just the elephant's voluntary departure. So b and w are candidate causes.

Can we find an INF favoring b over w?   Maybe: suppose that the boulder had not been left, and suppose that  -- as actually, due in the actual circumstances to the crushed rock --  the eggs had never came into contact with the trolley wheels. (This last is our INF.)  Then there would have been nothing to threaten the eggs, and so I assume they wouldn't have broken.  If on the other hand the elephant had not departed of its own accord, it would have still have been scared off by  the crunching noise, whence the eggs would still have been exposed to the trolley's full impact, exerted via crushed rock caught up in the wheels.

Can we find an INF favoring w over b?   Maybe.   Suppose that the elephant hadn't left of its own accord before the trolley arrived,  and suppose that  -- as actually -- it hadn't been scared off by the crunching noise.  (This last is our INF.)  Then the trolley would have been stopped by the elephant and the eggs would not have been smashed. If on the other hand the boulder hadn't been left, and the elephant hadn't been scared off by crunching sounds (there wouldn't have been any), the elephant would still have left of its own accord, and the trolley would have made it through to the eggs.

Which of these two INFs is the more natural: the one about the elephant's not being scared off by any crunching noises,  or the one about eggs never coming into contact with the trolley wheels?   Depending on how we answer, and on how we propose to deal with "ties,"  we will find ourselves with zero causes; two causes; one cause, involving a boulder;  or one cause, involving an armadillo.  Speaking for myself, I find "eggs and wheels never come into contact" at least as natural as, and probably more natural than, "elephant was not scared off by crunching sounds." (The first could be replaced by a positive description of how far exactly the wheels are from the eggs at particular times. The second by contrast seems essentially disjunctive, two of the disjuncts being  "there were crunching noises but the elephant was gone" and "there were no crunching noises.")   To that extent the theory rationalizes my indecision as between both, neither, and b alone.
 
 

XIII. Switching (with thanks to Carolina Sartorio)

The eggs have been cleaned up and the trolley is now bearing down on a tomato. The tomato lies 100 yards ahead on the track --  or rather tracks, for just ahead the track splits into two subtracks that reconverge before the tomato is reached.  Which subtrack the trolley takes is controlled by the position of a switch.  With the switch in its present position the trolley will reach the tomato via subtrack A. But someone pulls the switch (call the switch-pulling s) so as to divert it to track B.  As a result the trolley takes subtrack B to the tomato.

Now, is s a cause of the tomato's subsequently being squashed (t)?

Certainly t does not counterfactually depend on s.  What it does depend on are events that themselves counterfactually depend on s, for instance, the train's moving from track B back onto the main track.  If we go by Lewis's counterfactual theory, with its assumption of transitivity, this is enough to make s a cause.  But what should we de facto theorists say?

Given the role that transitivity played in reaching the "conclusion" that the switch-pulling is a cause of the squashing, it might seem at first that the de facto theory (which rejects transitivity) would pronounce differently.   But the matter is not so clear.  The first thing to notice is that the switch-pulling is a candidate cause.  The squashing depends on the switch-pulling together with the event r of track A being reconnected to the main line earlier in the day. (The two had been disconnected for repairs. If you like you can replace r with the "event" of the connection between A and the main line holding firm.)  Had neither s nor r occurrred, the trolley would have followed (unreconnected) track A into a ditch, and the tomato would have been spared.   So the squashing depends on the Ks = s and r.  Since it doesn't depend on any proper subset of the Ks, we move on to the question of whether INFs favoring s are less unnatural than INFs favoring r.

That the train never makes contact with track A might at first seem to favor the switching-to-B over the reconnecting-of-A.  Given that track A is out of the picture, one might think that if not for the switching the trolley would have had nowhere to go;  it would have stopped dead, maybe, or it would have jumped the tracks.  Either way,  the tomato wouldn't have been hit.

But is it really so obvious that the train would have (for no reason) jumped the tracks rather than (for no reason) switching to track B en route to the tomato?   I don't think it is obvious at all,  and so I'm reluctant to treat "trolley never comes into contact with track A" as an INF favoring the switching.  Nor does there seem to be any other natural INF favoring the switching.  (I can't claim to have conducted an exhaustive search.)   I tend to doubt then that we have de facto dependence here.   Which is good, or at least consistent, since I tend to doubt as well that the switch-pulling was a cause of the tomato's demise.

Other cases of switching, however, certainly do involve causation. Often these are cases where the available routes to the effect are importantly different, though as we'll see, that can't be all there is to it. Here is an intuitive example of Ned Hall's:
 

"The Kiss" One day, [Billy and Suzy] meet for coffee.  Instead of greeting Billy with her usual formal handshake,..Suzy embraces him and kisses him passionately, confessing that she is in love with him.  Billy is thrilled -- for he has long been secretly in love with Suzy, as well. Much later, as he is giddily walking home, he whistles a certain tune.  What would have happened had she not kissed him?  Well, they would have had their usual pleasant coffee together, and afterward Billy would have taken care of various errands, and it just so happends that in one of the stores he would have visited, he would have heard that very tune, and it would have stuck in his head, and consequently he would have whistled it on his way home. ...even though thereis the failure of counterfactual dependence typical of switching cases (if Suzy had not kissed Billy, he still would have whistled), there is, of course, no question whatsoever that as things stand, the kiss is among the causes of the whistling ("Causation and the Price of Transitivity")

No, because there are relevant differences between the two cases.  The trolley in "The Engineer" has a lot of momentum; something will have to be done with that momentum if it's supposed that the trolley never proceeds down track A despite that the switching mechanism was set for A.   A crazy story will have to be told, and the story whereby the trolley jumps spontaneously to track B seems not obviously crazier than the competition.  Billy's progress towards the store where he hears the tune, however, is driven not by any sort of momentum but just a series of random accidents.  This makes it easier to find an INF favoring the kiss than it was to find an INF favoring the switch-pulling.

Suppose we take as our candidate causes, first, the kiss, and second, the tune coming on in that store.  Had neither of these occurred, there would have been no humming, but the absence of either by itself is not enough to take the humming away.  Let INF be the fact of Billy's not entering the store.  (As always, this can be stated more positively in terms of his distance from the store.) Imagine that Billy had not been kissed; then holding fixed that he never enters the store, he would not have heard the tune and would not have wound up humming.  If on the other hand the tune had not come on in that store, then, still holding fixed the fact that he never enters the store, he would still have gone on to hum, due to the kiss.

Once again, the reason this INF works while "the trolley never makes contact with track A" doesn't has to do with momentum.  It's because the trolley has so much steam behind it that we are driven to invent crazy stories about why it never makes it, thus opening ourselves up to the crazy possibility that the train switches spontaneously to track B. It's because Billy has so little steam behind him that we can imagine him away from the store without resorting to anything that far out.  (E.g., without resorting to a kiss on the sidewalk from Suzy's equally thrilling cousin that gets Billy humming anyway, as a spontaneous switch to track B gets the tomato squashed anyway.)

All right, you say, but we can give Billy momentum.  Leaving the coffee shop unkissed, he would have been strongly motivated to visit his favorite bookstore, which specializes in poetry about unrequited love.  That is the store where he would have heard the tune.  Don't we still want to say that the kiss was a cause of Billy's humming?  We undoubtedly do.

This example helps us to home in on "factor X": the factor that distinguishes switches that are not causes (like the switch-pulling) from those that are  (like the kiss)?  The modified Billy example shows that factor X involves more than the fact that there would have been momentum toward the effect whether the switching event had occurred or not.  The other thing required is that it be the same momentum.   The momentum that would have been there, absent the switching event, has got to be the momentum we actually have, given that the switching event did occur.  A switch fails to be a cause, in other words,  when all it does is divert preexisting momentum down a different path.  The switch-pulling only diverts preexisting momentum, so it is not a cause.  The kiss does much more than that; it provides momentum of its own.  So the kiss can be and (it seems to me) is a cause.

None of this is part of the official theory, of course.   But it tells us something about what a theory had better say if it wants to match intuition.   Switches that do no more than divert preexisting momentum it should classify as not causes.  Other switches it should classify as causes.  Applied to our theory, this means that if x is a "merely momentum-diverting" switch, there should not be a relatively natural INF favoring x, whereas if it is any other kind of switch, such an INF should exist.

So far as I'm aware (I haven't looked at a huge range of examples), this is how it works out.  There may even be reason to expect it to work out this way, or at least not to be greatly surprised if it does.  Suppose that x is merely momentum-diverting. Then in x's absence it would seem natural for the momentum to revert to its "original" route, with the result that e.   If e is nevertheless to INF-depend on x, then INF takes on the burden of blocking the reverted momentum or at least channeling it away from e.  This I submit will be hard for any natural INF to do.  If on the other hand x is not merely momentum-diverting, then there is no reverted momentum to neutralize, and INF can just say that a certain road was not taken.