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COUNTERFACTUALS, CAUSATION, AND PREEMPTION

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The philosophical importance of counterfactuals stems from the fact that they seem to be closely connected to the concept of causation . Thus it seems that the truth of the above conditional is just what is required for Suzy's throw to count as a cause of the bottle's shattering. If philosophers were reluctant to exploit this idea prior to 1970, it was because of a widespread feeling that the truth-conditions of the counterfactual conditional were not sufficiently well understood. The development of a formal semantics for counterfactuals by Robert Stalnaker [ 1968 ] and David Lewis [ 1973b ] stands as a major recent achievement in philosophical logic.
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JOHN COLLINS
COUNTERFACTUALS, CAUSATION, AND
PREEMPTION
1
INTRODUCTION
A counterfactual is a conditional statement in the subjunctive mood. For example:
If Suzy hadn’t thrown the rock, then the bottle wouldn’t have shat-
tered.
The philosophical importance of counterfactuals stems from the fact that they seem
to be closely connected to the concept of causation. Thus it seems that the truth
of the above conditional is just what is required for Suzy’s throw to count as a
cause of the bottle’s shattering. If philosophers were reluctant to exploit this idea
prior to 1970, it was because of a widespread feeling that the truth-conditions of the
counterfactual conditional were not sufficiently well understood. The development
of a formal semantics for counterfactuals by Robert Stalnaker [1968] and David
Lewis [1973b] stands as a major recent achievement in philosophical logic.
Section 2 presents the standard Stalnaker–Lewis semantics for the counterfac-
tual conditional and develops some of the logical features of counterfactuals. Sec-
tion 3 presents Lewis’s original counterfactual theory of causation [1973a], and
explains the problems that eventually led him to abandon the theory in its original
form. The remainder of the article surveys the current state of counterfactual theo-
ries of causation, by presenting, in Sections 4, 5, and 6, three recent contending ac-
counts, due to Lewis [2000; 2004a], Yablo [2000; 2004] and Hall [2004b; 2004a].
The discussion does not aim to be exhaustive, but focuses on the central issue of
preemption, which has proved to be the major hurdle for counterfactual theories.
For a more complete picture, the reader is referred to the papers in the collection
edited by Collins, Hall, and Paul [2004]. The present article aims to provide an
brief introduction to this currently very lively area of applied philosophical logic.
For a more comprehensive introductory survey, Hall and Paul (forthcoming) is also
highly recommended.
The contemporary literature on counterfactuals and causation includes a vast
and potentially bewildering collection of examples and counterexamples. The
present article focuses on six central types of example, which are labeled (E1)–
(E6) so that the reader may more easily distinguish them from other less important
particular cases mentioned in passing.

2
JOHN COLLINS
2
THE LOGIC OF COUNTERFACTUALS
The counterfactual ‘If A were true, then C would be true’ with antecedent A and
consequent C is sometimes written ‘A2 → C’ in order to distinguish it from
other kinds of conditional statement. For example the counterfactual must be dis-
tinguished from both the material conditional of first-order logic, and the “strict
conditional” of entailment. The truth-functional material conditional ‘A → C’ is
logically equivalent to ‘∼A ∨ C’ and thus has truth-conditions weaker than those
of the corresponding counterfactual conditional. Not every counterfactual with a
false antecedent or a true consequent is true. The strict conditional, on the other
hand, is too strong; it might be true that:
If this match had been struck, it would have lighted.
But the lighting of the match is not logically entailed by its being struck.
A correct account of the semantics of the counterfactual will then, presumably,
locate it somewhere between these two extremes. But where? One obvious thought
is that ‘A2 → C’ is true if and only if C is entailed not by A alone, but by A in
conjunction with certain other truths, including, perhaps, the laws of nature. Thus
it might well be that the lighting of the match is entailed by its being struck, in
conjunction with the laws of nature, and certain other true matters of fact — for
example the presence of sufficient oxygen in the atmosphere.
The problem with this idea is that there is no single fixed set of truths that will
do the job for all A and C. That is because counterfactuals are non-monotonic. In
other words the inference pattern:
A2 → C
so:
(A & B)2 → C
is invalid. It may be true, for example, that the match would light if struck, and yet
not true that it would light if struck in the absence of oxygen.
But the counterfactual conditional will only serve as a fit tool for philosophical
analysis if we have a firm grasp of its logic and truth-conditions. In a famous essay
critical of philosophical use of the counterfactual idiom, Nelson Goodman framed
the challenge this way. In evaluating the truth of the counterfactual ‘A2 → C’
we want to hold fixed all those truths that are “cotenable” with the truth of the an-
tecedent A. Yet what might it mean for a proposition to be cotenable with A other
than that the proposition is one that would still be true even if A were true? If one
hopes to provide a semantics for counterfactuals in terms of what is cotenable with
a given antecedent, then one had better not rest this on a counterfactual analysis of
cotenability [Goodman, 1947].
It was the work of Stalnaker [1968] and Lewis [1973b], which in turn grew out
of the development of possible world semantics for modal logic in the 1960s, that
first convinced some of the skeptics that counterfactual conditionals were indeed
philosophically respectable.

COUNTERFACTUALS, CAUSATION, AND PREEMPTION
3
The Stalnaker–Lewis approach starts from the assumption that the set of all pos-
sible worlds may be weakly ordered with respect to the “comparative similarity”
of those world to the actual world. Since this ordering is transitive and connected,
it is useful heuristically to think of it as a comparative “closeness” relation. If A
is any proposition, call a world at which A is true an A-world. Then, in Lewis’s
formulation, the truth-condition for the counterfactual may be stated in this way:
‘A2 → C’ is true if and only if some (A & C)-world is more similar
to the actual world than any (A & ∼C)-world is.
If we simplify things by assuming that for each A there is always a single A-
world most similar to the actual world, then the condition becomes:
‘A2 → C’ is true if and only if C is true at the most similar A-world
to the actual world.
Now similarity is itself a philosophically problematic concept, but even without
settling on any particular criteria for making comparative judgments, we can see
how much of the logic of counterfactuals follows simply from the fact that any
candidate similarity ordering, like a closeness ordering, must be connected and
transitive.
We can see, for example, why strengthening the antecedent of a counterfactual
may lead from truth to falsehood. The fallacy is akin to that made a person who
infers from the fact that there is no bank in the closest town to here, that there is
no bank in the closest town to here with a restaurant.
We can also see immediately that contraposition fails for the counterfactual
conditional, i.e., that the inference
A2 → C
so:
∼C2 → ∼A
is fallacious. It doesn’t follow from the fact that the closest town with a restaurant
has no bank that the closest town with a bank has no restaurant.
A similar exploitation of the analogy with spatial closeness will convince us that
counterfactuals are not transitive. That is, we can add to our list of counterfactual
fallacies the following:
A2 → B
B2 → C
so:
A2 → C.
Moving beyond these purely logical points, however, more will need to be said
about the comparative similarity relation. The problem is that similarity admits of
different respects. One possible state of affairs may be more similar to actuality
than another in one respect, and less similar to actuality in another. Often it is
the context of utterance that determines the particular similarity ordering that a

4
JOHN COLLINS
speaker has in mind. But insofar as we want counterfactuals to provide objective
truth-conditions for causal statements, this question of context-dependence is fairly
pressing. How are the various respects of similarity to be weighed in order to
enable a single overall comparison?
It seems clear that similarities with respect to the laws of nature should generally
outweigh similarities with respect to accidental matters of fact. But this cannot be
an invariable rule. For if, as we are assuming, the laws of nature are deterministic
in both temporal directions, any supposed change in the way things presently are,
will be propagated, via the laws, into a divergent past as well as a divergent future.
Matching the past history of the actual world is very important for similarity, it
appears, even if the match can be obtained only by allowing a minor localized
violation of the laws of nature (a “small miracle”).
This is not to deny that we are sometimes prepared to speak as though things
would have been different in the past had they been different now. For example I
might say: “If I had jumped out the window just now, there would have to have
been a safety net in place, I’m not crazy!” But note the “have to have been” con-
struction in that sentence that serves as a syntactic indicator of the appropriateness
of the backtracking interpretation (see [Lewis, 1979]). The point is simply that this
backtracking interpretation is non-standard. Causes always, or at least typically,
precede their effects. Surely this asymmetry should be reflected by a correspond-
ing asymmetry in the counterfactuals.
But since the temporal asymmetry of causation seems to be a merely contingent
feature of the actual world, it seems wrong to build this asymmetry into the anal-
ysis of counterfactuals by stipulation. Lewis [1979] pursues the more ambitious
goal of identifying criteria for a comparative similarity relation that rule out back-
tracking in worlds like the actual world, but without making backward causation
an a priori impossibility.
Here is the rough idea. Suppose that c is some event that actually occurred at
time t, and consider a counterfactual whose antecedent asks us to suppose that c
hadn’t occurred. A non-backtracking reading of this counterfactual will be ensured
provided that all the closest possible worlds to the actual world at which c doesn’t
occur are worlds in which (i) past history up until some point shortly before time t
perfectly matches the history of the actual world, and (ii) this perfect match results
from a small “divergence” miracle. And this is correct, Lewis suggests, because
our world happens to be such that there is no possible world w in which (i) c fails
to occur, (ii) the future of w after time t exactly matches the actual future, and (iii)
this match results from a small “reconvergence” miracle.
An important recent paper by Adam Elga has raised a serious difficulty for
Lewis’s attempt to rule out backtracking contingently. Elga argues that statistical
mechanics provides examples which demonstrate that reconvergence to the actual
world requires no greater violation of the laws than divergence from it does. See
[Elga, 2000; Albert and Loewer, 2005].

COUNTERFACTUALS, CAUSATION, AND PREEMPTION
5
3
THE COUNTERFACTUAL THEORY OF CAUSATION
An early statement of a counterfactual analysis of causation can be found in the
work of David Hume [1902, § VII], where he writes:
. . . we may define a cause to be an object followed by another, and
where all the objects, similar to the first are followed by objects simi-
lar to the second
. Or, in other words, where, if the first object had not
been, the second never had existed
.
In fact, of course, the passage just quoted contains two quite different accounts
of the causal relation. While the second sentence treats causation in counterfactual
terms, the first expresses an idea that is closer to what has become known as a
regularity theory of causation.
Regularity theories tended to be the more favored of these two broad approaches
prior to the development of the Stalnaker–Lewis semantics for the counterfactual.
See, e.g., [Mackie, 1965].
A regularity analysis of causation states, roughly, that:
One event c is a cause of another event E just in case c is a member
of some minimal set of actual events that are jointly sufficient, given
the laws of nature, for the occurrence of the effect.
But regularity theories of causation have always faced difficulties. One is the
problem of distinguishing cause from effect. For if E is caused by c, there will
often be a set of conditions including e which, in conjunction with the laws, entail
that c occurs.
Another difficulty is the problem of distinguishing genuine causes from ineffi-
cacious epiphenomenal by-products of a causal process. If d and e are independent
effects of some common cause c, then it may well be the case that d belongs to a
minimal set of conditions which, along with the laws, are sufficient for e. Then d,
which is a epiphenomenal by-product of the process by which c caused e, will be
incorrectly counted as a cause of e.
In his seminal article [1973a], Lewis pointed out that a counterfactual analysis
of the causal relation is at a distinct advantage over a regularity theory in both
of these cases. Given an appropriate ban on a backtracking interpretation of the
relevant counterfactuals, one may simply deny the truth of the counterfactuals that
might otherwise cause problems. Suppose that e wouldn’t have occurred if c hadn’t
occurred. Might it also be the case that if e hadn’t occurred then c wouldn’t have?
Not unless the latter conditional is understood in the backtracking sense. Similarly,
if d and e are independent effects of a common cause c, then the only reason one
might be tempted to believe that if d hadn’t occurred then e wouldn’t have occurred
is if one thinks that if d hadn’t occurred then neither would c. Once again, this
would involve an illicit backtracking reading of the latter conditional.

6
JOHN COLLINS
Say that one event e is counterfactually dependent on another event c when e
wouldn’t have occurred if c hadn’t. The simplest counterfactual analysis of causa-
tion would simply construe causation as counterfactual dependence.
But this simplest account cannot be correct, as is demonstrated by the following
example:
(E1) Early Preemption: Suzy throws a rock at a bottle. The rock hits the bottle
and shatters it. Billy was standing by with a second rock. Had Suzy not
thrown her rock, Billy would have shattered the bottle by throwing his rock.
The problem here is that the shattering of the bottle is indeed caused by Suzy’s
throw, despite the fact that the shattering is not counterfactually dependent on the
throw. According to Lewis, the key to reinstating the throw as cause is to recog-
nize that there is a chain of events initiated by the throw and terminating in the
shattering, such that each event in the chain is counterfactually dependent on the
one before it. Call such a chain of events a chain of counterfactual dependence. In
this case, each event in the chain is the event of Suzy’s rock being located in mid-
flight at some point between her hand and the bottle. Although Billy was standing
by with appropriate intent and deadly aim, no such chain of events connects his
standing by with the shattering.
In cases like Early Preemption we have a chain of counterfactual dependence
leading from a cause to an effect, without the effect being counterfactually depen-
dent on the cause. The problem with preemption, then, seems to stem directly
something we noted in the previous section: the non-transitivity of the counterfac-
tual conditional. While causation appears to be a transitive relation, counterfactual
dependence is not.
If this diagnosis of the problem is correct, then the fix is very straightforward.
We should simply take causation to be the ancestral of the counterfactual depen-
dence relation. Thus we arrive at Lewis’s [1973a] counterfactual analysis of cau-
sation.
(C1)
Suppose that c and e are distinct events, and let C and E respectively be
the propositions that the events c and e occur. Then e is counterfactually
dependent
on c when the following two counterfactual conditionals are true:
(i)
C2 → E
(ii)
∼C2 → ∼E
(C2)
A causal chain is a finite sequence of actual events such that each event in
the sequence is counterfactually dependent on the previous event.
(C3)
One event is a cause of another if and only if there is a causal chain leading
from the first to the second.
Note that it is precisely the failure of contraposition for the counterfactual con-
ditional that leaves room for the ban on backtracking, by allowing that e may be

COUNTERFACTUALS, CAUSATION, AND PREEMPTION
7
counterfactually dependent on c without c also being counterfactually dependent
on e.
This original analysis was designed to deal with cases of preemption like (E1).
However the phenomenon of preemption has proved to be a far more serious dif-
ficulty for the counterfactual theory of causation than Lewis believed it to be in
1973.
The problem is that there are varieties of preemption that cannot be dealt with
by the transitivity strategy. Here is one such:
(E2)
Late Preemption: Billy and Suzy both throw rocks at a bottle. Suzy’s rock
gets there first, hitting the bottle and shattering it. Billy’s rock flies through
the now empty space where the bottle was standing.
This example differs in one key respect from the previous case. In Late Pre-
emption no stepwise dependent chain of events can be traced from Suzy’s throw to
the shattering of the bottle. For consider all the events in the process initiated by
Suzy’s throw prior to the shattering, that is: Suzy’s rock being located at various
positions at various times in mid-flight. The shattering fails to depend on any such
event, because, had Suzy’s rock failed to be there, the bottle would still have been
shattered by Billy’s rock.
Here is another way of seeing the difference between the Early and Late Pre-
emption examples. In cases like Early Preemption, the possible process initiated
by the preempted standby that would otherwise have led to the effect, is cut off by
the process leading from the preempting cause to the effect, at some time before
the effect occurs. In Late Preemption, this “cutting-off ” takes place only when the
effect occurs. (This way of looking at things explains the “early/late” terminology.)
Furthermore, as Jonathan Schaffer discovered, there are possible cases of pre-
emption that do not involve any kind of cutting-off at all. In Schaffer’s example of
“Trumping Preemption” the absence of any cutting, either early or late, is guaran-
teed simply by stipulating that the causal process in question works by action at a
distance.
(E3)
Trumping Preemption: The laws of magic are such that what happens at
midnight is determined by the first spell cast the previous day. At noon
Merlin casts the first spell of the day: a spell to turn the Prince into a frog.
At six that evening Morgana casts a spell to turn the Prince into a frog. At
midnight the Prince turns into a frog.
This example also causes problems for Lewis’s 1973 theory. The transfiguration
of the prince is not counterfactually dependent on Merlin’s spell, since if Merlin
had not cast the spell, the prince would still have been turned into a frog by Mor-
gana’s spell. In addition, there is no chain of counterfactual dependence leading
from Merlin’s spell to the transfiguration, since the example stipulates that none is
required.
It is tempting to dismiss such fantastic cases as being too far-fetched to be rel-
evant to any discussion of causation as it is in the actual world. But this would be

8
JOHN COLLINS
too hasty a dismissal. The fanciful nature of Schaffer’s story merely helps make
the causal structure of the example clear. The important point is that it seems per-
fectly possible that, for all we know, some actual cases of causation work that way.
Since the possibility of causation by trumping preemption cannot be ruled out a
priori
, a theory of causation will only be adequate if it can deal with such cases.
4
CAUSATION AS INFLUENCE
One tempting thought is that the problems facing the counterfactual theory might
be solved by taking events to be modally “fragile”, i.e., by claiming that any dif-
ference in time, or manner of occurrence makes for a numerically distinct event.
In our Late Preemption example, the bottle would still have been shattered had
Suzy not thrown her rock, but it would have been shattered a moment later by
Billy’s rock, and, presumably, shattered in a slightly different way. Thus, if the
actually occurring shattering is construed as a modally fragile event (i.e., an event
with a particularly rich essence) then the counterfactual dependence of the actual
shattering on Suzy’s throw is restored, since, had Suzy’s throw not taken place, the
shattering that would have occurred instead would have been a different shattering;
a numerically distinct event.
This “fragility strategy” is discussed by Lewis in various places (e.g., [1986b,
pp. 193–199] and [2004a, pp. 85–90]. As Lewis recognizes, there are various
reasons for rejecting the approach.
For one thing, the “uncommonly stringent” conditions of occurrence that the
fragility strategy imposes are at odds with our ordinary standards of event identity.
Lewis points out that we are usually quite happy to allow that one and the same
event might have been delayed, as, for example, when a seminar talk is postponed
rather than cancelled [Lewis, 2004a, p. 86].
Secondly, the fragility strategy produces spurious cases of causation, as in Lewis’s
example of the “Poison and the Pudding”. Suppose a poison kills its victim more
slowly and painfully when taken on a full stomach. The victim eats some pudding
and then drinks the poison. If the victim’s actual death is construed as modally
fragile, then it is an event that would not have occurred had the pudding not been
eaten. Yet the eating of the pudding was not a cause of his death [Lewis, 1986b,
pp. 198–199].
Still, the central idea of fragility has a role to play in Lewis’s [2000; 2004a]
revised account of “causation as influence”. While remaining neutral about the
ordinary identity conditions for events, Lewis proposes that we introduce a new
technical term to refer to events construed as modally fragile. Say that an alter-
ation
of an event is either the very fragile version of the event that actually occurs,
or a fragile alternative to it that differs slightly with respect to time or manner of
occurrence. Lewis then suggests that we “look at the pattern of counterfactual de-
pendence of alterations of the effect upon alterations of the cause”. Say that event
c influences event e when there is a substantial range c1, c2, c3, ... of alterations

COUNTERFACTUALS, CAUSATION, AND PREEMPTION
9
of c, and a substantial range e1, e2, e3, ... of alterations of e, such that if c1 had
occurred then e1 would have occurred, and if c2 had occurred then e2 would have
occurred, and so on [Lewis, 2004a, p. 91].
Our original notion of counterfactual dependence was a notion of “whether-
whether” dependence. One event is dependent on another is this sense just in case
whether or not the event occurs depends on whether or not the other occurs. But
there are other varieties of dependence. Lewis’s idea is that we should think of
degree of causal influence as being determined by the extent to which whether,
when, and how one thing happens depends counterfactually on whether, when,
and how, something else happens. Hall and Paul have usefully labeled this idea
the “counterfactual covariation” theory of causation (Hall and Paul, forthcoming).
As in Lewis’s original counterfactual theory, causation is now taken to be the
ancestral of the influence relation. One event c causes another event e if and only
if there is a chain of stepwise influence from c to e.
Let’s see how this idea works in our case of Late Preemption. If we consider
small alterations to the time of Suzy’s throw, we obtain corresponding alterations
to the time of shattering (provided of course that the throw is not so much delayed
that Billy’s rock gets there first). Or consider small alterations to the manner in
which Suzy throws her rock; alterations, perhaps, to the velocity and direction of
her throw. Throughout a range in which the velocity is still great enough for her
rock to beat Billy’s to the bottle, and Suzy’s aim is still accurate enough to score a
fairly direct hit, alterations to the throw will produce a counterfactually covarying
range of alterations to the shattering. So the shattering is influenced by Suzy’s
throw.
Not so for Billy’s throw. Unless we imagine Billy’s throw sufficiently altered in
time or manner so that his rock reaches the target before Suzy’s does, the extent to
which the time and manner of the shattering depends on alterations to the time and
manner of the throw are completely negligible. This, claims Lewis, is why Suzy’s
throw counts as a cause of the shattering and Billy’s does not.
Lewis also suggests that the analysis of causation as influence provides a solu-
tion to the problem of Trumping Preemption. Although there is no influence of the
whether-whether or when-when variety, the transformation of the prince is never-
theless influenced by Merlin’s spell since the manner of transformation covaries
with the kind of spell cast. For example: had Merlin uttered “Presto! Prince-to-
possum!” instead of “Presto! Prince-to-frog!” at noon, then the prince would have
turned into a possum, rather than a frog, at midnight. On the other hand, what hap-
pened at midnight was in no way dependent on whether, when, or how Morgana
acted in the late afternoon.
However this solution to the Trumping problem seems to turn on inessential
and eliminable features of Schaffer’s original example. If we suppose that Merlin’s
options are limited to a single spell that he may cast (standard prince-to-frog) and a
single time of day he may cast it (noon), then the transfiguration of the prince is no
longer influenced, in Lewis’s sense, by Merlin’s spell, though of course Merlin’s
spell is still its cause. (See [Collins, 2000], in [Collins et al., 2004, p. 114]. The

10
JOHN COLLINS
idea is due to Jacob Rosen.) This is a counterexample to the necessity of the
influence theory. Other counterexamples to the necessity of the influence theory
can be constructed from cases of Early or Late Preemption by ensuring that the
preempted backup would have brought about the effect in exactly the same manner
and at exactly the same time. See Strevens [2003] and Yablo’s “Smart Rock”
example reported in [Hall, 2004b, p. 237].
Schaffer has argued that the idea behind our first counterexample to necessity
can also be developed into a counterexample to sufficiency. Add to the story of
Merlin’s limited options the fact that Morgana has a vast range of possible spells
to cast, and available times at which to cast them. And now suppose Morgana
stands by, silently watching, just before noon as Merlin prepares to cast his spell.
Then, claims Schaffer, the transfiguration at midnight is influenced by Morgana’s
silent watching (given her vast range of options) though her watching is not among
its causes. The point here seems less clear than in the previous case, since Schaffer
must counter the suggestion that Morgana’s watching is a cause — by omission
of the prince’s becoming a frog. For more details, see [Schaffer, 2001]. Collins
suggests that Lewis’s own “Poison and Pudding” example was already a coun-
terexample to the sufficiency of the causation as influence account [2000].
5
DE FACTO DEPENDENCE
In our Early Preemption example, the shattering of the bottle was not counterfac-
tually dependent on Suzy’s throw, which, nevertheless, caused it. Yet note this:
holding fixed the fact that Billy does not throw his rock, the shattering does de-
pend on Suzy’s throw. That is, if Suzy hadn’t thrown her rock and Billy had still
not thrown his rock either, then the bottle would not have shattered. Stephen Yablo
[2000; 2004] suggests we say that the shattering has a “de facto dependence” on
Suzy’s throw, and count Suzy’s throw as a cause of the shattering in virtue of
this de facto dependence. A version of Yablo’s proposal will be developed in this
section. Closely related accounts have been proposed by Judea Pearl [2000] and
Christopher Hitchcock [2001].
In Late Preemption the same idea seems to work. Holding fixed the fact that
Billy’s rock doesn’t hit the bottle, the shattering depends on Suzy’s throw. That
is, if Suzy hadn’t thrown her rock, and yet Billy’s rock still hadn’t hit the bottle,
then the bottle would not have shattered. Admittedly, this last counterfactual is
a little weird. Since no backtracking is allowed, and we are holding fixed the
fact that Billy’s rock doesn’t hit the bottle, the antecedent is asking us to suppose
that Billy’s rock is thrown just as it actually was, that it follows the same deadly-
accurate trajectory toward the bottle (which is still there when it arrives, since Suzy
hasn’t thrown her rock) but then, somehow — miraculously! — fails to hit it.
Clearly some restrictions will have to be placed on the kind of proposition that
may be held fixed, otherwise everything will turn out to depend on everything else
given some suitably cooked-up background condition. For consider any simple

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