## 1. Lewis's 1973 Counterfactual Analysis



The guiding idea behind counterfactual analyses of causation is the
thought that - as David Lewis puts it - "We think of
a cause as something that makes a difference, and the difference it
makes must be a difference from what would have happened without it.
Had it been absent, its effects - some of them, at least, and
usually all - would have been absent as well" (1973b,
161).



The first explicit definition of causation in terms of counterfactuals
was, surprisingly enough, given by Hume, when he wrote: "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 similar to
the second*. Or, in other words, *where, if the first object had
not been, the second never had existed*" (1748, Section VII).
It is difficult to understand how Hume could have confused the first,
regularity definition with the second, very different counterfactual
definition (though see Buckle 2004: 212-13 for a brief
discussion).



At any rate, Hume never explored the alternative counterfactual
approach to causation. In this, as in much else, he was followed by
generations of empiricist philosophers. The chief obstacle in
empiricists' minds to explaining causation in terms of
counterfactuals was the obscurity of counterfactuals themselves, owing
chiefly to their reference to unactualised possibilities. The true
potential of the counterfactual approach to causation did not become
clear until counterfactuals became better understood through the
development of possible world semantics in the early 1970s.



The best known and most thoroughly elaborated counterfactual theory of
causation is David Lewis's theory in his (1973b). Lewis's
theory was refined and extended in articles subsequently collected in
his (1986a). In response to doubts about the theory's treatment
of preemption, Lewis subsequently proposed a fairly radical revision
of the theory (2000/2004a). In this section we shall confine our
attention to the original 1973 theory, deferring the later changes he
proposed for consideration below.


* 1.1 Counterfactuals and Causal Dependence
* 1.2 The Temporal Asymmetry of Causal Dependence
* 1.3 Transitivity and Preemption
* 1.4 Chancy Causation


### 1.1 Counterfactuals and Causal Dependence



Like most contemporary counterfactual theories, Lewis's theory
employs a possible world semantics for counterfactuals. Such a
semantics states truth conditions for counterfactuals in terms of
similarity relations between possible worlds. Lewis famously espouses
realism about possible worlds, according to which non-actual possible
worlds are real concrete entities on a par with the actual world
(Lewis 1986e). However, most contemporary philosophers would seek to
deploy the explanatorily fruitful possible worlds framework while
distancing themselves from full-blown realism about possible worlds
themselves (see the entry on
 possible worlds).



The central notion of a possible world semantics for counterfactuals
is a relation of *comparative similarity* between worlds (Lewis
1973a). One world is said to be *closer to actuality* than
another if the first resembles the actual world more than the second
does. In terms of this similarity relation, the truth condition for
the counterfactual "If *A* were (or had been) the case,
*C* would be (or have been) the case" is stated as
follows:



(1)
"If *A* were the case, *C* would be the
case" is *true* in the actual world if and only if either
(i) there are no possible *A*-worlds; or (ii) some
*A*-world where *C* holds is closer to the actual world
than is any *A*-world where *C* does not hold.


We shall ignore the first case in which the counterfactual is
vacuously true. The fundamental idea of this analysis is that the
counterfactual "If *A* were the case, *C* would be
the case" is true just in case it takes less of a departure from
actuality to make the antecedent true along with the consequent than
to make the antecedent true without the consequent.



In terms of counterfactuals, Lewis defines a notion of causal
dependence between events, which plays a central role in his theory of
causation (1973b).



(2)
Where *c* and *e* are two distinct possible
events, *e* *causally depends* on *c* if and only
if, if *c* were to occur *e* would occur; and if
 *c* were not to occur *e* would not occur.

This condition states that whether *e* occurs or not depends on
whether *c* occurs or not. Where *c* and *e* are
events that actually occur, this truth condition can be simplified
somewhat. For in this case it follows from the second formal condition
on the comparative similarity relation that the counterfactual
"If *c* were to occur *e* would occur" is
automatically true: this formal condition implies that a
counterfactual with true antecedent and true consequent is itself
true. Consequently, the truth condition for causal dependence
becomes:



(3)
Where *c* and *e* are two distinct *actual*
events, *e* *causally depends* on *c* if and only
 if, if *c* were not to occur *e* would not occur.

There are three important things to note about the definition of
causal dependence. First, it takes the primary relata of causal
dependence to be *events*. Lewis's own theory of events
(1986b) construes events as classes of possible spatiotemporal
regions. However, different conceptions of events are compatible with
the basic definition (Kim 1973a; for an alternative broadly Lewisian
take on events see McDonnell 2016 and Kaiserman 2017). Indeed, it even
seems possible to formulate it in terms of facts rather than events
(Mellor 1995, 2004).



Second, the definition requires the causally dependent events to be
*distinct* from each other. Distinctness means that the events
are not identical, neither is part of the other, and neither implies
the other. This qualification is important if spurious non-causal
dependences are to be ruled out. (For this point see Kim 1973b and
Lewis 1986b.) For while you would not have written 'Larry'
if you had not written 'rr'; and you would not have said
'Hello' loudly if you had not said 'Hello',
neither dependence counts as a causal dependence since the paired
events are not distinct from each other in the required sense.



Convinced by the need to make room in his analysis for causation by
(and of) absence - as when the gardener's failure to water
the plants causes their death - Lewis later amended his view to
the view that causal dependence is a matter of counterfactual
dependence between events or their absences (Lewis 2000: SSX;
2004b). We shall largely ignore this complication in what follows; for
some discussion of causation by absence see Schaffer 2000b, Beebee
2004b, McGrath 2005, Livengood and Machery 2007, Dowe 2009.



Third, the counterfactuals that are employed in the analysis are to be
understood according to what Lewis calls the standard interpretation.
There are several possible ways of interpreting counterfactuals; and
some interpretations give rise to spurious non-causal dependences
between events. For example, suppose that the events *c* and
*e* are effects of a common cause *d*. It is tempting to
reason that there must be a causal dependence between *c* and
*e* by engaging in the following piece of counterfactual
reasoning: if *c* had not occurred, then *d* would not
have occurred; and if *d* had not occurred, *e* would
not have occurred. But Lewis says the former counterfactual, which he
calls a *backtracking* counterfactual, is not to be used in the
assessment of causal dependence. The right counterfactuals to be used
are *non-backtracking counterfactuals* that typically hold the
past fixed up until the time (or just before the time) at which the
counterfactual's antecedent is supposed to obtain. Thus if
*c* had not occurred, *d* - which in fact occurred
before *c* - would have occurred anyway; so on the
standard interpretation, where backtracking counterfactuals are false,
the inference to the claim that *e* causally depends on
*c* is blocked.


### 1.2 The Temporal Asymmetry of Causal Dependence



What constitutes the direction of the causal relation? Why is this
direction typically aligned with the temporal direction from past to
future? In answer to these questions, Lewis (1979) argues that the
direction of causation is the direction of causal dependence; and it
is typically true that events causally depend on earlier events but
not on later events. He emphasises the contingency of the latter fact
because he regards backwards or time-reversed causation as a
conceptual possibility that cannot be ruled out *a priori*.
Accordingly, he dismisses any analysis of counterfactuals that would
deliver the temporal asymmetry by conceptual fiat.



Lewis's explanation of the temporal asymmetry of counterfactual
dependence comes from a combination of his analysis of the similarity
relation together with the (alleged) 'asymmetry of
overdetermination' - a contingent feature of the world.
According to this analysis, there are several respects of similarity
to be taken into account in evaluating non-backtracking
counterfactuals: similarity with respect to laws of nature and also
similarity with respect to particular matters of fact. Worlds are more
similar to the actual world the fewer miracles or violations of the
actual laws of nature they contain. Again, worlds are more similar to
the actual world the greater the spatio-temporal region of perfect
match of particular fact they have with the actual world. If the laws
of the actual world are deterministic, these rules will clash in
assessing which counterfactual worlds are more similar to the actual
world. For a world that makes a counterfactual antecedent true must
differ from the actual world either in allowing some violation of the
actual laws (a 'divergence miracle'), or in differing from
the actual world in particular fact. Lewis's analysis allows a
tradeoff between these competing respects of similarity in such cases.
It implies that worlds with an extensive region of perfect match of
particular fact can be considered very similar to the actual world
provided that the match in particular facts with the actual world is
achieved at the cost of a small, local miracle, but not at the cost of
a big, diverse miracle.



Taken by itself, this account contains no built-in time asymmetry.
That comes only when the account is combined with the asymmetry of
overdetermination: the (alleged) fact that effects are rarely
overdetermined by their causes, but causes are very often
overdetermined by their effects. Taking an example from Elga (2000):
suppose that Gretta cracks an egg at 8.00 (event *c*), pops it in
the frying pan, and eats it for her breakfast. What would have
happened had *c* not occurred? The right answer (Answer 1) is
that the egg would not then have been fried and Gretta would not have
eaten it - and not (Answer 2) that she would still have fried
and eaten the egg, but these events would somehow have come about
despite her failing to crack it in the first place. The question is:
how does Lewis's analysis of the similarity relation deliver Answer
1 and not Answer 2? In particular, consider worlds where there is
perfect match of particular fact until just before 8.00, and then a
miracle, and then no perfect match of particular fact thereafter. Call
the closest such world World 1. Now consider worlds where there is no
perfect match of particular fact before 8.00 (and in particular,
Gretta does not crack the egg), a miracle just *after* 8.00, and
then perfect match of particular fact thereafter. Call the closest
such world World 2. (Intuitively, in the first case we keep the past
fixed, insert a miracle just before 8.00 so that *c* doesn't
occur, and the future unfolds thereafter according to the (actual)
laws. In the second case, we keep the future fixed, insert a miracle
just *after* 8.00 so that *c* doesn't occur, and the past
unfolds according to the (actual) laws.) Why is World 1 closer to
actuality than is World 2?



Lewis's answer to that question comes from the fact that
*c* leaves very many traces: at 8.02, for example, there is the
egg cooking in the pan, the cracked empty shell in the bin, traces of
raw egg on Gretta's fingers, her memory of having just now cracked
it, and so on. So in World 2, Gretta fails to crack the egg but then,
shortly thereafter, seems to remember cracking it, there is the egg in
the pan, the empty shell in the bin, and so on. So World 2 -
since it contains all of these events without the egg being cracked in
the first place - needs to contain not just one miracle but
several: one to take care of each of these effects. World 1, by
contrast, requires just the one small miracle to stop Gretta cracking
the egg. Hence World 2 contains a 'big, diverse' miracle while
World 1 contains just one small miracle; hence World 1 is closer to
actuality than is World 2; hence Lewis's analysis yields the correct
result that had Gretta not cracked the egg, she would not have eaten
it.



The result in Gretta's case generalises to the extent that causes
are overdetermined by their effects but effects are not overdetermined
by their causes. Overdetermination of effects by causes does of course
happen - as when the victim is simultaneously shot by several
assassins - but it is relatively rare, and even when it happens
the effect is overdetermined only by a handful of events. By contrast,
the leaving of traces is ubiquitous - and (or so Lewis needs to
think) the extent of overdetermination, in any given case, is much
greater than in cases of cause-to-effect overdetermination. Both of
these, however, are contingent features of the actual world (or so
Lewis claims; but see SS2.1 below).



In general, then, the symmetric analysis of similarity and the *de
facto* asymmetry of overdetermination together imply that worlds
that accommodate counterfactual changes by preserving the actual past
and allowing for divergence miracles are more similar to the actual
world than worlds that accommodate such changes by allowing for
convergence miracles that preserve the actual future. This fact in
turn implies that, where the asymmetry of overdetermination obtains,
the present counterfactually depends on the past, but not on the
future.


### 1.3 Transitivity and Preemption



As Lewis notes (1973b), causal dependence between actual events is
sufficient for causation, but not necessary: it is possible to have
causation without causal dependence. A standard case of
'pre-emption' will illustrate this. Suppose that two shooters
conspire to assassinate a hated dictator, agreeing that one or other
will shoot the dictator on a public occasion. Acting side by side,
assassins *A* and *B* find a good vantage point, and,
when the dictator appears, both take aim (events *a* and *b*
respectively). *A* pulls her trigger and fires a shot that hits
its mark, but *B* desists from firing when he sees *A*
pull her trigger. Here assassin *A*'s actions (such as
her taking aim) are causes of the dictator's death, while
*B*'s actions (such as his taking aim) are merely
preempted potential causes. (Lewis distinguishes such cases of
*preemption* from cases of *symmetrical overdetermination*
in which two processes terminate in the effect, with neither process
preempting the other. Lewis believes that these cases are not suitable
test cases for a theory of causation since they do not elicit clear
judgements.) The problem raised by this example of preemption is that
both actions are on a par from the point of view of causal dependence:
had neither *A* nor *B* acted, then the dictator would
not have died; and if either had acted without the other, the dictator
would have died.



To overcome this problem Lewis extends causal dependence to a
transitive relation by taking its ancestral. He defines a *causal
chain* as a finite sequence of actual events *c*,
*d*, *e*, ... where *d* causally depends on
*c*, *e* on *d*, and so on throughout the
sequence. Then causation is finally defined in these terms:



(4)
*c* is *a cause* of *e* if and only if there
 exists a causal chain leading from *c* to *e*.

Given the definition of causation in terms of causal chains, Lewis is
able to distinguish preempting actual causes (such as *a*) from
preempted potential causes (such as *b*). There is a causal chain
running from *a* to the dictator's death, but no such
chain running from *b* to the dictator's death. Take, for
example, as an intermediary event occurring between *a* and the
dictator's death, the bullet from *A*'s gun
speeding through the air in mid-trajectory. The speeding bullet
causally depends on *a*, since that particular bullet would not
have been in mid-trajectory had *A* not taken aim; and the
dictator's death causally depends on the speeding bullet, since
by the time the bullet is in mid-trajectory *B* has refrained
from firing so that the dictator would not have died without the
presence of the speeding bullet. (Recall that we are not allowed to
'backtrack': it is not true that if the bullet had not been
mid-trajectory *A* would not have taken aim, and hence it is not
true that had the bullet not been mid-trajectory *B* would have
fired after all.) Hence, we have a causal chain, and so causation. But
no corresponding intermediary can be found between *b* and the
dictator's death; hence *b* does not count as causes of
the death.



Lewis's definition of causation also delivers the result that
causation is a transitive relation: whenever *c* causes *d*
and *d* causes *e*, it will also be true that *c*
causes *e*. The transitivity of causation fits with at least some
of our explanatory practices. For example, historians wishing to
explain some significant historical event will trace the explanation
back through a number of causal links, concluding that the event at
the beginning of the causal chain is responsible for the event being
explained. As we shall see later, however, some authors have claimed
that causation is not in fact transitive.


### 1.4 Chancy Causation



So far we have considered how the counterfactual theory of causation
works under the assumption of determinism. But what about causation
when determinism fails? Lewis (1986c) argues that chancy causation is
a conceptual possibility that must be accommodated by a theory of
causation. Indeed, contemporary physics tells us the actual world
abounds with probabilistic processes that are causal in character. To
take a familiar example (Lewis 1986c): suppose that you mischievously
hook up a bomb to a radioactive source and Geiger counter in such a
way that the bomb explodes if the counter registers a certain number
of clicks within ten minutes. If it happens that the counter registers
the required number of clicks and the bomb explodes, your act caused
the explosion, even though there is no deterministic connection
between them: consistent with the actual past and the laws, the Geiger
counter might not have registered sufficiently many clicks.



In principle a counterfactual analysis of causation is well placed to
deal with chancy causation, since counterfactual dependence does not
require that the cause was sufficient, in the circumstances, for the
effect - it only requires that the cause was necessary in the
circumstances for the effect. The problem posed by abandoning the
assumption of determinism, however, is that pervasive indeterminism
undermines the plausibility of the idea that - preemption and
overdetermination aside - effects generally counterfactually
depend on their causes. In the Geiger counter case above, for example,
suppose that the chance of the bomb exploding can be altered by means
of a dial. (A low setting means the Geiger counter needs to register a
lot of clicks in order for the bomb to go off in the next ten minutes,
thus making the explosion very unlikely; a high setting means it needs
to register very few clicks, thus making the explosion very likely.)
The dial is on a low setting; I increase the chance of the bomb
exploding by turning it up. My act was a cause of the explosion, but
it's not true that, had I not done it, the bomb would not have
exploded; it would merely have been very unlikely to do so. 



In order to accommodate chancy causation, Lewis (1986c) defines a more
general notion of causal dependence in terms of chancy
counterfactuals. These counterfactuals are of the form "If
*A* were the case *Pr* (*C*) would be
*x*", where the counterfactual is an ordinary
would-counterfactual, interpreted according to the semantics above,
and the *Pr* operator is a probability operator with narrow
scope confined to the consequent of the counterfactual. Lewis
interprets the probabilities involved as temporally indexed
single-case chances. (See his (1980) for the theory of single-case
chance.)



The more general notion of causal dependence reads:



(5)
Where *c* and *e* are distinct actual events,
*e* *causally depends* on *c* if and only if, if
*c* had not occurred, the chance of *e*'s
occurring would be much less than its actual chance.

This definition covers cases of deterministic causation in which the
chance of the effect with the cause is 1 and the chance of the effect
without the cause is 0. But it also allows for cases of irreducible
probabilistic causation where these chances can take non-extreme
values, as in the Geiger-counter-with-dial example above. It is
similar to the central notion of probabilistic relevance used in
probabilistic theories of type-causation, except that it employs
chancy counterfactuals rather than conditional probabilities. (See the
discussion in Lewis 1986c for the advantages of the counterfactual
approach over the probabilistic one. Also see the entry
 probabilistic causation.)



The rest of the theory of chancy causation follows the outlines of the
theory of deterministic causation: again, we have causation when we
have one or more steps of causal dependence.


## 2. Problems for Lewis's Counterfactual Theory



In this section we consider the principal difficulties for
Lewis's theory that have emerged in discussion over the last
forty-five years.


* 2.1 Temporal Asymmetry
* 2.2 Transitivity
* 2.3 Preemption


### 2.1 Temporal Asymmetry



There have been several important critical discussions of
Lewis's explanation of the temporal asymmetry of causation. (For
some early discussions see Horwich 1987: Chap. 10; Hausman 1998: Chap.
6; Price 1996: Chap. 6.) One important criticism concerns the
'asymmetry of miracles' that is central to Lewis's
account of the temporal asymmetry of causation: a miracle that
realises a counterfactual antecedent about particular facts at time
*t* by having a possible world diverge from the actual world
just before the time *t* is, Lewis claims, smaller and less
diverse than a miracle that realises the same counterfactual
antecedent and makes a possible world converge to the actual world
after the time *t*. Adam Elga (2000) has argued that the
asymmetry of miracles does not hold in many cases. 



Elga's argument proceeds by way of the example of Gretta cracking an
egg described earlier, and the basic idea is that in fact World 2
above - the closest world where particular fact in the past up
until shortly after 8.00 doesn't match actuality (and, in
particular, Gretta doesn't crack the egg), there is a miracle
shortly after 8.00, and thereafter World 2 evolves according to the
actual world's laws and matches actuality perfectly with respect to
particular fact. Thinking in the future-to-past direction for now,
consider what happens at the actual world from a time-reversed point
of view: Gretta transfers the egg from her plate to the hot pan, the
pan cools down and the egg uncooks, and then it leaps up into the
waiting shell, which closes neatly around it. Now (to get us to World
2) insert a *small* miracle - at 8.05, say, when the egg is
nicely cooked in the pan - altering just the positions of a few
molecules, say, so that what happens (again, proceeding from future to
past) is that (lawfully, once the miracle has occurred) the egg just
sits in the pan, cooling down and transferring heat to the pan in the
process, and then gradually rots in the way that eggs normally do
(except that they normally do that in the past-to-future direction).
The idea is that, while World 2 (viewed from past to future) looks
exceedingly strange - after all, it involves Gretta devouring
what was once a horrible rotten egg that somehow found its way into
her pan and then bizarrely de-rotted - none of that is unlawful.
It is just, thanks to the laws of thermodynamics, spectacularly
unlikely.



But how does this help us with the original problem with World 2,
namely the fact that all of the traces of Gretta's actual
egg-cracking, such as her remembering cracking it, the presence of the
empty, broken eggshell in the bin, and so on, would have to somehow be
individually be brought about in World 2 by additional miracles, in
order to preserve perfect match of particular fact from 8.05 onwards?
The short answer is that they don't. The 'traces' are there in
World 2 all right, and of course we would normally expect such
'traces' to point pretty conclusively to Gretta having recently
cracked an egg. But they don't lawfully *entail* that she did.
Again looking at World 2 from a time-reversed perspective, we take the
world as it is after 8.05, 'traces' and all, and run the laws
backwards (apart from the small miracle that makes the egg rot slowly
in the pan rather than leaping up into the waiting shell). What
'follows' (still going backwards in time) is anyone's guess, and
whatever it is it will doubtless look bizarre when viewed in the
usual, past-to-future direction. Be that as it may, World 2 is a world
with a single, small miracle at 8.05, with perfect match of particular
fact thereafter, which is just as close to the actual world as is
World 1, where there is a small miracle *before* Gretta cracks
the egg. So, Elga argues, there is no asymmetry of counterfactual
dependence, as Lewis defines it.



Several authors have come to the conclusion that a viable account of
the asymmetry of counterfactual dependence must derive not from the
asymmetry of miracles, as Lewis claims, but from thermodynamic
asymmetry: from the fact that entropy increases towards the future. In
particular, David Albert (2000) proposes that we need to assume the
'Past Hypothesis': the assumption that the universe began in an
extremely low entropy condition. The Past Hypothesis and its
relationship to the asymmetry of counterfactual dependence has been
much discussed. (See Frisch 2005, 2007; Loewer 2007; Price and Weslake
2009; Kutach 2002, 2013.)


### 2.2 Transitivity



As we have seen, Lewis builds transitivity into causation by defining
causation in terms of chains of causal dependence. However, a number
of alleged counter-examples have been presented which cast doubt on
transitivity. (Lewis 2004a presents a short catalogue of these
counterexamples.) Here is a sample of two counterexamples.



First, an unpublished but much-discussed example due to Ned Hall. A
hiker is walking along a mountain trail, when a boulder high above is
dislodged and comes careering down the mountain slopes. The hiker
notices the boulder and ducks at the appropriate time. The careering
boulder causes the hiker to duck and this, in turn, causes his
continued stride. (This second causal link involves *double
prevention*: the duck prevents the collision between hiker and
boulder which, had it occurred, would have prevented the hiker's
continued stride.) However, the careering boulder is the sort of thing
that would normally prevent the hiker's continued stride and so
it seems counterintuitive to say that it causes the stride.



Second, an example due to Douglas Ehring (1987). Jones puts some
potassium salts into a hot fire. Because potassium compounds produce a
purple flame when heated, the flame changes to a purple colour, though
everything else remains the same. The purple flame ignites some
flammable material nearby. Here we judge that putting the potassium
salts in the fire caused the purple flame, which in turn caused the
flammable material to ignite. But it seems implausible to judge that
putting the potassium salts in the fire caused the flammable material
to ignite.



Various replies have been made to these counterexamples. L.A. Paul
(2004) offers a response to the second example that involves
conceiving of the relata of causation as event *aspects*: she
argues that there is mismatch between the event aspect that is the
effect of the first causal link (the flame's being a purple
colour) and the event aspect that is the cause of the second causal
link (the flame's touching the flammable material). Thus, while
it's true that the purple flame did not cause the ignition,
there is no failure of transitivity after all. Maslen (2004) solves
the problem by appealing to a contrastivist account of causation (see
SS4 below): the contrast situation at the effect-end of the first
causal statement does not match up with the contrast situation at the
cause-end of the second causal statement. Thus, the first causal
statement should be interpreted as saying that Jones's putting
potassium salts in the fire rather not doing so caused the flame to
*turn purple rather than yellow*; but the second causal
statement should be interpreted as saying that *the purple
fire's occurring rather than not occurring* caused the
flammable material to ignite rather not to ignite. Where there is a
mismatch of this kind, we do not have a genuine counterexample to
transitivity. 



The first example cannot be handled in the same way. Some defenders of
transitivity have replied that our intuitions about the intransitivity
of causation in these examples are misleading. For instance, Lewis
(2004a) points out that the counterexamples to transitivity typically
involve a structure in which a *c*-type event generally
prevents an *e*-type but in the particular case the
*c*-event actually causes another event that counters the
threat and causes the *e*-event. If we mix up questions of what
is generally conducive to what, with questions about what caused what
in this particular case, he says, we may think that it is reasonable
to deny that *c* causes *e*. But if we keep the focus
sharply on the particular case, we must insist that *c* does in
fact cause *e*.



The debate about the transitivity of causation is not easily settled,
partly because it is tied up with the issue of how it is best for a
counterfactual theory to deal with examples of preemption. As we have
seen, Lewis's counterfactual theory relies on the transitivity
of causation to handle cases of preemption. If such cases could be
handled in some other way, that would take some of the theoretical
pressure off the theory, allowing it to concede the alleged
counterexamples to transitivity without succumbing to the difficulties
posed by preemption. (For more on this point see Hitchcock 2001. For
an extensive discussion of the issues around transitivity see Paul and
Hall 2013: Chap. 5.)


### 2.3 Preemption



As we have seen, Lewis employs his strategy of defining causation in
terms of chains of causal dependence not only to make causation
transitive, but also to deal with preemption examples. However, there
are preemption examples that this strategy cannot deal with
satisfactorily. Difficulties concerning preemption have proven to be
the biggest bugbear for Lewis's theory. (Paul and Hall 2013:
Chap. 3 contains an extensive discussion of the problems posed by
preemption and other kinds of redundant causation for counterfactual
theories.) 



In his (1986c: 200), Lewis distinguishes cases of *early* and
*late preemption*. In early preemption examples, the process
running from the preempted alternative is cut short before the main
process running from the preempting cause has gone to completion. The
example of the two assassins, given above, is an example of this sort.
The theory of causation in terms of chains of causal dependence can
handle this sort of example. In contrast, cases of late preemption are
ones in which the process running from the preempted cause is cut
short by the main process running to completion and bringing about the
effect before the preempted potential cause has the opportunity to do
so. The following is an example of late preemption due to Hall
(2004).



Billy and Suzy throw rocks at a bottle. Suzy throws first so that her
rock arrives first and shatters the glass; Billy's rock sails
through the air where the bottle had stood moments earlier. Without
Suzy's throw, Billy's throw would have shattered the
bottle. However, Suzy's throw caused the bottle to shatter,
while Billy's throw is merely a preempted potential cause. This
is a case of late preemption because the alternative process
(Billy's throw) is cut short by the main process (Suzy's
throw) running to completion.



Lewis's theory cannot explain the judgement that Suzy's
throw caused the shattering of the bottle. For there is no causal
dependence between Suzy's throw and the shattering, since even
if Suzy had not thrown her rock, the bottle would have shattered due
to Billy's throw. Nor is there a chain of stepwise dependences
running cause to effect, because there is no event intermediate
between Suzy's throw and the shattering that links them up into
a chain of dependences. Take, for instance, Suzy's rock in
mid-trajectory.  This event depends on Suzy's initial
throw, but the problem is that the shattering of the bottle does not
depend on it, because even without it the bottle would still have
shattered because of Billy's throw.



To be sure, the bottle shattering that would have occurred without
Suzy's throw would be different from the bottle shattering that
actually occurred with Suzy's throw. For a start, it would have
occurred later. This observation suggests that one solution to the
problem of late preemption might be to insist that the events involved
should be construed as fragile events. Accordingly, it will be true
rather than false that if Suzy had not thrown her rock, then the
actual bottle shattering, taken as a fragile event with an essential
time and manner of occurrence, would not have occurred. Lewis himself
does not endorse this response on the grounds that a uniform policy of
construing events as fragile would go against our usual practices, and
would generate many spurious causal dependences. For example, suppose
that a poison kills its victim more slowly and painfully when taken on
a full stomach. Then the victim's eating dinner before he drinks
the poison would count as a cause of his death since the time and
manner of the death depend on the eating of the dinner. (For
discussion of the limitations of this response see Lewis 1986c,
2000.)



The solution to the late preemption problem that Lewis cautiously
endorses in his 1986c appeals to the notion of quasi-dependence.
Consider a case that resembles the case of Billy and Suzy throwing
rocks at a bottle. Suzy throws a rock (*c*) and shatters the
bottle (*e*) in exactly the same way in which she does in the
original case. But in this case Billy and his rock are entirely
absent. In the original case, *e* is caused by but does not
counterfactually depend on *c*, whereas in this second case
*e* is caused by and does counterfactually depend on *c*.
But the intrinsic character of the process leading from *c* to
*e* is just the same in both cases. Thus, Lewis says, in the
original case (with Billy also throwing), *e* quasi-depends on
*c*. So "we could redefine a causal chain as a sequence of
two or more events, with either dependence or quasi-dependence at each
step. And as always, one event is a cause of another iff there is a
causal chain from one to the other" (1986c, 206). (A related
idea is pursued in Menzies 1996 and 1999.) Note that, this proposed
definition of a causal chain notwithstanding, the quasi-dependence
solution does not demand transitivity in the way that Lewis's
earlier solution to the problem of early preemption did: with back-up
potential causes safely out of the way, in all cases of preemption
(early and late), the effect should straightforwardly quasi-depend on
its cause.



Lewis's dissatisfaction with his own attempts to deal with the
problem of late preemption, as well as his theory's inability to
deal with 'trumping preemption' (Schaffer 2000a), led to
the development of his 2000 theory. A further problem relating to
preemption that arises for chancy causation - which the 2000
theory does not address - is discussed in
 SS5.4
 below.


## 3. Lewis's 2000 Theory



In an attempt to deal with the various problems facing his 1973
theory, Lewis developed a new version of the counterfactual theory,
which he first presented in his Whitehead Lectures at Harvard
University in March 1999. (A shortened version of the lectures
appeared as his 2000. The full lectures are published as his
2004a.)



Counterfactuals play a central role in the new theory, as in the old.
But the counterfactuals it employs do not simply state dependences of
*whether* one event occurs on *whether* another event
occurs. The counterfactuals state dependences of *whether*,
*when*, and *how* one event occurs on *whether*,
*when*, and *how* another event occurs. A key idea in the
formulation of these counterfactuals is that of an *alteration*
of an event. This is an actualised or unactualised event that occurs
at a slightly different time or in a slightly different manner from
the given event. An alteration is, by definition, a very fragile event
that could not occur at a different time, or in a different manner
without being a different event. Lewis intends the terminology to be
neutral on the issue of whether an alteration of an event is a version
of the same event or a numerically different event.



The central notion of the new theory is that of influence:



(6)
Where *c* and *e* are distinct events, *c*
*influences* *e* if and only if there is a substantial
range *c*1, *c*2, ... of different not-too-distant
alterations of *c* (including the actual alteration of
*c*) and there is a range *e*1, *e*2, ... of
alterations of *e*, at least some of which differ, such that if
*c*1 had occurred, *e*1 would have occurred, and if
*c*2 had occurred, *e*2 would have occurred, and so on.

Where one event influences another, there is a pattern of
counterfactual dependence of whether, when, and how upon whether,
when, and how. As before, causation is defined as an ancestral
relation:



(7)
*c* *causes* *e* if and only if there is a
 chain of stepwise influence from *c* to *e*.

One of the points Lewis advances in favour of this new theory is that
it handles cases of late as well as early preemption. (The theory is
restricted to deterministic causation and so does not address the
example of probabilistic preemption described below in
 SS5.4.)
 Reconsider, for instance, the example of late preemption involving
Billy and Suzy throwing rocks at a bottle. The theory is supposed to
explain why Suzy's throw, and not Billy's throw, is the
cause of the shattering of the bottle. If we take an alteration in
which Suzy's throw is slightly different (the rock is lighter,
or she throws sooner), while holding fixed Billy's throw, we
find that the shattering is different too. But if we make similar
alterations to Billy's throw while holding Suzy's throw
fixed, we find that the shattering is unchanged.



Another point in favour of the new theory is that it handles cases of
'trumping' preemption, first described by Jonathan
Schaffer (2000a). Lewis gives an example involving a major and a
sergeant who are shouting orders at the soldiers. The major and
sergeant simultaneously shout 'Advance!'; the soldiers
hear them both and advance. Since the soldiers obey the superior
officer, they advance because the major orders them to, not because
the sergeant does. So the major's command preempts or trumps the
sergeant's. Other theories have difficulty with trumping cases,
including - or so Lewis believes - his own attempt to
solve the late preemption problem by appealing to quasi-dependence
(2000, 184-5). The trumping case is one in which the causal chain
leading from the sergeant's shout to the soldiers'
advancing runs to completion - or at least, Lewis thinks, it is
epistemically possible that it does - just as the chain leading
from the major's shouting does. So it is an intrinsic duplicate
of the comparison case where the sergeant shouts but the major
doesn't; hence the soldiers' advancing quasi-depends on
the sergeant's shout, which is the wrong result. Lewis's
argues that his new theory handles trumping cases with ease. Altering
the major's command while holding fixed the sergeant's,
the soldier's response would be correspondingly altered. In
contrast, altering the sergeant's command, while holding fixed
the major's, would make no difference at all.



There is, however, some reason for scepticism about whether the new
theory handles the examples of late preemption and trumping completely
satisfactorily. In the example of late preemption, Billy's throw
has some degree of influence on the shattering of the bottle. For if
Billy had thrown his rock earlier (so that it preceded Suzy's
throw) and in a different manner, the bottle would have shattered
earlier and in a different manner. Likewise, the sergeant's
command has some degree of influence on the soldiers' advance in
that if the sergeant had shouted earlier than the major with a
different command, the soldiers would have obeyed his order. In
response to these points, Lewis must say that these alterations of the
events are too distant to be considered relevant. But some metric of
distance in alterations is required, since it seems that similar
alterations of Suzy's throw and the major's command are
relevant to their having causal influence.



It has also been argued that the new theory generates a great number
of spurious cases of causation (Collins 2000; Kvart 2001). The theory
implies that any event that influences another event to a certain
degree counts as one of its causes. But common sense is more
discriminating about causes. To take an example of Jonathan Bennett
(1987): rain in December delays a forest fire; if there had been no
December rain, the forest would have caught fire in January rather
than when it actually did in February. The rain influences the fire
with respect to its timing, location, rapidity, and so forth. But
common sense denies that the rain was a cause of *the fire*,
though it allows that it is a cause of the *delay in the fire*.
Similarly, in the example of the poison victim discussed above, the
victim's ingesting poison on a full stomach influences the time
and manner of his death (making it a slow and painful death), but
common sense refuses to countenance his eating dinner as a cause of
*his death*, though it may countenance it as a cause of *its
being a slow and painful death*. *Pace* Lewis, common sense
does not take just anything that affects the time and manner of an
event to be a cause of the event *simpliciter*.


## 4. Contextualism vs. Invariantism



A question that has received increasing attention in recent years is
whether causation is an 'invariant' relation or whether,
instead, the truth of a given causal claim varies according to the
context within which it is under discussion. (Note that
'invariant' is often used to describe a causal
relationship that is stable across a wide range of different
circumstances; this is not the meaning of 'invariant' as
it is being used here.) There is a wealth of evidence that
people's causal judgements are sensitive to contextual factors
(Hilton & Slugoski 1986; Cheng & Novick 1991; Knobe &
Fraser 2008; Hitchcock & Knobe 2009; Clarke et al. 2015; Kominsky
et al. 2015; Icard et al. 2017); however, in principle one might hold
the invariantist line and insist that what varies with context is not
truth but merely assertibility.



Consider a standard problem case: the gardener and the Queen equally
fail to water my flowers while I'm on holiday, and their
subsequent death (*e*) counterfactually depends equally on their
omissions. But many people's judgement is that only the
gardener's - and not the Queen's - omission is
a genuine cause of their death (Beebee 2004b, McGrath 2005, Livengood
& Machery 2007). We might accommodate that judgement by claiming
that it is false that the Queen's omission was a cause of
*e*, and conclude that we need some additional constraint on
causation aside from counterfactual dependence, e.g. that causes must
be 'deviant' or abnormal: the gardener's behaviour
was deviant (he was supposed to, or perhaps normally does, water my
flowers) but the Queen's was entirely normal (she never waters
my flowers, nor would anyone expect her to). Or we might accommodate
the judgement by appealing to pragmatic factors such as salience: the
gardener's and the Queen's omissions are both, equally,
causes of *e*, but in most conversational contexts the
Queen's omission simply isn't relevant. For example, if
I'm interested in finding out who is to blame for the
flowers' death, the Queen's neglect is simply irrelevant
to my investigation since it's obvious that she is not to
blame.



Lewis himself takes this latter, invariantist approach (Lewis 2004b).
The causes of any given event are many and varied, and in any given
explanatory context most of them fail to be salient; that is why it
strikes us as wrong to mention them (Lewis 1986d). Lewis's
approach is to explain (some of) the phenomena that motivate
contextualism by appealing to a broadly Gricean story about
conversational implicature. An advantage of invariantism is that it
allows us to conceive of causation as a fully objective,
mind-independent - or, as Menzies (2009: 342) puts it, a
'natural' - relation.



It is unclear, however, that all alleged cases of context-dependence
can be dealt with in this manner. Suzy's theft of the coconut
cake from the shop caused her subsequent illness: having stolen the
cake, she ate it - but (as she soon discovered) she's
allergic to coconut. Or did it? It depends, it would seem, on what we
contrast her theft of the coconut cake with. If she'd left the
shop empty-handed - or stolen a Bath bun instead - she
wouldn't have been ill. But if she'd paid for the cake
instead of stealing it, she still would have been ill. We sometimes
mark the intended contrast by, for example, stress: Suzy's theft
of the *coconut cake* caused her illness, but her *theft* of
the coconut cake didn't.



Lewis's original theory of events (1986b) was tailor-made to
deal with such cases - or so it might seem. According to that
theory, an event is a set of spatio-temporal regions of worlds. We can
distinguish between, for example, the event that is essentially
Suzy's theft of a cake (*e1*) and the event that
is essentially her acquiring (one way or another) a coconut cake
(*e2*): the two events consist in two different (but
overlapping) sets of spatio-temporal regions of worlds that share
their actual-world member, namely what actually happened in the cake
shop. And so - at least on the face of it - we can say
that *e2* was a cause of her illness but
*e1* was not (since had she not stolen a cake, she
would have bought the coconut cake instead).



It is unclear, however, that appeal to the essential features of
events successfully deals with the problem. After all, what if, had
Suzy not stolen a cake, the cake she would have bought was a Bath bun
and not the coconut cake she actually stole? (She really wanted a cake
but didn't have enough money for the coconut cake.) And in any case,
Lewis's own official view is that in supposing a putative cause
*c* absent we 'imagine that *c* is completely and cleanly
excised from history, leaving behind no fragment or approximation of
itself' (2004a: 90). So we don't appear to be able to recover the
truth of the claim that Suzy's *theft* of the cake was not a
cause of her subsequent illness. Moreover, Lewis's 2000 theory of
causation as influence abandons the distinction between the essences
of events to which the above response appealed: we have various
alterations of the theft of the coconut cake (*c*) -
including the purchase of a coconut cake and the theft of a Bath bun,
for example - some of which would have resulted in an alteration
of the effect *e* (Suzy's illness) and some of which would not
have. The degree of influence of *c* on *e* either is or is
not sufficient to make it the case that *c* was a cause of
*e*; either way, 'Suzy's theft of the coconut cake was a
cause of her illness' comes out either true or false
independently of context, which - according to the contextualist
- is the wrong result. (The invariantist, however, might insist
that there is no real problem here. 'Because she stole a coconut
cake' would be an inappropriate response to the question
'Why is Suzy ill?' if the request comes from the doctor,
who is not interested in how she procured the cake; but it would be an
appropriate response in the context of a discussion about, say, Suzy
getting her comeuppance from her shoplifting habit.)



Cei Maslen (2004), Jonathan Schaffer (2005) and Robert Northcott
(2008) all defend 'contrastive' accounts of causation. Schaffer
conceives causation as a four-place relation - *c* rather
than *c*\* caused *e* rather than *e*\* - and
claims that context (or other devices, such as stress on a particular
word) generally fixes the implied contrasts (*c*\* and *e*\*)
in our ordinary, two-place causal talk, thereby playing a role in the
truth or falsity of our (two-place) causal claims. Note that
contrastivism about causation is a distinct position from the view
that explanations are (always or sometimes) contrastive (see e.g.
Lewis 1986d, SSVI; Lipton 1991; Hitchcock 1999). On a contrastivist
view of explanation, explanations (always or sometimes) take the form
'Why *P* rather than *Q*?', where the contrast
(*Q*) may be explicitly stated or implied by the context in which
the question 'Why *P*?' is asked. Such a view is entirely
compatible with an invariantist view of causation, since the role of
the contrast may merely be to select which of *P*'s causes is
cited appropriately in answering the question. Note also that
contrastivism about explanation does not appear to solve the (alleged)
problem at hand. In the case of Suzy's theft of the cake, it is the
contrast on the side of causes (and hence explananda) that is at
issue, and not the contrast on the side of the effect (explanandum);
it is unclear how we might vary the contextually salient contrast to
'Suzy became ill' in such a way that different contrasts
deliver different verdicts on whether 'Suzy stole the coconut
cake' is an appropriate explanans.



While the contrastivist account of causation is generally considered
to be a version of causal contextualism, contrastivism is nonetheless
invariantist in the sense that there is a context-independent fact of the matter about
which four-place causal relations hold: it's just plain true that Suzy's stealing the cake
rather than leaving empty-handed caused her to be ill rather than
well, and false that her stealing rather than buying the cake caused
her to be ill rather than well. The contrastivist account might therefore be seen as a
kind of middle road between invariantism and contextualism. (See
Steglich-Petersen 2012 and Montminy & Russo 2016 for critical
discussions of contextualism/contrastivism.)



The contextualist/invariantist debate arises in a slightly different
form in much of the debate about the structural equations framework
(see
 SS5.5
 below), to which we now turn.


## 5. The Structural Equations Framework



A number of contemporary philosophers have explored an alternative
counterfactual approach to causation that employs the structural
equations framework. (Early exponents include Hitchcock 2001, 2007;
Woodward 2003; Woodward and Hitchcock 2003.) This framework, which has
been used in the social sciences and biomedical sciences since the
1930s and 1940s, received its state-of-the-art formulation in Judea
Pearl's landmark 2000 book. Hitchcock and Woodward acknowledge
their debt to Pearl's work and to related work on causal Bayes
nets by Peter Spirtes, Clark Glymour, and Richard Scheines (2001).
However, while Pearl and Spirtes, Glymour and Scheines focus on issues
to do with causal discovery and inference, Woodward and Hitchcock
focus on issues of the meaning of causal claims. For this reason,
their formulations of the structural equations framework are better
suited to the purposes of this discussion. The exposition of this
section largely follows that of Hitchcock 2001.


### 5.1 SEF: The Basic Picture



The structural equations framework describes the causal structure of a
system in terms of a causal model of the system, which is identified
as an ordered pair <*V*, *E*>, where *V* is a set
of variables and *E* a set of structural equations stating
deterministic relations among the variables. (We shall confine our
attention to deterministic systems here; see SS5.4 for a brief
discussion of chancy causation.) The variables in *V* describe
the different possible states of the system in question. While they
can take any number of values, in the simple examples to be considered
here the variables are binary variables that take the value 1 if some
event occurs and the value 0 if the event does not occur. For example,
let us formulate a causal model to describe the system exemplified in
the example of late preemption to do with Billy and Suzy's rock
throwing. We might describe the system using the following set of
variables:


* *BT* = 1 if Billy throws a rock, 0 otherwise;
* *ST* = 1 if Suzy throws a rock, 0 otherwise;
* *BH* = 1 if Billy's rock hits the bottle, 0
otherwise;
* *SH* = 1 if Suzy's rock hits the bottle, 0
otherwise;
* *BS* = 1 if the bottle shatters, 0 otherwise.



Here the variables are binary. But a different model might have used
many-valued variables to represent the different ways in which Billy
and Suzy threw their rocks, their rocks hit the bottle, or the bottle
shattered.



The structural equations in a model specify which variables are to be held
fixed at their actual values and how the values of other variables
depend on one another. There is a structural equation for each
variable. The form taken by a structural equation for a variable
depends on which kind of variable it is. The structural equation for
an *exogenous variable* (the values of which are determined by
factors outside of the model) takes the form of *Z* = *z*,
which simply states the actual value of the variable. The structural
equation for an *endogenous variable* (the values of which are
determined by factors within the model) states how the value of the
variable is determined by the values of the other variables. It takes
the form:



> 
> *Y* = *f*(*X*1,...,
> *X**n*)
> 



What does this structural equation mean? There are in fact competing
interpretations. Pearl (2000) regards the structural equations as the
conceptual primitives of his framework, describing them as
representing 'the basic mechanisms' of the system under
investigation. However, for the purposes of exposition, it is more
convenient to follow the interpretation of Woodward (2003) and
Hitchcock (2001), who think of the structural equations as expressing
certain basic counterfactuals of the following form:



> 
> If it were the case that *X*1 = *x*1,
> *X*2 = *x*2,...,
> *X**n* = *x**n*, then it
> would be the case that *Y* =
> *f*(*x*1,...,*x**n*).
> 



As this form of counterfactual suggests, the structural equations are
to be read from right to left: the antecedent of the counterfactual
states possible values of the variables *X*1 through
to *X**n* and the consequent states the
corresponding value of the endogenous variable *Y*. There is a
counterfactual of this kind for every combination of possible values
of the variables *X*1 through to
*X**n*. It is important to note that a structural
equation of this kind is therefore not, strictly speaking, an identity
since there is a right-to-left asymmetry built into it. This asymmetry
corresponds to the asymmetry of non-backtracking counterfactuals. For
example, supposing that the actual situation is one in which neither
Suzy nor Billy throws a rock so the bottle does not shatter, the
non-backtracking counterfactual "If either Suzy or Billy had
thrown a rock, the bottle would have shattered" is true. But the
counterfactual "If the bottle had shattered, either Suzy or
Billy would have thrown a rock" is false. 



As an illustration, consider the set of structural equations that
might be used to model the late preemption example of Billy and Suzy.
Given the set of variables *V* listed above, the members of the
set of the structural equations *E* might be stated as
follows:


* *ST* = 1;
* *BT* = 1;
* *SH* = *ST*;
* *BH* = *BT* & *~SH*;
* *BS* = *SH* v *BH*.



In these equations logical symbols are used to represent mathematical
functions on binary variables: *~X* = 1 [?] *X*;
*X* v *Y* = max{*X*, *Y*}; *X* & *Y*
= min{*X*, *Y*}. The first two equations simply state the
actual values of the exogenous variables *ST* and *BT*. The
third equation encodes two counterfactuals, one for each possible
value of *ST*. It states that if Suzy had thrown a rock (which in
fact she did), her rock would have hit the bottle; and if she
hadn't thrown a rock, it wouldn't have hit the bottle. The
fourth equation encodes four counterfactuals, one for each possible
combination of values for *BT* and *~SH*. It states that if
Billy had thrown a rock and Suzy's rock hadn't hit the
bottle, Billy's rock would have hit the bottle, but it
wouldn't have done so if one or more of these conditions had not
been met. The fifth equation also encodes four counterfactuals, one
for each possible combination of values for *SH* and *BH*.
It states that if one or other (or possibly both) of Suzy's rock
or Billy's rock had hit the bottle, the bottle would have
shattered; but if neither rock had hit the bottle, the bottle
wouldn't have shattered.



The structural equations above can be represented in terms of *a
directed graph*. The variables in the set *V* are represented
as nodes in the graph. An arrow directed from one node *X* to
another *Y* represents the fact that the variable *X*
appears on the right-hand side of the structural equation for
*Y*. In this case, *X* is said to be *a parent* of
*Y*. Exogenous variables are represented by nodes that have no
arrows directed towards them. A *directed path* from *X* to
*Y* in a graph is a sequence of arrows that connect *X* with
*Y*. The directed graph of the model described above of Billy and
Suzy example is depicted in Figure 1 below:



![Figure 1](figure1.jpg)
Figure 1.




The arrows in this figure tell us that the bottle's shattering
is a function of Suzy's rock hitting the bottle and
Billy's rock hitting the bottle; that Billy's rock hitting
the bottle is a function of Billy's throwing a rock and
Suzy's rock hitting the bottle; and that Suzy's rock
hitting the bottle is a function of her throwing the rock. It's
important to note that the nodes in the graph represent variables and
not - as in the case of 'neuron diagrams', such as
those found in Lewis 1986c - values of variables. Note also that
while the arrows tell us that there are counterfactual dependence
relations between the values of the variables - all of the
possible values, not just the actual ones - they don't
tells us what those dependence relations are; for that, you have to
look at the structural equations. For example, the directed graph only
tells us that the value of *BH* counterfactually depends somehow
or other on the values of *BT* and *SH*; e.g. it
doesn't tell us that if Billy had thrown and Suzy's rock
hadn't hit, Billy's rock would have hit.)



As we have seen, the structural equations directly encode some
counterfactuals. However, some counterfactuals that are not directly
encoded can be derived from them. Consider, for example, the
counterfactual "If Suzy's rock had not hit the bottle, it
would still have shattered". As a matter of fact, Suzy's
rock did hit the bottle. But we can determine what would have happened
if it hadn't done so, by replacing the structural equation for
the endogenous variable *SH* with the equation *SH* = 0,
keeping all the other equations unchanged. So, instead of having its
value determined in the ordinary way by the variable *ST*, the
value of *SH* is set 'miraculously'. Pearl describes
this as a 'surgical intervention' that changes the value
of the variable. In terms of its graphical representation, this
amounts to wiping out the arrow from the variable *ST* to the
variable *SH* and treating *SH* as if it were an exogenous
variable. After this operation, the value of the variable *BS*
can be computed and shown to be equal to 1: given that Billy had
thrown his rock, his rock would have hit the bottle and shattered it.
So this particular counterfactual is true. This procedure for
evaluating counterfactuals has direct affinities with Lewis's
non-backtracking interpretation of counterfactuals: the surgical
intervention that sets the variable *SH* at its hypothetical
value but keeps all other equations unchanged is similar in its
effects to Lewis's small miracle that realises the
counterfactual antecedent but preserves the past.



In general, to evaluate a counterfactual, say "If it were the
case that *X*1,...,*X**n*, then
...", one replaces the original equation for each variable
*X*i with a new equation stipulating its hypothetical
value,while keeping the other equations unchanged; then one computes
the values for the remaining variables to see whether they make the
consequent true. This technique of replacing an equation with a
hypothetical value set by a 'surgical intervention'
enables us to capture the notion of counterfactual dependence between
variables:



(8)
A variable *Y* *counterfactually depends* on a
variable *X* in a model if and only if it is actually the case
that *X* = *x* and *Y* = *y* and there exist
values *x*' [?] *x* and *y*' [?] *y*
such that replacing the equation for *X* with *X* =
 *x*' yields *Y* = *y*'.

Of course, so far we just have something we are calling a
'causal model', <*V, E*> ; we haven't
been told anything about how to extract causal information from it. As
should be obvious by now, the basic recipe is going to be roughly as
follows: the truth of '*c* causes *e*' (or
'*c* is an actual cause of *e*'), where *c*
and *e* are particular, token events, will be a matter of the
counterfactual relationship, as encoded by the model, between two
variables *X* and *Y*, where the occurrence of *c* is
represented by a structural equation of the form *X =
x1* and the occurrence of *e* is represented by a
structural equation of the form *Y = y1*. We can see
straightaway, however, that we can't straightforwardly identify
causation with counterfactual dependence as defined in (8) above.
That would get us the truth of "Suzy's throw caused her
rock to hit the bottle" (*ST* = 1 and *SH* = 1, and,
since *SH = ST* is a member of *E*, we know that if we
replace *ST* = 1 with *ST* = 0, we get *SH* = 0). But
it won't get us, for example, the truth of "Suzy's
throw caused the bottle to shatter", since if we replace
*ST* = 1 with *ST* = 0 and work through the equations we
still end up with *BS* = 1.



How, then, might we define 'actual causation' using the
structural equations framework? We'll get there by considering
how SEF deals with cases of late preemption such as the Suzy and Billy
case. Halpern and Pearl (2001, 2005), Hitchcock (2001), and Woodward
(2003) all give roughly the same treatment of late preemption. The key
to their treatment is the employment of a certain procedure for
testing the existence of a causal relation. The procedure is to look
for an intrinsic process connecting the putative cause and effect;
suppress the influence of their non-intrinsic surroundings by
'freezing' those surroundings as they actually are; and
then subject the putative cause to a counterfactual test. So, for
example, to test whether Suzy's throwing a rock caused the
bottle to shatter, we should examine the process running from
*ST* through *SH* to *BS*; hold fix at its actual value
(that is, 0) the variable *BH* which is extrinsic to this
process; and then wiggle the variable *ST* to see if it changes
the value of *BS*. The last steps involve evaluating the
counterfactual "If Suzy hadn't thrown a rock and
Billy's rock hadn't hit the bottle, the bottle would not
have shattered". It is easy to see that this counterfactual is
true. In contrast, when we carry out a similar procedure to test
whether Billy's throwing a rock caused the bottle to shatter,we
are required to consider the counterfactual "If Billy
hadn't thrown his rock and Suzy's rock had hit the bottle,
the bottle would not shattered". This counterfactual is false.
It is the difference in the truth-values of these two counterfactuals
that explains the fact that it was Suzy's rock throwing, and not
Billy's, that caused the bottle to shatter. (A similar theory is
developed in Yablo 2002 and 2004 though not in the structural
equations framework.)



Hitchcock (2001) presents a useful regimentation of this reasoning. He
defines *a route* between two variables *X* and *Z* in
the set *V* to be an ordered sequence of variables <*X*,
*Y*1,..., *Y**n*, *Z*> such
that each variable in the sequence is in *V* and is a parent of
its successor in the sequence. A variable *Y* (distinct from
*X* and *Z*) is *intermediate* between *X* and
*Z* if and only if it belongs to some route between *X* and
*Z*. Then he introduces the new concept of an active causal
route:



(9)
The route <*X*, *Y*1,...,
*Y**n*, *Z*> is *active* in the causal
model <*V*, *E*> if and only if *Z* depends
counterfactually on *X* within the new system of equations
*E'* constructed from *E* as follows: for all *Y*
in *V*, if *Y* is intermediate between *X* and *Z*
but does not belong to the route <*X*,
*Y*1,..., *Y**n*, *Z*>,
then replace the equation for *Y* with a new equation that sets
*Y* equal to its actual value in *E*. (If there are no
intermediate variables that do not belong to this route, then
*E'* is just *E*.) (Hitchcock 2001: 286).

This definition generalises the informal idea sketched in the example
of Suzy and Billy. <*ST, SH, BS*> is an active causal route
because when we hold *BH* fixed at its actual value
(Billy's rock doesn't hit the bottle), *BS*
counterfactually depends on *ST*. By contrast, the route
<*BT, BH, BS*> is not active because when we hold *SH*
fixed at its actual value (Suzy's rock does hit the bottle),
*BS* does not counterfactually depend on *BT*.



In terms of the notion of an active causal route, Hitchcock defines
actual or token causation in the following terms:



(10)
If *c* and *e* are distinct actual events and *X*
and *Z* are binary variables whose values represent the
occurrence and non-occurrence of these events, then *c* is *a
cause* of *e* if and only if there is an active causal route
from *X* to *Z* in an appropriate causal model <*V*,
 *E*>.

(We shall return to the notion of an 'appropriate causal
model' in
 SS5.3
 below.)



As stated, (10) doesn't handle cases of symmetric
overdetermination - as when Suzy and Billy both throw their
rocks independently, each throw is sufficient for the bottle to break,
and both rocks hit the bottle - so neither throw preempts the
other, since neither throw is on an active route as defined in (9).
To deal with such cases, Hitchcock weakens (10) by replacing the
'active route' in (10) with the notion of *a weakly
active route* (2001: 290). The essential idea here that there is a
weakly active route between *X* and *Z* just when *Z*
counterfactually depends on *X* under the freezing of some
possible, not necessarily actual, values of the variables that are not
on the route from *X* to *Z*. Intuitively, to recover
counterfactual dependence between Suzy's throw and the
shattering we hold fixed BT = 0: had Suzy not thrown *in the model
where Billy doesn't throw*, the bottle would not have
shattered. Similarly for Billy's throw.



The basic strategy deployed here to deal with both preemption and
symmetric overdetermination bears an obvious similarity to
Lewis's quasi-dependence solution to the late preemption
problem. Lewis resorts to quasi-dependence because the shattering of
the bottle (*e*) does not counterfactually depend on Suzy's
throw (*c*), thanks to what would have happened had she not
thrown (viz, Billy's rock would have shattered the bottle
instead). *e* quasi-depends on *c*, however, because of the
fact that in a possible world with the same laws where the intrinsic
character of the process from *c* to *e* is the same but
Billy doesn't throw, there is the required counterfactual
dependence. 'Freezing' variables that are not intrinsic to
the *c*-*e* process at their actual values (in late
preemption cases) - e.g. freezing *BH* at 0 - turns
roughly the same trick. The core difference is that Lewis's
solution involves appealing to the truth of a perfectly ordinary
counterfactual ("If Suzy had not thrown, ...") at a
possible world where some actual events (e.g. Billy's hit)
don't occur, while the structural-equations solution involves
appealing to the truth of a counterfactual with a special kind of
antecedent ("Had Suzy not thrown and Billy's rock still
not hit, ..."). Hitchcock calls these 'explicitly
nonforetracking' (ENF) counterfactuals. (Similarly for symmetric
overdetermination, where we 'freeze' *BT* at 0
- this time a non-actual value - to recover counterfactual
dependence between Suzy's throw and the shattering.)


### 5.2 SEF and Counterfactuals



Those who have pursued the SEF approach to providing an analysis of
'actual' causation - that is, the causal relation
between actual, particular events - have had very little to say
about the semantics of the counterfactuals that underpin SEF. Some
authors (e.g. Hitchcock 2001) explicitly - and many authors
implicitly - assume a broadly Lewisian approach to
counterfactuals, so that the structural equations are representations
of relations of facts about counterfactual dependence - as
described above - whose truth conditions are broadly
Lewisian.



On the other hand, one might go in the other direction, using the SEF
approach to deliver the truth conditions for counterfactuals by
treating the structural equations (such as *SH = ST*) as
representations of *causal* dependency relationships, which in
turn deliver those truth conditions (Galles & Pearl 1998, Woodward
and Hitchcock 2003, Schulz 2011, Briggs 2012). Relatedly, one might
eschew a Lewisian, miracle-based conception of an intervention and
define interventions in explicitly causal terms (see e.g. Woodward and
Hitchcock 2003: 12-13).



The choice between these two different ways of proceeding connects
with the broader debate about whether causation should be analysed in
terms of counterfactuals or vice versa. Lewis, of course, takes the
former approach. One attraction of doing so - at least for him
- is that it fits within a broadly Humean agenda: since
causation is a modal notion, it threatens the thesis of Humean
supervenience (Lewis 1986a, ix) unless it can somehow we cashed out in
terms of similarity relations between worlds, where those similarity
relations do not appeal in turn to causal (or other Humean
supervenience-violating) features of worlds. Lewis's analysis of
counterfactuals, together with his analysis of laws, turns that trick.
By contrast, other authors have argued that the trick simply cannot be
turned: we cannot analyse counterfactuals without appealing to
causation (Edgington 2011).



There are deep metaphysical issues at stake here, then: one might view
the SEF approach as offering a more sophisticated variant of
Lewis's approach that shares the reductionist aspirations of
that approach. Or one might - especially if one is sceptical
about the prospects for those reductionist aspirations - take
the SEF approach in anti-reductionist spirit, viewing it not as a way
of defining causation in non-causal terms but rather as a way of
extracting useful and sophisticated causal information from an
inherently causal model of a given complex situation.


### 5.3 Models and Reality



It is a general feature of the SEF approach that the model need not
include as variables all of the factors that are relevant to the
effect under consideration (and indeed no model never does -
there are just too many factors). In the Billy/Suzy model above, for
example, there are no variables describing the actual and possible
states corresponding to causal intermediaries between Billy's or
Suzy's throwing (or not throwing) and their respective rocks
hitting (or not hitting) the bottle. So what determines which
variables should and should not be included in the model in order to
uncover the causal relationships between the variables we're
interested in?



It's important to stress that there is no uniquely correct model
to be had for any given situation. A model that, for example,
interpolated large numbers of intermediaries between Suzy's
throw and her rock's hitting the bottle would reveal more of the
causal structure of both the actual situation and various different
counterfactual alternatives. But that doesn't make it the
'right' model for considering the causal status of
Billy's and Suzy's respective throws with respect to the
shattering of the bottle. Such a model would deliver the same result
as the simple one described above, and so the additional variables
would simply be an unnecessary complication. On the other hand, there
are limits on what we can leave out. For example, a causal model that
just included *ST* and *BS* as variables would not deliver
the result that Suzy's throw caused the bottle to shatter, since
that counterfactual is not true on this model. (To get it to come out
true, we need to include *BH* and hold it fixed at its actual
value, *BH* = 0.)



So what are the constraints on causal models, such that they
accurately represent the causal facts that we're interested in
(Halpern and Hitchcock 2010: SSSS4-5)? Various authors
have proposed constraints that tell us what count as (to use
Hitchcock's term) 'apt' models, many of which are
analogues of Lewis's constraints and for the same reasons,
namely to ensure that there is no spurious counterfactual dependence.
Thus Hitchcock (2001: 287) proposes that the values of variables
should not represent events that bear logical or metaphysical
relations to one another, and Blanchard and Schaffer (2017: 182)
propose that the values allotted should represent intrinsic
characterisations. Hitchcock (2001: 287) also proposes that the
variables should not be allotted values 'that one is not willing
to take seriously' (about which more below). Halpern and
Hitchcock (2010) add a 'stability' constraint: adding
additional variables should not overturn the causal verdicts. (This
constraint addresses the problem of the 'model' described
above that just includes *ST* and *BS*; that model delivers
a verdict, namely that Suzy's throw doesn't cause the
bottle to shatter, which is overturned by adding additional
variables.) And Hitchcock (2007: 503) proposes the constraint that the
model "should include enough variables to capture the essential
structure of the situation being modeled". (Though if one had
reductionist aspirations, this constraint would appear to render
one's analysis of causation viciously circular, since the
'essential structure' of the situation is presumably its
essential *causal* structure - just what a causal model is
supposed to deliver.)



Precisely what the constraints should be on 'apt' or
'appropriate' models is a matter of ongoing philosophical
debate (Blanchard and Schaffer 2017: SS1.3). The focus here is on
constraints that guarantee that a model does not deliver spurious
results (e.g. that Suzy's throw doesn't cause the bottle
to shatter). However, SEF is also used as a practical tool in
scientific inquiry, and this brings additional normative questions
into play concerning the choice of variables and their range of
permitted values. In the context of assigning blame for the broken
bottle, for example, it would not be relevant to include the strength
of the bottle's glass as a variable; by contrast, a local shop
owner - tired of Suzy and Billy's vandalism (they
routinely break his shop window as well as any bottles they come
across) - might be very interested in the question of what
strength of glass might suffice to withstand their rock-throwing. (See
e.g. Woodward 2016, Hitchcock 2017, and - for a practical
example in the context of causal inference in machine learning -
Chalupka, Eberhardt & Perona 2017.) 


### 5.4 SEF and Chancy Causation



As we saw in
 SS1.4
 above, Lewis revised his 1973 account of causation to take account of
chancy causation. Any account of causation that is based on the idea
that causes increase the chances of their effects encounters two main
problems: chance-increase is neither necessary nor sufficient for
causation. (Case 1: The doctor reduces the patient's very high
chance of a heart attack by drastic surgery. Unfortunately, the
surgery itself causes the patient to have a heart attack. The surgery
lowers the chance of, but causes, the heart attack. Case 2: Billy and
Suzy are throwing rocks at the bottle again. Each of their throws
increases the chance of the bottle shattering, but Suzy's throw
pre-empts Billy's. Billy's throw raises the chance of, but
does not cause, the bottle's shattering.)



The first kind of case can (perhaps) be dealt with by Lewis's
modified 1973 account, by finding some intermediate event *d*
such that the surgery raises the chance of *d* and *d* in
turn raises the chance of the heart attack. But it can't deal
with the second kind of case: Billy's throw meets Lewis's
sufficient condition for chancy causation and so the modified 1973
account erroneously counts it as a cause (Menzies 1996). This is a
problem that Lewis saw for his own account and never solved: his
later, 2000 account of causation as 'influence' assumes
determinism (2000: n.1; Lewis 2004a: 79-80) and so ignores the
problem. Examples of both kinds have been the subject of extensive
discussion in the context of both counterfactual and probabilistic
theories of causation. (For discussions about how best to deal with
them within theories that don't assume determinism, see Barker
2004; Beebee 2004a; Dowe 2000, 2004; Hitchcock 2004; Kvart 2004;
Noordhof 1999, 2004; Ramachandran 1997, 2004.)



SEF accounts similarly overwhelmingly assume determinism: as with
Lewis's original 1973 account, the basic building block of such
accounts is non-chancy counterfactual dependence. However there have
been recent attempts to extend SEF-based analyses to cover chancy
causation by Fenton-Glynn (Glynn 2011, Fenton-Glynn 2017; see also the
entry on
 Probabilistic Causation,
 SS4.4).


### 5.5 Defaults and Deviants



In
 SS4
 we saw two examples of the kinds of case that have motivated some
authors to endorse contextualism. The examples exhibit different
features. In the case of the theft of the coconut cake, the idea is
that in different contexts of utterance, the very same causal claim
- "Suzy's theft of the coconut cake from the shop
caused her subsequent illness" - can vary in truth value,
depending on whether the context determines that what is at issue is
Suzy's criminality (theft rather than purchase) or, instead,
which item she stole (coconut cake rather than Bath bun). The case of
the gardener and the Queen, by contrast, is a case where the (alleged)
cause (the gardener's omission) and non-cause (the
Queen's) stand in the same counterfactual relationship to the
effect, and yet judgements differ with respect to their causal
status.



A major focus of the debate within the SEF approach has been the
second kind of case - cases of what Menzies calls
'counterfactual isomorphs' (Menzies 2017), where two
different scenarios have isomorphic causal models and yet our
judgements about what causes what differs in the two different
scenarios. For example, consider a case of 'bogus
prevention' (Hiddleston 2005): believing that Killer has
previously poisoned Victim's coffee, Bodyguard puts an antidote
into it. However, in fact Killer had a change of heart and
didn't poison the coffee. Victim survives - but his
survival was not, surely, caused by Bodyguard's action, for
there was no threat to Victim's life that Bodyguard neutralized.
It is possible, however, to construct a causal model of this case that
is isomorphic to a standard case of symmetric overdetermination: Billy
and Suzy are throwing rocks again, but this time both of their throws
hit the bottle at the same time, so that each is sufficient for its
shattering and neither hit pre-empts the other. In that case
(allegedly), we identify both Billy's and Suzy's throws as
causes of the bottle's shattering (Blanchard and Schaffer 2017:
185-6).



One broad line of response to the problem of counterfactual isomorphs
has been to distinguish between 'default' (or
'normal') and 'deviant' events, and to build
this distinction into the way in which causal information is extracted
from the model. For example, Menzies' approach exploits the
machinery seen above with respect to Hithcock's solution to the
problem of symmetric overdetermination, which involves fixing
'off-path' variables at non-actual values. Menzies'
suggestion is that we fix those variables at their 'most
normal' values, so that in effect we evaluate the relevant
counterfactuals from the perspective of a world where those normal
values are actualised rather than from the perspective of the actual
world (Menzies 2004, 2007, 2009).



Intuitively, the basic idea is that (in the overdetermination case)
the 'most normal' value of each of *BT* and *ST*
is 0 (throwing rocks at bottles isn't normal!), so from the
perspective of a world where *ST* = 0, *BS* counterfactually
depends on *ST* (and similarly for *BT*). So both
Billy's and Suzy's throws count as causes of the
bottle's shattering. Poisoning someone's coffee is also
not normal. So we hold Killer's failure to poison the coffee
fixed (so in this case the 'most normal' world is just the
actual world as far as the poisoning is concerned), which delivers the
right result that Victim's survival does not counterfactually
depend on Bodyguard's administering the antidote. (See Hitchcock
2007 for a different solution that trades on the default/deviant
distinction.)



It is hard to see, however, how there could be a univocal and
reasonably well defined notion of 'normality' that would
deliver clear verdicts on what the 'normal' or
'default' values of variables are, and hence would deliver
an objectively 'correct' set of models, all of which
deliver the same verdict for the same situation (Blanchard and
Schaffer 2017: SSSS2 and 3). Blanchard and Schaffer argue that
'default-relativity' does not solve some of the problems
it was supposed to solve; more importantly, however, they argue that
the alleged cases of isomorphic causal models are not really genuine
cases in the first place: they arise because one (or both) of the
relevant models falls foul of the independently-motivated criteria for
'aptness' (see
 SS5.3
 above). For example, one such criterion is that the variables not be
allotted values 'that one is not willing to take
seriously' (Hitchcock 2001: 287). But the case of the gardener
and the Queen violates that criterion: the possibility of the Queen
watering my flowers is, precisely, one that we do not take seriously.
In other cases, they argue, the isomorphism results from deploying
'impoverished' models: models that fail to include enough
variables to adequately represent the 'essential causal
structure' of the situation being modelled (Blanchard and
Schaffer 2017: SS3).



Blanchard and Schaffer's own view is that, insofar as our causal
judgements do exhibit something like default-relativity, this is due
to cognitive heuristics that engender biases in our judgements.
Alternatives to 'deviant' events tend to 'leap
out' at us - they are salient because they are easy to
imagine - whereas alternatives to 'default' events
don't. Blanchard and Schaffer's view can thus be seen as a
version of invariantism, where the kinds of case that are supposed to
motivate contextualism are accommodated through features of our causal
talk and thought that are extraneous to the (non-norm-dependent)
concept of causation.



A host of issues remain in this area. One is whether or not we should
demand a uniquely 'correct' answer to the question of what
count as 'normal' or 'default' values of
variables in the first place; perhaps, if for example there are
different dimensions of 'normality' (statistical
likelihood, consonance with moral or legal norms, etc.), we should
embrace the idea that two apt models of the very same situation might
deliver different and equally correct results, depending on how the
normal or default values are set - which in turn would depend on
the context (e.g. the purpose for which the model is being
deployed).



More general issues that lie in the background of the debate about
'default-relativity' include whether the purpose of the
concept of causation is best served by an 'egalitarian' or
'invariantist' concept of causation or rather by one that
takes the concept to enshrine normative considerations. Hitchcock
(2017), for example, argues that since our interest in what causes
what is, in essence, an interest in what kinds of intervention would
bring about the kinds of results we want, we should take the latter
line. A still more general issue is whether there is just one concept
of causation at which all of the accounts on the table are or should
be aiming, or instead several (Hall 2004, McDonnell 2018). Or perhaps
causation is what Nancy Cartwright (following Neurath) calls a
'Ballung' concept: "a concept with rough, shifting,
porous boundaries, a congestion of different ideas and implications
that can in various combinations be brought into focus for different
purposes and in different contexts" (2017: 136).