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closest worlds in which that path is the only path between a and v (ibid.: 35). That
is, in worlds in which there was no way the victim could have died except by [the
novice s] bullet (ibid.: 35), the chance of the victim s death would be higher were
the trainee to shoot than it would be if he were to refrain from shooting. It should
now be clear that Dowe s proposal is really strikingly similar to my own, for one
class of possible worlds in which there was no way the victim could have died
except by [the novice s] bullet will be the class of worlds in which the back-up
assassin refrains from firing regardless of the trainee s action. That is, the counter-
factuals 2a and 2b that are to be compared on my account are precisely the sorts of
counterfactuals that are to be compared on Dowe s account as well.
The two accounts differ with respect to how the distinct paths that connect cause
and effect are to be delineated. According to my account, this is determined
entirely by the structure of the counterfactuals that characterize a given scenario.
According to Dowe s account, the decomposition of influence into distinct paths is
determined by the actual and potential causal processes that link the events in
question. To make this distinction vivid, note that on my analysis Figure 8.1 would
accurately characterize the structure of Back-up Assassin, even if the assassins
guns killed by some sort of unmediated action-at-a-distance. All that is required is
that the counterfactuals 1a 1b, and 2a 2d be true.
The complaint that I will be levelling against Dowe s account is that it does not go
far enough in telling us when two paths are genuinely distinct. Determining when
Routes, processes and chance-lowering causes 145
causal paths are genuinely distinct is essential if genuine cases of probability-
lowering causation are to be distinguished from imposters.
6 Delineating causal routes
In order to present this complaint more precisely, it will be helpful to return to the
positive account developed in Section 4 above. There we introduced two variables:
A, which takes the value 1 or 0 depending upon whether or not the novice shoots;
and B, which takes the value 1 or 0 depending upon whether or not the back-up
assassin shoots. But was this really necessary? As the scenario was described,
exactly one of the two assassins would shoot.11 So why not introduce a single vari-
able, S, which takes the value 1 if the trainee shoots (that is, if a occurs), and takes
the value 0 if the back-up shoots (b occurs)? Then we would arrive at a much
simpler description of the scenario, characterized by the following counterfactuals:
3a If a were to occur, then Ch(v) = 0.3
3b If b were to occur, then Ch(v) = 0.7
with corresponding equation:
(3) Ch(V = 1) = 0.3 + 0.4(1 S)
If we represent this graphically, we will have two nodes, S and V, with an arrow
running from S to V (see Figure 8.2). In this representation, there is only one route
from S to V. Along this route, a lowers the chance of v, thus reinstating the problem of
chance-lowering causes. If my solution is to succeed, some clarification is in order:
there must be a principled reason for taking Equations 1 and 2 to be the correct way to
characterize the scenario, and for taking Equation 3 to be inappropriate.
In order to address this question, we must ask what it means to represent two
events (such as a and b) as different values of the same variable, or as values of
different variables. When we represent two events as different values of the same
variable, we are representing those events as mutually exclusive. A variable is a func-
tion (over possible worlds, if you like), and hence it must be single-valued. More-
over, the two events will be exclusive, regardless of the equations that represent the
system. In particular, the exclusion of one event by the other will not correspond to
any of the arrows that figure in the corresponding graph. What this suggests is that
the relevant form of exclusion is not causal, but logical, conceptual or metaphysical.
In our example, the novice s shot prevents the back-up from shooting. This is a
causal relationship between the two events, corresponding to the arrow from A to B
in Figure 8.1. This causal relationship is concealed in Figure 8.2. We may thus offer
SV
Figure 8.2
146 Christopher Hitchcock
the following rule of thumb: two events are to be represented as values of different
variables if: (a) they are not mutually exclusive; or (b) they are mutually exclusive,
but the exclusion is causal one event prevents the other from occurring.
This rule is far from satisfactory: it appeals to causal facts, and the current
project is to recover causal facts from algebraic and graphical structures that are
defined in terms of counterfactuals only.12 Nonetheless, the foregoing consider-
ations point us in the right direction. We noted in Section 3 that within counter-
factual theories of causation, causation can only hold between distinct events. The
reason for the restriction to distinct events is familiar by now: we don t want to say
that my raising my arm caused my arm to go up, that my saying hello caused me
to say hello loudly, that my stroll caused my first fifty steps, and so on. In each
case, there is counterfactual dependence if I hadn t raised my arm, it wouldn t
have gone up but not causation. This problem was posed by Kim (1973). The
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