A Criterion of Probabilistic Causation*

Charles R. Twardy and Kevin B. Korbyz

The investigation of probabilistic causality has been plagued by a variety of misconceptions

and misunderstandings. One has been the thought that the aim of the probabilistic account of

causality is the reduction of causal claims to probabilistic claims. Nancy Cartwright (1979)

has clearly rebutted that idea. Another ill-conceived idea continues to haunt the debate,

namely the idea that contextual unanimity can do the work of objective homogeneity. It

cannot. We argue that only objective homogeneity in combination with a causal interpre-

tation of Bayesian networks can provide the desired criterion of probabilistic causality.

1. Introduction. The early work on probabilistic causality, especially that
of Salmon (1980) and Suppes (1970), made clear that we need to dis-
criminate among purported causal factors, all of which have a probabi-
listic impact on an effect, ruling out those which are screened off by
common ancestors. Subsequently, it also became clear that such a degree
of discrimination was insufficient, that the context in which probabilistic
impacts are found must be more acutely observed. The Contextual
Unanimity Thesis (CUT) emerged as the leading proposal for sifting out
those factors which promote (or prevent) some effect. CUT is the idea that
causes are context-independent. It claims that C causes E only if C raises
the probability of E across all homogeneous reference classes created by a
partition using all other factors causally relevant for E.1 This thesis has
been endorsed by many of those attempting to make sense of the concept
of probabilistic causality—among others, by Cartwright (1979), Skyrms

Philosophy of Science, 71 (July 2004) pp. 241–262. 0031-8248/2004/7103-0001$10.00

Copyright 2004 by the Philosophy of Science Association. All rights reserved.

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241

*Received March 2003; revised November 2003.

y To contact the authors write to Computer Science and Software Engineering, Monash
University, Clayton, VIC 3800, Australia; e-mail: {ctwardy, korb}@csse.monash.edu.au.

z We are grateful to Christoper Hitchcock, Dan Hausman, Peter Moulder, Michaelis Michael,
Lucas Hope and two anonymous referees for helpful comments. NSF grant SES 99-06565

provided partial support for this work.

1. Note that neither this nor any other probabilistic criterion of causality described in this

paper is an attempted reduction of causality to probability. In later sections we discuss what

else is needed to establish or identify causal relations.



(1980), Eells and Sober (1983), Humphreys (1989) and (in a weakened
form) Cartwright again (1989). According to this requirement, we should
not say that imbibing a poison causes death if, in fact, there is a perfect
antidote, for in the reference class of those treated with the antidote the
poison is inefficacious, which result defeats the causal claim. This may
seem like an odd requirement to impose upon causal talk, for there seems
nearly always to be available some reference class in which a putative
cause will fail to raise the probability of its effect: namely, the reference
class in which the object, thing, or space-time slice of interest is going to
be immediately obliterated, before the causal mechanism can operate to
completion. Thus, we should say that firing squads fail to cause the deaths
of their victims, since an imminent heart attack could preempt that
eventuality.2

A plausible response, and defense of CUT, could invoke conversa-
tional implicature, ruling out pre-emptive or other annoying causal factors
which are clearly, by conversational context, meant to be ruled out. But
such maneuvering aims to save unanimity theory in order to account for
ordinary language. Probabilistic causality is not primarily an ordinary
language investigation, as the long history in ordinary discourse of as-
suming that causes are sufficient for their effects surely demonstrates.
CUT is, at best, a poor aid in understanding ordinary language causal talk,
whereas the concept of objective homogeneity does the real work in
helping us make sense of probabilistic causality, if only in combination
with a known causal structure.3

First we need to clarify some of the terms of the discussion, before
turning to a critique of CUT. Our interest is the conditions for C being a
probabilistic cause of E. The relevant concept of probability is that of
physical probability or objective chance, presumably under a propensity
interpretation (or, perhaps, frequencies in an infinite hypothetical se-
quence of trials—the exact story is not the issue here). We are not directly
concerned with token causality—the conditions for a token event to be a
token cause of another token event—but with type causality, such as
whether smoking causes lung cancer.4

2. Skyrms’ formulation of ‘‘unanimity’’ escapes this kind of trivial counterexample by

allowing the cause to sometimes, but not always, fail to have a positive probabilistic impact

on the effect. Skyrms’ account nevertheless falls under every other objection we raise in this

discussion.

3. A forthcoming paper (‘‘Probabilistic Causality and Practical Causal Generalizations’’) by

Daniel Hausman makes this point in a different way.

4. The relation between type and token causality has not yet been fully explicated. Pearl

addresses this issue in terms of ‘‘causal beams’’ in Causality (2000). See also Halpern and

Pearl (2001).

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We discuss, and distinguish, two concepts that some may have con-
fused: contextual unanimity and objective homogeneity. Both appeal to
populations of events and their partitions. The cells in a partition are types
of events, in particular, events which share some fixed setof values.5 Both
contextual unanimity and objective homogeneity begin with a partition
where each cell (alternatively: subpopulation, reference class, or context)
is homogeneous for the effect. A cell Hi is homogeneous for an effect E if
there is no computable-beforehand property A which makes a difference
to the probability of E for events in the cell.6

There are two basic concepts of homogeneity, namely:

Objective Homogeneity: A population (or subpopulation) Hi is ob-
jectively homogeneous for an effect E if there is no computable-
beforehand property A which makes a difference to the probability of E.
That is, for any such A within Hi, PðE jAÞ¼ PðE jAÞ.

Epistemic Homogeneity: A population (or subpopulation) is episte-
mically homogeneous for an effect E if we do not know, of any computable-
beforehand property A, that it makes a difference to the probability of E.
When we are acting as practical investigators of causality, we are con-
strained by the limits of our knowledge, so epistemic homogeneity has a
role to play in accounting for scientific method and pragmatic judgment.
But, on our view, objective homogeneity is the regulative concept required
for analyzing the goal of our methodological efforts, namely causal
structure.

Employing causal Bayesian networks (treated in more detail subse-
quently), we can, in fact, introduce a third kind of homogeneity that
explicitly reflects this regulative role of objective homogeneity:

Model Homogeneity: Given a causal model M, a population (or
subpopulation) identified by a set of observed variables O in M is ho-
mogeneous for an effect E if there is no other variable in M which is a
non-descendent of E and the observation of which makes a difference to
the probability of E.

If our understanding of the causal structure of the world can be repre-
sented in a causal model, then our current best understanding of homo-
geneous contexts or subpopulations is given by its model homogeneities.
As we learn more about the world we develop and refine our best models,
and our epistemically homogeneous reference classes get refined in turn.

5. We represent possible events as variables whose values determine the type of that event.

6. Roughly, a property is computable-beforehand if it can be computed from properties

available to us up to and including the occurrence of C. See Church (1940).

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243probabilistic causation



We may reasonably hope that a sequence of such refinements is aimed at
the truth, which is the set of model homogeneities for the true causal
model. In the meantime, prior to Keynes’s limit, we work with models
that are false and our epistemic homogeneities may well mislead us. Such
is the fate of every inductive agent.

Note that, just as in Wesley Salmon’s early work on statistical rele-
vance (Salmon 1974), homogeneity (of every kind) is concerned with
quantities: it is essential for homogeneity to obtain that the probabilities
do not deviate. One of the attractions of CUT is that it has far laxer
standards—simply that differences in probability across contexts all pull
in the same direction, at whatever magnitude. We shall argue that the
attractiveness of laxity is strictly superficial.

2. Unanimity and Homogeneity. Contextual unanimity is properly de-
fined in terms of objective homogeneity (cf. Eells 1985):

Contextual Unanimity: A population is contextually unanimous for
C with respect to E if and only if in all cells (contexts) Hi in an objectively
homogeneous partition,7 PðE jCÞ > PðE jCÞ.
The corresponding thesis is the claim that causes are properly identified in
terms of unanimity. More formally,

Contextual Unanimity Thesis (CUT): C causes E only if the back-
ground population is unanimous for C with respect to E.

Whereas unanimity theory supposes that what is causally efficacious
(e.g., worthy of the rubric ‘‘causal power,’’ in Cartwright’s language)
must be so across contexts, homogeneity theory adopts the exactly op-
posite position: that causal factors are context-dependent and, to get the
causal story right, we must relativize causal claims to a specific context.
Not only need the causal powers push in the same general direction, but
they need to push with the very same strength. Anything less does in fact
average disparate powers, which is not good enough even for government
work.8 Hence, our initial thesis is:

Objective Homogeneity Thesis (OHT): C is causally relevant to E in
the homogeneous context Hi only if, within Hi; PðE jCÞ 6¼ PðE jCÞ.

Objective homogeneity was first championed by Salmon (1970) as a
key to statistical explanation. It takes seriously the relativization of causal

7. Of course, the homogeneity requirement here applies to all relevant properties except the

causal factor C in question and properties not fixed at the time of C.

8. Despite Cartwright’s remarks to the contrary (1989). Also, we shall henceforth use

‘‘causal power’’ and ‘‘causal capacity’’ interchangeably, as we do not believe that Cartwright

has a well-motivated distinction between them.

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244 charles r. twardy and kevin b. korb



claims to populations, advocated by Eells (1988), perhaps more seriously
than Eells did himself. The requirement of OHT is that causal claims be
relativized to objectively homogeneous reference classes (the cells Hi), so
that the introduction of any further putative causal factor cannot change
the probability of the effect to any degree whatsoever. If causal claims are
not so relativized, they can be arbitrarily wrong about the causal strength
(effect size).

Note that whereas CUT is specifically about a state C promoting the
outcome state E, OHT is about whether variable C is causally relevant to
variable E. But whenever one state of a variable is promoted, another
must be demoted, so when speaking of variables causing one another,
causality just is causal relevance. It is also worth noting that one can
always recover the CUT verdict from the OHT conditions, but not the
other way around.

There is a clear and direct relation between the properties of contextual
unanimity and objective homogeneity. Unanimity is defined in terms of
homogeneity, so there is a sense in which any case having the property of
being objectively homogeneous must trivially also have the property
of being contextually unanimous. That is, if we restrict our context to
the single cell satifying our objective homogeneity requirement, that
context will be unanimous for the effect. Of course, the aim of CUT is to
find a useful criterion that generalizes beyond such narrow contexts, so
the theses are distinct; OHT is quite significantly stricter than CUT.

It is worth noting that a weaker version of the homogeneity thesis can
be had by substituting epistemic for objective homogeneity—that is, re-
lying upon the model homogeneities of our currently best model. This
would produce a criterion far easier to meet: it would require only that we
not know of any causal factor which could partition the reference class to
produce distinct probabilities of the effect. And it is surely true that as a
practical matter we are often satisfied with epistemic homogeneity. In-
deed, Dupré and Cartwright appear to believe that epistemic homogeneity
is all that we can ever have:

But we have been arguing that however finely we partition the popu-
lation, we cannot know from the statistics that we have anything better
than an average causal upshot. (Dupré and Cartwright 1988, 530)

Science progresses, quite explicitly in the form of experimental methods,
from crude epistemically homogeneous classes to identifying inhomoge-
neities to more refined models and so improved epistemically homoge-
neous classes (as we pointed out in Korb 1999). And in some cases
science progresses to what is arguably objectively homogeneous (cf. Bell’s
theorem). Thus, objective homogeneity provides the metaphysical norm
toward which epistemic homogeneity can tend. Dupré loudly denounces

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such a view as hiding behind a ‘‘metaphysical fig leaf’’ (1993). But a de-
nunciation is no counter argument. If the history of science makes anything
clear, it is clear that a scientific explanation which omits unknown causal
factors may be wrong, discovered to be wrong, and properly be replaced by
an account which incorporates those variables. Dupré’s dismissal of meta-
physics would simply leave these known facts about science mysterious.

Objective homogeneity in OHT provides the stricter standard that can
make sense of our normative aspirations.

3. Problems with CUT. CUT fails to provide an adequate criterion for
probabilistic causality simply because it is not strict enough. If we partition
only as finely as required to satisfy contextual unanimity, we have effec-
tively declared that causation has only direction but not strength. CUT is
equivalent to creating a homogeneous partition and then agglomerating
disparate capacities to bring about an effect, on the grounds that they
operate in the same direction, regardless of their strengths. As Dan Haus-
man emphasizes,9 the appeal of CUT is that it appears to make causal talk
useful. Instead of requiring us to identify, even if only in principle, all the
factors needed for objective homogeneity, we can take the more relaxed
approach of identifying only such factors as are needed to force a common
causal bearing (direction of impact) on the effect. The difficulty with this
line of thought is two-fold. First, in non-microphysical cases we are no
closer practically to locating those factors needed for a common causal
bearing than we are to locating those needed for objective homogeneity
itself. If we can happily assert that smoking causes cancer, for example, it is
not because we know that there are no countervailing forces where smoking
is neutral or even preventative for cancer. So CUT fails even as an account
of ordinary causal talk. But second, this line of thought—aiming to account
for the utility of and/or the pragmatics of ordinary causal language—is a
diversion from giving a normative, metaphysical account of causation, and
not the very same thing. If we can connect the distinct metaphysics and
epistemics of causation, we will have achieved our goal.

Beyond the considerations about how causal talk works, CUT seems to
have little going for it. If some treatment raises the probability of survival
in one category of cancer patient by 300% and and in another by 5%, this
is not a trivial difference to be papered over by talk of ‘‘causal power.’’
Effect size matters! Indeed, the multivariate methods of statistics used to
analyze contingency tables and make sense out of such cases unsurpris-
ingly depend explicitly on exact estimates of just such differences. A
methodology for causal inference founded upon a metaphysics which
ignores them would simply be a non-starter.

9. In his forthcoming paper.

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246 charles r. twardy and kevin b. korb



3.1. Varieties of Populations. Taking particular values of C and E, there
are four kinds of population to consider (where C abbreviates C ¼ c for
some c, etc.):

1. Positive case: C raises the probability of E in all objectively ho-
mogeneous cells. In this case we can say C promotes E, or some-
times just C causes E.

2. Negative case: C lowers the probability of E in all objectively
homogeneous cells. In this case we can say C prevents E, and it is
strictly parallel to the positive case.

3. Mixed case: C raises the probability of E in some cells and lowers
the probability of E in others. Most real cases appear to be like this.

4. Neutral case: C is always irrelevant for E.

The neutral case has generally been taken to be non-problematic, with
the conclusion that C is not a cause of E. Later, we see where this can fail.

In the mixed case, an uncompromising CUT advocate would insist that
C is simply not a cause of E, regardless of it having measurable effects.10

Eells’ insistence on relativizing to contextually unanimous subpopu-
lations in fact puts him partway between that extremism and OHT. Eells’
view absorbs all mixed cases into one of the other three cases, by rela-
tivizing the causal claim to some, but not all, the factors which identify an
objectively homogeneous reference class. Under either interpretation,
CUT is mixing together distinct cells in the homogeneous partition. Dupré
and Cartwright correctly complain that, despite the unanimity require-
ment, all that we really have are ‘‘causal upshots’’—an average of a
variety of causal powers (even though they happen to point in the same
direction).

As we just mentioned, this is a problem if we should like to make sense
of multivariate methods of causal discovery, which crucially depend upon
effect sizes (see, e.g., Korb and Nicholson 2003; Spirtes et al. 2000). It is
also a problem when we attempt to make sense of causal interactions.

3.2. Mixtures and Interactions. In the linear case one cause interacts
with another when the effect is not simply the sum of independently
operating causal forces. Contextual unanimity does not have the tools to
make sense of these causal interactions. A strict CUT theorist must claim
that causes cannot interact, because according to CUT, neither positive
nor negative causes can be interacting, and mixed causes are not causes at
all. But we all know that causes do interact, so Eells and others have
concluded that interaction happens precisely in the mixed case: where the
valence of a cause reverses from positive to negative across contexts.

10. Glennan and others criticize CUT on these grounds, as we shall see later.

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247probabilistic causation



Yet mixed cases are neither necessary nor sufficient for causal inter-
action. They are not necessary because causes may interact by reinforcing
each other beyond their individual effects (synergy or potentiation), by
preempting one another, as well as by cancelling each other out in an
exclusive-or relation (as, for example, alkali and acid can do). They are
not sufficient because you can get a valence-reversal without interaction,
via dual (multiple) causal pathways: for example, oral contraceptives do
not interact with pregnancy to provide a mixed case for thrombosis—
there is merely a multiple-path dependency (see Figure 1).

4. Problems with OHT. We shall now consider some objections to OHT.
Glennan has recently objected that homogeneity forces us only to treat
fully specified complexes of causal factors, which are unnatural and un-
wieldy; Dupré and Cartwright contend that homogeneity does not work in
any case, since it fails for dual capacity scenarios. Neither objection can
be sustained—although the possibility of dual capacities (in the form of
precisely counteracting multiple pathways) will in the end force us to
elaborate OHT with explicit reference to causal structure.

4.1. Glennan’s Objection. In his discussion of contextual unanimity in
the units of selection debate in evolutionary theory, Glennan argues:

if one requires context independence [i.e., contextual unanimity] for
an entity to be a unit of selection, the unit of selection will inevitably
be the entire genome. (2002, 122)

This for the reason that there is very widespread interaction across loci,
with alleles at some potentially negating or reversing the developmental
effects at others, and all in such a complex interrelation that we cannot
seriously expect the causal impact on reproduction to be unanimous
without accounting for the entire nexus. So we will not get unanimity
until we conjoin a great many background conditions into the cause.
Glennan considers this an unwelcome result since it appears to trivialize
the debate on units of selection.

But perhaps that debate should be trivialized. The same criticism must
apply to OHT, as it is the more demanding requirement. If fixing the entire
genomic background is required to achieve contextual unanimity, it will
certainly also be required to find homogeneous reference classes for se-
lection. To be sure, OHT here has a purely linguistic advantage over
CUT: it asserts causality of the single gene (so long as there is any Hi
where it makes a difference), whereas CUT must assert causality only of
the conjunction of all factors required to achieve unanimity.

The same point can be put in a simpler domain than the units of
selection debate. Glennan also says,

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248 charles r. twardy and kevin b. korb



Now it is possible to accept this consequence of contextual una-
nimity—to say that generalizations like the one that smoking causes
cancer are false. (2002, 124)

Presumably, if we homogenists are constrained to deny such an evident
truth as that smoking causes cancer, then we are in trouble. But we are
not. Taken as a general claim, homogenists can readily affirm the efficacy
of smoking because there is at least one context in which it has the
requisite probabilistic impact. Taken as a specific claim, we can say it is
ambiguous and ask for a context. Within a context its truth or falsity will
be clear. In support of the homogenist’s treatment of the general claim,
some plausible account, under implicature, can be made ruling out known
or easily anticipated cases where the causal power of smoking is negated
or reversed. The same defense applies to Eells, though not to strict
unanimists (if there are any). But this line of consideration is irrelevant to
the metaphysics of causation.

Returning to the genome, it is entirely likely that a less trivializing
account of the units of selection debate can be made out of pragmatic
considerations. In any case, as with smoking, we can certainly say that
particular alleles cause particular characteristics, although no doubt only
and always in particular circumstances which may never be fully
accounted for. Objective homogeneity provides a regulative, normative
account of causality and not necessarily a pragmatic account accessible in
the short term. But rather than dismissing this verdict as a metaphysical
pipe dream, why shouldn’t we believe that a Bayesian network that fully
depicts the causal structure of, say, human ontogeny would require var-
iables representing every locus in the human genome? It would surely
require that—and more, since humans do not develop in a vacuum.

4.2. Dupré and Cartwright’s Objection. Dupré and Cartwright (1988,
section 4) argue that Eells’ relativization of causal claims to populations
(and, by extension, ours to homogeneous reference classes) fails in the
case of dual capacities. We can use Hesslow’s famous example (Hesslow
1976) (Figure 1): oral contraceptives increase the risk of thrombosis in
women who would not otherwise have become pregnant, but decrease
that risk in women who would otherwise have become pregnant.11

Of this example, Dupré and Cartwright say:

[This] criterion [unanimity and/or homogeneity] . . . yields the
conclusion that in this population contraceptives do not prevent

11. This is because pregnancy increases the risk twelve-fold and second-generation oral

contraceptives by three-fold. Third-generation oral contraceptives increase the risk six-fold

(Development and Evaluation Service 1996).

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249probabilistic causation



thrombosis (via prevention of pregnancy). The reason, we may all
agree, is trivial. We have held pregnancy fixed . . . (1988, 529)

If we hold pregnancy fixed, then contraceptives can only reinforce the
tendency that pregnancy has (in one subpopulation) to produce throm-
bosis (and with similar result in the other subpopulation). But this is
obviously wrong, as Dupré and Cartwright immediately observe:

[Many] women will in fact have been saved from thrombosis by the
pills; without the pills they would probably have become pregnant
and run a high risk of thrombosis.

Well, of course! If you hold fixed those who actually do become pregnant,
etc., then you will run a very high risk of confusion in interpreting your
experimental results. That is, in fact, the whole point behind Cartwright’s
refinement of her causality condition, CC*, elaborated in her Nature’s
Capacities and their Measurement (1989). A homogeneous reference
class traditionally does not refer to intermediate effects (or any effects in
the future of C), though it does require reference to the subjunctively
identified subpopulations (captured retrospectively by CC*), such as
those women who otherwise would have become pregnant—as we have
discussed at length elsewhere (Korb 1999).

By confusing the subjunctive and indicative moods, Dupré and
Cartwright have wrongly concluded that relativizing causal claims to
homogeneous classes continues to mix causal capacities. Nevertheless,
there are cases of dual capacities where OHT fails.

4.3. Causal Neutrality in Simple Worlds. When we identify homoge-
neous reference classes we are considering holding fixed all causally rel-
evant factors prior to the one in question. Pregnancy comes after the factor

Figure 1. Multiple paths to thrombosis.

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250 charles r. twardy and kevin b. korb



in question (the pill). Of course, the subjunctive property comes before,
which is what we were relying upon. We do not know precisely how to
analyze such subjunctive properties, but presumably it is to be done in
reference to a richer causal nexus than that described in Figure 1, with
multiple hidden causes prior to taking the pill fixing an objective proba-
bility of falling pregnant without the pill. But what if the world is simple?
What if there really are only three variables that describe the world?

If the world is as simple as Figure 1, and if the parameters exactly
balance, so that the pill has no net probabilistic impact on thrombosis,
then OHT advises that there is no causal relation between them. But that
is wrong—we are assuming that Figure 1 is the right causal model, after
all! So, it turns out that in the end, objective homogeneity alone cannot do
the work we require of it.12

We might be tempted simply to extend the partitioning conditions of
OHT to incorporate downstream events, since, after all, observing Preg-
nancy in the neutral Hesslow case reveals the causal connection between
the pill and thrombosis. Furthermore, allowing downstream variables
need not resurrect the Dupré–Cartwright objection about falsely con-
cluding that contraceptives only promote thrombosis, because it is a
principle of homogeneity theory to require specification of the context,
explicitly or implicitly. It is no error to say that, given that we know
someone is pregnant, the pill does not prevent pregnancy; nor in that case
can it be neutral for thrombosis.

Nevertheless, this future-fixing OHT is inadequate for another reason:
it sanctions the ascription of causality to non-causes. Consider the network
of Figure 2. If we are allowed to observe the effect A, then we induce a
probabilistic dependency between C and E (because the path from C to X
becomes active, as explained in Section 5.1). But there is no sense in

Figure 2. Failure of future-fixing.

12. Neither, of course, can CUT. And note that the real issue here is the neutralizing effect of

multiple pathways, not the simplicity of the world per se. In complex worlds, however, it

seems likely that the invocation of subjunctive properties will be available to imbalance such

pathways. We cannot escape by observing that such neutral cases have measure zero (since

they depend upon exact parameterizations): we understand perfectly well what the causal

story is here, so we ought to be able to codify that understanding.

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251probabilistic causation



which C causes E here, even if it is known in advance that its effect A will
hold.

4.4. Intransitivity. On reflection OHT has an obvious problem: it
ignores causal structure. Being exclusively probabilistic (and temporal),
it cannot but mishandle exact balances. But if we move beyond proba-
bilities to appeal to causal structure itself, we may wonder why we should
bother with the probabilities at all. Indeed, a plausible conjecture for a
satisfactory criterion of causality is the transitive closure over directed
causal links in a causal graph. In that case, we could deploy our best
causal discovery programs, for example, and simply examine the result-
ant graph to see whether X and Y are connected by a directed chain.
Unfortunately, transitivity over direct causal links does not hold (Cooper
1999; Hitchcock 2001b; Pearl 2000). There are strange cases where,
although X causes Y and Y causes Z, it just is not the case that X causes
Z. For example, suppose Y describes how a terrorist pushes a detonator
(with his left hand or his right hand) and Z an explosion. Then if X in-
dicates whether a dog bites the terrorist’s right hand, it may well have
an impact on Y: if the dog bites the right hand, the terrorist will push
with the left and otherwise with the right. But in either case the bomb
will explode, so the dog’s bite has no impact whatsoever on that out-
come. We need a criterion of causality where X does not here cause Z.13

It is only in cooperation that causal structure and probabilistic impact
can identify general causal structure.

5. Wiggle Logic. On OHT, the most natural kind of causation is similar
to what Eells calls ‘‘causal relevance.’’ We approach the problem by

13. A more mundane example was described by Richard Neapolitan (2003) in a presen-

tation: finesteride reduces DHT (a kind of testosterone) levels in rats; and low DHT can

cause erectile dysfunction. However, finesteride didn’t reduce DHT levels sufficiently for

erectile dysfunction to ensue, in one study.

Figure 3. Representing perfect intervention on C.

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252 charles r. twardy and kevin b. korb



considering what happens probabilistically when we intervene upon
(wiggle) the causal variable C in an attempt to fix its value to some c,
which we represent IC=c (or simply IC).

14

In the spirit of OHT we could say that one variable C causes another E
in a causal Bayesian network precisely when there is some causal context
and some possible manipulation of C which is followed by a change in the
distribution on E. In particular, a direct link should guarantee causation,
given a causal Bayesian network which is also a perfect map.15 A natural
way to interpret causal relevance is via manipulations of C given obser-
vations of other variables O:

Causal Relevance (CR): C causes E with respect to G=O if and only if

9O0 � O s:t: 9c PG=O 0ðE jIC¼cÞ 6¼ PG=O0ðEÞ:

Here, G=O means we have a causal graph G in which the variables O
have been observed. O therefore identifies the background in which the
causality question is raised, and O0 the context in which it is answered.16

5.1. Intervention and Activation. We have begun talking about inter-
vening on variables rather than just observing (or conditioning on)
them. Most of the literature on probabilistic causality has used notions
of intervention, especially in reference to experimental procedure, but
without clarifying the relation between intervention and observation.
Recent work in Bayesian networks has developed tools to represent
both formally.

Technically, the best way to represent IC¼c in a Bayesian network is to
add a parent node IC to C, allowing for any kind of success rate in
manipulating the value of C (Korb and Nicholson 2003). It is best in the
sense that it is the most flexible, for it can represent imperfect, as well as
perfect, control—that is, it can represent manipulations that may fail to fix
the value of C, as well as those which are guaranteed to succeed re-
gardless of the state of the parents of C. Adding the intervention variable
IC also allows us to represent any kind of interaction between the

14. C = c denotes a particular intervention rather than a particular value that C adopts. In

general, an intervention need not select a single state.

15. A perfect map is one in which each missing arc implies a probabilistic independence

(the Markov property) and each existing arc implies a probabilistic dependence (called

‘‘faithfulness’’ by Spirtes et al. 2000; cf. Pearl 1988). A causal Bayesian network is a

Bayesian network which can model deterministic interventions on C by removing arcs into C.

16. Menzies (2002, forthcoming) has a similar approach which he calls ‘‘difference-making

in context,’’ although he develops it with respect to deterministic causal models.

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253probabilistic causation



manipulation and any alternative causes.17 Regardless, in the case usually
imagined (if unusually realized), IC indeed exerts perfect control over C.
In that case, it is easiest to represent the intervention by cutting all arcs
into C, and then fixing C = c.18

Under perfect control, the two representations are equivalent, although
only the arc-cutting representation will be a perfect map. However, since
they are equivalent, it follows that observations of IC (i.e., manipulations
of C ) cannot induce a probabilistic dependency with anything causally
upstream from C.19 In particular, C and E cannot then have any common
cause inducing a probabilistic dependency between them. Nothing in the
discussion will hang on the difference between perfect and imperfect
control, so we shall assume perfect control, while continuing to refer to IC
as the potential controlling variable.

Interventions and observations will have no probabilistic impact upon
downstream events if they are blocked (d-separated) in the context O.20

Blocking: �2PathsðC; EÞ; is blocked in G=O if and only if O
d-separates C from E in G,

where Paths(C, E ) is the set of all directed paths from C to E. Equiva-
lently, � is blocked if and only if 9Z Z2�^Z2Oð Þ.21

What is doing the blocking (or not) in these cases is a set of observed
variables O. What we specifically need for our Wiggle Logic is the
concept of paths being ‘‘blocked’’ or not either because of such obser-
vations or because a direct connection between two variables X ! Y
2 PathsðC; EÞ has been otherwise blocked—for example, by hypo-
thetically introducing a previously unknown variable between X and Yand
observing its value. Let R stand for a set of restrictions on G, meaning a
set of observations O together with a set of arcs which are restricted or

17. There is another kind of intervention which adding IC allows us to represent, namely IC
deterministically forcing C to take a new non-degenerate probability distribution, inde-

pendent of its other parents.

18. This is a graphical representation of Pearl’s ‘‘do calculus’’ (Pearl 2000).

19. Note that this is only true with perfect control and generally false otherwise. Since

imperfect control introduces such complications, without any compensating benefit for our

purposes here, we shall ignore it hereafter.

20. For a formal definition of d-separation, see Pearl (1988). Informally, it is the graphical

counterpart to ‘‘screening off,’’ or conditional independence: observing a common ancestor

(or intermediate node in a chain) blocks a path relating two events and observing a common

descendent induces a probabilistic dependency. Sewell Wright (1934) was the first to rep-

resent these relationships graphically.

21. Note that this is only equivalent because Paths(C, E ) consists exclusively of directed

paths from C to E.

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254 charles r. twardy and kevin b. korb



blocked by such a hypothetical observation. When describing a set of
restrictions, we will by convention identify them with the variables at the
head of the arcs being restricted. For example, R ¼ O[ParentsðEÞ
would mean that all parents of E are either observed or their arcs into E
are otherwise restricted. Then,

Activation: � 2 Paths(C, E) is active in G=R where R�O if and only if
(a). � is not blocked in G=O and
(b). No X ! Y 2 � is restricted in R.

5.2. Paths Plus Probabilities. Probabilistic impact is insufficient as a
criterion for causality because of odd cases of neutrality. Transitive clo-
sure along a path is insufficient because of odd cases of intransitivity of
the probabilistic dependency. We are now finally ready to consider how to
articulate these two in a sufficient criterion for causal relevance. The most
direct attempt to marry causal structure with probabilistic impact is to find
probabilistic dependency along single paths:

Wiggle Logic I: C causes E with respect to G=R if and only if

9� 2 PathsðC; EÞ and
9c PG=RðEjIC¼cÞ 6¼ PG=RðEÞ

The criterion, illustrated in Figure 4, asserts that causal intervention
reveals causal relationships when and only when downstream effects

Figure 4. Wiggle Logic I. R fixes a background wherein one or more paths �i allow
intervention on C to make a probabilistic difference.

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255probabilistic causation



reflect the impact probabilistically. This rules out intransitivity cases, as
we want. Note that the issues concerning screening off that vexed early
attempts to provide a probabilistic criterion for causality do not even
arise: observing IC cuts all other arcs into C, preventing any probabilistic
influence from going backwards through a common cause.

On the other hand, this formulation fails to cope with accidental neu-
trality in small worlds: in the neutral Hesslow case, there just is no
probabilistic dependency between the pill and thrombosis, because the
dual pathways neutralize each other. But this is exactly and only because
there are neutralizing multiple pathways. There is no difficulty in ex-
tending WL I to cope with multiple paths:

Wiggle Logic II: C causes E with respect to G=R if and only if

9R0 � R s:t: : 9!� 2 PathsðC; EÞ s:t: � is active in G=R0and
9c PG=R0 ðEjIC¼cÞ 6¼ PG=R0 ðEÞ

where 9!� 2 Paths(C, E) means there is a unique directed path � from C
to E (see Figure 5).

The uniqueness requirement for � implies that any alternative directed
paths from C to E are blocked by R0. The causality criterion, then, just
combines transitivity (i.e., there being a directed path) with probabilistic
dependence under intervention, but in such a way that any neutralizing

Figure 5. Wiggle Logic II. R0 fixes a background and foreground wherein exactly one path
�i allows an intevention on C to make a probabilistic difference.

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256 charles r. twardy and kevin b. korb



alternative pathways are discounted. WL II trivially subsumes WL I; that
is, if WL I is satisfied then WL II must be satisfied. For that reason also,
any directly linked cause and effect will be identified by WL II.

With this in mind, let us revise CR to talk of restrictions:

Causal Relevance (CR II): C causes E with respect to G=R iff

9R0 � R s:t: 9c PG=R0 ðE jIC¼cÞ 6¼ PG=R0 ðEÞ:

Clearly if WL II holds, then CR II holds: there is some state of the
network such that wiggling C makes a difference to the distribution over
E. We conjecture that CR II implies WL II: if the second clause holds
without R0 intersecting all but one directed path from C to E, then it also
holds under some such circumstance. The advantage of WL II, in any
case, lies simply in making the joint dependency on transitive closure and
probabilistic dependency explicit.

R identifies the context within which the causality question is raised. R0

identifies some context within which it can be answered. R0 specifies
some state of the Bayesian network G as a whole, with all unnamed
variables left unobserved and unnamed arcs unrestricted. It may well be
that R0 shall need to fix one or more parents of E to particular values
before an intervention on C shows any probabilistic impact: that is, it may
be that C affects E only in some specific interaction involving other
causes. This criterion allows for that.

5.3. Objective Homogeneity. So what about objective homogeneity—
or, the model homogeneity which is its representative? In complex
worlds, where subjunctive properties are in principle computable be-
forehand from the rich causal background, objective homogeneity may be
enough. However, as we have seen, in simple worlds we need more.
Wiggle Logic (and CR) obtain this ‘‘more’’ by reference to any context R0

(or O) in a causal model, homogeneous or not. But in applying these
criteria, if we are looking at an inhomogeneous context, we are looking at
a wrong context. Suppose, for example, that there is only a single, direct
link between C and E. In that case, we can leave R0 ¼ O in WL II, since
there are no alternative paths to intercept. But if O fails to specify a
homogeneous background, if there are other causes of E that are not held
constant, then whether an intervention on C induces a change in the
distribution over E may depend upon the prior probability distribution
over those unfixed alternative causes. In short, WL II will only provide us
with a criterion for belief in causality—even relative to the model at hand,
rather than the normative, metaphysical criterion we were after. Without
objective homogeneity, WL II is satisfied too easily.

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257probabilistic causation



In order to deal with this problem we need to revert to the original idea
of OHT, requiring O to be model homogeneous in the first place.22 Thus,
objective homogeneity refuses to go away.

Because it works thread-by-thread, this forward-looking homogeneity
does not allow a direct answer to the question, ‘‘What is the overall effect
of contraceptive pills on thrombosis?’’ But then we already knew how to
get the right answer to that question: use the traditional computable-
beforehand homogeneous reference classes at C. What we answer here is
how to determine whether there is in fact some causal relation between C
and E—but not what measurable strength that relation has nor how it
contributes to an overall impact on E. We leave the proper accounting of
the notion of causal strength to future work.

5.4. Simple Simple Worlds. We suggest that WL II provides a fully
satisfactory accounting for the neutral thrombosis case. But, it could well
be objected that by introducing restriction and control variables we have
violated our own assumption of the world being simple: we have added
new variables to the model that was supposedly an exhaustive description
of the causal world. In short, we have cheated. We unashamedly confess
to cheating. Not being permitted this cheat is equivalent to restricting
consideration to a world where there is no such thing as causal inter-
vention. But it is arguable that in such a world there is no such thing as
causality, and the only legitimate considerations are probabilistic. In that
world there is no sense in which the original thrombosis model is superior
to the simpler v-structure Pill ! Pregnancy  Thrombosis—indeed,
there is a sense in which the latter, being a simpler representation of the
very same probability structure, is superior. We are not interested in
worlds which are non-causal, and our criterion of causality cannot be
faulted for failing to find any causality in them.

6. An INUS Condition for Probabilistic Causality. John Mackie (1993)
proposed the INUS account of causality: that a condition C is a cause of E
just in case it is an Insufficient, but Necessary, component of an Un-
necessary, but Sufficient, condition for E. This fails, if only because it
presupposes a deterministic universe and so rules upon synthetic matters
with a priori hubris. What we here offer is quite different, namely an
INUS condition for probabilitic causality:

Probabilistic dependency in an objectively homogeneous context is an
Insufficient (in that it requires a causal model to provide the directed

chain connecting C and E as well as the parent set of E) but

22. Here is one such O: Parents(E )\�i plus the parents of every intermediate node on �i, ex-
cluding the members of �i. This is sufficient, but not necessary, to obtain model homogeneity.

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258 charles r. twardy and kevin b. korb



Necessary part of WL II, which is an
Unnecessary (because of quantum mechanical cases where the

Markov property fails), but
Sufficient condition for causality.

7. Conclusion. Unanimity and homogeneity are, of course, related. For
one, unanimity is properly defined in terms of homogeneity, as we ob-
served above. But also they both make some attempt to relativize causal
claims and avoid the mistake of mixing together distinct effects. Thus,
some objections made to relativization apply to both. Those objections
succeed, at best, in pointing out that homogeneity theory does not provide
much of an account of ordinary causal talk. We shall leave such worries to
those who take ordinary language philosophy seriously.

Contextual unanimity theory for probabilistic causality is a red herring:
it fails to find the true causal capacities hidden within the mass of arti-
factual causal claims based upon averages; it also fails to account for
ordinary causal talk. Being easily impressed by causal factors that merely
push the effect in the same direction, the unanimity account distracts from
what is important. It is only from our best estimates of precise causal
strengths that we can hope to learn causal structure, as is clearly revealed
in the recent literature on causal discovery.

Objective homogeneity, properly understood within a causal context,
does not and can not mix distinct causal powers. It handles interactions
and allows us to make sense of the learning of causal structure, whether
by human or machine. Although causal structure alone does not suffice to
identify causality in Bayesian networks (because of intransitivity), causal
structure combined with objective homogeneity in Wiggle Logic does.

Finally, Wiggle Logic provides a clear account of the relation between
epistemics and metaphysics. Wiggle Logic provides a clear criterion of
causal relevance relative to a causal model—just as model homogeneity
provides a clear criterion of homogeneity relative to a model (if we replace
observations O with restrictions). As we learn more about the world we are
in, we refine our causal model of that world, by adding missing nodes and
arcs (or, perhaps, deleting some). Our account of causal relevance, and our
best guess of what the objectively homogeneous reference classes look
like, thereby become more sophisticated. In that way, our model homo-
geneities may tend toward what are truly the objective homogeneities.

Addendum I: More Pragmatics.
We have repeatedly pushed aside ordinary causal talk, except to say that

there is an obvious way in which the homogenist does not do violence
to ordinary causal talk. But if a homogenist were to estimate the actual risk
of smoking for a particular agent, that homogenist would have to specify a

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259probabilistic causation



reference class. Yet this homogenist would certainly know that the refer-
ence class is not truly objectively homogeneous. So, what should we say?

Instead of relying on implicature, one could also give an account of
actual causal talk as offering ‘‘best recommendations’’ given an episte-
mically homogeneous reference class. Such an account might use decision
theory to choose the optimal plan given an agent’s uncertainty about the
homogeneous reference classes. Christopher Hitchcock has developed
such an account (Hitchcock 2001a, 2002).23 Although his account is
presented for a particular agent choosing its own actions, it is easily
generalized. This is not our project, but we are sympathetic to such a
decision-theoretic approach.

Addendum II: How Objective Homogeneity Is Not Just
about Probabilities.

Christopher Hitchcock has asked us why probabilities are the right way
to discriminate capacities.24 Suppose, he says, there is a population with
three different subpopulations, P1, P2, and P3 relevant to smoking S and
death D. Smoking always increases the probability of death, but it can do
so either via lung cancer L or heart disease H. Smoking increases the
probability of death among P1 and P2 via H, and among P3 via L. By
strange coincidence it happens, however, that PðDjSP2Þ¼ PðDjSP3Þ,
and analogously for :S. Hitchcock says, ‘‘Then, according to homoge-
neity, populations P2 and P3 should be joined to form one homogeneous
sub-population,’’ which would aggregate different capacities.

If we model the situation in just the three variables S,P,D, then we have
the model shown in Figure 6. In this case, objective homogeneity requires
us to combine the populations P2 and P3, because of their identical
probabilistic consequences for other variables in the model.25

23. This is also the orientation of Daniel Hausman’s forthcoming paper.

24. In private communication.

25. This is not strictly speaking true, however Salmon’s appeal for always adopting the

broadest available homogeneous reference class (Salmon 1974) appears to us to be well

justified, if only on the grounds of strengthening the statistical basis for inductive inference.

Figure 6

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260 charles r. twardy and kevin b. korb



But what happens if we decide to model the story as Hitchcock has
presented it? In that case our causal model would be as shown in Figure 7.
Here we are no longer required to merge the subpopulations of P—rather,
we are obliged not to do so, since that would violate the story, mis-
describing causal effects on H and L. As its reformulation in WL II
makes clear, objective homogeneity discriminates capacities relative to a
causal model and not in vacuo. In short, the wrong model leads to wrong
conclusions.

references

Cartwright, Nancy (1979), ‘‘Causal Laws and EAective Strategies’’, Noûs 13: 419–437.
Reprinted and expanded in her 1983 book of the same title.

——— (1989), Nature’s Capacities and Their Measurement. Oxford: Clarendon Press.
Church, Alonzo (1940), ‘‘On the Concept of Random Sequence’’, Bulletin of the American

Mathematical Society 46: 130–135.
Cooper, Gregory (1999), ‘‘An Overview of the Representation and Discovery of Causal

Relationships Using Bayesian Networks’’, in Clark Glymour and Gregory F. Cooper
(eds.), Computation, Causation and Discovery. Cambridge, MA: MIT Press.

Development and Evaluation Service (1996), Factor V Leiden screening in oral contra-
ceptive users. Technical Report 58, R&D Directorate, South and West Regional Health
Authority, Canynge Hall, Whiteladies Road, Bristol BS8 2PR. http://www.doh.gov.uk/
research/swro/rd/publicat/dec/dec58.htm, using data from the 1994 Leiden thrombo-
philia study and others.

Dupré, John (1993), The Disorder of Things: Metaphysical Foundations of the Disunity of
Science. Cambridge, MA: Harvard University Press.

Dupré, John, and Cartwright, Nancy (1988), ‘‘Probability and Causality: Why Hume and
Indeterminism Don’t Mix.’’, Noûs 22(4): 521–536.

Eells, Ellery (1985), ‘‘Probabilistic Causality: Reply to John Dupré’’, Philosophy of Science
54: 105–114.

——— (1988), ‘‘Probabilistic Causal Levels’’, in Brian Skyrms and William L. Harper
(eds.), Causation, Chance and Credence. Dordrecht: Kluwer, 109–133.

Eells, Ellery, and Elliott Sober (1983), ‘‘Probabilistic Causality and the Question of Tran-
sitivity’’, Philosophy of Science 50: 35–57.

Glennan, Stuart (2002), ‘‘Contextual Unanimity and the Units of Selection Problem’’,
Philosophy of Science 69:118–137.

Halpern, Joseph Y., and Judea Pearl (2001), ‘‘Causes and Explanations: A Structural-Model
Approach-Part II: Explanations’’, in Proceedings of the seventeenth International

Figure 7

#04353 UCP: PHOS article # 710301

261probabilistic causation



Conference on Artificial Intelligence – IJCAI-01. San Francisco: Morgan Kaufmann,
27–34.

Hesslow, Germund (1976), ‘‘Discussion: Two Notes on the Probabilistic Approach to
Causality’’, Philosophy of Science 43:290–292.

Hitchcock, Christopher R. (2001a), ‘‘Causal Generalizations and Good Advice’’, Monist 84:
222–246.

——— (2001b), ‘‘The Intransitivity of Causation Revealed in Equations and Graphs’’,
Journal of Philosophy 158(6): 273–299.

——— (2002), ‘‘Good Advice’’, in Henry Kyburg, Jr. and Mariam Thalos (eds.), Proba-
bility is the Very Guide of Life: A Poor Folks’ Guide to the Uses of Probability.
Chicago: Open Court.

Humphreys, Paul (1989), The Chances of Explanation. Princeton: Princeton University
Press.

Korb, Kevin B. (1999), ‘‘Probabilistic Causal Structure’’, in Howard Sankey (ed.),
Causation and Laws of Nature, Dordrecht: Kluwer, 265–311.

Korb, Kevin B., and Ann E. Nicholson, (2003), Bayesian Artificial Intelligence. Boca Raton,
FL: CRC Press.

Mackie, John (1993), ‘‘Causes and Conditions’’, in Ernest Sosa and Michael Tooley (eds.),
Causation. Oxford Readings in the Philosophy of Science. New York: Oxford Uni-
versity Press, 33–55.

Menzies, Peter (2002), ‘‘Difference Making in Context’’, in John Collins, Ned Hall, and
L. A. Paul (eds.), Counterfactuals and Causation. Cambridge, MA: MIT Press.

——— (forthcoming), ‘‘The Causal EDcacy of Mental States’’, in Jean-Maurice Monnoyer
(ed.), The Structure of the World: the Renewal of Metaphysics in the Australian School.
Paris: Vrin Publishers.

Neapolitan, Richard E. (2003), ‘‘Stochastic Causality’’, paper presented at the International
Conference on Cognitive Science, Sydney, Australia, July 2003. Slides at http://
www.neiu.edu/reneapol/chapters.htm.

Pearl, Judea (1988), Probabilistic Reasoning in Intelligent Systems. San Mateo, CA: Morgan
and Kaufman.

——— (2000), Causality. New York: Cambridge University Press.
Salmon, Wesley C. (1970), ‘‘Statistical Explanation’’, in Robert G. Colodny (ed.), The

Nature and Function of Scientific Theories. Pittsburgh: University of Pittsburgh Press,
173–231. Reprinted in Salmon et al. 1971.

——— (1974), Statistical Explanation and Statistical Relevance. Pittsburgh: University of
Pittsburgh Press.

——— (1980), ‘‘Probabilistic Causality’’, Pacific Philosophical Quarterly 61:50–74.
Skyrms, Brian (1980), Causal Necessity. New Haven: Yale University Press.
Spirtes, Peter, Clark Glymour, and Richard Scheines (2000), Causation, Prediction, and

Search, 2d ed. Cambridge, MA: MIT Press
Suppes, Patrick (1970), A Probabilistic Theory of Causality. Amsterdam: North Holland.
Wright, Sewall (1934), ‘‘The Method of Path CoeDcients’’, Annals of Mathematical

Statistics 5(3):161–215.

#04353 UCP: PHOS article # 710301

262 charles r. twardy and kevin b. korb