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作者：Anonymous 在 罕见奇谈 发贴, 来自 http://www.hjclub.org
或然性因果关系－－对马奇(J. Mackie)理论的批评和修正
近日网友们关于“原因”和“因果关系”辩论得十分热烈。旁观了几天终于忍不住也来说几句。
关于事件A被称为事件B的原因，马奇提出“A必须是使B发生的一个充分而不必要条件的一个必要但不充分的组件”。然而，在现实世界中，有很多事件B是不存在使其发生的充分条件的；这样一来，对这些indeterministic事件，按马奇的说法，就没有任何事件称得为它们的原因。例如，虽然大家都知道，抽烟是人得肺癌的“原因”之一；但是由于我们无法找到“人得肺癌”的任何“充分（而不必要）条件”，因而就无法使“抽烟”成为这充分条件之“必要而不充分的组件”。所以按照 马奇的说法，抽烟或任何其他东西，都不能成为得肺癌之“原因”。更为一般的，所有indeterministic 事件都不可能有原因了 －－ 这恰恰是马奇理论的致命弱点！
作为理科爱好者，本人最欣赏的的是或然性因果关系理论；下面是它的一个简介（Stanford Encyclopedia of Philosophy， 附图可见于：http://plato.stanford.edu/entries/causation-probabilistic/）：
Probabilistic Causation
“Probabilistic Causation” designates a group of philosophical theories that aim to characterize
the relationship between cause and effect using the tools of probability theory. The central idea
behind these theories is that causes raise the probabilities of their effects, all else being equal. A
great deal of the work that has been done in this area has been concerned with making the
ceteris paribus clause more precise. This article traces these developments, as well as recent,
related developments in causal modeling. Issues within, and objections to, probabilistic theories
of causation will also be discussed.
1. Introduction and Motivation
1.1 Regularity Theories
1.2 Imperfect Regularities
1.3 Indeterminism
1.4 Asymmetry
1.5 Spurious Regularities
2. Preliminaries
3. Main Developments
3.1 The Central Idea
3.2 Spurious Correlations
3.3 Asymmetry
4. Counterfactual Approaches
5. Causal Modeling and Probabilistic Causation
5.1 Causal Modeling
5.2 The Markov and Minimality Conditions
5.3 What the Arrows Mean
5.4 The Faithfulness Condition
6. Further Issues and Problems
6.1 Contextual-unanimity
6.2 Potential Counterexamples
6.3 Singular and General Causation
6.4 Reduction and Circularity
Bibliography
Other Internet Resources
Related Entries
1. Introduction and Motivation
1.1 Regularity Theories
According to David Hume, causes are invariably followed by their effects: “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.” (1748, section VII.) Attempts to analyze causation
in terms of invariable patterns of succession are referred to as “regularity theories” of
causation. There are a number of well-known difficulties with regularity theories, and these
may be used to motivate probabilistic approaches to causation.
Suggested Readings: Hume (1748), especially section VII.
1.2 Imperfect Regularities
The first difficulty is that most causes are not invariably followed by their effects. For example,
it is widely accepted that smoking is a cause of lung cancer, but it is also recognized that not all
smokers develop lung cancer. (Likewise, not all non-smokers are spared the ravages of that
disease.) By contrast, the central idea behind probabilistic theories of causation is that causes
raise the probability of their effects; an effect may still occur in the absence of a cause or fail
to occur in its presence. Thus smoking is a cause of lung cancer, not because all smokers
develop lung cancer, but because smokers are more likely to develop lung cancer than
non-smokers. This is entirely consistent with there being some smokers who avoid lung cancer,
and some non-smokers who succumb.
The problem of imperfect regularities does not tell decisively against the regularity approach to
causation. Successors of Hume, especially John Stuart Mill and John Mackie, have attempted to
offer more refined accounts of the regularities that underwrite causal relations. Mackie
introduced the notion of an inus condition: an inus condition for some effect is an insufficient
but non-redundant part of an unnecessary but sufficient condition. Suppose, for example, that
a lit match causes a forest fire. The lighting of the match, by itself, is not sufficient; many
matches are lit without ensuing forest fires. The lit match is, however, a part of some
constellation of conditions that are jointly sufficient for the fire. Moreover, given that this set of
conditions occurred, rather than some other set sufficient for fire, the lighting of the match
was necessary: fires do not occur in such circumstances when lit matches are not present.
There are, however, disadvantages to this type of approach. The regularities upon which a
causal claim rest now turn out to be much more complicated then we had previously realized.
In particular, this complexity raises problems for the epistemology of causation. One appeal of
Hume's regularity theory is that it seems to provide a straightforward account of how we come
to know what causes what: we learn that A causes B by observing that As are invariably
followed by Bs. Consider again the case of smoking and lung cancer: on the basis of what
evidence do we believe that the one is a cause of the other? It is not that all smokers develop
lung cancer, for we do not observe this to be true. But neither have we observed some
constellation of conditions C, such that smoking is invariably followed by lung cancer in the
presence of C, while lung cancer never occurs in non-smokers meeting condition C. Rather,
what we observe is that smokers develop lung cancer at much higher rates than
non-smokers; this is the prima facie evidence that leads us to think that smoking causes lung
cancer. This fits very nicely with the probabilistic approach to causation.
As we shall see in Section 3.2 below, however, the basic idea that causes raise the probability
of their effects has to be qualified in a number of ways. By the time these qualifications are
added, it appears that probabilistic theories of causation have to make a move that is quite
analogous to Mackie's appeal to constellations of background conditions. Thus it is not clear that
the problem of imperfect regulaties, by itself, offers any real reason to prefer probabilistic
approaches to causation over regularity approaches.
Suggested Readings: Refined versions of the regularity analysis are found in Mill (1843),
Volume I, chapter V, and in Mackie (1974), chapter 3. The introduction of Suppes (1970)
presses the problem of imperfect regularities.
1.3 Indeterminism
While Mackie's inus condition approach can rule that smoking causes lung cancer even if there
are smokers who do not develop lung cancer, it does require that there be some conjunction
of conditions, including smoking, upon which lung cancer invariably follows. But even this more
specific regularity may fail if the occurrence of lung cancer is not physically determined by
those conditions. More generally, the regularity approach makes causation incompatible with
indeterminism: if an event is not determined to occur, then no event can be a part of a
sufficient condition for that event. (An analogous point may be made about necessity.) The
recent success of quantum mechanics -- and to a lesser extent, other theories employing
probability -- has shaken our faith in determinism. Thus it has struck many philosophers as
desirable to develop a theory of causation that does not presuppose determinism.
Many philosophers find the idea of indeterministic causation counterintuitive. Indeed, the word
“causality” is sometimes used as a synonym for determinism. A strong case for
indeterministic causation can be made by considering the epistemic warrant for causal claims.
There is now very strong empirical evidence that smoking causes lung cancer. Yet the question
of whether there is a deterministic relationship between smoking and lung cancer is wide open.
The formation of cancer cells depends upon mutation, which is a strong candidate for being an
indeterministic process. Moreover, whether an individual smoker develops lung cancer or not
depends upon a host of additional factors, such as whether or not she is hit by a bus before
cancer cells begin to form. Thus the price of preserving the intuition that causation
presupposes determinism is agnosticism about even our best supported causal claims.
Since probabilistic theories of causation require only that a cause raise the probability of its
effect, these theories are compatible with indeterminism. This seems to be a potential advantage
over regularity theories. It is unclear, however, to what extent this potential advantage is
actual. In the realm of microphysics, where we have strong (but still contestable) evidence of
indeterminism, our ordinary causal notions do not easily apply. This is brought out especially
clearly in the famous Einstein, Podolski and Rosen thought experiment. On the other hand, it is
unclear to what extent quantum indeterminism ‘percolates up’ to the macroworld of
smokers and cancer victims, where we do seem to have some clear causal intuitions.
Suggested Readings: Humphreys (1989), contains a sensitive treatment of issues involving
indeterminism and causation; see especially sections 10 and 11. Earman (1986) is a thorough
treatment of issues of determinism in physics.
1.4 Asymmetry
If A causes B, then, typically, B will not also cause A. Smoking causes lung cancer, but lung
cancer does not cause one to smoke. In other words, causation is usually asymmetric. This may
pose a problem for regularity theories, for it seems quite plausible that if smoking is an inus
condition for lung cancer, then lung cancer will be an inus condition for smoking. One way of
enforcing the asymmetry of causation is to stipulate that causes precede their effects in time.
Both Hume and Mill explicitly adopt this strategy. This has several systematic disadvantages.
First, it rules out the possibility of backwards-in-time causation a priori, whereas many believe
that it is only a contingent fact that causes precede their effects in time. Second, this approach
rules out the possibility of developing a causal theory of temporal order (on pain of vicious
circularity), a theory that has seemed attractive to some philosophers. Third, it would be nice if
a theory of causation could to provide some explanation of the directionality of causation,
rather than merely stipulate it.
Some proponents of probabilistic theories of causation follow Hume in identifying causal
direction with temporal direction. Others have attempted to use the resources of probability
theory to articulate a substantive account of the asymmetry of causation, with mixed success.
We will discuss these proposals at greater length in Section 3.3 below.
Suggested Readings: Hausman (1998) contains a detailed discussion of issues involving the
asymmetry of causation. Mackie (1974), chapter 3, shows how the problem of asymmetry can
arise for his inus condition theory. Lewis (1986) contains a very brief but clear statement of
the problem of asymmetry.
1.5 Spurious Regularities
Suppose that a cause is regularly followed by two effects. For instance, suppose that whenever
the barometric pressure in a certain region drops below a certain level, two things happen.
First, the height of the column of mercury in a particular barometer drops below a certain
level. Shortly afterwards, a storm occurs. This situation is shown schemaatically in Figure 1.
Then, it may well also be the case that whenever the column of mercury drops, there will be a
storm. (More plausibly, the dropping of the barometer will be an inus condition for the storm.)
Then it appears that a regularity theory would have to rule that the drop of the mercury
column causes the storm. In fact, however, the regularity relating these two events is
spurious; it does not reflect the causal influence of one on the other.
Figure 1
The ability to handle such spurious correlations is probably the greatest success of probabilistic
theories of causation, and remains a major source of attraction for such theories. We will
discuss this issue in greater detail in Section 3.2 below.
Suggested Readings: Mackie (1974), chapter 3, shows how the problem of spurious
regularities can arise for his inus condition theory. Lewis (1986) contains a very brief but clear
statement of the problem of spurious regularities.
2. Preliminaries
Before preceding to the formal development of a probablistic theory of causation in the next
section, it will be helpful to address a few preliminary points. First, a given event may have
many different causes. A match is struck and it lights. The striking of the match is a cause of its
lighting, but the presence of oxygen is also a cause, and there will be many others besides.
Sometimes, in casual conversation, we refer to one or another of these as “the cause” of the
match's lighting. Which cause we single out in this manner may depend upon our interests, our
expectations, and so on. Philosophical theories of causation normally attempt to analyze the
notion of “a cause.” Note also that causes may be standing conditions -- such as the presence
of oxygen -- as well as changes.
Second, it is common to distinguish two different kinds of causal claim. Singular causal claims,
such as “Jill's heavy smoking during the ‘80's caused her to develop lung cancer,” relate
particular events that have spatiotemporal locations. (Some authors claim that singular causal
claims relate facts instead.) When used in this way, cause is a success verb: the singular causal
claim implies that Jill smoked heavily during the ‘80's and that she developed lung cancer.
Note that this usage is at odds with the usage of “probabilistic causation” in the legal
literature. This phrase is used when an individual is exposed to a risk (such as a carcinogen)
regardless of whether one in fact succumbs to that risk. (The legal issue is whether an
individual who is exposed to a risk is thereby harmed, and can receive compensation for the
exposure.) General causal claims, such “smoking causes lung cancer” relate repeatable event
types or properties. Some authors have put forward probabilistic theories of singular causation,
others have advanced probabilistic theories of general causation. The relationship between
singular and general causation is discussed in Section 6.3 below; as we shall see, there seems to
be some reason to think that probabilistic theories of causation are better suited to analyzing
general causation. The causal relata -- the entities that stand in causal relations -- are
variously thought to be facts, events, properties, and so on. I will not try to adjudicate between
these different approaches, but will use the generic term “factor.” Note, however, that
probabilistic theories of causation do require that causal relata be broadly “propositional” in
character: they are the sorts of things that can be conjoined and negated.
Suggested Readings: Mill (1843) contains the classic discussion of “the cause” and “a
cause.” Bennett (1988) is an excellent discussion of facts and events.
3. Main Developments
3.1 The Central Idea
The central idea that causes raise the probability of their effects can be expressed formally
using the apparatus of conditional probability. Let A, B, C, … represent factors that
potentially stand in causal relations. Let P be a probability function, satisfying the normal rules
of the probability calculus, such that P(A) represents the empirical probability that factor A
occurs or is instantiated (and likewise for the other factors). The issue of how empirical
probability is to be interpreted will not be addressed here. Using standard notation, we let P(B
| A) represent the conditional probability of B, given A. Formally, conditional probability is
standardly defined as a certain ratio of probabilities:
P(B | A) = P(A & B)/P(A).
As an illustration, suppose that we toss a fair die. Let A represent the die's landing with an
even number (2, 4 or 6) showing on the topmost face. Then P(A) is one-half. Let B represent
the die's landing with a prime number (2, 3 or 5) showing on the topmost face (on that same
roll). Then P(B) is also one-half. Now the conditional probability P(B | A) is one-third. It is the
probability that the number on the die is both even and prime, i.e., that the number is 2,
divided by the probability that the number is even. The numerator is one-sixth, and the
denominator is one-half; hence that conditional probability is one-third. The concept of
conditional probability does not have any notion of temporal or causal order built into it.
Suppose, for example, that the die is rolled twice. It makes sense to ask about the probability
that the first roll is a prime number, given that the first roll is even; the probability that the
second roll is a prime number, given that the first roll is even; and the probability that the
first roll is a prime number, given that the second roll is even.
If P(A) is 0, then the ratio in the definition of conditional probability is undefined. There are,
however, other technical developments that will allow us to define P(B | A) when P(A) is 0.
The simplest is simply to take conditional probability as a primitive, and to define unconditional
probability as probability conditional on a tautology.
One natural way of understanding the idea that A raises the probability of B is that P(B | A)
> P(B | not-A). Thus a first attempt at a probabilistic theory of causation would be:
PR: A causes B if and only if P(B | A) > P(B | not-A).
This formulation is labeled PR for “Probability-Raising.” When P(A) is strictly between 0 and
1, the inequality in PR turns out to be equivalent to P(B | A) > P(A) and also to P(A & B) >
P(A)P(B). When this last relation holds, A and B are said to be positively correlated. If the
inequality is reversed, they are negatively correlated. If A and B are either positively or
negatively correlated, they are said to be probabilistically dependent. If equality holds, then A
and B are probabilistically independent or uncorrelated.
PR addresses the problems of imperfect regularities and indeterminism, discussed above. But it
does not address the other two problems discussed in section 1 above. First, probability-raising
is symmetric: if P(B | A) > P(B | not-A), then P(A | B) > P(A | not-B). The causal relation,
however, is typically asymmetric.
Figure 2
Second, PR has trouble with spurious correlations. If A and B are both caused by some third
factor, C, then it may be that P(B | A) > P(B | not-A) even though A does not cause B. This
situation is shown schematically in Figure 2. For example, let A be an individual's having
yellow-stained fingers, and B that individual's having lung cancer. Then we would expect that
P(B | A) > P(B | not-A). The reason that those with yellow-stained fingers are more likely to
suffer from lung cancer is that smoking tends to produce both effects. Because individuals with
yellow-stained fingers are more likely to be smokers, they are also more likely to suffer from
lung cancer. Intuitively, the way to address this problem is to require that causes raise the
probabilities of their effects ceteris paribus. The history of probabilistic causation is to a large
extent a history of attempts to resolve these two central problems.
Suggested Readings: For a primer on basic probability theory, see the entry for “probability
calculus: interpretations of.” This entry also contains a discussion of the intperpretation of
probability claims.
3.2 Spurious Correlations
Hans Reichenbach introduced the terminology of “screening off” to apply to a particular type
of probabilistic relationship. If P(B | A & C) = P(B | C), then C is said to screen A off from
B. (When P(A & C) > 0, this equality is equivalent to P(A & B | C) = P(A | C)P(B | C).)
Intuitively, C renders A probabilistically irrelevant to B. With this notion in hand, we can
attempt to avoid the problem of spurious correlations by adding a ‘no screening off’
condition to the basic probability-raising condition:
NSO: Factor A occurring at time t, is a cause of the later factor B if and only if:
1.P(B | A) > P(B | not-A)
2.There is no factor C, occurring earlier than or simultaneously with A, that
screens A off from B.
We will call this the NSO, or ‘No Screening Off’ formulation. Suppose, as in our example
above, that smoking (C) causes both yellow-stained fingers (A) and lung cancer (B). Then
smoking will screen yellow-stained fingers off from lung cancer: given that an individual
smokes, his yellow-stained fingers have no impact upon his probability of developing lung
cancer.
The second condition of NSO does not suffice to resolve the problem of spurious correlations,
however. This condition was added to eliminate cases where spurious correlations give rise to
factors that raise the probability of other factors without causing them. Spurious correlations
can also give rise to cases where a cause does not raise the probability of its effect. So genuine
causes need not satisfy the first condition of NSO. Suppose, for example, that smoking is highly
correlated with exercise: those who smoke are much more likely to exercise as well. Smoking is
a cause of heart disease, but suppose that exercise is an even stronger preventative of heart
disease. Then it may be that smokers are, over all, less likely to suffer from heart disease than
non-smokers. That is, letting A represent smoking, C exercise, and B heart disease, P(B | A)
< P(B | not-A). Note, however, that if we conditionalize on whether one exercises or not, this
inequality is reversed: P(B | A & C) > P(B | not-A & C), and P(B | A & not-C) > P(B |
not-A & not-C). Such reversals of probabilistic inequalities are instances of “Simpson's
Paradox.”
The next step is to replace conditions 1 and 2 with the requirement that causes must raise the
probability of their effects in test situations:
TS: A causes B if P(B | A & T) > P(B | not-A & T) for every test situation T.
A test situation is a conjunction of factors. When such a conjunction of factors is conditioned on,
those factors are said to be “held fixed.” To specify what the test situations will be, then, we
must specify what factors are to be held fixed. In the previous example, we saw that the true
causal relevance of smoking for lung cancer was revealed when we held exercise fixed, either
positively (conditioning on C) or negatively (conditioning on not-C). This suggests that in
evaluating the causal relevance of A for B, we need to hold fixed other causes of B, either
positively or negatively. This suggestion is not entirely correct, however. Let A and B be
smoking and lung cancer, respectively. Suppose C is a causal intermediary, say the presence of
tar in the lungs. If A causes B exclusively via C, then C will screen A off from B: given the
presence (absence) of carcinogens in the lungs, the probability of lung cancer is not affected by
whether those carcinogens got there by smoking (are absent despite smoking). Thus we will not
want to hold fixed any causes of B that are themselves caused by A. Let us call the set of all
factors that are causes of B, but are not caused by A, the set of independent causes of B. A
test situation for A and B will then be a maximal conjunction, each of whose conjuncts is
either an independent cause of B, or the negation of an independent cause of B.
Note that the specification of factors that need to be held fixed appeals to causal relations. This
appears to rob the theory of its status as a reductive analysis of causation. We will see in
Section 6.4 below, however, that the issue is substantially more complex than that. In any
event, even if there is no reduction of causation to probability, a theory detailing the
systematic connections between causation and probability would be of great philosophical
interest.
The move from the basic idea of PR to the complex formulation of TS is rather like the move
from Hume's original regularity theory to Mackie's theory of inus conditions. In both cases, the
move substantially complicates the epistemology of causation. In order to know whether A is a
cause of B, we need to know what happens in the presence and absence of B, while holding
fixed a complicated conjunction of further factors. The hope that a probabilistic theory of
causation would enable us to handle the problem of imperfect regularities without appealing to
such constellations of background conditions seems not to have been borne out. Nonetheless,
TS does seem to provide us with a theory that is compatible with indeterminism and that can
distinguish causation from spurious correlation.
TS can be generalized in at least two important ways. First, we can define a ‘negative cause’
or ‘preventer’ or ‘inhibitor’ as a factor that lowers the probability of its ‘effect’ in all
test situations, and a ‘mixed’ or ‘interacting’ cause as one that affects the probability of
its ‘effect’ in different ways in different test situations. It should be apparent that when
constructing test situations for A and B one should also hold fixed preventers and mixed
causes of B that are independent of A. Generalizing even further, one could define causal
relationships between variables that are non-binary, such as caloric intake and blood pressure.
In evaluating the causal relevance of X for Y, we will need to hold fixed the values of
variables that are independently causally relevant to Y. In principle, there are infinitely many
ways in which one variable might depend probabilistically on another, even holding fixed some
particular test situation. Thus, once the theory is generalized to include non-binary variables, it
will not be possible to provide any neat classification of causal factors into causes and
preventers.
These two generalizations bring out an important distinction. It is one thing to ask whether A
is causally relevant to B in some way; it is another to ask in which way is A causally
relevant to B. To say that A causes B is then potentially ambiguous: it might mean that A is
causally relevant to B in some way or other; or it could mean that A is causally relevant for
B in a particular way, that A promotes B or is a positive factor for the occurrence of B. For
example, if A prevents B, then A will count as a cause of B in the first sense, but not in the
second. Probabilistic theories of causation can be used to answer both types of question. A is
causally relevant to B if A makes some difference for the probability of B in some test
situation; whereas A is a positive or promoting cause of B if A raises the probability of B in
all test situations.
The problem of spurious correlations also plagues certain versions of decision theory. This can
happen when one's choice of action is symptomatic of certain good or bad outcomes, without
causing those outcomes. (The best-known example of this sort is Newcomb's Problem.) In cases
like this, some versions of decision theory appear to recommend that one act so as to receive
good news about events beyond one's control, rather than act so as to bring about desirable
events that are within one's control. In response, many decision theorists have advocated
versions of causal decision theory. Some versions closely resemble TS.
Suggested Readings: This section more or less follows the main developments in the history
of probabilistic theories of causation. Versions of the NSO theory are found in Reichenbach
(1956, section 23), and Suppes (1970, chapter 2). Good (1961, 1962) is an early essay on
probabilistic causation that is rich in insights, but has had surprisingly little influence on the
formulation of later theories. Salmon (1980) is an influential critique of these theories. The first
versions of TS were presented in Cartwright (1979) and Skyrms (1980). Eells (1991, chapters
2, 3, and 4) and Hitchcock (1993) carry out the two generalizations of TS described. Skyrms
(1980) presents a version of causal decision theory that is very similar to TS. See also the
entry for “decision theory: causal.”
3.3 Asymmetry
The second major problem with the basic probability-raising idea is that the relationship of
probability-raising is symmetrical. Some proponents of probabilistic theories of causation simply
stipulate that causes precede their effects in time. As we saw in Section 1.4 above, this strategy
has a number of disadvantages. Note also that while assigning temporal locations to particular
events is entirely coherent, it is not so clear what it means to say that one property or event
type occurs before another. For example, what does it mean to say that smoking precedes lung
cancer? There have been many episodes of smoking, and many of lung cancer, and not all of
the former occurred prior to all of the latter. This will be a problem for those who are
interested in providing a probabilistic theory of causal relations among properties or event
types.
Some defenders of manipulability or agency theories of causation have argued that the
necessary asymmetry is provided by our perspective as agents. In assessing whether A is a
cause of B, we must ask whether A increases the probability of B, where the relevant
conditional probabilities are agent probabilities: the probabilities that B would have were A
(or not-A) to be realized by the choice of a free agent. Critics have wondered just what these
agent probabilities are.
Other approaches attempt to locate the asymmetry between cause and effect within the
structure of the probabilities themselves. One very simple proposal would be to refine the way
in which the test situations are constructed. (See the previous section for discussion of test
situations.) In evaluating whether A is a cause of B, we should hold fixed not only the
independent causes of B, but also the causes of A. Thus if B is a cause of A, rather than vice
versa, A will not raise the probability of B in the appropriate test situation, since the presence
or absence of B will already be held fixed. This idea is built into the Causal Markov Condition
discussed in Section 5 below. Proponents of traditional probabilistic theories of causation have
not adopted this strategy. This may be because they feel that this refinement would take the
theory too close to vicious circularity: in order to assess whether A causes B, we would need
to know already whether B causes A.
A more ambitious approach to the problem of causal asymmetry is due to Hans Reichenbach.
Suppose that factors A and B are positively correlated:
1. P(A & B) > P(A)P(B)
It is easy to see that this will hold exactly when A raises the probability of B and vice versa.
Suppose, moreover, that there is some factor C having the following properties:
2. P(A & B | C) = P(A | C)P(B | C)
3. P(A & B | not-C) = P(A | not-C)P(B | not-C)
4. P(A | C) > P(A | not-C)
5. P(B | C) > P(B | not-C).
In this case, the trio ACB is said to form a conjunctive fork. Conditions 2 and 3 stipulate that
C and not-C screen off A from B. As we have seen, this sometimes occurs when C is a
common cause of A and B. Conditions 2 through 5 entail 1, so in some sense C explains the
correlation between A and B. If C occurs earlier than A and B, and there is no event
satisfying 2 through 5 that occurs later than A and B, then ACB is said to form a
conjunctive fork open to the future. Analogously, if there is a future factor satisfying 2
through 5, but no past factor, we have a conjunctive fork open to the past. If a past factor C
and a future factor D both satisfy 2 through 5, then ACBD forms a closed fork. Reichenbach's
proposal was that the direction from cause to effect is the direction in which open forks
predominate. In our world, there are many forks open to the future, few or none open to the
past. This proposal is closely related to Reichenbach's Common Cause Principle, which says that
if A and B are positively correlated (i.e., they satisfy condition 1), then there exists a C,
which is a cause of both A and B, and which screens them off from each other. (By contrast,
common effects do not in general screen off their causes.)
It is not clear, however, that this asymmetry between forks open to the past and forks open to
the future will be as pervasive as this proposal seems to presuppose. In quantum mechanics,
there are correlated effects that are believed to have no common cause that screens them off.
Moreover, if ACB forms a conjunctive fork in which C precedes A and B, but C has a
deterministic effect D which occurs after A and B, then ACBD will form a closed fork. A
further difficulty with this proposal is that since it provides a global ordering of causes and
effects, it seems to rule out a priori the possibility that some effects might precede their
causes. More complex attempts to derive the direction of causation from probabilities have been
offered; the issues here intersect with the problem of reduction, discussed in Section 6.4 below.
Suggested Readings: Suppes (1970, chapter 2) and Eells (1991, chapter 5) define causal
asymmetry in terms of temporal asymmetry. Price (1991) defends an account of causal
asymmetry in terms of agent probabilities; see also the entry for “causation and
manipulation.” Reichenbach's proposal is presented in his (1956, chapter IV). Some difficulties
with this proposal are discussed in Arntzenius (1993); see also his entry to this encylopedia
under “physics: Reichenbach's common cause principle.” Papineau (1993) is a good overall
discussion of the problem of causal asymmetry within probabilistic theories. Hausman (1998) is
a detailed study of the problem of causal asymmetry.
4. Counterfactual Approaches
A leading approach to the study of causation has been to analyze causation in terms of
counterfactual conditionals. A counterfactual conditional is a subjunctive conditional sentence,
whose antecedent is contrary-to-fact. Here is an example: “if the butterfly ballot had not been
used in West Palm Beach, then Albert Gore would be the president on the United States.” In
the case of indeterministic outcomes, it may be appropriate to use probabilistic consequents:
“if the butterfly ballot had not been used in West Palm Beach, then Albert Gore would have
had a .7 chance of being elected president.” A probabilistic counterfactual theory of causation
(PC) aims to analyze causation in terms of these probabilistic counterfactuals. The event B is
said to causally depend upon the distinct event A just in case both occur and the probability
that B would occur, at the time of As occurrence, was much higher than it would have been
at the corresponding time if A had not occurred. This counterfactual is to be understood in
terms of possible worlds: it is true if, in the nearest possible world(s) where A does not occur,
the probability of B is much lower than it was in the actual world. On this account, the
relevant notion of `probability-raising' is not understood in terms of conditional probabilities,
but in terms of unconditional probabilities in different possible worlds. The test situation is not
some specified conjunction of factors, but the sum total of all that remains unchanged in
moving to the nearest possible world(s) where A does not occur. Note that PC is intended
specifically as a theory of singular causation between particular events, and not as a theory of
general causation.
Causal dependence, as defined in the previous paragraph, is sufficient, but not necessary, for
causation. Causation is defined to be the ancestral of causal dependence; that is, A causes B
just in case there is a sequence of events C1, C2, …, Cn, such that C1 causally depends
upon A, C2 causally depends upon C1, …, B causally depends upon Cn. This modification
guarantees that causation will be transitive: if A causes C, and C causes B, then A causes B.
This modification is also useful in addressing certain problems discussed in Section 6.2 below.
Proponents of counterfactual theories of causation attempt to derive the asymmetry of
causation from a corresponding asymmetry in the truth values of counterfactuals. For instance,
it may be true that if Mary had not smoked, she would have been less likely to develop lung
cancer, but we would not normally agree that if Mary had not developed lung cancer, she
would have been less likely to smoke. Ordinary counterfactuals do not ‘backtrack’ from
effects to causes. This proscription against backtracking also solves the problem of spurious
correlations: we would not say that if the column of mecury had not risen, then the drop in
atmospheric pressure would have been less likely, and so the storm would have been less likely
as well.
One important question is whether the counterfactuals that appear in the analysis of causation
can be characterized without reference to causation. In order to do this, one would have to say
what makes some worlds closer than others without making reference to any causal notions.
Despite some interesting attempts, it is not clear whether this can be done. If not, then it will
not be possible to provide a reductive PC analysis of causation, although it may still be possible
to articulate interesting interconnections between causation, probability and counterfactuals.
The Philosopher Igal Kvart has been a persistent critic of the claim that it is possible to analyze
counterfactuals without using causation. He has developed a probabilistic theory of singular
causation that does not use counterfactuals. Nonetheless, his theory has a number of features
in common with counterfactual theories: it is an attempt to analyze singular causation among
events; it elaborates on the basic probability-raising idea in an attempt to avoid some of the
problems raised in Section 6.2 below; and it aspires to be a reductive analysis of causation,
making no reference to causal relations in the analysans.
Suggested Readings: Lewis (1986a) is the locus classicus for PC. Lewis (1986b) is an
attempt to explicate the notion of proximity among possible worlds. Recent attempts to analyze
causation in terms of probabilistic counterfactuals have become quite intricate; see for example
Noordhof (1999). For further discussion of counterfactual theories of causation, see the entry
under “causation, counterfactual theories.” For Kvart's theory, see for example Kvart (1997).
5. Causal Modeling and Probabilistic Causation
5.1 Causal Modeling
‘Causal modeling’ is a new interdisciplinary field devoted to the study of methods of causal
inference. This field includes contributions from statistics, artificial intelligence, philosophy,
econometrics, epidemiology, and other disciplines. Within this field, the research programs that
have attracted the greatest philosophical interest are those of the computer scientist Judea
Pearl and his collaborators, and of the philosophers Peter Spirtes, Clark Glymour, and Richard
Scheines (SGS). Not coincidentally, these two programs are the most ambitious in their claims to
have developed algorithms for making causal inferences on the basis of statistical data. These
claims have generated a great deal of controversy, often quite heated. Specfically, there seems
to be a great deal of resistance to the idea that automated procedures can take the place of
subject-specific background knowledge and good experimental design, the things that causal
inference has always depended on. To some extent, this debate is one over emphasis and
advertising. Both Pearl and SGS state explicit assumptions that must be made before their
procedures can yield results. Critics charge, first, that these assumptions are buried in fine
print while the automated procedures are advertised in bold; and second, that the required
assumptions are rarely satisfied in realistic cases, rendering the new procedures virtually
useless. These charges are orthogonal to the issue of whether the techniques perform as
advertised when the necessary assumptions do hold.
Our concern here will not be with the efficacy of these methods of causal inference, but rather
with their philosophical underpinnings. We will here follow the developments of SGS, as these
bear a stronger resemblance to the probabilistic theories of causation described in Section 3
above. (Pearl's approach, at least in its more recent development, bears a stronger connection
to counterfactual approaches.)
Suggested Readings: Pearl (2000) and Spirtes, Glymour and Scheines (2000) are the most
detailed presentations of the two research programs discussed. Both works are quite technical,
although the epilogue of Pearl (2000) provides a very readable historical introduction to Pearl's
work. Pearl (1999) also contains a reasonably accessible introduction to some of Pearl's more
recent developments. Scheines (1997) is a non-technical introduction to some of the ideas in
SGS (2000). McKim and Turner (1997) is a collection of papers on causal modeling, including
some important critiques of SGS.
5.2 The Markov and Minimality Conditions
We can present here only a very rudimentary overview of the SGS framework. We begin with
a set V of variables. The set may, for instance, include variables representing the
education-level, income, parental income, et al, of individuals in a population. These variables
are different from the factors that normally figure in probabilistic theories of causation. Factors
stand to variables as determinates to determinables. "Income" is a variable; "having an income
of $40,000 per year" is a factor. Given a set of variables, we may define two different
mathematical structures over this set. First, a directed graph G on V is a set of directed edges,
or ‘arrows’, having the variables in V as their vertices. The variable X is a ‘parent’ of Y
just in case there is an arrow from X to Y. X is an ‘ancestor’ of Y (equivalently, Y is a
‘descendant’ of X) just in case there is a ‘directed path’ from X to Y consisting of
arrows linking intermediate variables. The directed graph is acyclic if there are no loops, that
is, if no variable is an ancestor of itself. In addition to a directed acyclic graph over V, we also
have a probability distribution P over the values of variables in V.
The directed acyclic graph G over V may be related to the probability distribution in a number
of ways. One important condition that the two might satisfy is the so-called Markov Condition:
MC: For every X in V, and every set Y of variables in V \ DE(X), P(X | PA(X) &
Y) = P(X | PA(X)); where DE(X) is the set of descendants of X, and PA(X) is the
set of parents of X.
The notation needs a little clarification. Consider, for example, the first term in the equality.
Since X is a variable, it doesn't really make sense to talk about the probability of X, or of the
conditional probability of X. It makes sense to talk about the probability of having an income of
$40,000 per year (at least if we are talking about members of some well-defined population),
but it makes no sense to talk about the probability of "income". (Note that we do not mean here
the probability of having some income or other. That probability is one, assuming we allow
zero to count as a value of income.) This formulation of MC uses a common notational
convention. Whenever a variable, or set of variables appears, there is a tacit universal
quantifier ranging over values of the variable(s) in question. Thus MC should be understood as
asserting an equality between two conditional probabilities that holds for all values of the
variable X, and for all values of the variables in Y and PA(X). In words, the Markov condition
says that the parents of X screen X off from all other variables, except for the descendents of
X. Given the values of the variables that are parents of X, the values of the variables in Y
(which includes no descendents of X), make no further difference to the probability that X will
take on any given value.
As stated, the Markov Condition describes a purely formal relation between abstract entities.
Suppose, however, that we give the graph and probability distribution empirical
interpretations. The graph will represent the causal relationships among the variables in a
population, and the probability distribution will represent the empirical probability that an
individual in the population will possess certain values of the relevant variables. When the
directed graph is given a causal interpretation, it is called a causal graph. We will return
shortly to the question of what, exactly, the arrows in a causal graph represent.
The Causal Markov Condition (CMC) asserts that MC holds of a population when the directed
graph and probability distribution are given these interpretations. CMC does not hold in
general, but only when certain further conditions are satisfied. For instance, V must include all
common causes of variables that are included in V. Suppose, for example, that V = {X, Y}, that
neither variable is a cause of the other, and that Z is a common cause of X and Y (the true
causal structure is shown in Figure 3 below). The correct causal graph on V will include no
arrows, since neither X nor Y cause the other. But X and Y will be probabilistically
correlated, because of the underlying common cause. This is a violation of CMC. Since the
correct causal graph on {X, Y} has no arrows, X has no parents or descendents; thus CMC
entails that P(X | Y) = P(X). This equality is false, since X and Y are in fact correlated. CMC
can also fail for certain types of heterogeneous populations composed of subpopulations with
differenct causal structures. And CMC will fail for certain quantum systems. One area of
controversy concerns the extent to which actual populations satisfy CMC with respect to the
sorts of variable sets that are typically employed in empirical investigations. For purposes of
further discussion, we will assume that CMC holds.
Figure 3
The Causal Markov Condition is a generalization of Reichenbach's Common Cause Principle,
discussed in Section 3.3 above. Here are a few illustrations of how it works.
Figure 4
In Figures 3 and 4, CMC entails that the values of Z screen off the values of X from the
values of Y.
Figure 5
Figure 6
In Figures 5 and 6, CMC again entails that the values of Z screen off the values of X from the
values of Y. However, CMC does not entail that the values of W screen off the values of X
from the values of Y in Figure 5, whereas it does entail that the values of W screen off the
values of X from the values of Y in Figure 6. This shows that being a common cause of X and
Y is neither necessary nor sufficient for screening off the values of those variables.
Figure 7
In Figure 7, both Z and W are common causes of X and Y, yet CMC does not entail that
either one of them, by itself, suffices to screen off the values of X and Y. This seems
reasonable: if we hold fixed the value of Z, we should expect X and Y to remain correlated
due to the action of W. CMC does entail that Z and W jointly screen off X and Y; that is,
when we condition on the values of Z and W, there will be no residual correlation between X
and Y.
A second important relation between a directed graph and probability distribution is the
Minimality Condition. Suppose that the directed graph G on variable set V satisfies the Markov
condition with respect to the probability distribution P. The Minimality Condition asserts that
no sub-graph of G over V also satisfies the Markov Condition with respect to P. The Causal
Minimality Condition asserts that the Minimality Condition holds when G and P are given their
empirical interpretations. As an illustration, consider the variable set {X, Y}, let there be an
arrow from X to Y, and suppose that X and Y are probabilistically independent of each other
in P. This graph would satisfy the Markov Condition with respect to P: none of the
independence relations mandated by MC are absent (in fact, MC mandates no independence
relations). But this graph would violate the Minimality Condition with respect to P, since the
subgraph that omits the arrow from X to Y would also satisfy the Markov Condition.
Suggested Readings: Spirtes, Glymour and Scheines (2000) and Scheines (1997). Hausman
and Woodward (1999) provide a detailed discussion of the Causal Markov Condition.
5.3 What the Arrows Mean
We are now in a better position to say something about what the arrows in a causal graph
mean. First consider a simple graph with two variables X and Y and an arrow from X to Y.
The Minimality Condition requires that the two variables not be probabilistically independent.
This means that there must be values x and x of X and y of Y, such that
P(Y = y | X = x) P(Y = y | X = x).
This says nothing about how X bears on Y. Suppose for example, that we have a three
variable model, including the variables smoking, exercise, and heart disease. The causal graph
would (presumably) include an arrow from smoking to heart disease, and an arrow from
exercise to heart disease. Nothing in the graph indicates that increased levels of smoking
increase the risk and severity of heart disease, whereas increased levels of exercise (up to a
point, anyway) decrease the risk and severity of heart disease. Thus an arrows in a causal
graph indicates only that one variable is causally relevant to another, and says nothing about
the way in which it is relevant (whether it is a promoting, inhibiting, or interacting cause, or
stands in some more complex relation).
Figure 8
Consider Figure 8. Note that it differs from Figure 4 in that there is an additional arrow
running directly fron X to Y. What does this arrow from X to Y indicate? It does not merely
indicate that X is causally relevant to Y; in Figure 4, it is natural to expect that X will
relevant to Y via its effect on Z. Applying the Causal Markov and Minimality Conditions, the
arrow from X to Y indicates that Y is probabilistically dependent on X, even when we hold
fixed the value of Z. That is, X makes a probabilistic difference for Y, over and above the
difference it makes in virtue of its effect on Z. Figure 8 thus indicates that X has an effect on
Y via two different routes: one route that runs through the variable Z and the other route
which is direct, i.e., unmediated by any other variable in V. As an illustration, consider a
well-known example due to Germund Hesslow. Consumption of birth control pills (X) is a risk
factor for thrombosis (Y). On the other hand, birth control pills are an effective preventer of
pregnancy (Z), which is in turn a powerful risk factor for thrombosis. The use of birth control
pills may thus affect one's chances of suffering from thrombosis in two different ways, one
'direct', and one via the effect of pills on one's chances of becoming pregnant. Whether birth
control pills raise or lower the probability of thrombosis overall will depend upon the relative
strengths of these two routes. The probabilistic theories of causation described in Section 3
above are suited to analyze the total or net effect of one factor or variable on other, whereas
the causal modeling techniques discussed in this section are primarily geared toward
decomposing a causal system into individual routes of causal influence.
Suggested Readings: The birth control pill example was originally presented in Hesslow
(1976). Hitchcock (2001a) discusses the distinction between total or net effect, and causal
influence along individual routes.
5.4 The Faithfulness Condition
One final condition that SGS make extensive use of is the Faithfulness Condition. (I will dispense
with the distinction between the causal and non-causal versions.) The Faithfulness Condition
says that all of the (conditional and unconditional) probabilistic independencies that exist among
the variables in V are required by the Causal Markov Condition. For example, suppose that V
= {X, Y, Z}. Suppose also that X and Y are unconditionally independent of one another, but
dependent, conditional upon Z. (The other two variables pairs are dependent, both conditionally
and unconditionally.) The graph shown in Figure 8 does not satisfy the faithfulness condition
with respect to this distribution (colloquially, the graph is not faithful to the distribution). CMC,
when applied to the graph of Figure 8, does not imply the independence of X and Y. By
contrast, the graph shown in Figure 9 is faithful to the described distribution. Note that Figure
8 does satisfy the Minimality Condition; no subgraph satisfies CMC with respect to the
described distribution. (The graph in Figure 9 is not a subgraph of the graph in Figure 8.)
Figure 9
The Faithfulness Condition implies that the causal influences of one variable on another along
multiple causal routes does not ‘cancel’. For example, suppose that Figure 8 correctly
represents the underlying causal structure. Then the Faithfulness Condition implies that X and
Y cannot be unconditionally independent of one another in the empirical distribution. In
Hesslow's example, this means that the tendency of birth control pills to cause thrombosis along
the direct route cannot be exactly canceled by the tendency of birth control pills to prevent
thrombosis by preventing pregnancy. This ‘no canceling’ condition seems implausible as a
metaphysical or conceptual constraint upon the connection between causation and probabilities.
Why can't competing causal paths cancel one another out? Indeed, Newtonian physics provides
us with an example: the downward force on my body due to gravity triggers an equal and
opposite upward force on my body from the floor. My body responds as if neither force were
acting upon it. The Faithfulness Condition seems rather to be a methodological principle. Given
a distribution on {X, Y, Z} in which X and Y are independent, we should infer that the causal
structure is that depicted in Figure 9, rather than Figure 8. This is not because Figure 8 is
conclusively ruled out by the distribution, but rather because it is gratuitously complex: it
postulates causal connections that are not necessary to explain the underlying pattern of
probabilistic dependencies. The Faithfulness Condition is thus a formal version of Ockham's
razor.
SGS use the Causal Markov, Minimality, and Faithfulness Conditions to prove a variety of
statistical indistinguishability theorems. These theorems tell us when two distinct causal
structures can or cannot be distinguished on the basis of the probability distributions to which
they give rise. We will return to this issue in Section 6.4 below.
Suggested Readings: Spirtes, Glymour and Scheines (2000) and Scheines (1997).
6. Further Issues and Problems
6.1 Contextual-unanimity
According to TS, a cause must raise the probability of its effect in every test situation. This
has been called the requirement of contextual-unanimity. This requirement is vulnerable to
the following sort of counterexample. Suppose that there is a gene that has the following effect:
those that possess the gene have their chances of contracting lung cancer lowered when they
smoke. This gene is very rare, let us imagine (indeed, it need not exist at all in the human
population, so long as humans have some non-zero probability of possessing this gene, perhaps
as a result of a very improbable mutation). In this scenario, there would be test situations
(those that hold fixed the presence of the gene) in which smoking lowers the probability of lung
cancer: thus smoking would not be a cause of lung cancer according to the context-unanimity
requirement. Nonetheless, it seems unlikely that the discovery of such a gene (or of the mere
possibility of its occurrence) would lead us to abandon the claim that smoking causes lung
cancer.
This line of objection is surely right about our ordinary use of causal language. It is nonetheless
open to the defender of context-unanimity to respond that she is interested in supplying a
precise concept to replace the vague notion of causation that corresponds to our everyday
usage. In a population consisting of individuals lacking the gene, smoking causes lung cancer. In
a population consisting entirely of individuals who possess the gene, smoking prevents lung
cancer.
Note that this dispute only arises in the context of a heterogeneous population. Restricting
ourselves to one particular test situation, both parties can agree that smoking causes lung
cancer in that test population just in case it increases the probability of lung cancer in that test
situation.
One's position in this debate will depend, in part, on how one wants to use general causal claims
such as “smoking causes lung cancer”. If one conceives of them as causal laws, then the
contextual-unanimity requirement may seem attractive. If “smoking causes lung cancer” is a
kind of law, then its truth should not be contingent upon the scarcity of the gene that reverses
the effects of smoking. By contrast, one may understand the causal claim in a more practical
way, by treating it as a kind of policy-guiding principle. Since the gene in question is very rare,
it would still be rational for public health organizations to promote policies that would reduce
the incidence of smoking.
Suggested Readings: Dupré; (1984) presents this challenge to the context-unanimity
requirement, and offers an alternative. Eells (1991, chapters 1 and 2), defends
context-unanimity using the idea that causal claims are made relative to a population. Hitchock
(2001b) contains further discussion and develops the idea of treating general causal claims as
policy-guiding principles.
6.2 Potential Counterexamples
Given the basic probability-raising idea, one would expect putative counterexamples to
probabilistic theories of causation to be of two basic types: cases where causes fail to raise the
probabilities of their effects, and cases where non-causes raise the probabilities of non-effects.
The discussion in the literature has focused almost entirely on the first sort of example.
Consider the following example, due to Deborah Rosen. A golfer badly slices a golf ball, which
heads toward the rough, but then bounces off a tree and into the cup for a hole in one. The
golfer's slice lowered the probability that the ball would wind up in the cup, yet nonetheless
caused this result. One way of avoiding this problem is to attend to the probabilities that are
being compared. If we label the slice A, not-A is a disjunction of several alternatives. One such
alternative is a clean shot -- compared to this alternative, the slice lowered the probability of a
hole-in-one. Another alternative is no shot at all, relative to which the slice increases the
probability of a hole-in-one. By making the latter sort of comparison, we can recover our
original intuitions about the example.
A different sort of counterexample involves causal preemption. Suppose that an assassin puts a
weak poison in the king's drink, resulting in a 30% chance of death. The king drinks the poison
and dies. If the assassin had not poisoned the drink, her associate would have spiked the drink
with an even deadlier elixir (70% chance of death). In the example, the assassin caused the king
to die by poisoning his drink, even though she lowered his chance of death (from 70% to 30%).
Here the cause lowered the probability of death, because it preempted an even stronger cause.
One approach to this problem, built into the counterfactual approach described in Section 4
above, is to invoke the principle of the transitivity of causation. The assassin's action increased
the probability of, and hence caused, the presence of weak poison in the king's drink. The
presence of weak poison in the king's drink raised the probability of, and hence caused, the
king's death. (By this time, it is already determined that the associate will not poison the drink.)
By transitivity, the assassin's action caused the king's death. The claim that causation is
transitive is highly controversial, however, and is subject to many persuasive counterexamples.
Another approach would be to invoke a distinction introduced in Section 5.3 above. The
assassin's action affects the king's chances of death in two distinct ways: first, it introduces the
weak poison into the king's drink; second, it prevents the introduction of a stronger poison. The
net effect is to reduce the king's chance of death. Nonetheless, we can isolate the first of these
effects (which would be indicated by an arrow in a causal graph). We do this by holding fixed
the inaction of the associate: given that the associate did not in fact poison the drink, the
assassin's action increased the king's chance of death (from near zero to .3). We count the
assassin's action as a cause of death because it increased the chance of death along one of the
routes connecting the two events.
For a counterexample of the second type, suppose that two gunmen shoot at a target. Each has
a certain probability of hitting, and a certain probability of missing. Assume that none of the
probabilities are one or zero. As a matter of fact, the first gunman hits, and the second
gunman misses. Nonetheless, the second gunman did fire, and by firing, increased the
probability that the target would be hit, which it was. While it is obviously wrong to say that
the second gunman's shot caused the target to be hit, it would seem that a probabilistic theory
of causation is committed to this consequence. A natural approach to this problem would be to
try to combine the probabilistic theory of causation with a requirement of spatiotemporal
connection between cause and effect, although it is not at all clear how this hybrid theory
would work.
Suggested Readings: The example of the golf ball, due to Deborah Rosen, is first presented in
Suppes (1970) Salmon (1980) presents several examples of probability-lowering causes.
Hitchcock (1995) presents a response. Lewis (1986a) discusses cases of preemption, see also the
entry for “causation: counterfactual theories.” Hithcock (2001a) presents the solution in
terms of decomposition into component causal routes. Woodward (1990) describes the
structure that is instantiated in the example of the two gunmen. Humphreys (1989, section
14) responds. Menzies (1989, 1996) discusses examples involving causal pre-emption where
non-causes raise the probabilities of non-effects. Hitchcock (2002) provides a general discussion
of these counterexamples. For a discussion of attempts to analyze cause and effect in terms of
contiguous processes, see the entry for “causation: causal processes.”
6.3 Singular and General Causation
We noted in section 2 above that we make at least two different kinds of causal claim, singular
and general. With this distinction in mind, we may note that the counterexamples mentioned in
the previous section are all formulated in
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