The operationalizing assumption (OA) linking causal connections and probabilistic dependencies appears to be false – probabilistic dependencies seem to be neither sufficient nor necessary for nomic connections. In chapter 4,1 urged the reader instead to regard OA as a useful approximation. But perhaps appearances are misleading. Is it truly impossible to reduce claims about causation to claims about probabilistic dependencies? Chapter 11 suggested that a reduction may be possible within the context of path models. Could OA and the independence theory provide a reductive account of causation? And if they cannot, just how closely linked are lawful connections and probabilistic dependencies? To what extent can the independence theory of causation be made operational?

Do Probabilistic Dependencies Imply Causal Connections?

There seem to be cases in which a is not causally connected to b – that is, that there is no nomological link between a and b – even though a and b are distinct events and in the background circumstances the kinds a and b are probabilistically dependent. This section will be concerned with three difficulties for the claim that probabilistic dependencies are sufficient for causal connections.

The first concerns “spurious” probabilistic dependencies, like those between water levels in Venice and bread prices in England. Papineau has argued that “correlations between the stages of different time series are not to be counted as causally significant unless they display co-variation beyond that due to co-variation within each time series” (1993, p. 243).

This chapter tells you where I started, explains how my initial hunches led to difficulties, and lays bare a fundamental ambivalence that nags the discussions in this book. It exposes unclarities in what I was looking for in a theory of causation.

Metaphysical Theories

In this century, metaphysics has been in ill repute. The word “metaphysics” conjures up an image of a philosopher envisioning “the essential features of reality” and then confidently dictating to artists, moralists, and scientists. Does metaphysics have any genuine content? Are there metaphysical questions that do not collapse into either empirical questions subject to scientific inquiry or semantic questions subject to conceptual analysis?

I sympathize with this “positivistic” skepticism concerning the possibility of conjuring substantive knowledge out of pure contemplation, and there is, I hope, no conjuring in this book. Although questions such as “What are the differences between causes and effects?” are more general than the questions scientists ask, they are not of a different kind. Metaphysical questions concerning causation are continuous with scientific questions. Although more abstract than theories in geology or sociology, the account of causation I defend in this book is intended as an empirical theory (jointly of nature and of human explanatory practices). This theory, like scientific theories generally, is acceptable only if it helps our beliefs and practices to fit together coherently.

The Question

One might say, “The explosion was caused by the foreman's striking a match,” or “Margaret's hitting the tomato with a hammer smashed it.” The second claim does not use the word “cause,” but smashing is a kind of causing.

It is fitting for both historical and philosophical reasons to begin the discussion of causation with David Hume's theory and its contemporary developments. Hume's theory is the starting point for most modern treatments of causation, and the problems his theory must surmount are problems for all theories of causation. Some philosophers believe that Hume was right, though they would add that his account of causation needs refinement. Others, including me, would reject Hume's theory. Nevertheless, my views owe a great deal to Hume.

Hume's Theory

Hume argues that causation involves a regular association between cause and effect with the cause contiguous with its immediate effects and preceding them. He also argues that the human psychological propensity to pass from an “impression” (perception) of a cause to an idea of its effects (or from an impression of an effect to an idea of its cause) leads people mistakenly to believe that there is a necessary connection between cause and effect. This account takes the mystery out of causation, while retaining the all-important links between causation and regularities and explaining the mistaken belief that there is more to causation than regularity, contiguity, and time order.

Hume claims to clarify the content of our “ideas” (mental contents) and of the relations among them, and he does not clearly demarcate philosophy from introspective psychology. His account of causation is constrained by his empiricism – that is, his view that all our ideas derive from preceding impressions of sense or reflection. He is saying that nonlogical terms have meaning only if one can tell by observation whether they apply to things and that the evidence for all claims concerning matters of fact derives from observation.

Token and Type Causation

How should one interpret claims that a property A, a variable x, or kind of event a is a cause of another? In chapter 2,1 argued that cause and effect are events. Since properties, variables, and kinds are not events, they cannot be causes or effects. Either they stand in some different kind of causal relation, or the truth of the claim that a causes b derives from the causal relations obtaining among tokens. My view is that claims concerning causal relations among types, properties, and variables are generalizations concerning causal relations among tokens.

Many authors have held, in contrast, that there are two varieties of causation, one relating tokens and the other relating types. Peter Menzies, for example, writes, “What kinds of entities do causal sentences relate? This question is readily answered in the case of general causation, since a distinguishing mark of a general causal sentence is that its causal relata are properties. The question is much harder to answer in the case of singular causation” (1989b, p. 59). In Probabilistic Causality (1991), Ellery Eells develops different theories of “type-level” and “token-level” causation. Eells argues extensively for countenancing two distinct kinds of causation, and I will accordingly focus on his case (see also Sober 1985, 1986).

Eells presents his argument for the existence of a distinct type-level causal relation in the context of a theory of probabilistic causation, and some of his concerns cannot be addressed until §9.2.

Independence Implies Screening-Off

If causes are INUS conditions, the following two theorems (which are sketched by Papineau (1985b, p. 63 and 1985a, p. 279)) establish that effects are probabilistically dependent on their causes and that common causes screen off their effects.

Theorem 10.1: If is probabilistically independent of S and Z, and

Theorem 10.2: If (3) C is probabilistically independent of S, Z, Y, and W, (4) S, Z, Y, and W are probabilistically independent of one another, and (5) all probabilities are intermediate, then C and ∈C screen off A and B.

Proof: Given premises 1–3 and 5, Given premise which by 1, 2, 3, and 5 equals. Since Z and W are independent, and.

In Papineau's view, screening-off is explained by the independence among different deterministic causal facts.

Causal Graphs and Probability Distributions – Some Formal Results

In this section I shall sketch the proof of the striking theorem discussed in §10.3.

Theorem 10.3 CM and F imply

1. (Direct causal connection) For all x andy in V, x and y are adjacent if and only if they are probabilistically dependent conditional on every subset of <I>V that does not include them, and

2. (Direct causation) For all x, y and z in V, if JC and y and y and z are adjacent, and JC and z are not adjacent, then V causally depends on x and z if and only if x and z are probabilistically dependent conditional on every subset of Fthat contains y but not y or z.

Chapter 3 explored the possibilities of explaining the asymmetry of causation in terms of the relations between causation and time. Chapter 5 explored the possibilities of generating a theory of causal asymmetry from the connections between causation and agency. This chapter explores another attractive set of connections -those between causation and counterfactuals. When one asserts that one event a causes another distinct event b, then it seems that one is committed to the counterfactual: “If a had not occurred, then b would not have occurred either.” Hume himself wrote “ … we may define a cause to be an object, followed by another, and where all the objects similar to the first, are followed by objects similar to the second. Or in other words, where if the first object had not been, the second had never existed’ (Inquiry p. 51). The “other words” here are indeed “other words.” Hume seems mid-definition to leap from a regularity to a counterfactual theory of causation. This chapter explores the possibility of constructing a counterfactual theory of causation.

Lewis's Theory

According to David Lewis, if a and b are distinct events that actually occur, then b causally depends on a if and only if, if a were not to occur, then b would not occur either (1973a). If the match had not been struck, then it would not have lit. In Lewis's usage (in contrast to mine), “causal dependence” is a relation among token events, which is sufficient for causation. It is not the same thing as causation, because in cases of preemption it is not transitive.

One crucial asymmetry of causation concerns action. Knowing the causes of an event helps one to bring it about. Knowing its effects does not. Knowing what things are related to an event as effects of a common cause does not help one to bring it about either. Causes are means and tools. People can use them to bring about their effects. This observation suggests a theory of causal asymmetry – something like: a causes b if and only if a can be used as a means to bring about b. This view has appealed to scientists and to statisticians. For example, the economist Guy Orcutt writes, “… we see that the statement that … z1 is a cause of z2, is just a convenient way of saying that if you pick an action which controls z1, you will also have an action which controls z2” (1952, p. 307).

Such a theory links the asymmetry of causation to a method of determining which way the causal arrow points: If there is a correlation between a and b and one wants to know what the causal connection is, intervene to find out. a's cause b's just in case human interventions that bring about tokens of kind a lead tokens of kind b to occur as well, while human interventions that bring about b's do not lead to tokens of a occurring. This chapter will be concerned with this theory of causal asymmetry. Following Price and Menzies, I shall call it “the agency theory.” It might also be called “the manipulability theory.”

Causation and Causal Connection: A Graphical Exposition

The relations between causation and causal connection (and ultimately probabilistic dependencies) can be clarified with the help of the three kinds of graphs. Figure 12.3a, 12.3b, and 12.3c show respectively the normal, path, and connection graphs that correspond to the causal relations among the salt concentrations of the five salt basins in figure 6.3, which is reproduced in figure 12.3d. To each normal graph, there corresponds a single path graph, but a path graph may correspond to more than one normal graph. The connection principle implies that for each path graph, there is a unique connection graph: For all vertices u and v in a connection graph Gc, there is an undirected edge between u and v if and only if in a corresponding path graph Gp with the same vertices there is an edge from u to v or from v to u or there is an edge from some vertex to both u and v. If one substitutes the word “path” for the word edge, this same claim defines the unique connection graph corresponding to every normal graph.

If I (or Ig) is true, then there is a unique path graph corresponding to each connection graph that correctly represents the causal relations: For all vertices u and v in a path graph Gp, there is a directed edge from u to v if and only if in the corresponding connection graph Gc with the same vertices there is an undirected edge between u and v and for all vertices w other than v if there is an edge in Gc between w and u, then there is an edge between w…

In the discussion above I postponed addressing the problems of preemption and overdetermination as well as the difficulties raised by arbitrary event fusions and complex facts. In this chapter, I confront these problems and then end by drawing overall conclusions in which the pieces examined in the previous chapters will, I hope, find their places.

Overdetermination

Suppose an event e is causally overdetermined by two events, a and c. For example, suppose a and c are simultaneous, a involves pushing button one, which closes a circuit and causes a light to go on. c involves pushing button two, which closes the same circuit at exactly the same time and causes the same light to go on. e is the light going on. If a had not occurred, c alone would have caused e. If c had not occurred, a would have caused e. As things are, however, it is not the case that only one of a or c caused e. Either they both caused e or neither caused e. Each of them is a “causal overdeterminer” of e.

What is this relation of causal overdetermination? The most common approach in the literature, which I will endorse, denies that causal overdeterminers are causes. If a and c are causal overdeterminers of e, then neither a nor c is a cause of e. Instead one takes the instantiation of the disjunctive property, A or C at the relevant place and time as a cause of e.

There are two main motivations for developing a theory of probabilistic causation. The first is esoteric. According to contemporary physics, many occurrences are not determined. A complete specification of the state of some ensemble only determines a set of probabilities. Despite the temptation to maintain that what happens by chance is not caused, causal questions remain. When one sends a photon through a slit and illuminates a particular square on a grid behind the slit, one wants to say that sending the photon through the slit caused the square to be illuminated, even when the transmission of the photon created only a small probability that the particular square would be illuminated. There are also probabilistic phenomena involving large numbers of individuals, like those in statistical mechanics, that apparently call for probabilistic explanation, even though the relations among the individuals may be deterministic.

The second motivation is more prosaic. As is argued with special vividness by Elizabeth Anscombe (1971), causal attributions in everyday life often appear nondeterministic. People say that punishments deter theft even though they do not believe that the deterrence is perfect. People say that dropping a glass caused it to break, even though they are aware of similar circumstances in which similar glasses were dropped and did not break (see also Rosen 1982).

These prosaic cases do not force one to conclude that the causal relation is not deterministic. Many causal factors influence thefts in addition to the threat of punishment. One can conjecture that the threat of punishment completely deters theft given certain arrays of these other factors, and that it does not deter theft given other arrays.

This chapter will clarify details, especially concerning logical relations among CC, I, and CP.

The Operationalizing Assumption and the Connection Principle

Recall that the connection principle says:

“Causal connection” is a primitive – but I defended the following approximation:

As I shall explain at greater length in chapter 5*, generalizations about relations among types or properties, whether these are deterministic like DC or probabilistic like OA, must carry a reference to the “circumstances.” For example, in most circumstances there is a correlation between consuming highly acidic foods and stomach pains. In circumstances in which one has already ingested an alkali, this correlation can be reversed. The overall association depends on the de facto frequency of different circumstances. Although the unconditional probabilistic dependence between A and B is a reasonable guide to whether a and b are causally connected, one's operationalization of the notion of causal connection ought not to depend on de facto frequencies. So one should link causal connection to probabilistic dependence “in the background circumstances.”

This chapter presents what I believe to be the central asymmetry of causation. The ideas first occurred to me early in 1981 and derive from work by Herbert Simon (1953) and Douglas Ehring (1982). The central intuition is that causal priority consists in the causal connection among effects of a common cause and the causal independence among the causes of a given effect. For this reason I call this view the independence theory of causal priority.

Causal Connection and Probabilistic Dependency

J. L. Mackie suggests that one can factor the causal relation into a symmetrical relation of “causal connection” and an asymmetrical relation of “causal priority”: For all events a and b, a causes b iff a is causally connected to b and a is causally prior to b. In Mackie's view, to say that a and b are causally connected is to say “a and b are connected by some fact of causation” (Mackie 1980, p. 85). Causal connection is the topic of this section and of chapter 12, while causal priority is the topic of the whole book. Humeans, for example, take a to be causally prior to b if a precedes b, and they take a to be directly causally connected to b iff a and b are distinct and contiguous and one is necessary and sufficient in the circumstances for the other.

I share Mackie's intuition that causation has both a symmetrical and an asymmetrical aspect. The symmetrical part is akin to Hume's notion of a necessary connection.

After briefly discussing two other counterfactual theories of causation, this chapter formulates the account of similarity among possible worlds employed in chapter 6 and proves the claims it makes.

Mackie's Counterfactual Theory

John Mackie argued,

For example, suppose one breaks the connection between a car's engine and its wheels: the engine continues turning while the wheels stop.

Let us call the event of A's doing X on the particular occasion “a” and the event of B's doing Y on the occasion “b” Suppose that in the closest possible world in which some event c failed to occur, a would not be causally connected to b. If the causal connection between a and b depends on the existence of c, then c must be a cause of at least one of a or b. On Mackie's view a is causally prior to b if and only if, in the absence of some minimal difference c, a still occurs, but b does not (B does not do Y). So c causes b only. Mackie thus indirectly assumes that there are causes of b that are not causes of a, and this assumption is, of course, an immediate implication of both CP and agency views.

Conditioning on a variable is not the same thing as intervening to fix the value of a variable, and the conditional probability of y given some particular value of x is not the same thing as the probability of y conditional on an intervention that “sets” this value of x. When one conditions, one takes as given the probability distribution. When one intervenes, one changes the probability distribution (but see pp. 229–30). These observations suggest a problem. The difference in the probability of b with and without an intervention that brings about a seems a surer guide to causal relations than does a comparison of Pr(B/A) and Pr(B/∼A). Do attempts to reduce causation to facts about probability distributions or merely to operationalize them in those terms rest on a confusion of conditional probabilities and probabilities given interventions? Do they collapse when this distinction is drawn? For example, suppose (as Ronald Fisher hypothesized) that tokens of smoking, s, and lung cancer, c, are related as effects of some common cause and not as cause and effect. In that case, even though Pr(C/S) > Pr(C/∼S), the probability of lung cancer is unaffected by interventions that bring about or prevent smoking. Following Pearl (1993, p. 267; 1995, p. 673), let us indicate the conditional probability that B will be instantiated given an intervention that brings about a token of kind a as Pr(B/set-A) and the conditional probability that y = y* given an intervention that causes the value of x to be x* as Pr(y=y*/set-x=x*) or less precisely as Pr(y/set-x).

Theorem 4.1 (p. 81) T and CC entail that if a causes b, then everything causally connected to a and distinct from b is causally connected to b.

Theorem 4.2 (p. 81) Given T and CC, if a is distinct from b and not causally connected to b, then it is not causally connected to any cause of b.

Theorem 4.3 (p. 82) CC, I, and T entail A.

Theorem 4.4 (p. 84) CC and I imply that if a is causally connected to b and everything casually connected to a and distinct from b is causally connected to b, then a causes b.

Theorem 4.5 (p. 84) T, CC, and I entail CP.

Theorem 4.6 (p. 84) T, CC, and I entail CP.

Theorem 4.7 (p. 85) T, CC, CP, and A (asymmetry) imply I.

Theorem 5.1 (p. 107) DHIg, PIg, Tg, and PPg imply ATg.

Theorem 5.2 (p. 108) DHIg, CCg, Tg, and ATg entail CPg.

Theorem 5.3 (p. 108) DIg, PIg, Tg, and PPg imply ATg.

Theorem 5.4 (p. 108) DIg, CCg, Tg, and ATg entails CPg.

Theorem 5.5 (p. 108) DIg, PIg, CCg, and Tg entail CPg.

Theorem 5.6 (p. 109) Given CCg and NICg, Igg is entailed by DIg and PIg.

Theorem 5.7 (p. 110) CCg, PIg, DIg, and NICg entail Ig.

Theorem 6.1 (p. 135) SIM, CDCC, and I imply that individual causes will not be counterfactually dependent on individual effects and effects of a common asymmetric cause will not be counterfactually dependent on one another.

In chapter 4 I made the operationalizing assumption that there are causal connections among events if and only if there are probabilistic dependencies in the background circumstances among the relevant properties or variables. If this assumption is a reasonable approximation, then the claims about causal asymmetry made in chapters 4–7 have implications concerning probability distributions, which deserve further scrutiny. This chapter begins this scrutiny by exploring David Papineau's theory relating causation to conditional probabilistic dependencies.

The Fork Asymmetry

Suppose event types a and b are probabilistically dependent on one another:

1. 1. Pr(AB) > Pr(A)·Pr(B).

Suppose further that

1. 2. a and b are not related as cause and effect.

Then, according to Hans Reichenbach, there exists an event type c satisfying the following conditions:

1. 3. Pr(AC) > Pr(A)·Pr(C)

2. 4. Pr(BC) > Pr(B)·Pr(C)

3. 5. Pr(AB/C) = Pr(A/C)·Pr(B/C)

4. 6. Pr(AB/~C) = Pr(A/~C)·Pr(B/~C)

5. 7. Tokens of kind cprecede tokens of kinds a and b.

6. 8. Tokens of kind c are common causes of tokens of kinds a and b.

The probabilistic dependence between a and b is explained by the existence of the common cause c. Reichenbach proves that if none of the probabilities is one or zero, then 3–6 entail 1. This proof is given in chapter 4*, pp. 76–7.

7 or 8 is called in the literature “the fork asymmetry”: the causal arrows point away from the event type that does the screening-off and toward the future. 7 might be called the temporal fork asymmetry, while 8 might be called the causal fork asymmetry. There can be events of kind c satisfying 3–6 that are common effects of a and b and that succeed them in time, but only when there are also common (preceding) causes.

Several features of the causal relation suggest a close connection between causation and explanation.

1. The relevant aspects of cause and effect make reference to properties and do not appear to be fully concrete entities.

2. The relations between causation and laws and the scientific centrality of type-causal claims force one to emphasize the role of properties in the causal relation.

3. The importance of absences and nonoccurrences in causal relations and the fact that the same pair of substances can be the locus of causal relations in either direction, depending on context, all call into question a view of causation as a purely physical relation of the sort suggested by transfer theorists.

4. The relativization of causal relations to particular fields or systems points to links between causation and explanation.

5. A connection between independence and explanation is implicit in the thought explored at the ends of chapters 5 and 7 that independence should be regarded as specifying the circumstances in which causal relations obtain – or in which causal explanations are appropriate.

So something must be said about the intimate relations between explanation and causation. For millennia, citing a cause has been taken to be explaining. When Aristotle presents his doctrine of the four kinds of causes, he is developing an account of four different kinds of explanations one may give. Material causes of objects are explanations of their properties in terms of the material of which they consist. Formal causes are explanations in terms of structure. Final causes are explanations in terms of ends. Proximate causes – causes in our modern sense – are apparently linked to explanation as well.

Independence and a Token Version of Agency Theory

Analogous to the conditions in chapter 5 let us define:

If one wants to make the circularity less blatant, one can replace the last nine words with “then that intervention would be nomically connected to b,” and all the same results hold. As was the case with ATg, the quantification over a set of events is necessary, because the notion of a direct manipulation is relativized to a set of events.