Introduction

Causation is central to our understanding of the world and ourselves. The sciences provide causal explanations. Geology and physics explain the mechanisms that cause earthquakes and the causal processes by which earthquakes cause tsunamis. Biology explains the causes of the appearance of new biological species and of the extinction of other species. Historians explore the causes of antisemitism and the Holocaust. Our understanding of all these cases of causation is itself causally structured, and psychologists study the causal processes of learning, remembering and reasoning. Scientists and philosophers have also practical motivations for improving our understanding of causation. It is after all crucial for improving our means of changing things. Understanding the causes of the increase in economical inequality is inseparable from finding out by what means we might slow it down or reverse it. Understanding the causes of cancer, multiple sclerosis, or Covid-19 is required for conceiving of means for diminishing the incidence of these diseases, for soothing their symptoms and treating them. Understanding causation is also fundamental in the law. Individuals and firms are condemned on the basis of the courts’ judgment that they have caused harm. Causation poses a challenge both to philosophy of science and to metaphysics. What do scientists do when they look for and provide causal explanations? The challenge for metaphysics is to understand what reality must be like if causal explanations are true.

Before we look at philosophical theories of causation, we may start with the observation that the concept of causation that is used both in common sense and in science is structured around the following “central connotations of causation” (Reference SchafferSchaffer 2000a, p. 289).

1. 1. The causal relation holds between distinct events.Footnote ¹

2. 2. The causal relation is intrinsic to these events. In particular, whether a causes b is determined locally, at the spatiotemporal location of a, b and in the region between them. It does not depend on what happens at other times and places.

3. 3. Causes raise the chance of their effects.

4. 4. Causes precede their effects, or at least don’t follow them, in time.

5. 5. Asymmetry: if a causes b, it is not the case that b also causes a.

6. 6. Objectivity: whether a causes b does not depend on us or any other observers or explanation-seekers.

7. 7. The cause can be used as a means for making the effect happen.

8. 8. One can infer the effect from knowledge of the cause, and one can infer the cause from knowledge of the effect.

9. 9. Agents are morally and legally responsible for what they cause.

10. 10. Effects depend on their causes. This dependence can be expressed in counterfactuals.

This Element presents the major attempts to account for causation as a metaphysical concept, in terms of (1) regularities and laws of nature, (2) probabilities and Bayes nets, (3) necessitation between universals and causal powers, (4) counterfactual dependence, (5) interventions, (6) causal models, and (7) processes or mechanisms. I will argue that causal models and process accounts are complementary. Causal models provide an account of causation at the level of types, whereas processes account for causation between particular events. The last part presents some contemporary debates in the metaphysics of causation: on the relation between grounding and causation, eliminativism with respect to causation in physics, the challenge against “downward” causation from the Closure and Exclusion principles, robust and proportional causation, and degrees of causation.

1 Regularities and Laws of Nature

David Hume’s account of causation is one of the main sources of contemporary attempts to analyze causation in terms of regularity. Hume himself did not offer a metaphysical theory of causation, as it is independently of us, because he was skeptical about the possibility of knowing what it takes, in a metaphysical sense, for one event to cause another.Footnote ² Rather, he offers an epistemological theory of what makes us judge that one event causes another. There are two necessary conditions for a causing b (where a and b are particular eventsFootnote ³) that Hume claims we can know by observation: that a is earlier than b and that a and b touch each other (in technical jargon, that they are contiguous). Both can be challenged, especially on the background of contemporary physics,Footnote ⁴ but let us leave this to one side here. According to Hume, the origin of the difference between a causing b and a merely preceding and touching b, is psychological. When we judge that a causes b, rather than just following it, this judgment is based on an expectation that is the result of a psychological process of habituation.

2 Probabilities and Bayes Nets

We have seen that (DN-C) does not provide a sufficient condition for causation: facts that are related by valid nomological deduction are not always causally related. In statistics, great efforts have been deployed to develop an account of the conditions under which causal conclusions can be derived directly from observations (or data) without any use of the controversial notion of a law of nature. The idea is to construct criteria for causal dependence on the basis of conditional probabilities, which are estimated on the basis of the observation of relative frequencies.Footnote ²⁷ The central hypothesis of statistical analyses of causation is that the fact that the probability of factor B is higher (or lowerFootnote ²⁸) in the presence of factor A than in its absence, is a (fallible) indicator of the existence of a causal influence from factor A on factor B.

The attempt of finding statistical methods for discovering relations of causal influence has one big advantage over (DN-C), over and above avoiding the notion of law. It naturally accounts for the fact that causal influence manifests in observable data only statistically, not universally. Smoking causes lung cancer, but not every smoker gets lung cancer, and some nonsmokers do. In complex situations, which are characteristic of all sciences outside fundamental physics, each case is different and a large number of factors contribute positively or negatively to such an eventuality as the development of lung cancer in a particular person. Furthermore, contemporary fundamental physics contains irreducibly probabilistic processes such as radioactive decay, where only a certain percentage of nuclei undergo the process of decomposition. The effect of the radioactive decay of a given particular ¹⁴C nucleus, that is, the coming into existence of a ¹⁴N nucleus, cannot be deduced by a DN argument from the presence of the ¹⁴C nucleus, together with laws. Only its conditional probability can be deduced.

Most proposals for analyzing causation in terms of conditional probabilities do not even attempt to develop an analysis of causation at the level of particular events.Footnote ²⁹ Even if it could be established on purely statistical grounds (which is not the case, as we will see), that “smoking causes lung cancer,” such a causal influence at the level of populations does not entail that a particular smoker will get lung cancer, or if she gets cancer, that this cancer was caused by her smoking. The causal influence of S on C, at the level of population level factors, is often called “type-level causation,” or “causation between types.” The relation between type-level causation and causation between particular events falling under these types, or “token-level causation,” is controversial. The type-level causal influence of S on C is neither necessary nor sufficient for the token-level fact that person a’s smoking causes her to get cancer.

All attempts to analyze causation in terms of conditional probabilities start with the requirement that the cause be correlated with the effect. A necessary condition for S to be a (type-level) cause of C, is that P(C/S)>P(C) or, equivalently, P(C/S)>P(C/non-S). However, correlation is not sufficient for causation. If both lung cancer (C) and cardiovascular disease (D) are due to a common cause, such as smoking (S), then C and D can be correlated although none of C and D has any direct causal influence on the other. Reference ReichenbachReichenbach (1956, p. 159) has proposed to characterize the concept of common cause in purely probabilistic terms, with the help of the concept of a screening factor. If C and D are correlated so that P(C/D)>P(C/non-D), but if there is no correlation once the probabilities are evaluated conditional on factor S, to that P(C/S&D)=P(C/S&non-D), S is said to “screen off” C from D.Footnote ³⁰ This is indeed an indicator of the fact that S is a common cause of C and D, which explains C and D’s correlation. However, it is only a fallible indicator, in the sense that the fact that C is a screening factor for A and B is not sufficient for C being a common cause of A and B.Footnote ³¹

The inference from correlation to causation requires the absence of screening factors. However, it is not possible to obtain a sufficient condition for causation in purely probabilistic terms by using the criterion of the absence of screening factors. Either the absence of screening factors is explicitly expressed as a causal condition or not. In the first case, as in Reference Cartwright and CartwrightCartwright’s (1979, p. 423) principle: (CC) C causes E iff P(E/C&K)>P(E/K) for all other causes K of E that are not intermediate between C and E, we don’t have an analysis of causation in terms of probabilities alone because the analysans contains information about causes.

In the second case, the requirement that there are no screening factors is expressed in purely probabilistic terms. This doesn’t provide an appropriate analysis of causation either, for the following reasons. (1) It yields a requirement that is empirically unrealistic: conditionalizing on all other factors means comparing populations that are homogeneous with respect to all other factors.Footnote ³² Such populations do not exist outside of fundamental physics. (2) If we could somehow observe homogenous populations (homogenous except for the factors C and E), we wouldn’t find probabilities different from 0 and 1 except if factors C and E belong to fundamental physics.Footnote ³³ Intermediate probabilities (e.g., in medicine or social sciences) stem from the fact that empirically accessible populations are not homogeneous.

Now, here is the main problem we face in all situations where frequencies are measured in populations that are not homogeneous: as long as conditional probabilities are estimated from the observation of non-homogeneous populations, in other words, as long as not all factors different from C and E are held fixed, measures of conditional probabilities are always provisional. Judgments of comparative conditional probability are always at risk of being reversed once new factors are taken into account. Such reversal of conditional probabilities is known as Simpson’s reversal or Simpson’s paradox. Here is an example.Footnote ³⁴ A study on thyroid disease found that the survival rate over twenty years (L) of smokers (S) was higher than that of non-smokers. However, it would be mistaken to conclude from P(L/S)>P(L/non-S) that smoking is a causal factor increasing the chance of survival. Age turns out to be a so-called confounding factor. Once frequencies are calculated conditional on age, the inequality is reversed: In six out of seven age groups, the rate of survival was lower for smokers than for non-smokers, and the difference was minimal in the seventh.

This means that we can safely judge whether the higher conditional probability of E given C than given not-C is due to the causal influence of C on E, only if we know all confounding factors. However, confounding factors are typically common causes of C and E. In other words, it is possible to extract information about causal influence from information about conditional probabilities only on the basis of causal knowledge. This is often expressed by the slogan “no causes in, no causes out.”Footnote ³⁵

The development of Bayesian networks from the 1980s has much improved statistical methods for discovering (type-level) causal relations between variables. Bayes nets provide powerful means for estimating conditional probabilities on the basis of other probabilities. A Bayes net is a model, which can be represented as a graph that consists of (1) nodes, occupied by variables representing features of target systems, (2) directed edges linking some of these nodes to others, and (3) a probability distribution of each variable conditional on its “parents” in the graph. The set of parents of a variable A is the set of variables that are at the origin of an edge pointing toward A. Bayes nets are constructed on the basis of that probability distribution, together with a fundamental assumption, the “Markov condition.” The Markov condition says that each variable is probabilistically independent of its non-descendants, conditional on its parents in the graph. In other words, all correlations between variables that are not related as cause and effect are due to parent variables, and disappear once the probabilities are calculated conditional on these parent variables. Bayes nets are required to be acyclic, in the sense that they must not contain any circular path, where a path is a sequence of adjacent edges whose directions are aligned.

Here is an example.Footnote ³⁶

In Figure 1, variable X₅ has, for example, only parent variable, X₄. By construction of the net, the probability of X₅, given X₄, is independent of all other variables, that is, X₁, X₂, and X₃.

Figure 1 A Bayesian network representing dependencies among five variables. The running of the engine (X₁) causes both friction (X₂) and the working of the cooler (X₃). Both friction and the working of the cooler influence the temperature (X₄) of the engine, which influences the reading of the thermometer (X₅).

Although the links in a Bayes net are oriented (from parents to descendants or “children”), they become causal only by virtue of an interpretation. A Bayes net becomes a causal net by interpreting the arrows as representing direct causal influence between variables. In such a causal interpretation, the Markov condition becomes the causal Markov condition, according to which every variable in the net is probabilistically independent of its non-effects, conditional on its direct causes.Footnote ³⁷ If a causal net is interpreted as representing a certain system or type of systems, one can read off from that causal net the relations of causal influence between variables. Whether the results are correct of some real system depends on several hypotheses: that the set of variables V represents all causally relevant factors, that the probability distribution P over these variables represents the physical probabilities of the factors represented by these variables, and that the causal net represents the smallest directed acyclic graph on V that satisfies the Markov condition with respect to P.

From a metaphysical point of view, what we want to know is whether causal nets are a reliable tool in the sense of representing relations of causal influence in reality. First, causal nets have been introduced as tools for discovering and representing causal influence between general factors, which provides no direct information about causation between particular events.Footnote ³⁸ Second, powerful as they are, they constitute a tool that is as fallible as the hypotheses that are presupposed by their construction.

The Causal Markov Condition implies the following version of Reichenbach’s Principle of the Common Cause (Reference WilliamsonWilliamson 2009, p. 200):

(PCC) If variables A and B are probabilistically dependent then one causes the other or there is a set U of common causes in V which screens off A and B, that is, renders them probabilistically independent.

There seem to be many systems for which PCC is not correct because they contain factors that stand in non-causal dependence relations. Such dependence can be the consequence of logical, mathematical, or semantical constraints or of non-causal association laws; correlations may also exist in the absence of any common causes and any other constraint (Reference SoberSober 2001). Variables representing aspects of physical systems that are non-causally related can be probabilistically correlated although neither is a cause of the other and there is no characteristic that plays the role of a common cause so that it could be represented by a variable that screens them off from each other.Footnote ³⁹

Examples may be found in quantum theory. It has been argued that EPR experiments in quantum mechanics give rise to correlations between the states of entangled particles that are not screened off by common causes yet cannot be causal either.Footnote ⁴⁰ However, we can sidestep the controversial interpretation of quantum correlations. Here is Reference CartwrightCartwright’s (2007, p. 122) example of non-causally correlated variables whose correlation is not screened off by any common causes. Z is a chemical reaction that produces substance X and as a by-product substance Y. The probability of getting X (the valueFootnote ⁴¹ of variable X is 1; in short X=1) from Z (Z=1) is 0.8. We get Y (Y=1) if and only if we get X (X=1). Z is the common cause of X and Y. X and Y are correlated, yet the correlation does not vanish upon conditionalizing on the common cause (Z=1). P(X=1/Y=1&Z=1)=1> P(X=1 / Z=1)=0.8.

In Cartwright’s example, X=1 is not a cause of Y=1 although the variables X and Y are probabilistically correlated and although their correlation is not screened off by their common cause Z. Such examples show that probabilistic correlation that persists, once the probabilities are calculated conditional on common causes, is not sufficient for causation.

However, it would be a mistake to conclude that they refute the Causal Markov Condition (CMC). The CMC is a condition that holds by convention, in the sense that it is an assumption that is used to construct causal nets.Footnote ⁴² What Cartwright’s case shows is that the assumptions built into causal nets, and the CMC in particular, make it problematic to draw realistic conclusions on the causal influence relations in real systems, from causal nets built on the basis of those assumptions. In other words, if Cartwright is correct, then our conventions may prohibit us from constructing a causal net which models the world’s actual causal structure.

Maybe a model such as Cartwright’s, containing only variables X, Y, and Z, which does not satisfy the CMC, could in principle be completed (Reference PearlPearl 2000, p. 62) by the addition of some hitherto unknown and unobserved variable intermediate between Z, X, and Y, to yield a model that satisfies CMC. However, there is no guarantee that one can always find such variables that can be interpreted as representing features of the real systems that are the target of the model.

A similar reply is that one should avoid choosing variables that are non-causally related.Footnote ⁴³ The PCC holds of all models that are constructed without such variables. However, such a rule would undermine the prospect of using causal nets for the search for causal relations in real systems. For it would mean that we could use the PCC as a criterion for causal relations in real systems only if we knew beforehand (1) which variables to include in the model because they might be causes of the variables whose causes we are looking for and (2) which dependence relations are causal and which are not. If we have that knowledge before we even start building our models, we can build models using only causally related variables (satisfying the CMC). Statistical correlation without any common cause represented in the model can then be used as a sufficient condition for causation in the systems represented by the model.Footnote ⁴⁴

The hypothesis that probabilistic correlation is a necessary condition for causation is problematic too. Some real systems contain causal influence relations that are not reflected in the corresponding conditional probabilities. We will look at an example illustrating this in a moment. First note that although, in general, positive causal influence of factor C on factor E increases the probability of E, given C, that is, P(E/C)>P(E/not-C), and negative causal influence of C on E manifests as P(E/C)<P(E/not-C), this is not always the case. Take the radioactive decay of nuclei of the isotope ²²³Fr of Francium,Footnote ⁴⁵ represented in Figure 2.Footnote ⁴⁶

Figure 2 Cascade of radioactive decomposition of ²²³Fr.

99.994% of ²²³Fr decay into ²²³Ra, whereas the rest (0.006%) decays into ²¹⁹At. Both ²²³Ra nuclei and 3% of ²¹⁹At nuclei then decay into ²¹⁹Rn. Take a nucleus of ²¹⁹At which is at the causal origin of a nucleus of ²¹⁹Rn. It is much less likely to obtain a ²¹⁹Rn, given a ²¹⁹At nucleus, than it would have been if it had been a ²²³Ra nucleus. After all, only a small portion of ²¹⁹At become ²¹⁹Rn (97% of ²¹⁹At become ²¹⁵Bi), whereas all ²²³Ra become ²¹⁹Rn. This is a case where ²¹⁹At has a positive causal influence on ²¹⁹Rn, in the sense that the former nucleus is the causal source of the latter. However, this positive causal influence is not reflected in a higher probability of the effect given the cause, so that probability raising is not necessary for positive causal influence: The probability of having an effect of type ²¹⁹Rn is lower given the cause (a nucleus of type ²¹⁹At) than given the alternative possibility that the nucleus be of type ²²³Ra.Footnote ⁴⁷

It is neither necessary that the probability of the effect is higher given the cause (as we have just seen) nor that it is lower given the cause. Some causes are probabilistically independent of their effects. Take a roulette wheel without a “0.” B is the event of throwing a ball in the roulette; B takes value b₀ if the ball had fallen on black after the preceding throw; B takes value b₁ if it had fallen on red after the preceding throw; variable C takes value c if the ball falls on a pair number. Within a framework specified by these variables, B is the only possible cause of C=c, with P(c/b₀)=P(c/b₁)=1/2. Here B is a cause of C but B and C are probabilistically independent.Footnote ⁴⁸

Moreover, many (and probably most) causal interactions at the level of particular events are conceived by concepts which do not give rise to any correlations. Here is Dowe’s example: “Two men collide in the corridor. One gets a bruise on his arm, the other drops his papers. […] No one would expect to find any useful statistical correlations between bruises and sets of disordered papers” (Reference DoweDowe 1992, p. 206).Footnote ⁴⁹

Causal influence relations between variables do not always manifest in the form of the raising or lowering of the probability of the effect, conditional on the cause. Is this an obstacle to the use of causal nets to represent causal relations in real systems? The situation is similar to the problem raised by non-causal relations that refute the PCC. Causal influence without probability raising is excluded from causal nets by stipulation, by means of the so-called Causal Faithfulness Condition (CFC). A model satisfies the CFC if and only if every conditional independence relation true in the probability distribution P corresponding to the graph (V,E), where V is a set of variables V and E a set of edges, is entailed by the CMC (Reference Spirtes, Glymour and ScheinesSpirtes et al. 2000, p. 31).

The CFC is another condition that is, in addition to the Causal Markov condition, stipulated as part of the standard algorithm for the construction of causal nets from data on probability distributions over a set of variables. The Causal Markov condition roughly says that all statistical correlations between variables X and Y are either due to the fact that X causally influences Y or to the fact that X and Y have common causes. The Causal Faithfulness Condition roughly says that, if variables X and Y are probabilistically independent, neither causally influences the other and they do not have common causes. The Faithfulness condition rules out the possibility, among others,Footnote ⁵⁰ that X causally influences Y through two different paths whose effects exactly cancel each other out, in such a way that the total effect of X on Y is null. It is controversial whether this can happen in real systems. On one hand, external perturbations (i.e., noise) that influence each variable independently of the others make violations of the CFC by exact cancelation very improbable.Footnote ⁵¹ On the other hand, given that the values of the variables characterizing real systems can be measured only with finite precision, the cancelation of measured values seems possible. Indeed, many biological and artificial systems possess features that are designed to remain stable under external influences.Footnote ⁵² Let T represent the temperature of an animal’s body, H the heat in the environment, and C the state of a cooling mechanism. If C works well, the causal influences of H and C on T will cancel out, so that T remains stable under variations of H. If the values of these variables are discrete and correspond to measures with finite precision, it may well be that the cancelation is exact, so that H and T are not correlated.Footnote ⁵³

Let me conclude this section on the link between probability raising and causation with the following case.Footnote ⁵⁴ Consider an atom that has a very low probability, say .01, of decaying during a given time interval. The atom is hit by a neutron, which raises its probability of decaying in that interval to .99. The probabilities can be interpreted both as epistemic and, by the lights of contemporary physics, as physical chances independent of our knowledge. The collision of the atom with the neutron raises its probability of decaying although the collision is neither necessary nor sufficient for its decaying. The probabilities alone provide no reason why the fact that the collision raises the probability of the decay should entail that it causes the decay. True, the collision made the decay more probable, yet nothing precludes the possibility that the decay was exclusively due to factors internal to the atom, and thus that the collision was not the cause of the decay.

The fact that one factor A raises the probability of a second factor B, independently of the background as in the case just considered, or against the background of a causal net, is a strong but fallible indicator of the fact that A causally influences B. It is fallible indicator in the sense that A may not have any causal influence on B, although A raises B’s probability, and A may causally influence B although this isn’t manifest in the probabilities.

3 Necessitation between Universals and Causal Powers

If regularities and statistical correlation are only fallible indicators of causation, what is causation itself? Let us look at attempts to analyze causation with the help of specifically metaphysical concepts, such as necessitation between universals and causal powers.Footnote ⁵⁵

David Armstrong analyses singular causation in terms of the instantiation of structural universals.Footnote ⁵⁶ Every case of causation results from the instantiation of a causal law; and a causal law is a relation of necessitation of one universal by another. “The fundamental causal relation is a nomic one, holding between state-of-affairs types, between universals. Singular causation is no more than the instantiation of this type of relation in particular cases” (Reference ArmstrongArmstrong 1997, p. 227). Armstrong gives the example of “guillotining (that state-of-affairs type) causing immediate decapitation (a further state-of-affairs type). It is a second order state of affairs, a relation holding between the universals involved. This second order state of affairs must itself be a universal, a structural universal involving a certain linking of universals, a linking of state-of-affairs-types” (Reference ArmstrongArmstrong 1997, p. 226/7).

The postulate of universals is a controversial metaphysical hypothesis, which we cannot evaluate here. Let me mention three difficulties that stand in the way of the possibility of explaining causation in terms of universals. (1) Even if universals are accepted, the postulate of a second order universal of causal necessitation relating one universal (guillotining) to another (decapitating) raises the worry that Armstrong’s theory explains something familiar, that is, causation between states of affairs, in terms of something that is much more mysterious than what it is supposed to explain. If we don’t know what causation is at the level of singular events or states of affairs, does it help to learn that it is the instantiation of a causal relation between universals? (2) The “instantiation” of causation between universals by causation between instances of those universals raises what van Fraassen has called the “inference problem” (Reference LewisLewis 1983, p. 366; Reference van Fraassenvan Fraassen 1989, p. 96). How can laws, as relations between universals, explain regularities at the level of singular instances of these universals? (3) Armstrong’s theory also suffers from a more specific problem that it shares with the DN account: given that it takes all laws to be causal, it cannot account for non-causal association laws.

Others have suggested explaining causation in terms of the manifestation of causal powers (Reference Bird and MarmodoroBird 2010; Reference Mumford, Lill Anjum and MarmodoroMumford and Anjum 2010, Reference Mumford and Lill Anjum2011). The metaphysical theory of causal powers (Reference EllisEllis 2001; Reference MolnarMolnar 2003) challenges the assumption that events are metaphysically independent of one another, and that causation is an external relation between them. Rather, causation is built into the natural properties constituting events. These properties are causal powers that are, by their very nature, disposed to manifest in specific ways, according to the circumstances. According to the power theory of causation, the relation between cause and effect can be analyzed in terms of the relation between a power and its manifestation. Given that powers are causal by definition, such an analysis is not reductive. Mumford and Anjum’s version of this theory makes several original and controversial claims. Causes do not necessitate their effects, in the sense of being sufficient conditions for them, although they necessarily give rise to a tendency toward their effects. This tendency is a sui generis form of modality, irreducible to contingency and necessity (Reference Anjum and StephenAnjum and Mumford 2018a). Causes and effects are always simultaneous, and causation is non-symmetric and non-transitive (Reference Mumford and Lill AnjumMumford and Anjum 2011). The combined action of several powers can be represented in terms of the addition of vectors. Causation can be directly perceived, in particular in proprioception. I can do justice here neither to the metaphysics of powers nor to these specific theses about causation. However, it can be challenged whether it is possible to account for causation that extends over time, on the basis of the thesis that all causation is simultaneous (Reference ChakravarttyChakravartty 2013; Reference McKitrick, Mumford and AnjumMcKitrick 2013). It can also be challenged whether the analogy between the combination of causes and vector addition is satisfactory, given that causes often interact, in the sense that the effect of several causes acting together is not simply the cumulative result of the addition of the effects of each of them taken individually (Reference Fenton-GlynnFenton-Glynn 2012; Reference ChakravarttyChakravartty 2013).

4 Counterfactual Dependence

Some correlations do not correspond to any causal influence (and some cases of causal influence do not manifest in correlations). What distinguishes those correlations that are grounded on causal influence from those that are not?

David Hume developed the first regularity theory of causation. However, in one of his definitions he characterizes the relation between cause and effect in counterfactual form. “We may define a cause to be an object, followed by another, and where all the objects similar to the first are followed by objects similar to the second. Or in other words where, if the first object had not been, the second never had existed” (Reference Hume, Selby-Bigge and NidditchHume 1777/1975, p. 76, italics in original). Reference Lewis and LewisDavid Lewis (1973b/1986) and othersFootnote ⁵⁷ have developed this counterfactual condition, which Hume presents as an alternative but equivalent formulation of the regularity theory, into a new theory of causation. Counterfactual theories of causation are motivated by the strong intuitive link between causation and certain counterfactual statements. Event c is a cause of event e if and only if c makes some difference to e, and difference-making can be expressed in counterfactual form: If c had not occurred, e would not have occurred.Footnote ⁵⁸

The counterfactual approach to causation between particular events takes causation to be equivalent to a certain form of dependence. True, causation is a form of dependence. However, not every dependence is causal. Counterfactual theories of causation have attempted to find appropriate additional constraints on dependence in order to obtain a sufficient condition for causation. These constraints result from a reflection on cases of dependence that seem clearly non-causal; the aim is to construct an account that avoids those counterexamples.

One constraint on cause and effect is that they are distinct particular events in the sense that they do not share any part. This allows for avoiding the result that the following cases of (counterfactual) dependence be wrongly considered as causal, which they are not. If I had not spoken this sentence, I would not have spoken the first half of it.Footnote ⁵⁹ The events of my speaking this sentence and of my speaking the first half of it are not distinct because the second is a part of the first. Without the constraint according to which cause and effect must be distinct events, this would be a “false positive,” a counterexample, which the counterfactual theory of causation wrongly categorizes as causal.

Other constraints have been introduced to avoid that later events be wrongly considered as causes of earlier events although there seem to be many cases of so-called “backtracking” counterfactual dependence.

1. (1) Let us suppose that there are situations in which e₁ causes e₂ in such a way that the occurrence of e₁ is a sufficient condition for the occurrence of e₂. Nothing could prevent e₂ once e₁ has happened. A plausible case may be two successive events e₁ and e₂ on the trajectory of a particle traveling in a straight line through empty space. If e₁ is a sufficient condition for e₂, e₂ is a necessary condition for e₁: if e₂ hadn’t happened, that can only be because e₁ has not happened. So it would seem that the backtracking counterfactual ¬e₂→¬e₁ is true. Without additional constraints, this is a counterexample that refutes the analysis because e₂ is not a cause of e₁.

2. (2) Cases of counterfactual dependence between so-called epiphenomena are another source of counterexamples. If c is a common cause of e₁ and e₂, then there can be counterfactual dependence of one effect, e₂, on the other e₁. Let c represent a drop in atmospheric pressure, e₁ a drop in my barometer’s reading, and e₂ a storm. If c is a sufficient condition for e₁ and c is a necessary condition of e₂, then ¬e₁→¬c and ¬c→¬e₂, and it may also be the case (though there is no valid inference to this effect) that ¬e1→¬e₂. The storm may be counterfactually dependent on the fall of the barometer but it is not caused by it.

Both problems can be solved if backtracking counterfactuals are excluded. This requirement may be added as an additional constraint, or it may be argued that it results from the semantics of counterfactuals, that is, that it is implicit in the meaning of counterfactual statements.Footnote ⁶⁰

Additional constraints are also required to yield a form of counterfactual dependence that is necessary for causation. Without such constraints, there are cases of causation where the effect does not counterfactually depend on its cause. These are “false negatives,” that is, counterexamples that the counterfactual theory of causation wrongly categorizes as not causal.

If a cause is accompanied by a backup event, the effect is not counterfactually dependent on the cause, for the effect would have occurred in the absence of the cause, thanks to the presence of the backup. Redundant causation corresponds to situations where both events are effective in bringing out the effect. In situations where the backup cause is not effective, the effective cause is said to “preempt” the backup cause. The backup cause has the potential to cause the effect in the absence of the effective, preemptive cause. There is counterfactual dependence of the effect on the cause neither in cases of redundant causation nor in cases of preemption. Redundant and backup causes are widespread in biology. If the causal role of some organ A is vital, organisms will have higher fitness if they have a redundant or backup organ B that takes over whenever A is not well functioning. Each of our kidneys is redundant given the presence and functioning of the other. Here is a biological example of preemption.Footnote ⁶¹ Even in a healthy brain, neurons die at a certain rate, which increases with age. As dead cells and cellular debris accumulate, they harm surrounding cells, which in turn accelerates neuron death and causes neurodegenerative diseases such as Alzheimer’s disease. Microglia are cells that remove cellular debris from the brain. However, if microglia fail to accomplish their function, astrocytes, another kind of brain cell, step in to remove debris as a backup to microglia.

Cases of redundant causes and preemption point to a deep problem with all counterfactual theories. According to counterfactual theories, whether c causes e depends not only on c, e, and what happens in the spatiotemporal region between these events, but also on what goes on elsewhere. This contradicts the intuition that causation is intrinsic to the relata and the region between them.Footnote ⁶²

Reference Lewis and LewisLewis (1973b/1986) has adapted the counterfactual analysis to this type of situation by requiring that there exist a series of intermediate events between the cause and effect so that each of the events in the series is counterfactually dependent on its predecessor. In cases of so-called “early preemption,” there is indeed no stepwise counterfactual dependence of the effect on the preempted (i.e., non-effective) backup cause. However, this solution raises the problem that it builds transitivity into the concept of causation. It is controversial whether transitivity is part of the concept of causation.Footnote ⁶³

Anyway, the requirement of the existence of an intermediate chain with stepwise counterfactual dependence cannot account for some types of preemption, in particular “late pre-emption” and “trumping pre-emption” (Reference Schaffer, Collins, Hall and PaulSchaffer 2000/2004),Footnote ⁶⁴ which corresponds to the overdetermination of an event by two independent causes. To take Schaffer’s example of trumping preemption, if both the Major and the Sergeant give the Corporal at the same time the same order, for example, the order to advance, the behavior of the Corporal is overdetermined, in the sense that it is caused twice over. Given that the Major’s rank is higher, it is plausible that his order is the cause of the Corporal’s moving, but this effect does not counterfactually depend on its cause.

The counterfactual approach has mostly been interpreted by its advocates as belonging to conceptual analysis. The intuition that our concept of causation has at least a strong counterfactual component provides enough motivation to develop the account so as to cope with counterexamples. It is indeed plausible that the semantics of causal statements can be analyzed in terms of counterfactuals, whose truth conditions can in turn be analyzed in the framework of possible worlds. The psychological mechanisms of reasoning about causation also seem to involve reasoning about counterfactual situations.Footnote ⁶⁵ However, the role of counterfactuals for metaphysics is more controversial. Other possible worlds can be interpreted not only as a semantic tool but also as what makes true counterfactual statements in a metaphysical sense.Footnote ⁶⁶ Alternatively, the truth-makersFootnote ⁶⁷ of counterfactuals may be taken to be part of the actual world: either laws of nature, or powers or potentialities.Footnote ⁶⁸

5 Interventions and Causal Models

A strong motivation for searching knowledge about causation is that such knowledge enables us to control things. If I know that A causes B I can exploit that knowledge to obtain B by manipulating A. The same connection allows us, in the other direction, to obtain causal knowledge by manipulation. Observing the effects of our own manipulations plays an important role in the psychological mechanisms that allow humans to acquire causal knowledge (Reference Gopnik and SchulzGopnik and Schulz 2007; Reference WaldmannWaldmann 2017). In experimental sciences, interventions are a crucial method for discovering causal influences, in addition to the observation of correlations. Under certain conditions (to which we return shortly), if an experimenter manipulates a variable C and observes a subsequent variation in variable E, this indicates that C causally influences E.

Structural equations (SE), a formal tool first introduced in genetics (Reference WrightWright 1921) and econometrics (Reference HaavelmoHaavelmo 1943), can be used to build models of causal structures on the basis of information obtained in this way, completing the models built in the framework of Bayes nets.Footnote ⁶⁹ The philosophical analysis of such causal models integrates insights of older philosophical theories of causation in terms of manipulation by a free agent, according to which “an event A is a cause of a distinct event B just in case bringing about the occurrence of A would be an effective means by which a free agent could bring about the occurrence of B.”Footnote ⁷⁰ However, in that form, such an account suffered from two major defects: circularity and anthropocentrism. The latter is implicit in the thesis that an event can be a cause only if its occurrence can be the result of the decision of a free agent.Footnote ⁷¹ We will see shortly that recent accounts in terms of causal models replace the concept of manipulation by the concept of intervention, which is not tied to free action, thus avoiding anthropocentrism. As for circularity, it seems impossible to build a non-circular analysis of causation that is grounded on manipulation or intervention, insofar as both are forms of causation. Recent interventionist theories of causation such as Reference WoodwardWoodward’s (2003) are explicitly intended not to be reductionist.Footnote ⁷²

Each of the directed edges in a causal net (as introduced earlier) represents the direct influence of a “parent” variable on a “child” variable. The dependence of a variable on its parent variables can also be represented, in a more precise way, by a structural equation. SEs provide an alternative and complementary tool for the construction of causal models, aside from probability distributions, which we saw earlier. Contrary to ordinary equations, SE are defined to be asymmetrical: the SE of an “endogenous” variable V expresses the functional dependence of V on other variables in the model. A variable is called “exogenous” if its value is determined by factors external to the causal system whose model is being built.

SE have an interventionist interpretation and imply counterfactuals.Footnote ⁷³ Let Y be a function of X, and let X and Y have the values X=x₁ and Y=y₁ in some actual situation. The SE Y=f(X) entails that if X had some value ≠ x₁, for example, x₂, Y would have the value Y=y₂=f(x₂), and it entails that if an intervention set X to value x₂, Y would take value y₂.

Let me illustrate the construction of a causal net on the basis of structural equations with the biological case of preemption mentioned earlier, which presents a challenge to earlier counterfactual accounts.

A simple deterministic model can be built with the following binary variables:Footnote ⁷⁴

• MA=1 microglia active; MA=0 microglia not active;

• MR=1 microglia remove debris; MR=0 microglia don’t remove debris;

• AA=1 astrocytes active; AA=0 astrocytes not active;

• AR=1 astrocytes remove debris; AR=0 astrocytes don’t remove debris;

• BW=1 brain without debris; BW=0 brain with debris.

Each endogenous variable is associated with a SE. To apply the model to a situation, one needs to specify the values of the exogenous variables. Set MA=1 and AA=1. The value of the endogenous variable MR is then determined by the value of MA, according to the SE MR=f(MA)=MA. If the microglia are active, they remove the debris (MA=1 and MR=1) and if they aren’t, they don’t (MA=0 and MR=0).

The preemption of the process beginning with active astrocytes is expressed by the SE for AR: AR = min (AA, 1-MR). The astrocytes remove the debris only if (1) the astrocytes are active and if (2) the microglia haven’t removed the debris. The variable BW representing the state of the brain is also endogenous: BW = max (MR, AR). The brain is free of debris either because the microglia have removed the debris or because the astrocytes have removed them.

The content of the set of SE can be represented in a graph (Figure 3), where each variable corresponds to a node in the graph. An arrow from variable X to variable Y represents the fact that the value of Y depends on the value of X; in this case, X is called a “parent” of Y. A directed path from X to Y is a sequence of adjacent arrows leading from X to Y.

Figure 3 Graph representing a causal net.

Each arrow and each SE represents a set of counterfactual conditionals. Once a model is constructed, it can be used to determine the truth-value of new counterfactuals that do not simply correspond to one arrow. Say we want to know what would have happened if the microglia, though active, would somehow not have removed the debris. To find this out, one sets the variable corresponding to the antecedent of the counterfactual to the value it would have if the antecedent were true. In this case, one sets MR=0. This represents an “atomic intervention” (Reference PearlPearl 2000, p. 70). One does not take into consideration the past that might have led to the truth of the antecedent. Rather, the value of the antecedent (here, MR) is set while the values of all variables corresponding to the past of the antecedent keep the values they actually have. In the graphical representation, this means that all arrows leading to the variable MR are erased, which is equivalent to transforming MR into an exogenous variable.

In the interventionist interpretation of this formalism, this corresponds to a localized experimental intervention on variable MR, which originates from outside the system and is direct in the sense that it is not obtained indirectly by intervening on factors that influence MR within the system.Footnote ⁷⁵

Just as Lewis’ miracles, this conception of interventions guarantees that no “backtracking” counterfactual can be true. When the value of variable X is modified, the variables situated upstream from X are left untouched. In the standard representation, these are the variables figuring at the left of X. The values that the variables downstream from (i.e., to the right of) X take in a situation in which X takes the stipulated value can then be determined on the basis of the equations corresponding to the arrows starting at X.

Reference PearlPearl (2000, p. 70) defines the causal effect of X on Y, written “P(y/do(x)),” as the probability distribution of the different values y of Y, given that an intervention (“do”) has set variable X to the value x. This has the consequence that all factors different from X that also influence Y are included in X’s impact on Y. To avoid this result, Reference WoodwardWoodward (2003, p. 98) imposes additional constraints on interventions I appropriate to determining whether X causes Y.Footnote ⁷⁶ (1) I causes X. (2) I is the only cause of X, in the sense that all other influences on X are cut. (3) I does not cause Y through any paths that do not go through X. Say we want to find out whether the calculating activity of my computer (X) heats it up (Y). If I, the intervention of switching the computer on, not only triggers X but also turns on the cooler Z, I is not an intervention on X in the technical sense of Woodward’s conditions, for whether the value of Y is changed by I does not only depend on X but also on Z. In particular, the fact that Y doesn’t vary when I varies (because the heating influence of X on Y is offset by the cooling influence of Z on Y) does not justify the conclusion that X doesn’t influence Y. (4) I is statistically independent of all variables Z that influence Y through paths that do not go through X. If, in order to find out whether the indication X of a barometer causes thunderstorms Y, my interventions I on X depend on (my knowledge of) air pressure Z, then Y may vary as a function of the values that I imposes on X, whereas X does of course not cause Y.

Causal models built from SE can be used to define different concepts of causal influence. The concept of an actual cause is illuminating in situations of preemption.Footnote ⁷⁷ In the situation represented by Figure 3, BW depends counterfactually neither on MA nor on AA. However, it would be wrong to conclude that neither MA nor AA causes BW. Indeed, in each situation, either one or the other causes BW. The concept of actual cause can be used to justify the intuition that MA is the cause of BW in situations where MA=1, although BW doesn’t depend counterfactually on MA.

MA taking value 1 (MA=1) is an actual cause of BW=1. X=1 is an actual cause of Z=1 if and only if there is an “active causal route” (Reference HitchcockHitchcock 2001, p. 287) from X to Z in an appropriate causal model <V, E>. A “route” between X and Z in the set V is an ordered sequence of variables <X, Y₁, … , Y_n, Z> such that each variable in the sequence is in V and is a parent of its successor in the sequence. The route <X, Y₁, … , Y_n, Z> is active in the causal model <V, E> if and only if Z depends counterfactually on X within the new system of equations E’ constructed from E, by setting the values of all variables that do not lie on the route to their actual values (Reference HitchcockHitchcock 2001, p. 286).

In our example, MA=1 comes out as an actual cause of BW=1 because, if AR, which does not lie on the route from MA to BW is set to its actual value, that is, 0, BW counterfactually depends on MA.Footnote ⁷⁸

The structural equations framework provides the means for distinguishing different causal notions that can all be expressed by the common-sense word “cause.” This shows the fecundity of this approach, although it cannot provide a non-circular analysis of causation. As we have seen earlier, a variable can influence another variable in two independent ways in such a way that these influences cancel each other out. Switching on (S) the calculating activity X of my computer raises its temperature Y,Footnote ⁷⁹ but it also causes the onset of the cooling system Z, which lowers its temperature. It is possible that the positive direct influence of X on Y is exactly compensated by the negative influence of Z on Y, so that S has zero net influence on Y.Footnote ⁸⁰ In such a case, it seems intuitively both correct to say that switching the computer on heats it up and that it doesn’t heat it up. However, this involves no paradox insofar as the two judgments contain different notions of cause.Footnote ⁸¹ The former is correct if “heats it up” is taken to express the concept of being a contributing cause, the latter is correct if “heats it up” is taken to express the concept of being a total cause. In the situation sketched, S is not a “total cause” of Y but S is a “contributing cause” of Y.

(TC) X is a total cause of Y if and only if there is a possible intervention on X that will change Y or the probability distribution of Y (Reference WoodwardWoodward 2003, p. 51).

(CC) X is a contributing cause of Y with respect to a set of variables V, if and only if (1) there is a directed path from X to Y so that each variable on the path directly influences its immediate descendant, and (2) there is an intervention on X that changes the value of Y if the values of all variables in V that do not lie on that path are held fixed at some value (Reference WoodwardWoodward 2003, p. 57).

Indeed, if we hold Z in our example fixed, we find that an intervention on X modifies the value of Y, so that S is a contributing cause of Y, although the application of condition (TC) shows that it is not a total cause of Y: if Z is not held fixed, switching on the computer does not make its temperature rise.

Older versions of the manipulability theory make the judgment “X causes Y” depend on the possibility of acting on X. This seems to make it impossible to apply the concept of causation to events that are in principle outside the sphere of influence of human interventions. However, eruptions of volcanoes and explosions of supernovae are causes although no possible human action could ever bring them about or modify them. The notion of intervention solves this problem because it is defined without any reference to human action. Analyses of causation in terms of structural equations and directed graphs avoid anthropocentrism because the intervention that sets the value of the putative cause need not be the result of a human action. Natural events entirely independent of intentional actions can satisfy the formal conditions on an intervention modifying the value of the putative cause. Such “natural experiments”Footnote ⁸² provide just as good a basis for judging causal influence as intentional interventions by human experimenters. In neuropsychology, the hypothesis that the activation X of one brain region causally influences the activation Y of another brain region is confirmed by the observation that a modification of X due to accident or illness is systematically followed by a modification of Y.

However, there seem to be causal relations on which even interventions as defined in this sense seem to be impossible. To judge whether the gravitational attraction of the moon causally influences the tides, one must examine the consequences of an intervention on the position or the mass of the moon. It can be doubted whether interventions on the moon are physically possible: such an intervention would require modifying the position or the mass of the moon by some means that does not also directly influence the tides.Footnote ⁸³

The clearest limit to the applicability of the concept of intervention concerns cosmology. The possibility of an intervention on a system requires the system under study to be smaller than the whole universe because the origin of an intervention on some system must lie outside that system.Footnote ⁸⁴

Interventions provide a strong epistemic criterion for causal influence. However, we have seen that it may not always be possible, even in principle, to carry out interventions in the required sense. Even if interventionist conditions are sufficient, they are not necessary for causation. In this respect, the interventionist account does not provide a complete metaphysical theory of causation.

6 Methodological Interlude

We have had a look at four reductive approaches to causation and at the framework of causal models, which can be interpreted with the conceptual tools of earlier accounts (lawful regularity, conditional probability, counterfactual dependence, intervention) so as to yield a powerful formal method for representing causal knowledge and the search for causes. Before I introduce and evaluate the approach to causation in terms of physical processes linking spatiotemporally localized particular events, let me make some methodological remarks.

7 Processes and Mechanisms

We have examined accounts of causation in terms of regularity, probability raising, counterfactual dependence, intervention, and causal models. The first three of these suffer from counterexamples and the fourth cannot be applied in all situations. The approach in terms of causal models, which analyzes causation in terms of difference-making at the level of types of events, presupposes the distinction between causal and non-causal dependence.Footnote ⁹⁵ Causal models are built under the constraint of using only variables that are “independently fixable,” that is, that do not stand in non-causal dependence relations.Footnote ⁹⁶ Therefore, they do not provide a framework that can be used to explain the difference between causal and non-causal dependence relations.

This distinction may be made at the level of particular events, on the basis of the empirical hypothesis that causation is a local physical process that stretches between two events that are localized in space and time. We will see that this empirical hypothesis can be used as a complement to accounts of causal influence in terms of causal models.

One historical source of process accounts of causation is Reference RussellRussell’s (1948) analysis of causation in terms of “causal lines,” which is inspired by the physical notion of a world line (Reference ReichenbachReichenbach 1928/1958). The world line of an object or process is its spatiotemporal trajectory. Russell defines a causal line as a world line that possesses qualities or structures that are either constant or change only smoothly (Reference RussellRussell 1948, p. 477). However, being a causal line is neither necessary nor sufficient for being a real causal process. It is not sufficient because the continuity of structure or quality can also characterize what Reichenbach calls an “unreal sequence” (Reference ReichenbachReichenbach 1928/1958, §23, p. 148) and Reference SalmonSalmon (1984, p. 141ff) a “pseudo-processes.” Pseudo-processes are world lines that give human observers the illusory impression of a causal process: they are not causal, but their qualitative continuity qualifies them as Russellian causal lines. Take Reference ReichenbachReichenbach’s (1928/1958, §23, p. 148) fast rotating projector casting a spot of light on a distant wall so that the spot sweeps over the wall. The world line consisting of the series of places on the wall at the times at which the light spot appears on them is a causal line without being a causal process. It exhibits qualitative continuity but cannot be causal. This can be seen from the fact that its speed can exceed the speed of light if the distance between the projector and the wall is sufficiently large and the speed of rotation of the projector sufficiently fast. Each spot at x at t is the end point of a causal process originating in the projector, without having any influence on the adjacent spot. Being a causal line is not necessary for being a causal process either. Some causal processes lack continuity of structure and exhibit large and fast qualitative changes, for example, when several particles of different types follow each other in a “cascade” of radioactive decomposition (see Section 2).

Reference SalmonSalmon (1984) has combined Russell’s and Reichenbach’s analyses, suggesting that a causal process is a process that (1) has structure or qualities that are either permanent or only changing continuously and (2) is capable of transmitting a mark, which is defined as a local modification of structure.Footnote ⁹⁷ The light spot gliding along the wall is not a causal process because, if one modifies its color by inserting a red filter between the projector and the wall at one point, this modification will not propagate to the subsequent world line of the spot.

This analysis in terms of continuity of structure and mark transmission raises several difficulties.Footnote ⁹⁸ The mark transmission criterion is shown to be insufficient by pseudo-processes capable of transmitting marks. Kitcher (1989, p. 463) mentions derivative marks: when a passenger in a car holds a flag out of the window, the shadow cast by the car as it passes along a wall bears the mark of the flag. Moreover, the analysis of the notions of mark and of causal interaction seems to be circular:Footnote ⁹⁹ A mark is a modification of structure introduced into a process by a causal interaction, but an interaction is causal if it leads to the introduction of a mark.

An alternative proposal characterizes a causal process as a world line along which some physical quantity is transmitted or transferred, such as energy, momentum,Footnote ¹⁰⁰ or more generally, some conserved quantity.Footnote ¹⁰¹ Processes construed in terms of transference can be used to justify causal judgments that challenge difference-making accounts. If active microglia preempt active astrocytes in cleaning the brain of debris, there is no (chain of) counterfactual dependence between active microglia and the cleaning of debris. Transference accounts suggest that what grounds the fact that the microglia’s activity causes the cleaning of debris is that there is a physical process transmitting energy and momentum to the debris in the microglia’s case, but not in the astrocytes’. However, the latter claim can be challenged. Given that there is physical transmission everywhere (except for events that are spacelike separate in the sense of relativity theory), the astrocytes also transmit physical quantities to the debris.

Or take the radioactive decay of nuclei of the isotope ²²³Fr of Francium (see Section 2). This case refutes accounts of causation in terms of probability raising because the probability of having an effect of type ²¹⁹Rn is lower given the cause (there being a nucleus of type ²¹⁹At) than given the alternative possibility that it be of type ²²³Ra. It seems doubtful whether interventions can be used to overcome this difficulty. But transference of energy and other conserved quantities provides a ground for the judgment that the decay of a ²¹⁹At nucleus causes the coming into being of a ²¹⁹Rn nucleus. More generally, transference justifies causal judgments in cases where (1) the cause belongs to a type that lowers the probability of events of the type of the effect, (2) cause and effect are categorized in terms of types that have only one member, so that probabilistic approaches are inapplicable.

The hypothesis that causation is grounded on a process of transmission fits the intuitions of locality and intrinsicality, according to which the existence of a causal relation between a and b only depends on processes stretching from a to b. Difference-making accounts, in terms of nomological dependence, probability raising, counterfactuals, or in terms of causal models, run counter that intuition. They all construe causation as non-intrinsic.

Challenges against the hypothesis that causation is grounded on transference can be grouped in two sorts. Objections of the first sort are based on considerations having to do with the content of physical theory; they accept the project of finding out what causation is in the real world according to science, but object to the hypothesis that it is transference of an amount of a conserved quantity. The second type of challenge is conceptual and based on counterexamples: cases of causation without transference or cases of transference without causation. Let me look at the former challenges first.Footnote ¹⁰²

1. 1. In order to give sense to the idea that something x is transferred from a to b, x must persist over time. It can be questioned whether particular quantities of conserved quantities, such as energy, can persist through time like substances.Footnote ¹⁰³ However, the statement that an amount q of energy (or some other conserved quantity) is transferred between space-time regions A and B, where events a and b are located, can be justified without requiring that q is a substance. A necessary and sufficient condition for the transference of energy (or another conserved quantity) between A and B is the existence of a time-like or light-like curve connecting these regions along which a field carrying energy (or another conserved quantity) is propagated.

2. 2. In the framework of the theory of general relativity, it is not possible to derive a generally valid form of the conservation of energy. However, the notion of transmission presupposes that of conservation: only what is conserved can be transmitted. Therefore, transference of energy cannot be what makes true causal propositions bearing on large scale cosmological events, for example, that the big bang is the distant cause of the present expansion of the Milky Way, or on events in regions with non-negligible space-time curvature (Reference CurielCuriel 2000; Reference LamLam 2005). That leaves open the possibility that transference provides a necessary and sufficient local condition for causation between events that are situated in approximately flat regions of space-time.

3. 3. A third argument against the transference hypothesis relies on the existence of non-local dependencies in entangled systems in quantum mechanics. In a variant of the experiment first conceived by Reference Albert, Boris and RosenEinstein, Podolsky and Rosen (1935), two electrons are prepared in an entangled singlet state of spin ½, and then move in opposite directions. According to quantum mechanics, measurements of a given component of the spins of such entangled pairs of particles are strictly correlated. Whether or not the results of the measurements are determined before the measurements (as hidden-variable theories assume) or not (as the orthodox version of quantum mechanics assumes), the result of one measurement seems to have an instantaneous causal influence at a distance on the other measurement. If causation is grounded on transference, the dependence of one measurement on the other in an EPR-style experiment cannot be causal. The non-causal character of the dependence of one result of measurement on the other can also be brought out by the fact that it is impossible to manipulate one result by intervening on the other.Footnote ¹⁰⁴ The correlations between measurement events on entangled pairs of particles that are spacelike separated may be cases of non-local and non-causal determination.Footnote ¹⁰⁵

4. 4. A necessary and sufficient condition for the transference of energy (or other conserved quantities) between A and B (according to the transference hypothesis, itself necessary and sufficient for causation) is the existence of a time-like or light-like curve connecting these regions along which a field carrying energy (or another conserved quantity) is propagated. This seems to make the account circular because “carry” and “propagate” are causal notions. Attempts to analyze the meaning of these causal notions in terms of transmission would either lead to an infinite regress or a vicious circle. It seems more promising to analyze the meaning of these expressions in terms of laws of nature. This yields a theory in two parts: (1) Causation is grounded on transference. (2) Transference of amounts of conserved quantities is grounded on laws of physics. The transference theory is a variant of the nomological theory because, at bottom, laws of physics ground causation. However, not any law of any type, or even any type of law of evolution, is sufficient for causation, as it is according to (DN-C). The only laws that ground causation are the laws of propagation of energy and other conserved quantities along time-like or light-light curves.

   The fundamental role of laws raises two problems. (1) It threatens to make the account lose its locality and intrinsicality, which seemed to give it an advantage over non-local theories, such as causal models.Footnote ¹⁰⁶ (2) If the relevant laws are reversible, the asymmetry of causation is not a consequence of the physical transmission process that grounds causation in the actual world, but has an independent ground,Footnote ¹⁰⁷ such as the fact that our region of the universe contains many irreversible processes that are all oriented in the same direction, according to the second law of thermodynamics.Footnote ¹⁰⁸

   Now let us look at challenges that have been raised against process accounts from a conceptual point of view.

5. 5. Process theories take causation to be a type of relation or process whose characteristics can only be discovered by empirical research; they contain empirical hypotheses about what it is, physically, that makes a relation or process causal. Such hypotheses are fallible, that is, may turn out to be false on empirical grounds. That is a strength: after all, the fallibility of a theory proves that it is not tautological (Reference Popper KarlPopper 1934). But the fact that these theories have empirical content can also been seen as a weakness: Contrary to theories that aim at analyzing causation in an a priori manner by pure conceptual analysis, an empirical theory aims only at correctly characterizing causation as it is in the real world.Footnote ¹⁰⁹ One reply to this worry is that it is inappropriate to criticize process theories for not satisfying criteria that are not appropriate for them. An empirical theory of causation in reality has no ambition to account for all possible sorts of causation, in particular to those featuring in science fiction novels or fairy tales. Another reply is that process theory, although fallible as an empirical hypothesis, might well be, if it is true, necessarily true, in analogy with Kripke’s thesis that the statement that water is H₂O is a posteriori, but nevertheless, if true, necessarily true.Footnote ¹¹⁰

6. 6. Another objection is that transmission processes are everywhere. Events that are spatiotemporally sufficiently close to each other are, for example, often linked by transmissions of photons. Therefore, transmission theory seems condemned to lead to a plethora of true causal judgments.Footnote ¹¹¹ It can be replied first that those plethoric causal judgments are true but lack communicational relevance.Footnote ¹¹² Second, the relevant causal processes can be chosen on perfectly objective grounds, on the basis of the properties of the effect that is indicated in the explanandum of the causal explanation one is looking for. If one asks for the cause of Peter’s standing up, the relevant causal process is at the physiological and psychological level and leads to the instantiation of the physiological and psychological properties constitutive of standing up.

7. 7. An objection to this latter reply is that the transference hypothesis applies only to physical causal processes and is thus inadequate to ground the truth of ordinary causal judgments involving non-physical, for example, psychological, properties. Mary asks Peter to stand up, and he stands up. The cause of Peter’s standing up has to do with his understanding of the content of Mary’s request but apparently it does not seem to be relevant to consider the underlying causal process from the point of view of energy transmission.Footnote ¹¹³ The transference hypothesis entails indeed that all causes and effects are physical. What the objection shows is that a satisfactory account of causation must take the influence of properties into account, over and above the mere existence of causal relations based on transference. We will look at proposals for integrating the process account with causal models, in which the relevant properties are represented by variables, next.

8. 8. Process accounts seem to be refuted by omissions and preventions. If I kill a plant by omitting to water it, it seems that I have caused its death without having transmitted anything to it.Footnote ¹¹⁴ If on the contrary I prevent the plant’s death by watering it, the event of the plant’s death does not take place and cannot therefore be the target of any transmission. Reference Schaffer, Collins, Hall and PaulSchaffer (2000/2004) argues that there are many common-sense causal propositions bearing on situations in which no transmission seems to be involved. Striking cases are propositions expressing double prevention (Reference HallHall 2004, p. 241), in which the cause prevents the prevention of an effect. In the operon theory of gene expression,Footnote ¹¹⁵ gene expression is triggered by an “inductor” molecule that prevents a “repressor” molecule from preventing it. Omission and prevention are forms of difference-making. In the next section, we will look at attempts to integrate processes with difference making.

8 Integrating Causal Models with Processes

We have seen that accounts in terms of (1) nomological deducibility, (2) conditional probabilities, (3) counterfactual dependence, (4) interventions, all yield fallible criteria for causation.Footnote ¹¹⁶ The framework of causal models can be interpreted using the conceptual tools of (1) – (4) so as to solve many of the problems of earlier accounts. However, causal models provide only a part of a complete theory of causation. The construction of a model presupposes the distinction between causal and non-causal dependence. A set of variables is appropriate for the construction of a causal model only if it doesn’t contain any variables related by forms non-causal dependence. Otherwise, the variables do not respect “independent fixability.”Footnote ¹¹⁷ Causal models do not provide a framework within which one could represent the question which dependence relations are causal, and which are logical, conceptual or metaphysical, because their construction presupposes an answer to that question.

Accounts in terms of mechanisms can be seen as complementary to difference-making accounts, and in particular to causal models. This complementarity can be seen from two sides. Pearl says that “each parent-child relationship in the network represents a stable and autonomous physical mechanism” (Reference PearlPearl 2000, p. 22). These physical mechanisms are not characterized within causal models; rather, they can be interpreted as constituting the metaphysical reality that explains the functional dependence between variables expressed in structural equations and causal models. See Section 5.

Mechanisms have been introduced in the philosophy of causal explanation and causation as a reaction to the inadequacy of the DN model for the analysis of the scientific analysis of causation in the life sciences.Footnote ¹¹⁸ Accounts in terms of probability raising, counterfactuals, interventions, or causal models all seem inappropriate for causation in biology and neuroscience, where causal explanation does not in general rely on the discovery of exceptionless regularities, or manipulability at the level of variables, but on the discovery of mechanisms. There are at least two ways of analyzing the concept of a mechanism. One is Salmon’s and Dowe’s in terms of processes (see Section 7). A second analysis takes mechanisms to be systems of interacting parts.Footnote ¹¹⁹ “A mechanism underlying a behavior is a complex system which produces that behavior by the interaction of a number of parts according to direct causal laws” (Reference GlennanGlennan 1996, p. 52). The direct links between variables in causal models can be interpreted in terms of mechanisms, which motivates the mechanistic theory of token level causation: “two events are causally related when and only when they are connected by an intervening mechanism” (Reference Glennan, Beebee, Hitchcock and MenziesGlennan 2009, 316).Footnote ¹²⁰

So here is one aspect of the complementarity of causal models and mechanistic accounts: Causal models and other difference-making theories represent networks of causal links between variables, and mechanistic theories aim at identifying the metaphysical processes that underlie each direct link between variables (Reference GlennanGlennan 2017, p. 168).

Here is a second way in which mechanisms and difference-making relations (corresponding to direct links between variables in causal models) complement each other. Influence relations between variables representing higher-level features of, for example, biological systems, are “mechanically explicable” (Reference GlennanGlennan 1996, pp. 61–3). They are the result of interactions between parts of the system. The interactions between parts that help explain a given causal link at level n lie at lower levels. The concept of levels of properties is controversial and can receive different interpretations. In a mereological interpretation, the properties of a whole object lie at a higher level than the properties of its parts.Footnote ¹²¹ In general, interactions at level n-1 are themselves mechanically explicable in terms of still lower levels, n-2 and beyond. One example is the multilevel mechanism of long-term memory (Reference CraverCraver 2007, pp. 165–70). The mechanistic account of causation is not reductive and must be completed by some complementary account of what makes true the most fundamental interactions. The account of the fundamental account may be in terms of laws of nature, as Reference GlennanGlennan (1996) and Reference PsillosPsillos (2004) suggest, or in terms of direct causal influence between variables in causal models (Reference WoodwardWoodward 2011). Fundamental influence relations may be grounded on transference.

The complementarity of difference-making and mechanistic/process accounts must not be confused with pluralism, according to which “cause” is a polysemic word that is ambiguous between different concepts. For pluralism, what is meant by a causal statement depends on the context. Many varieties of pluralism are conceivable and many forms have been explored.Footnote ¹²² It is undeniable that there exist several causal concepts, all useful in certain contexts and irreducible to one another, such as direct and indirect cause, or contributing and total cause. The complementarity thesis just sketched is very different from the pluralist claim that causal statements like “smoking causes cancer” are ambiguous and express either the concept of causation as difference-making or the concept of causation as process or mechanism.Footnote ¹²³

The application to causation of the so-called “Canberra plan”Footnote ¹²⁴ provides another important way to conceive of the complementarity of difference-making accounts (such as causal models) and process accounts (Reference LewisLewis 1994a, Reference Lewis2004; Reference MenziesMenzies 1996; Reference BontlyBontly 2006). The Canberra plan is a general strategy for the naturalization of philosophical concepts, that is, for integrating common-sense notions with scientific knowledge. It is in part conceptual and in part empirical. In a first a priori step, the content of a concept is derived from ordinary common-sense intuitions or “platitudes” about the correct application of the concept: The concept is characterized in terms of a “functional role.” In a second, empirical step, scientific knowledge is used to produce a hypothesis about what in reality plays the functional role identified in the first step. Kim offers the example of the biological notion of a gene. In the first step the functional role corresponding to the notion of gene is identified as “that mechanism in a biological organism causally responsible for the transmission of heritable characteristics from parents to offsprings” (Reference KimKim 1998, p. 25). In the second step, (parts of) DNA molecules are identified as what fills the role.

Applying this strategy to causation, a first step identifies the conceptual role corresponding to the notion of cause. It is plausible that this conceptual role can be characterized by virtue of all or some subset of “the central connotations of causation” (Reference SchafferSchaffer 2000a, p. 289) I have mentioned at the very beginning of this Element.Footnote ¹²⁵ The second step consists in making a hypothesis, informed by science, about what fills the role identified in the first step. Menzies mentions as one promising hypothesis for the filler of the role described by the folk concept of causation, Reference FairFair’s (1979) proposal that “the causal relation is the relation of transfer of energy-momentum from cause to effect” (Reference MenziesMenzies 1996, p. 104).

One worry is that the Canberra plan takes the functional role identified in the first step to be a causal role.Footnote ¹²⁶ This makes it seem impossible to apply the reduction procedure to the concept of causation itself. But the role identified by a concept need not be a causal role. More substantial worries are that there are no clear-cut and universally shared intuitions about causation,Footnote ¹²⁷ and that these intuitions are not purely a priori because they are in part shaped by what science tells us about possible role-fillers at the second step.

Another proposal is that causation at the level of particular events is analyzed in a first step in physical terms, in terms of mechanisms or processes of transference, and that we select, in a second step, from the vast number of physical causal relations, those that make a difference, in terms of probability raising, counterfactuals or causal models. According to “causal foundationalism” (Reference NeyNey 2009), difference-making causation between events at the level of types results from our selecting those among the physical causal relations that are important for our purposes of explanation, prediction and action.

According to the complementarity thesis, there are two components in what makes true ordinary causal statements such as “Bob’s coughing caused Carol’s waking” (Reference GlennanGlennan 2010, p. 365). (1) A physical process between the relevant particular events and (2) a difference-making relation at the level of relevant types under which these events fall, that is, in this case coughing and waking.Footnote ¹²⁸ Both aspects of causation are required: on one hand “the spatiotemporal aspects of causation” (Reference Hitchcock, Machamer and WoltersHitchcock 2007, p. 214), on the other hand, the fact “that effects depend upon their causes, or that causes make a difference for their effects” (Reference Hitchcock, Machamer and WoltersHitchcock 2007, p. 214). What makes it true that c’s being F is causally responsibleFootnote ¹²⁹ for e’s being G, has two parts: there is a physical process linking c and e; 2) F makes a difference to G. The types by which events make a difference are in general not physical, but in a physicalist framework, it is supposed that all facts superveneFootnote ¹³⁰ on the set of physical facts. If this is correct, the process of Bob’s coughing waking Carol up supervenes on a physical process.Footnote ¹³¹

9 Contemporary Debates

10 Concluding Remarks

This Element began with the observation that causation is crucial for our conception of the world and ourselves, from the perspective of both common sense and science, but that its philosophical analysis remains controversial. So far, all efforts to elaborate a reductive account able to account for all uses of the concept of causation have failed. This has led many authors to abandon the attempt at finding a unique reductive analysis. Various forms of pluralism have been explored according to which there are many concepts of causation. One form of pluralism consists in interpreting various concepts of causation as so many conceptual tools that are useful in limited contexts. The strategy of elaborating causal concepts as tools has proven very fruitful in the framework of formal methods of causal modeling. These formal methods have allowed making much progress in elaborating different concepts of causation, such as contributing and total cause, direct and indirect cause, and actual cause. It has become clear that the failure to elaborate a reductive account is no obstacle to our improving our understanding of causation. In particular, it has turned out that one gets a better grasp on the concept of cause by clearly understanding various types of situations that have been obstacles to all simple reductive accounts, such as cases involving omissions, preventions, and cases where causation doesn’t seem to be transitive. Just as important is progress that brings into sharp focus unresolved issues that constitute challenges for future philosophical research on causation. Here are two of them.

Most contemporary philosophers accept both that causation, understood with the help of formal models such as structural equations and causal nets, is crucial for science and common sense and that causation has no place in physics. But how can there be causation in our ordinary life, and in chemistry, biology, psychology, economy and the law, if there is no physical causation? It seems difficult to reconcile this in particular with the widely accepted, and often presupposed, physicalist worldview, according to which all events and processes described in terms of the concepts of common sense or in the terms of some science other than physics, depend on physical events and processes. The tension is certainly alleviated once reductionism is abandoned: if one abandons the thesis of logical empiricism that all other sciences are in principle reducible to physics, there is no contradiction in holding both that there is no causation in physics and that there is causation everywhere else although everything depends in some sense on physics. But there remains a tension. This tension is apparent in arguments for eliminativism in the philosophy of mind, with respect to cognitive states and events. According to the so-called exclusion argument, cognitive states and events cannot be causes because their causal powers are excluded by the causal powers of the physiological and physical states of the cognitive systems entertaining these cognitive states. This argument presupposes that those underlying physical states have causal powers. Therefore, it is incompatible with the thesis that there is no physical causation.

Another topic that awaits further exploration is the distinction between causal and non-causal dependence. The construction of causal models presupposes this distinction because causal nets can only be built with variables that are chosen so that, for each pair of variables, there are only two possibilities: either they are independent of each other or they stand in a relation of causal influence. Variables must be chosen so that they do not depend on each other in any other way: a causal model must not contain variables that stand in logical, conceptual or supervenience relations, or that are related by non-causal association laws. Thus, the possibility of analyzing causation in terms of causal models depends on the distinction between causal and non-causal dependence. However, no generally accepted account of this distinction has yet emerged. From a metaphysical point of view, what is needed is a theory of what makes properties or events independent enough of each other so as to be able to stand in causal relations. Only variables that represent properties or events of that sort are appropriate for use in structural equations and causal nets. The challenge is to find a non-circular way of distinguishing causal and non-causal dependence and of characterizing various non-causal forms of dependence. As long as the choice of variables that are appropriate for causal models remains outside of philosophical accounts of the elaboration and use of such models, our understanding of causation remains incomplete.