Causal exclusion and causal Bayes nets

Causal exclusion and causal Bayes nets∗ AF T Alexander Gebharter Abstract: In this paper I reconstruct and evaluate the validity of two versions of causal exclusion arguments within the theory of causal Bayes nets. I argue that supervenience relations formally be- have like causal relations. If this is correct, then it turns out that both versions of the exclusion argument are valid when assuming the causal Markov condition and the causal minimality condition. I also investigate some consequences for the recent discussion of causal exclusion arguments in the light of an interventionist theory of causation such as Woodward’s (2003) and discuss a possible objection DR to my causal Bayes net reconstruction. 1 Introduction Causal exclusion arguments, most famously advanced by Kim (1989, 2000, 2003, 2005), can be used as arguments for epiphenomenalism or as arguments against non-reductive physicalism. Epiphenomenalism is the view that “mental events ∗ This Causal is the accepted version of the following article: exclusion and causal Bayes nets. Philosophy and Gebharter, A. (2017). Phenomenological Re- search, 95(2), 353–375. doi:10.1111/phpr.12247, which is published in final form at: http://onlinelibrary.wiley.com/journal/10.1111/(ISSN)1933-1592. 1 are caused by physical events in the brain, but have no effects upon any physical events” (Robinson, 2015). Non-reductive physicalism, on the other hand, basically consists of three assumptions: Mental properties supervene on physical properties, mental properties cannot be reduced to physical properties, and mental properties are causally efficacious (cf. Kim, 2005, p. 33). AF T In a nutshell, exclusion arguments assume non-reductive physicalism and conclude from several premises that mental properties supervening on physical properties cannot cause physical or other mental properties. The notion of causation used in these arguments is, however, typically somewhat vague and not specified in detail. Because of this, the validity of these arguments may depend on the specific theory of causation endorsed (cf. Hitchcock, 2012). Throughout the paper I treat theories of causation as tools for providing information about the world’s true causal structure and about the causal efficacy of properties on other properties. So a theory of causation may be better w.r.t. providing such information than another theory. If two such theories lead to different results about the validity of exclusion arguments, then the one providing more DR information relevant for exclusion arguments should be favored when evaluating the validity of such arguments. In this paper I reconstruct two versions of exclusion arguments and eval- uate their validity within a particular theory of causation, viz. the theory of causal Bayes nets. The theory of causal Bayes nets (CBNs) evolved from the Bayes net formalism (Neapolitan, 1990; Pearl, 1988). It was elaborated in de- tail by researchers such as Pearl (2000) and Spirtes, Glymour, and Scheines (2000). The theory connects causal structures to probability distributions and provides powerful methods for causal discovery, prediction, and testing of causal hypotheses. Furthermore, its core axioms can be justified by an inference to the best explanation (see Schurz, 2008 for a general approach) of certain statistical 2 phenomena, and several versions of the theory can be proven to have empirical content, by whose means not only the theory’s models, but also the theory as a whole becomes empirically testable (Schurz & Gebharter, 2015). So the theory of CBNs probably gives us the best empirical grasp on causation we have so far. Hence, it allows for an empirically informed treatment of causation in causal AF T exclusion arguments, and thus, also for an empirically informed evaluation of the validity of such arguments. Another strong motivation for this endeavor is that causal exclusion arguments have recently been intensively discussed (cf., e.g., Baumgartner, 2009, 2010; Eronen, 2012; Raatikainen, 2010; Shapiro, 2010; Shapiro & Sober, 2007; Woodward, 2008, 2014) within an interventionist framework of causation à la Woodward (2003), and that interventionist accounts do have a natural counterpart within the theory of CBNs (cf., e.g., Gebharter & Schurz, 2014 or Zhang & Spirtes, 2011). So the hope is that we can draw as of yet unconsidered conclusions for the interventionist debate surrounding causal exclusion arguments from a reconstruction on the basis of the theory of CBNs. This seems especially DR promising since one of the main problems interventionists have when testing causal efficacy of properties standing in supervenience relationships to other properties is that these properties cannot be simultaneously manipulated by interventions (for details, see section 4). So the interventionist account seems to have some kind of a blind spot when it comes to testing causal efficacy of such properties. The theory of CBNs, on the other hand, provides a neat and simple test for causal efficacy not requiring fixability by means of interventions. The paper is structured as follows: In section 2 I briefly introduce two vari- ants of the causal exclusion argument. In section 3, which is the main section of the paper, I reconstruct these two variants within the theory of CBNs and evaluate their validity. This requires an answer to the question of how super- 3 venience relationships should be represented in CBNs and a test for evaluating whether the instantiation of a property X at least sometimes contributes something to the occurrence of another property Y . I will argue that supervenience relationships can be treated similar to a CBN’s causal arrows. This assumption will be crucial for my argumentation in subsequent sections. A method for AF T testing a property’s causal efficacy is already implemented in the productivity condition, which can be proven to be equivalent to one of the theory of CBN’s core axioms, viz. the causal minimality condition (cf. Spirtes et al., 2000, p. 31). I conclude section 3 by demonstrating that mental properties supervening on physical properties cannot be causally efficacious if causal as well as supervenience relations are assumed to obey the core axioms of the theory of causal nets. In section 4 I investigate the consequences of these findings for the interventionist debate on the causal exclusion argument. In section 5 I defend my suggestion to treat supervenience relationships similar to causal arrows against an objection raised by Woodward (2014). I conclude in section 6. The causal exclusion argument DR 2 Causal exclusion arguments (cf. Kim, 1989, 2000, 2003, 2005) typically come in two variants (cf. Harbecke, 2013): (i) arguments against the causal efficacy of mental properties on physical properties, and (ii) arguments against the causal efficacy of mental properties on other mental properties. In this paper we will have a look at both variants. The diagram in Figure 1 (which is adapted from Kim, 2005) can be used to illustrate both versions of the causal exclusion argument. P1 and P2 stand for physical properties, while M1 and M2 stand for mental properties. P1 , P2 , M1 , and M2 are assumed to be pairwise non-identical. Furthermore, we also assume that there is no spatio-temporal overlap of P1 and P2 . Double-tailed arrows 4 AF T Figure 1: Diagram for illustrating the two versions of the causal exclusion argument. Single-tailed arrows stand for direct causal relations, while double-tailed arrows indicate supervenience relationships. (Ô⇒) represent relationships of supervenience. So M1 supervenes on P1 , and M2 supervenes on P2 , meaning that every change in M1 and M2 is necessarily associated with a change in P1 and P2 , respectively (cf. McLaughlin & Bennett, 2011). In addition, we assume that P1 and P2 fully determine M1 and M2 , respectively. So the occurrence of P1 and P2 suffices for the instantiation of M1 and M2 , respectively. Alternatively we can say that P1 constitutes M1 and that DR P2 constitutes M2 .1 Single-tailed arrows (Ð→) represent direct causal relationships. So P1 is a direct cause of P2 . Because we assume the completeness of the physical domain, i.e., that every physical property has a sufficient physical cause, also P2 has a sufficient physical cause.2 We assume P1 to be that cause, and hence, P1 ’s occurrence determines P2 ’s occurrence. Now the question is whether we can 1 The properties I called supervenience and constitution here are typically combined by assuming strong supervenience (cf. Kim, 2003, p. 151). However, I prefer to separate them in this paper. 2 According to the completeness of the physical, also P1 will have a sufficient physical cause. Since P1 ’s physical causes, however, will not be relevant for the argument, we do not represent them in the diagram. 5 draw single-tailed arrows from M1 to P2 and from M1 to M2 , i.e., whether M1 can cause P2 or M2 . This question is represented by the question marks over the single-tailed arrows M1 Ð→ P2 and M1 Ð→ M2 in the diagram. Version (i) of the causal exclusion argument roughly goes as follows: P1 , P2 , M1 , and M2 are instantiated. Now let us ask why P2 is instantiated. Because AF T of the causal completeness of the physical, P1 ’s instantiation suffices for P2 ’s instantiation. So P2 is instantiated because P1 is. Since P1 ’s occurrence necessitates P2 ’s occurrence, there is nothing the instantiation of M1 could contribute to P2 ’s occurrence. Hence, M1 has no causal influence on P2 . Version (ii) of the argument roughly goes as follows: P1 , P2 , M1 , and M2 are instantiated. Now the crucial question is why M2 is instantiated. M2 is constituted, and thus, fully determined by its physical supervenience base P2 . So P2 ’s occurrence suffices for M2 ’s occurrence. Hence, M2 is instantiated because P2 is. Since P2 ’s occurrence necessitates M2 ’s occurrence, there is nothing left the instantiation of M1 could contribute to M2 ’s occurrence. Thus, M1 cannot cause M2 .3 DR One assumption both versions of the exclusion argument require is that a cause’s instantiation contributes at least sometimes something to the occurrence of its direct effects. This assumption is highly plausible in the light of Occam’s razor, which states that one should assume theoretical entities (e.g., direct causal relations) only when they are required to explain otherwise unexplainable empirical facts. It will play a major role in the reconstruction of the 3I am indebted to Wlodek Rabinowicz for pointing out to me that one could conclude that P1 cannot be a cause of M2 by a similar argumentation, which even epiphenomenalists might find counterintuitive. However, the epiphenomenalist could solve this problem by interpreting constitution as a causal relation: She could then conclude that P1 cannot be a direct cause of M2 , but that P1 can be an indirect cause of M2 . P1 first directly causes P2 , which then directly causes M2 . Note that M1 —contrary to P1 —cannot even be an indirect cause of M2 . 6 causal exclusion argument in terms of causal Bayes nets. 3 Causal exclusion and causal Bayes nets In this section I reconstruct both versions of the causal exclusion argument and evaluate their validity on the basis of the empirically well-informed theory of AF T causal Bayes nets. I start with introducing important notions and the core axioms of the theory of CBNs. A causal model is a triple ⟨V, E, P ⟩ in which V is a set of variables, ⟨V, E⟩ is a directed acyclic graph (DAG) over V, and P is a probability distribution over V. The DAG ⟨V, E⟩ represents the modeled system’s causal structure, where Xi Ð→ Xj means that Xi is a direct cause of Xj (w.r.t. V). The set of all direct causes of a variable Xi in a causal model is called the set of Xi ’s parents Par(Xi ). The union of the set of all effects of a variable Xi (i.e., the set of all Xj with Xi Ð→ ... Ð→ Xj ) in a causal model and {Xi } is called the set of Xi ’s descendants Des(Xi ). A causal model’s probability distribution P represents the causal strengths of the causal influences propagated along the causal arrows. We define probabilistic DR dependence of a variable X on another variable Y conditional on a variable (or a set of variables) Z—Dep(X, Y ∣Z) for short— as P (x∣y, z) =/ P (x∣z) ∧ P (y, z) > 0 for some X-, Y -, and Z-values x, y, and z, respectively. X’s probabilistic independence from Y conditional on Z—Indep(X, Y ∣Z) for short—is defined as the negation of Dep(X, Y ∣Z), i.e., as P (x∣y, z) = P (x∣z) ∨ P (y, z) = 0 for all X-, Y -, and Z-values x, y, and z, respectively. The first axiom of the theory of CBNs is the causal Markov condition (CMC). A causal model ⟨V, E, P ⟩ satisfies CMC if and only if every variable Xi in V is probabilistically independent of its non-descendants conditional on its direct causes (Spirtes et al., 2000, p. 29). CBNs are causal models that satisfy CMC. 7 The DAG of a CBN determines the following Markov factorization: n P (X1 , ..., Xn ) = ∏ P (Xi ∣Par(Xi )) (1) i=1 Another important axiom is the causal minimality condition (Min). A CBN ⟨V, E, P ⟩ satisfies (Min) if and only if there is no CBN ⟨V, E′ , P ⟩ with E′ ⊂ E AF T (cf. Spirtes et al., 2000, p. 31). In other words: If deleting some causal arrow of the CBN’s graph would lead to a causal model that violates CMC, then the CBN is minimal, i.e., every arrow is required to prevent some (conditional) independence relation. This means that every arrow is responsible for some probabilistic dependence between the variables it connects. The idea that every causal arrow leaves some probabilistic footprints can also be expressed by the following productivity condition (Prod), which can be proven to be equivalent to Min for CBNs (Gebharter & Schurz, 2014, Theorem 1): Prod A causal model ⟨V, E, P ⟩ satisfies Prod if and only if Dep(Xj , Xi ∣Par(Xj )/{Xi }) holds for all Xi , Xj ∈ V with Xi Ð→ Xj . DR Recall from section 2 that causal exclusion arguments presuppose that a cause contributes at least sometimes something to the occurrence of its direct effects. This somewhat vaguely formulated requirement can be stated more precisely by means of a condition implemented in Prod above: A causal arrow Xi Ð→ Xj in a CBN is productive (or, equivalently, Xi contributes at least sometimes something to Xj ) if and only if Dep(Xj , Xi ∣Par(Xj )/{Xi }) holds, i.e., if there are some Xi - and Xj -values xi and xj , respectively, such that Xi ’s taking value xi makes a probabilistic difference for Xj ’s taking value xj w.r.t. some fixed context Par(Xj )/{Xi } = r. We can use this insight to test specific causal arrows for whether they are productive. Now we also see how Prod can be used to nicely reflect the assumption 8 (made in both versions of the causal exclusion argument) that properties can cause other properties only if they can contribute something to the occurrence of the latter. In causal Bayes nets terminology this assumption means to only allow for minimal CBNs, i.e., for CBNs that satisfy Prod. Or in other words: Only productive arrows can represent real causal relations. AF T Let us try to reconstruct both versions of the causal exclusion argument on the basis of the theory of CBNs next. To this end, let our CBN’s variable set V be identical to {P1 , P2 , M1 , M2 }. How does our CBN’s graph have to look like? We have to draw an arrow from P1 to P2 , since P1 is assumed to directly cause P2 . We also have to draw arrows from M1 to P2 and from M1 to M2 . These latter two arrows are the ones we want to test for productivity. But how should we represent the double-tailed arrows (indicating supervenience relationships) in our CBN? As Woodward (2014, sec. 1) remarks, there has been little discussion in the causal modeling literature on how to represent supervenience relationships (or other non-causal relationships). The answer to this question is not trivial. I will suggest an answer that requires that we take a closer look at how DR arrows work in causal Bayes nets. In particular, I will suggest to treat supervenience relationships similar to causal arrows in CBNs. In my argumentation I will not make use of intuitions about how interventions should work together with supervenience relationships. The reason for this is that there is still no consensus about that. It is still unclear whether directly intervening on mental properties is possible, or whether mental properties can only be manipulated by intervening on their physical supervenience bases, whether interventions on mental properties have to be common causes of these mental properties and their supervenience bases, etc. One of the goals of this paper is to implement supervenience dependencies in causal models to answer questions like these. Hence, I cannot let intuitions about how interventions should work together 9 with supervenience relations enter my argumentation for how to represent them in CBNs. A nice feature of CBNs is that, due to Equation 1, the conditional probabilities P (Xi ∣Par(Xi )) corresponding to the CBN’s arrows—these conditional probabilities are also called the CBN’s parameters—are stable in the sense AF T that they do not vary when one changes the prior distribution of some nondescendants of Xi . Let me illustrate this by means of the following simple example: Assume a CBN with DAG X Ð→ Y , where X and Y are binary variables. According to Equation 1, the CBN’s probability distribution factors as P (X, Y ) = P (Y ∣X) ⋅ P (X). Let us assume P (Y ∣X) and P (X) are specified as follows: P (x1 ) = 0.25 P (y1 ∣x1 ) = 0.75 P (x0 ) = 0.75 P (y0 ∣x1 ) = 0.25 P (y1 ∣x0 ) = 0.5 P (y0 ∣x0 ) = 0.5 Changing X’s prior distribution will not have an influence on the model’s DR parameters P (Y ∣X). It will, however, typically change conditional probabilities 0.75 ⋅ 0.25 P (y1 ∣x1 ) ⋅ P (x1 ) = = 0.3̇ 0.5625 ∑xi P (y1 ∣xi ) ⋅ P (xi ) P (x1 ∣y1 ) = 0.75 ⋅ 0.5 P (y1 ∣x1 ) ⋅ P (x1 ) = 0.6 = 0.625 P (y ∣x ) ⋅ P (x ) ∑xi 1 i i which are non-parameters. P (x1 ∣y1 ), for example, is such a non-parameter conditional probability. It can be computed as follows: P (x1 ∣y1 ) = (2) Now, if we change P (x1 ) from 0.25 to 0.5, for example, we get a different conditional probability P (x1 ∣y1 ): (3) The stability of a CBN’s parameters explained above corresponds to the intu10 ition that each subsystem of a CBN consisting of a variable Xi and its direct causes Par(Xi ) represents an autonomous causal mechanism (cf. Pearl, 2000, p. 22). Supervenience relationships, as assumed in the causal exclusion argument, seem to also possess this stability property. Recall that every mental property Mi is constituted by some physical property (or properties) Pi . This means that for every Pi -value pi there has to be exactly one Mi -value mi such that AF T P (mi ∣pi ) = 1 holds, where the conditional probabilities P (mi ∣pi ) = 1 cannot be changed by changing the prior distribution of some non-descendants of Mi . Moreover, the conditional probabilities P (mi ∣pi ) = 1 cannot even be changed by modifying the prior distribution of non-descendants of Mi in any possible expansion of our CBN. Otherwise Pi would not constitute Mi . Constitution is a metaphysical notion. If we would find that Pi does not determine Mi anymore when modifying the prior distribution of some non-descendants of Mi in an expansion of the CBN, we would conclude that we falsely took Mi as constituted by Pi . We would conclude that we found some kind of dependence of Mi on Pi that just looks like constitution in certain circumstances. DR There are (at least) two further reasons for treating supervenience relationships Pi Ô⇒ Mi like causal arrows in a CBN. The first one is that this represen- tation fits the idea of multiple realizability nicely, which typically goes hand in hand with the assumption that mental properties supervene on physical properties. For our causal diagram supervenience means that every Mi -value change has to lead to some probability change for some Pi -value. But the conditional probabilities P (Pi ∣Mi ) do not have to equal 1 or 0. They can vary when Mi ’s value is fixed. This directly corresponds to the multiple realizability intuition. Sometimes Pi can be changed while Mi is fixed, or in other words: There may be several instantiations of Pi that all constitute a certain instantiation of Mi . The last point speaking for treating a supervenience relationship Pi Ô⇒ Mi 11 similar to a causal relationship in a CBN is that micro properties are oftentimes understood as causes of macro properties. Friends of non-reductive physicalism could, for example, describe the temperature of a gas in a tank as the effect of the behavior of the gas particles in the tank, etc. Summarizing, we found some good reasons to treat double-tailed arrows AF T (Ô⇒) standing for supervenience relationships like a CBN’s causal arrows (Ð→).4 Thus, our CBN also has to feature a double-tailed arrow from P1 to M1 and from P2 to M2 . So the DAG of our CBN ultimately turns out to be the graph depicted in Figure 1, where the double-tailed arrows technically work exactly like single-tailed arrows. Note that I do not want to claim that supervenience is a special kind of causation here. I rather prefer to stay neutral on that question. My claim is that supervenience relationships, since they have the same formal properties as causal relations, can be modeled and formally represented in CBNs similar to causal relations. Woodward (2014) raised an objection from the perspective of an interventionist theory of causation to treating supervenience relationships like causal arrows. I illustrate his objection and defend my DR suggestion of how to model supervenience in CBNs in section 5. 4I would like to acknowledge that there may still be good reasons for not treating super- venience relationships like causal relationships in CBNs. So the adequacy of my suggestion is still debatable. Also note that within an interventionist framework (such as Woodward’s 2003) the consequences of the exclusion argument follow straightforwardly if one accepts that supervenience relations behave like ordinary causal relations in CBNs. Woodward (2014) is aware of this fact and no very elaborate further analysis is required to show this. (See section 4 for details.) I am indebted to an anonymous referee for this remark. However, within the CBN framework one would need to add something like Spirtes and Zhang’s (2011, p. 338) inter- vention principle to get these consequences similarly and in a straightforward interventionist fashion. The alternative way to do it consists in using the productivity test I introduced earlier. Using this productivity test has the advantage over introducing an additional intervention principle that it does not make the theory stronger than it has to be. 12 We now know how our CBN’s DAG has to look like. But how should we specify its probability distribution P ? From the considerations above we also already know that: Assuming the completeness of the physical domain means that there is a sufficient physical cause for every physical property. Physical properties are represented by the variables P1 and P2 in our CBN. We assume AF T P1 to represent P2 ’s sufficient physical cause, meaning that every P1 -value p1 is sufficient for P2 taking a certain value p2 . This probabilistic constraint that comes with assuming the completeness of the physical is labeled “physical completeness” below. Our next constraint comes with assuming that every mental property supervenes on some physical property. Mental properties are represented by the variables M1 and M2 in our CBN. M1 is assumed to supervene on P1 , and M2 is assumed to supervene on P2 . That Mi supervenes on Pi means in probabilistic terms that every value change of Mi has to lead to a probability change for some Pi -values. This constraint is labeled “supervenience” below. The last constraint comes with the assumption that mental properties are DR constituted by physical properties. We assume that P1 constitutes M1 , and that P2 constitutes M2 . Since the constituting property determines the constituted property, we have to assume for our CBN’s probability distribution that every value p1 of P1 determines M1 to take a certain value m1 , and that every value p2 of P2 determines M2 to take a certain value m2 . This constraint is labeled “constitution” below. Summarizing, our CBN’s probability distribution P has to satisfy the fol- lowing probabilistic requirements, where i ∈ {1, 2}: Physical completeness ∀p1 ∃p2 ∶ P (p2 ∣p1 ) = 1 Supervenience ∀mi ∀m′i ∃pi ∶ mi =/ m′i → P (pi ∣mi ) =/ P (pi ∣m′i ) Constitution ∀pi ∃mi ∶ P (mi ∣pi ) = 1 13 Now version (i) of the causal exclusion argument concludes that M1 cannot cause P2 , since P2 is fully determined by P1 and M1 has nothing to contribute to whether P2 occurs. We get the same result from our CBN. We can test the causal productiveness of the arrow M1 Ð→ P2 by checking whether Dep(P2 , M1 ∣Par(P2 )/{M1 }) holds. Let p1 be an arbitrarily chosen P1 -value. Due to the completeness of the physical for every p1 there is exactly one p2 such AF T that P (p2 ∣p1 ) = 1 holds, while P (p′2 ∣p1 ) = 0 holds for all p′2 =/ p2 . Now for every m1 there are two possible cases. Case 1: m1 and p1 are compatible, i.e., P (m1 , p1 ) > 0 holds. In that case conditionalizing on m1 will not change the conditional probabilities of p2 or p′2 given p1 , i.e., also P (p2 ∣m1 , p1 ) = 1 and P (p′2 ∣m1 , p1 ) = 0 will hold, meaning that no P2 -value depends on m1 conditional on p1 . Case 2: m1 and p1 are incompatible, i.e., P (m1 , p1 ) = 0 holds. It then follows from the definition of probabilistic independence introduced earlier that no P2 -value depends on m1 conditional on p1 . It follows that conditionalizing on p1 will render P2 independent from M1 . DR Since p1 was arbitrarily chosen, we can generalize this result: Conditionalizing on any P1 -value p1 will render P2 independent from M1 , i.e., P2 and M1 are independent conditional on Par(P2 )/{M1 } = P1 , meaning that the arrow M1 Ð→ P2 is unproductive. Since P1 is assumed to physically (or nomologically) determine P2 , this result can be generalized for every possible expansion of our CBN, meaning that M1 cannot cause P2 in any circumstances. Version (ii) of the exclusion argument says that M1 cannot cause M2 , since M2 is fully determined by P2 and, hence, M1 cannot contribute anything to whether M2 occurs. This claim is also provable within our CBN. Again, we can test the causal productiveness of the arrow M1 Ð→ M2 by checking whether 14 Dep(M2 , M1 ∣Par(M2 )/{M1 }) holds. Let p2 be an arbitrarily chosen P2 -value. Due to the fact that P2 constitutes M2 , there is exactly one M2 -value m2 for every P2 -value p2 such that P (m2 ∣p2 ) = 1 holds, while P (m′2 ∣p2 ) = 0 holds for all m′2 =/ m1 . Now for every M1 -value m1 there are two possible cases. Case 1: m1 and p2 are compatible, meaning that P (m1 , p2 ) > 0 holds. Then AF T P (m2 ∣m1 , p2 ) = 1 and P (m′2 ∣m1 , p2 ) = 0 will hold. Hence, no M2 -value depends on m1 conditional on p2 . Case 2: m1 and p2 are incompatible, meaning that P (m1 , p2 ) = 0 holds. From this it follows, again by the definition of probabilistic independence, that no M2 -value depends on m1 conditional on p2 . Therefore, conditionalizing on p2 renders M2 probabilistically independent from M1 . Recall that p2 was arbitrarily chosen. Hence, we can generalize our result: Conditionalizing on any P2 -value will render M2 probabilistically independent from M1 , meaning that M2 and M1 are independent conditional on Par(M2 )/{M1 } = P2 . Thus, the arrow M1 Ð→ M2 is unproductive. Since P2 is assumed to constitute M2 , this result can be generalized for every DR possible expansion of our CBN. This means, again, that M1 cannot cause M2 in any circumstances. The argumentation for the unproductiveness of the causal arrow M1 Ð→ M2 basically follows the same pattern as the one for the unproductiveness of the causal arrow M1 Ð→ P2 : In both cases we have a substructure X Ð→ Z ←Ð Y such that for every Y -value y there is exactly one Z-value z such that P (z∣y) = 1 holds, while P (z ′ ∣y) = 0 holds for all z ′ =/ z. For every Y -value y we distinguish the X-values x which are compatible with y from the ones which are not. Conditionalizing on compatible X-values in addition to y—this is case 1 above—will not change the probabilities of z and z ′ , meaning that also P (z∣x, y) = 1 and P (z ′ ∣x, y) = 0 will hold. So conditionalizing on a Y -value y 15 AF T Figure 2: Our productivity test reveals that the grey arrows M1 Ð→P2 and M1 Ð→M2 cannot propagate probabilistic dependence between the variables at their heads and tails. renders Z independent of those X-values x compatible with y. But Z will also be rendered independent of those X-values x which are incompatible with y—this is case 2 above. The latter follows directly from the definition of probabilistic independence and P (x, y) = 0. Figure 2 summarizes our findings in this section: The grey arrows M1 Ð→P2 and M1 Ð→M2 turn out to be unproductive, meaning that they cannot propagate DR probability between M1 and P2 or M2 . This result holds for every possible expansion of our CBN. If one accepts the plausible assumption that only those properties X are causes of a property Y , which have at least sometimes some probabilistic influence on Y (i.e., if one assumes Prod), then we have to delete the two grey arrows.5 Our result may be interpreted as empirically informed support for epiphe- nomenalism or as evidence against non-reductive physicalism: If causation is 5 Note that it is well-known that the causal minimality condition (and, hence, also Prod) may be violated in CBNs whose probability distributions feature deterministic dependencies (cf. Zhang & Spirtes, 2011). The unproductiveness of the causal arrows M1 Ð→ P2 and M1 Ð→ M2 in our causal exclusion CBN is basically such a violation of the causal minimality condition due to deterministic dependencies. 16 characterized by means of the causal Markov condition and the causal minimality condition, we assume that mental properties are non-identical to their physical supervenience bases, and that every physical property has a sufficient physical cause, then mental properties cannot act as causes for physical properties or as causes for other mental properties—they possess no causal power. This AF T result may, however, also be interpreted as evidence against non-reductive physicalism: By dropping the assumption that mental properties are non-identical to their supervenience bases we can restore mental properties’ causal efficacy. In that case we would not represent mental and physical properties by different variables in our causal model, and hence, our causal graph would become “flat”: M1 would be identical to P1 and M2 would be identical to P2 and we can, of course, have a productive arrow from M1 = P1 to M2 = P2 . 4 Consequences for the causal exclusion debate within the interventionist framework DR Shapiro and Sober (2007) and others (e.g., Raatikainen, 2010; Shapiro, 2010; Woodward, 2008) have recently argued that causal exclusion arguments are not valid in the light of an interventionist theory of causation such as Woodward’s (2003). This started a still ongoing debate within the interventionist framework about whether mental properties can (in principle) be efficacious causes of physical and other mental properties and about whether the causal efficacy of such mental properties can be accounted for on empirical grounds if the answer to the former question is a positive one. Within an interventionist theory of causation such as Woodward’s (2003), the efficacy of single causal arrows X Ð→ Y can only be tested by means of inter- 17 ventions.6 The method used for testing whether X Ð→ Y is causally efficacious is basically the same as the method for testing whether X is a direct cause of Y : One has to fix all elements of one’s set of variables V of interest different from X and Y by interventions and check whether Y would change when manipulating X (cf. Woodward, 2003, p. 59). Baumgartner (2010, 2009) convincingly showed AF T that one gets problems with this test for causal efficacy in the presence of supervenience relationships since P1 ’s value cannot be fixed by an intervention while M1 ’s value is changed by another intervention, simply because M1 supervenes on P1 . Hence, Woodward’s (2003) interventionist account would not only yield that M1 cannot be causally efficacious, but also that M1 cannot even be a direct cause of P2 or M2 . Baumgartner discusses several possibilities to modify Woodward’s (2003) interventionist theory to solve this problem, but concludes that none of these modifications would provide empirical evidence for mental properties’ causal efficacy, which is what the non-reductive physicalist wants. Later on Woodward (2014) made explicit that the version of his interventionist theory of causation he presented in (Woodward, 2003) was not intended to be DR applied to sets containing variables standing in non-causal relationships (such as supervenience relationships). For such variable sets Woodward (2014) proposes a modified interventionist theory that does not require to fix variables standing in non-causal dependencies to X or Y by interventions when testing for whether X is a direct cause of Y . He argues that this move in principle allows mental properties to be causally efficacious w.r.t. physical properties or other mental properties. 6 One may argue that Woodward’s (2003) interventionist theory does not exclude tests for whether X is a direct cause of Y based on observational data. But this would require additional principles, such as the causal Markov condition etc., not included in Woodward’s original interventionist theory. I take it that adding such principles would commit one to a CBN framework. 18 Baumgartner (2013) highlights the following shortcoming of Woodward’s (2014) modified interventionist theory: One consequence of this theory is that any intervention IM1 = iM1 on the mental property M1 has to cause both M1 and its supervenience base P1 over two different causal paths (M1 ←Ð IM1 Ð→ P1 ).7 So we get M1 as a direct cause of P2 if some intervention IM1 = iM1 leads to a AF T change in P2 . But can we also show that M1 Ð→ P2 is a productive causal relation—which is what the non-reductive physicalist wants—within Woodward’s modified account? The only possibility to test M1 ’s direct causal efficacy on P2 we have is to block all directed paths from IM1 to P2 different from IM1 Ð→ M1 Ð→ P2 by interventions and check whether some intervention IM1 = iM1 leads to a change in P2 . But we already know what happens if we carry out this test: Intervening on P1 freezes M1 to a certain value and no intervention IM1 = iM1 can lead to a change in P2 anymore. There is no other way to test whether M1 Ð→ P2 is productive within an interventionist framework. So it is, at least in principle, possible that the change in P2 associated with IM1 = iM1 is solely due to the causal path IM1 Ð→ P1 Ð→ P2 , while the arrow M1 Ð→ P2 is DR unproductive. Hence, though Woodward’s modified interventionist theory leads to the consequence that M1 is a direct cause of P2 , it cannot lend any support to the efficacy of M1 on P2 . One can formulate an analogous argument for the causal arrow M1 Ð→ M2 . So Woodward’s modified interventionist theory seems to have some kind of a blind spot when it comes to determining whether arrows exiting variables whose supervenience bases are also included in the variable set of interest are causally efficacious. However, one may argue (as Woodward, 2014 does) that the modified inter- ventionist account still renders causal exclusion arguments invalid, since it (at 7 Such common cause interventions are called fat-handed interventions in (Baumgartner & Gebharter, 2015). 19 least in principle) allows for M1 to be causally efficacious. It is at this point where we can enter the debate. We can say more about M1 ’s causal efficacy than Woodward can: The productivity test carried out in section 3 yields that both arrows are unproductive. So it turns out that both versions of the exclusion argument are valid when modeled in a CBN framework, while they are invalid AF T when applying Woodward’s modified interventionist theory of causation. Which consequences should we draw from this observation for our two versions of the exclusion argument? Recall from section 1 that both theories are treated as tools for providing information about the world’s true causal structure as well as the causal efficacy of properties on other properties. Such tools may be better or not so good in providing such information. In case the set of variables V whose causal structure should be analyzed contains variables standing in relationships of supervenience, it turned out that the theory of CBNs provides more information about the causal efficacy of some variables. It can be used for determining the causal efficacy of variables whose supervenience bases are included in V by means of the productivity test introduced in section 3, while DR Woodward’s (2014) modified interventionist theory keeps silent about the causal efficacy of such variables. But in evaluating the validity of the two versions of the exclusion argument we should use as much relevant information as possible. Hence, we should conclude that both argument versions are invalid within Woodward’s modified interventionist framework only because this framework does not come with a test for causal efficacy applicable to mental properties supervening on physical properties. But if we apply the CBN framework we get the relevant information that the arrows M1 Ð→ P2 and M1 Ð→ M2 are unproductive. Thus, we should conclude that both argument versions are valid. Now, as a final step, let us see what happens if we assume, as interventionists do, that there exists an intervention variable IM1 for M1 that is correlated with 20 P2 within our CBN representation. We add this intervention variable as a direct cause of M1 to our model. Now, since M1 Ð→ P2 is unproductive, we can conclude that IM1 can definitely not influence P2 over path IM1 Ð→ M1 Ð→ P2 . Hence, IM1 must influence P2 over another path. Thus, IM1 must be a direct common cause of M1 and of at least one of the variables P1 , P2 , or AF T M2 . Since P2 is fully determined by P1 (completeness of the physical) and M2 is fully determined by P2 (constitution), the argumentation pattern described in section 3 can be applied and our productivity test will lead to the result that arrows IM1 Ð→ P2 and IM1 Ð→ M2 would be unproductive. Hence, the correlation between IM1 and P2 can only be due to a causal path IM1 Ð→ P1 Ð→ P2 , and thus, IM1 must be a (direct) common cause of M1 and P1 such that IM1 Ð→ P1 is productive. So up to this point, we arrive at a consequence close to Baumgartner’s (2013). The main difference is that Baumgartner only showed that it is possible within the modified interventionist framework that IM1 influences P2 solely over path IM1 Ð→ P1 Ð→ P2 . We were also able to show that IM1 Ð→ P1 Ð→ P2 is the only path over which IM1 can be efficacious DR w.r.t. P2 . But we can say even more: The argumentation pattern used to show the unproductiveness of the arrows M1 Ð→ P2 and M1 Ð→ M2 in section 3 can also be applied to IM1 Ð→ M1 : Let p1 be an arbitrarily chosen P1 -value. Since M1 is constituted by P1 , for every P1 -value p1 there is exactly one M1 -value m1 such that P (m1 ∣p1 ) = 1, while P (m′1 ∣p1 ) = 0 holds for all m′1 =/ m1 . Now for every IM1 -value iM1 there are two possible cases. Case 1: iM1 and p1 are compatible, which means that P (iM1 , p1 ) > 0. If this is the case, then also P (m1 ∣iM1 , p1 ) = 1 and P (m′1 ∣iM1 , p1 ) = 0 will hold. Therefore, no M1 -value will depend on iM1 conditional on p1 . Case 2: iM1 and p1 are incompatible, i.e., P (iM1 , p1 ) = 0 holds. Then it 21 directly follows from the definition of probabilistic independence that no M1 value probabilistically depends on iM1 conditional on p1 . Therefore, M1 is independent from IM1 when conditionalizing on p1 . Since p1 was arbitrarily chosen, we can, again, generalize this result: Conditionalizing on any P1 -value will render M1 independent from IM1 , meaning that the causal AF T arrow IM1 Ð→ M1 is unproductive.8 This result can be generalized for every possible expansion of our CBN simply because P1 is assumed to constitute M1 and constitution is a metaphysical notion. Hence, IM1 cannot be a productive direct cause of M1 in any circumstances. Figure 3 graphically illustrates our findings, where single-tailed grey arrows, again, represent unproductive (possible) causal relations. For testing M1 ’s causal efficacy on P2 and M2 interventionists require interventions IM1 = iM1 on M1 which at least sometimes make a difference for M1 . But, as demonstrated, an intervention variable IM1 for M1 can only stand to P1 in a productive direct causal relationship. It follows that if causation is characterized by CMC and may object that it seems that the productivity test suggested will not work in case DR 8 One M1 and P1 are perfectly correlated, meaning that every M1 -value determines a certain P1 - value (with probability 1) and vice versa. In such a scenario one could ask why the arrow IM1 Ð→ M1 and not the arrow IM1 Ð→ P1 should be regarded as unproductive. If M1 and P1 are perfectly correlated, not only conditionalizing on P1 will render M1 independent of IM1 , but also conditionalizing on M1 will render P1 independent of IM1 . Here is my response: Who argues in such a way seems to have overlooked that the productivity test suggested also makes use of the system of interest’s underlying structure. For testing whether IM1 Ð→ M1 is productive we have, according to our productivity test, to check whether M1 depends on IM1 conditional on its parents different from IM1 , i.e., conditional on P1 . For testing whether IM1 Ð→ P1 is productive, on the other hand, we have to check whether P1 depends on IM1 unconditionally (since P1 does not have any parents different from IM1 ). We find Indep(M1 , IM1 ∣P1 ) and Dep(P1 , IM1 ) and conclude that IM1 Ð→ M1 is unproductive while IM1 Ð→ P1 is productive. 22 AF T Figure 3: The grey arrows indicate (possible) direct causal relations that cannot propagate dependence between the variables at their heads and tails. So P1 is the only variable in {M1 , M2 , P1 , P2 } that can be directly intervened on in a productive way. Prod (which seems to be reasonable when aiming at finding empirical evidence for causal relations M1 Ð→ M2 and M1 Ð→ P2 ), it is impossible to test for whether the arrows M1 Ð→ P2 and M1 Ð→ M2 are productive by means of interventions, even when allowing for common cause interventions IM1 of M1 and P1 . DR Note that IM1 , though inefficacious w.r.t. M1 over the arrow IM1 Ð→ M1 , can still be understood as an intervention on M1 (since IM1 might influence M1 over path IM1 Ð→ P1 Ô⇒ M1 ). So our findings still allow for a change in M2 or P2 induced by an intervention on M1 , and we can still interpret the experimental result that intervening on M1 leads to a change in M2 or P2 as evidence that by intervening on M1 we can bring about (or at least influence) M2 or P2 , respecitvely. All the causal work, however, is done by the path IM1 Ð→ P1 Ð→ P2 , and we are not allowed to interpret changes in M2 or P2 induced by an intervention IM1 = iM1 as evidence for the presence of an efficacious direct causal connection M1 Ð→ M2 or M1 Ð→ P2 , respectively. This will, of course, not distress scientists doing experiments too much, since 23 whether M1 or P1 does all the causal work will not make any difference for the experiment’s outcome. Summarizing, our findings strengthen Baumgartner’s (2013) results. It is not only the case that until now we do not know how to find empirical evidence for M1 ’s causal efficacy on P2 or M2 within an interventionist framework; rather AF T it seems generally (or theoretically) impossible that M1 has a causal influence on P2 or M2 . In addition, a common cause (or fat-handed) intervention IM1 for M1 and P1 cannot directly influence M1 . This means that attempts to render the causal effectiveness of mental properties on physical properties or on other mental properties plausible on empirical grounds within an interventionist framework seemed to be deemed to failure ab initio. 5 Woodward’s objection to treating supervenience relations like causal arrows In this section I defend the suggestion to treat supervenience relationships like DR causal arrows in a CBN (for which I argued in section 3) against an objection raised by Woodward (2014). Woodward’s objection is that treating supervenience like a causal relation would lead to absurd consequences and contradicts experimental practice. Woodward (2014, sec. 6) comes up with the following example to demonstrate this: Assume that high density cholesterol (HDC) and low density cholesterol (LDC) are both causes of having a heart disease (D). While high density cholesterol lowers the probability of heart disease, low density cholesterol raises the probability for heart disease. Let T C be a variable for total cholesterol that is defined as T C = HDC +LDC. Hence, T C will supervene on HDC and LDC. Now Woodward assumes that HDC, LDC, and T C are causes of D. If we want to represent all of these variables in a single CBN, this 24 AF T Figure 4: A CBN including D, T C, and T C’s supervenience base. Double-tailed arrows, again, stand for supervenience relations. CBN’s graph would—following my suggestion to treat supervenience relations like causal relations—look like the one in Figure 4. Again, double-tailed arrows are assumed to technically work exactly like single-tailed arrows. Now Woodward’s (2014) objection against treating double-tailed arrows like causal arrows roughly goes as follows: For testing whether LDC is a direct (and efficacious) cause of D, one has to fix all remainder variables by means of interventions and check whether intervening on LDC leads to a change in D. DR But when we run this test, then, since T C is defined as T C = HDC + LDC, we are not able to manipulate LDC when HDC and T C are both fixed by interventions. Thus, the interventionist theory of causation would tell us that LDC has no effect on D, and, moreover, that LDC is not even a cause of D. Similarly it can be shown that neither HDC nor T C would have an effect on D and that neither HDC nor T C would turn out to be causes of D within the interventionist framework. All of these consequences contradict the assumptions made above. Furthermore, treating supervenience relationships similar to causal arrows does not conform to experimental practice. No researcher would seriously consider to fix T C’s supervenience base LDC and HDC when testing T C’s causal efficacy w.r.t. D. According to Woodward, this would amount to double counting the effect of T C on D. Similar worries apply w.r.t. LDC and HDC. 25 Woodward (2014) interprets this observation as support for his claim that double-tailed arrows standing for supervenience relationships should not be treated like causal arrows and as motivation for modifying his interventionist theory of causation in such a way that it is no longer required to hold fixed variables stnading in non-causal relationships (such as supervenience relation- AF T ships) when testing for causal dependence and efficacy. But does Woodward’s observation really threaten my suggestion to treat supervenience relations like causal arrows in CBNs? I will argue that this is not the case. Actually, the problems Woodward describes arise only within an interventionist framework, but not within the CBN framework. Let me illustrate this by reconstructing the scenario described in the first paragraph of this section as a CBN. This CBN’s graph would, of course, again be the one depicted in Figure 4. Since the possibility to intervene on LDC and induce changes on D by means of this intervention when fixing HDC and T C by additional interventions is not required within the CBN framework for direct causation, we do not have to infer that LDC is not a direct cause of D. The same holds for HDC and T C. So we can DR avoid this problem. The second problem Woodward (2014) sees is that neither LDC, nor HDC or T C would turn out as efficacious w.r.t. D. Can we also avoid this problem? We can test each of the causal arrows LDC Ð→ D, HDC Ð→ D, and T C Ð→ D for productiveness in our CBN. If we do this, we find—by using the argumentation pattern described in section 3—that none of these arrows is productive, simply because every variable’s value is fully determined by the values of the other two variables. What should we make of this observation? First of all, let me emphasize that this situation is not so special that it can only occur in the presence of supervenience relationships. One can easily construct an equivalent (purely) causal model with different variables but with the same topological 26 structure and the same dependencies in which the double-tailed arrows are replaced by ordinary single-tailed causal arrows. More generally, the productivity test suggested in section 3 tells us that all of a variable’s causes are causally inafficacious if every one of these direct causes is fully determined by the other direct causes. What the productivity test would indicate in such a situation is AF T that at least one of the causal arrows should be deleted. (Recall from section 3 that the productivity condition is equivalent with the causal minimality condition.) After deleting one arrow, the productiveness of the remaining arrows may be restored. The same holds for our CBN. So which arrow(s) should we delete? If we delete LDC Ð→ D, then HDC Ð→ D and T C Ð→ D become productive in the resulting model. If we delete HDC Ð→ D, then LDC Ð→ D and T C Ð→ D become productive. And if we delete T C Ð→ D, then LDC Ð→ D and HDC Ð→ D become productive in the resulting model. Every one of these possible deletions of arrows would result in a causal model that still satisfies the causal Markov condition. If we delete more than one arrow, then the causal Markov condition would be violated DR (since the resulting graph would imply more probabilistic independencies than featured by our example). So, to account for all the (conditional and unconditional) dependencies among our four variables, we should only delete one arrow. Now we have two possibilities: We delete (i) one of the arrows exiting one of the variables of T C’s supervenience base, or we delete (ii) the arrow T C Ð→ D. If we decide in favor of (i), then we could ask ourselves why we should delete LDC Ð→ D rather than HDC Ð→ D (or vice versa). Which one of the two arrows we delete seems quite arbitrary. In addition, it would be strange to assume that the macro property T C and only one of its constituting properties is causally efficacious, while the other one is not. So it seems much more natural to decide in favor of (ii) and delete the arrow T C Ð→ D instead. If we do this, 27 then the resulting CBN gives us everything Woodward (2014) requested except that T C is causally efficacious w.r.t. D: LDC and HDC are causally efficacious w.r.t. D, and also T C-changes are associated with D-changes, simply because T C is constituted by LDC and HDC. Our CBN even mirrors scientific practice and provides the correct results AF T about the effects of interventions: An intervention on LDC, for example, can be modeled by adding an intervention variable ILDC , which is a direct cause only of LDC. Intervening on LDC corresponds to conditionalizing on one of ILDC ’s on-values and will have an effect on D. This effect solely arises due to the path ILDC Ð→ LDC Ð→ D. There is no double counting involved here. The same holds for an intervention on HDC: Every effect of an intervention IHDC = iHDC on D will solely arise due to the causal path IHDC Ð→ HDC Ð→ D. We can also handle an intervention on the constituted variable T C. Such an intervention could be represented by an intervention variable IT C . We add IT C as a direct cause of T C and assume that T C can be influenced by IT C . By means of the same argumentation pattern applied earlier in such situations, it turns out DR that the arrow IT C Ð→ T C is unproductive, since T C is constituted by LDC and HDC and, hence, determined by LDC and HDC. It follows that IT C must influence T C over another path. The only possibility to do so is over one of the paths IT C Ð→ LDC Ô⇒ T C or IT C Ð→ HDC Ô⇒ T C. So IT C has to be a common cause of T C and at least one of the variables LDC or HDC. Now a change of IT C ’s value may, of course, not only influence T C over one of these paths, but also D over one of the paths IT C Ð→ LDC Ð→ D or IT C Ð→ HDC Ð→ D. So, again, we get everything Woodward requested except the productive causal arrow T C Ð→ D. We can even say that an intervention on T C leads to a change in D, which might be something we find out in doing an experiment. But as in the case of the causal exclusion scenario, we should 28 AF T Figure 5: Grey arrows stand for (possible) direct causal relations that cannot propagate dependence between the variables at their heads and tails. IT C is a fat-handed intervention, i.e., a common cause of T C and at least one of the variables LDC or HDC. This is indicated by the dashed arrows. not read this as evidence for T C being causally efficacious w.r.t. D. The whole causal work is done by one or both of the causal paths IT C Ð→ LDC Ð→ D and IT C Ð→ HDC Ð→ D. But again, this should not be too cumbersome for the scientist. Whether the intervention is directly efficacious w.r.t. T C or effects T C only over one or both of its supervenience bases will not be of much interest to DR her, simply because it will not make any difference for the experiment’s outcome. Note that also testing whether D can be influenced by manipulating T C does not involve any double counting. Summarizing, the problems Woodward (2014) describes arise only within an interventionist framework and not when treating supervenience relationships like causal arrows in a CBN, as I have suggested in section 3. These findings are graphically illustrated in Figure 5. There may be other objections against my suggestion to treat supervenience relationships like causal arrows in CBNs. But as far as I can see such objections still wait to be discovered and formulated. 29 6 Conclusion In this paper I reconstructed two variants of the causal exclusion argument within the theory of CBNs. This seems promising since the theory of CBNs probably gives us the best grasp on causation from an empirical point of view we have so far. The reconstruction required to represent Kim’s (2005) diagram as AF T a CBN. Causal relations in Kim’s diagram can straightforwardly be represented by a CBN’s causal arrows. I argued that since relationships of supervenience behave exactly like causal arrows in CBNs, the double-tailed arrows standing for such relationships can be treated like ordinary single-tailed causal arrows in a CBN. The CBN’s probability distribution is constrained by the assumptions that P1 fully determines P2 (completeness of the physical), that every change of Mi ’s value leads to a probability change of at least one Pi -value (supervenience), and that Mi is fully determined by Pi (constitution). For this CBN it turned out that both causal arrows M1 Ð→ P2 and M1 Ð→ M2 are unproductive, meaning that they cannot transport probabilistic dependence. Because of the very nature of how P2 depends on P1 (completeness of the physical) and of how Mi depends DR on Pi (constitution), this result generalizes to all expansions of the CBN. Thus, both variants of the exclusion argument are valid under the proviso that causes contribute at least sometimes something to the occurrence of their effects. In section 4 I discussed the consequences of these findings for the discussion of causal exclusion arguments in the light of an interventionist theory of causation. Our findings strengthen Baumgartner’s (2013) criticism of Woodward (2014). Baumgartner concludes that it is unclear how one could provide empirical evidence for a mental property’s causal efficacy on physical properties within an interventionist framework. We could show that such a mental property M1 ’s causal efficacy on a physical property P2 or on another mental property M2 cannot be empirically supported at all, simply because causal arrows M1 Ð→ P2 30 and M1 Ð→ M2 are always unproductive, meaning that they do not imply any correlation between M1 and P2 and between M1 and M2 , respectively, in any circumstances. Moreover, it could be shown that it is generally impossible to have a causally productive direct intervention on a mental property, and thus, that attempts to investigate whether mental properties can be causally efficacious AF T within an interventionist framework were somehow unlucky from the beginning. In the last section of this paper I discussed an objection against modeling supervenience relationships similar to causal arrows raised by Woodward (2014). I argued that Woodward’s objection does not pose a threat to my suggestion of how to represent supervenience in CBNs. His objection works only within an interventionist framework. The problems he highlights do not appear in CBNs in which double-tailed arrows indicating supervenience relationships are treated similar to single-tailed arrows standing for direct causal dependencies. Acknowledgements: This work was supported by Deutsche Forschungsgemeinschaft (DFG), research unit FOR 1063. My thanks go to Christopher DR R. Hitchcock, Andreas Hüttemann, Markus Schrenk, and Gerhard Schurz for important discussions. Thanks also to Alexander G. Mirnig, Christian J. Feldbacher, Wlodek Rabinowicz, and an anonymous referee for helpful comments on earlier versions of this paper. References Baumgartner, M. (2009). 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