Is determinism more favorable than indeterminism for the causal Markov condition? Isabelle Drouet Abstract The present text comments on a paper by Daniel Steel, in which the author claims to extent from the deterministic to the general case the result according to which the causal Markov condition is satis�ed by systems with jointly indepen- dent exogenous variables. I show that Steel's claim cannot be accepted unless one is prepared to abandon standard causal modeling terminology. Correla- tively, I argue that the most fruitful aspect of Steel's paper consists in a realist conception of error terms and I show how this conception sheds new light on the relationship between determinism and the causal Markov condition. 1 Introduction Despite the stir that Spirtes et al. 1993 or Pearl 2000 caused in the philo- sophical community, it must be recognized that, in one way or another (see Williamson 2002 for a suggestion to that e�ect), Bayesian networks can con- tribute to infer causal knowledge from statistical data. Still they can be used for causal inference purposes only when the causal Markov condition (�CMC" from now onwards) holds. Roughly, this condition states that every phenomenon is probabilistically independent from all its non-e�ects conditional on its direct causes. Whether, or rather when, it holds remains a debated question. It is usually mentioned in favor of the CMC that it is true for deterministic systems with jointly independent exogenous variables.1Reciprocally, counterex- amples to the CMC often involve indeterminism. Now, Steel 2005 claims to establish that the determinism clause of the case for the CMC is unnecessary � that the CMC is true as soon as exogenous variables are jointly independent. If 1 correct, Steel's thesis should constitute an important contribution to the CMC debate, and to the larger debate on the possibility to infer causal knowledge from statistical data. In the present paper, I ponder the correctness of Steel's claim and the con- tribution of Steel 2005 to the CMC debate. A general presentation of the CMC- determinism issue is given in Section 2. Then, Section 3 sets out Steel's ar- gument. Section 4 assesses it, and leads to conclude that Steel's claim cannot be accepted unless standard terminology is abandoned. Nevertheless, Section 5 supports that Steel makes an interesting suggestion as to the representation of causal systems, and shows how this suggestion sheds new light on the relation- ship between determinism and the CMC. 2 Determinism and the CMC The CMC is the causal version of the more general Markov condition: De�nition 1 (Markov condition) Let V be a �nite set of variables, G a directed acyclic over V and p a probability distribution over V. (G, p) satis�es the Markov condition if and only if every variable in V is prob- abilistically independent from all its non-descendants in G conditional on its parents in G. Let us now consider a �system" � that is, in the present context, any group of causally interrelated phenomena. A simple example introduced by Halpern and Pearl (2005, 848) consists in a forest that can catch �re because of either lightning or a match lit by an arsonist. This system can be represented by: 1. the variables L, A, F respectively representing whether lightning strikes, whether an arsonist lights a match in the forest, and whether there is a forest �re.2More precisely, L is the binary variable which takes the value 1 if and only if lightning strikes and 0 otherwise, and so on for A and F ; 2. the directed graph L HHj F ©©¼ A representing the direct causal relationships among the variables L, A and F . This graph is �the causal graph" over {L, A, F} � or, equivalently, the causal graph for the system under consideration; 3. the probability distribution over {L, A, F}. More generally, any system can be represented by a set of variables and the causal graph and probability distribution over this set. Correlatively, the causal version of the Markov condition can be de�ned as a property of systems: De�nition 2 (Causal Markov condition) Let S be a system represented by the set of variables V, the acyclic causal graph CG and the probability distribu- tion p over V. S satis�es the causal Markov condition if and only if (CG, p) satis�es the Markov condition. 2 Let me introduce some vocabulary before I set out the usual result concerning the CMC and determinism: - a system S whose relevant observable aspects are represented by the vari- ables in V is: - acyclic if and only if CG is acyclic; - deterministic if and only if the values of the V variables with direct causes in V are functionally determined by the values of these direct causes; - the exogenous variables of S are the variables in V not having direct causes in V; - the variables in V are jointly independent if any two non-empty distinct subsets of V are probabilistically independent. Now the classic result beyond which Steel pretends to go can be stated: Theorem 1 (Classic result) Acyclic deterministic systems with jointly inde- pendent exogenous variables satisfy the CMC. Here is a proof for Theorem 1: Let S be an acyclic deterministic system represented by {V, G, p}. Let us assume that the exogenous variables in V are jointly independent. S satis�es the CMC if and only if any variable in V is independent for p of its non-descendants in G conditional on its parents in G. Then, let us consider any variable V in V and show that it is indeed independent of its non-descendants in G conditional on its parents in G. - if V is endogenous, then its value is functionally determined by the values of its parents in G. Therefore it is independent of any of its non-descendants in G when one conditionalizes on the set of its parents in G; - if V1 is exogenous, the set of its parents in G is empty. Therefore one must simply show that V is (unconditionally) independent of any of its non-descendants in G. Let V2 be such a variable. � if V2 is also exogenous, the independence of V1 and V2 stems from the hypothesis of joint independence of exogenous vari- ables; � if V2 is endogenous, by the determinism hypothesis, its value is determined by the value of the set of its direct causes in V. Beyond, and by acyclicity, this value is determined by that of a set of exogenous variables of S, say W. V1 does not belong to W since V2 is not a descendant of V1. Then the joint independence hypothesis entails that V1 is independent from W and therefore from V2. As a consequence, any variable in V is independent of its non-descendants in G conditional on its parents in G: S satis�es the CMC. 3 3 Steel's argument What is claimed in Steel 2005 is that the determinism clause in Theorem 1 is super�uous. In other words, a system would satisfy the CMC as soon as its exogenous variables are jointly independent. The present section sets out Steel's argument in favor of this claim. Steel's argument is stated in the framework of �causal functional models" (CFMs for short). Let me explain what they are by �rst introducing �functional models" (FMs). A FM is de�ned over a couple of sets of variables, say X = {X1, X2, . . . , Xn} and U = {U1, U2, . . . , Uk}. A functional model over (X, U) is (E, p) where: - E is a set of n equations such that each Xi appears as a function fi of a non-empty subset of ((X ∪ U) \ {Xi}); - p is a probability distribution over U. An example of an FM over ({X1, X2, X3}, {U1, U2}) is composed by the equa- tions: X1 = f1(U1, U2) X2 = f2(X1, U2) X3 = f3(X2) together with a probability distribution p over {U1, U2}.3Now Steel (2005, 9) describes (E, p) �causal" if 1) all the equations in E are causal generalizations and 2) any Xi in X is such that the set DirectCauses(Xi) of its direct causes in (X ∪ U) is included in (but not necessarily equal to) the set of its functional parents in (X ∪ U). Three remarks must be made before I can actually come to the result estab- lished by Steel: - as noticed by Steel himself (2005, 7), there is a �straightforward corre- spondence between functional models and directed graphs": for an FM M = (E, p) over (X, U), there is a unique directed graph GM over (X∪U) such that graphical parents in GM exactly correspond to functional par- ents in E. For our example FM, the corresponding directed graph is as follows: U1 - X1 ? X2 ? X3 ¾ U2 ¡¡ª If GM is acyclic (as is the case here), M as well as systems represented by M will themselves be labeled �acyclic"; - if M is a causal functional model, then GM is an over-graph of the causal graph CGM over DCM = ⋃ Xi∈X DirectCauses(Xi). As will be clearer soon, CGM is the causal graph for systems represented by M; 4 - if GM is acyclic, then the equations in E can be restated in such a way that X variables are functions only of the U variables from which they descend (Steel 2005, 8). As a consequence, p univocally extends to a probability distribution p′ over (U ∪ X), which itself univocally restricts to p′′ over DCM . In terms of CFMs and following the notations already introduced, the result established by Steel is the following: Theorem 2 (Steel's result) Let M = (E, p) be a CFM over (X, U). If GM is acyclic and variables in U are jointly independent for p, then (CGM , p′′) satis�es the Markov condition. In terms of systems4, it can be formulated as: Theorem 3 Let S be a system represented by the CFM M = (E, p) over (X, U). If S is acyclic and the variables in U are jointly independent for p, then S satis�es the CMC. A proof which di�ers slightly from the one given by Steel but makes it particularly clear what the rationale for the result is, is made up of: 1. a proof of the fact that (GM , p′) satis�es the Markov condition � in other words: a proof of the fact that (GM , p′) is a Bayesian network. This proof runs exactly as the proof that was given for Theorem 1 but with the role of the determinism clause played by functional determination; 2. a derivation of the probabilistic conditional independencies that are re- quired in order for (CGM , p′′) to satisfy the Markov condition. Those independencies are established by d-separation in the Bayesian network (GM , p′). The relationship between Steel's result and the determinism / indeterminism - CMC issue consists exactly in the following: CFMs can represent indetermin- istic as well as deterministic systems. In order to make it clear, Steel gives the following example: Imagine a special type of car, the quantum car. The ignition of the quan- tum car works by means of a fundamentally indeterministic process: when the key is turned, there is an irreducible probability of .85 that the car will start.(Steel 2005, 13) Then Steel (2005, 13) explains that the system constituted by the �quantum car" can be represented by a CFM M over ({X1, X2}, {U1, U2}), where - X1 is �a binary variable indicating whether the key is turned (X1 = 1 indicates that it has been)"; - X2 is �a binary variable representing whether the car starts (X2 = 1 indicates that it does)"; - (as far as I understand Steel's treatment of the example) U1 represents the reasons why the car may be started; - U2 is a binary variable whose possible values are 0 and 1 and which rep- resents whether the car starts once the key has been turned. 5 M is composed of the equations X1 = f1(U1) X2 = U2.X1 together with a probability distribution p over {U1, U2} which is such that p(U2 = 1) = .85. With p de�ned in this way, it becomes clear that U2 rep- resents the probabilistic nature of the action of X1 on X2. More generally, the example makes it clear how any indeterministic system can be represented by a causal functional model, with U variables representing the probabilistic na- ture of the action of indeterministic causes on their e�ects. Consequently, Steel considers that Theorem 2 implies that the CMC is true for any acyclic system with jointly independent exogenous variables, be it deterministic or not. This is what I bring into question. More precisely, I question neither the truth of theorem 2, nor the validity of Steel's proof, but rather the way Steel interprets it. 4 Assessment of Steel's argument Obviously, a necessary condition for Steel's result being interpretable in terms of any acyclic system with jointly independent exogenous variables sat- isfying the CMC is that the U variables of a CFM are the exogenous variables of systems represented by that CFM. This is assumed by Steel. Yet he has an hesitation when introducing them, referring to �a set of exogenous variables or error terms" (Steel 2005, 5). It seems to me that this hesitation is meaning- ful. But this can be justi�ed only after I have told the standard story about exogenous variables, error terms, and the way they di�er. The standard de�nition of exogenous variables was introduced in Section 2. One has 0) a system S, 1) a set V = {V1, V2, . . . Vn} of variables representing its relevant observable aspects, 2) a graph CG representing the direct causal relations on V. The exogenous variables of S are those variables in V that do not have any parents in CG. All this has already been stated. But from there on one can go one step further in the representation of S and consider a set of n equations such that each Vi constitutes the left-hand-side of exactly one equation and is a function of its parents in CG plus one variable Ti. This set of equations together with the probability distribution over T constitutes a representation of the system under consideration which is common in the �eld of causal modeling � see for instance Pearl's �causal models" (Pearl 2000, 27). Therefore this pattern of representation will be referred to as that of �standard causal functional models" (�standard CFMs" for short). The T variables of standard CFMs have a precise function: they enable a functional representation of the relations between V variables. Indeed, each Ti represents everything that contributes to the determination of the value of Vi and yet is not represented by the variables in V \ {Vi}. More precisely, and following Cartwright's analysis, T variables �represent, in one heap, measurement errors, omitted factors, and whatever truly probabilistic element there may be in the determination of an e�ect by its causes" (Cartwright 1999, 9). T variables are those usually called �error terms". This was the standard story. Let us now come back to Steel's one. Are his U variables �exogenous variables or error terms" (Steel 2005, 5)? Stated in other 6 words: what do Steel's U variables represent? It is clear from Steel's paper that U variables can represent two very di�erent kinds of things: - some of them, like U1 in the quantum car example, represent relevant observable aspects of the system under consideration that happen not to have direct causes in the system. Those variables are exogenous variables in the standard sense of the term; - the other ones, like U2, represent the way probabilistic causes act on their e�ects. Those variables are not exogenous variables since they do not represent observable relevant aspects of the system under consideration. Moreover, they represent one of the three kinds of in�uences that error terms in the standard sense represent. But they represent only one of the three kinds of in�uences that are represented �in one heap" by error terms. As a consequence, they are not error terms standardly speaking. This analysis easily accounts for the hesitation of Steel when introducing U: U is composite object whose elements either are exogenous variables in the standard sense, or represent part of what standard error terms represent. Consequently, U cannot be called �the set of exogenous variables" if one conforms to standard terminology. Stated in another way, Steel's result can be interpreted in terms of truth of the CMC for any acyclic system with jointly independent exogenous variables only if one abandons the standard pattern of functional representation of systems and the sense �exogenous variables" has in this context. Therefore it cannot be said without quali�cation that Steel established the super�uity of the determinism clause in Theorem 1. Omitting quali�cation, as Steel does, is seriously misleading. It could be that all this does not matter much. This would be the case in particular if Steel showed the way towards a proof of the truth of the CMC for any system with jointly independent exogenous variables in the standard sense). More likely, it may be that this can be proved in a way similar to the proof proposed for Steel's result. This seems all the more likely since the given proof-sketch is modular and has already been used to produce a proof of a result which is quite di�erent from Theorem 2 (Pearl 2000, 30). Yet a quick examination of the sketch reveals that one can obtain a �nal result in terms of joint independence of exogenous variables (standardly de�ned) only if one adopts a pattern of representation from which error terms are absent. But we saw that the �rst part of the proof relies in an essential way on the functional determination of the values of the e�ects by those of their direct causes. In the absence of error terms, such functional determination is exactly equivalent to determinism. In other words, the proof-sketch is no use for one who wants to establish that all acyclic systems with jointly independent exogenous variables satisfy the CMC. Of course this does not imply that the claim is false. What implies it by contrast is the fact that, under standard terminology, it remains possible for an acyclic system with jointly independent exogenous variables to fail to satisfy the CMC � provided it is indeterministic. This is the case of Nancy Cartwright's classic example: Cheap-but-Dirty employs a genuinely probabilistic process to produce the chemical [a chemical that is consumed in a given sewage plant]. The 7 probability of getting the desired chemical on any day the factory operates is eighty percent. [...] [Moreover] pollutants are emitted as a by-product whenever the chemical is produced.(Cartwright 1999, 7) Relevant observable aspects of the systems are represented by three binary vari- ables with value 0 or 1: O indicating whether Cheap-but-Dirty operates, S indicating whether sewage is produced, and P indicating whether pollutant is produced. It is easily seen that only one of them is exogenous in the standard sense: O. Hence, trivially, exogenous variables are jointly independent. More- over, the system is clearly acyclic. And yet the system does not satisfy the CMC since O does not screen o� P from S: p(P = 1, S = 1|O = 1) = .8, whereas p(P = 1|O = 1) × p(S = 1|O = 1) = .8 × .8 = .64. As a consequence, the result Steel claims to have established remains false under standard terminology. Acyclicity and joint independence of exogenous variables are su�cient for the CMC only if one accepts as exogenous variables, variables that represent the way probabilistic causes act on their e�ects. It could be argued that there is nothing wrong with this and, quite the opposite, that this modeling proposition constitutes the very innovation in Steel's paper. This is precisely the position adopted by Steel himself: �The basic insight is that exogenous variables in an FM can be interpreted either as representing causes or genuine indeterminism" (Steel 2005, 4). The matter is that variables representing the way probabilistic causes act on their e�ects already existed before Steel's paper, and that they were never called �exogenous variables". Conversely, �exogenous variables" and �systems with jointly independent exogenous variables" already had a meaning before Steel's paper, and this meaning is di�erent from the one they have in Steel 2005. Moreover, Steel does not give any independent justi�cation for substituting his interpretation of exogenous variables to the usual one. Then, his modeling innovation should not be accepted unless the associated terminology were carefully distinguished from the usual one and the author were careful not to claim to enter a pre-existing debate � two things that Steel fails to do. 5 What Steel's paper suggests I would like to end with more positive considerations. Indeed I think that Steel's paper actually contributes to the CMC debate, in spite of the di�culties hitherto highlighted. As the di�culties, the contribution lies in Steel's U vari- ables � and, to be more precise, in those U variables that are neither exogenous variables nor error terms in the standard sense of these terms. I have already stated that those variables represent only one of the three kinds of in�uences that standard error terms represent �in one heap", and this appeared to be problematic. But there is another way of looking at this: while standard error terms clearly fail to represent any real entity, Steel's U variables lean towards realism in the use of error terms � by which is meant that error terms are used in such a way that the structure of a causal model matches the real causal struc- ture it represents. More speci�cally, Steel's U variables suggest to disjoin the in�uences that are represented �in one heap" by standard error terms, and to represent distinct in�uences by distinct error terms. 8 Following this suggestion, to each variable V representing a relevant observ- able aspect of the system under consideration would correspond: - an error term representing the forgotten direct causes of what V repre- sents; - an error term representing the possible errors in the measurement of V ; - for each of V 's probabilistic direct causes, an error term representing the way it acts on V . These non-standard error terms lead to de�ne non-standard causal functional models. Given a system S whose relevant observable aspects are represented by V = {V1, V2, . . . , Vn}, a non-standard CFM for S is (E, p) where: - E is a set of n equations such that each Vi is the left-hand-side of exactly one equation and appears as a function of 1) its direct causes in V and 2) non-standard error terms Ti,1, Ti,2, . . . , Ti,m. Observe that m depends on which variable Vi is considered; - p is the probability distribution over the set T of non-standard error terms. Non-standard CFMs complete Steel's CFMs so as to ensure that everything standard CFMs represent is actually represented. This is enough for non- standard CFMs not implying the important rede�nition of usual terminology that Steel's CFMs did convey, and that was identi�ed as problematic. But as for the rest, the two patterns of representation are largely similar. First and fore- most, non-standard CFMs allows a characterization of determinism that relies on the very intuition conveyed by Steel's �De�nition of Deterministic Functional Models" (Steel 2005, 9): De�nition 3 (Characterization of Determinism) A system is determinis- tic if it is represented by a non-standard causal functional model with no error term representing the way a probabilistic cause acts on one of its e�ects. Then, as Steel's CFMs, non-standard CFMs are in �straightforward correspon- dence" (Steel 2005, 7) with directed graphs. Accordingly, they and the systems they represent will also be labeled �acyclic" when the corresponding directed graph is acyclic. Moreover, acyclicity of the non-standard CFM M = (E, p) over (V, T) remains su�cient for the probability distribution p over T to uni- vocally extend to p′ over (T ∪ V). Finally, and still as with Steel's CFMs, the graph GM corresponding to M is an over-graph of the causal graph for systems represented by M. Notice that this causal graph is now univocally determined by M (as the directed graph representing the direct causal relations amongst V variables that are depicted by E); therefore it can and will be noted: �CGM ". Does this tell anything new about the relationship between determinism and the CMC? Here, one may �rst notice that, under the notations that have just been introduced, the following result holds: Theorem 4 Let M = (E, p) be an acyclic non-standard CFM over (V, T) and p′′ the restriction of p′ to V. If variables in T are jointly independent, then (CGM , p′′) satis�es the Markov condition. 9 Theorem 4 is the equivalent, in the framework of non-standard CFMs, of Steel's result. It holds for exactly the same reasons and can also be stated in terms of systems: Theorem 5 Let S be a system represented by the non-standard CFM M over (V, T). If M is acyclic and variables in T are jointly independent, then S satis�es the CMC. By way of illustration, let us consider a classic (non quantum) car, by which is meant a car that starts each time the key is turned and there is petrol in the tank. Let S be this system. Three variables are needed in order to represent S: V1 representing whether the key is turned, V2 representing whether there is petrol in the car and V3 representing whether the car starts. Let M = (E, p) be the non-standard CFM representing S. Equations in E are as follows: V1 = g1(T1,1, T1,2) V2 = g2(T2,1, T2,2) V3 = g3(Y1, Y2, T3,1, T3,2) with T1,1 representing the omitted causes of V1, T1,2 the possible errors in the measurement of the value of V1, and correspondingly for V2 and V3. Given acyclicity of S, Theorem 5 states that a su�cient condition for it to satisfy the CMC is that the variables in {T1,1, T1,2, T2,1, T2,2, T3,1, T3,2} are jointly indepen- dent. Now, this condition can be re�ned. Indeed, and as already stressed by Cartwright (2001, 18) in another context, what is needed for a proof in the style of Pearl's one to be possible is only the joint independence of the �net e�ects" of error terms. In our example, �net e�ects" of error terms correspond to {T1,1, T1,2}, {T2,1, T2,2} and {T3,1, T3,2}. Then, following Cartwright, it is su�cient that any way of combining them into two non-empty sets is in two independent sets of variables. To be explicit, it is su�cient that: (1) {T1,1, T1,2}, {T2,1, T2,2} and {T3,1, T3,2} are pairwise independent; (2) {T1,1, T1,2} is independent from {T2,1, T2,2, T3,1, T3,2}; (3) {T2,1, T2,2} is independent from {T1,1; T1,2, T3,1, T3,2} and (4) {T3,1, T3,2} is independent from {T1,1, T1,2, T2,1, T2,2}. Joint independence of {T1,1, T1,2, T2,1, T2,2, T3,1, T3,2} is too strong condition for a proof following the sketch given above to be available. In particular it is not necessary for T1,1 to be independent from T1,2, for T2,1 to be independent from T2,2, or for T3,1 to be independent from T3,2. Stated in terms of a non-standard CFM M over (V, T), the proof only requires the following: for any two distinct non-empty V′, V′′ ⊂ V, the sets of error terms respectively corresponding to the variables in V′ and to the variables in V′′ are independent. More rigorously, and with ϕ(i) the number of error terms in M for Vi ∈ V, the result is as follows: Theorem 6 Let M = (E, p) be an acyclic non-standard CFM over V. If for any two V' = {Vi, . . . , Vj} and V� = {Vk, . . . , Vl} non empty disjoint 10 subsets of V, {Ti,1, . . . , Ti,ϕ(i), . . . , Tj,1, . . . , Tj,ϕ(j)} is independent from {Tk,1, . . . , Tk,ϕ(k), . . . , Tl,1, . . . , Tl,ϕ(l)}, then V satis�es the CMC. For simplicity's sake, let me refer to the antecedent of the implication stated by Theorem 6 as: �V satis�es JI". Now, in terms of systems represented by non-standard CFMs, we have: Theorem 7 Let S be a system represented by the non-standard CFM M. If M is acyclic and satis�es JI, then S satis�es the CMC. Under De�nition 3, Theorem 7 reveals a sense in which determinism is more favorable than indeterminism for the CMC. Indeed, while I agree with Steel writing that It is hard to see why deterministic causal systems would be more likely acyclic than indeterministic ones (Steel 2005, 16), I will explain why deterministic systems are more likely than indeterministic ones to have non-standard CFMs satisfying JI. To that e�ect, let me introduce a system S′ which is identical to the previously introduced S except for the fact that turning the key has �an irreducible probability of .85" of doing its job (exactly as in Steel's quantum example). S′ is represented by non-standard CFM M′ = (E′, q). Under previous notations, equations in E′ are: V1 = g1(T1,1, T1,2) V2 = g2(T2,1, T2,2) V3 = g3(T3,3 × V1, V2, T3,1, T3,2) with T3,3 a binary variable with possible values 0 and 1 and such that q(T3,3 = 1) = .85. As a result, M′ satis�es JI if: (1′) {T1,1, T1,2}, {T2,1, T2,2} and {T3,1, T3,2, T3,3} are pairwise independent; (2′) {T1,1, T1,2} is independent from {T2,1, T2,2, T3,1, T3,2, T3,3}; (3′) {T2,1, T2,2} is independent from {T1,1; T1,2, T3,1, T3,2, T3,3} and (4′) {T3,1, T3,2, T3,3} is independent from {T1,1, T1,2, T2,1, T2,2}. Now suppose that all these independencies hold. Then, the independencies stated by (1) to (4) all hold. This stems from the simple probabilistic following fact: for any two sets of variables Y and Z and any variable V , if Y∪{V } is in- dependent from Z, then Y is independent from Z. Then, if M′ satis�es JI, then M does too. Now, the converse is not true: {T1,1, T1,2} (for instance) can be in- dependent from {T3,1, T3,2} while not being independent from {T3,1, T3,2, T3,3}. These are no facts particular the chosen example. Indeed, the example makes clear that for any two S and S′ di�ering only by the probabilistic nature of some causal relations in S′, the non-standard CFMs M and M′ representing them di�er only by the absence from M of the non-standard error terms standing for indeterminism in M′. In such a case, M satis�es JI if M′ does while the converse does not hold. In this sense, non-standard CFMs representing deterministic systems are more likely than non-standard CFMs representing indeterministic systems to satisfy JI. To this exact � and acknowledgedly narrow � extent, determinism is more favorable a context than indeterminism for the truth of the CMC. 11 6 Conclusion Steel's claim to enlarge the usual result concerning the relationship between determinism and the CMC from the deterministic to the general case revealed unacceptable. More precisely, it was shown that Result 2 can be interpreted in terms of satisfaction of the CMC by any system with jointly independent exogenous variables only if one uses the expression �exogenous variables" in an uncommon way. Still, this way of using the expression suggested that error terms could be employed in a way more realist than the standard one. I de�ned the pattern of representation that stems from this suggestion, and explained how the determinism - CMC debate can take advantage of this new framework. Determinism was easily characterized in this framework and a sense in which determinism is more favorable than indeterminism for the CMC appeared: the su�cient condition we have for the truth of the CMC is such that if two systems di�er only by determinism, then the deterministic one satis�es the condition whenever the indeterministic one does � and the converse does not hold. Hinting at �non-standard causal functional models" and at De�nition 3 prob- ably constitute the most signi�cant contribution of Steel 2005 to the debate con- cerning the CMC in general, and its relationship to determinism in particular. But this de�nition, as well as the su�cient condition for the truth of the CMC that was derived from it, are valid only for systems represented by causal func- tional models � �non-standard" ones in the case in point. Hence, this de�nition and this condition are useful exactly in as much as causal functional models can represent real systems, and are there interesting exactly in as much as causal functional models can represent interesting real systems. Whether they can is a di�cult question that is not tackled here and would deserve more attention. 12 References Cartwright, Nancy (1999). Causal diversity and the causal Markov condition. Synthese 121(1): 3-27. Cartwright, Nancy (2001). What is wrong with Bayes nets?. The monist 84(2): 242-264. Halpern, Joe, and Judea Pearl (2005). Causes and explanations: A structural- model approach. Part I: causes. The British journal for the philosophy of science 56(4): 843-887. Pearl, Judea (2000). Causality. Models, reasoning, and inference. Cambridge university press. Spirtes, Peter, Clark Glymour, and Richard Scheines (1993). Causation, pre- diction, and search MIT press. Steel, Daniel (2005). Indeterminism and the causal Markov condition. The British journal for the philosophy of science 56(1): 3-26. Williamson, Jon (2002). Learning causal relationships. Technical report 02/02, LSE, CPNSS. 13 Footnotes 1. The claim is even broader, since it extends to �pseudo-indeterministic" systems. Yet these cases will not be discussed in the present paper. 2. These variables are meant to represent relevant observable aspects of the situ- ation under consideration. Still, other variables could be considered, giving rise to di�erent representations of one and the same system. This is not discussed in the present paper, but I will rather assume that the set of variables representing relevant observable aspects of a given system is univocally determined. 3. This example makes clear that there is no one-to-one correspondence between X and U variables of a given FM. This is an important di�erence between Steel's functional models and the more traditional causal models which will be consid- ered later on. 4. Steel does not make the distinction between systems and models representing them. This is unproblematic as long as one assumes that the correspondence between systems and models is functional, which he implicitly does. Yet for reasons that will become clear later on, I will keep systems distinct from models. 14