Causal Modeling and Transitivity

Transcription

1 Causal Modeling and Transitivity Joel Velasco Prelim Paper Submission Fall 2004

2 1 Traditional counterfactual views of causation make use of transitivity in solving problematic cases such as those of causal preemption. But apparent counterexamples to transitivity threaten those solutions. In The Intransitivity of Causation Revealed in Equations and Graphs Christopher Hitchcock argues that by using causal modeling methods, we can deal with cases of preemption without being forced to accept the transitivity of causation and thereby committing ourselves to unintuitive results. Hitchcock acknowledges that one of the many cases that he presents an example of a falling boulder due to Ned Hall (2004) might pose problems for his theory, but he feels that he has a good answer. Jim Woodward s Making Things Happen uses the same models method and defends Hitchcock s solution to the boulder problem. In this paper I will first attempt to clarify the method that they use to defend their solution and then argue that this solution cannot be generalized since two familiar examples of failures of transitivity cannot be accounted for on this theory. Examining causal switching cases will present us with an obvious fix to our theory, one which also resolves other failures of transitivity and turns out to be the perfect addition to the causal modelist s theory of causation. The basics of causal modeling What follows is an extremely simple introduction to causal modeling. It is intended to be just enough of an introduction so that we can advance to a discussion of the models that supposedly show failures of transitivity. For a much more complete, yet still simple explanation of causal modeling, see Hitchcock (2001). In each example we will produce a causal model consisting of a directed graph along with a system of structural equations. Formally, the causal model is an ordered pair (V,E) where V is a set of variables and E a set of structural equations relating these

3 2 variables. In the case of token causation, we will use events (or whatever your preferred causal relata) as the values of the variables in V. A directed graph is just a graph with the variables in V as vertices connected by directed edges that in our case will be arrows representing direct causal links between possible values of these variables. In this context, direct simply means unmediated by any other variable in V. In addition to the directed graph, we also have a system of structural equations in which each of the endogenous variables (a variable with an arrow into it) is written as a function of all and only its direct causes. Each function allows us to calculate the value of the endogenous variable from the values of its direct causes. Since we are assuming a deterministic framework, there is no need for a probability measure over variables or for the addition of error variables, both of which are commonly used with the same framework in the case of indeterministic causation. The values of the variables will represent various states of that variable. In ordinary cases, the only state we care about is whether or not it occurred and thus, we can take the values of the variables to be true and false I will use 1 and 0. If the variable X has the value 1, then X did occur. If X has the value 0, it did not occur. One of the advantages of these models is that the methods are easily adaptable to using non-binary variables (we could, for example, have a variable representing the mass of some object). In the case of exogenous variables (variables with no arrows pointing into them), we will simply set them equal to their actual values. For example, here is a directed graph: with the accompanying structural equations: A B C A = 1; B = A; C = B

4 3 This model represents A as actually occurring, B as directly caused by A, and C as directly caused by B. These equations are not identity relations in the ordinary sense. They tell us which variables are dependent on which others. They are uniquely read left-to-right as dependent-toindependent. For example, in the case of C=B, if we set the value of B to 0 (by causing B not to occur), the value of C will change to match it since C depends on B. If however, we set the value of C to 0 (by some form of external manipulation), then the value of B will not change correspondingly because B does not depend on the value of C, but rather depends only on the value of A. The structural equations in the case of token causation are just a way of encoding all of the basic counterfactual truths of the situation. They form the fundamental equations of the system from which all other counterfactuals can be derived. As in Lewis-style causal theories, all counterfactual judgments will be made in a non-backtracking way. We judge what would happen if B did not occur by setting the value of B to 0 and then solving the system of equations. When written correctly, these equations allow us to compute exactly what the value of any given variable would be under changes of various sorts. Philosophers of causation are likely to be familiar with neuron diagrams that may appear to be basically the same as these causal models and are capable of encoding quite complicated information (cf. Schaffer 2000). But these are traditionally limited in certain ways. Compare the previous causal model to the following one that is not amenable to neuron diagrams (at least in the way they are ordinarily portrayed): B A C A = 1; B = A; C = A & B

5 4 Here I use the truth-functional & to mean that C has the value 1 if and only if both A and B have the value 1. I will also use and v with their obvious interpretations. In the above model, A has an effect on C above and beyond that which it has through its link to B. It has both direct and indirect influence on C. In order for C to occur, both A and B have to occur. This may seem to be just the same as A occurring since B occurs when A does, but this model encodes counterfactual information such as If A had not occurred, but B were artificially made to occur, C would not occur and If A occurred, but the value of B were artificially set to 0, C would not occur. I mention setting variables to some given value artificially, and the reader may recognize that I am referring to what is called an intervention in the literature. Entire causal theories such as Woodward s manipulability view of causation (cf. Woodward 2003) are built upon the notion of interventions. For now, what is important is that when we judge counterfactuals, we must imagine the value of the antecedent variable being 1 (or 0) somehow independently of the value of its causes that are represented in the model. How to judge causal claims Now we have an obvious way of testing for counterfactual dependence: B depends on A if and only if some possible change in the value of A (through an intervention which does not affect any other variables directly) changes the value of B. But it is well known that we cannot simply declare that A causes B if and only if B counterfactually depends on A for then we run into familiar cases of preemption. Let s look at a standard case, Backup : Assassin 1 (A1) shoots and kills Victim. Assassin 2 (A2) witnesses the whole thing and does nothing. However, had A1 not shot, A2 would have shot and Victim would have died just the same. It is clear that A1 s shot caused the death. However, had it not occurred, the death still would have occurred, as A2

6 5 would have shot. Therefore the death does not counterfactually depend on A1 s shot. The Lewis-style standard answer is to define causation in terms of dependence* the ancestral of the dependence relation and then to interpolate a variable B representing the existence of a bullet in flight heading toward Victim. Had this (B) not occurred then A1 still would have shot (since we are not allowing backtracking) and A2 also would not have shot and so Victim would not have died. Since B causes the death and A1 s shot causes B, by transitivity (enforced by using the ancestral of a relation) we have that A1 s shot causes the death. On Hitchcock s modeling theory, we examine the obvious model for this scenario: A2 A1 D A1 = 1; A2 = A1; D = A1 v A2 Here the variable A1 represents Assassin 1 s shooting, A2 represents Assassin 2 s shooting and D represents Victim s death. In the actual case, it is easy to see that the solution to this system of equations is A1 = 1, A2 = 0, and D = 1. In the counterfactual scenario of A1 = 0, A2 would take the value 1 and the death (D=1) would still occur. So far everything is correct, but now we run into the problem of wanting to have A1 be a cause of D, but we have no counterfactual dependence between D and A1. Rather than defining causation in terms of the ancestral of dependence and thereby being forced to accept transitivity, Hitchcock s answer is that while it is important to a theory of causation that we use non-backtracking counterfactuals, it is also important to recognize the use of certain explicitly non-foretracking (ENF) counterfactuals. An example of this would be If c had not occurred, but d had occurred anyway, then Hitchcock remarks that this is similar to

7 6 a move that Lewis makes in Finkish Dispositions and a similar requirement is found in recent pieces by Stephen Yablo (2002, 2004). In Backup, we want to make use of the following ENF, If A1 had not shot and A2 still had not shot, then Victim would not have died. The truth of that ENF points to the existence of an active causal route between A1 and D. The official definition of an active causal route is the following: Act : The route <X, Y 1,, Y n, Z> is active in the causal model (V,E) if and only if Z depends counterfactually upon X within the new system of equations E constructed from E as follows: for all Y V, if Y is intermediate between X and Z, but does not belong to the route <X, Y 1,, Y n, Z>, then replace the equation for Y with a new equation that sets Y equal to its actual value in E. (If there are no intermediate variables that do not belong to this route, then E is just E.) This means that to check whether the route between X and Z is active, hold fixed the variables not on that route at their actual values, and then check whether or not Z depends on X. If it does, then the route is active. Now we have the official definition of cause: Hitchcock s definition of cause: Let c and e be distinct occurrent events, and let X and Z be variables such that the values of X and Z represent alterations of c and e respectively. Then c is a cause of e if and only if there is an active causal route from X to Z in an appropriate causal model (V,E).

8 7 The definition mentions an appropriate model so it is important to spell out what this means. Hitchcock says that there are at least three requirements for a model to be appropriate. The first two are that: 1) The equations in E must entail no false counterfactuals, and 2) they must not represent counterfactual dependence relations between events that are not distinct. These first two requirements are there for obvious reasons. The third requirement is that V should not contain any variables whose values correspond to possibilities that we consider to be too remote. We will discuss this third requirement at length later in the paper. For those who are concerned with broader issues, this is a useful place to point out that our models make no real distinction between events and omissions. That is, that fact that a variable has the value 1 or 0 (or some other number) is totally irrelevant to how it is treated. Our definition allows variables whose values are 0 to cause things, variables to cause other variables to have the value 0, and allows for the variables to represent basically any sort of thing that we like. For example, in Backup we could just as easily use a model with S for Victim survives instead of D for Victim s death and change the equation to S = A1 & A2 and we would get the same results about causation (treating S as D). Thus the model theorist is typical of the counterfactual theorist in treating omissions as events that can enter into causal relations. Now that we have our definitions in place, we can get to work checking for causation. In the case of Backup, to check whether A1 causes D, we ask if there is an active causal route between A1 and D. There are two possibilities to check here, A1 D, and the longer route, A1 A2 D. While the longer route is not active, the direct path is. A1 D is an active causal route because holding fixed the variables not on this route at their actual values (A2 = 0) we see that D does depend on A1. This is the ENF counterfactual had A1 not shot, then given that A2 didn t shoot, Victim would not have died which is just the one we wanted. Hitchcock points

9 8 out that if you want to model this scenario using B representing the existence of a bullet in flight from A1 to Victim it would look like this: A2 A1 B D A1 = 1; A2 = A1; B = A1; D = B v A2 On the current account, this is perfectly acceptable as now the causal route A1 B D is active. Hitchcock claims that it is a virtue of his theory that consideration of the esoteric and hard-tofind variable B is unnecessary, although harmless. In this case, it seems we should agree. One does not need to carefully consider the event of the bullet in flight to realize that A1 s shot is a cause of the death. Hitchcock also points out that the model theory s answer would work equally well if the assassin s guns did not work by firing bullets, but rather by some sort of unmediated action at a distance; again a benefit of our method. It seems as though the models method does work admirably well for basic preemption cases, but it is also important that it can give us the proper result in cases of intransitivity. 1 Hitchcock and Woodward believe that it can. I will attempt to clarify the solution that they suggest and then argue that, in general, it will fail to get the results that we want. The Boulder example Here is a (purported) example of intransitivity from Hall (2004): Boulder : a boulder falls causing a Hiker to duck. If he had not ducked, he would not have survived. Our intuitive judgments about this case are that although the fall does cause him to duck, and ducking does cause him to survive, the fall did not cause him to survive. Thus the failure of transitivity.

10 9 Hitchcock s answer (and Woodward s) is to model Boulder like this (F is for the fall of the boulder, D is for the hiker ducking, S is for the hiker surviving): D F S F = 1; D = F; S = Fv D In this scenario, if we hold D=1 fixed, we see that F S is not an active route. On the other hand, F D S is not active either for here there is nothing to hold fixed and S does not depend on F. So on this model, F is not a cause of S. This is the judgment that we were aiming for. What Hitchcock and Woodward rightly point out is that F D S is only inactive because of the lack of any intermediate variable (unrelated to D) on the F S chain. If we interpolated a variable, say the existence of a boulder two meters above the hiker s head, then holding that fixed, F D S is indeed active because if we drop F, we drop D and then given that there is such a boulder above his head, the hiker would not survive. Adding the variable B, our model would look like this: D F B S F = 1; D = F; B = F; S = B v D In this model, the falling boulder is a cause of the hiker s survival. This is obviously problematic as the definition of cause implies that something is a cause if there is an active causal route in any appropriate model. Hitchcock s response is that adding B to the diagram makes this an inappropriate model for the situation as the existence of an active causal route between F and S points us to ENFs of the sort, If the boulder had not fallen and there is a boulder 2 meters above

11 10 Hiker s head, whose antecedents are not possibilities that we should take seriously. As he says, We included the variables F and D in our original model of Boulder. This choice reflects our willingness to take seriously the possibility that the boulder does not fall, and the possibility that Hiker does not duck. Moreover, it reflects our willingness to take seriously the possibility Hiker does not duck even though the boulder falls. [However] we are not willing to take seriously the possibility that the boulder (or a boulder of similar size and shape) comes to be in that position even though the boulder does not fall in the first place. This possibility is just too far-fetched. (Hitchcock, 2001, pg. 298) So the official answer is that any model that includes B is not an appropriate model to use in testing for the existence of an active causal route. Examples like this one display the importance of the third requirement in our definition of appropriate model. While they have not provided an explicit guiding principle that spells out which variables in a given situation we are allowed to use in V, we can try to develop conditions based on two facts. D would have been inappropriate if not for the fact that it could reasonably have had a different value than it did, even though its direct cause F still occurred, and B is not appropriate because it couldn t reasonably retain its current value if the value of its direct cause had been different. It seems as though we are roughly working with something like the following: For each variable X in V, it must be the case that it is reasonable to imagine that the variable X has its value independently of whether or not its direct cause(s) occur. When do we have an inappropriate model? A few quick thoughts can convince you that this is much too strong of a condition. In fact, this guide to the acceptability of variables in appropriate causal models would be inconsistent with other models that they accept. Look at an example of late preemption, Bottle : Billy and Suzy throw rocks at a bottle. Suzy s rock gets there first shattering the

12 11 bottle. Billy s rock sails through the empty space where the bottle used to be. Halpern and Pearl (2001) model this case with BT for Billy throws his rock, ST for Suzy s throw, BH for Billy s rock hits the bottle, SH for Suzy s rock hits the bottle, and BS for the bottle s shattering. BT BH BS ST SH BT = 1; ST = 1; SH = ST; BH = BT & SH; BS = BH v SH. In this model, we have correctly represented that Suzy s throw is a cause of the bottle shattering but Billy s throw is not. Hitchcock does not display this solution, but he does mention and endorse it. (Hitchcock 2001, pg. 289) In addition, he expresses the importance of the following ENF counterfactual to this case: given that Billy s rock did not hit the bottle, if Suzy had not thrown, the bottle would have remained intact throughout the incident. At the very least, this seems to require the existence of the variable BH in any appropriate model. As Halpern and Pearl point out, it is essential that we have intermediate variables for Billy s rock hitting the bottle and Suzy s rock hitting the bottle. If we have just the throw variables along with the bottle shattering, this case cannot be distinguished from the case where Billy s rock hits first or the case where they hit at the same time. In the above model, it is holding BH = 0 fixed that makes ST SH BS an active route. To do this, we need the variable BH in addition to BT. But does this mean that we should take the counterfactual If BH & BT then seriously? No, of course not. Yet Hitchcock is clear

13 12 that we can somehow objectively determine which variables are allowed in an appropriate causal model simply by examining which possibilities we intend to take seriously. This shows that the proposal that we eliminate variables whose values could not in every case be reasonably be different than their direct causes would indicate is too strong. We can t eliminate all variables of that type from our models. To justify this, we need some difference between the variable B in Boulder and BH in Bottle. The obvious move to make here for the model theorist is to argue that different variables are required for events that might reasonably come apart. In the case of the boulder in the air, that variable will be 1 iff the boulder actually falls. No reasonable alterations to the story will destroy that link. On the other hand, Billy s throw and Billy s rock hitting the bottle are separable in at least one way. While it is unreasonable that Billy s rock hits the bottle even though he doesn t throw it, it is perfectly reasonable to imagine that Billy s rock doesn t hit the bottle even though he does throw it. In fact, this is what happens in the actual case. In the Boulder case, F=1 iff B=1 and any sort of manipulation to change this biconditional would have to drastically alter the causal setup of the story. In the Bottle case, there is no tight connection between BT and BH. Given that BT = 1, BH could easily be 1 (if SH had been 0) or it could be 0 as it is in the actual case. Incidentally, it is slightly harder to justify the variable SH in the model, though I think that our going theory can do so. We can argue that ST is not tightly connected to SH. Even though given the actual causal setup, SH = 1 iff ST = 1, it could have easily been otherwise for example, if Billy had thrown slightly faster and his rock had arrived first. In that case, SH could equal = 0 even though ST = 1. True, this is not part of the causal setup, but neither is the hiker not ducking even though the boulder falls. Yet it is clear that this should be thought of as a reasonable possibility in Boulder (as Hitchcock argues) and so it is

14 13 justifiable to use a variable D for the Hiker ducking, even though in the actual setup, D = 1 iff F = 1. Similarly, it is justifiable to use different variables ST and SH because in a slight alteration of the story, the tie between the values of these variables could come apart. Stated more precisely, the new proposal would be something along the lines of not allowing multiple variables whose values are too tightly connected for any reasonable intervention to break. In other words, it isn t that it is required that the effect be able to reasonably come apart from the cause in both the case where the cause occurs and when it doesn t, but that it could come apart in at least one of these two cases. In the case of Boulder, this allows us to justifiably eliminate the intermediate variable B, while in the Bottle case, both SH and BH are required as either could easily be separated from their respective partners ST and BT. It is interesting to note that requirement of appropriateness now forces our model in Backup not to include the variable B representing the bullet flying through the air. Hitchcock argues that it is ok, although harmless, to add the variable B to our model. Yet it seems clear that this variable is exactly analogous to the variable B in Boulder. It is just as unreasonable to imagine a bullet in flight from A1 to Victim even though A1 never fires as it is to imagine the boulder in a similar situation. Yet it is supposed to be a benefit of Hitchcock s theory that we can use such variables if we want. Given their answer to Boulder, it would seem that Hitchcock and Woodward would have to admit that it would be inappropriate after all to include B in Backup. Perhaps this is not so bad, but it is not worth squabbling over, as there are much more damaging examples to the models theory we have yet to examine. Failures of transitivity through complex routes

15 14 Even if we can give good arguments for why we should not include the variable B in the model for boulder, the model theorist faces the obvious problem of what to do in situations where it obviously is appropriate to add an intermediate variable. Boulder was an example where the fall of the boulder posed a fairly direct threat to the hiker s survival. But what if the path of influence went through a chain of events? Abstractly, it is clear such cases can occur. Imagine a button that when pushed has two effects. One is a threat to someone s life, and another provides a counter to that very threat. The button is pushed and the victim survives. Without even knowing much about this case, it seems clear that it will be a problem for the modeling theory as there will be an active route from the button pushing to the survival through the counter because we hold fixed the existence of the threat. In a later paper (Hitchcock, 2003) Hitchcock himself provides what he feels might be a counterexample to his theory (he states that his intuitions on the case are not clear). In the case that he gives, Captain and Assistant are on a mission to kill Victim. Upon spotting Victim, the Captain yells Fire! and the Assistant fires. Overhearing the order, Victim ducks and survives unscathed. The question is whether or not the Captain s order caused Victim s survival. The obvious model to use is D C A S C = 1; D = C; A = C; S = A v D I will call this story and the accompanying model Captain. Notice that this model is isomorphic to Boulder with the bad variable B replaced with A. As we know from that situation, in this model, C is a cause of S. Hitchcock reports various people as having conflicting intuitions in this case. I agree that it is problematic, but I think that most of the confusion can be attributed to trying to imagine what the situation would be had Captain not given the order to

16 15 fire. If they were really hunting for Victim, then it is possible that Assistant would have fired on his own when seeing Victim and if so, the loud call of fire did indeed save his life. Similarly, the order could be a cause if it was a likely alternative that the Captain simply signaled with his hand or had spoken more softly. On the other hand, if Victim s life was only in danger because of the Captain s order, and in all other reasonable alternatives the victim would have survived, then we should think of this case just like Boulder. In order to show that, I propose to change the scenario to one where the Captain is simply a crazed lunatic with no obvious plans or intentions or at least none that Assistant is aware of. As far as Assistant is concerned, they just happen to be wandering around a random town. They come across a complete stranger and for whatever reason, the Captain decided to yell, Kill him! and the assistant dutifully attempts to carry out the order by firing his gun. As in the original scenario, the hapless target hears the order and manages to scramble out of the way in time and run away. In this scenario, I feel that it is clear that the man s life was only placed in any danger because of the order and so we should not credit the order with saving his life. Yet the model for the situation seems as though it should be exactly the same. This reveals how insensitive models can be to contextual shifts. At any rate, I take it that we now have a counterexample to our theory since it is obviously essential that we add the variable A for the assistant firing for it is that and not just the Captain s order that places Victim s life in danger. Yet when we fix that A=1, the Captain s order is a cause of Victim s survival because without it, Victim could not have heard it and therefore would not have ducked and would not have survived.

17 16 Switching cases In addition to these kinds of cases, there are other kinds of cases that philosophers have cited as examples of failures of transitivity. One common type of case is that of switching. In these cases, the effect Z is going to happen through one of two causal processes and some event X determines which of the two processes occurs. The obvious model is something like this: B A C D B = A; C = A; D = B v C I will call this generic model Switch. In this generic model, I didn t include a value for A. The point is that if A were not to happen, that would cause C and C would cause D. If A did occur, it would merely turn off C and activate B which would cause D. If A does happen, it seems only to be a switching event, not a genuine cause of D. I give the case here abstractly because actual cases are often argued over. One possible example is again from Hall (2004): You are standing at a switch in the railroad tracks. Here comes the train: If you flip the switch, you ll send the train down the left-hand track; if you leave it where it is, the train will follow the right-hand track. Either way, the train will arrive at the same point, since the tracks reconverge up ahead. Hall uses this as a purported counterexample to transitivity since one might argue that your action of flipping the switch is not a cause of the trains arrival. 2 Perhaps another example is a switch that when up, allows a current to flow through circuit 1, lighting a light bulb. However, if the switch is down, the current is redirected to circuit 2 that ends up lighting the same light bulb. It seems to me that if the switch is down and the light is on, if Bob were to walk by and flip the switch up, we should not say that his flip caused the light to be on. After all, the light was already on before he even bothered to flip the switch. Now we have at least two types of

18 17 problem cases (switching and complex routes) with seemingly different structures. However, I will now point out a single fix that can cover both of these types of cases. The problem with switching The problem with calling A a cause in the switching example is that it has just as much of a claim to be a cause if it doesn t occur. That is, there is a sense in which it just doesn t matter whether A occurs or not. Given our model of switch, if A = 1, it would be a cause of D and if A=0, A s non-occurrence would be a cause of D. For many philosophers, this is problematic. The fact that we want to exclude as causes variables whose values would be a cause of the same effect regardless of what value they had suggests the obvious addition to our definition of cause: Not only must there be an active route from X to Z as in our previous definition, but there must be a possible value x of X such that if X were set to x, either Z would change its value or there would no longer be an active route from X to Z. This addition solves our problem in Switching. Imagine that A=1. Then A is a cause of D because there is an active causal route from A to D. But if it had been the case that A=0, then A s not occurring would have been a cause of D as there would still be an active route from A to D, it would just be a different route. Since there are no other possible values of A, our addition implies that A is not a cause of D. Of course our new addition to the definition faces at least two obvious questions that need to be answered. 1) This may solve our dilemma in the case of switching, but is it general enough to solve all of the flaws of our original definition? 2) Isn t this proposal just an ad-hoc solution due to a particular example and so is not a real solution?

19 18 I intend to argue that the answer to 1) is that yes, this solution is general enough to cover all of our problem cases (at least the ones we know of) while not creating any new problems of its own, and furthermore allows us to almost always get around the question of admissible variables. The answer to 2) is much harder. I will briefly say something about this to wrap up. Is this sufficiently general? In the case of Switching we have seen how our solution works. But one might not expect this solution to work in other cases. However, it does work in the cases that we have examined and I claim that it is in fact the perfect fit to our theory. As an example, let s look at the case of Captain. We already know that according to our previous definition of cause, C = 1 is a cause of the victim s survival. Now let s examine the scenario when C = 0. In this case, A = 0, D = 0, and S = 1. But now the causal route C A S is active since we hold D = 0 fixed. In other words, in the scenario where the Captain never gives an order, then given that the Victim did not duck, the Victim s survival does depend on the Captain not giving the order to fire. This shows us that given the model that we have, Hitchcock s definition implies that C did cause S, but that if C had not occurred, its non-occurrence would have caused S just the same! Therefore, our addition implies that C is not a cause of S. Of course just because our new solution solves Captain and Switching doesn t mean that it will solve all of our problems. Even in the unlikely scenario that our new theory yields the correct results in all cases, it is probably impossible to prove this. Instead, I will point out that our theory is certainly better off than it was before since it never makes any new events causes and among those events which previously counted as causes, it excludes as causes only events

20 19 that should genuinely be excluded. It has the additional benefit of solving tricky issues about admissible variables thus allowing the flexibility in our theory that we previously found to be beneficial. First let s look at examples of genuine causation to see that our definition doesn t overstep its usefulness and restrict too much. If Z does counterfactually depend on X, then our new definition changes nothing because there is an alteration of X that changes the value of Z. But what about cases of causation without counterfactual dependence such as Backup? Here we want A1 to cause D. We know that there is an active route between A1 and D. Our addition could go wrong if in the scenario where A1 = 0, there would still be an active route from A1 to D. Of course in this scenario, there is no such route. This is because D does not depend on A1=0 since altering that means that A1 does get to shoot and of course Victim still dies in that circumstance. Interestingly, this points us to a large class of examples of failures of transitivity. A1=0 would in fact cause A2=1 which causes D=1 but of course A1=0 does not cause D=1. If we admit the transitivity of causation, then A1 not shooting also causes the victim s death! Similarly, in a perfectly ordinary case of Billy throwing a rock at a bottle and breaking it with Suzy just standing there, we have a case of intransitivity. Had Suzy thrown her rock, she could have shattered the bottle and thus prevented Billy s rock from doing so. Thus her inaction causes Billy s rock to hit the bottle, which then causes the bottle to shatter. But it sure sounds funny to say that Suzy s not throwing her rock causes the bottle to shatter. If we accept the transitivity of causation (and we accept omissions as possible causes), we would be forced to have a different solution to this new problem. Back to admissible variables

21 20 As I said, new definition has the benefit of allowing us to almost never worry about what counts as an admissible variable. Just as it was a benefit that we could examine the variable B if we wanted to in the case of Backup, it seems to me that our theory would be better off if it allowed us to include the variable B in Boulder as well. Indeed it does. If we add B to our model, there is an active route between the fall of the boulder and the hiker s survival, but there also would have been an active route between the boulder not falling and his survival namely, F B S (here we hold fixed that he doesn t duck.) Our new restriction on causes allows us to be much more lenient with what variables we include in models. For example, in Backup we could add the variable HB representing the victim as being hit by a bullet and then set D = HB. Even if we felt that there was no reasonable way that the victim could have survived if he were hit by a bullet, still there is no harm in adding it. Similarly, in Boulder we could add HB representing the Hiker as being hit by the boulder. The reader can check that we still have A1 being a cause of D in Backup and F as not being a cause of S in Boulder. In fact, given our new definition of cause, we can answer the question of exactly when adding an interpolated variable between two events in a model will change which events are causes of which others and the answer is (almost) never. 3 This reflects the it doesn t matter if you add motto which is a major advantage of our theory which our previous definition didn t allow. But is it just ad hoc? In order to answer this, I would first like to point out that our theory couldn t possibly be worse off. Our addition can only turn causes into non-causes, so if it goes wrong, it must be because it implies that something that should be a cause (and would have been on our original definition) isn t. It seems as though what possibilities are genuinely reasonable is a modal notion and as

22 21 such should not depend on what events actually do occur. This implies that an appropriate model of a situation with some events occurring should also be an appropriate model of the same situation even if some of those events did not occur. This means that when we check whether an event s non-occurrence would have been a cause of the effect in question, it should be acceptable to use the same model and structural equations with the only change being the initial value of the potential cause. But if there is active causal route linking both the occurrence of the event to the effect and there would be such a route between the non-occurrence of the event and that same effect, we should not be willing to call that event a cause. Being a cause not only means being that which makes the effect happen but also implies something along the lines of that without which the effect would not have happened. Causes must be difference makers. (cf. Sartorio, forthcoming) The events that become non-causes with our new definition of cause were inappropriately called causes in the first place by the fact that there are often active routes between events without genuine causation. Indeed any example of intransitivity has A causing B and B causing C and so intuitively, in many cases there should be an active causal route between A and C even in cases where there is not genuine causation. The existence of active causal routes points to why we feel a pull to call these events causes in the first place. Other theories have notions of contributing cause (Woodward 2003, Pearl 2000) often pointing out that these are events that are mentioned in explanations of the effect and intuitively are somehow part of the causal history of the effect. Yet if we want to claim that this concept is more inclusive than causation simpliciter, then a restriction such as ours will certainly be required. Indeed, it seems to remove exactly the cases that we wanted to exclude, those that have active routes to the effect, but don t genuinely contribute anything because they aren t needed for the effect in any way.

23 22 REFERENCES: Hall, N. (2000). Causation and the Price of Transitivity, Journal of Philosophy, 97: Hall, N. (2004). Two Concepts of Causation, in J. Collins, N. Hall, L. Paul, Causation and Counterfactuals. MIT Press. Halpern, J., and Pearl, J. (2000). Causes and Explanations: A Structural Model Approach. Technical Report R-266, Cognitive Systems Laboratory, Los Angeles: University of California. Hitchcock, C. (2001). The Intransitivity of Causation Revealed in Equations and Graphs, Journal of Philosophy 98: Hitchcock, C. (2003). Of Humean Bondage, The British Journal for the Philosophy of Science 54: Lewis, D. (1973). Causation, Journal of Philosophy 70: Pearl, J. (2000). Causality: Models, Reasoning, and Inference. Cambridge University Press. Sartorio, C. (forthcoming). Causes as Difference-Makers, forthcoming in Philosophical Studies. Schaffer, J. (2000). Causation by Disconnection, Philosophy of Science 67: Woodward, J. (2001). Probabilistic Causality, Direct Causes and Counterfactual Dependence. In Stochastic Causality, ed. M. Galavotti, P. Suppes, and D. Costantini, Stanford: CSLI. Woodward, J. (2003). Making Things Happen. Oxford University Press. Yablo, Stephen (2002). De Facto Dependence, Journal of Philosophy 99: Yablo, Stephen (2004) Advertisement for a Sketch of an Outline of a Proto-Theory of Causation, in J. Collins, N. Hall, L. Paul, Causation and Counterfactuals. MIT Press.

24 23 NOTES: 1. It is also important to note that the theory as stated would not allow overdeterminers as causes. That is, if C1 and C2 are both sufficient for E and are not causally related to each other, then the theory implies that neither is a cause. Different model theorists react to this case differently. Pearl (2000) has a slightly different definition of cause which implies that overdeterminers are causes. Hitchcock (2001) gives an additional clause which determines whether a route is weakly active (so that there are weakly active routes between overdeterminers and their effects) and then states that it is up to the reader whether they want the definition of cause simpliciter to look for an active route or merely a weakly active route. Woodward (2003) argues that we should accept overdeterminers as causes and so his definition of cause just includes the extra clause of Hitchcock s and thus overdeterminers are causes on his view. Other than this note, I do not discuss overdetermination. 2. As I say, this actual example is quite controversial. Of course Hall himself does not accept this as a counterexample since he does accept the transitivity of causation. Halpern and Pearl (2001) deal with it in this way. We can model the situation with three variables: F for flip, T for track (with values 0 and 1 depending on whether it travels on the left or right track), and A for arrival. In this case, F will not be a cause of A. This treats the train as bound to travel on one and only one of the tracks and so the train will arrive no matter what. On the other hand, if there is a genuine possibility of one of the tracks breaking down (independently of the other) then each individual track should be treated as a separate mechanism. In the separate mechanism case, we should use separate variables for the left track and the right track and the event of

25 24 flipping the switch is and should be a cause of the arrival. Whether or not Train is a genuinely switching case in which the switch should not be treated as a cause of the effect, it seems clear to me that are other such cases. 3. I claimed above that adding an intermediate variable between two events would almost never change the causal relations between events. Of course there will be new relations between older variables and the new added variable. What I mean is that in almost every case the old relations that were there are still all there and that we haven t added any new ones. To add an intermediate variable do the following: take a causal model (V,E) in which a variable A is a direct cause of a variable B and form a new model (V,E ) by taking V and adding a new variable X directly caused by A and directly causing B and form E from E by replacing the equation for B with a new equation which has X substituted for A and adding the equation X = A. What we want to know are the possible counterexamples to the following claim: for any Y and Z in V, Y causes Z in (V,E) iff Y causes Z in (V,E ). It can be proven that there are only two kinds of cases that can change the causal relations. In both cases, we must have the following structure where the change is whether or not A causes B. Note that this may not be the exact model; there may be additional variables intermediate between A and D or D and B. What can change is whether or not the A D B path is active. D A X B In one kind of counterexample, we previously had A causing B and now we have D and X overdetermining B. On our original definition of cause, we would no longer have A causing B.

26 25 However, if we add the clause making overdeterminers causes, then we will still have A causing B and thus no overall change. In the other kind of case, A has to have at least three possible values where there is an active route from the actual value of A to B and at least one of the possible values of A is such that there would be no active route from this value of A to B. As for myself, I have no clear intuitions on such cases and I believe that these cases are complicated enough (and certainly rare enough) that we ought not to judge our theory deficient in this manner. At the very least we have improved how we deal with adding intermediate variables.