Causation and evidence

Causation

and Evidence

by

Melissa Renee Schumacher

MA-SSACHUSETTS INSTITUTE OF TECHNOLOGY

OCT 15 2015

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B.S. Philosophy, B.S. Electrical Engineering

North Carolina State University, 2009

SUBMITTED TO THE DEPARTMENT OF LINGUISTICS AND PHILOSOPHY

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN PHILOSOPHY

AT THE

MASSACHUSETTS INSTITUTE OF TECHNOLOGY

SEPTEMBER 2015

@ 2015 Massachusetts Institute Of Technology. All rights reserved.

Signature of Author:

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Certified by:

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Department of Linguistics and Philosophy

4 1August 26, 2015

ignature redacted

Signature redacted

t

₁₎

Professor of Philosophy

by

Melissa Ren6e Schumacher

Submitted to the Department of Linguistics and Philosophy on August 26, 2015, in Partial Fulfillment of the Requirements

for the Degree of Doctor of Philosophy in Philosophy

ABSTRACT

This work addresses questions about causation and evidence: How can we learn what causes what? Can we get evidence for objects that don't cause anything? And what is the evidential relationship between events in a causal loop?

Structural equations accounts of causation seem to provide a good basis for discovering causal relationships through observation. But these accounts can sometimes give the wrong verdict in cases that are structurally similar to cases that they do get right. Distinctions between default and deviant states, and between more and less normal worlds, have been introduced to solve this problem. In "Defaults, Normality, and Control" I argue that both of these kinds of solution introduce new problems without solving the old one. I propose a different theory of causation based on the structural equations account, designed to capture the intuition that the causes of an event are whatever could have, by not occurring, most easily prevented that event.

In the philosophical literature, Occam's Razor is standardly taken to be a constraint on the amount of (types of) objects a theory can be justifiably committed to. In "Occam's Razor and Philosophical Objects" I introduce an interpretation of Occam's Razor that doesn't fit that standard mold, but gives plausible answers to the questions "What is theoretical simplicity?" and "Why should we believe the simpler theory?". I then apply it to abstract and non-fundamental objects, and show that theories that include such objects need be no more complex than theories that don't. We can therefore be justified in believing such theories, even though they make the same predictions about observables as alternative theories.

In "Playing Dice With a Time Machine: A New Puzzle About Causal Loops", I use an original puzzle case to bring out the problem of calculating the probabilities of events in a causal loop, and I propose a solution. I also point to some difficulties involved in reaching that solution.

Thesis Supervisor: Bradford Skow Title: Associate Professor of Philosophy

I would like to thank my thesis committee - Brad Skow, Roger White, and Steve Yablo

-for their many insightful comments, and -for always challenging me to explain my ideas more clearly. None of these papers would be what they are without their help.

John Carroll, one of my mentors from my undergraduate days at NC State, provided wonderful and detailed comments on "Playing Dice With a Time Machine". That paper grew out of an overheard conversation between him and John Roberts of UNC, at a conference on time travel at NC State. John Roberts also provided very helpful comments on an early draft.

Judy Thomson, very kindly, insisted on reading "Occam's Razor and Philosophical Ob-jects", and had many thoughtful and encouraging things to say about it. I deeply appreciate her willingness - in spite of the fact that our philosophical interests don't have much overlap

- to meet with me during my time at MIT.

My fellow graduate students also helped me in innumerable ways, both large and small.

Without their kindness, encouragement, and unfailing enthusiasm for discussing weird ideas, grad school would have been much less fun and much less productive.

Defaults, Normality, and Control

Two concepts have recently been put forward as useful potential additions to a theory of causation: a distinction between default and deviant states, and a ranking of possible worlds according to their normality. I believe these additions won't solve the problems that their proponents think they will. In this paper I will discuss some difficulties for Structural Equations accounts (to be described below)- of actual causation', and some difficulties for attempts to address those difficulties using default and deviant states and relative normality. I will then propose a new theory of causation based on structural equations, and show how it compares with the others.

1

Structural Equations

Structural Equations (SE) accounts are a kind of counterfactual account of causation, the simplest version of which is: C causes E iff, had C not occurred, E would not have occurred. (This relationship between C and E is counterfactual dependence. Henceforth I'll just use "dependence" to mean "counterfactual dependence".) There's something right about this simple counterfactual account, but counterexamples abound. Here's one: an Assassin and

'An instance of actual or token causation is "Suzy's smoking caused her lung cancer" - as opposed to type causation, as in "Smoking causes lung cancer".

her Backup arrange to kill Victim. If Assassin fails to carry out the mission, Backup will kill Victim. Assassin kills Victim, and Backup doesn't have to act. Victim's death obviously was caused by Assassin, but Assassin doesn't pass the counterfactual test: had she not acted, Victim would have died anyway.

We can get out of counterexamples like these by modifying the simple account so it allows us to hold some things fixed. Holding fixed that Backup was at home all night in her pajamas watching Project Runway, it's true that Victim's death depends on Assassin. In general, if we can hold some things fixed, at either their actual values or non-actual ones, we can reveal the real causal structure of the cases we're interested in. The key question is, how do we do this in a principled way? What should we hold fixed? Which things should have to take their actual values, and which can take non-actual values? We will return to this question shortly.

The Structural Equations (SE) approach is a family of counterfactual theories of causation that use structural equations (naturally) to model causal situations. Finding a good model of a situation is tricky - it involves finding the best set of variables to represent what's going on. A variable can track things like the temperature of a room (continuum-many values), the number of pens on a desk (many discrete values), or whether a switch is on or off (binary values). The counterfactuals that are true in the situation of interest are translated into a set of equations that say how the value of each variable in the model depends on the values of the other variables. For instance, if we're using binary variables (which can only take the values 1 or 0), then the equation

E= A+C-AC

means that E would have the value '1' if A or C did, and would have the value '0' otherwise. There's always only one variable on the left-hand side of a structural equation, and its

value is set by the variables on the right-hand side. A variable that appears on the left-hand side of any equation is an endogenous variable. Its value depends on the values of other variables in the model, in the way indicated by the equation. A variable that only appears on the right-hand side of equations is an exogenous variable. Its value is taken as given - the way it gets its value isn't part of the model. There will nearly always be other exogenous variables that aren't explicitly represented in the model at all. For instance, in the Assassin/Backup case above, other unmentioned exogenous variables track whether Assassin's gun is working, where Victim is when he's killed, and whether there's enough oxygen in the atmosphere for all of them to stay alive until the murder is accomplished.

The equations determine the structure of the model, which can be represented visually

by drawing arrows from each variable to all the other variables whose values it influences. E.g., the above equation determines this structure:

A

E C

Figure 1

(Since equations can involve variables that take on a wide range of values, a simple diagram like this won't always capture all the counterfactual information contained in the equations. But in this paper we'll only see binary variables, so that feature of structural equations won't matter much here.)

Once we have a structure, and the actual values of all the variables in it, we can start testing for what actually caused what.

1.1

Hall

Hall (2007) considers two SE views, but I will focus on what I take to be the more plausible of the two, the HP-account2. The idea behind it is this: if we're testing for whether the variables X 's having the values x cause Y to be y, we need to look at all the paths that connect X and Y. A path is a chain of arrows from one variable to another. (If there aren't any paths between X and Y, then X can't have caused Y.) X causes Y iff there's some path such that, if we set X to some set of non-actual values, and possibly change the values of the off-path variables in an allowable way (to be explained below), then Y has a different value. This is how we incorporate the lesson we learned from the Assassin/Backup example: that sometimes we need to set certain off-path variables to certain values to get the real causal chain to reveal itself; if only they'd had those values in the first place, the effect would have depended on the cause outright.

Let's call the set of on-path variables (which include X) "Z" and the set of off-path variables "W". X = X causes Y = y if we can find at least one pair of variable settings '/

and TJ-/ that meets these criteria:

1. If X were to be x/ and W were to be Tw/, Y would not be y.

2. For all subsets W/ of W and ZI of Z, if everything in Z/ kept its actual value, everything in W/ were set to Ti_/, and everything in X stayed 4, Y would be y. 3. No subset of X satisfies the above conditions.

(1) and (2) pull in opposite directions. (1) expands the range of variable settings that we

can consider. Instead of changing X alone, we get to make changes to some other variables. (2) counteracts this permissiveness by making sure we hold enough things fixed. If giving

2

Named for Halpern and Pearl, who proposed it in Halpern and Pearl (2005). In my statement of the account below, I've removed a lot of the notation but have tried to keep the ideas the same. See Halpern and Pearl (2005) for the official version.

something in W its alternative value w/ would change Y, then (2) would kick that variable out of W. We wouldn't be able to consider it off-path, because it would be able to affect Y directly.

To get an idea of how this account works, consider an overdetermination example like the one above involving A, C, and E. Figure 1 indicates that all three variables take on the value '1' (if they didn't, their circles would be white instead of grey). The HP-account says that both A and C are causes of E. Let's look at A first. C is off the path between A and E, so we should be able to find some setting for it that will keep it from interfering with that path, and a setting for A that will change E to its non-actual value. And we can: a/ = 0

and c/ = 0. Let's make sure this pair meets the criteria.

1. If A were to be 0, and C were to be 0, E would not be 1.

2. Since A is the only thing on the path, and C is the only thing off it, this is pretty easy: if C were set to 0, and A kept its actual value, E would be 1.

3. There are no non-trivial subsets of A, so we don't have to worry about this one.

Thus A comes out as a cause of E. We can do the same thing for C (setting A, the off-path variable, to 0), so C is also a cause of E.

So far so good. But Hall thinks that neither of the SE accounts he considers will get all the right results, and nor will any SE account, unless it recognizes the importance of the difference between default and deviant behavior. Roughly speaking, default behavior is what something will do when it's not being acted on in any unusual way, and deviant behavior is what it does when it's affected by something.

We can see why this distinction might be important by considering this pair of causal structures3:

E G D :F tB Structure A G D F

C

B

More detail on how these pictures work is now needed in order to see what causal structures they represent. These are neuron diagrams. Each circle represents a binary variable, and each can be thought of as a neuron, which either "fires" (takes on the value '1') or doesn't (takes on the value '0'). An arrow from X to Y represents, as it did in the picture of the overdetermination case, that X stimulates Y - Y will fire iff X does (and no inhibitors interfere; see next sentence). If there is a line with a dot on the end from X to Y, it means that X inhibits Y - Y will not have the value '1' if X does, whether or not some other neuron stimulates Y. As before, a grey circle means that that neuron fires, and a white circle means it doesn't. A thickly-outlined circle (like the one for E in Structure B) means that that variable needs extra stimulation - it only fires when stimulated by two other neurons.

E = A(1- F)

(Eq-A) F = D(1 - B) D G + C - CG

B C

And the equations for Structure B are:

E AF

F D + B - DB (Eq-B)

D G(1 - C)

B =C

But that's not the only way to describe Structure B. According to standard SE views, we can symbolize the different values a variable can take in whatever way we want. In particular, instead of representing D as firing by saying "D=1", we can represent D as firing by saying

"D=O". Let's perform this switch for variables D, F, and G in Eq-B. We can think of this as

substituting in new variables D*, F*, and G*, which always have the opposite values of D, F, and G. We now have:

E = A(1 - F*)

F* = 1 -

1(1

(Eq-B *)

D* = 1 - [(1 - G*)(1 -

C)]

+

B = C

which look exactly like Eq-A, apart from the asterisks. Furthermore, D, F, and G have opposite values in structures A and B. So the asterisked variables have the same values as their counterparts. This means that any SE account that allows this sort of transformation will make the same causal judgements about Structure A as about Structure B.

on C (C prevents F, which would have prevented E); so C is a cause of E. In Structure B, if we hold fixed that D doesn't fire, E depends on C (C is needed to stimulate F, which is needed to stimulate E); so C is a cause of E here too. Hall thinks that this is the wrong judgement to make about Structure A, which makes sense, since C itself initiates the threat against E that it also cancels out. G could have initiated that threat too, but G doesn't actually fire, so E isn't actually in danger; as long as G doesn't fire, it doesn't matter what

C does. But he thinks it's the right judgement to make about Structure B.4 I think this is

supposed to be because C not only initiates a path to E, it cuts off G's path to E. G masks the causal route from C to E, as in standard overdetermination cases, because with G firing it doesn't matter what C does. But C does the same, and worse, to G. With C firing, it doesn't matter what G does, and C also pushes G aside, breaks up G's path, and forges its own.5 Still, whatever you think about these causal judgements, the two structures do seem different.

What we've done in replacing D, F, and G with D*, F*, and G* is to swap default and deviant behavior. As long as we can do that, structures in which an SE account seems to get the right results can be transformed into ones in which it gets the wrong results. Hall's solution is to build the default/deviant distinction into the test for causation. He says the only alternative variable settings that we should be allowed to consider are reductions of the actual situation, alternatives in which some variables, whether off-path or not, go from their deviant behavior to their default behavior. When we test for whether X causes Y, we check whether Y depends on X (without holding anything fixed) in some reduction in which X and Y both occur. If there is one, then X does cause Y, otherwise it doesn't.

Using Hall's criterion, we get that C causes E in Structure B, but not Structure A, as

4Hall (2007), 131-132.

5

1f this doesn't convince you, wait a bit; later we'll see some reasons to think that default and deviant behavior aren't symmetrical, that we should care more about dependence in default scenarios than in deviant ones.

desired. In Structure B, we can turn G off, and now we have a reduction in which C is needed to stimulate E. But in Structure A, the only variant of the actual situation in which

E depends on C is the one in which we turn G on - now C is needed to prevent F, which otherwise will prevent E. But this isn't a reduction, because we had to make G take on a deviant value.

1.2

Hitchcock

For the examples Hall gives, his criterion works fine. But Hitchcock (2009) argues that taking default and deviant behavior into account isn't as easy as this. Sometimes the default behavior of certain variables is what masks the real causal chain between other variables, or creates a misleading dependence between two variables. In cases like those, making more variables take on their default behavior won't help. In fact, we can take any of the examples Hall gave in which his account works and turn them into ones where his account fails, just

by adding in more variables in a certain way. For instance, here's a "switch" structure:

C

A

B

E

F D

Structure C

F is a switch that determines whether B will stimulate C or D - if F fires, as it does in the actual world, B stimulates C; if F doesn't fire, B stimulates D. In this case, there's no reduction that will make E depend on F. Hall's account therefore says that F is not a cause of E. This is exactly what we want, since whatever path B's stimulation takes, it will get to

C A B E F D H Structure C

In this case, again, F doesn't affect E. It merely controls which path B's stimulation will take, and both paths lead to E. However, Hall's account says that F is a cause of E in this case. We can get a reduction by turning off H, and in that reduction, if F hadn't fired, E wouldn't have fired (since without H firing, B alone isn't enough to make D fire).6

Just as Hall's technique can be generalized to create "dual" causal structures that the usual SE accounts will consider isomorphic to each other, Hitchcock's technique can be generalized to create causal structures in which setting variables to their default values will create the appearance of causation where there is none. So there must be a better theory of how default and deviant values are related to causation. We'll now turn to a proposal by Halpern and Hitchcock.

6

2

Incorporating the Distinction into the SE Approach

2.1

More Motivation

From Hall's examples, one might think that the default/deviant distinction is only operative or important in simple cases like neuron diagrams, where it's easy to see what behavior is default and what is deviant. Hitchcock and Knobe (2009) argue that there does seem to be a default/deviant distinction in our everyday judgements about causation, that it's not unsystematic or arbitrary, and that there's a good reason for it.

Consider this case7:

Only administrative assistants are supposed to take pens from the recep-tionist's desk, and professors have been repeatedly warned not to take those pens. During the course of the day, an administrative assistant takes a pen, and also a professor takes a pen. Later, the receptionist can't find a pen. Who caused the lack of pens?

If either the administrative assistant or the professor hadn't taken a pen, there would have been a pen left. Yet many people judge that the professor caused the lack of pens, and the administrative assistant didn't. Hitchcock and Knobe's explanation is that administrative assistants are allowed to take pens and professors aren't. Sure, the administrative assistant could have acted differently, but so could the professor, and the professor was supposed to. So it was her behavior that really mattered - we could take the administrative assistant's behavior for granted. His taking a pen is more like a background condition.

If you don't feel the pull of that thought, take this case: A doctor has been assigned to treat Billy, and if she gives him a dose of medicine on Monday, he'll recover on Tuesday. She forgets to give him the medicine on Monday, and he is still sick on Tuesday. Billy's doctor seems clearly to have caused his being sick on Tuesday (of course, Billy's illness is also a

7

cause). However, there were many other doctors in the hospital, any one of whom could have given Billy the medicine. It's true of them, as it's true of Billy's doctor, that had they given him the medicine on Monday, he wouldn't have been sick on Tuesday. Yet the other doctors, intuitively, aren't causes of Billy's being sick on Tuesday. Our intuition about this case is plausibly connected to the fact that the other doctors weren't supposed to treat Billy. It could be taken for granted that they wouldn't treat him.8

If that case doesn't move you either, don't worry: my main point is that Hitchcock and

Knobe don't accommodate these intuitions in the right way (and that Halpern and Hitchcock (2010), whose proposal we'll look at in the next section, don't do it right either). If you don't think those intuitions need to be accommodated in the first place, then you already agree with me about that (although you won't like my positive proposal).

Let's say they're right that a default/deviant distinction (or something like it) is operative in our everyday judgements about causation. Should we try to ignore it, or is it doing some useful work for us? Hitchcock and Knobe say it makes sense that we take default and deviant behavior into account, and the reason they give is that we're thinking about what we ought to do if we want to prevent the effect. What had the most control over whether the effect happened?

Our aim here is to construct an account that extends this basic insight, showing how people's concept of actual causation enables them to design effective interventions. In particular, we argue that information that has nothing to do with causal structure can sometimes prove helpful in

de-8

You might be worried that our intuition about this case is influenced by the blame that will attach to the

other doctors if they turn out to be causes of Billy's illness. We don't want to blame them, so we say they're not causes even though they really are. But we have the same kind of intuition about cases where the effect is something praiseworthy: by not pushing the button, Nixon averted a nuclear war. Also, I didn't push the button. Had I done so, nuclear war would have ensued. Do I deserve the Nobel Peace Prize? No. We also have this intuition about cases where the effect is neutral: had you picked out my clothes today, I would have been wearing blue socks instead of green ones. But you didn't cause me to be wearing green socks. It could be that we don't want to assign responsibility, whether positively-, negatively-, or neutrally-valenced, in any of these cases, but that if we weren't worried about the connection between responsibility and causation, we would say the doctors caused Billy's illness, I averted nuclear war, and you caused me to wear green socks.

termining which intervention would be most suitable and that people are actually making use of this information in deciding which factors to de-nominate as "causes". In general, while causal structure identifies all of the factors that could be intervened on (either singly or in combination) to ef-fect a change in the outcome, the actual causes are the factors that should be intervened on.9

This sounds very plausible to me. But they go on to draw a further conclusion: At the heart of our theory is the idea that people's judgments about the relevance of counterfactuals depend in an essential way on norms. The basic suggestion is that people classify events on a scale from "normal" to "abnormal". Then, when something abnormal occurs, they regard as relevant counterfactuals those that involve something more normal having occurred instead.'0

The proposal seems to be that we prefer to intervene on what's abnormal, and that we consider whatever we want to intervene on to be the cause(s). Hence our judgements about what's normal and abnormal (or default and deviant) greatly influence our causal

judge-9_{Hitchcock and Knobe (2009), 590.}

'0Hitchcock and Knobe (2009), 597. And for further support, see 599: "Looking through the psychological literature, one finds a number of different hypotheses about precisely how this reasoning process takes place, but the precise details of these hypotheses will not be important here. The key point for present purposes is just that, whatever else might be going on, people do generally show a tendency to direct their counterfactual thoughts toward ways in which the world could have been more normal - and there is considerable empirical support for this general claim. To take a case involving prescriptive norms: [Read (1985)] presented subjects with a vignette in which two people are playing cards. One person wins the hand, and subjects are invited to complete the sentence 'The outcome would have been different if...'. Subjects were much more strongly inclined to strengthen the losing hand than to weaken the winning hand. Or, for a case involving statistical norms: [Kahneman and Tversky (1982)] presented subjects with a vignette in which Mr. Jones was killed in a traffic accident on the way home from work when he was hit by a drug-crazed truck-driving teenager who ran a red light. In one version of the story, Mr. Jones left home from work early in order to run some errands; in another version, he deviated from his normal route home in order to enjoy the scenic drive along the lake. Subjects were informed of both possibilities; for example, subjects who were given the first version of the story were also told that Mr. Jones occasionally took a scenic route home, but that he took his normal route home that day. Subjects were then told that Jones' family and friends often found themselves thinking 'if only...'; the subjects were then told to complete the 'if only...' statements. Those subjects who were given the first version of the story were strongly inclined to say 'if only Jones had left at his regular time', rather than 'if only Jones had taken the scenic route'; whereas subjects who were given the second version of the story were inclined to say 'if only Jones had taken his regular route home' rather than 'if only Jones had left

ments. This accounts nicely for our intuitions in the doctor case above: doctors don't normally treat patients they're not assigned to, but do normally treat patients they are as-signed to. So we think it's most useful to intervene on Billy's doctor, who's not doing what

she's supposed to.

If that's the proposal, then the following case seems to be a clear counterexample: Suzy needs only one dose of a certain medicine to recover from her illness, but two doses won't do her any harm. Her doctor gives her a dose of the medicine, but also I just happen to be wandering through the hospital, and I too give her a dose of the medicine, despite the fact that I have no medical training and no idea what the medicine does. It happens to be true that if the doctor hadn't given her the medicine, I would have (holding fixed that I was there and did what I did), so her recovery doesn't depend on the doctor's action (nor on mine). However, not only was the doctor unlikely to need my help, it was very unlikely that I would be there in the first place, or give that exact medicine to that exact patient.

If we want to say that Billy's doctor was a cause of his being sick, but the other doctors in

the hospital weren't (even though they also failed to treat him), it seems like we can, and probably should, say in this case that Suzy's doctor was a cause of her recovery, and I wasn't

(even though I also treated her)."

Doctors giving medicine to their patients is extremely normal. But random untrained yobbos wandering around hospitals dispensing the right medicines by chance is highly

un-usual. If we always intervened on the abnormal thing, we'd want to intervene on me, i.e., take me to be the cause of Suzy's recovery. But we don't. And whether or not you think I'm

not a cause, Suzy's doctor certainly seems to be a cause. Yet, Suzy's doctor is just doing what she's supposed to.

"If your intuitions about this case depend partly on who gave Suzy the medicine first, consider this more

atemporal version instead: Suzy will get the medicine at noon if and only if her nurse has been given at least one prescription for the medicine by then. Both her doctor and I give prescriptions to the nurse before noon. (Thanks to Stephen Yablo for raising this worry.)

So there seems to be something wrong with this way of accounting for our intuitions. But let's stick with it for now, and see how Hitchcock and Knobe's general idea about norms can be incorporated into an SE approach.

2.2

Halpern and Hitchcock

Halpern and Hitchcock (2010) pick up where Hitchcock and Knobe left off:

While the definition of actual causation in terms of structural equations has been successful at dealing with many of the problems of causality, examples

of [Hall (2007)], [Hiddleston (2005)] and tHitchcock (2007)] show that it

gives inappropriate answers in cases that have structural equations isomor-phic to ones where it arguably gives the appropriate answer. This means that, no matter how we define actual causality in the structural-equations framework, the definition must involve more than just the structural

equa-tions. Recently, (Hall (2007)], [Halpern (2008)], and [Hitchcock (2007)] have

suggested that using defaults might be a way of dealing with the problem. As the psychologists Kahneman and Miller (1(1986)], p. 143) observe, "an event is more likely to be undone by altering exceptional than routine as-pects of the causal chain that led to it". This intuition is also present in the legal literature. [Hart and Honor6 (1985)] observe that the statement "It was the presence of oxygen that caused the fire" makes sense only if there were reasons to view the presence of oxygen as abnormal."

So their jumping-off point is the idea that our judgements about normality influence our causal judgements, and that normality should therefore be incorporated into theories of causation.

Their way of doing this within an SE framework is to add some new machinery to the HP-account: a ranking of worlds according to how normal they are. This can be thought of as assigning a number to each world, where lower numbers indicate less abnormality. On what do we base our judgements of normality? Drawing on the psychology literature on

counterfactual and causal reasoning, Halpern and Hitchcock point to statistical norms, moral norms, policies (as in the pen case above), and norms of "proper functioning" (governing organs, mechanical parts, and so on)13, but they don't intend this list to be exhaustive. And, just as models can be more or less apt, so can the norms we invoke, and the ranking of worlds we end up with.

According to the HP-account, we should look at worlds in which X has a non-actual value and some off-path variables are set to certain values, and see if Y also has a non-actual value.

If there's at least one world that does that (and meets the other criteria given in section 1.1),

then X caused Y. Halpern and Hitchcock's proposal is to restrict the worlds we're allowed to consider: only worlds that are at least as normal as the actual world count. If none of those worlds meet the criteria above, it doesn't matter whether a less normal world does; X didn't cause Y.

This account gives the right verdict in many cases. Let's see how it deals with the first hospital case (where we assume, for definiteness, that there are 100 doctors in the hospital):

Doctor 1 is assigned to treat Billy; the others are not. However, in fact, no doctor treats Billy. Further assume that, typically, no doctor is assigned to a given patient; if doctor i is not assigned to treat Billy, then typically doctor i does not treat Billy; and if doctor i is assigned to Billy, then typically doctor i treats Billy. We can capture this in an extended causal model where the world where no doctor is assigned to Billy and no doctor treats him has rank 0; the 100 worlds where exactly one doctor is assigned to Billy, and that doctor treats him, have rank 1; the 100 worlds where exactly one doctor is assigned to Billy and no one treats him have rank 2; and the 100x99 worlds where exactly one doctor is assigned to Billy but some other doctor treats him have rank 3. (The ranking given to other worlds is irrelevant.) In this extended model, in the context where doctor i is assigned to Billy but no one treats him, i is the cause of Billy's sickness (the world where i treats Billy has lower rank than the world where i is

13

assigned to Billy but no one treats him), but no other doctor is a cause of Billy's sickness. Moreover, in the context where i is assigned to Billy and treats him, then i is the cause of Billy's recovery (for [criterion (1) of the HP-account] consider the world where no doctor is assigned to Billy and none treat him).'

The idea is that for doctor 1 to be a cause of Billy's illness, we need to think about a world where doctor 1 (who's assigned to Billy) treats him, and see whether Billy recovers in that world (he does). This world has rank 1, and the actual world has rank 2, so it's normal enough, and doctor 1 comes out as a cause of Billy's illness. But now consider any other doctor in the hospital. In the world where that doctor treats Billy, Billy also recovers, but that world is one in which a doctor treats a patient they're not assigned to, so it has rank

3, which is too abnormal to consider.

Let's see what happens in my variant hospital case. This time, Suzy's doctor does treat her, and also a random non-doctor (me) treats her. This is pretty weird, weirder than it would have been if another doctor in the hospital had treated her, so let's give the actual world a ranking of 4. To check whether her doctor caused her recovery, we need to look at a world where the doctor doesn't treat her, and also where I (an off-path variable) don't treat her. In that world, she doesn't recover. That's a world where the doctor assigned to Suzy doesn't treat her, so it has a ranking of 2, which means it's more normal than the actual world, so Suzy's doctor is a cause of her recovery. But what about me? To check whether

I caused Suzy's recovery, we need to look at a world where I don't treat her, and also her

doctor (an off-path variable) doesn't treat her. In other words, the world we need to consider when testing for my causal power is the same as the world we need to consider when testing for her doctor's causal power. Again, it has ranking 2, so I also come out as a cause of Suzy's recovery. This isn't such a bad result - at least my abnormality doesn't make me the only, or primary, cause of Suzy's recovery. Maybe it's okay that both of us come out as causes.

14

But there remain two deeper worries.

One is that this proposal is too much like Hall's. Sometimes adding more default behavior would interfere with the real causes or mislead us about them, so in such cases we won't find the real causal chain by considering only worlds that are at least as normal as the actual one. To clarify this worry, let's look at a pair of cases like Structures C and C/ above: in both cases, there is a railroad track that forks and then reconverges before the next station.

A switch determines which fork the train takes.

(Open Tracks) Neither track is blocked. (Blocked Tracks) Both tracks are blocked.

Halpern and Hitchcock discuss Open Tracks.15 Intuitively, flipping the switch doesn't cause the train to arrive at the station, it only changes which path the train takes. And their proposal gets us that result: the switch makes a difference only in the world where one track is blocked (the one the train would have traveled on if the switch hadn't been flipped), but this is a more abnormal world than the actual one, where neither track is blocked, since it's abnormal for tracks to be blocked.

However, they don't consider Blocked Tracks. In this case, likewise, we think that flipping the switch doesn't cause the train to fail to arrive at the station. But now removing one blockage (the one on the track the train would have traveled on if not for the switch) gives us a more normal world, and in that more normal world, if the switch isn't flipped, the train will arrive. This seems exactly parallel to the switch case that Hitchcock used against Hall: the switch doesn't matter in the actual world, but removing a blockage (making the world more normal, putting something into its default state) makes the switch matter. We don't want a theory that counts the switch as a cause in this kind of case.

The other worry is about time. Recall that in Halpern and Hitchcock's discussion of the case where Billy's doctor treats him and he recovers, they explained that we need to consider

the world in which no one was assigned to treat Billy and no one treats him in order to get Billy's doctor to come out as a cause of his recovery. Yet when Billy's doctor is deciding whether to treat him, she's already been assigned to him, so it seems like the only relevant world in which she doesn't treat him is the world in which Doctor 1 has been assigned to treat Billy and no one treats him. Since that world is less normal than the actual one, we can't consider it, and she isn't a cause of his recovery.

Why should we care about what's already happened? We'll see some examples in the

next section (and see the Appendix for more), but for now it should seem at least minimally plausible given that the kind of theory we're after is an interventionist one. If we want to know what we'd need to do in order to change the actual outcome, we'll need to take into account what's already happened when it comes time to intervene.16

3

Where Do We Go From Here?

So, two desiderata for a theory of causation are: get the right verdict in cases where default

behavior is masking the causal chain (e.g., Blocked Tracks, and Hitchcock's counterexample to Hall), and get the right verdict when we incorporate information about the temporal order of events (such as Billy's doctor having been assigned to him already). What kind of theory would give us those?

For inspiration, I want to go back to some of the things Hitchcock and Knobe said: that the real cause is what had most control over the outcome, what could have prevented it most easily, what we think we could have intervened on most efficiently to prevent the effect. This thought seems to get something right, but spelling it out in terms of normal and abnormal situations, or default and deviant behavior, isn't the right way to go. I think we should just

6

And it does seem strange that Halpern and Hitchcock's theory says that 1) when considering whether Billy's doctor caused his recovery in the case where she treats him, we don't hold fixed that she was assigned to him, but 2) when considering whether the other doctors in the hospital caused Billy's illness in the case where his doctor doesn't treat him, we do hold fixed that those doctors aren't assigned to him.

stick with the thought we started with: our judgements about causation track something like which thing(s) we could have intervened on most easily to change the outcome, or which thing(s) would have had the best chance of changing the outcome. Often resetting things to normality is the easiest way to change the outcome, because it's easier to make something take its default value. But even if that were always the case, it's not so easy to build that into a theory of causation. As we've just seen, it might be overly simplistic to rank worlds

by normality and throw out the ones that are less normal than the actual world. Sometimes

worlds that are normal enough still shouldn't be considered, and sometimes weirder worlds should be.

So which counterfactuals are relevant? I think they all are, in a way. That is, if we want to know whether intervening on X would be a good way of preventing E, we shouldn't look for just one scenario in which X is different and E is prevented. We should compare X with the other potential causes, to see if it makes a relatively significant difference to whether

E happens. It might be possible to intervene on X in order to change E, even if it's not

advisable to do so.

Probabilities of counterfactuals might be useful here. E.g., in the random-stranger-at-the-hospital case, it seems more accurate to say "If the doctor hadn't given Suzy the medicine, she wouldn't have recovered" than to say "If the random untrained passerby hadn't given Suzy the medicine, she wouldn't have recovered". The doctor is a much more stable backup than me, so if I hadn't come through for Suzy, the doctor would have. But I was much less likely to be around and to grab the right bottle of medicine than the doctor was, so I'm not a good backup for the doctor.

In general, it might turn out that if the actual world is abnormal, the probability of

"If (normal event) hadn't occurred, then (effect) wouldn't have occurred" could be pretty

high (as it is in the doctor-vs.-random-stranger case), but that if the actual world is fairly normal, then the probability of "If (abnormal event) hadn't occurred, then (effect) wouldn't

have occurred" would be pretty high (as it is in the pen case). If so, looking for more normal worlds won't be helpful if the actual world is itself abnormal, and in any case, normality seems to be tracking the wrong thing. What we care about is what would be most likely to prevent the effect. It just so happens that in many toy cases, restricting our attention to more normal worlds will get us the right answer.

So far, we have a principle something like

A causes E only if: of all the things that could have prevented E by not

occurring, A's non-occurrence would have been most likely to prevent E. This could give us the first desideratum, depending on exactly how we spell it out.

To get the second, we need to take into account what's already happened when it comes time for A to take on its value. For concreteness, consider another train case. Again, the track forks and reconverges before the next station. The tracks are almost never blocked, and both tracks are equally likely to be blocked. But this time there's a blockage on the right-hand track, and the left-hand one is clear. The switch sends the train down the left track, and it arrives at the station. The switch seems clearly to be a cause of the train's arrival. So our theory shouldn't say: "In nearly all cases where the switch is flipped the other way, the train still arrives (because in nearly all cases, neither track is blocked). So the switch isn't a cause." Instead, we should say something like: "The worlds of interest aren't just any worlds in which the switch is flipped the other way. The worlds we care about are the ones in which, when the train is approaching the switch, the right track is already blocked, as was the case in the actual world. Among those worlds, the ones where the switch is flipped the other way are all worlds in which the train doesn't arrive. So the switch made a big difference, and is the cause of the train's arrival." This gives us another principle:

When determining what effect A's non-occurrence would have had on E, take for granted what happened before A's value was set.

proposal.

3.1

Towards a Theory

First, some machinery: when we're figuring out things like how likely it is that E will occur if C doesn't, what we want to know is how many non-C worlds are E worlds. So we need a set of possible worlds. Since the exogenous variables are what determine whether C and E occur, we'll use the exogenous variables to get the possible worlds. Each possible world is generated by a possible combination of allowable values of the exogenous variables. Some of the worlds in which each exogenous variable takes on one of its allowable values might still be too weird to consider, and if so, we eliminate those. We should also weight each world according to how likely it is. You can think of the weights as being encoded in the relative numbers of each type of world - e.g., if X is ten times more likely than Y, there will be ten X worlds for each Y world.

And a warning: we're going to want to know what counts as happening before C's value is set, and I won't have any very clear guidelines about this. Pinning down when C gets its value is itself not always straightforward, and even once we've decided that C gets its value at time t, deciding what to count as having happened already isn't as simple as holding fixed what happened before t and letting everything from that point on vary wildly1 7. Recall the train case at the end of the last section. Let's say the blockage on the right-hand track is a boulder that fell long after the switch was pulled. When considering whether the switch caused the train to arrive, should we hold fixed that the boulder was going to fall? The answer probably depends on the situation, and is certainly not a clear "no".18 In addition to

17

There are also some things that happened earlier that we don't want to hold fixed. If we hold fixed that a storm is coming, then in most worlds where the barometer falls, the lawn also gets wet. But the barometer isn't a cause of the wet lawn - they're both effects of the storm. A lot of these complications will be taken care of by the structural equations. We will start by narrowing down the potential causes to ones that are on a path to the effect, and there is no path from one effect of a common cause to another.

the aptness of the causal model itself, we can also argue over which things to count as being fixed at the time C takes on its value; this is part of the process of figuring out what caused what.

Now for some terminology: C=c has some positive control19 over whether E=e iff: holding fixed what happened before C's value was set, if C had not been c, E would have been less likely to be e. In other words, take the worlds in which things are the same as the actual world before C=c. Among those worlds, some go on to be C=c worlds and others don't (usually we're considering a specific salient alternative C=c/, but we needn't). Let's say n% of C-c worlds are E=e worlds, and m% of Cfc worlds are E=e worlds. C=c has some positive control over Eze iff n is larger than m.20

Now we can define

Actual Causation Among the variables V such that there's a path from V to E, the ones

that have the most positive control over E=e (or maybe the most above some thresh-old) are the actual causes of E=e.

(Note that if you're happy to allow causation to come in degrees, the amounts of positive control themselves will be most important: the more positive control V has over E, the more of a cause of E it is. If you want everything to be either a cause or a non-cause, then you can take the causes of E=e to be the variables that have the most positive control over it.)

I'll unimaginatively call the above theory controlism. Let's quickly see how it works with a simple example21_{: A hiker is hiking when suddenly a boulder dislodges itself from a cliff} is much more likely to be blocked. The switch sends the train down the left-hand track, and it arrives at the station. The more risky the right-hand track is, the more difference the switch makes, even though it actually turns out that both tracks are clear. If the right-hand track is risky enough (say, it's blocked 999,999 times out of a million), the switch does look like a cause of the train's arrival. So we shouldn't hold fixed that the right-hand track doesn't actually get blocked.

19

Positive as opposed to negative control. C=c has some negative control over whether E=e iff: had C not been c, E would have been more likely to be e. Negative control often doesn't come up - I just want

to flag the fact that even if you have a lot of control over whether a variable has a certain value, you can't be a cause of its taking on that value if you made it less likely that it would take on that value.

20

We can put a number on this if we want: C=c has degree x of positive control over whether E=e if n is x% larger than m. Numbers may not always be helpful though.

and hurtles toward her. Very luckily, she hears it falling and ducks, and her life is saved. This example has been used to make trouble for theories of causation that use chains of counterfactual dependence: had the boulder not fallen, the hiker wouldn't have heard it, and had she not heard it, she wouldn't have ducked. So on those theories it would seem that the boulder's falling is a cause of the hiker's living, when it's obviously not. Controlism doesn't have to deal with this problem. The boulder's falling decreases the chance that the hiker will live - she heard it by sheer luck and could easily have failed to get out of the way

- so it can't be a cause of her living." (The boulder's making a noise, however, increases the chance that she'll live, and it increases it a lot, since she wouldn't have ducked if she hadn't heard it. So the boulder's making a noise does count as a cause of the hiker's living, according to controlism. But this seems reasonable.)

Since this is a probabilistic theory of causation, let's get a traditional counterexample2 3 to such theories out of the way: A golfer makes a terrible shot, but the ball happens to bounce off a tree at just the right angle to roll back toward the green for a hole-in-one. The terrible shot was clearly a cause of the hole-in-one even though it vastly lowered its probability.

Controlism gets this result. Changing the way the golfer hit the ball would be a really good way to prevent the hole-in-one - in nearly all worlds where the golfer hits the ball differently, she doesn't get a hole-in-one. So her hit is definitely a cause. We think the terrible shot lowered the probability of the hole-in-one because terrible shots in general do

2 2

That is, the boulder's falling doesn't cause the hiker to live through that very falling-boulder incident.

It could, however, increase her chance of living through some other period of time, hence be a cause of her living through that period. It could be a cause of her living a longer life (because she realizes she really likes athletic jumping moves like the one she used to get out of the boulder's way, and she starts doing gymnastic exercises regularly, making her healthier), or a cause of her surviving the hiking trip (because the boulder scares away a wolverine that would have eaten her), or a cause of her surviving two minutes later (because she was about to eat a highly poisonous berry but she dropped it when she ducked). In a slightly different situation, the boulder could have not decreased her chance of living through that very falling-boulder incident - e.g., the boulder could have caused a rockslide that harmlessly pushed her out of the boulder's path - but then it would be a self-canceling threat, like C in Hall's Structure A, and would still not be a cause of her living through that moment, since without the boulder's falling, she would have

had the same chance of living through that moment. 2 3

vastly lower the probability of holes-in-one. This specific terrible shot, however, turned out not to be so terrible. It's true that every blade of grass and every gust of wind had to conspire to get the ball into the hole, so in that sense the shot didn't have that much control over whether a hole-in-one would happen. But what counts is that a) changing the way the ball is hit changes whether the hole-in-one happens, and b) the way the ball is hit has as much or more of an effect on whether the hole-in-one happens as any other potential cause. In fact, I think that every blade of grass and every gust of wind (not to mention the tree) involved in bringing the ball to the hole deserves equal credit for getting it there - all of them are causes of the hole-in-one. Controlism gets this result too: almost every world in which this blade of grass bent the other way, or that gust of wind was 0.2 km/h faster, or the tree was planted a bit to the left, is a world in which the ball doesn't get to the hole.2 4

3.2

Example

In the Appendix I'll cover many examples, and compare controlism not only with Halpern and Hitchcock's theory, but also with another SE account that involves default and deviant behavior, as well as one that doesn't. Here, as a way of summing up, I'll just run through the Assassin/Backup case from earlier, and compare Halpern and Hitchcock's verdict with mine.

In discussing cases involving murders and attempted murders, Halpern and Hitchcock say that a world is more normal to the degree that it involves fewer murders or attempts (even if the murderers are professional murderers and it's normal for people to do their jobs). So the world in which neither Assassin nor Backup attempts to kill Victim is more normal than the actual one. In this world, Assassin acts differently and Victim lives. So Assassin is

"You could also go the other way, and say that because there are so many equally good candidates for being causes of the hole-in-one, none of them get to count as a cause, even though they all made a big difference. This depends on whether you think lucky events have no real causes (so nothing is responsible for bringing them about) or many (so nothing stands out as being responsible for bringing them about).

a cause of his death.

I think this is the right verdict, but we reached it by the wrong means. Just as in the

case where Billy's doctor treated him, but we had to consider a world where no one was assigned to treat him in order to get her to come out as a cause of his recovery, in this case we have to consider a world where professional murderers just decide not to do their jobs in order to get Assassin to be a cause of Victim's death. From outside the theory, it's hard to see why such worlds are both more normal than the actual one and relevant to figuring out what caused what. Even if this tactic is plausible enough in toy cases like these, I think it will break down quickly once we move to real-world causation problems.

Controlism gets the same verdict, but I think for a better reason. When Assassin was deciding whether to kill Victim, she had a chance to change the odds of his dying. If she chose to kill him, he would die, and if she chose not to, there was a slight chance he would live, because there was a slight chance that something would go wrong and Backup would fail to act. Backup, on the other hand, never got a chance to act. Her chance would only have come if Assassin had failed. A slight change in the odds beats no chance to change them at all, so Assassin had more control over Victim's death.

What if we assume Backup is infallible? Then Assassin won't be changing the odds at all

by choosing whether to kill Victim herself. I have two points to make about this assumption: 1) It makes this case isomorphic to a game of "heads I win, tails you lose", where after the

coin lands heads we argue over whether that caused me to win. The way the coin lands isn't a cause of my winning; you and I, who agreed to the game, are the causes. Likewise, whoever set up the infallible assassination system is the real cause of Victim's death. Assassin and Backup are just acting out the consequences of that setup. 2) Even with this assumption, there's room to extend the notion of control in order to draw a distinction between Assassin and Backup: Assassin couldn't do anything to affect the odds of Victim's living because whatever she did, Victim would be dead by some means or other. Backup couldn't do

anything to affect the odds of Victim's living because she never got a chance to make a choice at all. In a case like this, where Victim is certainly dead no matter what happens,

if we want to count Assassin or Backup as an actual cause of the death anyway, getting to choose the means of death beats not getting to choose at all, so again Assassin has more control and is an actual cause of Victim's death. This isn't control in the probability-based sense I've characterized, but I think it's a legitimate extension of the concept, and in cases like this that involve extreme probabilities, we should feel free to extend it in this way.

4

Conclusion

Hall and Hitchcock et. al. have argued that unadorned structural equations accounts won't be able to accurately pick out the actual causes of events. Hall thinks we should reject structural equations accounts and adopt a different kind of theory involving default and deviant states. Hitchcock thinks we should stick with the structural equations approach but supplement it with default and deviant states in some way. I think the default/deviant distinction won't help, because it leads to new problems and fails to solve some of the old problems. If plain old structural equations accounts can't get us the results we want, I think it would be fruitful to pursue a theory in line with the idea that the cause(s) is(/are) whatever had the most control over whether the effect would occur.

Controlism gets a lot of advantages from being built on a structural equations account. To get a clear counterexample to controlism, we need a case where it seems right to model the situation in a certain way, and right to hold fixed that certain things happened or will happen when potential cause X gets its value, and also true that X has a lot of positive control over a certain event but isn't a cause of it, or vice versa. I think that in most potential counterexamples, the fact that something seems to have a lot of positive control and isn't a cause, or vice versa, will give us a good reason to rethink our model or the things

we hold fixed.

Appendix: Cases

As a kind of tutorial on how controlism works in practice, and to facilitate comparison with other theories, we'll see how it works in several cases. In each one, I'll say what verdict controlism gives and why, as well as what I think a few different SE theories would say.

One theory I'll be comparing is of course Halpern and Hitchcock's. From here on I'll call it normalism, since it has to do with the relative normality of worlds. To reiterate, it's this:

(Normalism) X is an actual cause of Y iff there is a way of setting the variables that meets

the criteria in section 1.1 and is such that the world in which the variables have those values and Y has a non-actual value, is at least as normal as the actual world. One of the other theories was introduced in Hitchcock (2007). It's based on the Principle of Sufficient Reason:

(PSR) When a set of variables all take their default value, they cannot by themselves cause

another variable to take a deviant value.

(I.e., when all the parents of X have default values, X does too). Models that satisfy PSR are self-contained - we don't need anything else to explain why variables in them take on deviant values. A causal network connecting X to Y is the set of all variables lying on any of the paths from X to Y. A causal network is self-contained iff every variable in it satisfies PSR (for parents that are also in the network) when all out-of-network variables take their actual values. (We'll see exactly what this means in practice when we get to the examples.) We can now state Hitchcock's principle for actual causation. He calls this principle "TC" for "token causation". I will call it geneticism, since it concerns whether child nodes match the

deviance, or lack thereof, of their parents:

(Geneticism) If the network between X and Y is self-contained, then X=x causes Y=y iff Y counterfactually depends on X.