Actual Causality

Joseph Y. Halpern

© 2016 Massachusetts Institute of Technology

All rights reserved. No part of this book may be reproduced in any form by any electronic or mechanical means (including photocopying, recording, or information storage and retrieval) without permission in writing from the publisher.

This book was set in L^AT_EX by Joseph Y. Halpern. Printed and bound in the United States of America.

Library of Congress Cataloging-in-Publication Data

Names: Halpern, Joseph Y., 1953

Title: Actual causality / Joseph Y. Halpern.

Description: Cambridge, MA : The MIT Press, [2016] — Includes bibliographical references and index.

Identifiers: LCCN 2016007205 — ISBN 9780262035026 (hardcover : alk. paper)

Subjects: LCSH: Functional analysis. — Probabilities. — Causation.

Classification: LCC QA320 .H325 2016 — DDC 122dc23 LC record available at http://lccn.loc.gov/2016007205

10987654321

For David, Sara, and Daniel (yet again). It’s hard to believe that you’re all old enough now to read and understand what I write (although I’m not sure you will want to!). The theory of causality presented here allows there to be more than one cause, and for blame to be shared. You’re all causes of my happiness (and have, at various times, been causes of other states of mind). I’m happy to share some blame with Mom for how you turned out.

Contents

Preface

1Introduction and Overview

Notes

2The HP Definition of Causality

2.1Causal Models

2.2A Formal Definition of Actual Cause

2.2.1A language for describing causality

2.2.2The HP definition of actual causality

2.3Examples

2.4Transitivity

2.5Probability and Causality

2.6Sufficient Causality

2.7Causality in Nonrecursive Models

2.8AC2(b^o) vs. AC2(b^u)

2.9Causal Paths

2.10Proofs

2.10.1Proof of Theorem 2.2.3

2.10.2Proof of Proposition 2.4.6

2.10.3Proof of Proposition 2.9.2

Notes

3Graded Causation and Normality

3.1Defaults, Typicality, and Normality

3.2Extended Causal Models

3.3Graded Causation

3.4More Examples

3.4.1Knobe effects

3.4.2Bogus prevention

3.4.3Voting examples

3.4.4Causal chains

3.4.5Legal doctrines of intervening causes

3.5An Alternative Approach to Incorporating Normality

Notes

4The Art of Causal Modeling

4.1Adding Variables to Structure a Causal Scenario

4.2Conservative Extensions

4.3Using the Original HP Definition Instead of the Updated Definition

4.4The Stability of (Non-)Causality

4.5The Range of Variables

4.6Dependence and Independence

4.7Dealing With Normality and Typicality

4.8Proofs

4.8.1Proof of Lemma 4.2.2

4.8.2Proof of Theorem 4.3.1

4.8.3Proofs and example for Section 4.4

Notes

5Complexity and Axiomatization

5.1Compact Representations of Structural Equations

5.2Compact Representations of the Normality Ordering

5.2.1Algebraic plausibility measures: the big picture

5.2.2Piggy-backing on the causal model

5.3The Complexity of Determining Causality

5.4Axiomatizing Causal Reasoning

5.5Technical Details and Proofs

5.5.1Algebraic plausibility measures: the details

5.5.2Proof of Theorems 5.3.1 (c) and 5.3.2 (b)

5.5.3Proof of Theorems 5.4.1, 5.4.2, and 5.4.4

Notes

6Responsibility and Blame

6.1A Naive Definition of Responsibility

6.2Blame

6.3Responsibility, Normality, and Blame

Notes

7Explanation

7.1Explanation: The Basic Definition

7.2Partial Explanations and Explanatory Power

7.3The General Definition of Explanation

Notes

8Applying the Definitions

8.1Accountability

8.2Causality in Databases

8.3Program Verification

8.4Last Words

Notes

References

Index

Preface

In The Graduate, Benjamin Braddock (Dustin Hoffman) is told that the future can be summed up in one word: “Plastics”. I still recall that in roughly 1990, Judea Pearl told me (and anyone else who would listen!) that the future was in causality. I credit Judea with inspiring my interest in causality. We ended up writing a paper on actual causality that forms the basis of this book. I was fortunate to have other coauthors who helped me on my (still ongoing) journey to a deeper understanding of causality and related notions, particularly Hana Chockler, Chris Hitchcock, and Tobi Gerstenberg. Our joint work features prominently in this book as well.

I thank Sander Beckers for extended email discussions on causality, Robert Maxton for his careful proofreading, and Chris Hitchcock, David Lagnado, Jonathan Livengood, Robert Maxton, and Laurie Paul for their terrific comments on earlier drafts of the book. Of course, I am completely responsible for any errors that remain.

My work on causality has been funded in part by the NSF, AFOSR, and ARO; an NSF grant for work on Causal Databases was particularly helpful.

Chapter 1

Introduction and Overview

Felix qui potuit rerum cognoscere causas. (Happy is one who could understand the causes of things.)

Virgil, Georgics ii.490

Causality plays a central role in the way people structure the world. People constantly seek causal explanations for their observations. Why is my friend depressed? Why won’t that file display properly on my computer? Why are the bees suddenly dying?

Philosophers have typically distinguished two notions of causality, which they have called type causality (sometimes called general causality) and actual causality (sometimes called token causality or specific causality; I use the term “actual causality” throughout the book). Type causality is perhaps what scientists are most concerned with. These are general statements, such as “smoking causes lung cancer” and “printing money causes inflation”. By way of contrast, actual causality focuses on particular events: “the fact that David smoked like a chimney for 30 years caused him to get cancer last year”; “the fact that an asteroid struck the Yucatan Peninsula roughly 66 million years ago caused the extinction of the dinosaurs”; “the car’s faulty brakes caused the accident (not the pouring rain or the driver’s drunkenness)”.

The reason that scientists are interested in type causality is that a statement of type causality allows them to make predictions: “if you smoke a lot, you will get (or, perhaps better, are likely to get) lung cancer”; “if a government prints $1 billion, they are likely to see 3% inflation”. Because actual causality talks about specific instances, it is less useful for making predictions, although it still may be useful in understanding how we can prevent outcomes similar to that specific instance in the future. We may be interested in knowing that Robert’s sleeping in caused him to miss the appointment, because it suggests that getting him a good alarm clock would be useful.

Actual causality is also a critical component of blame and responsibility assignment. That is perhaps why it arises so frequently in the law. In the law, we often know the relevant facts but still need to determine causality. David really did smoke and he really did die of lung cancer; the question is whether David’s smoking was the cause of him dying of lung cancer. Similarly, the car had faulty brakes, it was pouring rain, and the driver was drunk. Were the faulty brakes the cause of the accident? Or was it the rain or the driver’s drunkenness? Of course, as the dinosaur-extinction example shows, questions of actual causality can also be of great interest to scientists. And issues of type causality are clearly relevant in the law as well. A jury is hardly likely to award large damages to David’s wife if they do not believe that smoking causes cancer.

As these examples show, type causality is typically forward-looking and used for prediction. Actual causality, in contrast, is more often backward-looking. With actual causality, we know the outcome (David died of lung cancer; there was a car accident), and we retrospectively ask why it occurred. Type causality and actual causality are both of great interest and are clearly intertwined; however, as its title suggests, in this book, I focus almost exclusively on actual causality and notions related to it.

Most of the book focuses on how to define these notions. What does it mean to say that the faulty brakes were the cause of the accident and not the driver’s drunkenness? What does it mean to say that one voter is more responsible than another for the outcome of an election? Defining causality is surprisingly difficult; there have been many attempts to do so, going back to Aristotle. The goal is to get a definition that matches our natural language usage of the word “causality” (and related words such as “responsibility” and “blame”) and is also useful, so that it can be used, for example, to provide guidance to a jury when deciding a legal case, to help a computer scientist decide which line in a program is the cause of the program crashing, and to help an economist in determining whether instituting an austerity program caused a depression a year later.

The modern view of causality arguably dates back to Hume. (For the exact citation, see the notes at the end of this chapter.) Hume wrote:

We may define a cause to be an object followed by another, and where all the objects, similar to the first, are followed by objects similar to the second. Or, in other words, where, if the first object had not been, the second never had existed.

Although Hume says “in other words”, he seems to be conflating two quite different notions of causality here. The first has been called the regularity definition. According to this definition, we can discern causality by considering just what actually happens, specifically, which events precede others. Roughly speaking, A causes B if As typically precede Bs. (Note that this is type causality.)

By way of contrast, the second notion involves counterfactuals, statements counter to fact. It does not consider just what actually happened but also what might have happened, had things been different. Research in psychology has shown that such counterfactual thinking plays a key role in determining causality. People really do consider “what might have been” as well as “what was”.

A recent experiment illustrates this nicely. Compare Figures 1.1(a) and (b). In both, ball A is heading toward a goal. In Figure 1.1(a), its path is blocked by a brick; in Figure 1.1(b), it is not blocked. Now ball B hits A. As a result, instead of going straight toward the goal, A banks off the wall and goes into the goal. In both figures, the same thing happens: B hits A, then A banks off the wall and goes into the goal. But when people are asked whether B hitting A caused A to go into the goal, they typically answer “yes” for Figure 1.1(a) and “no” for Figure 1.1(b). Thus, they are clearly taking the counterfactual (what would have happened had A not hit B) into account when they answer the question.

Figure 1.1: Considering counterfactuals.

There have been many definitions of causality that involve counterfactuals. In this book, I consider only one of them (although it comes in three variants), which Judea Pearl and I developed; this has been called the Halpern-Pearl definition; I call it simply the “HP definition” throughout this book. I focus on this definition partly because it is the one that I like best and am most familiar with. But it has a further advantage that is quite relevant to the purposes of this book: it has been extended to deal with notions such as responsibility, blame, and explanation. That said, many of the points that I make for the HP definition can be extended to other approaches to causality involving counterfactuals.

Pretty much all approaches involving counterfactuals take as their starting point what has been called in the law the “but-for” test. A is a cause of B if, but for A, B would not have happened. Put another way, this says that the occurrence of A is necessary for the occurrence of B. Had A not occurred, B would not have occurred. When we say that A caused B, we invariably require (among other things) that A and B both occurred, so when we contemplate A not happening, we are considering a counterfactual. This kind of counterfactual reasoning clearly applies in the example above. In Figure 1.1(a), had A not hit B, B would not have landed in the goal, so A hitting B is viewed as a cause of B going in the goal. In contrast, in Figure 1.1(b), B would have gone in the goal even if it had not been hit by A, so A hitting B is not viewed as a cause of B going in the goal.

But the but-for test is not always enough to determine causality. Consider the following story, to which I will return frequently:

Suzy and Billy both pick up rocks and throw them at a bottle. Suzy’s rock gets there first, shattering the bottle. Because both throws are perfectly accurate, Billy’s would have shattered the bottle had it not been preempted by Suzy’s throw.

Here the but-for test fails. Even if Suzy hadn’t thrown, the bottle would have shattered. Nevertheless, we want to call Suzy’s throw a cause of the bottle shattering and do not want to call Billy’s throw a cause.

The HP definition deals with this problem by allowing the but-for test to be applied under certain contingencies. In the case of Suzy and Billy, we consider the contingency where Billy does not throw a rock. Clearly, if Billy does not throw, then had Suzy not thrown, the bottle would not have shattered.

There is an obvious problem with this solution: we can also consider the contingency where Suzy does not throw. Then the but-for definition would say that Billy’s throw is also a cause of the bottle shattering. But this is certainly not what most people would say. After all, it was Suzy’s rock that hit the bottle (duh … ). To capture the “duh” reaction, the definition of causality needs another clause. Roughly speaking, it says that, under the appropriate contingencies, Suzy’s throw is sufficient for the bottle to shatter. By requiring sufficiency to hold under an appropriate set of contingencies, which takes into account what actually happened (in particular, that Billy’s rock didn’t hit the bottle), we can declare Suzy’s throw to be a cause but not Billy’s throw. Making this precise in a way that does not fall prey to numerous counterexamples is surprisingly difficult.

I conclude this introduction with a brief overview of the rest of the book. In the next chapter, I carefully formalize the HP definition. Doing so requires me to introduce the notion of structural equations. The definition involves a number of technical details; I try to introduce them as gently as possible. I then show how the definition handles the rock-throwing example and a number of other subtle examples that have come up in the philosophy and law literatures. I also show how it deals with issues of transitivity, and how it can be extended to deal with situations where we have probabilities on outcomes.

There are several examples that the basic HP definition does not handle well. It is well known in the psychology literature that when people evaluate causality, they take into account considerations of normality and typicality. As Kahneman and Miller have pointed out, “[in determining causality], an event is more likely to be undone by altering exceptional than routine aspects of the causal chain that led to it”. In Chapter 3, I present an extension of the basic HP definition that takes normality into account and show that the extension deals with some of the problematic cases mentioned in Chapter 2.

The HP definition of causality is model-relative. A can be a cause of B in one model but not in another. As a result, two opposing lawyers can disagree about whether A is a cause of B even if they are both working with the HP definition; they are simply using different models. This suggests that it is important to have techniques for distinguishing whether one model is more appropriate than another. More generally, we need techniques for deciding what makes a model “good”. These issues are the focus of Chapter 4.

For an approach to causality to be psychologically plausible, humans need to be able to represent causal information in a reasonably compact way. For an approach to be computationally feasible, the questions asked must be computable in some reasonable time. Issues of compact representations and computational questions are discussed in Chapter 5. In addition, I discuss an axiomatization of the language of causality.

Causality is an all-or-nothing concept; either A is a cause of B or it is not. The HP definition would declare each voter in an 11–0 vote a cause of the outcome; it would also call each of the six people who voted in favor of the outcome a cause in a 6–5 victory. But people tend to view the degree of responsibility as being much lower in the first case than the second. In Chapter 6, I show how the HP definition can be extended to provide notions of responsibility and blame that arguably capture our intuitions in these cases.

Causality and explanation are clearly closely related. If A is a cause of B, then we think of A as providing an explanation as to why B happened. In Chapter 7, I extend the structural-model approach to causality to provide a definition of explanation.

I conclude in Chapter 8 with some discussion of where we stand and some examples of applications of causality.

I have tried to write this book in as modular a fashion as possible. In particular, all the chapters following Chapter 2 can be read independently of each other, in any order, except that Sections 4.4, 4.7, 5.2, and 6.3 depend on Section 3.2. Even in Chapter 2, the only material that is really critical in later chapters is Sections 2.1–2.3. I have tried to make the main text in each chapter as readable as possible. Thus, I have deferred technical proofs to a final section in each chapter (which can be skipped without loss of continuity) and have avoided in some cases a careful discussion of the many viewpoints on some of the more subtle issues. There are a few proofs in the main text; these are ones that I would encourage even non-mathematical readers to try reading, just to get comfortable with the notation and concepts. However, they can also be skipped with no loss of continuity.

Each chapter ends with a section of notes, which provides references to material discussed in the chapter, pointers to where more detailed discussions of the issues can be found, and, occasionally, more detail on some material not covered in the chapter.

There has been a great deal of work on causality in philosophy, law, economics, and psychology. Although I have tried to cover the highlights, the bibliography is by no means exhaustive. Still, I hope that there is enough material there to guide the interested reader’s exploration of the literature.

Notes

The hypothesis that the dinosaur extinction was due to a meteor striking the earth was proposed by Luis and Walter Alvarez [Alvarez et al. 1980]. While originally viewed as quite radical, this hypothesis is getting increasing support from the evidence, including evidence of a meteor strike in Chicxulub in the Yucatan Peninsula at the appropriate time. Pälike [2013] provides a summary of research on the topic.

The distinction between actual causality and type causality has been related to the distinction between causes of effects—that is, the possible causes of a particular outcome—and effects of causes—that is, the possible effects of a given event. Some have claimed that reasoning about actual causality is equivalent to reasoning about causes of effects, because both typically focus on individual events (e.g., the particular outcome of interest), whereas reasoning about type causality is equivalent to reasoning about effects of causes, because when thinking of effects of causes, we are typically interested in understanding whether a certain type of behavior brings about a certain type of outcome. Although this may typically be the case, it is clearly not always the case; Livengood [2015] gives a number of examples that illustrate this point. Dawid [2007] argued that reasoning about effects of causes is typically forward-looking and reasoning about causes of effects is typically backward-looking. The idea that type causality is forward-looking and actual causality is backward-looking seems to have been first published by Hitchcock [2013], although as the work of Dawid [2007] shows, it seems to have been in the air for a while.

The exact relationship between actual and type causality has been a matter of some debate. Some have argued that two separate theories are needed (see, e.g., [Eells 1991; Good 1961a; Good 1961b; Sober 1984]); others (e.g., [Hitchcock 1995]) have argued for one basic notion. Statisticians have largely focused on type causality and effects of causes; Dawid, Faigman, and Fienberg [2014] discuss how statistical techniques can be used for causes of effects. My current feeling is that type causation arises from many instances of actual causation, so that actual causation is more fundamental, but nothing in this book hinges on that.

Work on causality goes back to Aristotle [Falcon 2014]. Hume [1748] arguably started more modern work on causality. Robinson [1962] discusses Hume’s two definitions of causality in more detail. Perhaps the best-known definition involving regularity (in the spirit of the first sentence in the earlier Hume quotation) is due to Mackie [1974], who introduced what has been called the INUS condition: A is a cause of B if A is an insufficient but necessary part of a condition that is itself unnecessary but sufficient for B. The basic intuition behind this condition is that A should be a cause of B if A is necessary and sufficient for B to happen, but there are many counterexamples to this simple claim. Ball A hitting B is not necessary for B to go into the goal, even in Figure 1.1(b); removing the brick would also work. Mackie modified this basic intuition by taking A to be a cause of B if there exist X and Y such that adding (A∧X) ∨Y is necessary and sufficient for B, but neither A nor X by itself is sufficient to entail B. If this definition is taken apart carefully, it gives us INUS:

• A is, by assumption, insufficient for B (that’s the I in INUS).
• A is a necessary part of condition sufficient for B, namely, A ∧ X (that’s the N and the S)—A is necessary because X by itself is not sufficient. Note that A is necessary in this A ∧ X because, by assumption, X by itself is not sufficient.
• A ∧ X is unnecessary for B (that’s the U) because there is another cause of B, namely, Y.

Note that the notion of necessity and sufficiency used here is related to, but different from, that used when considering counterfactual theories of causality. While Mackie’s INUS condition is no longer viewed as an appropriate definition of causality (e.g., it has trouble with cases of preemption such as the rock-throwing example, where there is a backup ready to bring about the outcome in case what we would like to call the actual cause fails to do so), work on defining causality in terms of regularity still continues; Baumgartner [2013] and Strevens [2009] are recent exemplars.

I make no attempt to do justice to all the alternative approaches to defining causality; my focus is just the HP definition, which was introduced in [Halpern and Pearl 2001; Halpern and Pearl 2005a]. As I said, the HP definition is formalized carefully in Chapter 2. There is some discussion of alternative approaches to defining causality in the notes to Chapter 2. Paul and Hall [2013] provide a good overview of work on causality, along with a critical analysis of the strengths and weaknesses of various approaches.

The experiment with ball B hitting ball A is discussed by Gerstenberg, Goodman, Lagnado, and Tenenbaum [2014]. Figure 1.1 is taken from their paper (which also reports on a number of other related experiments).

The rock-throwing example is due to Lewis [2000]. It is but one of many examples of preemption in both the law and philosophy literatures.

Hart and Honoré [1985] discuss the but-for condition and, more generally, the use of actual causality in the law; it is the classic reference on the topic. Honoré [2010] and Moore [2009] provide more recent discussions. Lewis [1973a] initiated modern work in philosophy on counterfactual theories of causation; Menzies [2014] provides an overview of the work on the counterfactual approach to causation in philosophy. Sloman [2009] discusses how people ascribe causality, taking counterfactuals into account.

Although the focus in this book is on work on causality in philosophy, law, and psychology, causality also has a long tradition in other fields. The seminal work of Adam Smith [1994], considered a foundation of economics, is titled “An Inquiry into the Nature and Causes of the Wealth of Nations”. Hoover [2008] provides an overview of work on causality in economics. Statistics is, of course, very concerned with causal influence; Pearl [2009] provides an overview of work in statistics on causation. Discussion of causality in physics comes up frequently, particularly in the context of the Einstein-Podolsky-Rosen paradox, where the question is whether A can be a cause of B even though B happened so shortly after A that a signal traveling at the speed of light from A could not have reached B (see [Bell 1964]). More recently, (the HP definition of) causality has been applied in computer science; I discuss some of these applications in Chapter 8. The volume edited by Illari, Russo, and Williamson [2011] provides a broad overview of the role of causality in the social sciences, health sciences, and physical sciences.

Chapter 2

The HP Definition of Causality

The fluttering of a butterfly’s wing in Rio de Janeiro, amplified by atmospheric currents, could cause a tornado in Texas two weeks later.

Edward Lorenz

There is only one constant, one universal. It is the only real truth. Causality. Action, reaction. Cause and effect.

Merovingian, The Matrix Reloaded

In this chapter, I go through the HP definition in detail. The HP definition is a formal, mathematical definition. Although this does add some initial overhead, it has an important advantage: it prevents ambiguity about whether A counts as a cause of B. There is no need, as in many other definitions, to try to understand how to interpret the words. For example, recall the INUS condition from the notes in Chapter 1. For A to be a cause of B under this definition, A has to be a necessary part of a condition that is itself unnecessary but insufficient for B. But what is a “condition”? The formalization of INUS suggests that it is a formula or set of formulas. Is there any constraint on this set? What language is it expressed in?

This lack of ambiguity is obviously critical if we want to apply causal reasoning in the law. But as we shall see in Chapter 8, it is equally important in other applications of causality, such as program verification, auditing, and database queries. However, even if there is no ambiguity about the definition, it does not follow that there can be no disagreement about whether A is a cause of B.

To understand how this can be the case, it is best to outline the general approach. The first step in the HP definition involves building a formal model in which causality can be determined unambiguously. Among other things, the model determines the language that is used to describe the world. We then define only what it means for A to be a cause of B in model M. It is possible to construct two closely related models M₁ and M₂ such that A is a cause of B in M₁ but not in M₂. I do not believe that there is, in general, a “right” model; in any case, the definition is silent on what makes one model better than another. (This is an important issue, however. I do think that there are criteria that can help judge whether one model is better than another; see below and Chapter 4 for more on this point.) Here we already see one instance where, even if there is agreement regarding the definition of causality, we can get disagreements regarding causality: there may be disagreement about which model better describes the real world. This is arguably a feature of the definition. It moves the question of actual causality to the right arena—debating which of two (or more) models of the world is a better representation of those aspects of the world that one wishes to capture and reason about. This, indeed, is the type of debate that goes on in informal (and legal) arguments all the time.

2.1 Causal Models

The model assumes that the world is described in terms of variables; these variables can take on various values. For example, if we are trying to determine whether a forest fire was caused by lightning or an arsonist, we can take the world to be described by three variables:

• FF for forest fire, where FF = 1 if there is a forest fire and FF = 0 otherwise;
• L for lightning, where L = 1 if lightning occurred and L = 0 otherwise;
• MD for match dropped (by arsonist), where MD = 1 if the arsonist dropped a lit match and MD = 0 otherwise.

If we are considering a voting scenario where there are eleven voters voting for either Billy or Suzy, we can describe the world using twelve variables, V₁, …, V₁₁, W, where V_i = 0 if voter i voted for Billy and V₁ = 1 if voter i voted for Suzy, for i = 1,…, 11, W = 0 if Billy wins, and W = 1 if Suzy wins.

In these two examples, all the variables are binary, that is, they take on only two values. There is no problem allowing a variable to have more than two possible values. For example, the variable V_i could be either 0, 1, or 2, where V_i = 2 if i does not vote; similarly, we could take W = 2 if the vote is tied, so neither Billy nor Suzy wins.

The choice of variables determines the language used to frame the situation. Although there is no “right” choice, clearly some choices are more appropriate than others. For example, if there is no variable corresponding to smoking in model M, then in M, smoking cannot be a cause of Sam’s lung cancer. Thus, if we want to consider smoking as a potential cause of lung cancer, M is an inappropriate model. (As an aside, the reader may note that here and elsewhere I put “right” in quotes; that is because it is not clear to me that the notion of being “right” is even well defined.)

Some variables may have a causal influence on others. This influence is modeled by a set of structural equations. For example, if we want to model the fact that if the arsonist drops a match or lightning strikes then a fire starts, we could use the variables MD, FF, and L as above, with the equation FF = max(L, MD); that is, the value of the variable FF is the maximum of the values of the variables MD and L. This equation says, among other things, that if MD = 0 and L = 1, then FF = 1. The equality sign in this equation should be thought of more like an assignment statement in programming languages; once we set the values of MD and L, the value of FF is set to their maximum. However, despite the equality, a forest fire starting some other way does not force the value of either MD or L to be 1.

Alternatively, if we want to model the fact that a fire requires both a lightning strike and a dropped match (perhaps the wood is so wet that it needs two sources of fire to get going), then the only change in the model is that the equation for FF becomes FF = min(L, MD); the value of FF is the minimum of the values of MD and L. The only way that FF = 1 is if both L = 1 and MD = 1.

Just a notational aside before going on: I sometimes identify binary variables with primitive propositions in propositional logic. And, as in propositional logic, the symbols ∧,∨, and ¬ are used to denote conjunction, disjunction, and negation, respectively. With that identification, instead of writing max(L, MD ), I write L ∨ MD; instead of writing min(L, MD), I write L ∧ MD; and instead of writing 1 − L or 1 − MD, I write L or MD. Most people seem to find the logic notation easier to absorb. I hope that the intention will be clear from context.

Going on with the forest-fire example, it is clear that both of these models are somewhat simplistic. Lightning does not always result in a fire, nor does dropping a lit match. One way of dealing with this would be to make the assignment statements probabilistic. For example, we could say that the probability that FF = 1 conditional on L = 1 is .8. I discuss this approach in more detail in Section 2.5. It is much simpler to think of all the equations as being deterministic and then use enough variables to capture all the conditions that determine whether there is a forest fire. One way to do this is simply to add those variables explicitly. For example, we could add variables that talk about the dryness of the wood, the amount of undergrowth, the presence of sufficient oxygen (fires do not start so easily on the top of high mountains), and so on. If a modeler does not want to add all these variables explicitly (the details may simply not be relevant to the analysis), then another alternative is to use a single variable, say U, which intuitively incorporates all the relevant factors, without describing them explicitly. The value of U would determine whether the lightning strikes and whether the match is dropped by the arsonist.

The value of U could also determine whether both the match and lightning are needed to start a fire or just one of them suffices. For simplicity, rather than using U in this way, I consider two causal models. In one, called the conjunctive model, both the match and lightning are needed to start the forest fire; in the other, called the disjunctive model, only one is needed. In each of these models, U determines only whether the lightning strakes and whether the match is dropped. Thus, I assume that U can take on four possible values of the form (i, j), where i and j are each either 0 or 1. Intuitively, i describes whether the external conditions are such that the lightning strikes (and encapsulates all such conditions, e.g., humidity and temperature), and j describes whether the arsonist drops the match (and thus encapsulates all the psychological conditions that determine this). For future reference, let U₁ and U₂ denote the components of the value of U in this example, so that if U = (i, j), then U₁ = i and U₂ = j.

Here I have assumed a single variable U that determines the values of both L and MD. I could have split U into two variables, say U_L and U_MD, where U_L determines the value of L and U_MD determines the value of MD. It turns out that whether we use a single variable with four possible values or two variables, each with two values, makes no real difference. (Other modeling choices can make a big difference—I return to this point below.)

It is reasonable to ask why I have chosen to describe the world in terms of variables and their values, related via structural equations. Using variables and their values is quite standard in fields such as statistics and econometrics for good reason; it is a natural way to describe situations. The examples later in this section should help make that point. It is also quite close to propositional logic; we can think of a primitive proposition in classical logic as a binary variable, whose values are either true or false. Although there may well be other reasonable ways of representing the world, this seems like a useful approach.

As for the use of structural equations, recall that my goal is to give a definition of actual causality in terms of counterfactuals. Certainly if the but-for test applies, so that, but for A, B would not have happened, I want the definition to declare that A is a cause of B. That means I need a model rich enough to capture the fact that, but for A, B would not have happened. Because I model the world in terms of variables and their values, the A and the B in the definition will be statements such as X = x and Y = y. Thus, I want to say that if X had taken on some value x′ different from x, then Y would not have had value y. To do this, I need a model that makes it possible to consider the effect of intervening on X and changing its value from x to x′. Describing the world in terms of variables and their values makes it easy to describe such interventions; using structural equations makes it easy to define the effect of an intervention.

This can be seen already in the forest-fire example. In the conjunctive model, if we see a forest fire, then we can say that if the arsonist hadn’t dropped a match, there would have been no forest fire. This follows in the model because setting MD = 0 makes FF = min(L, MD ) = 0.

Before going into further details, I give an example of the power of structural equations. Suppose that we are interested in the relationship between education, an employee’s skill level, and his salary. Suppose that education can affect skill level (but does not necessarily always do so, since a student may not pay attention or have poor teachers) and skill level affects salary. We can give a simplified model of this situation using four variables:

• E for education level, with values 0 (no education), 1 (one year of education), and 2 (2 years of education);
• SL for skill level, with values 0 (low), 1 (medium), and 2 (high);
• S for salary, which for simplicity we can also take to have three values: 0 (low), 1 (medium), and 2 (high);
• U, a variable that determines whether education will have an impact on skill level.

There are two relevant equations. The first says that education determines skill level, provided that the external conditions (determined by U) are favorable, and otherwise has no impact:

The second says that skill level determines salary:

Although this model significantly simplifies some complicated relationships, we can already use it to answer interesting questions. Suppose that, in fact, we observe that Fred has salary level 1. We can thus infer that U = E = SL = 1. We can ask what Fred’s salary level would have been had he attained education level 2. This amounts to intervening and setting E = 2. Since U = 1, it follows that SL = 2 and thus S = 2; Fred would have a high salary. Similarly, if we observe that Fred’s education level is 1 but his salary is low, then we can infer that U = 0 and SL = 0, and we can say that his salary would still be low even if he had an extra year of education.

A more sophisticated model would doubtless include extra factors and more sensitive dependencies. Still, even at this level, I hope it is clear that the structural equations framework can give reasonable answers.

When working with structural equations, it turns out to be conceptually useful to split the variables into two sets: the exogenous variables, whose values are determined by factors outside the model, and the endogenous variables, whose values are ultimately determined by the exogenous variables. In the forest-fire example, the variable U, which determines whether the lightning strikes and whether the arsonist drops a match, is exogenous; the variables FF, L, and MD are endogenous. The value of U determines the values of L and MD, which in turn determine the value of FF. We have structural equations for the three endogenous variables that describe how this is done. In general, there is a structural equation for each endogenous variable, but there are no equations for the exogenous variables; as I said, their values are determined by factors outside the model. That is, the model does not try to “explain” the values of the exogenous variables; they are treated as given.

The split between exogenous and endogenous variables has another advantage. The structural equations are all deterministic. As we shall see in Section 2.5, when we want to talk about the probability that A is a cause of B, we can do so quite easily by putting a probability distribution on the values of the exogenous variables. This gives us a way of talking about the probability of there being a fire if lightning strikes, while still using deterministic equations.

In any case, with this background, we can formally define a causal model M as a pair , where is a signature, which explicitly lists the endogenous and exogenous variables and characterizes their possible values, and defines a set of structural equations, relating the values of the variables. In the next two paragraphs, I define and formally; the definitions can be skipped by the less mathematically inclined reader.

A signature is a tuple , where is a set of exogenous variables, is a set of endogenous variables, and associates with every variable Y a nonempty set of possible values for Y (i.e., the set of values over which Y ranges). As suggested earlier, in the forest-fire example, we can take ; that is, U is exogenous, consists of the four possible values of U discussed earlier, , and .

The function associates with each endogenous variable a function denoted F_X such that F_X maps to . (Recall that is the set consisting of all the variables in either or that are not in {X}. The notation denotes the cross-product of , where Z ranges over the variables in ; thus, if , then consists of tuples of the form (z₁, …, z_k), where z_i is a possible value of Z_i, for i = 1, …, k.) This mathematical notation just makes precise the fact that F_X determines the value of X, given the values of all the other variables in . In the running forest-fire example, F_FF would depend on whether we are considering the conjunctive model (where both the lightning strike and dropped match are needed for the fire) or the disjunctive model (where only one of the two is needed). In the conjunctive model, as I have already noted, F_FF (L, MD, U) = 1 iff (“iff” means “if and only if”—this standard abbreviation is used throughout the book) min(L, MD ) = 1; in the disjunctive model, F_FF (L, MD, U ) = 1 iff max(L, MD) = 1. Note that, in both cases, the value of F_FF is independent of U; it depends only on the values of L and MD. Put another way, this means that when we hold the values of L and MD fixed, changing the value of U has no effect on the value of F_FF. While the values of MD and L do depend on the value of U, writing things in this way allows us to consider the effects of external interventions that may override U. For example, we can consider F_FF (0, 1, (1, 1)); this tells us what happens if the lightning does not strike and the match is dropped, even if the value of U is such that L = 1. In what follows, we will be interested only in interventions on endogenous variables. As I said, we take the values of the exogenous variables as given, so do not intervene on them.

The notation F_X (Y, Y′, U ) for the equation describing how the value of X depends on Y, Y′, and U is not terribly user-friendly. I typically simplify notation and write X = Y + U instead of F_X (Y, Y′, U) = Y + U. (Note that the variable Y′ does not appear on the right-hand side of the equation. That means that the value of X does not depend on that of Y′.) With this simplified notation, the equations for the forest-fire example are L = U₁, MD = U₂, and either FF = min(L, MD ) or FF = max(L, MD), depending on whether we are considering the conjunctive or disjunctive model.

Although I may write something like X = Y + U, the fact that X is assigned Y + U does not imply that Y is assigned X − U; that is, F_Y (X, Y′, U ) = X − U does not necessarily hold. The equation X = Y + U implies that if Y = 3 and U = 2, then X = 5, regardless of how Y′ is set. Going back to the forest-fire example, setting FF = 0 does not mean that the match is “undropped”! This asymmetry in the treatment of the variables on the left- and right-hand sides of the equality sign means that (the value of) Y can depend on (the value of) X without X depending on Y. This, in turn, is why causality is typically asymmetric in the formal definition that I give shortly: if A is a cause of B, then B will typically not be a cause of A. (See the end of Section 2.7 for more discussion of this issue.)

As I suggested earlier, the key role of the structural equations is that they allow us to determine what happens if things had been other than they were, perhaps due to an external intervention; for example, the equations tell us what would happen if the arsonist had not dropped a match (even if in fact he did). This will be critical in defining causality.

Since a world in a causal model is described by the values of variables, understanding what would happen if things were other than they were amounts to asking what would happen if some variables were set to values perhaps different from their actual values. Setting the value of some variable X to x in a causal model results in a new causal model denoted M_X←x. In the new causal model, the equation for X is simple: X is just set to x; the remaining equations are unchanged. More formally, , where is the result of replacing the equation for X in by X = x and leaving the remaining equations untouched. Thus, if M^C is the conjunctive model of the forest fire, then in , the model that results from intervening in M ^C by having the arsonist not drop a match, the equation MD = U₂ is replaced by MD = 0.

As I said in Chapter 1, the HP definition involves counterfactuals. The equations in a causal model can be given a straightforward counterfactual interpretation. An equation such as x = F_X (u, y) should be thought of as saying that in a context where the exogenous variable U has value u, if Y were set to y by some means (not specified in the model), then X would take on the value x. As I noted earlier, when Y is set to y, this can override the value of Y according to the equations. For example, MD can be set to 0 even in a context where the exogenous variable U = (1, 1), so MD would be 1.

It may seem somewhat circular to use causal models, which clearly already encode causal relationships, to define causality. There is some validity to this concern. After all, how do we determine whether a particular equation holds? We might believe that dropping a lit match results in a forest burning in part because of our experience with lit matches and dry wood, and thus believe that a causal relationship holds between the lit match and the dry wood. We might also have a general theory of which this is a particular outcome, but there too, roughly speaking, the theory is being understood causally. Nevertheless, I would claim that this definition is useful. In many examples, there is general agreement as to the appropriate causal model. The structural equations do not express actual causality; rather, they express the effects of interventions or, more generally, of variables taking on values other than their actual values.

Of course, there may be uncertainty about the effects of interventions, just as there may be uncertainty about the true setting of the values of the exogenous variables in a causal model. For example, we may be uncertain about whether smoking causes cancer (this represents uncertainty about the causal model), uncertain about whether a particular patient Sam actually smoked (this is uncertainty about the value of the exogenous variables that determine whether Sam smokes), and uncertain about whether Sam’s brother Jack, who did smoke and got cancer, would have gotten cancer had he not smoked (this is uncertainty about the effect of an intervention, which amounts to uncertainty about the values of exogenous variables and possibly the equations). All this uncertainty can be described by putting a probability on causal models and on the values of the exogenous variables. We can then talk about the probability that A is a cause of B. (See Section 2.5 for further discussion of this point.)

Using (the equations in) a causal model, we can determine whether a variable Y is (in)dependent of variable X. Y depends on X if there is some setting of all the variables in other than X and Y such that varying the value of X in that setting results in a variation in the value of Y; that is, there is a setting of the variables other than X and Y and values x and x′ of X such that . (The represents the values of all the other variables. This vector notation is a useful shorthand that is used throughout the book; I say more about it for readers unfamiliar with it at the end of the section.) Note that by “dependency” here I mean something closer to “immediate dependency” or “direct dependency”. This notion of dependency is not transitive; that is, if X₁ depends on X₂ and X₂ depends on X₃, then it is not necessarily the case that X₁ depends on X₃. If Y does not depend on X, then Y is independent of X.

Whether Y depends on X may depend in part on the variables in the language. In the original description of the voting scenario, we had just twelve variables, and W depended on each of V₁, …, V₁₁. However, suppose that the votes are tabulated by a machine. We can add a variable T that describes that final tabulation of the machine, where T can have any value of the form (t₁, t₂), where t₁ represents the number of votes for Billy and t₂ represents the number of votes for Suzy (so that t₁ and t₂ are non-negative integers whose sum is 11). Now W depends only on T, while T depends on V₁, …, V₁₁.

In general, every variable can depend on every other variable. But in most interesting situations, each variable depends on relatively few other variables. The dependencies between variables in a causal model M can be described using a causal network (sometimes called a causal graph), consisting of nodes and directed edges. I often omit the exogenous variables from the graph. Thus, both Figure 2.1(a) (where the exogenous variable is included) and Figure 2.1(b) (where it is omitted) would be used to describe the forest-fire example. Figure 2.1(c) is yet another description of the forest-fire example, this time replacing the single exogenous variable U by an exogenous variable U₁ that determines L and an exogenous variable U₂ that determines MD. Again, Figure 2.1(b) is the causal graph that results when the exogenous variables in Figure 2.1(c) are omitted. Since all the “action” happens with the endogenous variables, Figure 2.1(b) really gives us all the information we need to analyze this example.

Figure 2.1: A graphical representation of structural equations.

The fact that there is a directed edge from U to both L and MD in Figure 2.1(a) (with the direction marked by the arrow) says that the value of the exogenous variable U affects the value of L and MD, but nothing else affects it. The directed edges from L and MD to FF say that only the values of MD and L directly affect the value of FF. (The value of U also affects the value of FF, but it does so indirectly, through its effect on L and MD.)

Causal networks convey only the qualitative pattern of dependence; they do not tell us how a variable depends on others. For example, the same causal network would be used for both the conjunctive and disjunctive models of the forest-fire example. Nevertheless, causal networks are useful representations of causal models.

I will be particularly interested in causal models where there are no circular dependencies. For example, it is not the case that X depends on Y and Y depends X or, more generally, that X₁ depends on X₂, which depends on X₃, which depends on X₄, which depends on X₁. Informally, a model is said to be recursive (or acyclic) if there are no such dependency cycles (sometimes called feedback cycles) among the variables.

The intuition that the values of exogenous variables are determined by factors outside the model is formalized by requiring that an exogenous variable does not depend on any other variables. (Actually, nothing in the following discussion changes if we allow an exogenous variable to depend on other exogenous variables. All that matters is that an exogenous variable does not depend on any endogenous variables.) In a recursive model, we can say more about how the values of endogenous are determined. Specifically, the values of some of the endogenous variables depend only on the values of the exogenous variables. This is the case for the variables L and MD in the forest-fire example depicted in Figure 2.1. Think of these as the “first-level” endogenous variables. The values of the “second-level” endogenous variables depend only on the values of exogenous variables and the values of first-level endogenous variables. The variable FF is a second-level variable in this sense; its value is determined by that of L and MD, and these are first-level variables. We can then define third-level variables, fourth-level variables, and so on.

Actually, the intuition I have just given is for what I call a strongly recursive model. It is easy to see that in a strongly recursive model, the values of all variables are determined given a context, that is, a setting for the exogenous variables in . Given , we can determine the values of the first-level variables using the equations; we can then determine the values of the second-level variables (whose values depend only on the context and the values of the first-level variables), then the third-level variables, and so on. The notion of a recursive model generalizes the definition of a strongly recursive model (every strongly recursive model is recursive, but the converse is not necessarily true) while still keeping this key feature of having the context determine the values of all the endogenous variables. In a recursive model, the partition of endogenous variables into first-level variables, second-level variables, and so on can depend on the context; in different contexts, the partition might be different. Moreover, in a recursive model, our definition guarantees that causality is asymmetric: it cannot be the case that A is a cause of B and B is a cause of A if A and B are distinct. (See the discussion at the end of Section 2.7.)

The next four paragraphs provide a formal definition of recursive models. They can be skipped on a first reading.

Say that a model M is strongly recursive (or acyclic) if there is some partial order ⪯ on the endogenous variables in M (the ones in ) such that unless X ⪯ Y, Y is not affected by X. Roughly speaking, X ⪯ Y denotes that X affects Y (think of “affects” here as the transitive closure of the direct dependency relation discussed earlier; X affects Y if there is some chain X₁, …, X_k such that X = X₁, Y = X_k, and X_i+1 directly depends on X_i for i = 1, …, k − 1. The fact that ⪯ is a partial order means that ⪯ is a reflexive, anti-symmetric, and transitive relation. Reflexivity means that X ⪯ X for each variable X—X affects X; anti-symmetry means that if X ⪯ Y and Y ⪯ X, then X = Y —if X affects Y and Y affects X, then we must have X = Y; finally, transitivity means that if X ⪯ Y and Y ⪯ Z, then X ⪯ Z—if X affects Y and Y affects Z, then X affects Z. Since ⪯ is partial, it may be the case that for some variables X and Y, neither X ⪯ Y nor Y ⪯ X holds; that is, X does not affect Y and Y does not affect X.

While reflexivity and transitivity seem to be natural properties of the “affects” relation, anti-symmetry is a nontrivial assumption. The fact that ⪯ is anti-symmetric and transitive means that there is no cycle of dependence between a collection X₁, …, X_n of variables. It cannot be the case that X₁ affects X₂, X₂ affects X₃, …, X_n−1 affects X_n, and X_n affects X₁. For then we would have X₁ ⪯ X₂, X₂ ⪯ X₃, …, X_n−1 X_n, and X_n ⪯ X₁. By transitivity, we would have X₂ ⪯ X₁, violating anti-symmetry. A causal network corresponding to a causal model where there is no such cyclic dependence between the variables is acyclic. That is, there is no sequence of directed edges that both starts and ends at the same node.

A model M is recursive if, for each context (setting of the exogenous variables) , there is a partial order of the endogenous variables such that unless , Y is independent of X in , where Y is independent of X in if, for all settings of the endogenous variables other than X and Y, and all values x and x′ of X, . If M is a strongly recursive model, then we can assume that all the partial orders are the same; in a recursive model, they may differ. Example 2.3.3 below shows why it is useful to consider the more general notion of recursive model.

As I said, if M is a recursive causal model, then given a context , there is a unique solution for all the equations. We simply solve for the variables in the order given by (where if and X ≠ Y). The value of the variables that come first in the order, that is, the variables X such that there is no variable Y such that , depend only on the exogenous variables, so their value is immediately determined by the values of the exogenous variables. The values of variables later in the order can be determined from the equations once we have determined the values of all the variables earlier in the order.

With the definition of recursive model under our belt, we can get back to the issue of choosing the “right” model. There are many nontrivial decisions to be made when choosing the causal model to describe a given situation. One significant decision is the set of variables used. As we shall see, the events that can be causes and those that can be caused are expressed in terms of these variables, as are all the intermediate events. The choice of variables essentially determines the “language” of the discussion; new events cannot be created on the fly, so to speak. In our running forest-fire example, the fact that there is no variable for unattended campfires means that the model does not allow us to consider unattended campfires as a cause of the forest fire.

Once the set of variables is chosen, the next step is to decide which are exogenous and which are endogenous. As I said earlier, the exogenous variables to some extent encode the background situation that we want to take as given. Other implicit background assumptions are encoded in the structural equations themselves. Suppose that we are trying to decide whether a lightning bolt or a match was the cause of the forest fire, and we want to take as given that there is sufficient oxygen in the air and the wood is dry. We could model the dryness of the wood by an exogenous variable D with values 0 (the wood is wet) and 1 (the wood is dry). (Of course, in practice, we may want to allow D to have more values, indicating the degree of dryness of the wood, but that level of complexity is unnecessary for the points I am trying to make here.) By making D exogenous, its value is assumed to be determined by external factors that are not being modeled. We could also take the amount of oxygen to be described by an exogenous variable (e.g., there could be a variable O with two values—0, for insufficient oxygen; and 1, for sufficient oxygen); alternatively, we could choose not to model oxygen explicitly at all. For example, suppose that we have, as before, a variable MD (match dropped by arsonist) and another variable WB (wood burning), with values 0 (it’s not) and 1 (it is). The structural equation F_WB would describe the dependence of WB on D and MD. By setting F_WB (1, 1) = 1, we are saying that the wood will burn if the lit match is dropped and the wood is dry. Thus, the equation is implicitly modeling our assumption that there is sufficient oxygen for the wood to burn. Whether the modeler should include O in the model depends on whether the modeler wants to consider contexts where the value of O is 0. If in all contexts relevant to the modeler there is sufficient oxygen, there is no point in cluttering up the model by adding the variable O (although adding it will not affect anything).

According to the definition of causality in Section 2.2, only endogenous variables can be causes or be caused. Thus, if no variables encode the presence of oxygen, or if it is encoded only in an exogenous variable, then oxygen cannot be a cause of the forest burning in that model. If we were to explicitly model the amount of oxygen in the air (which certainly might be relevant if we were analyzing fires on Mount Everest), then F_WB would also take values of O as an argument, and the presence of sufficient oxygen might well be a cause of the wood burning, and hence the forest burning. Interestingly, in the law, there is a distinction between what are called conditions and causes. Under typical circumstances, the presence of oxygen would be considered a condition, and would thus not count as a cause of the forest burning, whereas the lightning would. While the of oxygen would be considered a condition and would thus not count as a cause of the forest burning, while the lightning would. Although the distinction is considered important, it does not seem to have been carefully formalized. One way of understanding it is in terms of exogenous versus endogenous variables: conditions are exogenous, (potential) causes are endogenous. I discuss an alternative approach to understanding the distinction, in terms of theories of normality, in Section 3.2.

It is not always straightforward to decide what the “right” causal model is in a given situation. What is the “right” set of variables to use? Which should be endogenous and which should be exogenous? As we shall see, different choices of endogenous variables can lead to different conclusions, although they seem to be describing exactly the same situation. And even after we have chosen the variables, how do we determine the equations that relate them? Given two causal models, how do we decide which is better?

These are all important questions. For now, however, I will ignore them. The definition of causality I give in the next section is relative to a model M and context . A may be a cause of B relative to and not a cause of B relative to . Thus, the choice of model can have a significant impact in determining causality ascriptions. The definition is agnostic regarding the choice of model; it has nothing to say about whether the choice of variables or the structural equations used are in some sense “reasonable”. Of course, people may legitimately disagree about how well a particular causal model describes the world. Although the formalism presented here does not provide techniques to settle disputes about which causal model is the right one, at least it provides tools for carefully describing the differences between causal models, so that it should lead to more informed and principled decisions about those choices. That said, I do return to some of the issues raised earlier when discussing the examples in this section, and I discuss them in more detail in Chapter 4.

As promised, I conclude this section with a little more discussion of the vector notation, for readers unfamiliar with it. I use this notation throughout the book to denote sets of variables or their values. For example, if there are three exogenous variables, say U₁, U₂, and U₃, and U₁ = 0, U₂ = 0, and U₃ = 1, then I write U as an abbreviation of U₁ = 0, U₂ = 0, and U₃ = 1. This vector notation is also used to describe the values of a collection of endogenous variables. I deliberately abuse notation at times and view both as a set of variables and as a sequence of variables. For example, when I write , then I am thinking of as the sequence (U₁, U₂, U₃), and this equality means that U₁ = 0, U₂ = 0, and U₃ = 1. However, in expressions such as and , should be understood as the corresponding unordered set {U₁, U₂, U₃}. I hope that the intent is always clear from context. For the most part, readers who do not want to delve into details can just ignore the vector arrows.

2.2 A Formal Definition of Actual Cause

In this section, I give the HP definition of causality and show how it works in a number of examples. But before giving the definition, I have to define a formal language for describing causes.

2.2.1 A language for describing causality

To make the definition of actual causality precise, it is helpful to have a formal language for making statements about causality. The language is an extension of propositional logic, where the primitive events (i.e., primitive propositions) have the form X = x for an endogenous variable X and some possible value x of X. The primitive event MD = 0 says “the lit match is not dropped”; similarly, the primitive event L = 1 says “lightning occurred”. These primitive events can be combined using standard propositional connectives, such as ∧, ∨, and ¬. Thus, the formula MD = 0 ∨ L = 1 says “either the lit match is not dropped or lightning occurred”, MD = 0 ∧ L = 1 says “the lit match is not dropped and lightning occurred”, and ¬(L = 1) says “lightning did not occur” (which is equivalent to L = 0, given that the only possible values of L are 0 or 1). A Boolean combination of primitive events is a formula obtained by combining primitive events using ∧, ∨, and ¬. For example, ¬ (MD = 0 ∨ L = 1) ∧ WB = 1 is a Boolean combination of the primitive events MD = 0, L = 1, and WB = 1. The key novel feature in the language is that we can talk about interventions. A formula of the form says that after intervening to set the variables in to , X takes on the value x.

(A short aside: I am thinking of an event here as a subset of a state space; this is the standard usage in computer science and probability. Once I give semantics to causal formulas, it will be clear that we can associate a causal formula φ with a set of contexts—the set of context where φ is true—so it can indeed be identified with an event in this sense. In the philosophy literature, the use of the word “event” is somewhat different, and what counts as an event is contentious; see the notes at the end of this chapter and at the end of Chapter 4.)

Now for the formal details. Given a signature , a primitive event is a formula of the form X = x, for and . A causal formula (over ) is one of the form [Y₁ ← y₁, …, Y_k ← y_k]φ, where

• φ is a Boolean combination of primitive events,
• Y₁, …, Y_k are distinct variables in , and
• .

Such a formula is abbreviated as , using the vector notation. The special case where k = 0 is abbreviated as [ ]φ or, more often, just φ. Intuitively, [Y₁ ← y₁, …, Y_k ← y_k]φ says that φ would hold if Y_i were set to y_i, for i = 1, …, k. It is worth emphasizing that the commas in [Y₁ ← y₁, …, Y_k ← y_k]φ are really acting like conjunctions; this says that if Y₁ were set to y₁ and … and Y_k were set to y_k, then φ would hold. I follow convention and use the comma rather than ∧. I write Y_j ← y_j rather than Y_j = y_j to emphasize that here Y_j is assigned the value y_j, as the result of an intervention. For , let consist of all Boolean combinations of causal formulas, where the variables in the formulas are taken from and the sets of possible values of these variables are determined by .

The language above lets us talk about interventions. What we need next is a way of defining the semantics of causal formulas: that is, a way of determining when a causal formula is true. A causal formula ψ is true or false in a causal model, given a context. I call a pair consisting of a causal model M and context a causal setting. I write if the causal formula ψ is true in the causal setting . For now, I restrict attention to recursive models, where, given a context, there are no cyclic dependencies. In a recursive model, if the value of X is x once we set the exogenous variables to . Here I am using the fact that, in a recursive model, the values of all the endogenous variables are determined by the context. If ψ is an arbitrary Boolean combination of primitive events, then whether can be determined by the usual rules of propositional logic. For example, if either or .

Using causal models makes it easy to give the semantics of formulas of the form and, more generally, . Recall that the latter formula says that, after intervening to set the variables in to , ψ holds. Given a model M, the model that describes the result of this intervention is . Thus,

The mathematics just formalizes the intuition that the formula is true in the causal setting exactly if the formula ψ is true in the model that results from the intervention, in the same context .

For example, if M^d is the disjunctive model for the forest fire described earlier, then (M ^d, (1, 1)) ⊨ [MD ← 0](FF = 1): even if the arsonist is somehow prevented from dropping the match, the forest burns (thanks to the lightning); that is, . Similarly, (Md, (1, 1)) ⊨ [L ← 0](FF = 1). However, (M ^d, (1, 1)) ⊨ [L ← 0; MD ← 0](FF = 0): if the arsonist does not drop the lit match and the lightning does not strike, then the forest does not burn.

The notation is standard in the logic and philosophy communities. It is, unfortunately, not at all standard in other communities, such as statistics and econometrics. Although notation is not completely standard in these communities, the general approach has been to suppress the model M and make the context an argument of the endogenous variables. Thus, for example, instead of writing , in these communities they would write . (Sometimes the exogenous variables are also suppressed, or taken as given, so just X = x is written.) More interestingly, instead of , in these communities they would write . Although the latter notation, which I henceforth call statisticians’ notation (although it is more widespread), is clearly more compact, the compactness occasionally comes at the price of clarity. For example, in the disjunctive forest-fire example, what does FF ₀(1, 1) = 1 mean? Is it (M ^d, (1, 1)) ⊨ [MD ← 0](FF = 1) or (M^d, (1, 1)) = [L ← 0](FF = 1)? This problem can be solved by adding the variable being intervened on to the subscript if it is necessary to remove ambiguity, writing, for example, FF _L←0(1, 1) = 0. Things get more complicated when we want to write , where ϕ is a Boolean combination of primitive events, or when there are several causal models in the picture and it is important to keep track of them. That said, there are many situations when writing something like Y_x(u) = y is completely unambiguous, and it is certainly more compact. To make things easier for those used to this notation, I translate various statements into statisticians’ notation, particularly those where it leads to more compact formulations. This will hopefully have the added advantage of making both notations familiar to all communities.

2.2.2 The HP definition of actual causality

I now give the HP definition of actual causality.

The types of events that the HP definition allows as actual causes are ones of the form X₁ = x₁ ∧ … ∧ X_k = x_k—that is, conjunctions of primitive events; this is often abbreviated as . The events that can be caused are arbitrary Boolean combinations of primitive events. The definition does not allow statements of the form “A or A′ is a cause of B,” although this could be treated as being equivalent to “either A is a cause of B or A′ is a cause of B”. However, statements such as “A is a cause of either B or B′” are allowed; this is not equivalent to “either A is a cause of B or A is a cause of B′”.

Note that this means that the relata of causality (what can be a cause and what can be caused) depend on the language. It is also worth remarking that there is a great deal of debate in the philosophical literature about the relata of causality. Although it is standard in that literature to take the relata of causality to be events, exactly what counts as an event is also a matter of great debate. I do not delve further into these issues here; see the notes at the end of the chapter for references and a little more discussion.

Although I have been talking up to now about “the” HP definition of causality, I will actually consider three definitions, all with the same basic structure. Judea Pearl and I started with a relatively simple definition that gradually became more complicated as we discovered examples that our various attempts could not handle. One of the examples was discovered after our definition was published in a conference paper. So we updated the definition for the journal version of the paper. Later work showed that the problem that caused us to change the definition was not as serious as originally thought. However, recently I have considered a definition that is much simpler than either of the two previous versions, which has the additional merit of dealing better with a number of problems. It is not clear that the third definition is the last word. Moreover, showing how the earlier definitions deal with the various problems that have been posed lends further insight into the difficulties and subtleties involved with finding a definition of causality. Thus, I consider all three definitions in this book. Following Einstein, my goal is to make things as simple as possible, but no simpler!

Like the definition of truth on which it depends, the HP definition of causality is relative to a causal setting. All three variants of the definition consist of three clauses. The first and third are simple and straightforward, and the same for all the definitions. All the work is done by the second clause; this is where the definitions differ.

Definition 2.2.1 is an actual cause of φ in the causal setting if the following three conditions hold:

AC1. and .

AC2. See below.

AC3. is minimal; there is no strict subset of such that satisfies conditions AC1 and AC2, where is the restriction of to the variables in . ■

AC1 just says that cannot be considered a cause of φ unless both and φ actually happen. (I am implicitly taking to characterize the “actual world” here.) AC3 is a minimality condition, which ensures that only those elements of the conjunction that are essential are considered part of a cause; inessential elements are pruned. Without AC3, if dropping a lit match qualified as a cause of the forest fire, then dropping a match and sneezing would also pass the tests of AC1 and AC2. AC3 serves here to strip “sneezing” and other irrelevant, over-specific details from the cause. AC3 can be viewed as capturing part of the spirit of the INUS condition; roughly speaking, it says that all the primitive events in the cause are necessary for the effect.

It is now time to bite the bullet and look at AC2. In the first two versions of the HP definition, AC2 consists of two parts. I start by considering the first of them, denoted AC2(a). AC2(a) is a necessity condition. It says that for X = x to be a cause of ϕ, there must be a value x′ in the range of X such that if X is set to x′, φ no longer holds. This is the but-for clause; but for the fact that X = x occurred, φ would not have occurred. As we saw in the Billy-Suzy rock-throwing example in Chapter 1, the naive but-for clause will not suffice. We must be allowed to apply it under certain contingencies, that is, under certain counterfactual settings, where some variables are set to values other than those they take in the actual situation or held to certain values while other variables change. For example, in the case of Suzy and Billy, we consider a contingency where Billy does not throw.

Here is the formal version of the necessity condition:

AC2(a). There is a partition of (the set of endogenous variables) into two disjoint subsets and (so that ) with and a setting and of the variables in and , respectively, such that

Using statisticians’ notation, if φ is , then the displayed formula becomes .

Roughly speaking, AC2(a) says that the but-for condition holds under the contingency . We can think of the variables in as making up the “causal path” from to φ. Intuitively, changing the value of some variable in results in changing the value(s) of some variable(s) in , which results in the values of some other variable(s) in being changed, which finally results in the truth value of φ changing. (Whether we can think of the variables in as making up a causal path from some variable in to some variable in φ turns out to depend in part on which variant of the HP definition we consider. See Section 2.9 for a formalization and more discussion.) The remaining endogenous variables, the ones in , are off to the side, so to speak, but may still have an indirect effect on what happens.

Unfortunately, AC1, AC2(a), and AC3 do not suffice for a good definition of causality. In the rock-throwing example, with just AC1, AC2(a), and AC3, Billy would be a cause of the bottle shattering. We need a sufficiency condition to block Billy. The sufficiency condition used in the original definition, roughly speaking, requires that if the variables in and an arbitrary subset of other variables on the causal path (i.e., besides those in ) are held at their values in the actual context (where the value of a variable Y in the actual context is the value y^∗ such that (M, u) ⊨ Y = y^∗; the values of the variables in in the actual context is , by AC1), then φ holds even if is set to (the setting for used in AC2(a)). This is formalized by the following condition:

AC2(b^o). If is such that , then for all subsets of , we have

Again, taking φ to be , in statisticians’ notation this becomes “if , then for all subsets of , we have .” It is important to write at least the explicitly in the subscript here, since we are quantifying over it.

Note that, due to setting to , the values of the variables in may change. AC2(b^o) says that this change does not affect φ; φ continues to be true. Indeed, ϕ continues to be true even if some variables in are forced to their original values.

Before going on, I need to make a brief technical digression to explain a slight abuse of notation in AC2(b^o). Suppose that , , and . Then is intended to be an abbreviation for Z₁ ← 1; that is, I am ignoring the second component of here. More generally, when I write , I am picking out the values in that correspond to the variables in and ignoring those that correspond to the variables in . I similarly write if is a subset of .

For reasons that will become clearer in Example 2.8.1, the original definition was updated to use a stronger version of AC2(b^o). Sufficiency is now required to hold if the variables in any subset of are set to the values in (as well as the variables in any subset of being set to their values in the actual context). Taking and as in AC2(a) (and AC2(b^o)), here is the updated definition:

AC2(b^u). If is such that , then for all subsets of and subsets of , we have .

In statisticians’ notation, this becomes “if , then for all subsets of and v of , we have .” Again, we need to write the variables in the subscript here.

The only difference between AC2(b^o) and AC2(b^u) lies in the clause “for all subsets of ”: AC2(b^u) must hold even if only a subset of the variables in are set to their values in . This means that the variables in essentially act as they do in the real world; that is, their values are determined by the structural equations, rather than being set to their values in .

The superscripts o and u in AC2(b^o) and AC2(b^u) are intended to denote “original” and “updated”.

I conclude by considering the simpler modified definition. This definition is motivated by the intuition that only what happens in the actual situation should matter. Thus, the only settings of variables allowed are ones that occur in the actual situation. Specifically, the modified definition simplifies AC2(a) by requiring that the only setting of the variables in that can be considered is the value of these variables in the actual context. Here is the modified AC2(a), denoted AC2(a^m) (the m stands for “modified”):

AC2(a^m). There is a set of variables in and a setting of the variables in such that if , then

That is, we can show the counterfactual dependence of φ on by keeping the variables in fixed at their actual values. In statisticians’ notation (with φ being ), this becomes “if , then ”.

It is easy to see that AC2(b^o) holds if all the variables in are fixed at their values in the actual context: because records the value of the variables in in the actual context, if is (re)set to , its value in the actual context, then φ must hold (since, by AC1, and φ both hold in the actual context); that is, we have if is the value of the variables in in the actual context. For similar reasons, AC2(b^u) holds if all the variables in are fixed at their values in the actual context. Thus, there is no need for an analogue to AC2(b^o) or AC2(b^u) in the modified definition. This shows that the need for a sufficiency condition arises only if we are considering contingencies that differ from the actual setting in AC2(a). It is also worth noting that the modified definition does not need to mention (although can be taken to be the complement of ).

For future reference, for all variants of the HP definition, the tuple in AC2 is said to be a witness to the fact that is a cause of φ. (I take the witness to be in the special case that .)

As I said, AC3 is a minimality condition. Technically, just as there are three versions of AC2, there are three corresponding versions of AC3. For example, in the case of the modified definition, AC3 should really say “there is no subset of satisfying AC1 and AC2(a^m)”. I will not bother writing out these versions of AC3; I hope that the intent is clear whenever I refer to AC3.

If I need to specify which variant of the HP definition I am considering, I will say that is a cause of φ according to the original (resp., updated, modified) HP definition. If is a cause of φ according to all three variants, I often just say “ is a cause of φ in ”. Each conjunct in is called part of a cause of φ in context . As we shall see, what we think of as causes in natural language correspond to parts of causes, especially with the modified HP definition. Indeed, it may be better to use a term such as “complete cause” for what I have been calling cause and then reserve “cause” for what I have called “part of a cause”. I return to this point after looking at a few examples.

Actually, although is an abbreviation for the conjunction X₁ = x₁ ∧…∧X_k = x_k, when we talk about being a cause of φ (particularly for the modified HP definition), it might in some ways be better to think of the disjunction. Roughly speaking, is a cause of φ if it is the case that if none of the variables in had their actual value, then φ might not have occurred. However, if has more than one variable, then just changing the value of a single variable in (or, indeed, any strict subset of the variables in ) is not enough by itself to bring about ¬φ (given the context). Thus, there is a sense in which the disjunction can be viewed as a but-for cause of φ. This will be clearer when we see the examples.

Note that all three variants of the HP definition declare X = x a cause of itself in (M, u) as long as (M, u) = X = x. This does not seem to me so unreasonable. At any rate, it seems fairly harmless, so I do not bother excluding it (although nothing would change if we did exclude it).

At this point, ideally, I would prove a theorem showing that some variant of the HP definition of actual causality is the “right” definition of actual causality. But I know of no way to argue convincingly that a definition is the “right” one; the best we can hope to do is to show that it is useful. As a first step, I show that all the definitions agree in the simplest and arguably most common case: but-for causes. Formally, say that X = x is a but-for cause of φ in if AC1 holds (so that and there exists some x′ such that . Note here I am assuming that the cause is a single conjunct.

Proposition 2.2.2 If X = x is a but-for cause of Y = y in , then X = x is a cause of Y = y according to all three variants of the HP definition.

Proof: Suppose that X = x is a but-for cause of Y = y. There must be a possible value x′ of X such that . Then (∅, ∅, x′) (i.e., and X = x′) is a witness to X = x being a cause of φ for all three variants of the definition. Thus, AC2(a) and AC2(a^m) hold if we take . Since , if , where , then it is easy to see that : since X = x and in the (unique) solution to the equations in context , this is still the case in the unique solution to the equations in M_X←x. For similar reasons, for all subsets of . (See Lemma 2.10.2, where these statements are formalized.) Thus, AC2(b^o) holds. Because v, AC2(b^u) follows immediately from AC2(b^o). ■

Of course, the definitions do not always agree. Other connections between the definitions are given in the following theorem. In the statement of the theorem, the notation denotes the cardinality of .

Theorem 2.2.3

(a)If X = x is part of a cause of φ in according to the modified HP definition, then X = x is a cause of φ in according to the original HP definition.

(b)If X = x is part of a cause of φ in according to the modified HP definition, then X = x is a cause of φ in according to the updated HP definition.

(c)If X = x is part of a cause of φ in according to the updated HP definition, then X = x is a cause of φ in according to the original HP definition.

(d)If is a cause of φ in according to the original HP definition, then (i.e., is a singleton).

Of course, (a) follows from (b) and (c); I state it separately just to emphasize the relations. Although it may not look like it at first glance, part (d) is quite similar in spirit to parts (a), (b), and (c); an equivalent reformulation of (d) is: “If X = x is part of a cause of φ in according to the original HP definition, then X = x is a cause of φ in according to the original HP definition.” Call this statement (d′). Clearly (d) implies (d′). For the converse, suppose that (d′) holds, is a cause of φ in , and . Then if X = x is a conjunct of , by (d′), X = x is a cause of φ in . But by AC3, this can hold only if X = x is the only conjunct of , so .

Parts (a), (b), and (c) of Theorem 2.2.3 show that, in a sense, the original HP definition is the most permissive of the three and the modified HP definition is the most restrictive, with the updated HP definition lying somewhere in between. The converses of (a), (b), and (c) do not hold. As Example 2.8.1 shows, if X = x is a cause of φ according to the original HP definition, it is not, in general, part of a cause according to the modified or updated HP definition. In this example, we actually do not want causality to hold, so the example can be viewed as showing that the original definition is too permissive (although this claim is not quite as clear as it may appear at first; see the discussion in Section 2.8). Example 2.8.2 shows that part of a cause according to the updated HP definition need not be (part of) a cause according to the modified HP definition.

The bottom line here is that, although these definitions often agree, they do not always agree. It is typically on the most problematic examples on which they disagree. This is perhaps not surprising given that the original definition was updated and modified precisely to handle such examples. However, even in cases where the original definition seems to give “wrong” answers, there are often ways of dealing with the problem that do not require modifying the definition. Moreover, the fact that causes are always singletons with the original definition makes it attractive in some ways. The jury is still out on what the “right” definition of causality is. Although my current preference is the modified HP definition, I will consider all three definitions throughout the book. Despite the extra burden for the reader (for which I apologize in advance!), I think doing so gives a deeper insight into the subtleties of causality.

The proof of Theorem 2.2.3 can be found in Section 2.10.1. I recommend that the interested reader defer reading the proof until after going through the examples in the next section, since they provide more intuition for the definitions.

2.3 Examples

Because I cannot prove a theorem showing that (some variant of) the HP definition is the “right” definition of causality, all I can do to argue that the definition is reasonable is to show how it works in some of the examples that have proved difficult for other definitions to handle. That is one of the goals of this section. I also give examples that illustrate the subtle differences between the variants of the HP definition. I start with a few basic examples mainly intended to show how the definition works.

Example 2.3.1 For the forest-fire example, I consider two causal models, M^c and M^d, for the conjunctive and disjunctive cases, respectively. These models were described earlier; the endogenous variables are L, MD, and FF, and U is the only exogenous variable. In both cases, I want to consider the context (1, 1), so the lightning strikes and the arsonist drops the match. In the conjunctive model, both the lightning and the dropped match are but-for causes of the forest fire; if either one had not occurred, the fire would not have happened. Hence, by Proposition 2.2.2, both L = 1 and MD = 1 are causes of FF = 1 in (M^c, (1, 1)) according to all three variants of the definition. By AC3, it follows that L = 1 MD = 1 is not a cause of FF = 1 in (M ^c, (1, 1)). This already shows that causes might not be unique; there may be more than one cause of a given outcome.

However, in the disjunctive case, there are differences. With the original and updated definition, again, we have that both L = 1 and MD = 1 are causes of FF = 1 in (M ^d, (1, 1)). I give the argument in the case of L = 1; the argument for MD = 1 is identical. Clearly (M^d, (1, 1)) = FF = 1 and (M^d, (1, 1)) = L = 1; in the context (1, 1), the lightning strikes and the forest burns down. Thus, AC1 is satisfied. AC3 is trivially satisfied: since consists of only one element, L, must be minimal.

For AC2, as suggested earlier, let , , x′ = 0, and w = 0. Clearly, (M^d, (1, 1)) ⊨ [L ← 0, MD ← 0](FF = 1); if the lightning does not strike and the match is not dropped, the forest does not burn down, so AC2(a) is satisfied. To see the effect of the lightning, we must consider the contingency where the match is not dropped; AC2(a) allows us to do that by setting MD to 0. (Note that setting L and MD to 0 overrides the effects of U; this is critical.) Moreover,

in context (1, 1), if the lightning strikes, then the forest burns down even if the lit match is not dropped, so AC2(b^o) and AC2(b^u) are satisfied. (Note that since , the only subsets of are the empty set and the singleton set consisting of just FF; similarly, since , the only subsets of are the empty set and the singleton set consisting of MD itself. Thus, we have considered all the relevant cases here.)

This argument fails in the case of the modified definition because the setting MD = 0 is not allowed. The only witnesses that can be considered are ones where has the value it does in the actual context; in the actual context here, MD = 1. So, with the modified definition, neither L = 1 nor MD = 1 is a cause. However, it is not hard to see that L = 1 ∧ MD = 1 is a cause of FF = 1. This shows why it is critical to consider only single conjuncts in Proposition 2.2.2 and Theorem 2.2.3 is worded in terms of parts of causes. Although L = 1 ∧ MD = 1 is a cause of FF = 1 according to the modified HP definition, it is not a cause according to either the original or updated HP definition. In contrast, L = 1 and MD = 1 are both parts of a cause of FF = 1 according to all three definitions.

It is arguably a feature of the original and updated HP definition that they call both L = 1 and MD = 1 causes of FF = 1. Calling the conjunction L = 1 ∧ MD = 1 a cause of FF = 1 does not seem to accord with natural language usage. There are two ways to address this concern. As I suggested earlier, with the modified HP definition, it may be better to think of parts of causes as coming closer to what we call causes in natural language. That would already deal with this concern. A second approach, which I also hinted at earlier, is to observe that it may be better to think of the disjunction L = 1 ∨ MD = 1 as being the cause. Indeed, we can think of MD = 1 ∨ L = 1 as a but-for cause of FF = 1; if it does not hold, then it must be the case both L = 0 and MD = 0, so there is no fire. The reader should keep in mind both of these ways of thinking of conjunctive causes with the modified HP definition.

As we shall see, the notion of responsibility, discussed in Chapter 6, allows the original and updated HP definition to distinguish these two scenarios, as does the notion of sufficient cause, discussed in Section 2.6. ■

This simple example already reveals some of the power of the HP definition. The case where the forest fire is caused by either the lightning or the dropped match (this is said to be a case of overdetermination in the literature) cannot be handled by the simple but-for definition used in the law. It seems reasonable to call both the lightning and the dropped match causes, or at least parts of causes. We certainly wouldn’t want to say that there is no cause, as a naive application of the but-for definition would do. (However, as I noted above, if we allow disjunctive causes, then a case can be made that the disjunction L = 1 ∨ MD = 1 is a but-for cause of the forest fire.) Overdetermination occurs frequently in legal cases; a victim may be shot by two people, for example. It occurs in voting as well; I look at this a little more carefully in the following example.

Example 2.3.2 Consider the voting scenario discussed earlier where there are 11 voters. If Suzy wins 6–5, then all the definitions agree that each of the voters for Suzy is a cause of Suzy’s victory; indeed, they are all but-for causes. But suppose that Suzy wins 11–0. Here we have overdetermination. The original and updated HP definition would still call each of the voters a cause of Suzy winning; the witness involves switching the votes of 5 voters. Since the modified HP definition does not allow such switching, according to the modified HP definition, any subset of six voters is a cause of Suzy winning (and every individual is part of a cause). Again, if we think of the subset as being represented by a disjunction, it can be thought of as a but-for cause of Suzy winning. If all six voters had switched their votes to Billy, then Suzy would not have won. Minimality holds; if a set of fewer than six voters had voted for Billy, then Suzy would still have won. ■

We do have an intuition that a voter for Suzy is somehow “less” of a cause of the victory in an 11–0 victory than in a 6–5 victory. This intuition is perhaps most compatible with the modified HP definition. In this case, is a cause of φ if is a minimal set of variables whose values have to change for ¬φ to hold. The bigger is, the less of a cause each of its conjuncts is. I return to this issue when I discuss responsibility in Chapter 6.

Example 2.3.3 Now I (finally!) consider the rock-throwing example, where Suzy and Billy both throw rocks at a bottle, but Suzy’s hits the bottle, and Billy’s doesn’t (although it would have hit had Suzy’s not hit first). We get the desired result—that Suzy’s throw is a cause, but Billy’s is not—but only if we model the story appropriately. Consider first a coarse causal model, with three endogenous variables:

• ST for “Suzy throws”, with values 0 (Suzy does not throw) and 1 (she does);
• BT for “Billy throws”, with values 0 (he doesn’t) and 1 (he does);
• BS for “bottle shatters’, with values 0 (it doesn’t shatter) and 1 (it does).

For simplicity, assume that there is one exogenous variable u, which determines whether Billy and Suzy throw. (In most of the subsequent examples, I omit the exogenous variable in both the description of the story and the corresponding causal network.) Take the formula for BS to be such that the bottle shatters if either Billy or Suzy throws; that is, BS = BT ∨ ST. (I am implicitly assuming that Suzy and Billy never miss if they throw. Also, I follow the fairly standard mathematics convention of saying “if” rather than “if and only if” in definitions; clearly, the equation is saying that the bottle shatters if and only if either Billy or Suzy throws.) For future reference, I call this model (where RT stands for “rock throwing”).

BT and ST play symmetric roles in ; there is nothing to distinguish them. Not surprisingly, both Billy’s throw and Suzy’s throw are classified as causes of the bottle shattering in in the context u where Suzy and Billy both throw according to the original and updated HP definition (the conjunction ST = 1 ∧ BT = 1 is the cause according to the modified HP definition). The argument is essentially identical to that used for the disjunctive model of the forest-fire example, where either the lightning or the dropped match is enough to start the fire. Indeed, the causal network describing this situation looks like that in Figure 2.1, with ST and BT replacing L and MD. For convenience, I redraw the network here. As I said earlier, here and in later figures, I typically omit the exogenous variable(s).

Figure 2.2: M_RT —a naive model for the rock-throwing example.

The trouble with M_RT is that it cannot distinguish the case where both rocks hit the bottle simultaneously (in which case it would be reasonable to say that both ST = 1 and BT = 1 are causes of BS = 1) from the case where Suzy’s rock hits first. M_RT has to be refined to express this distinction. One way is to invoke a dynamic model. Although this can be done (see the notes at the end of the chapter for more discussion), a simpler way to gain the expressiveness needed here is to allow BS to be three-valued, with values 0 (the bottle doesn’t shatter), 1 (it shatters as a result of being hit by Suzy’s rock), and 2 (it shatters as a result of being hit by Billy’s rock). I leave it to the reader to check that ST = 1 is a cause of BS = 1, but BT = 1 is not (if Suzy doesn’t throw but Billy does, then we would have BS = 2). To some extent, this solves our problem. But it borders on cheating; the answer is almost programmed into the model by invoking the relation “as a result of” in the definition of BS = 2, which requires the identification of the actual cause.

A more useful choice is to add two new variables to the model:

• BH for “Billy’s rock hits the (intact) bottle”, with values 0 (it doesn’t) and 1 (it does); and
• SH for “Suzy’s rock hits the bottle”, again with values 0 and 1.

We now modify the equations as follows:

• BS = 1 if SH = 1 or BH = 1;
• SH = 1 if ST = 1;
• BH = 1 if BT = 1 and SH = 0.

Thus, Billy’s throw hits if Billy throws and Suzy’s rock doesn’t hit. The last equation implicitly assumes that Suzy throws slightly ahead of Billy, or slightly harder.

Call this model . is described by the graph in Figure 2.3 (where again the exogenous variables are omitted). The asymmetry between BH and SH (in particular, the fact that Billy’s throw doesn’t hit the bottle if Suzy throws) is modeled by the fact that there is an edge from SH to BH but not one in the other direction; BH depends (in part) on SH but not vice versa.

Figure 2.3: —a better model for the rock-throwing example.

Taking u to be the context where Billy and Suzy both throw, according to all three variants of the HP definition, ST = 1 is a cause of BS = 1 in , but BT = 1 is not. To see that ST = 1 is a cause according to the original and updated HP definition, note that it is immediate that AC1 and AC3 hold. To see that AC2 holds, one possibility is to choose , , and w = 0. When BT is set to 0, BS tracks ST: if Suzy throws, the bottle shatters, and if she doesn’t throw, the bottle does not shatter. It immediately follows that AC2(a) and AC2(b^o) hold. AC2(b^u) is equivalent to AC2(b^o) in this case, since is a singleton.

To see that BT = 1 is not a cause of BS = 1 in , we must check that there is no partition of the endogenous variables into sets and that satisfies AC2. Attempting the symmetric choice with , , and w = 0 violates AC2(b^o) and AC2(b^u). To see this, take . In the context where Suzy and Billy both throw, BH = 0. If BH is set to 0, then the bottle does not shatter if Billy throws and Suzy does not. It is precisely because, in this context, Suzy’s throw hits the bottle and Billy’s does not that we declare Suzy’s throw to be the cause of the bottle shattering. AC2(b^o) and AC2(b^u) capture that intuition by forcing us to consider the contingency where BH = 0 (i.e., where BH takes on its actual value), despite the fact that Billy throws.

Of course, just checking that this particular partition into and does not make BT = 1 a cause does not suffice to establish that BT = 1 is not a cause according to the original or updated definitions; we must check all possible partitions. I just sketch the argument here: The key is to consider whether BH is in or . If BH is in , then how we set BT has no effect on the value of BS: If BH is set to 0, then the value of BS is determined by the value of ST; and if BH is set to 1, then BS = 1 no matter what BT is. (It may seem strange to intervene and set BH to 1 if BT = 0. It means that somehow Billy hits the bottle even if he doesn’t throw. Such “miraculous” interventions are allowed by the definition; part of the motivation for the notion of normality considered in Chapter 3 is to minimize their usage.) This shows that we cannot have BH in . If BH is in , then we get the same problem with AC2(b^o) or AC2(b^u) as above; it is easy to see that at least one of SH or ST must be in , and must be such that whichever is in is set to 0.

Note that, in this argument, it is critical that in AC2(b^o) and AC2(b^u) we allow setting an arbitrary subset of variables in to their original values. There is a trivial reason for this: if we set all variables in to their original values in , then, among other things, we will set BS to 1. We will never be able to show that Billy’s throw is not a cause if BS is set to 1. But even requiring that all the variables in be set to their original values does not work. For suppose that we take and take such that ST = 0. Clearly AC2(a) holds because if BT = 0, then BS = 0. But if all the variables in are set to their original values, then SH is set to 1, so the bottle shatters. To show that BT = 1 is not a cause, we must be able to set BH to its original value of 0 while keeping SH at 0. Setting BH to 0 captures the intuition that Billy’s throw is not a cause because, in the actual world, his rock did not hit the bottle (BH = 0).

Finally, consider the modified HP definition. Here things are much simpler. In this case, taking does not work to show that ST = 1 is a cause of BS = 1; we are not allowed to change BT from 1 to 0. However, we can take . Fixing BH at 0 (its setting in the actual context), if ST is set to 0, then BS = 0, that is, (M, u) ⊨ [ST ← 0, BH ← 0](BS = 0). Thus, ST = 1 is a cause of BS = 1 according to the modified HP definition. (Taking would also have worked to show that ST = 1 is a cause of BS = 1 according to the original and updated HP definition. Indeed, by Theorem 2.2.3, showing that ST = 1 is a cause of BS = 1 according to the modified definition suffices to show that it is also a cause according to the original and updated definition. I went through the argument for the original and updated definition because the more obvious choice for in that case is {BT}, and it works under those definitions.) But now it is also easy to check that BT = 1 is not a cause of BS = 1 according to the modified HP definition. No matter which subset of variables other than BT are held to their values in u and no matter how we set BT, we have SH = 1 and thus BS = 1.

I did not completely specify in the example above. Specifically, I did not specify the set of possible contexts. What happens in other contexts is irrelevant for determining whether ST = 1 or BT = 1 is a cause of BS = 1 in . But once I take normality considerations into account in Chapter 3, it will become relevant. I described u above as the context where Billy and Suzy both throw, implicitly suggesting that there are four contexts, determining the four possible combinations of Billy throwing and Suzy throwing. So we can take to be the model with these four contexts.

There is arguably a problem with this definition of : it “bakes in” the temporal ordering of events, in particular, that Suzy’s rock hits before Billy’s rock. If we want a model that describes “rock-throwing situations for Billy and Suzy” more generally, we would not want to necessarily assume that Billy and Suzy are accurate, nor that Suzy always hits before Billy if they both throw. Whatever assumptions we make along these lines can be captured by the context. That is, we can have the context determine (a) whether Suzy and Billy throw, (b) how accurate they are (would Billy hit if he were the only one to throw, and similarly for Suzy), and (c) whose rock hits first if they both throw and are accurate (allowing for the possibility that they both hit at the same time). This richer model would thus have 48 contexts: there are four choices of which subset of Billy and Suzy throw, four possibilities for accuracy (both are accurate, Suzy is accurate and Billy is not, and so on), and three choices regarding who hits first if they both throw. If we use this richer model, we would also want to expand the language to include variables such as SA (Suzy is accurate), BA (Billy is accurate), SF (Suzy’s throw arrives first if both Billy and Suzy throw), and BF (Bily’s throw arrives first). Call this richer model . In , the values of ST, BT, SA, BA, SF, and BF are determined by the context. As with and , the bottle shatters if either Suzy or Billy hit it, so we have the equation BS = BH ∨ SH. But now the equations for BH and SH depend on ST, BT, SA, BA, SF, and BF. Of course SH = 0 if Suzy doesn’t throw (i.e., if ST = 0) or is inaccurate (SA = 0), but even in this case, the equations need to tell us what would have happened had Suzy thrown and been accurate, so that we can determine the truth of formulas such as [ST ← 1, SA ← 1](SH = 0). As suggested by the discussion above, the equation for SH is

In words, Suzy hits the bottle if she throws and is accurate, and either (a) Billy doesn’t throw, (b) Billy is inaccurate, or (c) Billy throws and is accurate but Suzy’s rock arrives at the bottle first (or at the same time as Billy’s); otherwise, Suzy doesn’t hit the bottle.

The context u in corresponds to the context u^∗ in where Billy and Suzy both throw, both are accurate, and Suzy’s throw hits first. Exactly the same argument as that above hows that ST = 1 is a cause of , and BT = 1 is not. In the model , the direction of the edge in the causal network between SH and BH depends on the context. In some contexts (the ones where both Suzy and Billy are accurate and Suzy would hit first if they both throw), BH depends on SH; in other contexts, SH depends on BH. As a result, is not what I called a strongly recursive model. is still a recursive model because in each context u′ in , there are no cyclic dependencies; there is still an ordering on the endogenous variables such that unless , Y is independent of X in . In particular, although and there is a context u″ in such that , is still considered recursive.

It is critical in this analysis of the rock-throwing example that the model includes the variables BH and SH, that is, that Billy’s rock hitting the bottle is taken to be a different event than Suzy’s rock hitting the bottle (although there is no problem if it includes extra variables). To understand the need for BH and SH (or some analogous variables), consider an observer watching the situation. Why would she declare Suzy’s throw to be a cause and Billy’s throw not to be a cause? Presumably, precisely because it was Suzy’s rock that hit the bottle and not Billy’s. If this is the reason for the declaration, then it must be modeled. Of course, using the variables BH and SH is not the only way of doing this. All that is needed is that the language be rich enough to allow us to distinguish the situation where Suzy’s rock hits and Billy’s rock does not from the one where Billy’s rock hits and Suzy’s rock does not. (That is why I said “or some analogous variables” above.) An alternative approach to incorporating temporal information is to have time-indexed variables (e.g., to have a family of variables BS _k for “bottle shatters at time k” and a family H_k for “bottle is hit at time k”). With these variables and the appropriate equations, we can dispense with SH and BH and just have the variables H₁, H₂, … that talk about the bottle being hit at time k, without the variable specifying who hit the bottle. For example, if we assume that all the action happens at times 1, 2, and 3, then the equations are H₁ = ST (if Suzy throws, then the bottle is hit at time 1), H₂ = BT ∧¬H₁ (the bottle is hit at time 2 if Billy throws and the bottle was not already hit at time 1), and BS = H₁ ∨ H₂ (if the bottle is hit, then it shatters). Again, these equations model the fact that Suzy hits first if she throws. And again, we could assume that the equations for H₁ and H₂ are context-dependent, so that Suzy hits first in u, but Billy hits first in u′. In any case, in context u where Suzy and Billy both throw but Suzy’s rock arrives at the bottle first, Suzy is still the cause, and Billy isn’t, using essentially the same analysis as that above.

To summarize the key point here, the language (i.e., the variables chosen and the equations for them) must be rich enough to capture the significant features of the story, but there is more than one language that can do this. As we shall see in Chapter 4, the choice of language can in general have a major impact on causality judgments. ■

The rock-throwing example emphasizes an important moral. If we want to argue in a case of preemption that X = x rather than Y = y is the cause of φ, then there must be a variable (BH in this case) that takes on different values depending on whether X = x or Y = y is the actual cause. If the model does not contain such a variable, then it will not be possible to determine which one is in fact the cause. The need for such variables is certainly consistent with intuition and the way we present evidence. If we want to argue (say, in a court of law) that it was A’s shot that killed C and not B’s, then, assuming that A shot from C’s left and B from the right, we would present evidence such as the bullet entering C from the left side. The side from which the shot entered is the relevant variable in this case. The variable may involve temporal evidence (if C’s shot had been the lethal one, then the death would have occurred a few seconds later), but it certainly does not have to.

The rock-throwing example also emphasizes the critical role of what happens in the actual situation in the definition. Specifically, we make use of the fact that, in the actual context, Billy’s rock did not hit the bottle. In the original and updated HP definition, this fact is applied in AC2(b): to show that Billy is a cause, we would have to show that the bottle shatters when Billy throws, even if BH is set to its actual value of 0. In the modified HP definition, it is applied in AC2(a^m) to show that Suzy’s throw is a cause: we can set ST = 0 while keeping BH fixed at 0. The analogous argument would not work to show that Billy’s throw is a cause.

Although what happens in the actual context certainly affects people’s causality judgments, why is the way that this is captured in AC2(a^m), AC2(b^o), or AC2(b^u) the “right” way of capturing it? I do not have a compelling answer to this question, beyond showing that these definitions work well in examples. I have tried to choose examples that, although not always realistic, capture important features of causality. For example, while the rock-throwing example may not seem so important in and of itself, it is an abstraction of a situation that arises frequently in legal cases, where one potential cause is preempted by another. Monopolistic practices by a big company cause a small company to go bankrupt, but it would have gone bankrupt anyway because of poor management; a smoker dies in a car accident, but he would have died soon due to inoperable lung cancer had there not been an accident. A good definition of causality is critical for teasing out what is a cause and what is not in such cases. Many of the other examples in this section are also intended to capture the essence of other important issues that arise in legal (and everyday) reasoning. It is thus worth understanding in all these cases exactly what the role of the actual context is.

Example 2.3.4 This example considers the problem of what has been called double prevention.

Suzy and Billy have grown up just in time to get involved in World War III. Suzy is piloting a bomber on a mission to blow up an enemy target, and Billy is piloting a fighter as her lone escort. Along comes an enemy fighter plane, piloted by Enemy. Sharp-eyed Billy spots Enemy, zooms in, and pulls the trigger; Enemy’s plane goes down in flames. Suzy’s mission is undisturbed, and the bombing takes place as planned.

Is Billy a cause of the success of the mission? After all, he prevented Enemy from preventing Suzy from carrying out the mission. Intuitively, it seems that the answer is yes, and the obvious causal model gives us this. Suppose that we have the following variables:

• BPT for “Billy pulls trigger”, with values 0 (he doesn’t) and 1 (he does);
• EE for “Enemy eludes Billy”, with values 0 (he doesn’t) and 1 (he does);
• ESS for “Enemy shoots Suzy”, with values 0 (he doesn’t) and 1 (he does);
• SBT for “Suzy bombos target”, with values 0 (she doesn’t) and 1 (she does);
• TD for “target destroyed”, with values 0 (it isn’t) and 1 (it is).

The causal network corresponding to this model is given in Figure 2.4.

In this model, BPT = 1 is a but-for cause of TD = 1, as is SBT = 1, so both BPT = 1 and SBT = 1 are causes of TD = 1 according to all the variants of the HP definition.

Figure 2.4: Blowing up the target.

We can make the story more complicated by adding a second fighter plane escorting Suzy, piloted by Hillary. Billy still shoots down Enemy, but if he hadn’t, Hillary would have. The natural way of dealing with this is to add just one more variable HPT representing Hillary’s pulling the trigger iff EE = 1 (see Figure 2.5), but then, using the naive but-for criterion, one might conclude that the target will be destroyed (TD = 1) regardless of Billy’s action, so BPT = 1 would not be a cause of TD = 1. All three variants of the HP definition still declare BPT = 1 to be a cause of TD = 1, as expected. We now take and fix HPT at its value in the actual context, namely, 0. Although Billy’s action seems superfluous under ideal conditions, it becomes essential under a contingency where Hillary for some reason fails to pull the trigger. This contingency is represented by fixing HPT at 0 irrespective of EE.

Figure 2.5: Blowing up the target, with Billy and Hillary.

So far, this may all seem reasonable. But now go back to the original story and suppose that, again, Billy goes up along with Suzy, but Enemy does not actually show up. Is Billy going up a cause of the target being destroyed? Clearly, if Enemy had shown up, then Billy would have shot him down, and he would have been a but-for cause of the target being destroyed. But what about the context where Enemy does not show up? It seems that the original and updated HP definition would say that, even in that context, Billy going up is a cause of the target being destroyed. For if we consider the contingency where Enemy had shown up, then the target would not have been destroyed if Billy weren’t there, but since he is, the target is destroyed.

This may seem disconcerting. Suppose it is known that just about all the enemy’s aircraft have been destroyed, and no one has been coming up for days. Is Billy going up still a cause? The concern is that, if so, almost any A can become a cause of any B by telling a story of how there might have been a C that, but for A, would have prevented B from happening (just as there might have been an Enemy that prevented the target being destroyed had Billy not been there).

Does the HP definition really declare Billy showing up a cause of the target being destroyed in this case? Let’s look a little more carefully at how we model this story. If we want to call Billy going up a cause, then we need to add to the model a variable BGU corresponding to Billy going up. Similarly, we should add a variable ESU corresponding to Enemy showing up. We have the obvious equations, which say that (in the contexts of interest) BPT = 1 (i.e., Billy pulls the trigger) only if Billy goes up and Enemy shows up, and EE = 0 only if either Enemy doesn’t show up or Billy pulls the trigger. This gives us the model shown in Figure 2.6.

Figure 2.6: Blowing up the target, where Enemy may not show up.

In the actual world, where Billy goes up and Enemy doesn’t, Billy doesn’t pull the trigger (why should he?), so BPT = 0. Now it is easy to see that Billy going up (i.e., BGU = 1) is not a cause of the target being destroyed if Enemy doesn’t show up. AC2(b^o) fails (and hence so does AC2(b^u)): if Enemy shows up, the target won’t be destroyed, even if Billy shows up, because in the actual world BPT = 0. The modified HP definition addresses the problem even more straightforwardly. No matter which variables we fix to their actual values, TD = 1 even if BGU = 0. ■

Cases of prevention and double prevention are quite standard in the real world. It is standard to install fire alarms, which, of course, are intended to prevent fires. When the batteries in a fire alarm become weak, fire alarms typically “chirp”. Deactivating the chirping sound could be a cause of a fire not being detected due to double prevention: deactivating the chirping sound prevents the fire alarm from sounding, which in turn prevents the fire from being detected.

Example 2.3.5 Can not performing an action be (part of) a cause? Consider the following story.

Billy, having stayed out in the cold too long throwing rocks, contracts a serious but nonfatal disease. He is hospitalized and treated on Monday, so is fine Tuesday morning.

But now suppose that the doctor does not treat Billy on Monday. Is the doctor’s not treating Billy a cause of Billy’s being sick on Tuesday? It seems that it should be, and indeed it is according to all variants of the HP definition. Suppose that is the context where, among other things, Billy is sick on Monday and the doctor forgets to administer the medication Monday. It seems reasonable that the model should have two variables:

• MT for “Monday treatment”, with values 0 (the doctor does not treat Billy on Monday) and 1 (he does); and
• BMC for “Billy’s medical condition”, with values 0 (recovered) and 1 (still sick).

Sure enough, in the obvious causal setting, MT = 0 is a but-for cause of BMC = 1, and hence a cause according to all three variants of the HP definition.

This may seem somewhat disconcerting at first. Suppose there are 100 doctors in the hospital. Although only one of them was assigned to Billy (and he forgot to give medication), in principle, any of the other 99 doctors could have given Billy his medication. Is the fact that they didn’t give him the medication also part of the cause of him still being sick on Tuesday?

In the causal model above, the other doctors’ failure to give Billy his medication is not a cause, since the model has no variables to model the other doctors’ actions, just as there was no variable in the causal model of Example 2.3.1 to model the presence of oxygen. Their lack of action is part of the context. We factor it out because (quite reasonably) we want to focus on the actions of Billy’s doctor.

If we had included endogenous variables corresponding to the other doctors, then they too would be causes of Billy’s being sick on Tuesday. The more refined definition of causality given in Chapter 3, which takes normality into account, provides a way of avoiding this problem even if the model includes endogenous variables for the other doctors. ■

Causation by omission is a major issue in the law. To take just one of many examples, a surgeon can be sued for the harm caused due to a surgical sponge that he did not remove after an operation.

The next example again emphasizes how the choice of model can change what counts as a cause.

Example 2.3.6 The engineer is standing by a switch in the railroad tracks. A train approaches in the distance. She flips the switch, so that the train travels down the right track instead of the left. Because the tracks reconverge up ahead, the train arrives at its destination all the same.

If we model this story using three variables—F for “flip”, with values 0 (the engineer doesn’t flip the switch) and 1 (she does); T for “track”, with values 0 (the train goes on the left track) and 1 (it goes on the right track); and A for “arrival”, with values 0 (the train does not arrive at the point of reconvergence) and 1 (it does)—then all three definitions agree that flipping the switch is not a cause of the train arriving.

But now suppose that we replace T with two binary variables, LB (for left track blocked), which is 0 if the left track is not blocked, and 1 if it is, and RB (for right track blocked), defined symmetrically. Suppose that LB, RB, and F are determined by the context, while the value of A is determined in the obvious way by which track the train is going down (which is determined by how the switch is set) and whether the track that it is going down is blocked; specifically, A = (F ∧¬LB) ∨ (¬F ∧ ¬RB). In the actual context, F = 1 and LB = RB = 0. Under the original and updated HP definition, F = 1 is a cause of A = 1. For in the contingency where LB = 1, if F = 1, then the train arrives, whereas if F = 0, then the train does not arrive. While adding the variables LB and RB suggests that we care about whether a track is blocked, it seems strange to call flipping the switch a cause of the train arriving when in fact both tracks are unblocked. This problem can be dealt with to some extent by invoking normality considerations, but not completely (see Example 3.4.3). With the modified definition, the problem disappears. Flipping the switch is not a cause of the train arriving if both tracks are unblocked, nor is it a cause of the train not arriving if both tracks are blocked. ■

Example 2.3.7 Suppose that a captain and a sergeant stand before a private, both shout “Charge!” at the same time, and the private charges. Some have argued that, because orders from higher-ranking officers trump those of lower-ranking officers, the captain is a cause of the charge, whereas the sergeant is not.

It turns out that which of the sergeant or captain is a cause depends in part on the variant of the HP definition we consider and in part on what the possible actions of the sergeant and captain are. First, suppose that the sergeant and the captain can each either order an advance, order a retreat, or do nothing. Formally, let C and S represent the captain’s and sergeant’s order, respectively. Then C and S can each take three values, 1, −1, or 0, depending on the order (attack, retreat, no order). The actual value of C and S is determined by the context. P describes what the private does; as the story suggests, P = C if C ≠ 0; otherwise P = S. In the actual context, C = S = P = 1.

C = 1 is a but-for cause of P = 1; setting C to −1 would result in P = −1. Thus, C = 1 is a cause of P = 1 according to all variants of the HP definition. S = 1 is not a cause of B = 1 according to the modified HP definition; changing S to 0 while keeping C fixed at 1 will not affect the private’s action. However, S = 1 is a cause of P = 1 according to the original and updated HP definition, with witness ({C}, 0, 0) (i.e., in the witness world, C = 0 and S = 0): if the captain does nothing, then what the private does is completely determined by the sergeant’s action.

Next, suppose that the captain and sergeant can only order an attack or a retreat; that is, the range of each of C and S is {−1, 1}. Now it is easy to check that C = 1 is the only cause of P = 1 according to all the variants of the HP definition. For the original and updated definition, S = 1 is not a cause because there is no setting of C that will allow the sergeant’s action to make a difference.

Finally, suppose that the captain and sergeant can only order an attack or do nothing; that is, the range of each of C and S is {0, 1}. C = 1 and S = 1 are both causes of P = 1 according to the original and updated HP definition, using the same argument used to show that S = 1 was a cause when it was also possible to order a retreat. Neither is a cause according to the modified HP definition, but C = 1 ∧ S = 1 is a cause, so both C = 1 and S = 1 are parts of a cause.

Note that if the range of the variables C and S is {0, 1}, then there is no way to capture the fact that in the case of conflicting orders, the private obeys the captain. In this setting, we can capture the fact that the private is “really” obeying the captain if both the captain and sergeant give orders by adding a new variable SE that captures the sergeant’s “effective” order. If the captain does not issue any orders (i.e., if C = 0), then SE = S. If the captain does issue an order, then SE = 0; the sergeant’s order is effectively blocked. In this model, P = C if C ≠ 0; otherwise, P = SE. The causal network for this model is given in Figure 2.7.

Figure 2.7: A model of commands that captures trumping.

In this model, the captain causes the private to advance, but the sergeant does not, according to all variants of the HP definition. To see that the captain is a cause according to the modified HP definition, hold SE fixed at 0 and set C = 0; clearly P = 0. By Theorem 2.2.3, C = 1 is a cause of P = 1 according to the original and updated definition as well. To show that S = 1 is not (part of) a cause according to all variants of the HP definition, it suffices to show it is not a cause according to the original HP definition. Suppose that we want to argue that S = 1 causes P = 1. The obvious thing to do is to take and . However, this choice does not satisfy AC2(b^o): if C = 0, SE = 0 (its original value), and S = 1, then P = 0, not 1. The key point is that this more refined model allows a setting where C = 0, S = 1, and P = 0 (because SE = 0). That is, despite the sergeant issuing an order to attack and the captain being silent, the private does nothing. ■

The final example in this section touches on issues of responsibility in the law.

Example 2.3.8 Suppose that two companies both dump pollutant into the river. Company A dumps 100 kilograms of pollutant; company B dumps 60 kilograms. The fish in the river die. Biologists determine that k kilograms of pollutant suffice for the fish to die. Which company is the cause of the fish dying if k = 120, if k = 80, and if k = 50?

It is easy to see that if k = 120, then both companies are causes of the fish dying, according to all three definitions (each company is a but-for cause of the outcome). If k = 50, then each company is still a cause according to the original and updated HP definition. For example, to see that company B is a cause, we consider the contingency where company A does not dump any pollutant. Then the fish die if company B pollutes, but they survive if B does not pollute. With the modified definition, neither company individually is a cause; there is no variable that we can hold at its actual value that would make company A or company B a but-for cause. However, both companies together are the cause.

The situation gets more interesting if k = 80. Now the modified definition says that only A is a cause; if A dumps 100 tons of pollutant, then what B does has no impact. The original and updated definition also agree that A is a cause if k = 80. Whether B is a cause depends on the possible amounts of pollutant that A can dump. If A can dump only 0 or 100 kilograms of pollutant, then B is not a cause; no setting of A’s action can result in B’s action making a difference. However, if A can dump some amount between 20 and 79 kilograms, then B is a cause.

It’s not clear what the “right” answer should be here if k = 80. The law typically wants to declare B a contributing cause to the death of the fish (in addition to A), but should this depend on the amount of pollutant that A can dump? As we shall see in Section 6.2, thinking in terms of responsibility and blame helps clarify the issue (see Example 6.2.5). Under minimal assumptions about how likely various amounts of pollutant are to be dumped, B will get some degree of blame according to the modified definition, even when it is not a cause. ■

2.4 Transitivity

Example 2.4.1 Now consider the following modification of Example 2.3.5.

Suppose that Monday’s doctor is reliable and administers the medicine first thing in the morning, so that Billy is fully recovered by Tuesday afternoon. Tuesday’s doctor is also reliable and would have treated Billy if Monday’s doctor had failed to. … And let us add a twist: one dose of medication is harmless, but two doses are lethal.

Is the fact that Tuesday’s doctor did not treat Billy the cause of him being alive (and recovered) on Wednesday morning?

The causal model M_B for this story is straightforward. There are three variables:

• MT for Monday’s treatment (1 if Billy was treated Monday; 0 otherwise);
• TT for Tuesday’s treatment (1 if Billy was treated Tuesday; 0 otherwise); and
• BMC for Billy’s medical condition (0 if Billy feels fine both Tuesday morning and Wednesday morning; 1 if Billy feels sick Tuesday morning, fine Wednesday morning; 2 if Billy feels sick both Tuesday and Wednesday morning; 3 if Billy feels fine Tuesday morning and is dead Wednesday morning).

We can then describe Billy’s condition as a function of the four possible combinations of treatment/nontreatment on Monday and Tuesday. I omit the obvious structural equations corresponding to this discussion; the causal network is shown in Figure 2.8.

Figure 2.8: Billy’s medical condition.

In the context where Billy is sick and Monday’s doctor treats him, MT = 1 is a but-for cause of BMC = 0; if Billy had not been treated on Monday, then Billy would not have felt fine on Tuesday. MT = 1 is also a but-for cause of TT = 0, and TT = 0 is a but-for cause of Billy’s being alive (BMC ≠ 3, or equivalently, BMC = 0 ∨ BMC = 1 ∨ BMC = 2). However, MT = 1 is not part of a cause of Billy’s being alive, according to any variant of the HP definition. It suffices to show that it is not a cause according the original HP definition. This follows because setting MT = 0 cannot result in Billy dying, not matter how we fix TT.

This shows that causality is not transitive according to the HP definition. Although MT = 1 is a cause of TT = 0 and TT = 0 is a cause of BMC = 0 ∨ BMC = 1 ∨ BMC = 2, MT = 1 is not a cause of BMC = 0 ∨ BMC = 1 ∨ BMC = 2. Nor is causality closed under right weakening; that is, replacing a conclusion by something it implies. If A is a cause of B and B logically implies B′, then A may not be a cause of B′. In this case, MT = 1 is a cause of BMC = 0, which logically implies BMC = 0 ∨ BMC = 1 ∨ BMC = 2, which is not caused by MT = 1. ■

Although this example may seem somewhat forced, there are many quite realistic examples of lack of transitivity with exactly the same structure. Consider the body’s homeostatic system. An increase in external temperature causes a short-term increase in core body temperature, which in turn causes the homeostatic system to kick in and return the body to normal core body temperature shortly thereafter. But if we say that the increase in external temperature happened at time 0 and the return to normal core body temperature happened at time 1, we certainly would not want to say that the increase in external temperature at time 0 caused the body temperature to be normal at time 1!

There is another reason that causality is intransitive, which is illustrated by the following example.

Example 2.4.2 Suppose that a dog bites Jim’s right hand. Jim was planning to detonate a bomb, which he normally would do by pressing the button with his right forefinger. Because of the dog bite, he presses the button with his left forefinger. The bomb still goes off.

Consider the causal model with variables DB (the dog bites, with values 0 and 1), P (the press of the button, with values 0, 1, and 2, depending on whether the button is not pressed at all, pressed with the right hand, or pressed with the left hand), and B (the bomb goes off). We have the obvious equations: DB is determined by the context, P = DB + 1, and B = 1 if P is either 1 or 2. In the context where DB = 1, it is clear that DB = 1 is a but-for cause of P = 2 (if the dog had not bitten, P would have been 1), and P = 2 is a but-for cause of B = 1 (if P were 0, then B woud be 0), but DB = 1 is not a cause of P = 1. Regardless of whether the dog had bitten Jim, the button would have been pressed, and the bomb would have detonated. ■

In retrospect, the failure of right weakening is not so surprising. Taking true to be a tautology, if A is a cause of B, then we do not want to say that A is a cause of true, although B logically implies true. However, the failure of transitivity is quite surprising. Indeed, despite Examples 2.4.1 and 2.4.2, it seems natural to think of causality as transitive. People often think in terms of causal chains: A caused B, B caused C, C caused D, and therefore A caused D, where transitivity seems natural (although the law does not treat causality as transitive in long causal chains; see Section 3.4.4). Not surprisingly, there are some definitions of causality that require causality to be transitive; see the notes at the end of the chapter for details.

Why do we feel that causality is transitive? I believe that this is because, in typical settings, causality is indeed transitive. I give below two simple sets of conditions that are sufficient to guarantee transitivity. Because I expect that these conditions apply in many cases, it may explain why we naturally generalize to thinking of causality as transitive and are surprised when it is not.

For the purposes of this section, I restrict attention to but-for causes. This is what the law has focused on and seems to be the situation that arises most often in practice (although I will admit I have only anecdotal evidence to support this); certainly the law has done reasonably well by considering only but-for causality. Restricting to but-for causality has the further advantage that all the variants of the HP definition agree on what counts as a cause. (Thus, in this section, I do not specify which variant of the definition I am considering.) Restricting to but-for causality does not solve the transitivity problem. As Examples 2.4.1 and 2.4.2 already show, even if X₁ = x₁ is a but-for cause of X₂ = x₂ and X₂ = x₂ is a but-for cause of X₃ = x₃, it may not be the case that X₁ = x₁ is a cause of X₃ = x₃.

The first set of conditions assumes that X₁, X₂, and X₃ each has a default setting. Such default settings make sense in many applications; “nothing happens” can be often taken as the default. Suppose, for example, a billiards expert hits ball A, causing it to hit ball B, causing it to carom into ball C, which then drops into the pocket. In this case, we can take the default setting for the shot to be the expert doing nothing and the default setting for the balls to be that they are not in motion. Let the default setting be denoted by the value 0.

Proposition 2.4.3 Suppose that

(a)X₁ = x₁ is a but-for cause of X₂ = x₂ in ,

(b)X₂ = x₂ is a but-for cause of X₃ = x₃ in ,

(c)x₃ ≠ 0,

(d), and

(e).

Then X₁ = x₁ is a but-for cause of X₃ = x₃ in .

Proof: If X₂ = 0 in the unique solution to the equations in the causal model in context and X₃ = 0 in the unique solution to the equations in in context , then it is immediate that X₃ = 0 in the unique solution to the equations in in context . That is, . It follows from assumption (a) that . We must thus have x₁ ≠ 0; otherwise, , so , which contradicts assumptions (b) and (c). Thus, X₁ = x₁ is a but-for cause of X₃ = x₃, since the value of X₃ depends counterfactually on that of X₁. ■

Although the conditions of Proposition 2.4.3 are clearly rather specialized, they arise often in practice. Conditions (d) and (e) say that if X₁ remains in its default state, then so will X₂, and if both X₁ and X₂ remain in their default states, then so will X₃. Put another way, this says that the reason for X₂ not being in its default state is X₁ not being in its default state, and the reason for X₃ not being in its default state is X₁ and X₂ both not being in their default states. The billiard example can be viewed as a paradigmatic example of when these conditions apply. It seems reasonable to assume that if the expert does not shoot, then ball A does not move; and if the expert does not shoot and ball A does not move (in the context of interest), then ball B does not move, and so on.

Of course, the conditions on Proposition 2.4.3 do not apply in either Example 2.4.1 or Example 2.4.2. The obvious default values in Example 2.4.1 are MT = TT = 0, but the equations say that in all contexts of the causal model M_B for this example, we have . In the second example, if we take DB = 0 and P = 0 to be the default values of DB and P, then in all contexts of the causal model M_D, we have .

While Proposition 2.4.3 is useful, there are many examples where there is no obvious default value. When considering the body’s homeostatic system, even if there is arguably a default value for core body temperature, what is the default value for the external temperature? But it turns out that the key ideas of the proof of Proposition 2.4.3 apply even if there is no default value. Suppose that X₁ = x₁ is a but-for cause of X₂ = x₂ in and X₂ = x₂ is a but-for cause of X₃ = x₃ in . Then to get transitivity, it suffices to find values , , and such that , , and . The argument in the proof of Proposition 2.4.3 (formalized in Lemma 2.10.2) shows that . It then follows that X₁ = x₁ is a but-for cause of X₃ = x₃ in . In Proposition 2.4.3, , , and were all 0, but there is nothing special about the fact that 0 is a default value here. As long as we can find some values , , and , these conditions apply. I formalize this as Proposition 2.4.4, which is a straightforward generalization of Proposition 2.4.3.

Proposition 2.4.4 Suppose that there exist values , , and such that

(a)X₁ = x₁ is a but-for cause of X₂ = x₂ in ,

(b)X₂ = x₂ is a but-for cause of X₃ = x₃ in ,

(c),

(d) (i.e., ), and

(e) (i.e., ).

Then X₁ = x₁ is a but-for cause of X₃ = x₃ in .

To see how these ideas apply, suppose that a student receives an A+ in a course, which causes her to be accepted at Cornell University (her top choice, of course!), which in turn causes her to move to Ithaca. Further suppose that if she had received an A in the course, she would have gone to university U₁ and as a result moved to city C₁, and if she had gotten anything else, she would have gone to university at U₂ and moved to city C₂. This story can be captured by a causal model with three variables: G for her grade, U for the university she goes to, and C for the city she moves to. There are no obvious default values for any of these three variables. Nevertheless, we have transitivity here: The student’s A+ was a cause of her being accepted at Cornell, and being accepted at Cornell was a cause of her move to Ithaca; it seems like a reasonable conclusion that the student’s A+ was a cause of her move to Ithaca. Indeed, transitivity follows from Proposition 2.4.4. We can take the student getting an A to be , the student being accepted at university U₁ to be , and the student moving to C₁ to be (assuming that U₁ is not Cornell and that C₁ is not Ithaca, of course).

The conditions provided in Proposition 2.4.4 are not only sufficient for causality to be transitive, they are necessary as well, as the following result shows.

Proposition 2.4.5 If X₁ = x₁ is a but-for cause of X₃ = x₃ in , then there exist values , , and such that , , and .

Proof: Since X₁ = x₁ is a but-for cause of X₃ = x₃ in , there must exist a value such that . Let and be such that . It easily follows that . ■

In light of Propositions 2.4.4 and 2.4.5, understanding why causality is so often taken to be transitive comes down to finding sufficient conditions to guarantee the assumptions of Proposition 2.4.4. I now present a set of conditions sufficient to guarantee the assumptions of Proposition 2.4.4, motivated by the two examples showing that causality is not transitive. To deal with the problem in Example 2.4.2, I require that for every value in the range of X₂, there is a value in the range of X₁ such that . This requirement holds in many cases of interest; it is guaranteed to hold if X₁ = x₁ is a but-for cause of X₂ = x₂ and X₂ is binary (since but-for causality requires that two different values of X₁ result in different values of X₂). But this requirement does not hold in Example 2.4.2; no setting of DB can force P to be 0.

Imposing this requirement still does not deal with the problem in Example 2.4.1. To do that, we need one more condition: X₂ must lie on every causal path from X₁ to X₃. Roughly speaking, this says that all the influence of X₁ on X₃ goes through X₂. This condition does not hold in Example 2.4.1: as Figure 2.8 shows, there is a direct causal path from MT to BMC that does not include TT. However, this condition does hold in many cases of interest. Going back to the example of the student’s grade, the only way that the student’s grade can influence which city the student moves to is via the university that accepts the student.

To make this precise, I first need to define causal path. A causal path in a causal setting is a sequence (Y₁, …, Y_k) of variables such that Y_j+1 depends on Y_j in context for j = 1, …, k − 1. Since there is an edge between Y_j and Y_j+1 in the causal network for M (assuming that we fix context ) exactly if Y_j+1 depends on Y_j, a causal path is just a path in the causal network. A causal path in from X₁ to X₂ is just a causal path whose first node is X₁ and whose last node is X₂. Finally, Y lies on a causal path in from X₁ to X₂ if Y is a node (possibly X₁ or X₂) on a causal path in from X₁ to X₂.

The following result summarizes the second set of conditions sufficient for transitivity. (Recall that denotes the range of the variable X.)

Proposition 2.4.6 Suppose that X₁ = x₁ is a but-for cause of X₂ = x₂ in the causal setting , X₂ = x₂ is a but-for cause of X₃ = x₃ in , and the following two conditions hold:

(a)for every value , there exists a value such that ;

(b)X₂ is on every causal path in from X₁ to X₃.

Then X₁ = x₁ is a but-for cause of X₃ = x₃.

The proof of Proposition 2.4.6 is not hard, although we must be careful to get all the details right. The high-level idea of the proof is easy to explain. Suppose that X₂ = x₂ is a but-for cause of X₃ = x₃ in . Then there must be some values and such that . By assumption, there exists a value such that . The requirement that X₂ is on every causal path from X₁ to X₃ guarantees that implies in (i.e., implies ). Roughly speaking, X₂ “screens off” the effect of X₁ on X₃, since it is on every causal path from X₁ to X₃. Now we can apply Proposition 2.4.4. I defer the formal argument to Section 2.10.2.

It is easy to construct examples showing that the conditions of Proposition 2.4.6 are not necessary for causality to be transitive. Suppose that X₁ = x₁ causes X₂ = x₂, X₂ = x₂ causes X₃ = x₃, and there are several causal paths from X₁ to X₃. Roughly speaking, the reason that X₁ = x₁ may not be a but-for cause of X₃ = x₃ is that the effects of X₁ on X₃ may “cancel out” along the various causal paths. This is what happens in the homeostasis example. If X₂ is on all the causal paths from X₁ to X₃, then as we have seen, all the effect of X₁ on X₃ is mediated by X₂, so the effect of X₁ on X₃ on different causal paths cannot “cancel out”. But even if X₂ is not on all the causal paths from X₁ to X₃, the effects of X₁ on X₃ may not cancel out along the causal paths, and X₁ = x₁ may still be a cause of X₃ = x₃. That said, it seems difficult to find a weakening of the condition in Proposition 2.4.6 that is simple to state and suffices for causality to be transitive.

2.5 Probability and Causality

In general, an agent trying to determine whether A is a cause of B may not know the exact model or context. An agent may be uncertain about whether Suzy and Billy both threw, or just Suzy threw; he may be uncertain about who will hit first if both Suzy and Billy throw (or perhaps they hit simultaneously). In a perhaps more interesting setting, at some point, people were uncertain about whether smoking caused cancer or whether smoking and cancer were both the outcome of the same genetic problem, which would lead to smoking and cancer being correlated, but no causal relationship between the two. To deal with such uncertainty, an agent can put a probability on causal settings. For each causal setting , the agent can determine whether A is a cause of B in , and so compute the probability that A is a cause of B. Having probabilities on causal settings will be relevant in the discussion of blame in Chapter 6. Here I focus on what seems to be a different source of uncertainty: uncertainty in the equations. In causal models, all the equations are assumed to be deterministic. However, as was observed, in many cases, it seems more reasonable to think of outcomes as probabilistic. So rather than thinking of Suzy’s rock as definitely hitting the bottle if she throws, we can think of her as hitting with probability .9. That is, rather than taking Suzy to always be accurate, we can take her to be accurate with some probability.

The first step in considering how to define causality in the presence of uncertain outcomes is to show how this uncertainty can be captured in the formal framework. Earlier, I assumed that, for each endogenous variable X, there was a deterministic function F_X that described the value of X as a function of all the other variables. Now, rather than assuming that F_X returns a particular value, I assume that it returns a distribution over the values of X given the values of all other variables. This is perhaps easiest to understand by means of an example. In the rock-throwing example, rather than assuming that if Suzy throws a rock, she definitely hits the bottle, suppose we assume that she only only hits it with probability .9 and misses with probability .1. Suppose that if Billy throws and Suzy hasn’t hit the bottle, then Billy will hit it with probability.8 and miss with probability .2. Finally, for simplicity, assume that, if hit by either Billy or Suzy, the bottle will definitely shatter.

The next two paragraphs show how the probabilistic version of this story can be modeled formally in the structural-equations framework and can be skipped on a first reading. Recall that for the more sophisticated model of the (deterministic) rock-throwing story, although I earlier wrote the equation for SH as SH = ST, this is really an abbreviation for the function F_SH (i₁, i₂, i₃, i₄, i₅) = i₂: F_SH is a function of the values of the exogenous variable U (which is not described in the causal network but determines the values of BT and ST) and the endogenous variables ST, BT, BH, and BS. These values are given by i₁, …, i₅. The equation SH = ST says that the value of SH depends only on the value of ST and is equal to it; that explains the output i₂. For the probabilistic version of the story, F_SH (i₁, 1, i₃, i₄, i₅) is the probability distribution that places probability .9 on the value 1 and probability .1 on the value 0. In words, this says that if Suzy throws (ST = 1), then no matter what the values i₁, i₃, i₄, and i₅ of U, BT, BH, and BS are, respectively, F_SH (i₁, 1, i₃, i₄, i₅) is the probability distribution that puts probability .9 on the event SH = 1 and probability .1 on the event SH = 0. This captures the fact that if Suzy throws, then she has a probability .9 of hitting the bottle (no matter what Billy does). Similarly, F_SH (i₁, 0, i₃, i₄, i₅) is the distribution that places probability 1 on the value 0: if Suzy doesn’t throw, then she certainly won’t hit the bottle, so the event SH = 0 has probability 1.

Similarly, in the deterministic model of the rock-throwing example, The function F_BH is such that F_BH (i₁, i₂, i₃, i₄, i₅) = 1 if (i₃, i₄) = (1, 0), and is 0 otherwise. That is, taking i₁, …, i₅ to be the values of U, ST, BT, SH, and BS, respectively, Billy’s throw hits the bottle only if he throws (i₃ = 1) and Suzy doesn’t hit the bottle (i₄ = 0). For the probabilistic version of the story, F_BH (i₁, i₂, i₃, i₄, i₅) is the distribution that puts probability .8 on 1 and probability .2 on 0; for all other values of i₃ and i₄, F_BH (i₁, i₂, 1, 0, i₅) puts probability 1 on 0. (Henceforth, I just describe the probabilities in words rather than using the formal notation.)

The interpretation of these probabilities is similar to the interpretation of the deterministic equations we have used up to now. For example, the fact that Suzy’s rock hits with probability.9 does not mean that the probability of Suzy’s rock hitting conditional on her throwing is .9; rather, it means that if there is an intervention that results in Suzy throwing, the probability of her hitting is .9. The probability of rain conditional on a low barometer reading is high. However, intervening on the barometer reading, say, by setting the needle to point to a low reading, does not affect the probability of rain.

There have been many attempts to give a definition of causality that takes probability into account. They typically take A to be a cause of B if A raises the probability of B. I am going to take a different approach here. I take a standard technique in computer science to “pull out the probability”, allowing me to convert a single causal setting where the equations are probabilistic to a probability over causal settings, where in each causal setting, the equations are deterministic. This, in turn, will allow me to avoid giving a separate definition of probabilistic causality. Rather, I will be able to use the definition of causality already given for deterministic models and talk about the probability of causality, that is, the probability that A is a cause of B. As we shall see, this approach seems to deal naturally with a number of problems regarding probabilistic causality that have been raised in the literature.

The assumption is that the equations would be deterministic if we knew all the relevant details. This assumption is probably false at the quantum level; most physicists believe that at this level, the world is genuinely probabilistic. But for the macroscopic events that I focus on in this book (and that most people are interested in when applying causality judgments), it seems reasonable to view the world as fundamentally deterministic. With this viewpoint, if Suzy hits the rock with probability .9, then there must be (perhaps poorly understood) reasons that cause her to miss: an unexpected gust of wind or a momentary distraction throwing off her aim. We can “package up” all these factors and make them exogenous. That is, we can have an exogenous variable U′ with values either 0 or 1 depending on whether the features that cause Suzy to miss are present, where U′ = 0 with probability .1 and U′ = 1 with probability .9. By introducing such a variable U′ (and a corresponding variable for Billy), we have essentially pulled the probability out of the equations and put it into the exogenous variable.

I now show in more detail how this approach plays out in the context of the rock-throwing example with the probabilities given above. Consider the causal setting where Suzy and Billy both throw. We may be interested in knowing the probability that Suzy or Billy will be a cause of the bottle shattering if we don’t know whether the bottle will in fact shatter, or in knowing the probability that Suzy or Billy is the cause if we know that the bottle actually did shatter. (Notice that in the former case, we are asking about an effect of a (possible) cause, whereas in the latter case, we are asking about the cause of an effect.) When we pull out the probability, there are four causal models that arise here, depending on whether Suzy’s rock hits the bottle and whether Billy’s rock hits the bottle if Suzy’s rock does not hit. Specifically, consider the following four models, in a context where both Suzy and Billy throw rocks:

• M₁: Suzy’s rock hits and Billy’s rock would hit if Suzy missed;
• M₂: Suzy’s rock hits but Billy’s rock would not hit if Suzy missed;
• M₃: Suzy’s rock misses and Billy’s rock hits;
• M₄: neither rock hits.

Note that these models differ in their structural equations, although they involve the same exogenous and endogenous variables. M₁ is just the model considered earlier, described in Figure 2.3. Suzy’s throw is a cause in M₁ and M₂; Billy’s throw is a cause in M₃.

If we know that the bottle shattered, Suzy’s rock hit, and Billy’s didn’t, then the probability that Suzy is the cause is 1. We are conditioning on the model being either M₁ or M₂ because these are the models where Suzy’s rock hits, and Suzy is the cause of the bottle shattering in both. But suppose that all we know is that the bottle shattered, and we are trying to determine the probability that Suzy is the cause. Now we can condition on the model being either M₁, M₂, or M₃, so if the prior probability of M₄ is p, then the probability that Suzy is a cause of the bottle shattering is .9/(1 − p): Suzy is a cause of the bottle shattering as long as she hits the bottle and it shatters (i.e., in models M₁ and M₂). Unfortunately, the story does not give us that the probability of M₄. It is .1 − q, where q is the prior probability of M₃, but the story does not tell us that either. We can say more if we make a reasonable additional assumption: that whether Suzy’s rock hits is independent of whether Billy’s rock would hit if Suzy missed. Then M₁ has probability .72 (= .9 × .8), M₂ has probability .18, M₃ has probability .08, and M₄ has probability .02. With this assumption and no further information, the probability of Suzy being a cause of the bottle shatter is .9/.98 = 45/49.

Although it seems reasonable to view the outcomes “Suzy’s rock hits the bottle” and “Billy’s rock would have hit the bottle if Suzy’s hadn’t” as independent, they certainly do not have to be. We could, for example, assume that Billy and Suzy are perfectly accurate when there are no wind gusts and miss when there are wind gusts, and that wind gusts occur with probability .7. Then it would be the case that Suzy’s rock hits with probability .7 and that Billy’s rock would have hit with probability .7 if Suzy’s had missed, but there would have been only two causal models: the one that we considered in the deterministic case, which would be the actual model with probability .7, and the one where both miss. (A better model of this situation might have an exogenous variable U′ such that U′ = 1 if there are wind gusts and U′ = 0 if there aren’t, and then have deterministic equations for SH and BH.) The key message here is that, in general, further information might be needed to determine the probability of causality beyond what is provided by probabilistic structural equations as defined above.

A few more examples should help illustrate how this approach works and how much we can say.

Example 2.5.1 Suppose that a doctor treats Billy on Monday. The treatment will result in Billy recovering on Tuesday with probability .9. But Billy might also recover with no treatment due to some other factors, with independent probability .1. By “independent probability” here, I mean that conditional on Billy being treated on Monday, the events “Billy recovers due to the treatment” and “Billy recovers due to other factors” are independent; thus, the probability that Billy will not recover on Tuesday given that he is treated on Monday is (.1)(.9). Now, in fact, Billy is treated on Monday and he recovers on Tuesday. Is the treatment the cause of Billy getting better?

The standard answer is yes, because the treatment greatly increases the likelihood of Billy getting better. But there is still the nagging concern that Billy’s recovery was not due to the treatment. He might have gotten better if he had just been left alone.

Formally, we have a causal model with three endogenous binary variables: MT (the doctor treats Billy on Monday), OF (the other factors that would make Billy get better occur), and BMC (Billy’s medical condition, which we can take to be 1 if he gets better and 0 otherwise). The exogenous variable(s) determine MT and OF. The probability that MT = 1 does not matter for this analysis; for definiteness, assume that MT = 1 with probability .8. According to the story, OF = 1 with probability .9. Finally, BMC = 1 if OF = 1, BMC = 0 if MT = OF = 0, and BMC = 1 with probability .9 if MT = 1 and OF = 0. Again, after we pull out the probability, there are eight models, which differ in whether MT is set to 0 or 1, whether OF is set to 0 or 1, and whether BMC = 1 when MT = 1 and OF = 0. Treating these choices as independent (as is suggested by the story), it is straightforward to calculate the probability of each of the eight models. Since we know that Billy actually did get better and was treated by his doctor, we ignore the four models where MT = 0 and the model where MT = 1, OF = 0, and BMC = 0 if MT = 1 and OF = 0. The remaining three models have probability .8(1 − (.9)(.1)). MT = 1 is a cause of BMC = 0 in two of the three models; it is not a cause only in the model where OF = 1, MT = 1, but BMC would be 0 if OF were set to 0. This latter model has prior probability .8(.1)(.1). Straightforward calculations now show that the treatment cause of Billy’s recovery with probability . Similarly, the probability that Billy’s recovery was caused by other factors is . These probabilities sum to more than 1 because with probability , both are causes; Billy’s recovery is overdetermined.

Now suppose that we can perform a (somewhat expensive) test to determine whether the doctor’s treatment actually caused Billy to get better. Again, the doctor treats Billy and Billy recovers, but the test is carried out, and it is shown that the treatment was not the cause. It is still the case that the treatment significantly raises the probability of Billy recovering, but now we certainly don’t want to call the treatment the cause. And, indeed, it is not. Conditioning on the information, we can discard three of the four models. The only one left is the one in which Billy’s recovery is due to other factors (with probability 1). Moreover, this conclusion does not depend on the independence assumption. ■

Example 2.5.2 Now suppose that there is another doctor who can treat Billy on Monday, with different medication. Both treatments are individually effective with probability .9. Unfortunately, if Billy gets both treatments and both are effective, then, with probability.8, there will be a bad interaction, and Billy will die. For this version of the story, suppose that there are no other factors involved; if Billy gets neither treatment, he will not recover. Further suppose that we can perform a test to see whether a treatment was effective. Suppose that in fact both doctors treat Billy and, despite that, Billy recovers on Tuesday. Without further information, there are three relevant models: in the first, only the first doctor’s medication was effective; in the second, only the second doctor’s medication was effective; in the third, both are effective. If further testing shows that both treatments were effective, then there is no uncertainty; we are down to just one model: the third one. According to the original and updated HP definition, both treatments are the cause of Billy’s recovery, with probability 1. (According to the modified HP definition, they are both parts of the cause.) This is true despite the fact that, given that the first doctor treated Billy, the second doctor treating Billy significantly lowers the probability of Billy recovering (and symmetrically with the roles of the doctors reversed). ■

Example 2.5.3 Now suppose that a doctor has treated 1,000 patients. They each would have had probability .9 of recovering even without the treatment. With the treatment, they recover with independent probability .1 In fact, 908 patients recovered, and Billy is one of them. What is the probability that the treatment was the cause of Billy’s recovery? Calculations similar to those in Example 2.5.1 show that it is 10/91. Standard probabilistic calculations show that there is quite a high probability that the treatment is a cause of at least one patient’s recovery. Indeed, there is quite a high probability that the treatment is the unique cause of at least one patient’s recovery. But we do not know which patient that is. ■

Example 2.5.4 Again, Billy and Suzy throw rocks at a bottle. If either Billy or Suzy throws a rock, it will hit the bottle. But now the bottle is heavy and will, in general, not topple over if it is hit. If only Billy hits the bottle, it will topple over with probability.2; similarly, if only Suzy hits the bottle, it will topple over with probability.2. If they hit the bottle simultaneously, it will topple with probability.7. In fact, both Billy and Suzy throw rocks at the bottle, and it topples over. We are interested in the extent to which Suzy and Billy are the causes of the bottle toppling over.

For simplicity, I restrict attention to contexts where both Billy and Suzy throw, both hit, and the bottle topples over. After pulling out the probability, there are four causal models of interest, depending on what would have happened if just Suzy had hit the bottle and if just Billy had hit the bottle:

• M₁: the bottle would have toppled over if either only Suzy’s rock had hit it or if only Billy’s rock had hit it;
• M₂: the bottle would have toppled over if only Suzy’s rock had hit it but not if only Billy’s rock had hit it;
• M₃: the bottle would have toppled over if only Billy’s rock had hit it but not if only Suzy’s rock had hit it;
• M₄: the bottle would not have toppled over if either only Billy’s rock or only Suzy’s rock had hit it.

I further simplify by treating the outcomes “the bottle would have toppled over had only Suzy’s rock hit it” and “the bottle would have toppled over had only Billy’s rock hit it” as independent (even though neither may be independent of the outcome “the bottle would have toppled over had both Billy and Suzy’s rocks hit it”). With this assumption, the probability of M₁ (conditional on the bottle toppling over if both Suzy’s and Billy’s rock hit it) is .04, the conditional probability of M₂ is .16, the conditional probability of M₃ is .16, and the conditional probability of M₄ is .64.

It is easy to check that Suzy’s throw is a cause of the bottle toppling in models M₁, M₂, and M₄, but not in M₃. Similarly, Billy’s throw is a cause in models M₁, M₃, and M₄. They are both causes in models M₁ and M₄ according to the original and updated HP definition, as well as in M₄ according to the modified HP definition; in M₁, the conjunction ST = 1 ∧ BT = 1 is a cause according to the modified HP definition. Thus, the probability that Suzy’s throw is part of a cause of the bottle shattering is .84, and likewise for Billy. The probability that both are parts of causes is .68 (since this is the case in models M₁ and M₄). ■

As these examples show, the HP definition lets us make perfectly sensible statements about the probability of causality. There is nothing special about these examples; all other examples can be handled equally well. Perhaps the key message here is that there is no need to work hard on getting a definition of probabilistic causality, at least at the macroscopic level; it suffices to get a good definition of deterministic causality. But what about if we want to treat events at the quantum level, which seems inherently probabilistic? A case can still be made that it is psychologically useful to think deterministically. Whether the HP definitions can be extended successfully to an inherently probabilistic framework still remains open.

Even if we ignore issues at the microscopic level, there is still an important question to be addressed: In what sense is the single model with probabilistic equations equivalent to the set of deterministic causal models with a probability on them? In a naive sense, the answer is no. The examples above show that the probabilistic equations do not in general suffice to determine the probability of the deterministic causal models. Thus, there is a sense in which deterministic causal models with a probability on them carry more information than a single probabilistic model. Moreover, this extra information is useful. As we have seen, it may be necessary to determine the answer to questions such as “How probable is it that Billy is the cause of the bottle shattering?” when all we see is a shattered bottle and know that Billy and Suzy both threw. Such a determination can be quite important in, for example, legal cases. Although bottle shattering may not be of great legal interest, we may well want to know the probability of Bob being the cause of Charlie dying in a case where there are multiple potential causes.

If we make reasonable independence assumptions, then a probabilistic model determines a probability on deterministic models. Moreover, under the obvious assumptions regarding the semantics of causal models with probabilistic equations, formulas in the language described in Section 2.2.1 have the same probability of being true under both approaches. Consider Example 2.5.4 again. What is the probability of a formula like [ST ← 0](BS = 1) (if Suzy hadn’t thrown, the bottle would have shattered)? Clearly, of the four deterministic models described in the discussion of this example, the formula is true only in M₁ and M₃; thus, it has probability .2. Although I have not given a semantics for formulas in this language in probabilistic causal models (i.e., models where the equations are probabilistic), if we assume that the outcomes “the bottle would have toppled over had only Suzy’s rock hit it” and “the bottle would have toppled over had only Billy’s rock hit it” are independent, as I assumed when assigning probabilities to the deterministic models, we would expect this statement to also be true in the probabilistic model. The converse holds as well: if a statement about the probability of causality holds in the probabilistic model, then it holds in the corresponding deterministic model.

This means that (under the independence assumption) as far as the causal language of Section 2.2.1 is concerned, the two approaches are equivalent. Put another way, to distinguish the two approaches, we need a richer language. More generally, although we can debate whether pulling out the probability “really” gives an equivalent formulation of the situation, to the extent that the notion of (probabilistic) causality is expressible in the language of Section 2.2.1, the two approaches are equivalent. In this sense, it is safe to reduce to deterministic models.

This observation leads to two further points. Suppose that we do not want to make the relevant independence assumptions. Is there a natural way to augment probabilistic causal models to represent this information (beyond just going directly to a probability over deterministic causal models)? See Section 5.1 for further discussion of this point.

Of course, we do not have to use probability to represent uncertainty; other representations of uncertainty could be used as well. Indeed, there is a benefit to using a representation that involves sets of probabilities. Since, in general, probabilistic structural equations do not determine the probability of the deterministic models that arise when we pull out the probability, all we have is a set of possible probabilities on the deterministic models. Given that, it seems reasonable to start with sets of probabilities on the equations as well. This makes sense if, for example, we do not know the exact probability that Suzy’s rock would hit the bottle, but we know that it would hit it with probability between .6 and .8. It seems that the same general approach of pulling out the probabilities works if we represent uncertainty using sets of probabilities, but I have not explored this approach in detail.

2.6 Sufficient Causality

Although counterfactual dependence is a key feature of causality, people’s causality judgments are clearly influenced by another quite different feature: how sensitive the causality ascription is to changes in various other factors. If we assume that Suzy is very accurate, then her throwing a rock is a robust cause of the bottle shattering. The bottle will shatter regardless of whether Billy throws, even if the bottle is in a slightly different position and even if it is slightly windy. On the other hand, suppose instead that we consider a long causal chain like the following: Suzy throws a rock at a lock, causing it to open, causing the lion that was in the locked cage to escape, frightening the cat, which leapt up on to the table and knocked over the bottle, which then shattered. Although Suzy’s throw is still a but-for cause of the bottle shattering, the shattering is clearly sensitive to many other factors. If Suzy’s throw had not broken the lock, if the lion had run in a different direction, or if the cat had not jumped on the table, then the bottle would not have shattered. People seem inclined to assign less blame to Suzy’s throw in this case. I return to the issue of long causal chains in Sections 3.4.4 and provide a solution to it using the notion of blame in Section 6.2. Here I define a notion of sufficient causality that captures some of the intuitions behind insensitive causation. Sufficient causality also turns out to be related to the notion of explanation that I define in Chapter 7, so it is worth considering in its own right, although it will not be a focus of this book.

The key intuition behind the definition of sufficient causality is that not only does suffice to bring about φ in the actual context (which is the intuition that AC2(b^o) and AC2(b^u) are trying to capture), but it also brings it about in other “nearby” contexts. Since the framework does not provide a metric on contexts, there is no obvious way to define nearby context. Thus, in the formal definition below, I start by considering all contexts. Since conjunction plays a somewhat different role in this definition than it does in the definition of actual causality (particularly in the modified HP definition), I take seriously here the intuition that what we really care about are parts of causes, rather than causes.

Again, there are three variants of the notion of sufficient causality, depending on which version of AC2 we use. The differences between these variants do not play a role in our discussion, so I just write AC2 below. Of course, if we want to distinguish, AC2 below can be replaced by either AC2(a) and AC2(b^o), AC2(a) and AC2(b^u), or AC2(a^m), depending on which version of the HP definition is being considered.

Definition 2.6.1 is a sufficient cause of φ in the causal setting if the following conditions hold:

SC1. and .

SC2. Some conjunct of is part of a cause of φ in . More precisely, there exists a conjunct X = x of and another (possibly empty) conjunction such that is a cause of φ in ; that is, AC1, AC2, and AC3 hold for .

SC3. for all contexts .

SC4. is minimal; there is no strict subset of such that satisfies conditions SC1, SC2, and SC3, where is the restriction of to the variables in . ■

SC3 is the key condition here; it says that suffices to bring about φ in all contexts. Thus, in the case of the 11–0 victory, any set of six voters becomes a sufficient cause (no matter which variant of the HP definition we use), assuming that there are contexts corresponding to all possible voting configurations. This suggests that sufficient causality is related to the modified HP definition, which, in this case, would also take any subset of six voters to be a cause. However, this is misleading, as the next example shows.

Consider the forest-fire example again. In the disjunctive model, each of L = 1 and MD = 1 is a sufficient cause; in the conjunctive model, L = 1 ∧ MD = 1 is a sufficient cause, assuming that there is a context where there is no lightning and another where the arsonist does not drop a match. This is the case for all variants of the HP definition. Recall that this is just the opposite of what the modified HP definition does with actual causality; it would declare L = 1 ∧ MD = 1 the cause in the disjunctive model and both L = 1 and MD = 1 individually causes in the conjunctive model.

I wrote SC2 as I did so as to take seriously the notion that all we really care about when using the modified (and updated) definition is parts of causes. Thus, in the disjunctive model for the forest fire example, L = 1 and MD = 1 are both sufficient causes of the forest fire in the context (1, 1) where there is both a lightning strike and a dropped match, even if we use AC2(a^m) in SC2. By assumption, SC3 holds: both [L ← 1](FF = 1) and [MD ← 1](FF = 1) hold in all contexts. Moreover, both L = 1 and MD = 1 are parts of a cause of FF = 1 (namely, L = 1 MD = 1), so SC2 holds.

The forest-fire example shows that sufficient causality lets us distinguish what can be called joint causes from independent causes. In the disjunctive forest-fire model, the lightning and the dropped match can each be viewed as independent causes of the fire; each suffices to bring it about. In the conjunctive model, the lightning and the dropped match are joint causes; their joint action is needed to bring about the forest fire. The distinction between joint and independent causality seems to be one that people are quite sensitive to. Not surprisingly, it plays a role in legal judgments.

Going on with examples, consider the sophisticated rock-throwing model from Example 2.3.3 again. Assume that Suzy is always accurate; that is, she is accurate in all contexts where she throws, and would be accurate even in contexts where she doesn’t throw if (counterfactually) she were to throw; that is, assume that [ST ← 1](BS = 1) holds in all contexts, even if Suzy does not actually throw. Then Suzy’s throw is a sufficient cause of the bottle shattering in the context that we have been considering, where both Billy and Suzy throw, and Suzy’s throw actually hits the bottle. Billy’s throw is not a sufficient cause in this context because it is not even (part of) a cause. However, if we consider the model , which has a larger set of contexts, including ones where Suzy might throw and miss, then ST = 1 is not a sufficient cause of BS = 1 in the context u^∗ where Suzy is accurate and her rock hits before Billy’s, but SH = 1 is. Recall that ST = 1 remains a cause of BS = 1 in . This shows that (part of) a cause is not necessarily part of a sufficient cause.

In the double-prevention example (Example 2.3.4), assuming that Suzy is accurate, SBT = 1 (Suzy bombing the target) is a sufficient cause of TD = 1 (the target being destroyed). In contrast, Billy pulling the trigger and shooting down Enemy is not, if there are contexts where Suzy does not go up, and thus does not destroy the target. More generally, a but-for cause at the beginning of a long causal chain is unlikely to be a sufficient cause.

The definition of actual causality (Definition 2.2.1) focuses on the actual context; the set of possible contexts plays no role. By way of contrast, the set of contexts plays a major role in the definition of sufficient causality. The use of SC3 makes sufficient causality quite sensitive to the choice of the set of possible contexts. It also may make it an unreasonably strong requirement in some cases. Do we want to require SC3 to hold even in some extremely unlikely contexts? A sensible way to weaken SC3 would be to add probability to the picture, especially if we have a probability on contexts, as in Section 2.5. Rather than requiring SC3 to hold for all contexts, we could then consider the probability of the set of contexts for which it holds. That is, we can take to be a sufficient cause of φ with probability α in if SC1, SC2, and SC4 hold and the set of contexts for which SC3 holds has probability at least α.

Notice that with this change, there is a tradeoff between minimality and probability of sufficiency. Consider Suzy throwing rocks. Suppose that, although Suzy is quite accurate, there is a (quite unlikely) context where there are high winds, so Suzy misses the bottle. In this model, a sufficient cause for the bottle shattering is “Suzy throws and there are no high winds”. Suzy throwing is not by itself sufficient. But it is sufficient in all contexts but the one in which there are high winds. If the context with high winds has probability .1, then Suzy’s throw is a sufficient cause of the bottle shattering with probability .9.

This tradeoff between minimality and probability of sufficiency comes up again in the context of the definition of explanation, considered in Section 7.1, so I do not dwell on it here. But thinking in terms of probability provides some insight into whether not performing an action should be treated as a sufficient cause. Consider Example 2.3.5, where Billy is hospitalized. Besides Billy’s doctor, there are other doctors at the hospital who could, in principle, treat Billy. But they are unlikely to do so. Indeed, they will not even check up on Billy unless they are told to; they presumably have other priorities.

Now consider a context where in fact no doctor treats Billy. In that case, Billy’s doctor not treating Billy on Monday is a sufficient cause for Billy feeling sick on Tuesday with high probability because, with high probability, no other doctor would treat him either. However, another doctor not treating Billy has only a low probability of being a sufficient cause for Billy feeling sick on Tuesday because, with high probability, Billy’s doctor does treat him. This intuition is similar in spirit to the way normality considerations are brought to bear on this example in Chapter 3 (see Example 3.2.2).

For the most part, I do not focus on sufficient causality in this book. I think that notions such as normality, blame, and explanation capture many of the same intuitions as sufficient causality in an arguably better way. However, it is worth keeping sufficient causality in mind when we consider these other notions.

2.7 Causality in Nonrecursive Models

Up to now, I have considered causality only in recursive (i.e., acyclic) causal models. Recursive models are also my focus throughout the rest of the book. However, there are examples that arguably involve nonrecursive models. For example, the connection between pressure and volume is given by Boyle’s Law, which says that PV = k: the product of pressure and volume is a constant (if the temperature remains constant). Thus, increasing the pressure decreases the volume, and decreasing the volume increases the pressure. There is no direction of causality in this equation. However, in any particular setting, we are typically manipulating either the pressure or the volume, so there is (in that setting) a “direction” of causality.

Perhaps a more interesting example is one of collusion. Imagine two arsonists who agree to burn down the forest, when only one match suffices to burn down the forest. (So we are essentially in the disjunctive model.) However, the arsonists make it clear to each other that each will not drop a lighted match unless the other does. At the critical moment, they look in each others’ eyes, and both drop their lit matches, resulting in the forest burning down. In this setting, it is plausible that each arsonist caused the other to throw the match, and together they caused the forest to burn down. That is, if MD_i stands for “arsonist i drops a lit match”, for i = 1, 2, and U is an exogenous variable that determines whether the arsonists initially intend to drop the match, we can take the equation for MD₁ to be such that MD ₁ = 1 if and only if U = 1 and MD₂ = 1, and similarly for MD₂. This model is not recursive; in all contexts, MD₁ depends on MD ₂ and MD ₂ depends on MD₁. In this section, I show how the HP definition(s) can be extended to deal with such nonrecursive models.

In nonrecursive models, there may be more than one solution to an equation in a given context, or there may be none. In particular, this means that a context no longer necessarily determines the values of the endogenous variables. Earlier, I identified a primitive event such as X = x with the basic causal formula [ ](X = x), that is, a formula of the form [Y₁ ← y₁, …, Y_k ← y_k](X = x) with k = 0. In recursive causal models, where there is a unique solution to the equations in a given context, we can define [ ](X = x) to be true in if X = x is the unique solution to the equations in context . It seems reasonable to identify [ ](X = x) with X = x in this case. But it is not so reasonable if there may be several solutions to the equations or none.

What we really want to do is to be able to say that X = x under a particular setting of the variables. Thus, we now take the truth of a primitive event such as X = x to be relative not just to a context but to a complete assignment: a complete description of the values of both the exogenous and the endogenous variables. As before, the context assigns a value to all the exogenous variables; assigns a value to all the endogenous variables. Now define if X has value x in . Since the truth of X = x depends on just , not , I sometimes write .

This definition can be extended to Boolean combinations of primitive events in the standard way. Define iff for all solutions to the equations in . Since the truth of depends only on the setting of the exogenous variables, and not on , I write to emphasize this.

With these definitions in hand, it is easy to extend the HP definition of causality to arbitrary models. Causality is now defined with respect to a tuple because we need to know the values of the endogenous variables to determine the truth of some formulas. That is, we now talk about being an actual cause of φ in . As before, we have conditions AC1–AC3. AC1 and AC3 remain unchanged. AC2 depends on whether we want to consider the original, updated, or modified definition. For the updated definition, we have:

AC2. There exists a partition of with and some setting of the variables in such that

In recursive models, the formula in the version of AC2(a) above is equivalent to , the formula used in the original formulation of AC2(a). Given a recursive model M, there is a unique solution to the equations in ; holds in that solution iff φ does not hold. However, with nonrecursive models, there may several solutions; AC2(a) holds if φ does not hold in at least one of them. By way of contrast, for AC2(b^u) to hold, φ must stay true in all solutions to the equations in in context . AC2(a) already says that there is some contingency in which setting to is necessary to bring about φ (since setting to something other than in that contingency may result in φ no longer holding). Requiring ¬φ to hold in only one solution to the equations seems in the spirit of the necessity requirement. However, AC2(b^u) says that setting to is sufficient to bring about φ in all relevant settings, so it makes sense that we want φ to hold in all solutions. That’s part of the intuition for sufficiency. Clearly, in the recursive case, where there is only one solution, this definition agrees with the definition given earlier.

Analogous changes are required to extend the original and modified HP definition to the nonrecursive case. For AC2(b^o), we require only that for all subsets of , we have , and do not require this to hold for all subsets of ; for AC2(a^m), we require that consist of the values of the variables in in the actual context. Again, these definitions agree with the earlier definitions in the recursive case.

Note that, with these definitions, all definitions agree that MD₁ = 1 and MD₂ = 1 are causes of the fire in the collusive disjunctive scenario discussed above. This is perhaps most interesting in the case of the modified HP definition, where in the original disjunctive scenario, the cause was the conjunction MD = 1 ∧ L = 1. However, in this case, each arsonist is a but-for cause of the fire. Although it takes only one match to start the fire, if arsonist 1 does not drop his match, then arsonist 2 won’t drop his match either, so there will not be a fire. The key point is that the arsonists do not act independently here, whereas in the original story, the arsonist and lightning were viewed as independent.

Interestingly, we can make even the collusion story recursive. Suppose that we add a variable EC for eye contact (allowing for the possibility that the arsonists avoid each other’s eyes), where the equations are such that each drops the match only if they do indeed have eye contact. Now we are back at a model where each arsonist is (separately) a cause of the fire (as is the eye contact) according to the original and updated HP definition, and MD₁ = 1 ∧ MD ₂ = 1 is a cause according to the modified HP definition. We can debate which is the “better” model here. But I have no reason to believe that we can always replace a nonrecursive model by a recursive model and still describe essentially the same situation.

Finally, it is worth noting that causality is not necessarily asymmetric in nonrecursive models; we can have X = x being a cause of Y = y and Y = y being a cause of X = x. For example, in the model of the collusive story above (without the variable EC), MD₁ = 1 is a cause of MD₂ = 1 and MD₂ = 1 is a cause of MD₁ = 1. It should not be so surprising that we can have such symmetric causality relations in nonrecursive models. However, it is easy to see that causality is asymmetric in recursive models, according to all the variants of the HP definition. AC2(a) and AC2(a^m) guarantee that if X = x is a cause of Y = y in (M, u), then Y depends on X in context u. Similarly, X must depend on Y if Y = y is a cause of X = x in (M, u). This cannot be the case in a recursive model.

Although the definition of causality in nonrecursive models given here seems like the most natural generalization of the definition for recursive models, I do not have convincing examples suggesting that this is the “right” definition, nor do I have examples showing that this definition is problematic in some ways. This is because all the standard examples are most naturally modeled using recursive models. It would be useful to have more examples of scenarios that are best captured by nonrecursive models for which we have reasonable intuitions regarding the ascription of causality.

2.8 AC2(b^o) vs. AC2(b^u)

In this section, I consider more carefully AC2(b^o) and AC2(b^u), to give the reader a sense of the subtle differences between original and updated HP definition. I start with the example that originally motivated replacing AC2(b^o) by AC2(b^u).

Example 2.8.1 Suppose that a prisoner dies either if A loads B’s gun and B shoots or if C loads and shoots his gun. Taking D to represent the prisoner’s death and making the obvious assumptions about the meaning of the variables, we have that D = (A ∧ B) C. (Note that here I am identifying the binary variables A, B, C, and D with primitive propositions in propositional logic, as I said earlier I would. I could have also written this as D = max(min(A, B), C).) Suppose that in the actual context u, A loads B’s gun, B does not shoot, but C does load and shoot his gun, so the prisoner dies. That is, A = 1, B = 0, and C = 1. Clearly C = 1 is a cause of D = 1. We would not want to say that A = 1 is a cause of D = 1, given that B did not shoot (i.e., given that B = 0). However, suppose that we take the obvious model with the variables A, B, C, D. With AC2(b^o), A = 1 is a cause of D = 1. For we can take and consider the contingency where B = 1 and C = 0. It is easy to check that AC2(a) and AC2(b^o) hold for this contingency, since (M, u) ⊨ [A ← 0, B ←1, C ← 0](D = 0), whereas (M, u) = [A ← 1, B ← 1, C ← 0](D = 1). Thus, according to the original HP definition, A = 1 is a cause of D = 1. However, AC2(b^u) fails in this case because (M, u) ⊨ [A ← 1, C ← 0](D = 0). The key point is that AC2(b^u) says that for A = 1 to be a cause of D = 1, it must be the case that D = 1 even if only some of the values in are set to their values . In this case, by setting only A to 1 and leaving B unset, B takes on its original value of 0, in which case D = 0. AC2(b^o) does not consider this case.

Note that the modified HP definition also does not call A = 1 a cause of D = 1. This follows from Theorem 2.2.3 and the observations above; it can also be seen easily by observing that if the values of any subset of variables are fixed at their actual values in context u, setting A = 0 will not make D = 0.

Although it seems clear that we do not want A = 1 to be a cause of D = 1, the situation for the original HP definition is not as bleak as it may appear. We can deal with this example using the original HP definition if we add extra variables, as we did with the Billy-Suzy rock-throwing example. Specifically, suppose that we add a variable B′ for “B shoots a loaded gun”, where B′ = A ∧ B and D = B′ ∨ C. This slight change prevents A = 1 from being a cause of D = 1, even according to the original HP definition. I leave it to the reader to check that any attempt to now declare A = 1 a cause of D = 1 according to the original HP definition would have to put B′ into . However, because B′ = 0 in the actual world, AC2(b^o) does not hold. As I show in Section 4.3, this approach to dealing with the problem generalizes. There is a sense in which, by adding enough extra variables, we can always use AC2(b^o) instead of AC2(b^u) to get equivalent judgments of causality. ■

There are advantages to using AC2(b^o). For one thing, it follows from Theorem 2.2.3(d) that with AC2(b^o), causes are always single conjuncts. This is not the case with AC2(b^u), as the following example shows.

Example 2.8.2 A votes for a candidate. A’s vote is recorded in two optical scanners B and C. D collects the output of the scanners; D′ records whether just scanner B records a vote for the candidate. The candidate wins (i.e., WIN = 1) if any of A, D, or D′ is 1. The value of A is determined by the exogenous variable. The following structural equations characterize the value of the remaining variables:

• B = A;
• C = A;
• D = B ∧ C;
• D′ = B ∧ ¬A; and
• WIN = A ∨ D ∨ D′.

Call this causal model M_V. Roughly speaking, D′ acts like BH in the rock-throwing example as modeled by . The causal network for M_V is shown in Figure 2.9.

In the actual context u, A = 1, so B = C = D = WIN = 1 and D′ = 0. I claim that B = 1 ∧ C = 1 is a cause of WIN = 1 in (M_V, u) according to the updated HP definition (which means that, by AC3, neither B = 1 nor C = 1 is cause of WIN = 1 in (M_V, u)). To see this, first observe that AC1 clearly holds. For AC2, consider the witness where (so ) and w = 0 (so we are considering the contingency where A = 0). Clearly, (M_V, u) ⊨ [A ← 0, B ← 0, C ← 0](WIN = 0), so AC2(a) holds, and (M_V, u) ⊨ [A ← 0, B ← 1, C ← 1](WIN = 1). Moreover, (M_V, u) ⊨ [B ← 1, C ← 1](WIN = 1), and WIN = 1 continues to hold even if D is set to 1 and/or D′ is set to 0 (their values in (M_V, u)). Thus, AC2 holds.

Figure 2.9: Another voting scenario.

It remains to show that AC3 holds and, in particular, that neither B = 1 nor C = 1 is a cause of WIN = 1 in (M_V, u) according to the updated HP definition. The argument is essentially the same for both B = 1 and C = 1, so I just show it for B = 1. Roughly speaking, B = 1 does not satisfy AC2(a) and AC2(b^u) for the same reason that BT = 1 does not satisfy it in the rock-throwing example. For suppose that B = 1 satisfies AC2(a) and AC2(b^u). Then we would have to have , and we would need to consider the contingency where A = 0 (otherwise WIN = 1 no matter how we set B). Now we need to consider two cases: and . If , then if we set D′ = 0, we have (M_V, u) ⊨ [A ← 0, B ← 1, D′ ← 0](WIN = 0), so AC2(b^u) fails (no matter whether C and D are in or ). And if we set D′ = 1, then AC2(a) fails, since (M_V, u) ⊨ [A ← 0, B ← 0, D′ ← 1](WIN = 1). Now if , note that (M_V, u) ⊨ D′ = 0. Since (M_V, u) ⊨ [A ← 0, B ← 1, D′ ← 0](WIN = 0), again AC2(b^u) fails (no matter whether C or D are in or ). Thus, B = 1 is not a cause of WIN = 1 in (M_V, u) according to the updated HP definition.

By way of contrast, B = 1 and C = 1 are causes of WIN = 1 in (M_V, u) according to the original HP definition (as we would expect from Theorem 2.2.3). Although B = 1 and C = 1 do not satisfy AC2(b^u), they do satisfy AC2(b^o). To see that B = 1 satisfies AC2(b^o), take and consider the witness where A = 0, C = 1, and D′ = 0. Clearly we have both that (M_V, u) ⊨ [B ← 0, A ← 0, C ← 1, D′←0](WIN = 0) and that (M_V, u) = [B ← 1, A ← 0, C ← 1, D′ ← 0](WIN = 1), and this continues to hold if D is set to 1 (its original value). A similar argument shows that C = 1 is a cause.

Interestingly, none of B = 1, C = 1, or B = 1 C = 1 is a cause of WIN = 1 in (M_V, u) according to the modified HP definition. This follows from Theorem 2.2.3; if B = 1 were part of a cause according to the modified HP definition, then it would be a cause according to the updated HP definition, and similarly for C = 1. It can also be seen directly: setting B = C = 0 has no effect on WIN if A = 1. The only cause of WIN = 1 in (M_V, u) according to the modified HP definition is A = 1 (which, of course, is also a cause according to the original and updated definitions). ■

The fact that causes may not be singletons has implications for the difficulty of determining whether A is an actual cause of B (see Section 5.3). Although this may suggest that we should then use the original HP definition, adding extra variables if needed, this conclusion is not necessarily warranted either. For one thing, it may not be obvious when modeling a problem that extra variables are needed. Moreover, the advantages of lower complexity may be lost if sufficiently many additional variables need to be added. The modified HP definition has the benefits of both typically giving the “right” answer without the need for extra variables (but see Section 4.3), while having lower complexity than either the original and modified HP definition (again, see Section 5.3).

I conclude this section with one more example that I hope clarifies some of the details of AC2(b^u) (and AC2(b^o)). Recall that AC2(b^u) requires that φ remain true if is set to , even if only a subset of the variables in are set to their values in and all the variables in an arbitrary subset of are set to their original values (i.e., the values they had in the original context, where ). Since I have viewed the variables in as being on the causal path to φ (although that is not quite right; see Section 2.9), we might hope that not only is setting to sufficient to make φ true, but it is also enough to force all the variables in to their original values. Indeed, there is a definition of actual causality that makes this requirement (see the bibliographic notes). Formally, we might hope that if , then the following holds:

(2.1)

It is not hard to show (see Lemma 2.10.2) that, in the presence of (2.1), AC2(b^u) can be simplified to

there is no need to include the subsets Z′. Although (2.1) holds in many examples, it seems unreasonable to require it in general, as the following example shows.

Example 2.8.3 Imagine that a vote takes place. For simplicity, two people vote. The measure is passed if at least one of them votes in favor. In fact, both of them vote in favor, and the measure passes. This version of the story is almost identical to the situation where either a match or a lightning strike suffices to bring about a forest fire. If we use V₁ and V₂ to denote how the voters vote (V_i = 0 if voter i votes against and V_i = 1 if she votes in favor) and P to denote whether the measure passes (P = 1 if it passes, P = 0 if it doesn’t), then in the context where V₁ = V₂ = 1, it is easy to see that each of V₁ = 1 and V₂ = 1 is part of a cause of P = 1 according to the original and updated HP definition. However, suppose that the story is modified slightly. Now assume that there is a voting machine that tabulates the votes. Let T represent the total number of votes recorded by the machine. Clearly, T = V₁ + V₂ and P = 1 iff T ≥ 1. The causal network in Figure 2.10 represents this more refined version of the story.

In this more refined scenario, V₁ = 1 and V₂ = 1 are still both causes of P = 1 according to the original and updated HP definition. Consider V₁ = 1. Take and , and consider the contingency V₂ = 0. With this witness, P is counterfactually dependent on V₁, so AC2(a) holds. To check that this contingency satisfies AC2(b^u) (and hence also AC2(b^o)), note that setting V₁ to 1 and V₂ to 0 results in P = 1, even if T is set to 2 (its current value). However, (2.1) does not hold here: T does not retain its original value of 2 when V₁ = 1 and V₂ = 0.

Figure 2.10: A more refined voting scenario.

Since, in general, one can always imagine that a change in one variable produces some feeble change in another, it seems unreasonable to insist on the variables in remaining constant; AC2(b^o) and AC2(b^u) require merely that changes in not affect φ. ■

2.9 Causal Paths

There is an intuition that causality travels along a path: A causes B, which causes C, which causes D, so there is a path from A to B to C to D. And, indeed, many accounts of causality explicitly make use of causal paths. In the original and updated HP definition, AC2 involves a partition of the endogenous variables into two sets, and . In principle, there are no constraints on what partition to use, other than the requirement that be a subset of . However, when introducing AC2(a), I suggested that can be thought of as consisting of the variables on the causal path from to φ. (Recall that the notion of a causal path is defined formally in Section 2.4.) As I said, this intuition is not quite true for the updated HP definition; however, it is true for the original HP definition. In this section, I examine the role of causal paths in AC2. In showing that is a cause of φ, I would like to show that we can take (or in the case of the modified HP definition) to consist only of variables that lie on a causal path from some variable in to some variable in φ. The following example shows that this is not the case in general with the updated HP definition.

Example 2.9.1 Consider the following variant of the disjunctive forest-fire scenario. Now there is a second arsonist; he will drop a lit match exactly if the first one does. Thus, we add a new variable MD′ with the equation MD′ = MD. Moreover, FF is no longer a binary variable; it has three possible values: 0, 1, and 2. We have that FF = 0 if L = MD = MD′ = 0; FF = 1 if either L = 1 and MD = MD′ or MD = MD′ = 1; and FF = 2 if MD = MD′. We can suppose that the forest burns asymmetrically (FF = 2) if exactly one of the arsonists drops a lit match (which does not happen under normal circumstances); one side of the forest is more damaged than the other (even if the lightning strikes). Call the resulting causal model . Figure 2.11 shows the causal network corresponding to .

Figure 2.11: The forest fire with two arsonists.

In the context u where L = MD = 1, L = 1 is a cause of FF = 1 according to the updated HP definition. Note that there is only one causal path from L to FF: the path consisting of just the variables L and FF. But to show that L = 1 is a cause of FF = 1 according to the updated HP definition, we need to take the witness to be ({MD}, 0, 0); in particular, MD′ is part of , not .

To see that this witness works, note that , so AC2(a) holds. Since we have both and , AC2(b^u) also holds. However, suppose that we wanted to find a witness where the set (so that can be the unique causal path from L to FF). What do we set MD and MD′ to in the witness? At least one of them must be 0, so that when L is set to 0, FF ≠ 1. Setting MD′ = 0 does not work: , so AC2(b^u) does not hold. And setting MD = 0 and MD′ = 1 does not work either, for then FF = 2, independent of the value of L, so again AC2(b^u) does not hold.

By way of contrast, we can take ({MD, MD′},, 0) to be a witness to L = 1 being a cause according to the original HP definition; AC2(b^o) holds with this witness because .

Finally, perhaps disconcertingly, L = 1 is not even part of a cause of FF = 1 in (M, u) according to the modified HP definition. That is because MD = 1 is now a cause; setting MD to 0 while keeping MD′ fixed at 1 results in FF being 2. Thus, unlike the original forest-fire example, L = 1 ∧ MD = 1 is no longer a cause of FF = 1 according to the modified HP definition. This problem disappears once we take normality into 5account; see Example 3.2.5. account (see Example 3.2.5). ■

For the original HP definition, we can take the set in AC2, not just in this example, but in general, to consist only of variables that lie on a causal path from a variable in to a variable in φ. Interestingly, it is also true of the modified HP definition, except in that case we have to talk about rather than because is the analogue of in the modified HP definition.

Proposition 2.9.2 If is a cause of φ in according to the original or modified HP definition, then there exists a witness to this such that every variable lies on a causal path in from some variable in to some variable in φ.

For definiteness, I take X be a variable “in φ” if, syntactically, X appears in φ, even if φ is equivalent to a formula that does not involve X. For example, if Y is a binary variable, I take Y to appear in X = 1 ∧ (Y = 1 ∨ Y = 1), although this formula is equivalent to X = 1. (The proposition is actually true even if I assume that Y does not appear in X = 1 ∧ (Y = 1 ∨ Y = 1).) The proof of the proposition can be found in Section 2.10.3.

2.10 Proofs

In this section, I prove some of the results stated earlier. This section can be skipped without loss of continuity.

2.10.1 Proof of Theorem 2.2.3

I repeat the statements of the theorem here (and in the later sections) for the reader’s convenience.

Theorem 2.2.3:

(a)If X = x is part of a cause of φ in according to the modified HP definition, then X = x is a cause of φ in according to the original HP definition.

(b)If X = x is part of a cause of φ in according to the modified HP definition, then X = x is a cause of φ in according to the updated HP definition.

(c)If X = x is part of a cause of φ in according to the updated HP definition, then X = x is a cause of φ in according to the original HP definition.

(d)If is a cause of φ in according to the original HP definition, then (i.e., is a singleton).

Proof: For part (a), suppose that X = x is part of a cause of φ in according to the modified HP definition, so that there is a cause such that X = x is one of its conjuncts. I claim that X = x is a cause of φ according to the original HP definition. By definition, there must exist a value and a set such that if , then . Moreover, is minimal.

To show that X = x is a cause according to the original HP definition, we must find an appropriate witness. If , then it is immediate that is a witness. If , suppose without loss of generality that , and X = X₁. In general, if is a vector, I write to denote all components of the vector except the first one, so that . I want to show that X₁ = x₁ is a cause of φ in according to the original HP definition. Clearly, , since is a cause of φ in according to the modified HP definition, so AC1 holds. The obvious candidate for a witness for AC2(a) is , where • is the operator that concatenates two vectors so that, for example, (1, 3) • (2) = (1, 3, 2); that is, roughly speaking, we move X₂, …, X_n into . This satisfies AC2(a), since by assumption. AC3 trivially holds for X₁ = x₁, so it remains to deal with AC2(b^o). Suppose, by way of contradiction, that . This means that satisfies AC2(a^m), showing that AC3 (more precisely, the version of AC3 appropriate for the modified HP definition) is violated (taking ) as the witness), and is not a cause of φ in according to the modified HP definition, a contradiction. Thus, .

This does not yet show that AC2(b^o) holds: there might be some subset of variables in that change value when is set to and is set to , and when these variables are set to their original values in does not hold, thus violating AC2(b^u). In more detail, suppose that there exist a set and values for each variable such that (i) and (ii) . But then is not a cause of φ in according the modified HP definition. Condition (ii) shows that AC2(a^m) is satisfied for , taking as the witness, so again AC3 is violated. It follows that AC2(b^u) holds, completing the proof.

The proof of part (b) is similar in spirit. Indeed, we just need to show one more thing. For AC2(b^u), we must show that if , and , then

(2.2)

(Here I am using the abuse of notation that I referred to in Section 2.2.2, where if and , I write , with the intention that the components of not included in are ignored.) It follows easily from AC1 that (2.2) holds if . And if (2.2) does not hold for some strict nonempty subset of , then is not a cause of φ according to the modified HP definition because AC3 does not hold; AC2(a^m) is satisfied for .

For part (c), the proof is again similar in spirit. Indeed, the proof is identical up to the point where we must show that AC2(b^o) holds. Now if and , then satisfies AC2(a) (taking as the witness). It also satisfies AC2(b^u), since satisfies AC2(b^u), by assumption. Thus, the version of AC3 appropriate for the updated HP definition is violated.

Finally, for part (d), the proof is yet again similar in spirit. Suppose that is a cause of φ in according to the original HP definition and . Let X = x be a conjunct of . Again, we can show that X = x is a cause of φ in according to the original HP definition, which is exactly what is needed to prove part (d), using essentially the same argument as above. I leave the details to the reader. ■

2.10.2 Proof of Proposition 2.4.6

In this section, I prove Proposition 2.4.6. I start with a simple lemma that states a key (and obvious!) property of causal paths: if there is no causal path from X to Y, then changing the value of X cannot change the value of Y. This fact is made precise in the following lemma. Although it is intuitively obvious, proving it carefully requires a little bit of work. I translate the proof into statisticians’ notation as well.

Lemma 2.10.1 If Y and all the variables in are endogenous, , and there is no causal path in from a variable in to Y, then for all sets of variables disjoint from and Y, and all settings and for , y for Y, and for , we have

and

Proof: Define the maximum distance of a variable Y in a causal setting , denoted maxdist (Y), to be the length of the longest causal path in from an exogenous variable to Y. We prove both parts of the result simultaneously by induction on maxdist (Y). If maxdist (Y) = 1, then the value of Y depends only on the values of the exogenous variables, so the result trivially holds. If maxdist (Y) > 1, let Z₁, …, Z_k be the endogenous variables on which Y depends. These are the parents of Y in the causal network (i.e., these are exactly the endogenous variables Z such that there is an edge from Z to Y in the causal network). For each Z ∈ {Z₁, …, Z_k}, maxdist (Z) < maxdist (Y): for each causal path in from an exogenous variable to Z, there is a longer path to Y, namely, the one formed by adding the edge from Z to Y. Moreover, there is no causal path in from a variable in to any of Z₁, …, Z_k nor is any of Z₁, …, Z_k in (for otherwise there would be a causal path in from a variable in to Y, contradicting the assumption of the lemma). Thus, the inductive hypothesis holds for each of Z₁, …, Z_k. Since the value of each of Z₁, …, Z_k does not change when we change the setting of from to , and the value of Y depends only on the values of Z₁, …, Z_k and (i.e., the values of the exogenous variables), the value of Y cannot change either. ■

I can now prove Proposition 2.4.6. I restate it for the reader’s convenience.

Proposition 2.4.6: Suppose that X₁ = x₁ is a but-for cause of X₂ = x₂ in the causal setting , X₂ = x₂ is a but-for cause of X₃ = x₃ in and the following two conditions hold:

(a)for every value , there exists a value such that ;

(b)X₂ is on every causal path in from X₁ to X₃.

Then X₁ = x₁ is a but-for cause of X₃ = x₃.

Proof: Since X₂ = x₂ is a but-for cause of X₃ = x₃ in , there exists a value such that (i.e., ). Choose such that (i.e., . By assumption, there exists a value such that (i.e., . I claim that (i.e., . This follows from a more general claim. I show that if Y is on a causal path from X₂ to X₃, then

(2.3)

(i.e., ).

Define a partial order ⪯ on endogenous variables that lie on a causal path from X₂ to X₃ by taking Y₁ ≺ Y₂ if Y₁ precedes Y₂ on some causal path from X₂ to X₃. Since M is a recursive model, if Y₁ and Y₂ are distinct variables and Y₁ ≺ Y₂, we cannot have Y₂ ≺ Y₁ (otherwise there would be a cycle). I prove (2.3) by induction on the ≺ ordering. The least element in this ordering is clearly X₂; X₂ must come before every other variable on a causal path from X₂ to X₃. (i.e., ) by assumption, and clearly (i.e., . Thus, (2.3) holds for X₂. Thus completes the base case of the induction.

For the inductive step, let Y be a variable that lies on a causal path in from X₂ and X₃, and suppose that (2.3) holds for all variables Y′ such that Y′? Y. Let Z₁, …, Z_k be the endogenous variables that Y depends on in M. For each of these variables Z_i, either there is a causal path in from X₁ to Z_i or there is not. If there is, then the path from X₁ to Z_i can be extended to a directed path P from X₁ to X₃ by going from X₁ to Z_i, from Z_i to Y, and from Y to X₃ (since Y lies on a causal path in from X₂ to X₃). Since, by assumption, X₂ lies on every causal path in from X₁ to X₃, X₂ must lie on P. Moreover, X₂ must precede Y on P. (Proof: Since Y lies on a path P′ from X₂ to X₃, X₂ must precede Y on P′. If Y precedes X₂ on P, then there is a cycle, which is a contradiction.) Since Z_i precedes Y on P, it follows that Z_i ≺ Y, so by the inductive hypothesis, iff (i.e., .

Now if there is no causal path in from X₁ to Z_i, then there also cannot be a causal path P from X₂ to Z_i (otherwise there would be a causal path in from X₁ to Z_i formed by appending P to a causal path from X₁ to X₂, which must exist because, if not, it easily follows from Lemma 2.10.1 that X₁ = x₁ would not be a cause of X₂ = x₂). Since there is no causal path in from X₁ to Z_i, we must have that iff (i.e., iff .

Since the value of Y depends only on the values of Z₁, …, Z_k and , and I have just shown that iff (i.e., iff it follows that iff (i.e., ).

This completes the proof of the induction step. Since X₃ is on a causal path in from X₂ to X₃, it now follows that iff (i.e., iff ). Since (i.e., ) by construction, we have that (i.e., ), as desired. Thus, X₁ = x₁ is a but-for cause for X₃ = x₃. ■

2.10.3 Proof of Proposition 2.9.2

In this section, I prove Proposition 2.9.2.

Proposition 2.9.2: If is cause of φ in according to the original or modified HP definition, then there exists a witness to this such that every variable lies on a causal path in from some variable in to some variable in φ.

To prove the result, I need to prove a general property of causal models, one sufficiently important to be an axiom in the axiomatization of causal reasoning given in Section 5.4.

Lemma 2.10.2 If , then if and only if .

Proof: In the unique solution to the equations when is set to , . So the unique solution to the equations when is set to is the same as the unique solution to the equations when is set to and is set to . The result now follows immediately. ■

With this background, I can prove Proposition 2.9.2.

Proof of Proposition 2.9.2: Suppose that is a cause of φ in with witness according to the original (resp., modified) HP definition. Let , let consist of the variables in that lie on a causal path in from some variable in to some variable in φ, and let . Notice that is a superset of . Let . The proof diverges slightly now depending on whether we consider the original or modified HP definition. In the case of the original HP definition, let y_j be the value of Y_j such that ; in the case of the modified HP definition, let y_j be the value of Y_j such that . I claim that is a witness to being a cause of φ in according to the original (resp., modified) HP definition. (Recall from the proof of Theorem 2.2.3 that denotes the result of concatenating the vectors and .) Once I prove this, I will have proved the proposition.

Since is a cause of φ by assumption, AC1 and AC3 hold; they are independent of the witness. This suffices to prove that AC2(a) and AC2(b^o) (resp., AC2(a^m)) hold for this witness.

Since is a witness to being a cause of φ in according to the original (resp., modified) HP definition, by AC2(a) (resp., AC2(a^m)), it follows that

(2.4)

Since no variable in is on a causal path in from a variable in to a variable in φ, for each , either there is no causal path in from a variable in to Y or there is no causal path in from Y to a variable in φ (or both). Without loss of generality, we can assume that the variables in are ordered so that there is no causal path in from a variable in to any of the first j variables, Y₁, …, Y_j, and there is no causal path in from any of the last k − j variables, Y_j+1, …, Y_k, to a variable in φ.

I first do the argument in the case of the modified HP definition. In this case, must be the value of the variables in in context ; by definition, is the value of the variables in . Thus, is a legitimate witness for AC2(a^m). Moreover, since , by Lemma 2.10.2, we have

Since there is no causal path in from to any of Y₁, …, Y_j, by Lemma 2.10.1, it follows that

(2.5)

Thus, by (2.4), (2.5), and Lemma 2.10.2, it follows that

(2.6)

Finally, since there is no causal path in from any of Y_j+1, …, Y_k to any variable in φ, it follows from (2.6) and Lemma 2.10.1 that

so AC2(a^m) holds, and is a cause of φ in according to the modified HP definition with witness .

We have to work a little harder in the case of the original HP definition (although the basic ideas are similar) because we can no longer assume that . By the definition of in this case, we have . Thus, by Lemma 2.10.2,

This shows that AC2(a) holds for the witness .

For AC2(b^o), we must show that . By assumption,

(2.7)

If Y ∈ {Y₁, …, Y_j}, then by choice of y, . Thus, by Lemma 2.10.1,

(2.8)

By (2.7), (2.8), and Lemma 2.10.2,

(2.9)

Finally, since there is no causal path in from any of Y_j+1, …, Y_k to a variable in φ, it follows from (2.9) and Lemma 2.10.1 that

Thus, is a cause of φ in according to the original HP definition with witness . ■

Notes

The use of structural equations in models for causality goes back to the geneticist Sewall Wright [1921] (see [Goldberger 1972] for a discussion). Herb Simon [1953], who won a Nobel Prize in Economics (and, in addition, made fundamental contributions to Computer Science), and the econometrician Trygve Haavelmo [1943] also made use of structural equations. The formalization used here is due to Judea Pearl, who has been instrumental in pushing for this approach to causality. Pearl [2000] provides a detailed discussion of structural equations, their history, and their use.

The assumption that the model is recursive is standard in the literature. The usual definition says that if the model is recursive, then there is a total order ⪯ of the variables such that X affects Y only if X ⪯ Y. I have made two changes to the usual definition here. The first is to allow the order to be partial; the second is to allow it to depend on the context. As the discussion of the rock-throwing example (Example 2.3.3) shows, the second assumption is useful and makes the definition more general; a good model of the rock-throwing example might well include contexts where Billy hits before Suzy if they both throw. Although it seems more natural to me to assume a partial order as I have done here, since the partial order can be read off the causal network, assuming that the order is partial is actually equivalent to assuming that it is total. Every total order is a partial order, and it is well known that every partial order can be extended (typically, in many ways) to a total order. If ⪯′ is a total order that extends ⪯ and X affects Y only if X ⪯ Y, then certainly X affects Y only if X ⪯′ Y. Allowing the partial order to depend on the context is also nonstandard, but a rather minimal change.

As I said in the notes to Chapter 1, the original HP definition was introduced by Halpern and Pearl in [Halpern and Pearl 2001]; it was updated in [Halpern and Pearl 2005a]; the modified definition was introduced in [Halpern 2015a]. These definitions were inspired by Pearl’s original notion of a causal beam [Pearl 1998]. (The definition of causal beam in [Pearl 2000, Chapter 10] is a modification of the original definition that takes into account concerns raised in an early version of [Halpern and Pearl 2001].) Interestingly, according to the causal beam definition, A qualifies as an actual cause of B only if something like AC2(a^m) rather than AC2(a) holds; otherwise it is called a contributory cause. The distinction between actual cause and contributory cause is lost in the original and updated HP definition. To some extent, it resurfaces in the modified HP definition; in some cases, what the causal beam definition would classify as a contributory cause but not an actual cause would be classified as part of a cause but not a cause according to the modified HP definition.

Although I focus on (variants of) the HP definition of causality in this book, it is not the only definition of causality that is given in terms of structural equations. Besides Pearl’s causal beam notion, other definitions were given by (among others) Glymour and Wimberly [2007], Hall [2007], Hitchcock [2001, 2007], and Woodward [2003]. Problems have been pointed out with all these definitions; see [Halpern 2015a] for a discussion of some of them. As I mentioned in Chapter 1, there are also definitions of causality that use counterfactuals without using structural equations. The best known is due to Lewis [1973a, 2000]. Paul and Hall [2013] cover a number of approaches that attempt to reduce causality to counterfactuals.

Proposition 2.2.2 and parts (a) and (b) of Theorem 2.2.3 are taken from [Halpern 2015a]. Part (d) of Theorem 2.2.3, the fact that with the original HP definition causes are always single conjuncts, was proved (independently) by Hopkins [2001] and Eiter and Lukasiewicz [2002].

Much of the material in Sections 2.1–2.3 is taken from [Halpern and Pearl 2005a]. The formal definition of causal model is taken from [Halpern 2000]. Note that in the literature (especially the statistics literature), what I call here “variables” are called random variables.

In the philosophy community, counterfactuals are typically defined in terms of “closest worlds” [Lewis 1973b; Stalnaker 1968]; a statement of the form “if A were the case then B would be true” is taken to be true if in the “closest world(s)” to the actual world where A is true, B is also true. The modification of equations may be given a simple “closest world” interpretation: the solution of the equations obtained by replacing the equation for Y with the equation Y = y, while leaving all other equations unaltered, can be viewed as the closest world to the actual world where Y = y.

The asymmetry embodied in the structural equations (i.e., the fact that variables on the left- and right-hand sides of the equality sign are treated differently) can be understood in terms of closest worlds. Suppose that in the actual world, the arsonist does not drop a match, there is no lightning, and the forest does not burn down. If either the match or lightning suffices to start the fire, then in the closest world to the actual world where the arsonist drops a lit match, the forest burns down. However, it is not necessarily the case that the arsonist drops a match in the world closest to the actual world where the forest burns down. The formal connection between causal models and closest-world semantics for counterfactuals is somewhat subtle; see [Briggs 2012; Galles and Pearl 1998; Halpern 2013; Zhang 2013] for further discussion.

The question of whether there is some circularity in using causal models, where the structural equations can arguably be viewed as encoding causal relationships, to provide a model of actual causality was discussed by Woodward [2003]. As he observed, if we intervene to set X to x, the intervention can be viewed as causing X to have value x. But we are not interested in defining a causal relationship between interventions and the values of variables intervened on. Rather, we are interested in defining a causal relation between the values of some variables and the values of otherwise; for example, we want to say that the fact that X = x in the actual world is a cause of Y = y in the actual world. As Woodward points out (and I agree), the definition of actual causality depends (in part) on the fact that intervening on X by setting it to x causes X = x, but there is no circularity in this dependence.

The causal networks that are used to describe the equations are similar in spirit to Bayesian networks, which have been widely used to represent and reason about (conditional) dependencies in probability distributions. Indeed, once we add probability to the picture as in Section 2.5, the connection is even closer. Historically, Judea Pearl [1988, 2000] introduced Bayesian networks for probabilistic reasoning and then applied them to causal reasoning. This similarity to Bayesian networks will be exploited in Chapter 5 to provide insight into when we can obtain compact representations of causal models.

Spirtes, Glymour, and Scheines [1993] study causality by working directly with graphs, not structural equations. They consider type causality rather than actual causality; their focus is on (algorithms for) discovering causal structure and causal influence. This can be viewed as work that will need to be done to construct the structural equations that I am taking as given here. The importance of choosing the “right” set of variables comes out clearly in the Spirtes et al. framework as well.

The difference between conditions and causes in the law is discussed by Katz [1987] and Mackie [1965], among others.

In the standard philosophical account, causality relates events, that is, for A to be a cause of B, A and B have to be events. But what counts as an event is open to dispute. There are different theories of events. (Casati and Varzi [2014] provide a recent overview of this work; Paul and Hall [2013, pp. 58–60] discuss its relevance to theories of causality.) A major issue is whether something not happening counts as an event [Paul and Hall 2013, pp. 178–182]. (This issue is also related to that of whether omissions count as causes, discussed earlier.) The As and Bs that are the relata of causality in the HP definition are arguably closer to what philosophers have called true propositions (or facts) [Mellor 2004]. Elucidating the relationship between all these notions is well beyond the scope of this book.

Although I have handled the rock-throwing example here by just adding two additional variables (BH and SH), as I said, this is only one of many ways to capture the fact that Suzy’s rock hit the bottle and Billy’s did not. The alternative of using variables indexed by time was considered in [Halpern and Pearl 2005a].

Examples 2.3.4 (double prevention), 2.3.5 (omissions as causes), and 2.4.1 (lack of transitivity) are due to Hall; their descriptions are taken from (an early version of) [Hall 2004]. Again, these problems are well known and were mentioned in the literature much earlier. See Chapter 3 for more discussion of causation by omission and the notes to that chapter for references.

Example 2.3.6 is due to Paul [2000]. In [Halpern and Pearl 2005a], it was modeled a little differently, using variables LT (for train is on the left track) and RT (for train is on the right track). Thus, in the actual context, where the engineer flips the switch and the train goes down the right track, we have F = 1, LT = 0, and RT = 0. With this choice of variables, all three variants of the HP definition declare F = 1 a cause of A = 1 (we can fix RT at its actual value of 0 to get the counterfactual dependence of A on F). This problem can be dealt with using normality considerations (see the last paragraph of Example 3.4.3). In any case, as Hitchock and I pointed out [Halpern and Hitchcock 2010], this choice of variables is somewhat problematic because LT and RT are not independent. What would it mean to set both LT and RT to 1, for example? That the train is simultaneously going down both tracks? Using LB and FB, as done in [Halpern and Hitchcock 2010], captures the essence of Hall’s story while avoiding this problem. The issue of being able to set variables independently is discussed in more detail in Section 4.6.

Example 2.3.8 is taken from [Halpern 2015a].

O’Connor [2012] says that, each year, an estimated 4,000 cases of “retained surgical items” are reported in the United States and discusses a lawsuit resulting from one such case.

Example 2.3.7 is due to Bas van Fraassen and was introduced by Schaffer [2000b] under the name trumping preemption. Schaffer (and Lewis [2000]) claimed that, because orders from higher-ranking officers trump those of lower-ranking officers, the captain is a cause of the charge and the sergeant is not. The case is not so clearcut to me; the analysis in the main text shows just how sensitive the determination of causality is to details of the model. Halpern and Hitchcock [2010] make the point that if C and S have range {0, 1} (and the only variables in the causal model are C, S, and P), then there is no way to capture the fact that in the case of conflicting orders, the private obeys the captain. The model of the problem that captures trumping with the extra variable SE, described in Figure 2.7, was introduced in [Halpern and Pearl 2005a].

Example 2.4.2 is due to McDermott [1995], who also gives other examples of lack of transitivity, including Example 3.5.2, discussed in Chapter 3.

The question of the transitivity of causality has been the subject of much debate. As Paul and Hall [2013] say, “Causality seems to be transitive. If C causes D and D causes E, then C thereby causes E.” Indeed, Paul and Hall [2013, p. 215] suggest that “preserving transitivity is a basic desideratum for an adequate analysis of causation”. Lewis [1973a, 2000] imposes transitivity in his definition of causality by taking causality to be the transitive closure (“ancestral”, in his terminology) of a one-step causal dependence relation. But numerous examples have been presented that cast doubt on transitivity; Richard Scheines [personal communication, 2013] suggested the homeostasis example. Paul and Hall [2013] give a sequence of such counterexamples, concluding that “What’s needed is a more developed story, according to which the inference from ‘C causes D’ and ‘D causes E’ to ‘C causes E’ is safe provided such-and-such conditions obtain—where these conditions can typically be assumed to obtain, except perhaps in odd cases … ”. Hitchcock [2001] argues that the cases that create problems for transitivity fail to involve appropriate “causal routes” between the relevant events. Propositions 2.4.3, 2.4.4, and 2.4.6 can be viewed as providing other sets of conditions for when we can safely assume transitivity. These results are proved in [Halpern 2015b], from where much of the discussion in Section 2.4 is taken. Since these definitions apply only to but-for causes, which all counterfactual-based accounts (not just the HP accounts) agree should be causes, the results should hold for almost all definitions that are based on counterfactual dependence and structural equations.

As I said, there are a number of probabilistic theories of causation that identify causality with raising probability. For example, Eells and Sober [1983] take C to be a cause of E if Pr(E| K_i ∧ C) > Pr(E|K_i ∧ ¬C) for all appropriate maximal specifications K_i of causally relevant background factors, provided that these conditionals are defined (where they independently define what it means for a background factor to be causally relevant); see also [Eells 1991]. Interestingly, Eells and Sober [1983] also give sufficient conditions for causality to be transitive according to their definition. This definition does not involve counterfactuals (so the results of Section 2.4 do not apply).

There are also definitions of probabilistic causality in this spirit that do involve counterfactuals. For example, Lewis [1973a] defines a probabilistic version of his notion of dependency, by taking A to causally depend on B if and only if, had B not occurred, the chance of A’s occurring would be much less than its actual chance; he then takes causality to be the transitive closure of this causal dependence relation. There are several other ways to define causality in this spirit; see Fitelson and Hitchcock [2011] for an overview. In any case, as Example 2.5.2 shows, no matter how this is made precise, defining causality this way is problematic. Perhaps the best-known example of the problem is due to Rosen and is presented by Suppes [1970]: A golfer lines up to driver her ball, but her swing is off and she slices it badly. The slice clearly decreases the probability of a hole-in-one. But, as it happens, the ball bounces off a tree trunk at just the right angle that, in fact, the golfer gets a hole-in-one. We want to say that the slice caused the hole-in-one (and the definition given here would say that), although slicing the ball lowers the probability of a hole-in-one.

The difference between the probability of rain conditional on a low barometer reading and the probability that it rains as a result of intervening to set the barometer reading to low has been called by Pearl [2000] the difference between seeing and doing.

Examples 2.5.1, 2.5.2, and 2.5.3 are modifications of examples in [Paul and Hall 2013], with slight modifications; the original version of Example 2.5.3 is due to Frick [2009]. As I said, the idea of “pulling out the probability” is standard in computer science; see [Halpern and Tuttle 1993] for further discussion and references. Northcott [2010] defends the view that using deterministic models may capture important features of the psychology of causal judgment; he also provides a definition of probabilistic causality. Fenton-Glynn [2016] provides a definition of probabilistic causality in the spirit of the HP definitions; unfortunately, he does not consider how his approach fares on the examples in the spirit of those discussed here. Much of the material in Section 2.5 is taken from [Halpern 2014b].

Example 2.5.4 is in the spirit of an example considered by Hitchcock [2004]. In his example, both Juan and Jennifer push a source of polarized photons, which greatly increases the probability that a photon emitted by the source will be transmitted by a polarizer. In fact, the polarizer does transmit a photon. Hitchcock says, “It seems clearly wrongheaded to ask whether the transmission was really due to Juan’s push rather than Jennifer’s push. Rather … each push increased the probability of the effect (the transmission of the photon), and then it was simply a matter of chance. There are no causal facts of the matter extending beyond the probabilistic contribution made by each. This is a very simple case in which our intuitions are not clouded by tacit deterministic assumptions.”

Hitchcock’s example involves events at the microscopic level, where arguably we cannot pull out the probability. But for macroscopic events, where we can pull out the probability, as is done in Section 2.5, I believe that we can make sense out of the analogous questions, and deterministic assumptions are not at all unreasonable.

Balke and Pearl [1994] give a general discussion of how to evaluate probabilistic queries in a causal model where there is a probability on contexts.

Representation of uncertainty other than probability, including the use of sets of probability measures, are discussed in [Halpern 2003].

Lewis [1986b, Postscript C] discussed sensitive causation and emphasized that in long causal chains, causality was quite sensitive. Woodward [2006] goes into these issues in much more detail and makes the point that people’s causality judgments are clearly influenced by how sensitive the causality ascription is to changes in various other factors. He points out that one reason some people tend to view double prevention and absence of causes as not quite “first-class” causes might be that these are typically quite sensitive causes; in the language of Section 2.6, they are sufficient causes with only low probability.

Pearl and I [2005a] considered what we called strong causality, which is intended to capture some of the same intuitions behind sufficient causality. Specifically, we extended the updated definition by adding the following clause:

AC2(c). for all settings .

Thus, instead of requiring to hold in all contexts, some of the same effect is achieved by requiring that φ holds no matter what values of are considered. The definition given here seems somewhat closer to the intuitions and allows the consideration of probabilistic sufficient causality by putting a probability on contexts. This, in turn, makes it possible to bring out the connections between sufficient causality, normality, and blame.

Although Pearl [1999, 2000] does not define the notion of sufficient causality, he does talk about the probability of sufficiency, that is, the probability that X = x is a sufficient cause of Y = y. Roughly speaking, this is the probability that setting X to x results in Y = y, conditional on X ≠ x and Y ≠ y. Ignoring the conditioning, this is very close in spirit to the definition given here. Datta et al. [2015] considered a notion close to the notion of sufficient causality defined here and pointed out that it could be used to distinguish joint and independent causality. Honoré [2010] discusses the distinction between joint and independent causation in the context of the law. Gerstenberg et al. [2015] discuss the impact of sufficient causality on people’s causality judgments.

The definition of causality in nonrecursive causal models is taken from the appendix of [Halpern and Pearl 2005a]. Strotz and Wold [1960] discuss recursive and nonrecursive models as used in econometrics. They argue that our intuitive view of causality really makes sense only in recursive models, and that once time is taken into account, a nonrecursive model can typically be viewed as recursive.

Example 2.8.1, which motivated AC2(b^u), is due to Hopkins and Pearl [2003]. Example 2.8.2 is taken from [Halpern 2008]. Hall [2007] gives a definition of causality that he calls the H-account that requires (2.1) to hold (but only for , not for all subsets ). Example 2.8.3, which is taken from [Halpern and Pearl 2005a], shows that the H-account would not declare V₁ = 1 a cause of P = 1, which seems problematic. This example also causes related problems for Pearl’s [1998, 2000] causal beam definition of causality.

Hall [2007], Hitchcock [2001], and Woodward [2003] are examples of accounts of causality that explicitly appeal to causal paths. In [Halpern and Pearl 2005a, Proposition A.2], it is claimed that all the variables in can be taken to lie on a causal path from a variable in to φ. As Example 2.9.1 shows, this is not true in general. In fact, the argument used in the proof of [Halpern and Pearl 2005a, Proposition A.2] does show that with AC2(b^o), all the variables in can be taken to lie on a causal path from a variable in to φ (and is essentially the argument used in the proof of Proposition 2.9.2). The definition of causality was changed from using AC2(b^o) to AC2(b^u) in the course of writing [Halpern and Pearl 2005a]; Proposition A.2 was not updated accordingly.

Chapter 3

Graded Causation and Normality

Perhaps the feelings that we experience when we are in love represent a normal state.

Anton Chekhov

In a number of the examples presented in Chapter 2, the HP definition gave counterintuitive ascriptions of causality. I suggested then that taking normality into account would solve these problems. In this chapter, I fill in the details of this suggestion and show that taking normality into account solves these and many other problems, in what seems to me a natural way. Although I apply ideas of normality to the HP definition, these ideas should apply equally well to a number of other approaches to defining causality.

This approach is based on the observation that norms can affect counterfactual reasoning. To repeat the Kahneman and Miller quote given in Chapter 1, “[in determining causality], an event is more likely to be undone by altering exceptional than routine aspects of the causal chain that led to it”. In the next section, I give a short discussion of issues of defaults, typicality, and normality. I then show how the HP definition can be extended to take normality into account. The chapter concludes by showing how doing this deals with the problematic examples from Chapter 2 as well as other concerns.

3.1 Defaults, Typicality, and Normality

The extended account of actual causation incorporates the concepts of defaults, typicality, and normality. These are related, although somewhat different notions:

• A default is an assumption about what happens, or what is the case, when no additional information is given. For example, we might have as a default assumption that birds fly. If we are told that Tweety is a bird and given no further information about Tweety, then it is natural to infer that Tweety flies. Such inferences are defeasible: they can be overridden by further information. If we are additionally told that Tweety is a penguin, we retract our conclusion that Tweety flies.
• To say that birds typically fly is to say not merely that flight is statistically prevalent among birds, but also that flight is characteristic of the type “bird”. Although not all birds fly, flying is something that we do characteristically associate with birds.
• The word normal is interestingly ambiguous. It seems to have both a descriptive and a prescriptive dimension. To say that something is normal in the descriptive sense is to say that it is the statistical mode or mean (or close to it). However, we often use the shorter form norm in a more prescriptive sense. To conform with a norm is to follow a prescriptive rule. Prescriptive norms can take many forms. Some norms are moral: to violate them would be morally wrong. For example, many people believe that there are situations in which it would be wrong to lie, even if there are no laws or explicit rules forbidding this behavior. Laws are another kind of norm, adopted for the regulation of societies. Policies that are adopted by institutions can also be norms. For instance, a company may have a policy that employees are not allowed to be absent from work unless they have a note from their doctor. There can also be norms of proper functioning in machines or organisms. There are specific ways that human hearts and car engines are supposed to work, where “supposed” here has not merely an epistemic force, but a kind of normative force. Of course, a car engine that does not work properly is not guilty of a moral wrong, but there is nonetheless a sense in which it fails to live up to a certain kind of standard.

Although this might seem like a heterogeneous mix of concepts, they are intertwined in a number of ways. For example, default inferences are successful to the extent that the default is normal in the statistical sense. Adopting the default assumption that a bird can fly facilitates successful inferences in part because most birds are able to fly. Similarly, we classify objects into types in part to group objects into classes, most of whose members share certain features. Thus, the type “bird” is useful partly because there is a suite of characteristics shared by most birds, including the ability to fly. The relationship between the statistical and prescriptive senses of “normal” is more subtle. It is, of course, quite possible for a majority of individuals to act in violation of a given moral or legal norm. Nonetheless, it seems that the different kinds of norms often serve as heuristic substitutes for one another. For example, there are well-known experiments showing that we are often poor at explicit statistical reasoning, employing instead a variety of heuristics. Rather than tracking statistics about how many individuals behave in a certain way, we might well reason about how people should behave in certain situations. The idea is that we use a script or a template for reasoning about certain situations, rather than actual statistics. Prescriptive norms of various sorts can play a role in the construction of such scripts. It is true, of course, that conflation of the different sorts of norm can sometimes have harmful consequences. Less than a hundred years ago, for example, left-handed students were often forced to learn to write with their right hands. In retrospect, this looks like an obviously fallacious inference from the premise that the majority of people write with their right hand to the conclusion that it is somehow wrong to write with the left hand. But the ease with which such an inference was made illustrates the extent to which we find it natural to glide between the different senses of “norm”.

When Kahneman and Miller say that “an event is more likely to be undone by altering exceptional than route aspects of the causal chain that led to it”, they are really talking about how we construct counterfactual situations. They are also assuming a close connection between counterfactuals and causality, since they are effectively claiming that the counterfactual situation we consider affects our causality judgment. The fact that norms influence causal judgment has been confirmed experimentally (although the experiments do not show that the influence goes via counterfactual judgments). To take just one example (see the notes at the end of the chapter for others), consider the following story from an experiment by Knobe and Fraser:

The receptionist in the philosophy department keeps her desk stocked with pens. The administrative assistants are allowed to take the pens, but faculty members are supposed to buy their own. The administrative assistants typically do take the pens. Unfortunately, so do the faculty members. The receptionist has repeatedly emailed them reminders that only administrative assistants are allowed to take the pens. On Monday morning, one of the administrative assistants encounters Professor Smith walking past the receptionist’s desk. Both take pens. Later that day, the receptionist needs to take an important message … but she has a problem. There are no pens left on her desk.

People are much more likely to view Prof. Smith as the cause of there not being pens than the administrative assistant. The HP definition does not distinguish them, calling them both causes. The extended HP definition given in the next section does allow us to distinguish them. Since the HP definition is based on counterfactuals, the key step will be to have the choice of witness be affected by normality considerations. Although the assumption that normality considerations affect counterfactual judgments does not follow immediately from the experimental evidence, it seems like a reasonable inference, given the tight connection between causality and counterfactuals.

3.2 Extended Causal Models

I add normality to causal models by assuming that an agent has, in addition to a theory of causal structure (as modeled by the structural equations), a theory of “normality” or “typicality”. This theory includes statements of the form “typically, people do not put poison in coffee”. There are many ways of giving semantics to such typicality statements (see the notes at the end of the chapter). All of them give ways to compare the relative “normality” of some objects. For now, I take those objects to be the worlds in a causal model M, where a world is an assignment of values to the endogenous variables; that is, a complete description of the values of the endogenous variables in M. I discuss after Example 3.2.6 why a world here is taken to be an assignment only to the endogenous variables and not an assignment to both exogenous and endogenous variables, as in the complete assignments considered in Section 2.7. A complete assignment is essentially equivalent to the standard philosophers’ notion of a world as a maximal state of affairs, so my usage of “world” here is not quite the same as the standard philosophers’ usage. In Section 3.5, I discuss an alternative approach, which puts the normality ordering on contexts, much in the spirit of putting a probability on contexts as was done in Section 2.5. Since (in a recursive model) a context determines the values of all endogenous variables, a context can be identified with a complete assignment, and thus with the philosophers’ notion of world. But for now I stick with putting a normality ordering on worlds as I have defined them.

Most approaches to normality/typicality implicitly assume a total normality ordering on worlds: given a pair of worlds w and w′, either w is more normal than w′, w is less normal than w′, or the two worlds are equally normal. The approach used here has the advantage of allowing an extra possibility: w and w′ are incomparable as far as normality goes. (Technically, this means that the normality ordering is partial, rather than total.)

We can think of a world as a complete description of a situation given the language determined by the set of endogenous variables. Thus, a world in the forest-fire example might be one where U = (0, 1), L = 0, MD = 1, and FF = 0; the match is dropped, there is no lightning, and no forest fire. As this example shows, a “world” does not have to satisfy the equations of the causal model.

For ease of exposition, I make a somewhat arbitrary stipulation regarding terminology. In what follows, I use “default” and “typical” when talking about individual variables or equations. For example, I might say that the default value of a variable is zero or that one variable typically depends on another in a certain way. I use “normal” when talking about worlds. Thus, I say that one world is more normal than another. I assume that it is typical for endogenous variables to be related to one another in the way described by the structural equations of a model, unless there is some specific reason to think otherwise. This ensures that the downstream consequences of what is typical are themselves typical (again, in the absence of any specific reason to think otherwise).

Other than this mild restriction, I make no assumptions on what gets counted as typical. This introduces a subjective element into a causal model. Different people might well use different normality orderings, perhaps in part due to focusing on different aspects of normality (descriptive normality, typicality, prescriptive normality, etc.). This might bother those who think of causality as an objective feature of the world. In this section, I just focus on the definitions and give examples where the normality ordering seems uncontroversial; I return to this issue in Chapter 4. I will point out for now that there are already subjective features in a model: the choice of variables and which variables are taken to be exogenous and endogenous is also determined by the modeler.

I now extend causal models to provide a formal model of normality. I do so by assuming that there is a partial preorder ⪰ over worlds. Intuitively, s ⪰ s′ means that s is at least as normal as s′. Recall from Section 2.1 that a partial order on a set S is a reflexive, anti-symmetric, and transitive relation on S. A partial preorder is reflexive and transitive but not necessarily anti-symmetric; this means that we can have two worlds s and s′ such that s ⪰ s′ and s′⪰ s without having s = s′. That is, we can have two distinct worlds where each is at least as normal as the other. By way of contrast, the standard ordering ≥ on the natural numbers is anti-symmetric; if x ≥ y and y ≥ x, then x = y. (The requirement of anti- symmetry is what distinguishes a (total or partial) order from a preorder.) I write s ≻ s′ if s ⪰ s′ and it is not the case that s′⪰ s, and s ≡ s′ if s ⪰ s′ and s′⪰ s. Thus, s ≻ s′ means that s is strictly more normal than s′, whereas s ≡ s′ means that s and s′ are equally normal. Note that I am not assuming that ⪰ is total; it is quite possible that there are two worlds s and s′ that are incomparable in terms of normality. The fact that s and s′ are incomparable does not mean that s and s′ are equally normal. We can interpret it as saying that the agent is not prepared to declare either s or s′ as more normal than the other and also not prepared to say that they are equally normal; they simply cannot be compared in terms of normality.

Recall that normality can be interpreted in terms of frequency. Thus, we say that birds typically fly in part because the fraction of flying birds is much higher than the fraction of non-flying birds. We might consider a world where a bird can fly to be more normal than one where a bird cannot fly. If we were to put a probability on worlds, then a world where a bird flies might well have a greater probability than one where a bird does not fly. Although we could interpret s ≻ s′ as meaning “s is more probable than s′”, this interpretation is not always appropriate. For one thing, interpreting ⪰ in terms of probability would make ⪰ a total order—all worlds would be comparable. There are advantages to allowing ⪰ to be partial (see the notes at the end of the chapter for more discussion of this point). More importantly, even when thinking probabilistically, I view s ≻ s′ as saying that s is much more probable than s′. I return to this point below.

Take an extended causal model to be a tuple , where is a causal model and ⪰ is a partial preorder on worlds, which can be used to compare how normal different worlds are. In particular, ⪰ can be used to compare the actual world to a world where some interventions have been made. Which world is the actual world? That depends on the context. In a recursive extended causal model, a context determines a world denoted . We can think of the world as the actual world in context ; it is the world that would occur given the setting of the exogenous variables in , provided that there are no external interventions.

I now modify the HP definitions slightly to take the ordering of worlds into account. I start with the original and updated definitions. In this case, define to be a cause of φ in an extended model M and context if is a cause of φ according to Definition 2.2.1, except that in AC2(a), a clause is added requiring that , where is the world that results by setting to and to in context . This just says that we require the witness world to be at least as normal as the actual world. Here is the extended AC2(a), which I denote AC2⁺(a).

AC2⁺(a). There is a partition of (the set of endogenous variables) into two disjoint subsets and with and a setting and of the variables in and , respectively, such that and .

This can be viewed as a formalization of Kahneman and Miller’s observation that we tend to consider only possibilities that result from altering atypical features of a world to make them more typical, rather than vice versa. In this formulation, worlds that result from interventions on the actual world “come into play” in AC2⁺(a) only if they are at least as normal as the actual world. If we are using the modified HP definition, then AC2(a^m) is extended to AC2⁺(a^m) in exactly the same way; the only difference between AC2⁺(a) and AC2⁺(a^m) is that, in the latter (just as in AC2(a^m)), is required to consist of the values of the variables in in the actual context (i.e., ).

Issues of normality are, by and large, orthogonal to the differences between the variants of the HP definition. For ease of exposition, in the remainder of this chapter, I use the updated HP definition and AC+(a), unless noted otherwise. Using AC2⁺(a) rather than AC2(a), with reasonable assumptions about normality, takes care of the problems pointed out in Section 2, although it raises other issues.

Example 3.2.1 As I observed, one way to avoid calling the presence of oxygen a cause of the forest burning is simply not to include a variable for oxygen in the model or to take the presence of oxygen to be exogenous. But suppose someone wanted to consider oxygen to be an endogenous variable. Then as long as a world where there is no oxygen is more abnormal than a world where there is oxygen, the presence of oxygen is not a cause of the forest fire; AC2⁺(a) fails.

However, suppose that the fire occurs when a lit match is dropped in a special chamber in a scientific laboratory that is normally voided of oxygen. Then there would presumably be a different normality ordering. Now the presence of oxygen is atypical, and the witness to oxygen being a cause is as normal as (or at least not strictly less normal than) the witness to the match being a cause. In this case, I suspect most people would call the presence of oxygen a cause of the fire. ■

Example 3.2.2 Recall Example 2.3.5, where there are potentially many doctors who could treat Billy besides his regular doctor. Although it seems reasonable to call Billy’s regular doctor a cause of him being sick on Tuesday if the doctor fails to treat him, it does not seem reasonable to call all the other doctors causes of him being sick. This is easy to deal with if we take normality into account. It seems perfectly reasonable to take the world where Billy’s doctor treats Billy to be at least as normal as the world where Billy’s doctor does not treat Billy; it also seems reasonable to take the worlds where the other doctors treat Billy as being less normal than the world where they do not treat Billy. With these assumptions, using AC2⁺(a), Billy’s doctor not treating Billy is a cause of him being sick, whereas the other doctors not treating him are not causes. Thus, using normality allows us to capture intuitions that people seem to have without appealing to probabilistic sufficient causality and thus requiring a probability on contexts (see, for comparison, the discussion in Section 2.6).

What justifies a normality ordering where it is more normal for Billy’s doctor to treat him than another random doctor? All the interpretations discussed in Section 3.1 lead to this judgment.

• Statistically, patients are far more likely to be treated by their own doctors than by a random doctor.
• We assume that, by default, a patient will be treated by his doctor.
• A patient’s doctor does typically treat the patient, in the sense that this is a characteristic of being someone’s doctor.
• It is a norm that a patient’s doctor will treat the patient.

There are philosophers who claim that an omission—the event of an agent not doing something—can never count as a cause. Although I do not share this viewpoint, it can be accommodated in this framework by taking not acting to always be more normal than acting. That is, given two worlds s and s′ that agree in all respects (i.e., in the values of all variables) except that in s some agents do not perform an act that they do perform in s′, we would have s ≻ s′. (As an aside, since ⪰ is allowed to be partial, if different agents perform acts in s and s′, then we can just take s and s′ to be incomparable.)

We can equally well accommodate a viewpoint that allows all omissions to count as causes. This is essentially the viewpoint of the basic HP definition, and we can recover the original HP approach (without normality), and hence drop the distinctions between omissions and other potential causes, by simply taking all worlds to be equally normal. (Indeed, by doing this, we are effectively removing normality from the picture and are back at the original HP definition.)

Of course, the fact that we can accommodate all these viewpoints by choosing an appropriate normality ordering raises the concern that we can reverse-engineer practically any verdict about causality in this way. Although it is true that adding normality to the model gives a modeler a great deal of flexibility, the key word here is “appropriate”. A modeler will need to argue that the normality ordering that she has chosen is reasonable and appropriate to capture the intuitions we have about the situation being modeled. Although there can certainly be some disagreement about what counts as “reasonable”, the modeler does not have a blank check here. I return to this issue in Section 4.7. ■

These examples already show that, once we take normality into account, causality ascriptions can change significantly. One consequence of this is that but-for causes may no longer be causes (according to all of the variants of the HP definition). The presence of oxygen is clearly a but-for cause of fire—if there is no oxygen, there cannot be a fire. But in situations where we take the presence of oxygen for granted, we do not want to call the presence of oxygen a cause.

Moreover, although it is still the case that a cause according to the updated HP definition is a cause according to the original HP definition, the other parts of Theorem 2.2.3 fail once we take normality into account.

Example 3.2.3 Suppose that two people vote in favor of a particular outcome and it takes two votes to win. Each is a but-for cause of the outcome, so all the definitions declare each voter to be a cause. But now suppose that we take the normality ordering to be such that it is abnormal for the two voters to vote differently, so a world where one voter votes in favor and the other votes against is less normal than a world where both vote in favor or both vote against. Taking V₁ and V₂ the variables describing how the voters vote, and O to describe the outcome, then with this normality ordering, neither V₁ = 1 nor V₂ = 1 is a cause of the outcome according to any of the variants of the HP definition, although V₁ = 1 ∧ V₂ = 1 is a cause. So, in particular, this means that causes are not necessarily single conjuncts according to the original HP definition once it is extended to take normality into account. Moreover, if the world where V₁ = V₂ = 0 is also taken to be abnormal, there is no cause at all of O = 1. ■

The fact that there are no causes in this case may not seem so unreasonable. With this normality setting, the outcome is essentially being viewed as a foregone conclusion and thus not in need of a cause. But there are other cases where having no causes seems quite unreasonable.

Example 3.2.4 Suppose that Billy is sick, Billy’s regular doctor treats Billy, and Billy recovers the next day. In this case, we would surely want to say that the fact that Billy’s doctor treated Billy is a cause of Billy recovering the next day. Indeed, it is even a but-for cause of Billy recovering. But if we assume that it is more normal for Billy’s regular doctor to treat Billy when he is sick than not to treat him, then taking normality into account, there is no cause of Billy recovering! It is easy to construct many other examples where there are no causes if the normal thing happens (a gardener watering her plants is not a cause of the plants surviving; if I normally drive home, then the fact that I drove home is not a cause of me being at home 20 minutes after I left work; and so on). This problem is dealt by the graded causality approach discussed in Section 3.3. ■

Example 3.2.5 Consider Example 2.9.1 again, where there is lightning and two arsonists. As we observed, the lightning is no longer a cause of the forest fire according to the modified HP definition, since each arsonist by himself is a cause. But suppose that we take a world where MD ≠ MD′ to be abnormal. This is consistent with the story, which says that one arsonist dropping a match “does not happen under normal circumstances”; in particular, MD ≠ MD′ is inconsistent with the equations. In that case, neither MD = 1 nor MD′ = 1 is a cause of FF = 1 according to modified HP definition in the extended causal model; neither satisfies AC2⁺(a^m). Thus, L = 1 ∧ MD = 1 becomes a cause of FF = 1 according to modified HP definition, and L = 1 is again part of a cause of FF = 1. ■

Example 3.2.6 Consider the sophisticated model of the rock-throwing example, and suppose that we declare the world where Billy throws a rock, Suzy doesn’t throw, and Billy does not hit abnormal. This world was needed to show that Suzy throwing is a cause according to the modified HP definition. Thus, with this normality ordering ST = 1 is not a cause; rather, ST = 1 BT = 1 is a cause, which seems inappropriate. However, ST = 1 remains a cause according to the original and updated HP definition, since we can now take the witness to be the world where ST = 0 and BT = 0; these variants allow the variables in to take on values other than their values in the actual world. Thus, BT = 1 is part of a cause according to the modified HP definition but not a cause according to the original or updated HP definition. ■

Example 3.2.6 shows in part why the normality ordering is placed on worlds, which are assignments of values to the endogenous variables only. The witness world where BT = 1, BH = 0, and ST = 0 does not seem so abnormal, even if it is abnormal for Billy to throw and miss in a context where he is presumed accurate.

Putting the normality ordering on worlds does not completely save the day, however. For one thing, it seems somewhat disconcerting that Billy’s throw should be part of the cause according to the modified HP definition in Example 3.2.6. But things get even worse. Suppose that we consider the richer model rock-throwing model , which includes the variable BA, for “Billy is an accurate rock thrower”. In , we have essentially moved the question of Billy’s accuracy out of the context and made it an explicit endogenous variable, that is, it is determined by the world. In the actual world, Billy is accurate; that is, BA = 1. (Note that we may even have observations confirming this; we could see Billy’s rock passing right over where the bottle was just after Suzy’s rock hit it and shattered it.) There is still no problem here for the original and updated HP definition. If s is the actual world, then the world s_BT=0,ST=0,u is arguably at least as normal as s_u; although BA = 1 in s_BT=0,ST=0,u, it is not abnormal for Billy not to hit the bottle if he doesn’t throw. On the other hand, in the world s_{BH =0,ST=0,u} which is needed to show that Suzy’s throw is a cause of the bottle shattering according to the modified HP definition, we have BT = 1, BA = 1, and BH = 0: although Billy throws and is accurate, he misses the bottle. This doesn’t seem like such a normal world. But if we declare this world to be abnormal, then Suzy’s throw is no longer a cause of the bottle shattering according to the modified HP definition.

It gets even worse (at least according to some people’s judgments of causality) if we replace Billy the person by Billy the automatic rock-throwing machine, which is programmed to throw rocks. Again, it does not seem so bad if BA is not part of the language; programming errors or other malfunctions can still result in a rock-throwing machine missing. But with BA in the language (in which case we take BA = 1 to mean that the machine doesn’t malfunction and is programmed correctly), it does not seem so reasonable to take a world where BA = 1, BT = 1, and BH = 0 to be at least as normal as the actual world.

Does this represent a serious problem for the modified HP definition (when combined with normality)? I would argue that, although at first it may seem reasonable (and is what was done in the literature), the real source of the problem is putting normality on worlds. I return to this point in Section 3.5, where I consider an alternative approach to defining normality. For now I continue with putting normality on worlds and consider an approach that deals with the problem in the rock-throwing example in a different way, which has the further advantage of providing a more general approach to combining normality with causality.

3.3 Graded Causation

AC2⁺(a) and AC2⁺(a^m), as stated, are all-or-nothing conditions; either it is legitimate to consider a particular world (the one that characterizes the witness) or it is not. However, in some cases, to show that A is a cause of B, we might need to consider an unlikely intervention; it is the only one available. This is exactly what happened in the case of the rock-throwing machine.

As observed in Examples 3.2.3 and 3.2.4, with certain normality orderings, some events can have no cause at all (although they would have a cause if normality constraints were ignored). We can avoid this problem by using normality to rank actual causes, rather than eliminating them altogether. Doing so lets us explain the responses that people make to queries regarding actual causation. For example, while counterfactual approaches to causation usually yield multiple causes of an outcome φ, people typically mention only one of them when asked for a cause. One explanation of this is that they are picking the best cause, where best is judged in terms of normality.

To make this precise, say that world s is a witness world (or just witness) for being a cause of φ in context if for some choice of , , and for which AC2(a) and AC2(b^u) hold (AC2(a) and AC2(b^o) if we are using the original HP definition; AC2(a^m) if we are using the modified HP definition), s is the assignment of values to the endogenous variables that results from setting and in context . In other words, a witness s is a world that demonstrates that AC2(a) holds. (Calling a world s a witness world is consistent with calling the triple a witness, as I did in Section 2.2. The triple determines a unique world, which is a witness world. I continue to use the word “witness” for both a witness world and a triple. I hope that the context will make clear which is meant.)

In general, there may be many witness worlds for being a cause of φ in . Say that s is a best witness for being a cause of φ if there is no other witness s′ such that s′ ≻ s. (Note that there may be more than one best witness.) We can then grade candidate causes according to the normality of their best witnesses. We would expect that someone’s willingness to judge an actual cause of φ in increases as a function of the normality of the best witness world for in comparison to the best witness for other candidate causes. Thus, we are less inclined to judge that is an actual cause of φ when there are other candidate causes with more normal witnesses.

Note that once we allow graded causality, we use AC2(a) (resp., AC2(a^m)) rather than AC2⁺(a) (resp., AC2⁺(a^m)); normality is used only to characterize causes as “good” or “poor”. In Example 3.2.2, each doctor other than Billy’s doctor not treating Billy is still a cause of Billy being sick on Tuesday, but is such a poor cause that most people are inclined to ignore it if there is a better cause available. On the other hand, in Example 3.2.4, Billy’s doctor treating Billy is a cause of Billy recovering the next day, and is the only cause, so it is what we take as “the” cause. In general, an event whose only causes are poor causes in this sense is one that we expected all along, so doesn’t require an explanation (see Example 3.4.1). However, if we are looking for a cause, then even a poor cause is better than nothing.

Before going on, I should note that it has been common in the philosophy literature on causation to argue that, although people do seem to grade causality, it is a mistake to do so (see the notes for more discusion on this point). As I said, one of my goals is to get a notion of causality that matches how people use the word and is useful. Since it is clear that people make these distinctions, I think it is important to get a reasonable definition of causality that allows the distinctions to be made. Moreover, I think that people make these distinctions because they are useful ones. Part of why we want to ascribe causality is to understand how to solve problems. We might well want to understand why Billy’s doctor did not treat Billy. We would not feel the need to understand why other doctors did not treat Billy.

3.4 More Examples

In this section, I give a few more examples showing the power of adding normality to the definition of causality and the advantages of thinking in terms of graded causality.

3.4.1 Knobe effects

Recall the vignette from Knobe and Fraser’s experiment that was discussed in Section 3.1, where Professor Smith and the administrative assistant both take pens from the receptionist’s desk. After reading this vignette, subjects were randomly presented with one of the following propositions and asked to rank their agreement on a 7-point scale from -3 (completely disagree) to +3 (completely agree):

Professor Smith caused the problem.
The administrative assistant caused the problem.

Subjects gave an average rating of 2.2 to the first claim, indicating agreement, but −1.2 to the second claim, indicating disagreement. Thus, subjects are judging the two claims differently, due to the different normative status of the two actions. (Note that subjects were only presented with one of these claims: they were not given a forced choice between the two.)

In another experiment, subjects were presented with a similar vignette, but this time both professors and administrative assistants were allowed to take pens. In this case, subjects tend to give intermediate values. That is, when the vignette is changed so that Professor Smith is not violating a norm when he takes the pen, not only are subjects less inclined to judge that Professor Smith caused the problem, but they are more inclined to judge that the administrative assistant caused the problem. The most plausible interpretation of these judgments is that subjects’ increased willingness to say that the administrative assistant caused the problem is a direct result of their decreased willingness to say that Professor Smith caused the problem. This suggests that attributions of actual causation are at least partly a comparative affair.

The obvious causal model of the original vignette has three endogenous variables:

• PT = 1 if Professor Smith takes the pen, 0 if she does not;
• AT = 1 if the administrative assistant takes the pen, 0 if she does not;
• PO = 1 if the receptionist is unable to take a message, 0 if no problem occurs.

There is one equation: PO = min(PT, AT) (equivalently, PO = PT ∧ AT). The exogenous variables are such that PT and AT are both 1. Therefore, in the actual world, we have PT = 1, AT = 1, and PO = 1.

Both PT = 1 and AT = 1 are but-for causes of PO = 1 (and so causes according to all three variants of the definition). The best witness for PT = 1 being a cause is the world (PT = 0, AT = 1, PO = 0); the best witness for AT = 1 being a cause is (PT = 1, AT = 0, PO = 0). (Here and elsewhere, I describe a world by a tuple consisting of the values of the endogenous variables in that world.) The original story suggests that the witness for PT = 1 being a cause is more normal than the witness for AT = 1 by saying “administrative assistants are allowed to take pens, but professors are supposed to buy their own”. According to the definition above, this means that people are more likely to judge that PT = 1 is an actual cause. However, if the vignette does not specify that one of the actions violates a norm, we would expect the relative normality of the two witnesses to be much closer, which is reflected in how subjects actually rated the causes.

This example, I believe, illustrates the power of allowing a “graded” approach to causality. As I said above, there have been claims that it is a mistake to focus on one cause and ignore others (which is an extreme of the graded approach that occurs often in practice). It seems to me that the “egalitarian” notion of cause is entirely appropriate at the level of causal structure, as represented by the equations of a causal model. These equations represent arguably objective features of the world and are not sensitive to factors such as contextual salience and human considerations of normality. Pragmatic, subjective factors then determine which actual causes we select and label “the” causes (or, at least, as “better” causes).

3.4.2 Bogus prevention

The initial impetus for work on adding normality considerations to basic causal models was provided by the examples like the following “bogus prevention” example.

Example 3.4.1 Assassin is in possession of a lethal poison but has a last-minute change of heart and refrains from putting it in Victim’s coffee. Bodyguard puts antidote in the coffee, which would have neutralized the poison had there been any. Victim drinks the coffee and survives. Is Bodyguard’s putting in the antidote a cause of Victim surviving? Most people would say no, but according to the original and updated HP definition, it is. For in the contingency where Assassin puts in the poison, Victim survives iff Bodyguard puts in the antidote. According to the modified HP definition, Bodyguard putting in the antidote is not a cause, but is part of a cause (along with Assassin not putting in the poison). Although the latter position is perhaps not completely unreasonable (after all, if Bodyguard didn’t put in the antidote and Assassin did put in the poison, Victim would have died), most people would not take either Bodyguard or Assassin to be causes. Roughly speaking, they would reason that there is no need to find a cause for something that was expected all along. ■

What seems to make Example 3.4.1 particularly problematic is that, if we use the obvious variables to describe the problem, the resulting structural equations are isomorphic to those in the disjunctive version of the forest-fire example where either the lit match or the lightning suffices to start the fire. Specifically, take the endogenous variables in Example 3.4.1 to be A (for “assassin does not put in poison”), B (for “bodyguard puts in antidote”), and VS (for “victim survives”). That is,

• A = 1 if Assassin does not put in the poison, 0 if he does;
• B = 1 if Bodyguard adds the antidote, 0 if he does not;
• VS = 1 if Victim survives, 0 if he does not.

Then A, B, and VS satisfy exactly the same equations as L, MD, and FF, respectively: A and B are determined by the environment, and VS = A ∨ B. In the context where there is lightning and the arsonist drops a lit match, both the lightning and the match are causes of the forest fire according to the original and updated HP definition (and parts of causes according to the modified HP definition), which seems reasonable. For similar reasons, in this model, the original and updated HP definition declare both A = 1 and B = 1 to be causes of VS = 1 (and the modified HP definition declares them to be parts of causes). But here it does not seem reasonable that Bodyguard’s putting in the antidote is a cause or even part of a cause. Nevertheless, it seems that any definition that just depends on the structural equations is bound to give the same answers in these two examples.

Using normality gives us a straightforward way of dealing with the problem. In the actual world, A = 1, B = 1, and VS = 1. The witness world for B = 1 being a cause of VS = 1 is the world where A = 0, B = 0, and VS = 0. If we make the assumption that A is typically 1 and that B is typically 0, this leads to a normality ordering in which the two worlds (A = 1, B = 1, VS = 1) and (A = 0, B = 0, VS = 0) are incomparable; although both are abnormal, they are abnormal in incomparable ways. Since the unique witness for B = 1 to be an actual cause of VS = 1 is incomparable with the actual world, using AC2⁺(a), we get that B = 1 is not an actual cause of VS = 1.

Note that if we require AC2⁺(a), A = 1 is not a cause of VS = 1 either; the world (A = 0, B = 0, VS = 0) is also the unique witness world for A = 1 being a cause of VS = 1. That means VS = 1 has no causes. Again, in this case, we would typically not look for an explanation or a cause of VS = 1 because it was expected all along. However, if we are looking for a cause, using graded causality, both A = 1 and B = 1 count as causes of VS = 1 according to the original and updated HP definition (extended to take normality into account, of course), although they get a “poor” grade; with the modified HP definition, the conjunction A = 1 B = 1 counts as a cause. Again, we are inclined to accept a “poor” cause as a cause if there are no better alternatives available.

Now suppose that we consider a world where assassination is typical; assassins are hiding behind every corner. Would we then want to take worlds where A = 0 to be more normal than worlds where A = 1, all else being fixed? Indeed, even if assassinations were not common-place, isn’t it typical for assassins to poison drinks? That is what assassins do, after all. Here we are dealing with two different interpretations of “normality”. Even if assassinations occur frequently, they are morally wrong, and hence abnormal in the moral sense. Still, perhaps we would be more willing to accept the assassin not putting in poison as a cause of the victim surviving in that case; we certainly wouldn’t take VS = 1 to be an event that no longer requires explanation. Indeed, imagine the assassin master sent his crack assassin out to murder the victim and then discovers that the victim is still alive. Now he would want an explanation, and the explanation that the assassin did not actually put in the poison now seems more reasonable.

Although (despite the caveats just raised) using normality seems to deal with the assassin problem, there is another, arguably better, way of modeling this example that avoids issues of normality altogether. The idea is similar in spirit with how the rock-throwing problem was handled. We add a variable PN to the model, representing whether a chemical reaction takes place in which poison is neutralized. Of course PN = 1 exactly if the assassin puts in the poison and Bodyguard adds the antidote. Thus, the model has the following two equations:

• PN = ¬A ∧ B (i.e., (1 − A) × B) and
• VS = max(A, PN ) (i.e., A ∨ PN).

Thus, in the actual world, where A = 1 and B = 1, PN = 0. The poison is not neutralized because there was never poison there in the first place. With the addition of PN, B = 1 is no longer a cause of VS = 1 according to the original or updated HP definitions, without taking normality into account. Clearly, if B = 1 were to be a cause, the set would have to include A, and we would have to set A = 0. The question is whether PN is in or . Either way, AC2(b^o) would not hold. For if , then since its original value is 0, if A is set to 0 and PN is set to its original value of 0, then VS = 0 even if B = 1. In contrast, if , we must set PN = 0 for AC2(a) to hold, and with this setting, AC2(b^o) again does not hold.

However, A = 1 is a cause of VS = 1, according to all variants of the HP definition. (If we fix PN at its actual value of 0 and set A = 0, then VS = 0.) It follows that B = 1 is not even part of a cause of VS = 1 according to the modified HP definition. (This also follows from Theorem 2.2.3.) Although it seems reasonable that B = 1 should not be a cause of VS = 1, is it reasonable that A = 1 should be a cause? Anecdotally, most people seem to find that less objectionable than B = 1 being a cause. Normality considerations help explain why people are not so enthusiastic about calling A = 1 a cause either, although it is a “better” cause than B = 1.

Interestingly, historically, this example was the motivation for introducing normality considerations. Although normality is perhaps not so necessary to deal with it, it does prove useful in a small variant of this example.

Example 3.4.2 Again, suppose that Bodyguard puts an antidote in Victim’s coffee, but now Assassin puts the poison in the coffee. However, Assassin would not have put the poison in the coffee if Bodyguard hadn’t put the antidote in. (Perhaps Assassin is putting in the poison only to make Bodyguard look good.) Formally, we have the equation A = ¬B. Now Victim drinks the coffee and survives.

Is Bodyguard putting in the antidote a cause of Victim surviving? It is easy to see that, according to all three variants of the HP definition, it is. If we fix A = 0 (its value in the actual world), then Victim survives if and only if Bodyguard puts in the antidote. Intuition suggests that this is unreasonable. By putting in the antidote, Bodyguard neutralizes the effect of the other causal path he sets in action: Assassin putting in the poison.

Here normality considerations prove quite useful. The world where Bodyguard doesn’t put in the antidote but Assassin puts in the poison anyway (i.e., A = B = 0) directly contradicts the story. Given the story, this world should be viewed as being less normal than the actual world, where A = 0 and B = 1. Thus, B = 1 is not a cause of V = 1 in the extended model, as we would expect. ■

As the following example shows, normality considerations do not solve all problems.

Example 3.4.3 Consider the train in Example 2.3.6 that can go on different tracks to reach the station. As we observed, if we model the story using only the variables F (for “flip”) and T (for “track”), then flipping the switch is not a cause of the train arriving at the station. But if we add variables LB and RB (for left-hand track blocked and right-hand track blocked, with LB = RB = 0 in the actual context), then it is, according to the original and updated HP definition, although not according to the modified HP definition. Most people would not consider the flip a cause of the train arriving. We can get this outcome even with the original and updated definition if we take normality into account. Specifically, suppose that the train was originally going to take the left track and took the right track as a result of the switch. To show that the switch is a cause, consider the witness world where LB = 1 (and RB = 0). If we assume, quite reasonably, that this world is less normal than the actual world, where LB = RB = 0, then flipping the switch no longer counts as a cause, even with the original and updated HP definition.

But normality doesn’t quite save the day here. Suppose that we consider the context where both tracks are blocked. In this case, the original and updated HP definitions declare the flip a cause of the train not arriving (by considering the contingency where LB = 0). Normality considerations don’t help; the contingency where LB = 0 is more normal than the actual situation, where the track is blocked. The modified definition again matches intuitions better; it does not declare flipping the switch a cause.

As pointed out in the notes to Chapter 2, there is yet another way of modeling the story. Instead of the variables LB and RB, we use the variables LT (“train went on the left track”) and RT (“train went on the right track”). In the actual world, F = 1, RT = 1, LT = 0, and A = 1. Now F = 1 is a cause of A = 1, according to all the variants of the HP definition. If we simply fix LT = 0 and set F = 0, then A = 0. But in this case, normality conditions do apply: the world where the train does not go on the left track despite the switch being set to the left is less normal than the actual world. ■

3.4.3 Voting examples

Voting can lead to some apparently unreasonable causal outcomes (at least, if we model things naively). Consider Jack and Jill, who live in an overwhelmingly Republican district. As expected, the Republican candidate wins with an overwhelming majority. Jill would normally have voted Democrat but did not vote because she was disgusted by the process. Jack would normally have voted Republican but did not vote because he (correctly) assumed that his vote would not affect the outcome. In the naive model, both Jack and Jill are causes of the Republican victory according to all the variants of the HP definition. For if enough of the people who voted Republican had switched to voting Democrat or abstaining, then if Jack (or Jill) had voted Democrat, the Democrat would have won or it would have been a tie, whereas the Republican would have won had they abstained.

We can do better by constructing a model that takes these preferences into account. One way to do so is to assume that their preferences are so strong that we may as well take them for granted. Thus, the preferences become exogenous; the only endogenous variables are whether they vote (and the outcome); if they vote at all, they vote according to their preferences. In this case, Jack’s not voting is not a cause of the outcome, but Jill’s not voting is.

More generally, with this approach, the abstention of a voter whose preference is made exogenous and is a strong supporter of the victor does not count as a cause of victory. This does not seem so unreasonable. After all, in an analysis of a close political victory in Congress, when an analyst talks about the cause(s) of victory, she points to the swing voters who voted one way or the other, not the voters who were taken to be staunch supporters of one particular side.

That said, making a variable exogenous seems like a somewhat draconian solution to the problem. It also does not allow us to take into account smaller gradations in depth of feeling. At what point should a preference switch from being endogenous to exogenous? We can achieve the same effect in an arguably more natural way by using normality considerations. In the case of Jack and Jill, we can take voting for a Democrat to be highly abnormal for Jack and voting for a Republican to be highly abnormal for Jill. To show that Jack (resp., Jill) abstaining is a cause of the victory, we need to consider a contingency where Jack (resp., Jill) votes for the Democratic candidate. This would be a change to a highly abnormal world in the case of Jack but to a more normal world in the case of Jill. Thus, if we use normality as a criterion for determining causality, Jill would count as a cause, but Jack would not. If we use normality as a way of grading causes, Jack and Jill would still both count as causes for the victory, but Jill would be a much better cause. More generally, the more normal it would be for someone who abstains to vote Democrat, the better a cause that voter would be. The use of normality here allows for a more nuanced gradation of cause than the rather blunt approach of either making a variable exogenous or endogenous.

Now consider a vote where everyone can vote for one of three candidates. Suppose that the actual vote is 17–2–0 (i.e., 17 vote for candidate A, 2 for candidate B, and none for candidate C). Then not only is every vote for candidate A a cause of A winning, every vote for B is also a cause of A winning according to the original and updated HP definition. To see this, consider a contingency where 8 of the voters for A switch to C. Then if one of the voters for B votes for C, the result is a tie; if that voter switches back to B, then A wins (even if some subset of the voters who switch from A to C switch back to A). According to the modified HP definition, each voter for B is part of a cause of A’s victory. The argument is essentially the same: had a voter for B switched to C along with eight of the voters for A, it would have been a tie, so A would not have won.

Is this reasonable? What makes it seem particularly unreasonable is that if it had just been a contest between A and B, with the vote 17–2, then the voters for B would not have been causes of A winning. Why should adding a third option make a difference? Arguably, the modified HP definition makes it clear why adding the third option makes a difference: the third candidate could win according to the appropriate circumstances, and those circumstances include the voters for B changing their votes.

Indeed, it is clear that adding a third option can well make a difference in some cases. For example, we speak of Nader costing Gore a victory over Bush in the 2000 election. But we don’t speak of Gore costing Nader a victory, although in a naive HP model of the situation, all the voters for Gore are causes of Nader not winning as much as the voters for Nader are causes of Gore not winning. The discussion above points to a way out of this dilemma. If a sufficiently large proportion of Bush and Gore voters are taken to be such strong supporters that they will never change their minds, and we make their votes exogenous, then it is still the case that Nader caused Gore to lose (assuming that most Nader voters would have voted for Gore if Nader hadn’t run), but not the case that Gore caused Nader to lose. Similar considerations apply in the case of the 17–2 vote. And again, we can use normality considerations to give arguably more natural models of these examples. As we shall see in Chapter 6, using ideas of blame and responsibility give yet another approach to dealing with these concerns.

3.4.4 Causal chains

As I argued in Section 2.4, causation seems transitive when there is a simple causal chain involving but-for causality, and the final effect counterfactually depends on the initial cause. By contrast, the law does not assign causal responsibility for sufficiently remote consequences of an action. For example, in Regina v. Faulkner, a well-known Irish case, a lit match aboard a ship caused a cask of rum to ignite, causing the ship to burn, which resulted in a large financial loss by Lloyd’s insurance, leading to the suicide of a financially ruined insurance executive. The executive’s widow sued for compensation, and it was ruled that the negligent lighting of the match was not a cause (in the legally relevant sense) of his death. By taking normality into account, we can make sense of this kind of attenuation.

We can represent the case of Regina v. Faulkner using a causal model with nine variables:

• M = 1 if the match is lit, 0 if it is not;
• R = 1 if there is rum in the vicinity of the match, 0 if not;
• RI = 1 if the rum ignites, 0 if it does not;
• F = 1 if there is further flammable material near the rum, 0 if not;
• SD = 1 if the ship is destroyed, 0 if not;
• LI = 1 if the ship is insured by Lloyd’s, 0 if not;
• LL = 1 if Lloyd’s suffers a loss, 0 if not;
• EU = 1 if the insurance executive was mentally unstable, 0 if not; and
• ES = 1 if the executive commits suicide, 0 if not.

There are four structural equations:

• RI = min(M, R);
• SD = min(RI, F);
• LL = min(SD, LI); and
• ES = min(LL, EU ).

This model is shown graphically in Figure 3.1. The exogenous variables are such that M, R, F, LI, and EU are all 1, so in the actual world, all variables take the value 1. Intuitively, the events M = 1, RI = 1, SD = 1, LL = 1, and ES = 1 form a causal chain. The first four events are but-for causes of ES = 1, and so are causes according to all variants of the HP definition.

Figure 3.1: Attenuation in a causal chain.

Let us now assume that, for the variables M, R, F, LI, and EU, 0 is the typical value, and 1 is the atypical value. Consider a very simple normality ordering, which is such that worlds where more of these variables take the value 0 are more normal. For simplicity, consider just the first and last links in the chain of causal reasoning: LL = 1 and M = 1, respectively. The world w = (M = 0, R = 1, RI = 0, F = 1, SD = 0, LI = 1, LL = 0, EU = 1, ES = 0) is a witness of M = 1 being a cause of ES = 1 (for any of the variants of the HP definition). This is quite an abnormal world, although more normal than the actual world, so M = 1 does count as an actual cause of ES = 1, even using AC2⁺(a). However, note that if any of the variables R, F, LI, or EU is set to 0, then we no longer have a witness. Intuitively, ES counterfactually depends on M only when all of these other variables take the value 1. Now consider the event LL = 1. The world w′ = (M = 0, R = 0, RI = 0, F = 0, SD = 0, LI = 0, LL = 0, EU = 1, ES = 0) is a witness for LL = 1 being an actual cause of ES = 1 according to the original and updated HP definition. World w′ is significantly more normal than the best witness for M = 1 being a cause. Intuitively, LL = 1 needs fewer atypical conditions to be present in order to generate the outcome ES = 1. It requires only the instability of the executive, but not the presence of rum, other flammable materials, and so on. So although both LL = 1 and M = 1 count as causes using AC2⁺(a), LL = 1 is a better cause, so we would expect that people would be more strongly inclined to judge that LL = 1 is an actual cause of ES = 1 than M = 1.

Once we take normality into account, we can also have a failure of transitivity. For example, suppose that we slightly change the normality ordering so that the world w = (M = 0, R = 1, RI = 0, F = 1, SD = 0, LI = 1, LL = 0, EU = 1, ES = 0) that is a witness of M = 1 being a cause of ES = 1 is less normal than the actual world, but otherwise leave the normality ordering untouched. Now M = 1 is no longer a cause of ES = 1 using AC2⁺(a). However, causality still holds for each step in the chain: M = 1 is a cause of RI = 1, RI = 1 is a cause of SD = 1, SD = 1 is a cause of LL = 1, and LL = 1 is a cause of ES = 1. Thus, normality considerations can explain the lack of transitivity in long causal chains.

Note that the world w′ is not a witness according to the modified HP definition since, in w′, the values of the variables R, F, and LI differ from their actual values. Normality does not seem to help in this example if we use the modified HP definition. However, as we shall see in Section 6.2, we can deal with this example perfectly well using the modified HP definition if we use ideas of blame.

It is worth noting that, with the original and updated HP definition, the extent to which we have attenuation of actual causation over a causal chain is not just a function of the number of links in the chain. It is, rather, a function of how abnormal the circumstances are that must be in place in order for the causal chain to run from start to finish.

3.4.5 Legal doctrines of intervening causes

In general, the law holds that one is not causally responsible for some outcome that occurred only due to either a later deliberate action by some other agent or some very improbable event. For example, if Anne negligently spills gasoline, and Bob carelessly throws a cigarette into the spilled gasoline, then Anne’s action is a cause of the fire. But if Bob maliciously throws a cigarette into the spilled gasoline, then Anne is not considered a cause. (This example is based on the facts on an actual case: Watson v. Kentucky and Indiana Bridge and Railroad.) This kind of reasoning can also be modeled using an appropriate normality ordering, as I now show.

To fully capture the legal concepts, we need to represent the mental states of the agents. We can do this with the following six variables:

• AN = 1 if Anne is negligent, 0 if she isn’t;
• AS = 1 if Anne spills the gasoline, 0 if she doesn’t;
• BC = 1 if Bob is careless (i.e. doesn’t notice the gasoline), 0 if not;
• BM = 1 if Bob is malicious, 0 otherwise;
• BT = 1 if Bob throws a cigarette, 0 if he doesn’t; and
• F = 1 if there is a fire, 0 if there isn’t.

We have the following equations:

• F = min(AS, BT ) (i.e., F = AS ∧ BT);
• AS = AN; and
• BT = max(BC, BM, 1 − AS ) (i.e., BT = BC ∨ BM ∨ ¬AS).

This model is shown graphically in Figure 3.2. (Note that the model incorporates certain assumptions about what happens in the case where Bob is both malicious and careless and in the cases where Anne does not spill gasoline; what happens in these cases is not clear from the usual description of the example. These assumptions do not affect the analysis.)

Figure 3.2: An intervening cause.

It seems reasonable to assume that BM, BC, and AN typically take the value 0. But more can be said. In the law, responsibility requires a mens rea—literally, a guilty mind. Mens rea comes in various degrees. Starting with an absence of mens rea and then in ascending order of culpability, these are:

• prudent and reasonable: the defendant behaved as a reasonable person would;
• negligent: the defendant should have been aware of the risk of harm from his actions;
• reckless: the defendant acted in full knowledge of the harm her actions might cause;
• criminal/intentional: the defendant intended the harm that occurred.

We can represent this scale in terms of decreasing levels of typicality. Specifically, in decreasing order of typicality, we have

1. BC = 1;
2. AN = 1;
3. BM = 1.

Thus, although “carelessness” is not on the scale, I have implicitly taken it to represent less culpability than negligence.

In all the examples up to now, I have not needed to compare worlds that differed in the value of several variables in terms of normality. Here, when comparing the normality of worlds in which two of these variables take the value 0 and one takes the value 1, we can capture legal practice by taking the world with BC = 1 to be most normal, the world with AN = 1 to be next most normal, and the world with BM = 1 to be least normal. Consider first the case where Bob is careless. Then in the actual world we have

In the structural equations, F = 1 depends counterfactually on both BC = 1 and AN = 1. Thus, these are both but-for causes, so are causes according to all the variants of the HP definition. The best witness for AN = 1 being a cause is

and the best witness for BC = 1 being a cause is

Both of these worlds are more normal than the actual world. The first is more normal because AN takes the value 0 instead of 1. The second world is more normal than the actual world because BC takes the value 0 instead of 1. Hence, according to AC2⁺(a), both are actual causes. However, the best witness for AN = 1 is more normal than the best witness for BC = 1. The former witness has BC = 1 and AN = 0, and the latter witness has BC = 0 and AN = 1. Since AN = 1 is more atypical than BC = 1, the first witness is more normal. This means that we are more inclined to judge Anne’s negligence a cause of the fire than Bob’s carelessness.

Now consider the case where Bob is malicious. The actual world is

Again, without taking normality into account, both BM = 1 and AN = 1 are actual causes. The best witness for AN = 1 being a cause is

and the best witness for BM = 1 is

However, now the best witness for AN = 1 is less normal than the best witness for BM = 1 because BM = 1 is more atypical than AN = 1. So using this normality ordering, we are more inclined to judge that Bob’s malice is an actual cause of the fire than that Anne’s negligence is.

Recall that in Example 3.4.1, judgments of the causal status of the administrative assistant’s action changed, depending on the normative status of the professor’s action. Something similar is happening here: the causal status of Anne’s action changes with the normative status of Bob’s action. This example also illustrates how context can play a role in determining what is normal and abnormal. In the legal context, there is a clear ranking of norm violations.

3.5 An Alternative Approach to Incorporating Normality

I now present an alternative approach to incorporating normality concerns in causal models, inspired by the discussion after Example 3.2.6. As suggested earlier, this approach is also more compatible with the way that probability was incorporated into causal models. The first step is to put the normality ordering on causal settings. Since, for the purposes of this discussion, the causal model is fixed, this amounts to putting a normality ordering on contexts, rather than on worlds. This means that normality can be viewed as a qualitative generalization of probability. But, as I suggested before, to the extent that we are thinking probabilistically, we should interpret as meaning “ is much more probable than ”.

Putting the normality ordering on contexts clearly does not by itself solve the problem. Indeed, as noted in the discussion immediately after Example 3.2.6, the motivation for putting the normality ordering on worlds rather than contexts was to solve problems that resulted from putting the normality ordering on context! There is an additional step: lifting the normality order on contexts to a normality ordering on sets of contexts. Again, this is an idea motivated by probability. We usually compare events, that is, sets of worlds, using probability, not just single worlds; I will also be comparing sets of contexts in the alternative approach to using normality.

When using probability (ignoring measurability concerns), the probability of a set is just the sum of the probabilities of the elements in the set. But with normality, “sum” is in general undefined. Nevertheless, we can lift a partial preorder on an arbitrary set S to a partial preorder on 2^S (the set of subsets of S) in a straightforward way. Given sets A and B of contexts, say A ⪰^e B if, for all contexts , there exists a context such that . (I use the superscript e to distinguish the order ⪰e on events—sets of contexts—from the underlying order ⪰ on contexts.) In words, this says that A is at least as normal as B if, for every context in B, there is a context in A that is at least as normal.

Clearly we have u ⪰ u′ iff {u} ⪰^e {u}, so ⪰e extends ⪰ in the obvious sense. The definition of ⪰e has a particularly simple interpretation if ⪰ is a total preorder on contexts and A and B are finite. In that case, there is a most normal context u_A in A (there may be several equally normal contexts in A that are most normal; in that case, let u_A be one of them) and a most normal context u_B in B. It is easy to see that A ⪰^e B iff u_A ⪰ u_B. That is, A is at least as normal as B if the most normal worlds in A are at least as normal as the most normal worlds in B. Going back to probability, this says that the qualitative probability of an event is determined by its most probable element(s). For those familiar with ranking functions (see the notes at the end of the chapter), this ordering on events is exactly that determined by ranking functions.

Taking the causal model M to be fixed, let ⟦φ⟧ denote the event corresponding to a propositional formula φ; that is, is the set of contexts where φ is true. I now modify AC2⁺(a) and AC2(a^m) so that instead of requiring that , I require that ; that is, the set of worlds where the witness holds is at least as likely as the set of worlds satisfying .

Doing this solves the problem in the rock-throwing example. For one thing, with this change, the addition of BA (the accuracy of Billy’s throw) to the language in the rock-throwing example has no impact. More generally, whether something like accuracy is modeled by an endogenous variable or an exogenous variable has no impact on normality considerations; in either case, we would still consider the same set of contexts when comparing the normality of ⟦BH = 0 ∧ ST = 0 ∧ BS = 0⟧ and ⟦ST = 1 ∧ BS = 1⟧. It seems reasonable to view BH = 0 ∧ ST = 0 ∧ BS = 0 as being relatively normal. Even if Billy typically throws and typically is an accurate shot, he does occasionally not throw or throw and miss. Thinking probabilistically, even if the probability of the event ⟦BH = 0 ∧ ST = 0 ∧ BS = 0⟧ is lower than that of ⟦ST = 1 ∧ BS = 1⟧, it is not sufficiently less probable to stop the former from being as normal as the latter. (Here I am applying the intuition that ≻ represents “much more probable than” rather than just “more probable than”.) This viewpoint makes sense even if Billy is replaced by a machine. Put another way, although it may be unlikely that the rock-throwing machine does not throw or throws and misses, it may not be viewed as all that abnormal.

Going back to Example 3.2.6, it is still the case that Suzy’s throw by itself is not a cause of the bottle shattering if ⟦BH = 0 ∧ ST = 0 ∧ BS = 0⟧ is not at least as normal as ⟦ST = 1 ∧ BS = 1⟧ according to the modified HP definition (extended to take normality into account). Similarly, Suzy’s throw is not a cause according to the original and updated HP definition if ⟦BT = 0 ∧ ST = 0 ∧ BS = 0⟧ is not at least as normal as ⟦ST = 1 ∧ BS = 1⟧. Since Billy not throwing is only one reason that Billy might not hit, ⟦BH = 0 ∧ ST = 0 ∧ BS = 0⟧ is at least as normal as ⟦BT = 0 ∧ ST = 0 ∧ BS = 0⟧. Thus, using the witness BT = 0 ∧ ST = 0 will not make Suzy’s throw a cause using the original and updated HP definition if it is not also a cause using the modified HP definition. Since we can always take BH = 0 to be the witness for the original and updated HP definition, all the variants of the HP definition will agree on whether ST = 1 is a cause of BS = 1.

Is it so unreasonable to declare both Billy and Suzy throwing a cause of the bottle shattering if ⟦BH = 0 ∧ ST = 0 ∧ BS = 0⟧ is not at least as normal as ⟦ST = 1 ∧ BS = 1⟧? Roughly speaking, this says that a situation where the bottle doesn’t shatter is extremely abnormal. Suzy’s throw is not the cause of the bottle shattering; it was going to happen anyway. The only way to get it not to happen is to intervene on both Suzy’s throw and Billy’s throw.

We can also apply these ideas to graded causality. Instead of comparing witness worlds, I now compare witness events. But rather than just considering the best witnesses, I can now consider all witnesses. That is, suppose that are the witnesses to being a cause of φ in , and are the witnesses to being a cause of φ in . Now we can say that is at least as good a cause of φ in as if

(3.1)

That is, is at least as good a cause of φ in as if the disjunction of the (formulas describing the) witness worlds for being a cause of φ in is at least as normal as the disjunction of the witness worlds for being a cause of φ in .

We can further extend this idea to parts of causes. (Recall that I earlier made the claim that it is arguably better to redefine cause to be what I am now calling part of a cause.) Suppose that are all the causes of φ in that include X = x as a conjunct, with the corresponding witnesses. (It may be the case that and although the corresponding witnesses are different; a single cause may have a number of distinct witnesses.) Similarly, suppose that are all the causes of φ in that include Y = y as a conjunct, with the corresponding witnesses. Again, we can say that X = x is at least as good a part of a cause for φ in as Y = y if (3.1) holds.

The analysis for all the examples in Section 3.4 remains unchanged using the alternative approach, with the exception of the causal chain example in Section 3.4.4. Now the analysis for the original and updated HP definition becomes the same as for the modified HP definition. To determine the best cause, we need to compare the normality of ⟦M = 1 ∧ ES = 0⟧ to that of ⟦LL = 1 ∧ ES = 0⟧. There is no reason to prefer one to the other; the length of the causal chain from M to ES plays no role in this consideration. As I said, this example can be dealt with by considering blame, without appealing to normality (see Section 6.2).

This modified definition of graded causality, using (3.1), has some advantages.

Example 3.5.1 There is a team consisting of Alice, Bob, Chuck, and Dan. In order to compete in the International Salsa Tournament, a team must have at least one male and one female member. All four of the team members are supposed to show up for the competition, but in fact none of them does. According to the original and updated definition, each of Alice, Bob, Chuck, and Dan is a cause of the team not being able to compete. However, using the alternative approach to graded causality, with (3.1), Alice is a better cause than Bob, Chuck, or Dan. Every context where Alice shows up and at least one of Bob, Chuck, or Dan shows up gives a witness world for Alice not showing up being a cause; the only witness worlds for Bob not showing up being a cause are those where both Alice and Bob show up. Thus, the set of witnesses for Alice being a cause is a strict superset of those for Bob being a cause. Under minimal assumptions on the normality ordering (e.g., the contexts where just Alice and Bob show up, the contexts where just Alice and Chuck show up, and the contexts where Alice and Dan show up are all mutually incomparable) a cause are incomparable both to the contexts where Chuck is a cause and the contexts where Dan is a cause, and similarly for Chuck and Dan), Alice is a better cause.

According to the modified HP definition, Alice by herself is not a cause for the team not being able to compete; neither is Bob. However, both are parts of a cause. The same argument then shows that Alice is a better part of a cause than Bob according to the modified HP definition.

With the normality definition in Section 3.2, each of Alice, Bob, Charlie, and Dan would be considered equally good causes according to the original and updated HP definition, and equally good parts of causes according to the modified HP definition. People do judge Alice to be a better cause; the alternative approach seems to capture this intuition better. (See also the discussion of Example 6.3.2 in Chapter 6.) ■

Although this alternative approach to AC2⁺(a) and graded causality deals with the rock-throwing example and gives reasonable answer in Example 3.5.1, it does not solve all the problems involving normality and causality, as the following example shows.

Example 3.5.2 A and B each control a switch. There are wires going from an electricity source to these switches and then continuing on to C. A must first decide whether to flip his switch left or right, then B must decide (knowing A’s choice). The current flows, resulting in a bulb at C turning on, iff both switches are in the same position. B wants to wants to turn on the bulb, so flips her switch to the same position as A does, and the bulb turns on.

Intuition suggests that A’s action should not be viewed as a cause of the C bulb being on, whereas B’s should. But suppose that we consider a model with three binary variables, A, B, and C. A = 0 means that A flips his switch left and A = 1 means that A flips his switch right; similarly for B. C = 0 if the bulb at C does not turn on, and C = 1 if it does. In this model, whose causal network is given in Figure 3.3, A’s value is determined by the context, and we have the equations

• B = A and
• C = 1 iff A = B.

Figure 3.3: Who caused the bulb to turn on?

Without taking normality considerations into account, in the context where A = 1, both A = 1 and B = 1 are causes of C = 1, according to all variants of the HP definition. B = 1 is in fact a but-for cause of C = 1; to show that A = 1 is a cause of C = 1 for all variants of the HP definition, we can simply hold B at its actual value 1 and set A = 0. Similarly, for all other contexts, both A and B are causes of the outcome. However, people tend to view B as the only cause in all contexts.

Turning on a light bulb is ethically neutral. Does the situation change at all if instead of a bulb turning on, the result of the current flowing is that a person gets a shock?

Before considering normality, it is worth noting that the causal network in Figure 3.3 is isomorphic to that given in Figure 2.8 for Billy’s medical condition. Moreover, we can make the stories isomorphic by modifying the story in Example 2.4.1 slightly. Suppose we say that Billy is fine on Wednesday morning if exactly one doctor treats him and sick otherwise (so now BMC becomes a binary variable, with value 0 if Billy is fine on Wednesday morning and 1 if he is sick). If both doctors treat Billy, then I suspect most people would say that Tuesday’s doctor is the cause of Billy being sick; but if neither does, I suspect that Monday’s doctor would also be viewed as having a significant causal impact. Moreover, in the normal situation where Monday’s doctor treats Billy and Tuesday’s does not, I suspect that Monday’s doctor would be viewed as more of a cause of Billy feeling fine on Wednesday than Tuesday’s doctor.

If the HP definition is going to account for the differences between the stories, somehow normality considerations should do it. They do help to some extent in the second story. In particular, normality considerations work just right in the case that both doctors treat Billy (if we take the most normal situation to be the one where Monday’s doctor treats him and Tuesday’s doesn’t). But if neither doctor treats Billy, the same normality considerations suggest that Monday’s doctor is a better cause than Tuesday’s doctor. Although this may not be so unreasonable, I suspect that people would still be inclined to ascribe a significant degree of causality to Tuesday’s doctor. I return to this point below.

Now suppose instead that (what we have taken to be) the normal thing happens: just Monday’s doctor treats Billy. As expected, Billy feels fine on Wednesday. Most people would say that Monday’s doctor is the cause of Billy feeling fine, not Tuesdays’s doctor. Unfortunately, AC2⁺(a) (or AC2⁺(a^m)) is violated no matter who we try to view as the cause. What about graded causality? The witness for Monday’s doctor being a cause is s_MT=0,TT=0,u; the witness for Tuesday’s doctor being a cause is s_TT=1,u. In the first witness, we get two violations to normality: Billy’s doctor doesn’t do what he is supposed to do, nor does Tuesday’s doctor. In the second witness, we get only one violation (Tuesday’s doctor treating Billy when he shouldn’t). However, a case can still be made that the world where neither doctor treats Billy (MT = 0, TT = 0) should be viewed as more normal than the world where both do (MT = 1, TT = 1). The former witness might arise if, for example, both doctors are too busy to treat Billy. If Billy’s condition is not that serious, then they might sensibly decide that he can wait. However, for the second witness to arise, Tuesday’s doctor must have treated Billy despite Billy’s medical chart showing that Monday’s doctor treated him. Tuesday’s doctor could have also easily talked to Billy to confirm this. Arguably, this makes the latter world more of a violation of normality than the former. (The alternative approach works the same way for this example.)

When it comes to determining the cause of the bulb turning on, the situation is more problematic. Taking u to be the context where A = 1, both s_A=0,B=1,u and s_B=0,u seem less normal than s_u, and there seems to be no reason to prefer one to the other, so graded causality does not help in determining a cause. Similarly, for the alternative approach, both ⟦A = 0 ∧ B = 1 ∧ C = 0⟧ and ⟦B = 0 ∧ C = 0⟧ are less normal than either ⟦A = 1 ∧ C = 1⟧ or ⟦B = 1 ∧ C = 1⟧. Again, there is no reason to prefer B = 1 as a cause. If instead we consider the version of the story where, instead of a light bulb turning on, the result is a person being shocked, we can view both ⟦A = 0 ∧ B = 1 ∧ C = 0⟧ and ⟦B = 0 ∧ C = 0⟧ as more normal than either ⟦A = 1 ∧ C = 1⟧ or ⟦B = 1 ∧ C = 1⟧, since shocking someone can be viewed as abnormal. Now if we take normality into account, both A = 1 and B = 1 become causes, but there is still no reason to prefer B = 1.

We can deal with both of these issues by making one further change to how we take normality into account. Rather than considering the “absolute” normality of a witness, we can consider the change in normality in going from the actual world to the witness and prefer the witness that leads to the greatest increase (or smallest decrease) in normality. If the normality ordering is total, then whether we consider the absolute normality or the change in normality makes no difference. If witness w₁ is more normal than witness w₂, then the increase in normality from the actual world to w₁ is bound to be greater than the increase in normality from the actual world to w₂. But if the normality ordering is partial, then considering the change in normality from the actual world can make a difference.

Going back to Billy’s doctors, it seems reasonable to take the increase in normality going from s_MT=1,TT=1,u to s_MT=1,TT=0,u to be much greater than that of going from s_MT=1,TT=1,u to s_MT=0,TT=1,u. Thus, if both doctors treat Billy, then Tuesday’s doctor is a much better cause of Billy feeling sick on Wednesday than Monday’s doctor. In contrast, the increase in normality in going from s_MT=0,TT=0,u to s_MT=1,TT=0,u is arguably greater (although not much greater) than that of going from s_MT=0,TT=0,u to s_MT=0,TT=1,u. Thus, if neither doctor treats Billy, then Monday’s doctor is a better cause of Billy feeling sick on Wednesday than Tuesday’s doctor, but perhaps not a much better cause. Note that we can have such a relative ordering if the normality ordering is partial (in particular, if the worlds s_MT=0,TT=0,u and s_MT=1,TT=1,u are incomparable), but not if the normality ordering is total.

Similar considerations apply to the light bulb example. Now we can take the change from the world where both A = 1 and B = 1 to the world where A = 1 and B = 0 to be smaller than the one to the world where A = 0 and B = 1, because the latter change involves changing what A does as well as violating normality (in the sense that B does not act according to the equations), while the former change requires only that B violate normality. This gives us a reason to prefer B = 1 as a cause. ■

Considering not just how normal a witness is but the change in normality in going from the actual world to the witness requires a richer model of normality, where we can consider differences. I do not attempt to formalize such a model here, although I believe that it is worth exploring. These considerations also arise when dealing with responsibility (see Example 6.3.4 for further discussion).

Notes

Much of the material in this chapter is taken from [Halpern and Hitchcock 2015] (in some cases, verbatim); more discussion can be found there regarding the use of normality and how it relates to other approaches. A formal model that takes normality into account in the spirit of the definitions presented here already appears in preliminary form in the original HP papers [Halpern and Pearl 2001; Halpern and Pearl 2005a] and in [Halpern 2008]. Others have suggested incorporating considerations of normality and defaults in a formal definition of actual causality, including Hall [2007], Hitchcock [2007], and Menzies [2004, 2007].

The idea that normality considerations are important when considering causality goes back to the work of psychologists. Exploring the effect of norms on judgments of causality is also an active area of research. I mention here some representatives of this work, although these references do not do anywhere close to full justice to the ongoing work:

• Kahneman and Tversky [1982] and Kahneman and Miller [1986] showed that both statistical and prescriptive norms can affect counterfactual reasoning.
• Alicke [1992] and Alicke et al. [2011] showed that subjects are more likely to judge that someone caused some negative outcome when they have a negative evaluation of that person.
• Cushman, Knobe, and Sinnott-Armstrong [2008] showed that subjects are more likely to judge that an agent’s action causes some outcome when they hold that the action is morally wrong; Knobe and Fraser [2008] showed that subjects are more likely to judge that an action causes some outcome if it violates a policy (the Prof. Smith example in Section 3.1 is taken from their paper); Hitchcock and Knobe [2009] showed that this effect occurs with norms of proper functioning.
• McGill and Tenbrunsel [2000] considered the effect of likelihood and what they call mutability and propensity on normality judgments (and hence on causality judgments). Roughly speaking, propensity would judge a witness world to be more normal if it had a higher prior probability of occurring; if we are trying to determine whether X = x is a cause of φ and the witness world has X = x′, then mutability takes into account how “easy” it is to change X from x to x′. These issues are best explained by an example. Mandel and Lehman [1996] considered a scenario where an executive who chose to take a different route home was hit by a drunk teenager. Drunk driving is more likely to cause accidents than taking nonstandard routes home. Thus, propensity considerations (as well as notions of normality involving conventions) would say that the teenager should be a cause. However, if the teenager were perceived to be incapable of changing his drinking behavior (perhaps he is under heavy social pressure), then the causality ascribed to the choice of route goes up. McGill and Tenbrunsel [2000] showed that the cause with higher propensity is typically chosen, provided it is perceived to be mutable.

Approaches to giving semantics to typicality statements include preferential structures [Kraus, Lehmann, and Magidor 1990; Shoham 1987], ϵ-semantics [Adams 1975; Geffner 1992; Pearl 1989], possibility measures [Dubois and Prade 1991], and ranking functions [Goldszmidt and Pearl 1992; Spohn 1988]. The details of these approaches do not matter for our purposes; see [Halpern 2003] for an overview of these approaches. The approach used in Section 3.2 is the one used in [Halpern and Hitchcock 2015]; it can be viewed as an instance of preferential structures, which allow a partial preorder on worlds. By way of contrast, possibility measures and ranking functions put a total preorder on worlds. The model of causality that took normality considerations into account that I gave in [Halpern 2008] used ranking functions and, as a consequence, made the normality ordering total. Using a total normality order causes problems in some examples (see below).

There has been a great deal of discussion in the philosophy community regarding whether omissions count as causes. Beebee [2004] and Moore [2009], for example, argue against the existence of causation by omission in general; Lewis [2000, 2004] and Schaffer [2000a, 2004, 2012] argue that omissions are genuine causes in such cases; Dowe [2000] and Hall [2004] argue that omissions have a kind of secondary causal status; and McGrath [2005] argues that the causal status of omissions depends on their normative status. For example, Billy’s doctor’s failure to treat Billy is a cause of Billy’s sickness because he was supposed to treat Billy; the fact that other doctors also failed to treat Billy does not make them causes. As I observed in the main text, by taking an appropriate normality ordering, we can capture all these points of view. Wolff, Barbey, and Hausknecht [2010] and Livengood and Machery [2007] provide some experimental evidence showing how people view causality by omission and when they allow for causes that are omissions. Their results support the role of normality in these causality judgments (although the papers do not mention normality).

As I said in the main text, there have been claims that it is a mistake to focus on one cause. For example, John Stuart Mill [1856, pp. 360–361] writes:

…it is very common to single out one only of the antecedents under the denomination of Cause, calling the others mere Conditions. … The real cause, is the whole of these antecedents; and we have, philosophically speaking, no right to give the name of cause to one of them, exclusively of the others.

Lewis [1973a, pp. 558–559] also seems to object to this practice, saying:

We sometimes single out one among all the causes of some event and call it “the” cause, as if there were no others. Or we single out a few as the “causes”, calling the rest mere “causal factors” or “causal conditions”. … I have nothing to say about these principles of invidious discrimination.

Hall [2004, p. 228] adds:

When delineating the causes of some given event, we typically make what are, from the present perspective, invidious distinctions, ignoring perfectly good causes because they are not sufficiently salient. We say that the lightning bolt caused the forest fire, failing to mention the contribution of the oxygen in the air, or the presence of a sufficient quantity of flammable material. But in the egalitarian sense of “cause”, a complete inventory of the fire’s causes must include the presence of oxygen and of dry wood.

Sytsma, Livengood, and Rose [2012] conducted the follow-up to the Knobe and Fraser [2008] experiment discussed in Example 3.4.1 and Section 3.1. They had their subjects rate their agreement on a 7-point scale from 1 (completely disagree) to 7 (completely agree) with the statements “Professor Smith caused the problem” and “The administrative assistant caused the problem”. When they repeated Knobe and Fraser’s original experiment, they got an average rating of 4.05 for Professor Smith and 2.51 for the administrative assistant. Although their difference is less dramatic than Knobe and Fraser’s, it is still statistically significant. When they altered the vignette so that Professor Smith’s action was permissible, subjects gave an average rating of 3.0 for Professor Smith and 3.53 for the administrative assistant.

The bogus prevention problem is due to Hitchcock [2007]; it is based on an example due to Hiddleston [2005]. Using a normality ordering that is total (as I did in [Halpern 2008]) causes problems in dealing with Example 3.4.1. With a total preorder, we cannot declare (A = 1, B = 0, VS = 0) and (A = 0, B = 1, VS = 1) to be incomparable; we must compare them. To argue A = 1 is not a cause, we have to assume that (A = 1, B = 1, VS = 1) is more normal than (A = 0, B = 0, VS = 0). This ordering does not seem so natural.

Example 3.4.2 is due to Hitchcock [2007], where it is called “counterexample to Hitchcock”. Its structure is similar to Hall’s short-circuit example [Hall 2007, Section 5.3]; the same analysis applies to both. The observation that normality considerations deal well with it is taken from [Halpern 2015a]. The observation that normality considerations do not completely solve the railroad switch problem, as shown in Example 3.4.3, is due to Schumacher [2014]. The version of the switch problem with the variables LT and RT is essentially how Hall [2007] and Halpern and Pearl [2005a] modeled the problem. The voting examples in Section 3.4.3 are due to Livengood [2013]; the analysis given here in terms of normality is taken from [Halpern 2014a]. The analysis of all the other examples in Section 3.4 is taken from [Halpern and Hitchcock 2015].

For the details of Regina v. Faulkner, see [Regina v. Faulkner 1877]. Moore [2009] uses this type of case to argue that our ordinary notion of actual causation is graded, rather than all-or-nothing, and that it can attenuate over the course of a causal chain. In the postscript of [Lewis 1986b], Lewis uses the phrase “sensitive causation” to describe cases of causation that depend on a complex configuration of background circumstances. For example, he describes a case where he writes a strong letter of recommendation for candidate A, thus earning him a job and displacing second-place candidate B, who then accepts a job at her second choice of institution, displacing runner-up C, who then accepts a job at another university, where he meets his spouse, and they have a child, who later dies. Lewis claims that although his writing the letter is indeed a cause of the death, it is a highly sensitive cause, requiring an elaborate set of detailed conditions to be present. Woodward [2006] says that such causes are “unstable”. Had the circumstances been slightly different, writing the letter would not have produced this effect (either the effect would not have occurred or it would not have been counterfactually dependent on the letter). Woodward argues that considerations of stability often inform our causal judgments. The extended HP definition allows us to take these considerations into account.

The Watson v. Kentucky and Indiana Bridge and Railroad can be found in [Watson v. Kentucky and Indiana Bridge and Railroad 1910].

Sander Beckers [private communication, 2015] pointed out that putting a normality ordering on worlds does not really solve the problem of making BT = 1 part of a cause of BS = 1 (see the discussion after Example 3.2.6), thus providing the impetus for the alternative definition of normality discussed in Section 3.5. Example 3.5.1 is due to Zultan, Gerstenberg, and Lagnado [2012], as are the experiments showing that people view Alice as a better cause than Bob in this example. The work of Zultan, Gerstenberg, and Lagnado related to this example is discussed further in the notes to Chapter 6. The example of C being shocked in Example 3.5.2 is due to McDermott [1995]. He viewed it as an example of lack of transitivity: A’s flipping right causes B to flip right, which in turn causes C to get a shock, but most people don’t view A’s flip as being a cause of C’s shock, according to McDermott. As I observed, normality considerations can also be used to give us this outcome. Various approaches to extending a partial ordering ⪰ on worlds to an ordering ⪰e on events are discussed in [Halpern 1997]; the notation ⪰ is taken from [Halpern 2003].

Chapter 4

The Art of Causal Modeling

Fashion models and financial models are similar. They bear a similar relationship to the everyday world. Like supermodels, financial models are idealized representations of the real world, they are not real, they don’t quite work the way that the real world works. There is celebrity in both worlds. In the end, there is the same inevitable disappointment.

Satyajit Das, Traders, Guns & Money: Knowns and Unknowns in the Dazzling World of Derivatives

Essentially, all models are wrong, but some are useful.

G. E. P. Box and N. R. Draper, Empirical Model Building and Response Surfaces

In the HP definition of causality, causality is relative to a causal model. X = x can be the cause of φ in one causal model and not in another. Many features of a causal model can impact claims of causality. It is clear that the structural equations can have a major impact on the conclusions we draw about causality. For example, it is the equations that allow us to conclude that lower air pressure is the cause of the lower barometer reading and not the other way around; increasing the barometer reading will not result in higher air pressure.

But it is not just the structural equations that matter. As shown by the Suzy-Billy rock-throwing example (Example 2.3.3), adding extra variables to a model can change a cause to a non-cause. Since only endogenous variables can be causes, the split of variables into exogenous and endogenous can clearly affect what counts as a cause, as we saw in the case of the oxygen and the forest fire. As a number of examples in Chapter 3 showed, if we take normality considerations into account, the choice of normality ordering can affect causality. Even the set of possible values of the variables in the model makes a difference, as Example 2.3.7 (where both the sergeant and captain give orders) illustrates.

Some have argued that causality should be an objective feature of the world. Particularly in the philosophy literature, the (often implicit) assumption has been that the job of the philosopher is to analyze the (objective) notion of causation, rather like that of a chemist analyzing the structure of a molecule. In the context of the HP approach, this would amount to designating one causal model as the “right” model. I do not believe that there is one “right” model. Indeed, in the spirit of the quote at the beginning of this chapter, I am not sure that there are any “right” models, but some models may be more useful, or better representations of reality, than others. Moreover, even for a single situation, there may be several useful models. For example, suppose that we ask for the cause of a serious traffic accident. A traffic engineer might say that the bad road design was the cause; an educator might focus on poor driver education; a sociologist might point to the pub near the highway where the driver got drunk; a psychologist might say that the cause is the driver’s recent breakup with his girlfriend. Each of these answers is reasonable. By appropriately choosing the variables, the structural-equations framework can accommodate them all.

That said, it is useful to have principles by which we can argue that one model is more reasonable/useful/appropriate than another. Suppose that a lawyer argues that, although his client was drunk and it was pouring rain, the cause of the accident was the car’s faulty brakes, which is why his client is suing GM for $5,000,000. If the lawyer were using the HP definition of causality, he would then have to present a causal model in which the brakes were the cause. His opponent would presumably then present a different model in which the drunkenness or the rain was a cause. We would clearly want to say that a model that made the faulty brakes a cause because it did not include the rain as an endogenous variable, or it took drunkenness to be normal, was not an appropriate model.

The structural equations can be viewed as describing objective features of the world. At least in principle, we can test the effects of interventions to see whether they are correctly captured by the equations. In some cases, we might be able to argue that the set of values of variables is somewhat objective. In the example above, if C can only report values 0 and 1, then it seems inappropriate to take {0, 1, 2} to be its set of values. But clearly what is exogenous and what is endogenous is to some extent subjective, as is the choice of variables, and the normality ordering. Is there anything useful that we can say about them? In this chapter, I look at the issue and its impact more carefully.

4.1 Adding Variables to Structure a Causal Scenario

A modeler has considerable leeway in choosing which variables to include in a model. As the Suzy-Billy rock-throwing example shows, if we want to argue that X = x is the cause of φ rather than Y = y, then there must be a variable that takes on different values depending on whether X = x or Y = y is the actual cause. If the model does not contain such a variable, then there is no way to argue that X = x is the cause. There is a general principle at play here. Informally, say that a model is insufficiently expressive if it cannot distinguish two different scenarios (intuitively, ones for which we would like to make different causal attributions). The naive rock-throwing model of Figure 2.2, which just has the endogenous variables ST, BT, and BS, is insufficiently expressive in this sense because it cannot distinguish between a situation where Suzy and Billy’s rock hit the bottle simultaneously (in which case we would want to call Billy’s throw a cause) and the one where Suzy hits first (so Billy’s throw is not a cause). What about the bogus prevention example from Section 3.4.2? As we saw, here too adding an extra variable can give us a more reasonable outcome (without appealing to normality considerations). I would argue that in this case as well, the problem is that the original model was insufficiently expressive.

The next four examples show the importance of having variables that bring out the causal structure of a scenario and to distinguish two causal scenarios. One way of understanding the role of the variables SH and BH in the rock-throwing example is that they help us distinguish the scenario where Suzy and Billy hit simultaneously from the one where Suzy hits first. The following examples show that this need to distinguish scenarios is quite widespread.

Example 4.1.1 There are four endogenous binary variables, A, B, C, and S, taking values 1 (on) and 0 (off). Intuitively, A and B are supposed to be alternative causes of C, and S acts as a switch. If S = 0, the causal route from A to C is active and that from B to C is dead; and if S = 1, the causal route from A to C is dead and the one from B to C is active. There are no causal relations between A, B, and S; their values are determined by the context. The equation for C is C = (¬S ∧ A) ∨ (S ∧ B).

Suppose that the context is such that A = B = S = 1, so C = 1. B = 1 is a but-for cause of C = 1, so all the variants of the HP definition agree that it is a causes, as we would hope. Unfortunately, the original and updated definition also yield A = 1 as a cause of C = 1. The argument is that in the contingency where S is set to 0, if A = 0, then C = 0, whereas if A = 1, then C = 1. This does not seem so reasonable. Intuitively, if S = 1, then the value of A is irrelevant to the outcome. Considerations of normality do not help here; all worlds seem to be equally normal. Although the modified HP definition does not take A = 1 to be a cause, it is part of a cause: A = 1 S = 1 is a cause of C = 1 because if A and S are both set to 0, C = 0.

This might seem to represent a victory for the modified HP definition, but the situation is a little more nuanced. Consider a slightly different story. This time, we view B as the switch, rather than S. If B = 1, then C = 1 if either A = 1 or S = 1; if B = 0, then C = 1 only if A = 1 and S = 0. That is, C = (B ∧ (A ∨ S)) ∨ (¬B ∧ A ∧ ¬S). Although this is perhaps not as natural a story as the original, such a switch is surely implementable. In any case, a little playing with propositional logic shows that, in this story, C satisfies exactly the same equation as before: (¬S ∧ A) ∨ (S ∧ B) is equivalent to (B ∧ (A ∨ S)) ∨ (¬B ∧ A ∧ ¬S). The key point is that, unlike the first story, in the second story, it seems to me quite reasonable to say that A = 1 is a cause of C = 1 (as is B = 1). Having A = 1 is necessary for the first “mechanism” to work. But as long as we model both stories using just the variables A, B, C, and S, they have the same models with the same equations, so no variant of the HP definition can distinguish them.

Given that we have different causal intuitions for the stories, we should model them differently. One way to distinguish them is to add two more endogenous random variables, say D and E, that describe the ways that C could be 1. In the original story, we would have the equations D = ¬S ∧ A, E = S ∧ B, and C = D ∨ E. In this model, since D = 0 in the actual context, it is not hard to see that all variants of the HP definition agree that A = 1 is not a cause of C = 1 (nor is it part of a cause), whereas B = 1 and S = 1 are causes, as they should be. Thus, in this model, we correctly capture our intuitions for the original story.

To capture the second story, we can add variables D′ and E′ such that D′ = B ∧ (A ∨ S), E′ = ¬B ∧ A ∧ S, and C = D′ ∨ E′. In this model, it is not hard to see that according to the original and updated definition, all of A = 1, B = 1, and S = 1 are causes of C = 1; according to the modified HP definition, the causes stay the same: B = 1 and A = 1 ∧ S = 1.

Now, as I said above, we can certainly create these setups in the real world. Consider the original story. We can design a circuit where there is a source of power at A and B, a physical switch at S, and a bulb at C that turns on (C = 1) if either the circuit is connected to A (A = 1) and the switch is turned left (S = 0) or the circuit is connected to B (B = 1) and the switch is turned right (S = 1). We can similarly design a circuit that captures the second story. With this circuit, we can think of D as representing “there is current flowing between power source A and bulb C”; there are similar physical analogues of E, D′, and E′. But even without physical analogues, these “structuring” variables can play an important role. Suppose that the agent does not see the underlying physical setup. All he sees are switches corresponding to A, B, and S, which can be manipulated. In the first setup, flipping the switch for A connects or disconnects the circuit and the A battery, and similarly for B, while flipping the switch for S connects C either to the A or B battery. In the second story, the switches for A, B, and S work differently, but as far as the observer is concerned, all interventions lead to the same outcome on the bulb C in both cases. A modeler could usefully introduce D and E (resp., D′ and E′) to disambiguate these models. These variables can be viewed as the modeler’s proxies for the underlying physical circuit; they help describe the mechanisms that result in C being 1.

Note that we do not want to think of D as being defined to take the value 1 if A = 1 and S = 0. For then we could not intervene to set D = 0 if A = 1 and S = 0. Adding a variable to the model commits us to being able to intervene on it. (I return to this point in Section 4.6.) In the real world, setting D to 0 despite having A = 1 and S = 0 might correspond to the connection being faulty when the switch is turned left. ■

Example 4.1.2 A lamp L is controlled by three switches, A, B, and C, each of which has three possible positions, −1, 0, and 1. The lamp switches on iff two or more of the switches are in the same position. Thus, L = 1 iff (A = B) ∨ (B = C) ∨ (A = C). Suppose that, in the actual context, A = 1, B = −1, and C = −1. Intuition suggests that although B = −1 and C = −1 should be causes of L = 1, A = 1 should not: since the setting of A does not match that of either B or C, it has no causal impact on the outcome.

Since B = −1 and C = −1 are but-for causes of L = 1, all three variants of the HP definition declare them to be causes. Unfortunately, the original and updated definition also declare A = 1 to be a cause. For in the contingency where B = 1 and C = −1, if A = 1, then L = 1, whereas if A = 0, then L = 0. Adding defaults to the picture does not solve the problem. Again, the modified HP definition does the “right” thing here and does not declare A = 1 to be a cause or even part of a cause.

However, just as in Example 4.1.1, another story can be told here, where the observed variables have the same values and are connected by the same structural equations. Now suppose that L = 1 iff either (a) none of A, B, or C is in position −1; (b) none of A, B, or C is in position 0; or (c) none of A, B, or C is in position 1. It is easy to see that the equations for L are literally the same as in the original example. But now it seems more reasonable to say that A = 1 is a cause of L = 1. Certainly A = 1 causes L = 1 as a result of no values being 0; had A been 0, then the lamp would still have been on, but now it would be as a result of no values being −1. Considering the contingency where B = 1 and C = −1 “uncovers” the causal impact of A.

Again, the distinction between the two stories can be captured by adding more variables. For the second story, add the variables NOT (−1), NOT (0), and NOT (1), where NOT (i) is 1 iff none of A, B, or C is i. Then L = NOT (−1) ∨ NOT (0) ∨ NOT (1). Now all three variants of the HP definition agree that A = 1 is a cause of L = 1 (as well as B = −1 and C = 1). For the original story, add the variables TWO (−1), TWO (0), and TWO (1), where TWO (i) = 1 iff at least two of A, B, and C are i, and take L = TWO (−1) ∨ TWO (0) ∨ TWO(1). In this case, all three variants agree that A = 1 is not a cause of L = 1 (whereas B = −1 and C = −1 continue to be causes).

Once again, I think of the variables NOT (−1), NOT (0), and NOT (1) (resp., TWO (−1), TWO (0), and TWO (1)) as “structuring” variables that help the modeler distinguish the two scenarios and, specifically, the mechanisms that lead to the lamp turning on.

This example (and the others in this section) show that causality is to some extent dependent on how we describe a situation. If we describe the rule for the lamp being on as “it is on if two or more of the switches are in the same position”, we get a different answer than if we say “it is on if there is a position such that none of the switches are in that position”, despite the fact that these two rules lead to the same outcomes. This seems to me compatible with human ascriptions of causality. But even if accept that, where does this leave a modeler who is thinking in terms of, say, the first rule, and to whom the second rule does not even occur? My sense is that the description of the first rule naturally suggests a “structuring” in terms of the variables TWO (−1), TWO (0), and TWO (1), so it is good modeling practice to add these variables even if no other way of structuring the story occurs to the modeler. ■

Example 4.1.3 Consider a model M with four endogenous variables, A, B, D, and E. The values of A and D are determined by the context. The values of B and E are given by the equations B = A and E = D. Suppose that the context u is such that A = D = 1. Then clearly, in (M, u), A = 1 is a cause of B = 1 and not a cause of E = 1, whereas D = 1 is a cause of E = 1 and not of B = 1. The problem comes if we replace A in the model by X, where, intuitively, X = 1 iff A and D agree (i.e., X = 1 if the context would have been such that A = D = 1 or A = D = 0). The value of A can be recovered from the values of D and X. Indeed, it is easy to see that A = 1 iff X = D = 1 or X = D = 0. Thus, we can rewrite the equation for B by taking B = 1 iff X = D = 1 or X = D = 0. Formally, we have a new model M ′ with endogenous variables X, B, D, and E. The value of D and X is given by the context; as suggested above, equations for the B and E are

The context u is such that D = X = 1, so B = E = 1. In (M′, u), it is still the case that D = 1 is a cause of E = 1, but now D = 1 is also a cause of B = 1.

Is it so unreasonable that D = 1 is a cause of B = 1 in M′ but is not a cause of B = 1 in M? Of course, if we have in mind the story represented by model M, then it is unreasonable for D = 1 to be a cause. But M′ can also represent quite a different situation. Consider the following two stories. We are trying to determine the preferences of two people, Betty and Edward, in an election. B = 1 if Betty is recorded as preferring the Democrats and B = 0 if Betty is recorded as preferring the Republicans, and similarly for E. In the first story, we send Alice to talk to Betty and David to talk to Edward to find out their preferences (both are assumed to be truthful and good at finding things out). When Alice reports that Betty prefers the Democrats (A = 1), then Betty is reported as preferring the Democrats (B = 1); similarly for David and Edward. Clearly, in this story (which is modeled by M), D = 1 causes E = 1 but not B = 1.

But now suppose instead of sending Alice to talk to Betty, Xavier is sent to talk to Carol, who knows only whether Betty and Edward have the same preferences. Carol tells Xavier that they indeed have the same preferences (X = 1). Upon hearing that X = D = 1, the vote tabulator correctly concludes that B = 1. This story is modeled by M′. But in this case, it strikes me as perfectly reasonable that D = 1 should be a cause of B = 1. This is true despite the fact that if we had included the variable A in M′, it would have been the case that A = B = 1.

Again, we have two different stories; we need to have different models to disambiguate them. ■

At the risk of overkill, here is yet one more example:

Example 4.1.4 A ranch has five individuals: a₁,…, a₅. They have to vote on two possible outcomes: staying at the campfire (O = 0) or going on a round-up (O = 1). Let A_i be the random variable denoting a_i’s vote, so A_i = j if a_i votes for outcome j. There is a complicated rule for deciding on the outcome. If a₁ and a₂ agree (i.e., if A₁ = A₂), then that is the outcome. If a₂,…, a₅ agree and a₁ votes differently, then the outcome is given by a₁’s vote (i.e., O = A₁). Otherwise, majority rules. In the actual situation, A₁ = A₂ = 1 and A₃ = A₄ = A₅ = 0, so by the first mechanism, O = 1. The question is what were the causes of O = 1.

Using the naive causal model with just the variables A₁,…, A₅, O, and the obvious equations describing O in terms of A₁,…, A₅, it is almost immediate that A₁ = 1 is a but-for cause of O = 1. Changing A₁ to 0 results in O = 0. Somewhat surprisingly, in this naive model, A₂ = 1, A₃ = 0, A₄ = 0, and A₅ = 0 are also causes according to the original and updated HP definition. To see that A₂ = 1 is a cause, consider the contingency where A₃ = 1. Now if A₂ = 0, then O = 0 (majority rules); if A₂ = 1, then O = 1, since A₁ = A₂ = 1, and O = 1 even if A₃ is set back to its original value of 0. The argument that A₃ = 0, A₄ = 0, and A₅ = 0 are causes according to the original and updated definition is the same; I will give the argument for A₃ = 0 here. Consider the contingency where A₂ = 0, so that all voters but a₁ vote for 0 (staying at the campsite). If A₃ = 1, then O = 0 (majority rules). If A₃ = 0, then O = 1, by the second mechanism (a₁ is the only vote for 0), whereas if A₂ is set to its original value of 1, then we still have O = 1, now by the first mechanism. Although the modified HP definition does not declare any of A₂ = 1, A₃ = 0, A₄ = 0, or A₅ = 0 to be causes, it does declare each of the conjunctions A₂ = 1 ∧ A₃ = 0, A₂ = 1 ∧ A₄ = 0, and A₂ = 1 ∧ A₅ = 0 to be a cause; for example, if the values of A₂ and A₃ are changed, then O = 0. So, according to the modified HP definition, each of A₂ = 1, A₃ = 0, A₄ = 0, and A₅ = 0 is part of a cause.