# Actual Causality

## Joseph Y. Halpern

### Reviewed by Ian Rosenberg and Clark Glymour

*Actual Causality
*Joseph Y. Halpern

Cambridge, MA: MIT Press, 2016, £40

ISBN 9780262035026

**1 Introduction**

Theories of ‘actual causation’ aim to provide an informative guide for assessing which events cause which others in circumstances where almost everything else is known: which other events occurred or did not occur, and how (if at all) the occurrence or non-occurrence of a particular event (regarded as values of a variable) can depend on specific other events (or their absence), also regarded as values of variables. The ultimate aim is a theory that can agreeably be applied in causally fraught circumstances of technology, the law, and everyday life, where the identification of relevant features is not immediate and judgements of causation are entwined with judgements of moral or legal responsibility. Joseph Halpern’s *Actual Causality* is the latest and most extensive addition to this effort, carried out in a tradition that holds causation to be difference making.

*Actual Causality* is notably valuable in providing a wealth of examples and carefully tracing their provenance. Beyond articulation of (several) theories of actual causation, the book includes guidance on how to model situations, and on legal and moral applications. A reader with a critical eye will learn a great deal from it about the possibilities and difficulties of giving a clear, quasi-formal account of judgements of causal relations among particular events. For those interested in the subject, those are reasons enough to study and value the book. They will not, unfortunately, find in it a satisfactory theory of actual causation, but they will find a challenging, and in places frustrating, exploration of possibilities.

Any such theory meets two kinds of test cases, those from scientific and mechanical contexts where the pertinent events, the dependencies, and what differences are made by specific interventions on component parts, are settled. Fault detection problems in engineering provide such examples: when a machine of known design fails to produce the right output for an input, there is a fact of the matter as to what component failures caused the wrong output, and what interventions to replace them will restore the intended functioning. The very cover of *Actual Causality*—depicting a Rube Goldberg-ish machine—suggests mechanical applications.

A formula for assessing actual causation that works smoothly for mechanics can grind against intuition in examples drawn from the law and from everyday life, where judgements can vary with descriptions of events and are confounded with moral sensibilities and epistemological issues. It would be too much to expect a theory to handle all such cases in a manner that prompts unanimous agreement, but we do expect a theory that unambiguously solves the cases it is intended to treat and does not yield results that violate some of our most fundamental judgements connecting causality and responsibility.

**2 The Modified Definition of Causality**

Halpern provides a formal language for actual causation. He offers three distinct definitions of causality, two based on previous proposals in collaboration with Pearl, and a modified HP definition. Halpern presents the modified version as his best, so we will here ignore the others. In its crudest form, the idea is that event *x*—a happening or non-happening—that is a value of variable *X*, is a cause of event y, a value of variable *Y,* just if, were the value of *X* to be different, the value of *Y* would be different, provided some things were to stay the same. That basic inspiration is compromised at various points in Halpern’s intricate discussion of ‘normality’.

Halpern gives a formal definition of ‘*x* actually causes* φ *(according to a model *M* with variables *V*, for a context* u*)’:

## AC1: (

M, u) |=(X=x) and (M, u) |=φ.(p. 23)## AC2(a

^{m}): There is a setWof variables inVand a settingx′of the variables inX′such that if (M, u) |=W=w*, then (M, u) |= [X←x′,W ←w*] ¬φ. (p. 25)## AC3:

Xis minimal; there is no strict subsetX′ofXsuch thatX′=x′satisfies the conditions AC1 and AC2, wherex′is the restriction ofxto the variables inX.(p. 23)

Here *M*, the model, is a pair, , where G is a directed (usually, acyclic) graph each of whose vertices is a variable ranging over possible values of a particular event—for example, ‘happens’ coded as 1 or ‘does not happen’ coded as 0—and F is a set of functions specifying the value of each variable for each assignment of values to its parents in the graph. Function *u* assigns values to the zero indegree variables in G, which, with F, determine unique values for all other variables in G. (*M, u*) is the tuple of the causal model and the context, or total assignment of values to the zero indegree variables and thus indirectly to all other variables in accord with F.^{[1]} This can be understood as the ‘actual world’. *X *and *W* are sets of variables, and *x* and *w* are value assignments that correspond to them. *w** is the value of *W* in the actual world. The stylus, ‘|=’, indicates semantic entailment. The formulae in square brackets indicate interventions that fix the values of the upper-case variables to the lower-case values and should be read as if they were on the left hand side of the entailment symbol. The interventions and assignments that satisfy AC2(a^{m}) are said to be a ‘witness’ to the actual causality of *x* provided *x* satisfies the other two requirements. Note that while we will almost always discuss cases in which variables range over two values, Halpern allows variables to have a larger range.

AC1 just says that the actual world satisfies the value assignment of the proposed cause and effect. In other words, *X *= *x* cannot be a cause of *φ* unless *x* is the actual value of *X* and *φ* is true. AC3 is a straightforward minimality constraint on the cause: *X* and * x* can be vectors (or sets or lists), and minimality requires that no vector whose components are a proper subset of those of

*x*satisfies the other conditions. AC2(a

^{m}) is the real substance of the definition, and is the most complex criterion. In Halpern’s words it says, ‘we can show the counterfactual dependence of

*φ*on

*X*by keeping the variables in

*W*fixed at their actual values’ (p. 25). We just need to find some (possibly empty) set of variables to hold at their actual values. With this set of variables fixed,

*x*must be a ‘but-for’ cause of

*φ*. In that circumstance, change in the value of

*X*would change the truth value of

*φ.*When

*x*has more than one component but satisfies AC1–AC3, each component of

*x*is a ‘partial cause’ of

*φ.*Note that AC2(a

^{m}) does not allow fixing any variable other than

*X*at any value other than its actual one.

Halpern extends his theory to include probability relations. He takes probabilities that might be given by a contingency table specifying the probability of each value of each variable, *X*, conditional on each assignment of values to the parents of *X* in the graph, as measured on deterministic cases—or as Halpern says, ‘pulled out’—and his deterministic account applied in each case, with resulting probabilities.^{[2]}

**3 Mechanics and Overdetermination**

The familiar cases of overdetermination given by (i) and (ii) in Figure 1 involve at least two potential causes and an effect, each with possible values in {0, 1}; *Z* = *X* + *Y*, where the addition is Boolean; equivalently, *Z* occurs if and only if *X* occurs or *Y* occurs (see Figure 1).

Halpern addresses Figure 1(i) with the notion of a partial cause. Neither *X* = 1 nor *Y* = 1 is a cause of *Z* = 1, only their conjunction is, but each is a partial cause of *Z* = 1. Figure 1(ii), known in the literature as backup, is handled nicely by Halpern’s modified account: *X* = 1 is the actual cause, *Y* = 0 is not a cause. Figure 1(iii) has no name, and it appears not to have been discussed in the ingenious literature of cases of actual causation; Halpern does not consider it. Halpern’s theory, however, is unequivocal: *X* = 1 is a cause of *Z* = 1, but *Y* = 1 is not a cause, not even a partial cause. Halpern (personal communication) endorses this result. In mechanical cases, Halpern’s solution to Figure 1(iii) yields what would sometimes be called in the fault detection literature a ‘root cause’. If *X* were a component stuck at 1 with the result that *Y* = 1 and *Z* = 1, and the desired behaviour is that *Z* = 0, a repair would be to replace *X* with a component that is not stuck at 1. In this mechanical case, Halpern’s theory gets things right.** **

**4 Life, Death, and Probability**

Consider another interpretation of the causal system of Figure 1(iii). An obedient gang is ordered by its leader to join him in murdering someone, and does so, all of them shooting the victim at the same time or all of them together pushing the plunger connected to a bomb. The action of any one of the gang would suffice for the victim’s death. If responsibility implies causality, whom among them is responsible? Were you among the jury, whom would you convict? What ought The Hague court do in cases of subordinates sure to obey orders? Halpern’s theory says the gang leader and only the gang leader is a cause of the victim’s death. This is a morally intolerable result; absent a plausible general principle severing responsibility from causation, any theory that yields such a result should be rejected.

What if the action of the superior does not necessitate the action of the subordinate, but only makes it probable? In that case, the result on Halpern’s theory of probabilistic causality is that the probability that *Y* = 1 is a cause of *Z* = 1 is zero! Halpern’s strategy is:

## […] to convert a single causal setting where the equations are probabilistic to a probability over 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 use the definition of causality already given for deterministic models and talk about the probability of causality, that is, the probability that

Ais a cause ofB. (p. 48)

Suppose then that we alter Figure 1(iii) so that the probability that *Y* = 1 when *X* = 1 is *p*, leaving the dependence of *Z* on *X* and *Y*unchanged. Then, following Halpern’s probability recipe, we have for *p* fraction of the cases *X* = *Y* = *Z* = 1, and (1 – *p*) fraction of cases in which *X* = *Z* = 1 and *Y* = 0. Applying Halpern’s theory to each of the *p* cases yields that *Y* = 1 is not a cause of *Z* = 1, as before. But the (1 – *p*) cases where *X* = 1 and Y = 0 and Z = 1 are instances of backup, Figure 1(ii), for which Halpern’s deterministic theory also yields that the value of *Y* is not a cause of *Z* = 1. Thus, whatever *p* may be, Halpern’s theory says the probability that the value of *Y* is an actual cause of *Z*= 1 is zero.

We will consider whether any of Halpern’s proposed normality conditions rescue his theory from this and other consequences.

**5 The Curious Case of Billy and Suzy**

Halpern analyses the following example repeatedly throughout his book to compare different formulations of causation and normality, so we will give it in full detail. The story is simple: Billy and Suzy both throw rocks at a bottle, Suzy just before Billy. Suzy’s rock hits the bottle, smashing it, but had her rock not hit it, Billy’s would have, to the same effect. The variables ‘Suzy Throws’, ‘Billy Throws’, and ‘Bottle Smashes’ are represented as *ST*, *BT*, and *BS*, respectively. Halpern adds variables *BH* and *SH* for Billy or Suzy’s rock actually hitting the bottle. Whether Suzy’s rock hits depends only on her throwing since she is a good shot, but Billy’s rock will only hit the bottle if he throws and Suzy’s rock doesn’t hit. Halpern refers to this causal model as *M**′ _{RT}*, and it appears in the form of Figure 2.

*SH = ST*

*BH* = *BT* ∧ ¬*SH*

*BS* = *BH* ∨ *SH*

The intuition here is that Suzy should be counted as a cause but not Billy, since her throw actually broke the bottle. Halpern’s modified account, given above, correctly captures this result. To see this, we take the witness in which Suzie does not throw, or *X *=* ST* = 0, and we set *w *equal to the actual values of [*BT*, *BH*]. Although Billy throws, he does not hit, as in the real world. So we obtain *BS* = 0 as desired. Following Halpern’s lead, let us denote this [*ST* = 0, *BT* = 1, *BH* = 0, *BS* = 0]. We will refer to this witness later.

**6 Normality**

Halpern develops various examples that challenge his modified theory, and in response he investigates supplementing the theory with considerations of ‘normality’. The difficulties include the subjectivity of creating causal models and determining which potential witnesses decide the question of actual causality. The introduction of normality also serves the ambition to make finer distinctions, for example, between a cause and a ‘background condition’. In situations where his original definition cannot differentiate among possible causes, normality allows Halpern to select among them in order to gain better agreement with intuition. He offers in sequence several accounts of normality, each designed to deal with a counterexample to the previous one. In the course of these changes, some of the revisions violate the restrictions of the modified account and some lose the basic intuition of the modified account—an actual cause is an event that were it not to happen and things otherwise stay pretty much as they are, the effect would not happen. His discussion is intricate, and inevitably so are our assessments.

Halpern recognizes that ‘normality’ is often vague, or at least ambiguous. A situation would be perfectly normal if all of the variables took on their default values, or those that we expect with no other information. Related to normality is the notion of typicality, which can be understood as statistical prevalence or as a characteristic trait of an object. Moral or legal norms can also be used to decide which value a variable should ‘normally’ take. Halpern uses all of these ideas to help evaluate the normality of a world or variable assignment (p. 78). Beyond these mostly qualitative considerations, Halpern acknowledges that measuring normality cannot be done easily, although in the end he offers a counting principle.

Halpern’s allows that there may be two worlds, *s* and *s′*, whose relative normality is incomparable:

## The fact that

sands′are incomparable does not mean thatsands′are equally normal. We can interpret it as saying that the agent is not prepared to declare eithersors′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. (p. 81)

Even worlds that are intuitively comparable might be so close to each other in normality that the judgement of equi-normality, or not, would be entirely subjective. To avoid this issue, Halpern requires that an inequality in normality should be interpreted as a very large difference: ‘to the extent that we are thinking probabilistically, we should interpret *u > u′* as meaning “*u *is much more probable than *u′* ”’ (p. 97).

Halpern’s normality conditions are constraints on witnesses to an actual cause, that is, on counterfactual situations in which a potential causal variable and a potential effect variable both change value. Since he proposes various combinations of accounts, counting the number of proposed theories of normality is not straightforward, but the following seem basic:

- A witness is a circumstance at least as normal as the actual circumstance.
- A witness is a circumstance of which there is no other more normal for all possible alternative circumstances.
- A witness is a circumstance whose events are the most probable.
- A witness is a circumstance that is produced with the fewest number of interventions on the actual circumstance.

An intervention on a variable changes its value or fixes its value at its actual value, possibly violating some of the constraints, *F*. Values of variables ‘downstream’ from an intervened value (graphically, descendants) take on new values, or remain the same, in accordance with *F*and the values specified by the interventions.

Observe that variants (2), (3), and (4) change the framing from the modified criterion without normality. In the modified criterion, and with normality condition (1), candidate witnesses were assessed for a candidate actual cause without regard to candidate witnesses for other candidate causes. But in (2), (3), and (4), potential witnesses for different potential causes are in competition, and only a witness and actual cause of which there is no more normal witness for any potential cause are allowed.

**6.1 Halpern’s first approach to normality**

Consider the case of a camper dropping a match and starting a forest fire. The fire would not start unless there were oxygen available, but we would not ordinarily cite the presence of oxygen as a cause. Halpern deals with this sort of case by appeal to normality. His first approach to incorporating normality involves replacing the second condition, AC2(a^{m}), with:

## AC2

^{+}(a^{m}): There is a partition ofV(the set of endogenous variables) into two disjoint subsetsZandWwithX⊆Zand a settingx′andwof the variables inXandW, respectively, such that (X=x′, W=w, u) ≥ (u) and (M, u) |= [X←x′,W←w]¬φ.(p. 81)

This seems merely to add to the ‘modified’ version of the condition that ‘we require the witness world to be at least as normal as the actual world’ (p. 81). Halpern defines a world as ‘an assignment of values to the endogenous variables’ in the model, *M* (p. 79).^{[3]} The witness world is the one that satisfies *X *=* x, W *=* w, *and ¬*φ*. In the case of a camper dropping a match and starting a forest fire, we can construct the witness for oxygen as a cause by intervening and eliminating the oxygen, but an Earth now without oxygen is not at least as normal as reality (p. 82). So, by the normality condition, we cannot find that oxygen was a cause of the forest fire. Thus normality helps us limit the scope of our causal attributions and thereby helps us select the more intuitively reasonable cause.

There is a troubling ambiguity in AC2^{+}(a^{m}). In AC2(a^{m}), *w* is *w**, the actual value of *W. *There is no asterisk on *w* in AC2^{+}(a^{m}) and Halpern does not comment on the absence. So we do not know whether in producing a witness for *x* as an actual cause, interventions are allowed that fix other variables at other than their actual values. However, in Example 3.26 (p. 84), Halpern contrasts this normality condition for his modified definition, AC2(a^{m}), with the same normality condition applied to Halpern–Pearl theories, observing that the latter give different results than the modified definition because they do not restrict *w* to actual values. So we take it that in AC2^{+}(a^{m}), *w* should be*w**, the actual values. In later versions of normality Halpern seems to forget this condition.

If we incorporate normality as Halpern initially proposes—the witness case must be at least as normal as the actual case—we lose the result that Suzy is a cause of the bottle breaking and Billy is not. Halpern writes:

## Suppose 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 isnota cause; rather,ST= 1 ∧BT= 1 is a cause, which seems inappropriate. (p. 84)

Halpern’s supposition that it would be abnormal for Suzy not to throw, Billy to throw, and Billy not to hit seems quite reasonable. He has thus far assumed that Billy would hit the bottle had not Suzy hit it first. This relationship between the two throws is, in fact, the essential feature of this example, and the reason it is an interesting scenario to explore in the first place. But Halpern seems to immediately reverse course to preserve his interpretation’s validity. Just lines later he claims, ‘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’ (p. 84). We seem to have a case of the philosophical argument, ‘I need it to be the case that *p*; therefore, p’.

But Halpern is not so sanguine: the first normality condition is consistent with none of the events being actual causes if no candidate witness is as normal as the actual case. And Halpern notes correctly that this first treatment of normality delivers unintuitive results in straightforward scenarios where only the most normal series of events plays out. In the case of a gardener watering her plants, the gardener not caring for her garden is less normal than reality, so that her watering of the plants is not a cause of the plants’ survival. So, to graded causation.

**6.2 Graded causation**

Halpern introduces a new notion, ‘graded causation’. Instead of judging whether a variable is the cause of an outcome in a binary fashion, potential causes can be compared based on the normality of their witness worlds, and the potential cause (or causes) with the most normal witness is selected. Since the world in which the gardener does not water her plants is more normal than, say, the worlds in which soil does not provide nutrients or the sun does not shine, the gardener watering her plants is the cause of their survival. Notice that we are now in a different regime from the plain modified account: in the plain modified account, and in the first version of normality, each potential cause is assessed by itself; with graded causation, potential causes are in competition.

Returning to Billy and Suzy, Halpern’s graded approach also fails. Suzy throws first and breaks the bottle, but the graded approach to normality again fails to deem her a cause. The most normal witness to any cause of the bottle breaking is the circumstance in which neither of them throw; as before, by the original formulation of the scenario, the witness to Suzy’s throw as an actual cause of the bottle breaking is abnormal. We have the same problem as with the first normality proposal: Billy’s and Suzy’s throws are the conjunctive cause. Halpern goes on to introduce a third alternative approach to incorporating normality.

**6.3 Graded causation with probabilities of events**

This approach differs from the first primarily in that Halpern puts the normality ordering on *sets* of contexts instead of worlds. A context ‘can be identified with a complete assignment’ (p. 80), that is, an assignment of values to all of the variables. Here is the formalization, fixing the causal model, *M*, and using a pre-ordering of normality of contexts for *M* and interventions on variables in *M*:

## There is a partition of

V(the set of endogenous variables) into two disjoint subsetsZandWwithX⊆Zand a settingx′andwof the variables inXandW, respectively, such that [[X=x′, W=w, u]] ≥ [[u]] and (M, u) |= [X ←x′,W ←w] ¬φ. (p. 97)

Here [[k]] denotes the event corresponding to *k*, that is, the set of contexts in which *k* is true. This new condition says, ‘the set of worlds where the witness *X *=* x′* ∧ *W *=* w* ∧ ¬*φ* holds is at least as likely as the set of contexts satisfying *X *=* x *∧* φ′* (p. 97).

Informally, Halpern says more on the same page: the context with the maximum probability among worlds where the witness holds must be at least as probable as the actual context (p. 97). He suggests that this proposal be combined with graded normality, which would imply that all actual causes must have equiprobable witnesses greater than the probability of any other candidate witnesses for any other causes. Halpern has previously defined probabilities for contexts given a probabilistic model, that is, one that assigns probabilities to each value of a child variable for each assignment of values to its parent variables. But now he requires a probability over sets of such assignments. Since the contexts are mutually exclusive, the probability of a set of contexts should be the sum of the probabilities of the contexts. Halpern thinks this takes care of Billy and Suzy. Let’s see. We assume that if neither throw then neither hit and the bottle does not break, and if either hit then the bottle breaks.

*ST *= 0 and *BS *= 0 are necessary elements of the witness context for Suzy’s throw to be a cause. If the fixed values, *w*, of the witness must be the actual values of *W*, then we are restricted once more to the context in the sixth row of Table 1, where the bottle shatters, and, as before, we do not have a witness to Suzy as a cause: if Suzy does not throw, the probability is one that the bottle shatters. If *W* is not required to have its actual value, the probability that Suzy does not throw and the bottle does not shatter is *v* + *z*, which must be equal to or greater than *p* for Suzy’s throw to cause the bottle to shatter. It comes out right for Halpern only if probably Billy does not throw or Billy is a lousy shot. The second disjunct rather loses the tension of the example.

Halpern claims his reasoning extends even to a modified scenario in which Billy is a rock-throwing machine: ‘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*’ (p. 98). We are forced to ask how unlikely an event must be for it to cross the threshold into abnormality. Halpern himself admits that the elements of the set violate statistical typicality, and they certainly go against any kind of defaults that a reasonable person might hold. In his own explanation of this approach, he says the set in which the witness holds ‘must be at least as likely’ (p. 97) as the set in which reality holds. His consideration of the rock-throwing machine seems to directly contradict this.

Halpern’s normality considerations so far actually make his definition perform worse, and on an example he himself has used as a key test of his own definition. The Billy and Suzy case comes out neatly on the plain modified account; it comes out as an *ad hoc* mess on the graded probability account, and only gets that far by ignoring the letter of AC2(a^{m}). But if we ignore the letter of AC2(a^{m}) and allow interventions that do not fix variables at their actual values, backup no longer turns out right on the modified account and only turns out right on the graded and graded probability accounts if we arbitrarily add *ad hoc* assumptions about normality or probability.

**6.4 The ABC switch and counting interventions**

Halpern presents another example, the ABC switch, for which he allows that his previous normality considerations fail, and then introduces still another method of considering normality.

The new example involves a system of two switches connected to a light bulb, operated by two individuals, *A* and *B* (see Figure 3). We will again give the situation in full detail, starting with Halpern’s description:* *

AandBeach control a switch. There are wires going from an electricity source to these switches and then continuing on toC.Amust first decide whether to flip his switch left or right, thenBmust decide (knowingA’s choice). The current flows, resulting in a bulb atCturning on, iff both switches are in the same position.Bwants to turn on the bulb, so flips her switch to the same position asAdoes, and the bulb turns on. (p. 100)

Schematically:

*B* = *A *= 1; *C *= 1 if and only if *A* = *B*.

Halpern claims ‘intuition suggests that *A*’s actions should not be viewed as a cause of the *C* bulb being on, whereas *B*’s should’ (p. 100). We cannot achieve this result with the modified HP definition alone. If we hold *B* constant at its actual value (1) and intervene on *A*, we get *A*is a cause. Similarly for *B*. So we obtain both *A* and *B* as actual causes of the outcome. Looking to the original approach to normality, we get no help: ‘Taking *u* to be the context where *A* = 1, both [*A* = 0, *B *= 1, *u*] and [*B *= 0, *u*] seem less normal than *u*, and there seems to be no reason to prefer one to the other, so graded causality does not help in determining a cause’ (p. 101). The third approach also fails: ‘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’ (p. 101). So Halpern concludes that his previous approaches to incorporating normality offer no help in this scenario.

In response to this new difficulty, Halpern introduces a technique for incorporating normality: ‘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’ (p. 102). To measure this change in normality, Halpern relies on the structural equations and the number of interventions required to create the witness context. He does not say whether we are to weight these two kinds of interventions differently or equally:

## Now we take the change from the world where both

A= 1 andB= 1 to the world whereA= 1 andB= 0 to be smaller than the one to the worlds whereA= 0 andB= 1 because the latter change involves changing whatAdoes as well as violating normality (in the sense thatBdoes not act according to the equations), while the former change requires only thatBviolate normality. This gives us a reason to preferB= 1 as a cause. (p. 102)

Halpern’s story is not entirely clear because there is no initial state given for *A*, *B*, and *C*. So suppose that initially *A *and *B* have some third neutral value, there is no current, and the light is off. A is set to 1 and *B* is set to 1 and the light turns on. If *A* had been set to 0 then—keeping the value of *B* at its actual value, 1—in accord with AC2(a^{m}) the light would not be on and we would have a witness to *A* = 1 as a cause. If *B* had been set to 0, then the light would not be on, also in accord with AC2(a^{m}), and we would have a witness to *B* = 1 as a cause. The former requires two interventions, the latter one, so *B* = 1 wins the causality challenge.

We have now lost the entire spirit of the plain modified condition. Not only are candidate causes competing, but application of the very idea of AC2(a^{m}), that other variables may be fixed at their actual values, counts against a witness: each variable fixed at its actual value counts as an intervention, so the more the witness is required to agree with the actual case, the worse the witness!

What does this new account of normality do for previous examples? There are two possible witnesses for Suzy alone being an actual cause: [*ST* = 0, *BT* = 1, *BH* = 0, *BS* = 0] requires two interventions on the actual state of affairs; and [*ST* = 0, *BT* = 0, *BH* = 0, *BS* = 0] also requires two interventions. But the latter is also a witness for Billy being the cause. Either both throws are causes or there is no decision.

If we use only the number of interventions to compare normality, we create all sorts of confusing results. Consider the forest fire example again. Removing the oxygen from the forest or preventing the hiker from dropping a match both only take one intervention. Of course, one of these ‘interventions’ is more plausible than the other, and we might say more probable than the other. So is the idea that we count interventions and then use only the most probable?

**6.5 The first example, with normality**

We consider whether normality helps with the example with which we began. Our model is graphically similar to that of the light switch example but with a different functional relation. We have variables ‘gang leader shoots’ (*GS*), ‘henchman shoots’ (*HS*), ‘enemy dies’ (*ED*). The enemy dies if either the gang leader or the henchman shoots, but the henchman will shoot if and only if the gang leader does (see Figure 4).

*HS *=* GS*

*ED *=* GS *∨* HS*

Using the plain modified definition, the gang leader caused the death of the enemy, as expected. We cannot, on the other hand, create a witness for the henchman being a cause. If we set *HS *= 0, we still get *ED* = 1 unless we also intervene to change *GS *= 1 to* GS *= 0, which the modified *HP* definition does not allow*. *We can construct a witness for both being a cause, but given the minimality condition, we must reject this as a cause. So the henchman cannot even be considered part of the cause. We have seen that probability does not prevent this result. In Section 6.2, Halpern proposes a measure of degree of blame, invoking his probabilistic account. By that measure, the henchman is blameless.

Working through the normality considerations, we get results that are unacceptable or various, although they are a bit difficult because, well, killing people is not normal. With both the initial approach and the graded approach, we still obtain *GS *= 1 is a cause, since, given the moral and legal norms concerning murder, the world in which the gang leader doesn’t shoot is at least as normal as the supposed actual case. We have no clear idea what to do for a witness for the gang leader as a cause. There are worlds in which he and the gang member both shoot and miss, worlds in which the gang member does not shoot but the gang leader does and misses, and the world in which neither of them shoot. We have no idea as to the probabilities. The last—neither of them shoot—is the same witness as for the gang leader. Are they both then causes, or only the gang leader? What happens when we count interventions? An intervention that changes the gang leader’s order and firing, while retaining all of the functional dependencies (*F*)—so that none of the gang, including the leader, fire and the intended victim survives—is a witness to the gang leader as a cause with one intervention. In order to provide a witness for the gang member as cause, both the gang member and the gang leader must be intervened on. But maybe not—maybe the single intervention on the gang leader, which results in the gang member not shooting, counts as a witness for the gang member and the gang leader as the cause, so the gang member is a partial cause. But that would seem to violate the minimality requirement of AC^{m}, and would also violate a strict reading of AC(2^{m}) that requires in a witness an intervention to directly change the actual cause. Halpern’s brief discussion (p. 102) provides us with no guidance.

**7 Results**

If the reader is lost in the tangle of normalities, our sympathies. We have now considered several examples, each with a distinct causal structure and applicable norms. The results often disagree. Table 2 is our summary of the examples and their results for the several accounts of normality. We count a criterion as a success (marked with an O) for a case if its application unequivocally agrees with intuition, as defined within the example.

**8 Conclusion**

There are some serious issues with Halpern’s theory. Although the modified definition is clean and (given the presuppositions of actual causality theories) unambiguous, and captures intuitions in some difficult cases, it fails as a general tool to determine actual causality. Normality, as a qualitative consideration, seems at first to help, and Halpern is able to create a seemingly intuitive formalization of its application to causal models. Indeed, it does put useful restrictions on some examples, restrictions that allow us to differentiate between confusing potential causes. But when this formalization fails to correct some problem cases, normality turns into a quagmire. Halpern responds by creating another proposal and then another and another and another. These successive considerations, with increasing vagueness or ambiguity, fail to correctly interpret the examples that their predecessors were created to address, and, what’s more, all of them fail on the canonical rock-throwing example that the plain modified definition successfully interpreted. On the simple, morally salient example with which we began, the basic modified criterion fails, recourse to probability fails, and supplements with normality conditions fail or are obscure. Worse perhaps is that without notice, the framework of the plain modified account is eroded and, in a central respect, even reversed. What we are left with is an imperfect definition of actual causality, together with a set of *ad hoc *corrections, none of them satisfactory. The discussions of responsibility and blame signal that Halpern intends his theory to be a serious guide in law and moral appraisal. For the reasons indicated, we think that would be a very bad thing.

Notwithstanding our hope that the theory never reaches into the practice of justice, and our conclusion that no coherent, adequate theory of actual causation emerges, a lot can be learned from Halpern’s book. The variety of examples and proposals considered, and the subtleties of their problems, should be a caution to anyone attempting a theory of actual causality. In that service, *Actual Causality *deserves to become a standard but not an icon.

*Ian Rosenberg
Department of Philosophy
Carnegie Mellon University
rosenbergian15@gmail.com*

and

*Clark Glymour
Department of Philosophy*

Carnegie Mellon University

*cg09@andrew.cmu.edu*

**References**

Baez, J. C. and Shulman, M. [unpublished]: ‘Lectures on n-Categories and Cohomology’.

Benini, M., Schenkel, A. and Szabo, R. J. [2015]: ‘Homotopy Colimits and Global Observables in Abelian Gauge Theory’, *Letters in Mathematical Physics*, 105, pp. 1193–222.

Borceux, F. [1994a]: *Handbook of Categorical Algebra, 1: Basic Category Theory*, Cambridge: Cambridge University Press.

Borceux, F. [1994b]: *Handbook of Categorical Algebra, 2: Categories and Structures,* Cambridge: Cambridge University Press.

Borceux, F. [1994c]: *Handbook of Categorical Algebra, 3: Categories of Sheaves*, Cambridge: Cambridge University Press.

Dougherty, J. [2017]: ‘Sameness and Separability in Gauge Theories’, *Philosophy of Science*, 84, pp. 1189–201.

Lurie, J. [2009]: *Higher Topos Theory*, Princeton, NJ: Princeton University Press.

MacLane, S. [1971]: *Categories for the Working Mathematician*, New York: Springer.

Nguyen, J., Teh, N. J. and Wells, L. [forthcoming]: ‘Why Surplus Structure Is Not Superfluous’, *British Journal for the Philosophy of Science*.

nLab [2018]: ‘Principle of Equivalence‘.

Schreiber, U. and Shulman, M. [unpublished]: ‘Quantum Gauge Field Theory in Cohesive Homotopy Type Theory’.

**Notes**

^{1} A word on terminology. Halpern sometimes treats *u* as an assignment of a value to an exogenous variable not represented in his or our causal graphs. This value then determines the values of all zero-indegree variables in the graph, which in turn, through *F*, determine the values of all other variables in the graph. Halpern omits this variable for situations that can be adequately described within the graph. Thus when he writes ‘endogenous’, what is intended may vary depending on the use of this variable. Where *u *is included, he means all variables represented in the graph. Where it is excluded, he follows the more usual terminology, and refers to only those variables in the graph that have positive indegree. His use of ‘exogenous’ varies similarly, meaning either just *u *or those variables in the graph with zero indegree, respectively. Technically, the difference makes no difference, but a careful reading of each scenario must be given in order to correctly interpret Halpern’s meaning.

^{2} We do not understand one of Halpern’s remarks about his probabilistic version. Turning the joint probabilities of events contingent on values of their direct graphical parents into measures on deterministic models is straightforward, but Halpern (p. 47) says the resulting probabilities are *not* to be understood as probabilities conditional on models in which the parents have such-and-such values, but rather are to be understood as ‘interventions’, writing: ‘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 Suzy’s rock hitting is .9’. He justifies this remark with the following: ‘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’. The first quoted sentence is odd, since in the example the probabilities are the same in either case, intervention or conditioning. The second quoted sentence is of course true; we are perplexed as to what it has to do with Halpern’s claim that the probability for any event is to be understood as an intervention.

^{3} Reminder: Technically, Halpern introduces a single unrepresented variable that he regards as the only ‘exogenous variable’ and whose values specify the values of the substantive variables that would otherwise be exogenous. Thus his ‘endogenous variables’ are all the variables represented in the causal graph, including the variables of zero indegree (those that would customarily be deemed ‘exogenous’).