A Causal Analysis of Harm
Abstract
As autonomous systems rapidly become ubiquitous, there is a growing need for a legal and regulatory framework that addresses when and how such a system harms someone.
There have been several attempts within the philosophy literature to deﬁne harm, but none of them has proven capable of dealing with the many examples that have been presented, leading some to suggest that the notion of harm should be abandoned and “replaced by more well-behaved notions”.
As harm is generally something that is caused, most of these deﬁnitions have involved causality at some level.
Yet surprisingly, none of them makes use of causal models and the deﬁnitions of actual causality that they can express.
In this paper we formally deﬁne a qualitative notion of harm that uses causal models and is based on a well-known deﬁnition of actual causality [14].
The key features of our deﬁnition are that it is based on contrastive causation and uses a default utility to which the utility of actual outcomes is compared.
We show that our deﬁnition is able to handle the examples from the literature, and illustrate its importance for reasoning about situations involving autonomous systems.
Introduction
The notion that one should not cause harm is a central tenet in many religions; it is enshrined in the medical profession’s Hippocratic Oath, which states explicitly “I will do no harm or injustice to [my patients]” [22] it is also a critical element in the law.
Not surprisingly, there have been many attempts in the philosophy literature to deﬁne harm.
Motivated by the observation that we speak of “causing harm”, most of these have involved causality at some level.
All these attempts have encountered difﬁculties.
Indeed, Bradley [4] says:
Unfortunately, when we look at attempts to explain the nature of harm, we ﬁnd a mess.
The most widely discussed account, the comparative account, faces counterexamples that seem fatal.
But no alternative account has gained any currency.
My diagnosis is that the notion of harm is a
Frankensteinian jumble ...
It should be replaced by other more well-behaved notions. The situation has not improved much since Bradley’s paper (see, e.g., recent accounts like [5, 8]). Yet the legal and regulatory aspects of harm are becoming particularly important now, as autonomous systems become increasingly more prevalent. In fact, the new proposal for Europe’s AI act [7] contains over 25 references to “harm” or “harmful”, saying such things as “...it is appropriate to classify [AI systems] as high-risk if, in the light of their intended purpose, they pose a high risk of harm to the health and safety or the fundamental rights of persons ...”
[7, Proposal preamble, clause (32)].
Moreover, the European Commission recognized that if harm is to play such a crucial role, it must be deﬁned carefully, saying “Stakeholders also highlighted that ...
it is important to deﬁne ...‘harm’ [7, Part 2, Section 3.1].
Legislative bodies in the UK are also discussing the question of harm and who caused harm in the case of accidents involving autonomous vehicles.
The Law
Commission of England and Wales and the Scottish Law Commission are recommending that drivers of self-driving cars should not be legally responsible for crashes; rather, the onus should lie with the manufacturer [6].
In particular, if there is harm then this is caused by the manufacturers.
The manufacturers translate this recommendation to a standard according to which the driver does not even have to pay attention while at the wheel.
If a complex situation arises on the road requiring the driver’s attention, the car will notify the driver, giving them 10 seconds to take control.
If the driver does not react in time, the car will ﬂash emergency lights, slow down, and eventually stop [20].
Consider the following example (to which we return later).
Example 1 (Autonomous Car)
An autonomous car detects an unexpected stationary car in front of it on a highway.
It could alert the driver Bob, who would then have to react within 10 seconds.
However, 10 seconds is too long: the car will crash into the stationary car within 8 seconds.
The autonomous car’s algorithm directs it to crash into the safety fence on the side of the highway, injuring Bob.
Bob claims that he was harmed by the car.
Moreover, he also claims that, if alerted, he would have been able to ﬁnd a better solution that would not have resulted in his being injured (e.g., swerving into the incoming trafﬁc then back to his own lane after passing the stationary car).
We assume that if the autonomous car had done nothing and collided with the stationary car, both drivers would have been injured much more severely.
(cid:117)(cid:116)
While the causal model depicting this story is fairly straightforward, the decision on whether harm was caused to Bob, and if yes, who or what caused the harm, is far less clear.
Indeed, the philosophy literature seems to suggest that trying to determine this systematically is a lost cause.
But as this example illustrates, the stakes of having a well-deﬁned notion of harm have become much higher with the advent of automated decision-making.
In contrast to human agents, such systems do not have an informal understanding of harm that informs their actions; so we need a formal deﬁnition.
Situations like that described in Example 1 are bound to arise frequently in the interaction of autonomous systems with human users, in a variety of domains.
We brieﬂy outline two of those.
Imagine a UAV used by the military has to decide whether or not it should bomb a suspected enemy encampment.
The problem is that the target is not clearly identiﬁed, because there are two camps close to each other: one consisting of civilian refugees, another consisting of a rebel group that is about to launch a deadly attack on the refugee camp, killing all of its inhabitants.
The UAV’s decision is based only on the expected utility of the refugees, and therefore it bombs the camp.
Tragically, as it turns out, the camp was that of the refugees.
Here we have the intuition that the UAV harmed these refugees, despite the fact that both actions would have led to all the refugees being killed.
Examples in which one event (the bombing) preempts another event (the attack) from causing an outcome are known as Late Preemption examples in the causality literature; we discuss them later in the paper.
In the healthcare domain, autonomous systems are used for, among other things, classifying MRI brain images suspected of containing a tumor.
If an image is classiﬁed as having a tumor, the system decides whether to recommend a surgery.
While the overall accuracy of the system is superior to that of humans, in some instances the system overlooks an operable tumor.
Imagine a patient who has such a tumor and dies from brain cancer as the result of not undergoing surgery, leading to a dispute between the patient’s family and the hospital regarding whether the patient was harmed.
Even if both parties agree that the patient would probably have been alive if the diagnosis had been performed by a human, the hospital might claim that using the system is the optimal policy, and therefore one should compare the actual outcome only to those that could have occurred under the policy.
Fortunately, the formal tools at our disposal to develop a formal notion of harm have also improved over the past few years; we take full advantage of these developments in this paper.
Concretely, we provide a formal deﬁnition of harm that we believe deals with all the concerns that have been raised, seems to match our intuitions well, and connects closely to work on decision theory and utility.
Here we brieﬂy give a high-level overview of the key features of our approach and how they deal with the problems raised in the earlier papers.
There is one set of problems that arise from using counterfactuals that also arise with causality, and can be dealt with using the by-now standard approaches in deﬁning causality.
For example, Carlson,
Johansson, and Risberg [5] raise a number of problems with deﬁning harm causally that are solved by simply applying the deﬁnition of actual causality given by Halpern [13, 14].
The issue of whether failing to take an action can be viewed as causing harm (e.g., can failing to water a neighbor’s plants after promising to do so be viewed as causing harm) can also be dealt with by using the standard deﬁnition of causality (which allows lack of an action to be a cause).
We remark that Richens, Beard, and Thompson [24] (RBT from now on) also recently observed that using causality appropriately could deal with some of the problems raised in the harm literature. They also give a formal deﬁnition of harm that uses causal models, but it does not make use of a sophisticated deﬁnition of actual causality such as the one given by Halpern [13, 14].
(See Sectionfor a comparison of our approach to theirs and more discussion of this issue.)
RBT focus on the more quantitative, probabilistic aspects of harm.
We also believe that a quantitative account is extremely important; we offer such an account in [2]. Conceptually though, the qualitative notion comes ﬁrst: only after establishing whether or not there was harm does it make sense to ask how much harm occurred. Indeed, our quantitative account generalizes the qualitative account we develop here in several ways (see Section 6).
In any case, just applying the deﬁnition of causality does not deal with all problems.
The other key step that we take is to assume that there exists a default utility.
Roughly speaking, we deﬁne an event to cause harm whenever it causes the utility of the outcome to be lower than the default utility.
The default may be context-dependent, and there may be disagreement about what the default should be.
We view that as a feature of our deﬁnition.
For example, we can capture the fact that people disagree about whether a doctor euthanizing a patient in great pain causes harm by taking it to be a disagreement about what the appropriate default should be.
Likewise, the dispute between the family and the hospital described above can be modeled as a disagreement about the right default.
Moreover, by explicitly bringing utility into the picture, we can connect issues that that have been discussed at length regarding utility (e.g., what the appropriate discount factor to apply to the utility of future generations is) to issues of harm.
Causal Models and Actual Causality
We start with a review of causal models [16], since they play a critical role in our deﬁnition of harm.
The material in this section is largely taken from [14].
We assume that the world is described in terms of variables and their values.
Some variables may have a causal inﬂuence on others.
This inﬂuence is modeled by a set of structural equations .
It is 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.
The structural equations describe how these values are determined.
Formally, a causal model M is a pair ( S , F ) , where S is a signature , which explicitly lists the endogenous and exogenous variables and characterizes their possible values, and F deﬁnes a set of (modiﬁable) structural equations , relating the values of the variables.
A signature S is a tuple ( U , V , R ) , where U is a set of exogenous variables, V is a set of endogenous variables, and R associates with every variable Y ∈
U ∪ V a nonempty set R ( Y ) of possible values for Y (i.e., the set of values over which Y ranges ).
For simplicity, we assume here that V is ﬁnite, as is R ( Y ) for every endogenous variable Y ∈ V .
F associates with each endogenous variable X ∈ V a function denoted F X (i.e., F X = F ( X ) )
such that F X : ( × U ∈U R ( U )) ×
( × Y ∈V−{ X } R ( Y ))
→ R ( X ) .
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 U ∪ V .
The structural equations deﬁne what happens in the presence of external interventions.
Setting the value of some set (cid:126)X of variables to (cid:126)x in a causal model
M = ( S , F ) results in a new causal model, denoted M (cid:126)X ← (cid:126)x , which is identical to M , except that the equations for (cid:126)X in F are replaced by (cid:126)X = (cid:126)x . Note that the causal models we consider here are deterministic. In general, one can also consider probabilistic causal models . A probabilistic causal model is a tuple M =
( S , F , Pr) , where ( S , F ) is a causal model, and Pr is a probability on contexts.
Deterministic models sufﬁce for offering a qualitative notion of harm, but we use probabilistic causal models for our quantitative generalization [2].
The dependencies between variables in a causal model M =
(( U , V , R ) , F ) can be described using a causal network (or causal graph ), whose nodes are labeled by the endogenous and exogenous variables in M , with one node for each variable in U ∪ V .
The roots of the graph are (labeled by) the exogenous variables.
There is a directed edge from variable X to Y if Y depends on X ; this is the case if there is some setting of all the variables in U ∪ V 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 (cid:126)z of the variables other than X and Y and values x and x (cid:48) of X such that F Y ( x,(cid:126)z ) (cid:54) =
F Y ( x (cid:48) ,(cid:126)z ) .
A causal model M is recursive (or acyclic ) if its causal graph is acyclic.
It should be clear that if M is an acyclic causal model, then given a context , that is, a setting (cid:126)u for the exogenous variables in U , the values of all the other variables are determined (i.e., there is a unique solution to all the equations).
We can determine these values by starting at the top of the graph and working our way down.
In this paper, following the literature, we restrict to recursive models.
We call a pair ( M, (cid:126)u ) consisting of a causal model M and a context (cid:126)u a (causal) setting .
A causal formula ψ is true or false in a setting.
We write ( M, (cid:126)u ) |
= ψ if the causal formula ψ is true in the setting ( M, (cid:126)u ) .
The | = relation is deﬁned inductively.
( M, (cid:126)u ) | =
X = x if the variable X has value x in the unique (since we are dealing with acyclic models) solution to the equations in M in context (cid:126)u (that is, the unique vector of values for the exogenous variables that simultaneously satisﬁes all equations in M with the variables in U set to (cid:126)u ).
Finally, ( M, (cid:126)u ) | =
[ (cid:126)Y ← (cid:126)y ] ϕ if ( M (cid:126)Y ← (cid:126)y , (cid:126)u ) | = ϕ .
A standard use of causal models is to deﬁne actual causation : that is, what it means for some particular event that occurred to cause another particular event.
There have been a number of deﬁnitions of actual causation given for acyclic models (e.g., [1, 10, 11, 16, 14, 17, 18, 29, 30]).
Although most of what we say in the remainder of the paper applies without change to other deﬁnitions of actual causality in causal models, for deﬁniteness, we focus here on what has been called the modiﬁed Halpern-Pearl deﬁnition [12, 14], which we brieﬂy review.
(See [14] for more intuition and motivation.)
The events that can be causes are arbitrary conjunctions of primitive events (formulas of the form
X = x ); the events that can be caused are arbitrary Boolean combinations of primitive events.
To relate the deﬁnition of causality to the (contrastive) deﬁnition of harm, we give a contrastive variant of the deﬁnition of actual causality; rather than deﬁning what it means for (cid:126)X = (cid:126)x to be an (actual) cause of φ , we deﬁne what it means for (cid:126)X = (cid:126)x rather than (cid:126)X = (cid:126)x (cid:48) to be a cause of φ rather than φ (cid:48) .
Deﬁnition 1 (cid:126)X = (cid:126)x rather than (cid:126)X = (cid:126)x (cid:48) is an actual cause of φ rather than φ (cid:48) in ( M, (cid:126)u ) if the following three conditions hold:
[Equation] ) | =
( (cid:126)X = (cid:126)x ) ∧ φ .
AC2.
There is a set (cid:126)W of variables in V and a setting (cid:126)w of the variables in (cid:126)W such that ( M, (cid:126)u ) | =
(cid:126)W = (cid:126)w and ( M, (cid:126)u ) | =
[ (cid:126)X ← (cid:126)x (cid:48) , (cid:126)W ← (cid:126)w ] φ (cid:48) , where φ (cid:48) ⇒ ¬ φ is valid.
AC3.
(cid:126)X is minimal; there is no strict subset (cid:126)X (cid:48)(cid:48) of (cid:126)X such that (cid:126)X (cid:48)(cid:48) =
(cid:126)x (cid:48)(cid:48) can replace (cid:126)X = (cid:126)x (cid:48) , where (cid:126)x (cid:48)(cid:48) is the restriction of (cid:126)x to the variables in (cid:126)X (cid:48)(cid:48) .
AC1 just says that (cid:126) X = (cid:126)x cannot be considered a cause of φ unless both (cid:126) X = (cid:126)x and φ actually happen.
AC3 is a minimality condition, which says that a cause has no irrelevant conjuncts.
AC captures the standard but-for condition ( (cid:126)X = (cid:126)x rather than (cid:126)X = (cid:126)x (cid:48) is a cause of φ if, had (cid:126)X beem (cid:126)x (cid:48) rather than (cid:126)x , φ would not have happened) but allows us to apply it while keeping ﬁxed some variables to the value that they had in the actual setting ( M, (cid:126)u ) .
If (cid:126)X = (cid:126)x is an actual cause of φ and there are two or more conjuncts in (cid:126)X = (cid:126)x , one of which is X = x , then X
= x is part of a cause of φ .
In the special case that (cid:126)W = ∅ , we get the standard but-for deﬁnition of causality: if (cid:126)X = (cid:126)x had not occurred (because (cid:126)X was (cid:126)x (cid:48) instead) φ would not have occurred (because it would have been φ (cid:48) ).
The reader can easily verify that (cid:126)X = (cid:126)x is an actual cause of φ according to the standard noncontrastive deﬁnition [14] iff there exist (cid:126)x (cid:48) and φ (cid:48) such that (cid:126)X = (cid:126)x rather than (cid:126)X = (cid:126)x (cid:48) is an actual cause of φ rather than φ (cid:48) according to our contrastive deﬁnition.
Deﬁning Harm
Many deﬁnitions of harm have been considered in the literature.
The ones most relevant to us are those involving causality and counterfactuals, which have been split into two groups, called the causal account of harm and the counterfactual comparative account account of harm .
Carlson et al.
[5] discuss many variants of the causal account; they all have the following structure:
An event e harms an agent ag
if and only if there is a state of affairs s such that (i) e causes s to obtain, and (ii) s is a harm for ag .
The deﬁnitions differ in how they interpret the second clause.
We note that although these deﬁnitions use the word “cause”, it is never deﬁned formally.
“Harm” is also not always deﬁned, although in some cases the second clause is replaced by phrases that are intended to be easier to interpret.
For example, what Suits [27] calls the causal-intrinsic badness account takes s to be a harm for ag if s is “intrinsically bad” for ag .
The causal-counterfactual account (see, e.g., [9, 21, 28, 9]) has the same structure; the ﬁrst clause is the same, but now the second clause is replaced by a phrase involving counterfactuals.
In its simplest version, this can be formulated as follows: s is a harm for ag if and only if ag would have been better off had s not obtained.
Even closer to our account is what has been called the contrastive causal-counterfactual account .
For example, Bontly [3] proposed the following:
An event e harms a person ag if and only if there is a state of affairs s and a contrast state of affairs s (cid:48) such that (i) e rather than a contrast event e (cid:48) causes s rather than s (cid:48) to obtain, and (ii) ag is worse off in s than in s (cid:48) .
Our formal deﬁnition of harm is quite close to Bontly’s.
We replace “state of affairs” by “outcomes”, and associate with each outcome a utility.
This is essentially the standard model in decision theory, where actions map states to outcomes, which have associated utilities.
Besides allowing us to connect our view to the standard decision-theoretic view (see, e.g., [23, 25]), this choice means that we can beneﬁt from all the work done on utility by decision theorists.
To deﬁne harm formally in our framework, we need to both extend and specialize causal models: We specialize causal models by assuming that they include a special endogenous variable O for outcome .
The various values of the outcome value will be assigned a utility.
We often think of an action as affecting many variables, whose values together constitute the outcome.
The decision to “package up” all these variables into a single variable O here is deliberate; we do not want to consider the causal impact of some variables that make up the outcome on other variables that make up the outcome (and so do not want to allow interventions on individual variables that make up an outcome; we allow only interventions on complete outcomes).
On the other hand, we extend causal models by assigning a utility value to outcomes (i.e., on values of the outcome variable), and by having a default utility.
We thus take a causal utility model to be one of the form M =
(( U , V , R ) , F , u , d ) , where ( U , V , R ) , F ) is a causal model one of whose endogenous variables is O , u : R ( O ) → [0 , 1] is a utility function on outcomes (for simplicity, we assume that utilities are normalized so that the best utility is 1 and the worst utility is 0), and d ∈ [0 , 1] is a default utility.
2 As before, we call a pair ( M, (cid:126)u ) , where now M is a causal utility model and (cid:126)u ∈ R ( U ) , a setting.
Just like causality, we deﬁne harm relative to a setting.
Whether or not an event (cid:126) X = (cid:126)x harms an agent in a given setting will depend very much on the choice of utility function and default value.
Thus, to justify a particular ascription of harm, we will have to justify both these choices.
In the examples we consider, we typically view the utility function to be ag ’s utility function, but we are not committed to this choice (e.g., when deciding whether harm is caused by a parent not giving a child ice cream, we may use the parent’s deﬁnition of utility, rather than the child’s one).
The choice of a default value is more complicated, and will be discussed when we get to examples; for the deﬁnition itself, we assume that we are just given the model, including utility function and default value.
The second clause of our deﬁnition is a formalization of Bontly’s deﬁnition, using the deﬁnition of causality given in Section 2, where the events for us, as in standard causal models, have the form (cid:126)X = (cid:126)x and the alternative events have the form (cid:126)X = (cid:126)x (cid:48) , and they cause outcomes O =
o and O = o (cid:48) , respectively. Unlike Bontly’s deﬁnition (and others), not only do we require that ag is worse off in outcome o
(the analogue of state of affairs s ) than in outcome o (cid:48)
(where “worse off” is formalized by taking the utility to be lower), we also require the utility of o to be lower than the default utility .
There is also an issue as to whether we consider there to be harm if (cid:126)X = (cid:126)x (cid:48) results in a worse outcome than o .
Since intuitions may differ here, we formalize this requirement in a third clause, H3, and use it to distinguish between harm and strict harm.
We will see the effects of our modiﬁcations to Bontly’s deﬁnition when we consider examples in Section 4.
Deﬁnition 2 (cid:126)X = (cid:126)x harms ag in ( M, (cid:126)u ) , where M = (( U , V , R ) , F , u , d ) , if there exist o ∈ R ( O ) and (cid:126)x (cid:48) ∈ R ( (cid:126)X ) such that
H1.
u ( O = o ) <
d ; and H2.
there exists o (cid:48)
∈ R ( O ) such that (cid:126)X = (cid:126)x rather than (cid:126)X = (cid:126)x (cid:48) causes O
= o rather than O = o (cid:48) and u ( O = o ) <
u ( O = o (cid:48) ) .
(cid:126)X = (cid:126)x strictly harms ag in ( M, (cid:126)u ) if, in addition, H3.
u ( O = o ) ≤
u ( O = o (cid:48)(cid:48) ) for the
unique o (cid:48)(cid:48) ∈
R ( O ) such that ( M, (cid:126)u ) | =
[ (cid:126)X ← (cid:126)x (cid:48) ]
( O = o (cid:48)(cid:48) ) .
In the special case where Deﬁnition 2 is satisﬁed for some value o (cid:48) appearing in H2 such that u ( O = o ) <
d ≤ u ( O = o (cid:48) ) , we say that (cid:126)X = (cid:126)x causes ag ’s utility to be lower than the default .
It is important to point out that it is quite rare for harm and strict harm to come apart. For one, it requires causation and but-for causation to come apart (otherwise o (cid:48) = o (cid:48)(cid:48) ).
In addition, it requires
O to have at least three values (otherwise again o (cid:48)
=
o (cid:48)(cid:48) ).
Lastly, even if these two conditions are met, we also need that u ( O = o (cid:48)(cid:48) ) < u ( O = o ) <
u
( O = o (cid:48) ) .
We discuss one example in the supplementary material (see Section A.1) in which these conditions are met. As with most concepts in actual causality, deciding whether harm occurred is intractable. Indeed, it is easy to see that it is at least as hard as causality, which is DP-complete [13]. However, this is unlikely to be a problem in practice, since we expect that the causal models that arise when we want to deal with harm will have few variables, which take on few possible values (or will involve many individuals that can all be described with by a small causal model), so we can decide harm by simply checking all possibilities.
It is useful to compare our deﬁnition with the counterfactual comparative account of harm.
Here it is, translated into our notation:
Deﬁnition 3 (cid:126)X = (cid:126)x counterfactually harms ag in ( M, (cid:126)u ) , where M =
(( U , V , R ) , F , u , d ) if there exist o, o (cid:48) ∈
R
( O ) and (cid:126)x (cid:48)
∈ R ( (cid:126)X ) such that
C1. ( M, (cid:126)u ) | = (cid:126)X = (cid:126)x ∧ O = o ; C2. ( M, (cid:126)u ) | =
[ (cid:126)X ← (cid:126)x (cid:48) ] ( O = o (cid:48) ) ; C3.
u ( O = o ) <
u ( O = o (cid:48) ) . That is, (cid:126)X = (cid:126)x counterfactually harms ag if, for some x (cid:48) and o (cid:48) , (cid:126)X = (cid:126)x is what actually happens (C1),
O = o (cid:48) would have happened had (cid:126)X been set to (cid:126)x (cid:48) (C2), and ag gets higher utility from o (cid:48) than from o (C3).
C1 and C2 together are equivalent to AC1 and AC2 in the special case that (cid:126)W = ∅ .
That is, C1 and C2 essentially amount to but-for causality.
C3 differs from our conditions by not taking into account the default value.
Note that Deﬁnition 3 has no analogue of AC3, but all the examples focus on cases where (cid:126)X is actually a singleton, so AC3 is trivially satisﬁed.
The key point from our perspective is that the counterfactual comparative account considers only but-for causality, and does not consider a default value.
The examples in the next section show how critical these distinctions are.
As mentioned earlier, RBT recently developed a formal account of harm using causal models.
While their account is probabilistic and quantitative, we can consider the special case where everything is deterministic and qualitative.
When we do this, their account reduces to a strengthening of Deﬁnition 3 that brings it somewhat closer to our account: they also suggest using defaults, but have default actions rather than default utilities.
In their version of Deﬁnition 3, (cid:126)X is taken to be the variable representing the action(s) performed and x (cid:48) is the default action.
In order to deal with the limitations of but-for causality, RBT offer a more general account (see their Appendix A) that uses path-speciﬁc causality, instead of actual causation. This makes their account different from ours in some signiﬁcant respects; see Section 5.
Examples
We now analyse several examples to illustrate how our deﬁnition handles the most prominent issues that have been raised in the literature on harm.
Bradley [4, p. 398] identiﬁes two such issues that strike him “as very serious”, namely the problem of preemption, and the problem of distinguishing harm from merely failing to beneﬁt.
These problems therefore serve as a good starting point.
4.1 Preemption
To anyone familiar with the literature on actual causation what follows will not come as a surprise.
Lewis used examples of preemption to argue that there can be causation without counterfactual dependence (i.e., we need to go beyond but-for causality); this conclusion is now universally accepted.
Essentially the same examples show up in the literature on harm: cases of preemption show that an event can cause harm even though the agent’s well-being does not counterfactually depend on it.
Thus, the counterfactual comparative account of harm fails for the same reason it failed for causality.
The good news is that the formal deﬁnition of causation (by design) handles problems like preemption well; moreover, the solution carries over directly to our deﬁnition of harm.
The following vignette is due to Bradley [4], but issues of preemption show up in many papers on causality [1, 11, 16, 14, 18, 29]; all can be dealt with essentially the same way.
Example 2 (Late Preemption) Suppose Batman drops dead of a heart attack.
A millisecond after his death, his body is hit by a ﬂaming cannonball.
The cannonball would have killed Batman if he had still been alive.
So the counterfactual account entails that the heart attack was not harmful to
Batman.
It didn’t make things go worse for him.
But intuitively, the heart attack was harmful.
The fact that he would have been harmed by the ﬂaming cannonball anyway does not seem relevant to whether the heart attack was actually harmful.
In terms of the formal deﬁnition, we take H to represent whether Batman has a heart attack ( H =
if he doesn’t ; H = 1 if he does), C to represent if Batman is hit by a cannonball, and D to represent whether Batman dies.
Let (cid:126)u be the context where H = 1 .
Even without describing the equations, according to the story, ( M, (cid:126)u ) | =
H = 1 ∧ D =
∧
[ H = 0]( D = 1) : Batman has a heart attack and he dies, but he would have died even if he did not have a heart attack (since he would have been hit by the cannon ball).
Thus, C3 does not hold, since o = o (cid:48) ; the outcome is the same whether or not
Batman has a heart attack.
The standard causal account handles this problem by introducing a new variable K , standing for “Batman is killed by the cannonball” to take into account the temporal asymmetry between death by heart attack and death by cannonball.
We could also deal with this asymmetry by having “time-stamped” variables that talk about when Batman is alive. For more details on incorporating temporal information by using time-stamped variables, see [14].)
The causal model has the following equations: D = S ∨ K (i.e., D = 1 if either S = 1 or K = 1 : Batman dies if his heart stops or the canonball kills him); S = H (Batman’s heart stops if he has a heart attack); and K =
¬ S ∧ C (Batman is killed by the canonball if the canonball hits him and his heart is still beating).
We now get that Batman’s heart attack rather than its absence is a cause of him being alive rather than dead.
Clearly ( M, (cid:126)u ) | =
H =
∧ D
= 1 .
If we ﬁx K =
(its actual value, since the cannonball in fact does not kill Batman; he is already dead by the time the cannonball hits him), then we have that ( M, (cid:126)u ) | =
[ H = 0 , K = 0]( D = 0) , so AC2 holds.
Thus, the causal part of H2 holds.
(See [14,
Example 2.3.3] for a detailed discussion of an isomorphic example.)
If we further assume, quite reasonably, that Batman prefers being alive to being dead (so the utility of being alive is higher than that of being dead) and that the default utility is that of him being alive, then
H1 and H2 hold.
Thus, our deﬁnition of harm avoids the counterintuitive conclusion by observing that Batman’s heart attack caused his death, thereby causing the utility to be lower than the default.
(cid:117)(cid:116)
Our analysis of preemption is indicative of the more general point that many of the issues plaguing the literature on harm can be resolved by making use of causal models and the deﬁnitions of causation that they allow.
Causal models allow a more precise and explicit representation of the relevant causal structure, thereby forcing a modeler to make modeling choices that resolve the inherent ambiguity that comes with an informal and underspeciﬁed causal scenario.
Obviously such modeling choices can be the subject of debate (see [15] for a discussion of these modeling choices).
The point is not that using causal models by itself determines a unique verdict on whether harm has occurred, but rather that such a debate cannot even be had without being explicit about the underlying causal structure.
4.2 Failing to Beneﬁt
One of the central challenges in deﬁning harm is to distinguish it from merely failing to beneﬁt.
Although most authors deﬁne beneﬁt simply as the symmetric counterpart to harm, we do not believe that this is always appropriate; we return to this issue in [2] where we consider more quantitative notions of harm.
But for the current discussion, we can set this issue aside: what matters is that merely failing to make someone better off does not in itself sufﬁce to say that there was harm.
Carlson et al.
[5] present the following well-known scenario to illustrate the point.
Example 3 (Golf Clubs) Batman contemplates giving a set of golf clubs to Robin, but eventually decides to keep them.
If he had not decided to keep them, he would have given the clubs to Robin, which would have made Robin better off.
By keeping the golf clubs, Batman clearly failed to make Robin better off.
The counterfactual account considers any such failure to result in harm.
Indeed, it is easy to see that C1–C3 hold.
If we take GGC to represent whether Batman gives the golf clubs to Robin ( GGC = 1 if he does; GGC =if he doesn’t) and the outcome O to represent whether Robin gets the golf clubs ( O = 1 if he does;
O = 0 if he doesn’t), then GGC = 0 is a but-for cause of GGC = 0 , so C1 and C2 hold.
If we further assume that Robin’s utility of getting the golf clubs is higher than his utility of not getting them, then C3 holds.
Yet it sounds counterintuitive to claim that Batman harmed Robin on this occasion.
(cid:117)(cid:116)
Although H2 holds in our account of harm (for the same reason that C1–C3 hold), we avoid the counterintuitive conclusion by assuming that the default utility is u ( O = 0) , so H1 does not hold.
This seems to us reasonable; there is nothing in the story that suggests that Robin is entitled to expect golf clubs.
On the other hand, if we learn that Batman is a professional golfer, Robin has been his reliable caddy for many years, and that at the start of every past season Batman has purchased a set of golf clubs for Robin, then it sounds quite plausible that the default is for Robin to receive a set of golf clubs.
With this default, H1 does hold, and our deﬁnition concludes that Robin has been harmed.
Thus our account can offer different verdicts depending on the choice of default utility.
As we said in the introduction, we view this ﬂexibility as a feature of our account.
This point is highlighted in the following, arguably more realistic, scenario.
(RBT make exactly the same point as we do when they analyze such examples [24, p. 15].)
Example
(Tip) Batman contemplates giving a tip to his waiter, but eventually decides to keep the extra money for himself.
If he had not decided to keep it, he would have given it to the waiter, which would have made the waiter better off.
To those living in the US, it does not at all sound counterintuitive to claim that Batman harmed the waiter, for his income substantially depends on receiving tips and he almost always does receive a tip.
Indeed, if we take the default utility to be that of receiving a tip, then in this example, the waiter is harmed by Batman not giving a tip.
By way of contrast, in countries in Europe where a tip would not be expected, it seems to us reasonable to take the default utility to be that of not receiving a tip.
In this case, the waiter would not be harmed.
(cid:117)(cid:116)
Examples 3 and 4 are isomorphic as far as the causal structure goes; we can take the utilities to be the same as well.
This means that we need additional structure to be able to claim that the agent is harmed in one case and not the other.
That additional structure in our framework, which we would argue is quite natural, is the choice of default utility.
Note that neither scenario explicitly mentions what the default utility should be.
We thus need to rely on further background information to make a case for a particular choice.
There can be many factors that go into determining a good default.
We therefore do not give a general recipe for doing so.
Indeed, as we pointed out in the introduction with the euthanasia example, reasonable people can disagree about the appropriate default (and thus reach different conclusions regarding harm).
4.3 Preventing Worse
There exist situations in which the actual event rather than an alternative event causes a bad outcome rather than a good outcome, but the alternative results in an even worse outcome.
Because of the latter, we do not consider these situations to be cases of strict harm, due to condition H3 in Deﬁnition 2.
From the perspective of the car manufacturer, this is precisely what is going on in our starting Example 1, but Bob might disagree.
We now take a closer look at this example to bring out the conﬂicting perspectives.
Example 5 (Autonomous Car) Let O be a three-valued variable capturing the outcome for Bob, with the utility deﬁned as equal to the value of O .
O = 0 .
5 stands for the injury resulting from crashing into the safety fence, and a potentially more severe injury resulting from crashing into the stationary car is captured by O = 0 .
Bob not being injured is O = 1 .
Recall that the system has the built-in standard that the driver’s reaction time is 10 seconds, which is too long to avoid colliding into the stationary car.
Imagine the manufacturer implemented this standard by restricting the system’s actions in such cases to two possibilities: do not intervene ( F = 0 ) or drive into the fence ( F = 1 ).
This means that the causal structure is very similar to our Late Preemption example (Example 2), for hitting the fence preempts the collision with the stationary car.
We therefore add a variable to capture the asymmetry between hitting the fence and hitting the stationary car: FH and CH respectively.
The equation for O is then such that O = 1 if
FH = CH
= 0 , O = 0 .
5 if FH = 1 , and O =
if CH = 1 and FH = 0 . [Figure]
As the autonomous car drives towards the fence only because there is a stationary car, the equation for F is F = C (where C represents the presence of the car).
The fact that hitting the fence prevents hitting the car is captured in the equation for CH : CH = C ∧ ¬ FH .
Lastly, we have FH = F .
The context is such that F = 1 and C = 1 , and thus FH = 1 , O = 0 . 5 , and CH = 0 .
Did the system harm Bob?
Carlson et al.
[5] answer this in the negative for an example that is essentially the same as this one (see their “Many Threats” example), and use this verdict to argue against the causal-counterfactual account, which reaches the opposite verdict.
According to them, the contrastive causal-counterfactual account does not reach this verdict because there is no contrastive causation here, but as they do not give a deﬁnition of causation, it is impossible to reconstruct how they arrive at this verdict.
In any case, what matters for our purposes is that our deﬁnition of causation does consider the system’s hitting the fence rather than not intervening to cause Bob being mildly injured rather than not being injured at all.
To see why, observe that taking (cid:126)W to be CH , we get that
F = 1 rather than F = 0 causes O = 0 .
5 rather than O = 1 : ( M, (cid:126)u ) | =
[ F ← 0 , CH ← 0] O = 1 .
Therefore, if we assume that the default utility is that of Bob not being injured, conditions H1 and H2 are satisﬁed and there is harm.
Notice though that F = 1 rather than F = 0 is also a but-for cause of O = 0 .
5 rather than O = 0 , that is, Bob’s being mildly injured rather than severely injured counterfactually depends on the system’s action.
This is where condition H3 kicks in: it ensures that we do not consider there to be strict harm caused if the alternative would have resulted in an even worse outcome. Thus, the car manufacturer could make the case that although their policy harmed
Bob, it was justiﬁed in doing so.
More generally, it is an easy consequence of our deﬁnitions that in cases where (cid:126)X = (cid:126)x rather than (cid:126)X = (cid:126)x (cid:48) causes harm but not strict harm, the alternative event would have resulted in strict harm, i.e., (cid:126)X = (cid:126)x (cid:48) rather than (cid:126)X = (cid:126)x would have caused strict harm.
As a result, it is sensible in such cases for someone to argue that they were justiﬁed in causing harm, as the alternative would have been worse.
Bob, on the other hand, believes he has been strictly harmed, because he claims that he could have prevented the collision if he had been alerted.
This disagreement can be captured formally by stating that Bob is using a three-valued variable F instead of a binary one, where the third option ( F = 2 ) corresponds to alerting Bob.
Incorporating this variable into the model (and assuming that Bob is correct regarding his driving skills) we would again get that F = 1 rather than F = 2 causes O = {+
+}
.rather than O = 1 , but with the important distinction that H3 is satisﬁed for these contrast values and thus the system’s action does strictly harm Bob.
Our analysis does not resolve the conﬂict (and it is not meant to do so), instead it allows for a precise formulation of the source of the disagreement.
(cid:117)(cid:116)
In the supplementary material, we describe four more examples illustrating multiple contrasts for the outcome, the role of the choice and the range of variables and the choice of default, and our rationale for considering a contrastive deﬁnition, rather than causal-counterfactual one.  Comparison to RBT In this section, we do a more careful comparison of our approach and that of RBT.
RBT focus on choices made by an agent, where these are choices of what action to take, and assume that there is a default action, to which they compare the choice made by the agent.
It follows easily from RBT’s deﬁnition that if the agent performs the default action, there is no harm.
Yet there are many instances in which doing what is morally preferable causes harm, albeit accidentally.
Simply imagine a doctor prescribing medication to a patient, and the patient unfortunately suffering a very rare allergic reaction to the medication, where the reaction is far worse than the initial condition that the patient had. Then clearly the doctor harmed the patient. The most obvious choice of default action here is the actual action (and, in fact, RBT themselves mention following “clinical guidelines” as an example of a default action in their Appendix C). But this means that according to RBT’s deﬁnition there would not be harm here.
Although we use a default utility, there is no need for this default utility to be the utility of a default action, so we do not have this problem.
Another signiﬁcant difference between our approach and that of RBT is that, although RBT use causal models, unlike us, they do not use a sophisticated deﬁnition of actual causality such as the one given by Halpern [13, 14].
In their Deﬁnition 3, RBT consider but-for causality.
Not surprisingly, this will not sufﬁce to deal with problematic examples where the more general notion of causality is needed (see, e.g., Example 2).
In their Deﬁnition 9, they generalize Deﬁnition 3 to allow for path-dependent causality.
They do not explain how they choose which paths to consider, but in both Example 2 and the corresponding example of late preemption they consider, by choosing the appropriate paths, they can simulate the effects of AC2.
(Speciﬁcally, they can simulate the effect of choosing a set (cid:126)W of variables and ﬁxing them to their actual values.)
As a consequence, they get the same results as those obtained by Halpern’s deﬁnition of actual causality.
It is not clear whether this will always be the case.
More importantly, the ability to determine harm relative to some choice of paths gives the modeler a signiﬁcant extra degree of freedom to tailor the results obtained.
We believe that if paths are going to be used, there needs to be a more principled analysis of how to go about choosing them.
Conclusion
We have deﬁned a qualitative notion of harm, and shown that it deals well with the many problematic examples in the philosophy literature.
We believe that our deﬁnition will be widely applicable in the regulatory frameworks that we expect to be designed soon in order to deal with autonomous systems.
Of course, having a qualitative notion of harm is only a ﬁrst step.
For practical applications, we often need to quantify harm; for example, we may want to choose the least harmful of a set of possible interventions. As we said, we develop a quantitative notion of harm in [2].
While one could just deﬁne a quantitive notion that considers the difference between the utility of the actual outcome and the default utility (this is essentially what RBT do), we believe that the actual problem is more nuanced.
For example, even if we can agree on the degree of harm to an individual, if there are many people involved and there is a probability of each one being harmed, should we just sum the individual harms, weighted by the probability?
We argue that this is not always appropriate, and discuss alternatives, drawing on work from the decision-theory literature.
Acknowledgements
The authors would like to thank the NeurIPS reviewers for their detailed comments and Jonathan
Richens for a fruitful discussion of a preliminary version of this paper. Sander Beckers was supported by the German Research Foundation (DFG) under Germany’s Excellence Strategy – EXC number 2064/1 – Project number 390727645, and by the Alexander von Humboldt Foundation. Hana Chockler was supported in part by the UKRI Trust-worthy Autonomous Systems Hub (EP/V00784X/1) and
the UKRI Strategic Priorities Fund to the UKRI Research Node on Trustworthy Autonomous Systems Governance and Regulation (EP/V026607/1).
Joe Halpern was supported in part by NSF grant
IIS-1703846, ARO grant W911NF-22-1-0061, and MURI grant W911NF-19-1-0217.
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Here we provide some additional details on three topics. First, we illustrate the role played by H3 in our deﬁnition of harm. Second, we discuss in more detail how our approach differs from that of RBT. Third, we present four more examples that illustrate how our deﬁnition handles issues which have been discussed in the harm literature.
As we mentioned, H3 is intended to capture the intuition that there is no strict harm if the alternative would have resulted in an even worse outcome. For example, following the reasoning of the car manufacturer in Example 5, the system’s decision to drive into the fence rather than doing nothing is not strictly harmful because Bob would have suffered even worse injuries had the system done nothing. Since H1 and H2 are satisﬁed for this particular contrastive event, our deﬁnition would reach the opposite verdict if it weren’t for H3.
Note that the counterfactual comparative account (Deﬁnition 3) also says that there is no harm: the alternative event under consideration would have given a worse outcome, so that C3 is not satisﬁed, and therefore there is no harm.
Considering H gives more insight into the differences between the counterfactual account and ours.
Suppose that we consider some contrastive event (cid:126)X = (cid:126)x (cid:48) such that ( M, (cid:126)u ) | = (cid:126)X = (cid:126)x ∧ O = o and ( M, (cid:126)u ) | =
[ (cid:126)X ← (cid:126)x (cid:48) ]
( O = o (cid:48)(cid:48) ) , so C1 and C2 hold, and the ﬁrst half of H2 holds if o (cid:48)(cid:48) (cid:54) =
o : (cid:126)X = (cid:126)x rather than (cid:126)X = (cid:126)x (cid:48) causes O
= o rather than
O =
o (cid:48)(cid:48) .
H3 plays no role if H1 is not satisﬁed, so for simplicity, suppose that H1 also holds.
Then it is easy to see that whenever u ( O = o ) (cid:54) =
u ( O =
o (cid:48)(cid:48) )
, our deﬁnition of strict harm gives the same verdict as the counterfactual comparative deﬁnition for this particular contrast (i.e., for this choice of (cid:126)x (cid:48) ): if u ( O = o ) <
u ( O =
o (cid:48)(cid:48) ) ,
then o (cid:48)(cid:48) (cid:54)
=
o , so H holds, as do H3 and C3; it follows that according to both deﬁnitions (cid:126)X = (cid:126)x harms the agent.
On the other hand, if u ( O = o ) > u ( O =
o (cid:48)(cid:48) )
, then neither C3 nor H3 hold (for this choice of (cid:126)x (cid:48) ).
What happens if u ( O = o )
=
u ( O =
o (cid:48)(cid:48) ) ?
This can happen for two reasons:
there is no but-for causation, that is, o =
o (cid:48)(cid:48) ; 2.
there is but-for causation but the counterfactual outcome O
=
o (cid:48)(cid:48) happens to have utility identical to the actual outcome.
Thus, roughly speaking (and ignoring the key role played by the default utility), our deﬁnition differs from the counterfactual comparative account only if (cid:126)X = (cid:126)x rather than (cid:126)X = (cid:126)x (cid:48)(cid:48) is not a but-for cause of the actual utility: changing (cid:126)x into (cid:126)x (cid:48)(cid:48) does not change the agent’s utility.
Examples in which the ﬁrst reason is relevant are widespread and crucial to our analysis, for those are precisely the examples in which actual causation (Deﬁnition 1) and but-for causation come apart.
Our Late Preemption example (Example 2) offers one illustration, the literature on actual causation contains many more.
An example where the second reason is relevant involves a more subtle way in which but-for causation comes apart from actual causation.
Consider a “Sophie’s choice” like setting:
An agent must choose whether X = 1 or X = 2 .
There are two children, who will either live or die depending on the choice: if X = i is chosen, then child i lives ( L i = 1 ) and child 3 − i dies ( L 3 − i = 0 ).
The possible outcomes are that both children live ( o 11 ), just child 1 lives ( o 10 ), just child 2 lives ( o 01 ), and neither child lives ( o 00 ), where d = u
( O = o 11 ) > u ( O = o 10 ) =
u ( O = o 01 ) > u ( O = o 00 ) .
In fact, X = 1 is chosen, so we get but-for causality, but switching from X = 1 to X = 2 gives an outcome of equal utility.
However, if we hold L 1 = 1 ﬁxed (which we can do in our framework to show causality) and switch to X = 2 , then we get the outcome O
= o 11 .
Thus, in our framework
X = 1 strictly harms the agent; in the causal counterfactual framework, it does not.
This emphasizes the point we (and RBT) made that one set of problems that occur in deﬁning harm is identical to the type of problems that occur in deﬁning causation, and can be solved in the same way.
A.2 More Examples
We present four further examples that illustrate how our approach deals with the difﬁculties of deﬁning harm that have been highlighted in the literature.
The cases in Sections 4.1 and 4.2 all involved a binary outcome; there were only two relevant events that could occur.
Carlson et al.
[5] discuss cases that involve more than two possible events in order to argue against existing causal accounts.
The following example forms one instance of their argument.
Example 6 (Tear Gas)
The Joker sprays tear gas in exactly one of Batman’s eyes.
If he had not done that, he would have sprayed tear gas in both of Batman’s eyes, which would have made Batman even worse off.
One of the alternatives available to the Joker, however, was to simply leave Batman alone.
Intuitively here Joker harms Batman when he sprays him.
To argue that the “incorrect” answer is obtained by the deﬁnition of harm they focus on, Carlson et al.
consider a speciﬁc alternative event, namely, that Joker sprays tear gas in both of Batman’s eyes, while observing that other alternatives (like leaving Batman alone) are also available.
Rather than existentially quantifying over (cid:126)x (cid:48) , as we have done, (both in Deﬁnition 2 and the gloss of the counterfactual harm deﬁnition given in
Deﬁnition 3), they take a version of counterfactual harm where (cid:126)X = (cid:126)x (cid:48) is taken to be the closest alternative to (cid:126)X = (cid:126)x (according to some implicit, but unspeciﬁed, notion of closeness).
Both our deﬁnition of harm and our gloss of the counterfactual deﬁnition (with the obvious assumptions about utility, and taking the default utility to be that of Batman being unharmed for our deﬁnition) agree that Joker did harm Batman in this case, as we would expect.
In this example, there are three events of interest (Joker sprays tear gas in one eye; Joker sprays tear gas in both eyes; Joker doesn’t spray tear gas at all).
We can model this using a variable TG that takes on three possible values (say, 0 , 1 , and 2 ).
According to Deﬁnition 3, as long as one of them leads to a better utility than what actually happened, there was harm.
But as the golf clubs example shows, this conclusion is not always justiﬁed; in general, we need to take defaults into account. [Equation] Now we present an example, due to Shiffrin [26], that illustrates the role of both the choice of the range of variables in the causal model and the choice of default. Example 7 Betty is drowning in a fast-moving river.
Veronica rescues her by grabbing her arm and pulling her out, accidentally fracturing Betty’s humerus.
Did Veronica’s rescue harm Betty?
Shiffrin claims it does because Veronica could have pulled her out without breaking her arm.
Indeed, Klocksiem [19], in his analysis, points out that “it seems possible to rescue someone from drowning without breaking her arm”.
The ﬁrst step in our analysis is to decide whether we should allow this possibility.
That is, suppose that we have a variable P that describes how and whether Veronica pulls out Betty.
{+We can take P = 
+}
if Veronica does not pull out
Betty, P = 1 if she pulls her out by grabbing (and breaking) her arm.
The modeler must then decide whether to allow P to take a value, say 2, where P = 2 if Veronica rescues Betty in such a way that
Betty’s arm is not broken.
Reasonable people might disagree whether such an event is possible.
First suppose we decide that P can take only values 0 and 1.
Then the possible outcomes are that Betty drowns ( O = 0 ) or Betty is saved ( O = 1 ).
In this model, any utility function that makes the utility of drowning worse than that of being saved would result in Veronica’s rescue not harming Betty.
Now suppose that we allow P = 2 .
Then we would take O = 1 to represent Betty being saved but her arm being broken, and O = 2 to represent Betty being saved without her arm being broken.
In that case, whether Veronica harms Betty depends on the default.
If we take the default utility to be u ( O = 2) then Veronica does cause Betty harm, while if we take it to be u ( O = 0) , she does not.
Note that the latter choice is quite defensible.
Given Betty’s situation, making it out alive in whatever way possible would presumably be all that matters to her.
(cid:117)(cid:116)
This example clearly shows that to apply our framework in practice, it is important to have some guidelines on what count as a reasonable choice, both in the choice of variables and values and the choice of default value.
As we mentioned in the introduction, Halpern and Hitchock [15] discuss this issue in the context of causal models; to the best of our knowledge, this issue has not been discussed in the context of default values.
While this issue is beyond the scope of the current paper, we should make clear that we would not, in general, expect there to be a unique “correct” model.
As we have said repeatedly, reasonable people can disagree about these choices.
There is one ﬁnal issue we would like to address: why we consider a contrastive deﬁnition rather than just giving a deﬁnition in the spirit of the causal-counterfactual account.
Deﬁnition 2 explicitly invokes a contrastive outcome o (cid:48) whose utility is better than that of the actual outcome.
We could have instead just deﬁned harm as the result of causing an outcome whose utility is worse than the default.
One reason why we did not do so is that the default utility is not always achievable, and it would be counterintuitive to say that the agent was harmed if the outcome has a utility lower than the default, even though it is the best possible outcome.
For example, there are diseases for which a surgery can only provide a temporary cure; in this case, a successful surgery gives the patient a temporary relief, and an unsuccessful surgery results in the patient’s death.
While the default utility for the patient, as for all people, is to be alive and healthy, saying that a successful surgery harmed the patient seems wrong.
In fact, deﬁning harm as the result of causing an outcome with the utility worse than the default provides counterintuitive results even when the default utility is achievable, as the following example demonstrates.
Example 8 (Pills) Consider the following vignette, again taken from [5] (where it is presented as a problem for both the causal-counterfactual and contrastive causal-counterfactual accounts): Barney suffers from a painful disease. On Monday, he can either take Pill A or not. On Tuesday, he will have another choice, between taking Pill B or not.
Barney believes that he will be completely cured just in case he takes only Pill A, and partially cured just in case he takes both pills.
Accordingly, he takes Pill A on Monday and does not take Pill B on Tuesday ... He is, however, misinformed about the effects of the pills. Taking only Pill A causes his disease to be merely partially cured.
If he had taken both pills, he would have been completely cured.
Had he not taken Pill A on Monday, on the other hand, nothing he could have done later would have produced even a partial cure.
To capture this in our framework, let O be a three-valued variable that captures Barney’s health:
O = 2 if he is fully cured, O = 1 if he is partially cured, and O = 0 if he is not cured at all.
A and
B capture whether or not Barney takes pills A and B respectively.
{+The equation for O is then such that O = 
+}
if A = B = 1 , O = 1 if A = 1 and B = 0 , and O = 0 otherwise.
As Barney considers taking pill B only if he fails to take pill A, the equation for B is B = ¬ A .
The context is such that
A = 1 ; therefore, B = 0 and O = 1 .
Carlson et al.
claim that taking the pill does not harm Barney; we agree.
Yet it easy to see that
A = 1 does cause O = 1 .
Indeed it is a but-for cause: had Barney not taken the pill, O would have been 0 .
It is easy to see why this is a problem for the causal-counterfactual account: Barney would have been better off if O = 1 had not obtained; speciﬁcally, he would be better off if O had been(although this is not the outcome that results when changing A to 0 and therefore is not a problem for the counterfactual comparative account).
Carlson et al.
also view it as a problem for the contrastive causal-counterfactual account, because in applying it, they compare O = 1 to the outcome O = 2 , (which, again, is not the outcome that obtains by switching A to 0 ), since they take the closest world to the one where Barney takes just one pill to be the world where he takes both pills.
Our deﬁnition avoids this problem.
We do not consider the “closest” state of affairs.
Rather, we compare O = 1 to the outcome O = 0 caused by switching to A = 0 .
O = 0 has utility worse than that of the outcome obtained from A = 1 , so there is no harm according to our deﬁnition, for what we view as the “right” reasons.
Assuming that the default utility is u ( O = 2) , A = 1 does cause an outcome whose utility is worse than the default and therefore a non-contrastive version of our deﬁnition would not have given the desired result.
(cid:117)(cid:116)