doctor i treats Billy. We can capture this in an extended causal model where the world where no doctor is assigned to Billy and no doctor treats him has rank 0; the 100 worlds where exactly one doctor is assigned to Billy, and that doctor treats him, have rank 1; the 100 worlds where exactly one doctor is assigned to Billy and no one treats him have rank 2; and the 100 × 99 worlds where exactly one doctor is assigned to Billy but some doctor treats him have rank 3.

(The ranking given to other worlds is irrelevant.) In this extended model, in the context where doctor i is assigned to Billy but no one treats him, i is the cause of Billy's sickness
(the world where i treats Billy has lower rank than the world where i is assigned to Billy but no one treats him), but no other doctor is a cause of Billy's sickness. Moreover, in the context where i is assigned to Billy and treats him, then i is the cause of Billy's recovery (for AC2(a), consider the world where no doctor is assigned to Billy and none treat him).

I consider one more example here, due to Hitchcock
[2007], that illustrates the interplay between normality and causality.

Example 4.2: Assistant Bodyguard puts a harmless antidote in Victim's coffee. Buddy then poisons the coffee, using a type of poison that is normally lethal, but is countered by the antidote. Buddy would not have poisoned the coffee if Assistant had not administered the antidote first. (Buddy and Assistant do not really want to harm Victim. They just want to help Assistant get a promotion by making it look like he foiled an assassination attempt.) Victim drinks the coffee and survives.

Is Assistant's adding the antidote a cause of Victim's survival? Using the preliminary HP definition, it is; if Assistant does not add the antidote, Victim survives. However, using an extended causal model with the normality assumptions implied by the story, it is not. Specifically, suppose we assume that if Assistant does not add the antidote, then Buddy does not normally add poison. (Buddy, after all, is normally a law-abiding citizen.) In the corresponding extended causal model, the world where Buddy poisons the coffee and Assistant does not add the Antidote has a higher rank (i.e., is less normal than) the world where Buddy poisons the coffee and Assistant adds the antidote. This is all we need to know about the ranking function to conclude that adding the antidote is not a cause. By way of contrast, if Buddy were a more typical assassin, with reasonable normality assumptions, the world where he puts in the poison and Assistant puts in the antidote would be less normal than then one Buddy puts in the poison and Assistant does not put in the antidote, so Assistant would be a cause of Victim being a alive.

Interestingly, Hitchcock captures this story using structural equations that also make Assistant putting in the antidote a *cause* of Buddy putting in the poison. This is the device used to distinguish this situation from one where Buddy is actually means Victim to die (in which case Buddy would presumably have put in the poison even if Assistant had not added the antidote). However, it is not clear that people would agree that Assistant putting in the antidote really caused Buddy to add the poison; rather, it set up a circumstance where Buddy was willing to put it in. I would argue that this is better captured by using the normality statement
"If Assistant does not put in the antidote, then Buddy does not normally add poison." As this example shows, there is a nontrivial interplay between statements of causality and statements of normality.

I leave it to the reader to check that reasonable assumptions about typicality can also be used to deal with the other problematic examples for the HP definition that have been pointed out in the literature, such as Larry the Loanshark [Halpern and Pearl 2005, Example 5.2] and Hall's [2007]
watching police example. (The family sleeps peacefully through the night. Are the watching police a cause? After all, if there had been thieves, the police would have nabbed them, and without the police, the family's peace would have been disturbed.)
This is not the first attempt to modify structural equations to deal with defaults; Hitchcock [2007] and Hall [2007] also consider this issue. Neither adds any extra machinery such as ranking functions, but both assume that there is an implicitly understood notion of normality. Roughly speaking, Hitchcock [2007] can be understood as giving constraints on models that guarantee that the answer obtained using the preliminary HP definition agrees with the answer obtained using the definition in extended causal models. I do not compare my suggestion to that of Hall [2007], since, as Hitchcock [2008] points out, there are a number of serious problems with Hall's approach. It is worth noting that both Hall and Hitchcock assume that a variable has a "normal" or "default" setting; any other setting is abnormal. However, it is easy to construct examples where what counts as normal depends on the context. For example, it is normal for doctor i to treat Billy if i is assigned to Billy; otherwise it is not.

## 5 The Ness Approach

In this section I provide a sufficient condition to guarantee that a single conjunct is a cause. Doing so has the added benefit of providing a careful comparison of the NESS test and the HP approach. Wright does not provide a mathematical formalization of the NESS test; what I give here is my understanding of it.

A is a cause of B according to the NESS test if there exists a set S = {A1**, . . . , A**k} of events, each of which actually occurred, where A = A1, S is sufficient for for B, and S − {A1} is not sufficient for B. Thus, A is an element of a sufficient condition for B, namely S, and is a necessary element of that set, because any subset of {A1**, . . . , A**k} that does not include A is not sufficient for B.

4 The NESS test, as stated, seems intuitive and simple.

Moreover, it deals well with many examples. However, although the NESS test looks quite formal, it lacks a definition of what it means for a set S of events to be *sufficient* for B
to occur. As I now show, such a definition is sorely needed.

4The NESS test is much in the spirit of Mackie's INUS test
[Mackie 1965], according to which A is a cause of B if A is an insufficient but necessary part of a condition which is unnecessary but sufficient for B. However, a comparison of the two approaches is beyond the scope of this paper.