A new principle of plural harm

According to the counterfactual comparative account (CCA), an event harms a person if and only if it makes things worse for her. Cases of overdetermination and preemption pose a serious challenge to CCA since, in these cases, although it is evident that people are harmed, there are no individual events that harm them. However, while there are no individual events that make people worse off in cases of overdetermination and preemption, there are pluralities of events that do so. In light of this feature of these cases, several philosophers have suggested that it is these pluralities that do the harming. In this article, I will argue that although the most prominent accounts of plural harm – e.g., Neil Feit’s account – fare better than one might initially think, they fail to deal adequately with a number of intriguing cases of preemption first introduced by Alastair Norcross. I will also introduce a new view on plural harm and argue that this view, apart from dealing with the cases of overdetermination and preemption that the other accounts of plural harm handle, also deals adequately with Norcross’s cases.


Introduction
The most widely discussed account of harm is arguably the counterfactual comparative account of harm: CCA: An event E harms S if and only if S would have been better off if E had not occurred. 1 Apart from being simple, CCA has several other attractive features. It captures the intuitively appealing idea that harming someone is to make a difference for the worse for that person. Furthermore, CCA has the intuitively correct implications in standard cases of harm. For example, CCA explains why a person's being stabbed normally harms her: She would have been better off if she had not been stabbed. CCA also handles the harm of prevention. If a person is prevented from receiving necessary medical care, CCA plausibly implies that she is harmed since she would have been better off if she had received that care. One of the most attractive features of CCA is perhaps that it explains the harm of death. If a person's death deprives her of a valuable future, CCA plausibly implies that the person's death harms her since it would have been better for the person if her death had not occurred. Importantly, CCA also explains why death might not always be harmful. If a person's death prevents her from experiencing a future full of suffering, CCA credibly implies that the person's death does not harm her since it would not have been better for her if her death had not occurred.
However, CCA has been subjected to several difficult objections. The most pressing one might well be that CCA does not have the intuitively correct implications in cases of overdetermination and preemption. Consider the following overdetermination case, for example: Shooters. X and Y simultaneously [and independently] shoot and kill me. Either shot by itself would have killed (Parfit, 1984, 70). 2 According to CCA, neither X's act nor Y's act harms me since neither X's act nor Y's act makes things worse for me. If X had not shot me, Y would nevertheless have done so, and the other way around. However, if neither X's act nor Y's act harms me, then how am I harmed? There has to be some event or events that harm me. I am clearly harmed and just as there cannot be an effect without a cause, there cannot be a harm without a harmer.
Several philosophers have developed views based on the attractive idea that the plurality consisting of X's act and Y's act harms me, that is to say, that X's act and Y's act together harm me. 3 For example, Derek Parfit has suggested that X and Y together harm me ''because if both had acted differently, I would not have died' ' (1984, 71). However, as many commentators have noted, Parfit's view is 1 CCA and the other principles of harm discussed in this article are principles of overall harm, which means that ''better off'' in CCA and the other principles should be read as ''on balance better off.'' 2 Parfit calls this case ''Case One. '' inadequate. To deal with the difficulties associated with Parfit's view, Neil Feit has revised and developed Parfit's view in various ways (2015). However, although Feit's proposal handles several of the difficulties confronting Parfit's view, it fails (or so I shall argue) to deal adequately with a couple of intriguing cases first introduced by Alastair Norcross, where one of these cases involves a counterfactually dependent event and the other involves an unavailable event (2005). In this article, I shall develop a new view that apart from handling cases like Shooters, handles Norcross's cases too. This is my plan: In Sects. 2-3, I discuss Parfit's view. I suggest that when Parfit's view is suitably revised, it is better than one might think. However, I accept that we might want to revise it further in light of its implications in some challenging cases. In Sect. 4, I introduce Feit's proposal. I note some minor problems with its formulation and show that Feit's proposal handles the cases that motivated its introduction. In Sects. 5-8, I discuss Alastair Norcross's cases and I also introduce my view (Sect. 6). I argue that although Feit's view is better in some respects than what Feit himself seems to note, in contrast to my view, it still fails all things considered. During the course of the discussion, I also reply to some potential counterexamples to my view (Sect. 7).

Parfit's view
As I noted, to handle Shooters Parfit suggests that X and Y together harm me. According to Parfit, they do so ''because if both had acted differently, I would not have died' ' (1984, 71). 4 According to Parfit, then, some people together harm someone if they together make things worse for that person, or more precisely, if it would have been better for the person had they all acted differently. However, Parfit immediately goes on to note that this cannot be all there is to it: Dancer. This case is exactly like Shooters with the addition that Fred Astaire dances nearby.
As Parfit notes, it is ''true that if X, Y and Fred Astaire all had acted differently, I would not have been harmed' ' (1984, 71). However, according to Parfit, ''this does not make Fred Astaire a member of a group that together harm me' ' (1984, 71). To avoid this implication, Parfit adds the following claim: (C8) When some group together harm … other people, this group is the smallest group of whom it is true that, if they had all acted differently, the other people would not have been harmed (1984,71).
Parfit's principle is about groups of people and how they act. However, CCA and other contemporary views on harm are about events. In light of this consideration and others, Neil Feit has formulated Parfit's view in terms of pluralities of events.
According to Feit, in a case like Shooters, it is the plurality of X's shooting and Y's shooting that harms me. Importantly, according to Feit, by speaking of the plurality of these events, he does not intend to introduce any metaphysically contentious entities into the world -as for example, sets of events or big spread-out events -he is rather making a plural reference to them (2015, 370). Furthermore, although it might seem natural to think that for something to count as a plurality it must consist of at least two events, Feit stipulates that a plurality may consist of a single event (2015,370). Feit ends up formulating the following Parfitian principle of plural harm: PH: E harms S if and only if E is the smallest plurality of events such that if none of them had occurred, S would have been better off (2015,371).
A note of clarification is called for. The harming plurality obviously cannot be the smallest plurality of all pluralities that make things worse for a person. If it were, I would not be harmed by the plurality consisting of X's act and Y's act since I have been made worse off by smaller pluralities than that one. In fact, PH would imply that there is no plurality that harms me since I have been made worse off by several pluralities consisting of just one event. In light of these considerations, Jens Johansson and Olle Risberg have suggested that we read ''smallest plurality'' as ''smallest plurality that includes some event in E' ' (2019, 356).
As I noted, Feit stipulates that a plurality may consist of a single event. This means that if an event harms someone according to CCA, it also harms that person according to PH (and vice versa). A plurality of one event is obviously the smallest plurality that includes that event. In short, PH entails CCA (but not the other way around since CCA only applies to pluralities of single events). 5 PH has the intuitively correct implications in the cases I have discussed. It implies, both in Shooters and Dancer, that the plurality consisting of X's shooting and Y's shooting harms me. In both cases, this is the smallest plurality including some of their acts that makes me worse off. Furthermore, in Dancer, PH implies that the plurality consisting of X's shooting, Y's shooting, and Fred's dancing does not harm me, since the plurality consisting of X's shooting and Y's shooting is smaller.

The ''No-Smallest-Plurality'' problem
Quite a few philosophers have noted that there are pertinent cases where there is no smallest plurality (Feit, 2015;Norcross, 2005;Parfit, 1984;Sandberg, 2011;Talbot, 2018). For example, consider the following case: Overkill. If two people or more stand on a platform, this will kill a hundred miners. If one person stands on the platform, the miners won't be killed. X, Y, and Z stand on this platform. 6 According to PH, the plurality consisting of X's act, Y's act, and Z's act does not harm the miners since there are three smaller pluralities including some of their acts that make the miners worse off: the plurality consisting of X's act and Y's act; the plurality consisting of X's act and Z's act; the plurality consisting of Y's act and Z's act. For example, the miners would not have been killed if neither X nor Y had stood on the platform since at least two people have to stand on the platform for the miners to be killed.
Moreover, according to PH, none of these three smaller pluralities harms the miners either. For example, consider the plurality consisting of X's act and Y's act. This plurality does not harm the miners since it is not the smallest plurality that includes some of their acts (i.e., that includes either X's act or Y's act) that make the miners worse off. The plurality consisting of X's act and Z's act and the plurality consisting of Y's act and Z's act are equally small and each of them makes the hundred miners worse off as well. The miners would not have been killed if neither X nor Z had stood on the platform; nor would the miners have been killed if neither Y nor Z had stood on the platform. Again, at least two persons have to stand on the platform for the miners to be killed. Consequently, according to PH, there is no plurality that harms the miners that involves the acts of X, Y and Z. However, the miners are clearly harmed by what they do.
There is a fairly obvious way around this difficulty. For example, consider again the plurality consisting of X's act and Y's act. Although it is true that this plurality is not the smallest plurality including any of their acts that makes the miners worse off, there is no smaller plurality including any of their acts that does; rather, there are two equally large: i.e., the plurality consisting of X's act and Z's act and the plurality consisting of Y's act and Z's act. In light of these observations, we may revise PH as follows: PH*: A plurality of events E harms S if and only if S would have been better off if none of the events of E had occurred, and there is no smaller plurality that includes some event in E such that if none of the events of that plurality had occurred, S would have been better off.
It is worth mentioning that PH* handles an objection that Brian Talbot has raised against similar views. Talbot notes that ''[o]ne can amend Parfit's view so that it does not require that there be a unique smallest necessary group'' (2017, 2256, fn. 21). Talbot goes on to claim that ''[c]ounterexamples to these variants can be generated by changing the ability of different actors to cause harm'' (2017, 2256, fn. 21). To illustrate this point, he imagines a case where two machines, machine 1 and 2, together with a person, Barry, shoot and kill a person, Abel, and where machine ''1's shot would be sufficient to kill Abel by itself, but it would take 2 and Barry shooting together to kill Abel'' (2017, 2256, fn. 21). According to Talbot, ''[i]t's still intuitively wrong for Barry to shoot Abel … although he is not part of the smallest group necessary to kill Abel'' (2017, 2256, fn. 21). 7 However, according to PH*, Barry's act belongs to a plurality that harms Abel. Consider the plurality consisting of machine 1's shooting Abel and Barry's shooting Abel. This plurality makes Abel worse off and there is no smaller plurality including any of these two acts that does. If neither of these events had occurred Abel would not have been killed. Furthermore, if machine 1 had not shot Abel, he would still have been killed by Barry and machine 2; and if Barry had not shot Abel, he would still have been killed by machine 1. So, Talbot's example does not pose a challenge to PH*.
It might be argued that PH* nevertheless is false. Although PH* implies that there are several pluralities that harm the miners in Overkill, as we saw, it also implies that the plurality consisting of X's act, Y's act, and Z's act does not harm them. But it might seem clear that this plurality harms the miners. For example, according to Feit ''In this case, we should want to say that the harm consists in the [three] people together standing on the platform' ' (2015, 375).
Moreover, PH* might not seem to have the intuitively correct implications in cases like the following case: Two Buttons. There are two buttons. I die if both buttons are pressed, but not if only one or neither of the buttons is pressed. X and Y press a button each, resulting in my death.
According to PH*, the plurality consisting of X's pressing a button and Y's pressing a button does not harm me since there are smaller pluralities that include one of these events that make me worse off. For instance, I would have been better off if Y had refrained from pressing a button. However, one might think that it is obvious that the plurality consisting of X's pressing a button and Y's pressing a button harms me. 8 In my view, a proponent of PH* might reasonably explain away our intuitions in Overkill and Two Buttons. For example, in Two Buttons, it is not implausible that the reason we are inclined to think that the plurality consisting of X's act and Y's act harms me is that we conflate this plurality with the individual acts of this plurality (i.e., X's act and Y's act, respectively), which clearly do harm me. However, although it might seem natural to think that if each of two pluralities exemplifies the same harm related property -i.e., if each of two pluralities harms someone, benefits someone, neither harms nor benefits someone, and so on -the two of them together also exemplify that property, this clearly does not follow. For example, in Shooters, according to PH*, while neither X's shooting me nor Y's shooting me harms me, the plurality consisting of their acts does. Of course, this attempt of explaining away the pertinent intuitions would be futile if PH* were to imply that there are no Footnote 7 continued a smallest group necessary to kill Abel.'' (It is true that Barry is not part of a smallest group necessary to kill Abel since the group consisting of machine 1 is smaller.) 8 Two Buttons has the same structure as a case originally presented by Johansson and Risberg (2019, 360). According to Johansson and Risberg, if you are inclined to say that some events together harm someone, it is particularly natural to do so in cases like Two Buttons. But as they go on to note, principles like PH* do not allow us to do that. harming pluralities in Overkill and Two Buttons (since there clearly are), or if PH* were to imply that events that clearly belong to a harming plurality in these two cases, are not included in one. However, as I have demonstrated, this is not the case.
As I noted, according to Feit, we should say that the plurality consisting of X's act, Y's act, and Z's act harms the miners. This is why he rejects PH and develops his own proposal. However, it is not clear that Feit can reject the reply that I have given in defense of PH*. For example, Feit cannot reject my reply for reasons of principle since he essentially uses the same strategy to defend his own view. 9 Indeed, anyone defending a view that implies that neither X's act nor Y's act harms me in Shooters has to adopt this kind of strategy. Having said that, it is difficult to see that a view would be disadvantaged by accommodating our intuitions in Overkill and Two Buttons. We should therefore consider Feit's view on plural harm.

Feit's proposal
Feit has proposed another solution to the ''no-smallest-plurality'' problem. According to Feit, we can solve this problem ''by appealing to the notion of a super-plurality' ' (2015, 375). A super-plurality is a plurality that contains all the members of other pluralities. According to Feit, the ''rough idea is that when multiple pluralities are ''tied for smallest,'' the harm is the smallest super-plurality of all these pluralities'' (2015,375). In Overkill, the plurality consisting of X's standing on the platform, Y's standing on the platform, and Z's standing on the platform is the smallest super-plurality of the three pluralities that are tied for smallest: namely, X's standing on the platform and Y's standing on the platform, X's standing on the platform and Z's standing on the platform, and Y's standing on the platform and Z's standing on the platform.
Feit proposes the following general principle of plural harm: GPH: E harms S if and only if E is the smallest super-plurality of every plurality of events P such that (1) if none of the events in P had occurred, S would have been better off; and (2) there is no smaller sub-plurality of P such that if none of the events in it had occurred, S would have been better off (2015,376). 10 In my discussion of PH, I noted that PH must be clarified. More precisely, I noted that the harming plurality cannot be the smallest plurality of every plurality that makes things worse for a person. Following Johansson and Risberg, I suggested that we read ''smallest plurality'' as ''smallest plurality that includes some event in E.'' A similar clarification is called for as regards GPH. The harming plurality cannot be the smallest super-plurality of every plurality that satisfies conditions (1) and (2) of GPH. If it were, the plurality consisting of X's, Y's act, and Z's act might not harm 9 See Sect. 5 and footnote 22 below. 10 Feit subsequently revises GPH to make room for degrees of harm, which I won't go into here. (But see footnote 19.) the miners since it might not be the smallest super-plurality of every plurality that satisfies these conditions. Suppose that some weeks before X, Y and Z stood on the platform, the miners were trapped in the mine for a couple of days without food and water due to some mistake that their supervisor made. If the harming plurality were the smallest super-plurality of every plurality that satisfies conditions (1) and (2) of GPH, the supervisor's mistake should also be included in the plurality that harms the miners since it satisfies these conditions: it makes the miners worse off and it does not have a smaller sub-plurality that does. In fact, under these circumstances, there would be one and only one (very large) plurality that harms the miners. In light of these considerations, I propose that we read ''every plurality of events P'' as ''every sub-plurality of events P of E.'' 11 According to GPH, the plurality consisting of X's act, Y's act, and Z's act harms the miners. Each of the three sub-pluralities that are ''tied for smallest'' satisfies (1) and (2). Consider for example the plurality consisting of X's act and Y's act. The miners would have been better off if neither of these acts were performed, and this plurality does not have a smaller sub-plurality such that the miners would have been better off if that plurality had not been performed. For example, if X had not stood on the platform, the miners would still have been killed by the plurality consisting of Y's act and Z's act. Furthermore, according to GPH, each of the three subpluralities that are ''tied for smallest'' also harms the miners. For example, the plurality consisting of X's act and Y's act has a sub-plurality that satisfies the two conditions: namely, itself. Just as a set is a subset of itself, a plurality is a subplurality of itself.
Apart from having the intuitively correct implications in Overkill, GPH also has the intuitively correct implications in Shooters, Dancer and Two Buttons. Consider first Shooters. The plurality consisting of X's shooting and Y's shooting has one sub-plurality that satisfies conditions (1) and (2): namely, itself. I would have been better off if neither X's act nor Y's act had been performed and the plurality consisting of their acts does not have a sub-plurality that makes me worse off. For example, if Y had not shot me, X would still have done so. Consequently, according to GPH, the plurality consisting of X's shooting and Y's shooting harms me.
Consider next Dancer. According to GPH, the plurality consisting of X's shooting, Y's shooting, and Fred's dancing does not harm me. The only subplurality of this plurality that satisfies (1) and (2) is the plurality consisting of X's shooting and Y's shooting. The plurality consisting of all three acts, makes me worse off but it has a smaller sub-plurality that does so too (i.e., the plurality consisting of X's shooting and Y's shooting). A plurality consisting of either X's act or Y's act (but not both) and Fred Astaire's act, does not make me worse off. For example, I would still have been killed by Y, if X and Fred Astaire had acted differently. So, Fred's act is not a member of a sub-plurality of the pertinent plurality (i.e., the one that consists of all three acts) that satisfies the two conditions. Finally, in Two Buttons the plurality consisting of X's pressing a button and Y's pressing a button harms me, according to GPH. Both the sub-plurality consisting of X's act and the one consisting of Y's act satisfy (1) and (2). For example, consider the one consisting of X's act. This plurality makes things worse for me: I would have been better off if X had not pressed the button. However, this plurality obviously does not have a smaller sub-plurality that makes things worse for me.
I said above that it is difficult to see that a view would be disadvantaged by accommodating our intuitions in Overkill and Two Buttons. But it is worth noting that GPH implies that many weird pluralities that might not seem to be harming pluralities, are harming pluralities. For example, according to GPH, the plurality of my accidentally closing a door on my finger as a child and my getting a parking ticket as an adult harms me. Both events respectively satisfy conditions (1) and (2). For instance, getting the parking ticket made me worse off and this plurality does not have smaller sub-plurality that does. In fact, as this example illustrates, according to GPH, any number of pluralities that satisfy conditions (1) and (2) with respect to a harm subject, compose a further harming plurality no matter how different they are and no matter how far apart in space and time they are. 12 To defend her own view, a proponent of PH* might point to this feature of GPH.
It should be mentioned that Feit may not accept my reading of his view. He does not seem to regard pluralities that are ''tied for smallest'' as harming pluralities. For example, as I noted in the first paragraph of this section, he claims that ''when multiple pluralities are ''tied for smallest,'' the harm is the smallest super-plurality of all these pluralities'' (2015, 375, my emphasis). Furthermore, in Overkill, he suggests that there is an unique super-plurality that harms the miners (2015, 376). In my view, anyone attracted to a view on plural harm should want her view to yield the result that each of the three pluralities ''tied for smallest'' in Overkill harms the miners. These pluralities make things worse for the miners and they do not have superfluous members, that is to say, members like Fred Astaire's act. However, as we shall see, even if Feit would insist on a reading of his view that is consistent with the claim that there is a unique harming plurality in cases like Overkill, this would not make his view better equipped to handle the coming objections.

Counterfactually dependent events
I said in the introduction that Feit's proposal fails to deal adequately with one of Norcross's cases involving a counterfactually dependent event. Feit acknowledges that one of Norcross's cases, in which one of the acts counterfactually depends on 12 In fairness, I should mention that Feit realizes this. While Feit does not develop an account that relativizes harm to time in his paper (i.e., an account that takes ''E harms S at t'' to be the fundamental unit of analysis), his view is that a fully adequate account of harm should do so, and that such an account would also handle the ''weird plurality '' problem (2015, 378-379). The fact that Feit holds this view might also explain why he does not clarify PH and GPH along the lines suggested by Johansson and Risberg. I thank a reviewer for pointing this out. the other, poses a challenge to his view. According to Feit, the following of Norcross's cases is problematic: Joke Teller. X tells me a particularly funny joke. I laugh so much that I become hoarse, and visit the soda machine to purchase a Coke. Y, … lurking to practice his assassination skills, shoots and kills me painlessly. The Coke that I had purchased, and was about to drink, had been infected with a deadly poison in a freak undetected soda canning accident (2005, 156-157). 13 According to Feit, the problem is that his view ''implies that Y's shooting does not harm Norcross, and is not even one of the events that together harm him'' (2015,382). According to Feit this is so because, ''[i]f a plurality of two or more events, one of which is Y's shooting, satisfies condition (1) -that is, if Norcross would have been better off if none of the events in it had occurred -it will fail to satisfy condition (2). There will be a smaller sub-plurality, which does not include Y's shooting, that will satisfy (1)'' (2015,382). According to Feit, this implies that the harm on his view ''does not include Y's shooting at all. It will include only events in the causal sequence that contains X's telling the joke, Norcross' visiting the soda machine, and Norcross' purchasing the coke' ' (2015, 382).
However, Feit argues that this implication can be defended. According to Feit, the best way to do so is to maintain that ''although there are events that harm Norcross to a significant degree, the assassination is not one of them'' (2015, 382). Feit claims that the strategy just mentioned is plausible because ''there are independent reasons to think that Y's shooting of Norcross does not harm him'' (2015,382). He motivates this claim as follows: Suppose for example that we replace the assassin, Y, with an impersonal process. In this version of the case, Y is absent, and instead Norcross is killed, painlessly, by a freak electrical discharge as he pulls the Coke from the machine. Given our knowledge that Norcross was just about to drink the poison, the inclination to say that the discharge harms him is much weaker. In fact, there is a good reason to say that the discharge is a benefit, insofar as it prevents an imminent unpleasant death by Coke poisoning. The same goes for Y in the original example (2015, 382). But Feit's strategy is not convincing. It is important to note that Feit's problem is not just explaining why it might be plausible to deny that Y's shooting harms Norcross. It is also to explain why it might be plausible to deny that Y's shooting of Norcross is a member of a plurality that harms Norcross. And it is well-known that there are cases where a plurality of acts harms someone but where the individual acts of that plurality benefit that person. 14 The fact that an act benefits someone 13 Norcross calls this case ''Case-Two-and-a-Half.'' 14 So-called ''Coordination cases'' have this feature. Here is an example: X and Y press a button each. As a result, I feel a horrible pain. If only one of them had pressed a button, my pain would have been a lot worse, but if neither of them had pressed a button, I would have felt no pain at all. (This case is an adaption of a case presented by Frank Jackson, 1987, 98-99). simply does not indicate that it isn't a member of a plurality of acts that harms that person.
However, Joke Teller does not actually pose a challenge to Feit's view. It is simply not correct that GPH implies that Y's shooting ''is not even one of the events that together harm [Norcross].'' Consider the plurality consisting of Norcross's purchasing the Coke and Y's shooting. According to GPH, this plurality harms Norcross. It makes Norcross worse off, and it does not have a smaller sub-plurality that does: If Norcross had not purchased the Coke, he would still have been killed by Y. (He is standing in front of the machine.) And if Y had not shot Norcross, he would have been killed by the poisoned Coke.
Feit is not alone in making this kind of mistake. For example, Norcross claims that on Parfit's view, ''Y isn't even a member of a group who harms me'' (2005,157). According to Norcross, this is so because the ''smallest group who harms me consists of just X' ' (2005, 157). And this is so because ''if X hadn't told me the joke, I wouldn't have needed to visit the soda machine'' (2005,157). However, according to PH* (and PH), Y's shooting is a member of a plurality that harms Norcross. Consider again the plurality consisting of Norcross's purchasing the Coke and Y's shooting. According to PH* (and PH), this plurality harms Norcross since it is the smallest plurality that includes a member of this plurality that makes Norcross worse off. Of course, the plurality consisting of Norcross's telling the joke is smaller but it does not include a member of the plurality consisting of Norcross's purchasing the Coke and Y's shooting.
Unfortunately, although this way of saving GPH seems to work in Joke Teller, Norcross discusses another case involving a counterfactually dependent event, but where this kind of maneuver does not seem to work: Poisoner and Shooter. X tricks me into drinking poison that will shortly result in my painful death. Before any of the real unpleasantness can set in, however, the slightly bitter taste of the poison prompts me to visit the nearest soda machine to purchase a Coke to take the taste away. Y is lurking by the soda machine waiting for a victim on whom to practice his assassination skills. Y shoots and kills me instantly and painlessly. If I hadn't drunk the poison, I wouldn't have visited the soda machine and wouldn't have been shot. In fact, if I hadn't visited the soda machine, Y would have grown tired of waiting for a victim, and would have decided to become an accountant instead of an assassin (2005,156). 15 According to GPH, Y's act is not a member of a plurality that harms Norcross. Consider first the plurality consisting of X's tricking Norcross into drinking poison and Y's shooting him. Although Norcross would have been better off if none of the acts of this plurality had been performed, this plurality has a smaller sub-plurality such that it would have been better for Norcross if that plurality had not been performed: If X had not tricked Norcross into drinking poison, he would obviously not have been poisoned but he would also not have visited the soda machine where he was shot. However, note that in this regard, Poisoner and Shooter is no different from Joke Teller. As we saw, according to GPH, the corresponding plurality in Joke Teller -i.e., the one consisting of X's telling Norcross a joke and Y's shooting Norcross -does not harm Norcross either.
Consider next the plurality consisting of Norcross's purchasing the Coke and Y's shooting Norcross. As we saw, GPH implies that this plurality harms Norcross in Joke Teller. However, according to GPH, this plurality does not harm him in Poisoner and Shooter. In contrast to Joke Teller, where Norcross would have been better off if neither he nor Y had acted as they did, in Poisoner and Shooter, Norcross would not have been better off if neither of them had acted as they did: If Norcross had declined to purchase the Coke and Y had abstained from shooting, Norcross would still have died from the poison.
To handle this difficulty, Feit might perhaps point out that GPH implies that the plurality consisting of X's tricking Norcross into drinking poison harms him. This plurality makes Norcross worse off and it does not have a smaller sub-plurality that does. So, it is not like there is no harming plurality at all in this case. Furthermore, Feit might point out that since the poison is already spreading through Norcross's body when Y shoots him, in contrast to Joke Teller, Norcross is as good as dead at that point. Give or take some, Y is crudely put practically putting a bullet through a dead body. However, consider this modified version of Poisoner and Shooter:

Poisoner and Shooter (Modified Version). This case is like Poisoner and
Shooter except that Norcross would have visited the soda machine even if X had not tricked him into drinking poison. In this case, Norcross is on his way to the soda machine because he is already thirsty. Before he gets there, he bumps into X who when hearing that Norcross is thirsty, seizes the opportunity to harm him by tricking him into drinking a glass of water spiked with poison. However, the glass of poisoned water does not quench Norcross's thirst, which prompts him to continue to the sodamachine where he is shot by Y.
If Norcross is as good as dead in the previous case when Y shoots him, this is true in this case too. However, according to GPH, the plurality consisting of X's tricking Norcross into drinking poison and Y's shooting him does harm Norcross in this case since he would have been better off if neither of these two acts had been performed. Moreover, this plurality does not have a smaller sub-plurality that makes Norcross worse off: If X had not tricked him into drinking poison he would still have been killed by Y's shooting him, and the other way around. 16 The difference between Poisoner and Shooter and the modified version of this case is that Y's act counterfactually depends on X's act in the former, but not the latter. However, this does not seem to be the kind of consideration that motivates excluding Y's act from being a member of a harming plurality. For example, according to Norcross, Y acts wrongly because ''[we] don't, in general, excuse behavior that appears to be wrong, if we discover that the agent wouldn't have had the opportunity to perform the wrong act, were it not for seemingly unrelated behavior of someone else' ' (2005, 156). Norcross's point is about wrongness, but it seems to apply equally well to harm. Why should the fact that Y would not have had the opportunity to kill Norcross unless X had acted as he did yield the result that Y's act is not even one of the acts that harm Norcross? 17 It is worth noting that there are further considerations that support the view that Feit's proposal does not have the intuitively correct implications in cases like Poisoner and Shooter. For example, according to Thomas D. Bontly, ''it really is plausible to say that [X and Y] ''together'' harm [Norcross]' ' (2016, 1240 ' (2016, 1240).
To sum up the discussion of this section: Although Feit's proposal handles one type of cases involving a counterfactually dependent event (i.e., cases like Joke Teller), it fails in another type of cases involving a counterfactually dependent event (i.e., cases like Poisoner and Shooter). In Poisoner and Shooter, Feit's proposal implies that Y's act is not included in a plurality that harms Norcross although there seem to be several considerations in favor of placing Y's act in such a plurality.

A new principle of plural harm
Feit's view thus struggles to handle a class of cases involving counterfactually dependent events. The problem seems to lie with condition (2), i.e., the requirement that ''there is no smaller sub-plurality of P such that if none of the events in it had occurred, S would have been better off.'' By including this condition, according to GPH, in Poisoner and Shooter we should consider what would have happened if X had not tricked Norcross into drinking poison. As we saw, since Y's shooting Norcross would not then have occurred (because it depends on X's act), GPH ends up excluding Y's act from a harming plurality.
However, there is another factor that we may appeal to. Instead of focusing on what would have happened if X had not tricked Norcross into drinking poison, we may appeal to what would have happened if X had not tricked Norcross into drinking poison but Y would have shot Norcross anyway. We would be justified in doing so because we arguably want to include Y's act in a harming plurality primarily because Y's shooting Norcross clearly in some sense is sufficient to kill Norcross. In my view, the fact that Y kills Norcross in the nearest world where Y still shoots Norcross although X does not trick him into drinking poison, captures 17 Parfit discusses a pair of cases that are partly similar to Poisoner and Shooter and the modified version of this case (1984,(70)(71). that sense. In light of these remarks, I suggest the following new view on plural harm: NPH: E harms S if and only if E is the smallest super-plurality of every subplurality of events P of E such that (1) if none of the events of P had occurred, S would have been better off; and (2) each event e of P is such that S would not have been better off if e, but none of the other events of P, had occurred.
I shall presently demonstrate that NPH handles the cases we have discussed. At this point it is sufficient to show that the pertinent pluralities in these cases satisfy condition (2) of NPH (or fail to satisfy it as in Dancer) since I have already shown that they satisfy the rest of this view. As the reader can see, aside from condition (2), NPH is identical with GPH.
As indicated in the previous paragraph, in Poisoner and Shooter, the plurality consisting of X's tricking Norcross into drinking poison and Y's shooting Norcross satisfies condition (2) of NPH. Norcross would not have been better off if X's tricking Norcross into drinking poison but not Y's shooting him had occurred since he would then have died from the poison. Similarly, Norcross would not have been better off if Y's shooting Norcross had occurred but not X's tricking him into drinking poison since he would then have died from the shot.
In Joke Teller, the plurality consisting of X's telling Norcross a funny joke and Y's shooting Norcross also satisfies condition (2) of NPH. Norcross would not have been better off if X's telling Norcross a funny joke had occurred but not Y's shooting him since he would then have died from the poison. (Recall that the joke prompts Norcross to visit the soda machine where he buys the poisoned coke.) Likewise, Norcross would not have been better off if Y's shooting Norcross had occurred but not X's telling him a funny joke since he would then have been killed by the shot.
Consider now Shooters. In this case, the plurality of X's shooting me and Y's shooting me satisfies condition (2) of NPH. I would neither have been better off if X's shooting me had occurred but not Y's shooting me nor if Y's shooting me had occurred but not X's shooting me since in either case, I would have been shot. In Dancer, the plurality of X's shooting me, Y's shooting me, and Fred Astaire's dancing does not satisfy condition (2) of NPH since I would have been better off if Fred Astaire's dance had occurred but not the rest of that plurality. (Recall that the pluralities consisting of Fred Astaire's act and either one of X's act or Y's act, but not both, fail to satisfy condition (1) of GPH and NPH.) In Overkill, each of the three pluralities ''tied for smallest'' satisfies condition (2) of NPH. For example, the miners would not have been better off if X's standing on the platform had occurred but not Y's standing on the platform since X would under those circumstances have killed the miners together with Z. Clearly, for the same reason, the miners would not have been better off if Y's standing on the platform had occurred but not X's standing on the platform. Note that this means that NPH (correctly) yields the result that the plurality consisting of X's act, Y's act, and Z's act also harms the miners since each of the three pluralities ''tied for smallest'' is a sub-plurality of this larger plurality.
Finally, in Two Buttons, although it might at a first glance be thought that neither X's pressing a button nor Y's pressing a button satisfies condition (2) of NPHsince this condition initially might seem applicable only to pluralities consisting of more than one event -each of their acts actually satisfies this condition. For example, I would not have been better off if Y's act had been performed but none of the other members of this plurality (there are none) since X's pressing together with Y's pressing the button would still have killed me. Again, note that this means that NPH (again correctly) yields the result that the plurality consisting of X's act and Y's act harms me since each act is a sub-plurality of the larger plurality consisting of X's act and Y's act.

Challenges to NPH
There is a class of cases that might be thought to present a serious challenge to NPH. These cases are like Shooters except that it would not only have been better for the victim of the case if neither of the culprits had acted as they did, but also if one of them had acted differently. For example, consider the following case: Torturers. X and Y, independently of one another, press a button, resulting in my feeling terrible pain. If only X had pressed a button, I would have felt the same amount of pain. However, if only Y had pressed a button, I would have felt less pain. If neither of them had pressed a button, I would have felt no pain at all. 18 It seems plausible that philosophers that appeal to the notion of plural harm should want to include Y's act in a plurality that harms me in this case. Just as Y would have harmed me in Shooters if X had not shot me, Y would have harmed me in Torturers if X had not pressed a button. In both cases, Y performs an act that is normally harmful. But according to NPH, the plurality of X's act and Y's act does not harm me. Although this plurality makes me worse off, I would have been better off if Y's pressing a button had occurred, but not X's pressing a button. Clause (2) of NPH is therefore not satisfied.
Although it isn't optimal that NPH has the implication that Y's act is not included in a plurality that harms me, I do not think that this implication is seriously disturbing. There are three considerations in particular that I would like to highlight. First, what would have been seriously disturbing would have been if NPH had yielded the result that there is no plurality that harms me in Torturers. However, according to NPH, X's pressing a button harms me since I would have been better off if none of the acts of this single event plurality had been performed, and since (obviously) I would not have been better off if X's pressing the button had occurred but not the rest of the members of that plurality (since it has none). Secondly, although it is true that Y's act would have harmed me in the absence of X's act in both Shooters and Torturers, there is an important difference between the cases. In Torturers, the harm Y's act would have caused me alone is not on par with the harm X's act would have caused me alone, whereas in Shooters, the harm Y's act would have caused me alone is on par with the harm X's act would have caused me alone. In light of this difference, it does not seem overly problematic that NPH does not include Y's act in a plurality that is responsible for the full harm that befalls me. Thirdly, in Torturers, GPH also implies that the plurality consisting of X's pressing a button and Y's pressing a button does not harm me. Although I would have been better off if neither X nor Y had pressed a button, I would have been better off if X had not pressed one. Clause (2) of GPH is therefore not satisfied. Consequently, GPH is not better off than NPH in this regard. 19 There is also another class of cases that might seem to pose a challenge to NPH. These cases involve events that have beneficial effects together with other events but that have negative effects by themselves. The following case belongs to cases of that kind: Poisoner and Saver. X tricks me into drinking poison that will shortly result in my painful death. As the real unpleasantness sets in, Y gives me an antidote that saves me, which he would not have done unless X had poisoned me. However, the antidote is peculiar: If I had taken it without also having taken the poison, it would have resulted in my painful death. 20 According to NPH, the plurality consisting of X's act and Y's act harms me. I would have been better off if neither of them had acted as they did since the real unpleasantness has already begun when Y gives me the antidote. Furthermore, I would obviously not have been better off if X had tricked me into drinking poison but Y had failed to give me the antidote. Finally, I would not have been better off if Y had given me the antidote and X had not tricked me into drinking poison since the antidote would have killed me in absence of the poison. However, since Y's act benefits me, one might think that Y's act should not be included in a plurality that 19 As I noted in footnote 10, Feit revises GPH to take account of degrees of harm. A reviewer has pointed out that the following principle, which is a revision of NPH along those lines, implies that the plurality consisting of X's act and Y's act harms me: QNPH: E harms S to degree n if and only if E is the smallest super-plurality of every sub-plurality of events P of E such that (1) if none of the events of P had occurred, S would have been better off to degree n; and (2) each event e of P is such that S would not have been better off to degree n if e, but none of the other events of P, had occurred.
To illustrate, the reviewer assumes that the plurality consisting of X's act and Y's act causes me to experience 100 units of pain, and that X's act by itself causes me to experience 100 units of pain but that Y's act by itself causes me to experience 75 units of pain. In that case (as the reviewer points out), clause (1) of QNPH is satisfied because I would have been better off to degree 100 if neither of them had pressed a button. Furthermore (as the reviewer also points out), clause (2) of QNPH is satisfied because I would not have been better off to degree 100 if either of them had pressed a button alone. I would not have been better off at all if X had pressed a button alone and I would have been better off to degree 25 if Y had pressed a button alone. According to QNPH, then, the plurality consisting of X's pressing a button and Y's pressing a button harms me. In light of this observation, the reviewer suggests that I adopt QNPH instead of NPH. Although this would be an attractive move, I am not convinced that it is really needed. In my view, it is not sufficiently clear (for the reasons I gave above) that the plurality consisting of X's act and Y's act really harms me. I hope to explore QNPH and its implications in future writings 20 Erik Carlson suggested this case to me. harms me. Furthermore, it is worth noting that GPH does not seem to include Y in a plurality that harms me. According to GPH, the plurality consisting of X's act and Y's act does not harm me since this plurality has a smaller sub-plurality that is such that I would have been better off if it had not occurred: I would have been better off if X had not poisoned me. Under those circumstances, I would not have felt any unpleasantness at all and Y would not have given me the antidote.
In my view, Poisoner and Saver does not pose a serious challenge to NPH. It simply does not seem very counterintuitive that the plurality consisting of X's act and Y's act harms me. As I noted above, the fact that an act benefits someone does not show that it is not a member of a plurality of acts that harms the person. Granted, in Poisoner and Saver, it might appear particularly odd that Y's act is included in a harming plurality. However, this might well be because the description of the case suggests that Y's act is well-intentioned, which should bear no weight in attributing harm. Imagine that Y's act isn't well-intentioned but that Y actually intends to kill me by giving the antidote (which it would do unless it is taken together with the poison). For example, we may imagine that Y thinks that X, who is supposed to trick me into drinking poison, has changed her mind at the last second and has given me a harmless substance instead. Under these circumstances, we might not be so hesitant to include Y's act in a harming plurality. So, it does not seem clear that it is particularly odd that Y's act is included in plurality that harms me.
In short, neither Torturers nor Poisoner and Saver motivates a rejection of NPH. Moreover, Torturers gives us no reason to prefer GPH over NPH, and although Poisoner and Saver might give us some reason to prefer GPH over NPH, it gives us no strong reason to do so. However, as I shall presently argue, there is a class of cases that seem fatal to GPH, but which NPH handles well.

Unavailable events
In the cases we have discussed up until now, the relevant events have all occurred. However, as indicated above, there are preemption cases where the preempted alternative or alternatives never occur. Norcross's notorious case involving the basketball coach Bobby Knight is an example of such a case (2005,(165)(166). Nathan Hanna has given a succinct presentation of Norcross's case: Choker. Bobby Knight gets mad at philosopher Phil, in response to a perceived slight. So, he chokes Phil. Luckily for Phil, Knight is in anger management. It's taught him to better control his behavior. He choked Phil because he applied anger management techniques. If he hadn't he would have dismembered Phil (Hanna, 2016, 263). 21 According to Feit, his view yields the result that there is a plurality that harms Phil, in which the choking is a member. However, since Bobby Knight doesn't actually dismember Phil, one might wonder how that can be. What other events together with the choking harm Phil? 22 According to Feit, ''the truth of counterfactuals is grounded in non-modal facts about the actual world' ' (2015, 380). In this case, the relevant counterfactual is ''If Bobby Knight had not choked Phil, he would have dismembered him.'' According to Feit, this counterfactual is grounded by ''certain of Knight's token psychological states (perhaps feeling of rage)' ' (2015, 381). And it is these events together with the choking that harm Phil since ''if none of these events had occurred, Phil would have been better off' ' (2015, 381 (2019,359). Now, although the plurality consisting of the choking and the M-events makes thing worse for Phil, this plurality obviously has a smaller sub-plurality that also makes things worse for Phil; namely the plurality consisting of the M-events. Consequently, on this construal, GPH yields the result that the plurality consisting of the choking and the M-events does not harm Phil.
On the second construal, Knight's attacking Phil at all does not counterfactually depend on the M-events. He would have done so even if the M-events had not occurred. On this construal, then, the M-events might be exclusively ''dismemberment oriented.'' However, according to Johansson and Risberg, on this construal there are plausibly other events -the ''M*-events'' -such that Knight's attacking Phil at all counterfactually depends on them. On this construal, it might initially seem that GPH implies that the plurality consisting of the choking and the M-events harms Phil since this plurality does not have a smaller sub-plurality that makes things worse for him.
However, according to Johansson and Risberg, Phil would not have been better off if neither the choking nor the M-events had occurred since the M*-events would still have occurred. And although the M*-events would not have caused the choking, ''in light of the significant amount of hostility that they display toward Phil, there is little reason to suppose that they would not have resulted in something equally detrimental to his well-being'' (2019,(359)(360). So, on the second construal, GPH also yields the result that the plurality consisting of the choking and the M-events does not harm Phil. Moreover, Johansson and Risberg point out that the plurality consisting of the choking, the M-events and the M*-events does not harm Phil according to Feit's view, since this plurality has a smaller sub-plurality that makes things worse for him: namely the M*-events (2019, 359-360).
As I said above, NPH does a better job in this case. According to this view, the choking belongs to a plurality that harms Phil on both ways to construe Choker. On the first construal, the plurality consisting of the choking and the M-events makes thing worse for Phil. Furthermore, on this construal, it is true that Phil would not have been better off if the choking but not the M-events had occurred since this means that he still gets choked. And he would not have been better off if the M-events had occurred but not the choking since Knight would then have dismembered him. On the second construal, the plurality consisting of the choking, the M-events and the M*-events makes thing worse for Phil. Furthermore, one quickly realizes that Phil would not have been better off if any of the events alone had happened without the others. For example, Phil would not have been better off if the M*-events had occurred without the other two since we may assume that they would have resulted in something equally detrimental to Phil's well-being.

Concluding remarks
In this article, I have argued that although the most prominent accounts of plural harm -i.e., the Parfitian accounts (PH and PH*) and Neil Feit's account (GPH)fare better than one might initially think, they fail to deal adequately with a number of fascinating cases of preemption first introduced by Alastair Norcross. I have also argued that there is a view of plural harm (NPH) that apart from dealing with the cases of overdetermination and preemption that the other accounts of plural harm handle, also deals adequately with Norcross's cases. Of course, some philosophers might be skeptical of views of plural harm and might not accept that NPH deals with Norcross's cases in a satisfactory way. Hopefully, they accept that I have at least shown that, on balance, NPH handles Norcross's difficult cases better than the other views discussed in this article.
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