I. Limited Aggregation

The intuitive idea behind Limited Aggregation is that interpersonal aggregation of harms and benefits should sometimes make a difference to what we ought to do and sometimes not. In Death vs. Headaches it should not make a difference, in Death vs. Paraplegias it should. Recent accounts of Limited Aggregation can be traced back to comments T. M. Scanlon made in What We Owe to Each Other. While justifying the “individualist restriction” of his Contractualism, which, in principle, forbids any form of interpersonal aggregation, he offers the following qualification:

Derek Parfit suggested a view that according to him captured the essence of Scanlon's remarks. He calls it the

This view allows us to capture the aforementioned pair of intuitions about Deaths vs. Headaches and Death vs. Paraplegias. It was recently spelled out in a systematic way by Alex Voorhoeve.Footnote ⁶ On his approach, which he calls

Voorhoeve explains that, in a pairwise comparison between two claims, the weaker claim is relevant to the stronger claim if and only if the person holding that weaker claim would not be required to give it up, if she herself were the one to make the choice. The approach thus takes into account the subjective viewpoint of individuals from which it seems permissible to favour oneself to a certain degree. For example, it seems permissible for me to save myself from paraplegia rather than a stranger from death, but impermissible to spare myself a minor headache at the cost of the death of a stranger.

In Death vs. Headaches, ARC says that we ought to save Ann irrespective of the number of people we could spare the headache, because the individual claims to be spared a headache are not close enough to Ann's claim to be saved from death. We therefore ought to satisfy the single strongest individual claim. In Death vs. Paraplegias on the other hand, the individual claims to be saved from paraplegia presumably are close enough to Ann's claim. For some number of people facing paraplegia, we therefore ought to save them rather than Ann from death, since this maximizes the sum of strength-weighted, relevant claims.

Before pointing out the proposal's problems, I want to pause to emphasize that ARC, as I understand it, is a view about moral choice, and not about axiology. ARC will sometimes select outcomes that are worse in terms of overall goodness than other available outcomes.Footnote ⁸ I take it that ARC and Limited Aggregation in general should only apply to the deontic assessment of choices and not to the evaluation of outcomes. As we will see later on, Limited Aggregation violates structural properties that, in my view, are inherent to the very concept of goodness.Footnote ⁹ To keep things simple, I will assume the utilitarian axiology according to which the best outcome is the one maximizing the overall unweighted sum of well-being. However, nothing hinges on this assumption, as long as we agree that axiology is a fully aggregative matter. If you prefer a prioritarian axiology, for example, then you can simply plug it in. Two further assumptions: First, I only consider choice under certainty, so each alternative act will be associated with exactly one outcome. And second, I only consider “same-people choices”, i.e. all available outcomes contain the same set of people.

With these assumptions in place, we can now move on to the first set of objections against Limited Aggregation that I want to consider.

II. The anchoring ambiguity

The problem I deal with in this section is due to Patrick Tomlin.Footnote ¹⁰ It stems from an ambiguity in Voorhoeve's account that had not been appreciated before. The ambiguity is the following. There are (at least) two ways to understand the “strongest competing claim” (which Tomlin calls the “anchoring claim”) to which any claim c must be close enough in order to be relevant. The first meaning is “the strongest claim with which c competes”. The second meaning is “the strongest claim in the competition”. These two meanings come apart when the two groups of claims that compete with each other are not homogeneous in the sense that it is not the case that all claims within each group are exactly equally strong.Footnote ¹¹ Take the following example.

What in this case determines the relevance of the claims of those who stand to lose a finger? Ann's claim or the claims of those who stand to lose a leg? Ann's “death-claim” (i.e. her claim to be saved from death) is the strongest claim in the competition, but this claim doesn't compete with the finger-claims. If we suppose that Ann's claim is the anchoring claim, then in Tomlin's terminology we “anchor by strength”. The strongest claims with which the finger-claims actually compete, however, are the leg-claims. If we take those claims to be the anchoring claims, then we “anchor by competition”. If we now plausibly assume that finger-claims are close enough to leg-claims, but not to death-claims, then they are irrelevant under the first interpretation, but relevant under the second.

Tomlin shows that both interpretations give rise to implausible results. First consider Anchor by Competition and the following two-stage case.Footnote ¹²

I will illustrate this and following cases with simple figures. Each column represents one group we can save. The numbers (1, n, m in this case) represent the number of people holding a claim of a given strength. The higher up the number is in a given column, the stronger the claim. To keep things simple, I will assume that there are only three strengths a claim can take. There will only be death-claims, leg-claims, and finger-claims. If claims are represented in parenthesis, that means that they are not present at the first stage but are added at the second stage. Two-Stage Case A can then be represented as follows.

Two-Stage Case A

Now suppose n is such that according to Limited Aggregation, at stage 1, we ought to save the people from losing a leg. Now we add m people who stand to lose a finger both to Ann's side and to the side of those people who stand to lose a leg. As before, we assume that finger-claims are close enough to leg-claims, but not to death-claims. According to Anchor by Competition, this means that the claims of the people we added to Ann's side are relevant, but the claims of the people we added to the other side are not. This seems counterintuitive, especially considering that at stage 2 we might have to reverse our judgement from stage 1 and should now save Ann from death and the m people on her side from losing a finger. This is because their relevant claims might turn the balance in Ann's favour, while the irrelevant claims on the other side carry no counterbalancing weight. The approach violates what Tomlin calls

Consider now Anchor by Strength by way of the following case, also adapted from Tomlin.Footnote ¹⁴

Two-Stage Case B

Stage 1. We can either save n people from losing a finger (A) or m people from losing a leg (B).

Stage 2. Ann, whom we can save from suffering a terminal disease, is added to A.

Suppose, again, that finger-claims are relevant to leg-claims, but not to death-claims, and that n and m are such that at stage 1, we ought to save the people from losing a finger. It is obvious that adding Ann to this group ought to strengthen the case for saving them. Not only can we now save n people from losing a finger, but we can also save Ann's life and we obviously ought to do so. Anchor by Strength, however, might tell us otherwise. Since the strongest claims in the overall competition now no longer are the leg-claims, but Ann's death-claim, the finger-claims are no longer relevant and should play no role in our decision anymore. This means that we now have to weigh Ann's single claim against m leg-claims and depending on how large m is, ARC might tell us to satisfy those claims, which is clearly the wrong result. Anchor by Strength violates what Tomlin calls the

This principle seems intuitively plausible enough to reject any approach that violates it.Footnote ¹⁶ In his article Tomlin goes over a number of possible, more complicated anchoring rules, which, however, suffer from even more devastating counterexamples.Footnote ¹⁷ This suggests that the approach of determining a set of relevant claims and then restricting aggregation to these claims is doomed to fail. In the following section I therefore develop an alternative approach at fleshing out Limited Aggregation.

III. Aggregation with Constraints

I suggest the following specification of Limited Aggregation. In a choice situation with competing claims, we should look at the single strongest individual claim. If this claim is very much stronger than the strongest claim with which it competes, we do whatever satisfies this claim. If it is only slightly stronger, we do whatever is best in aggregate. Put differently, there is a general rule to do whatever is best in aggregate, i.e. realize the best outcome. However, the strongest individual claim can raise a veto. We then need to see whether this veto is justified or not. If it is justified, we do whatever satisfies this claim. If not, we proceed to realize the best outcome. Call this

This principle does better than ARC in Tomlin's cases. Reconsider

Two-Stage Case A

On Aggregation with Constraints, in stage 1, Ann's claim isn't much stronger (given our assumptions) than any other person's competing claim, so we ought to do what realizes the best outcome. This depends on how large n is. The added finger-claims at stage 2, do not change this assessment. Aggregation with Constraints does not violate Equal Consideration for Equal Claims, at least not in this case. One case, in which one might think that it looks like it does violate the principle, is the following.Footnote ¹⁸

Two-Stage Case C

Stage 1. We can either save Ann from a terminal illness (A) or n different people from losing a finger (B).

Stage 2. m people whom we can save from losing a leg are added to each side.

At stage 1, Ann's claim is much stronger than any other claim, so by Aggregation with Constraints, we ought to save her. This judgement does not depend on how large n is. Now at stage 2 we may have to rethink our verdict. This is because the added leg-claim on the side of those threatened to lose a finger competes with Ann's death-claim and, by assumption, is close enough to her claim. On Aggregation with Constraints we therefore now ought to do what leads to the best outcome. If n is large enough, we will now have to let Ann die and satisfy the finger-claims and the leg-claims we added to their side.

One might think that this violates Equal Consideration for Equal Claims. This is because, after all, we added the same number of leg-claims to each side and one could hold that since they carry equal weight, they should somehow “cancel each other out” and therefore together should do nothing to overturn our previous judgement at stage 1. The fact, however, that they do not cancel each out in this way does not imply that we do not give the two claims equal consideration or even that we disregard the leg-claims we added to Ann's side. To see this, consider an instance of Two-Stage Case C, where n is such that on aggregation, the finger-claims together with the one leg-claim would just barely outweigh Ann's single death claim. The leg-claim we added to Ann's side now turns the tables in her favour. On aggregation it adds just as much force to her side as the other leg-claim added to the other side. It is only when n is large enough that saving these people from losing a finger would add more value to the outcome than saving Ann could that we should overturn our judgement from stage 1. Only then is it the case that the added leg-claim to Ann's side can do nothing to help her. For all other n, it will turn out that we ought to help her and the person who was added to her side.

What is true, though, in cases like Two-Stage Case C, is that adding relevant claims to either side can “activate” previously irrelevant claims. It is like turning a switch that takes us from a non-aggregative to an overall aggregative mode. This implication may seem especially troublesome, when more claims are added to the group that will not be chosen than to the group that will be chosen. Take the following variation.

Two-Stage Case C*

In this case, if n is very large, m could also be quite large, and we would still at stage 2 overturn our previous judgement. How problematic is this implication? As before, it does not violate Tomlin's Equal Consideration for Equal Claims. However, one might think that adding a claim to group B, while adding more claims of equal strength to an opposing group A, should not strengthen B*'s choiceworthiness compared to A.

I agree that at first blush this implication seems odd. However, on reflection, proponents of Limited Aggregation have a justification available. This is the thought that is at the very heart of Limited Aggregation. When we refuse to let a number of weaker claims outweigh one very strong claim, we do so because the person with the strong claim has so much more at stake than any person with a competing claim. Whenever one person (or a group of people) has much more at stake than any other person we could help instead, we ought to help this person irrespective of the numbers involved.

I take it that this is the driving intuition behind cases like Death vs. Headaches and it is fulfilled at the first stage, so we refuse the aggregation of the n finger-claims. The antecedent of this judgement no longer holds at the second stage, so by this judgement it is no longer true that we ought to help the person with the strongest claim, no matter what. There now is another person who has, by stipulation, a relevantly similar claim. We are going to help this person alongside many others with weaker claims instead, and so realize the best outcome possible.

It makes a difference whether a claim competes against relevantly similar claims or not. If it does not, then we refuse to aggregate. If it does, then what we ought to do depends on the specific case and the specific version of Limited Aggregation. However, the fact that the existence of such a claim could make a difference to our mode of practical reasoning is not surprising.

The implication of Aggregation with Constraints in Two-Stage Case C is thus fundamentally different from the implication of ARC in Tomlin's Two-Stage Case A. In this case, it was the addition of finger-claims that, by stipulation, are irrelevant to Ann's death-claim, which changed things. In Two-Stage Case C it is the addition of relevant leg-claims that changes things. This is a crucial difference, as we should expect relevant claims to have the power to change the normative situation in which we find ourselves. We shouldn't expect irrelevant claims to have that power, however.Footnote ¹⁹ In essence, this is just a restatement of what it means to buy into Limited Aggregation. In certain situations, only some claims matter, others don't. Proponents of the view should therefore not be unnerved by this implication in Two-Stage Case C. Rather, this is what we should expect of a view of Limited Aggregation.

So far, I have shown that Aggregation with Constraints does not seem to run into the same problems as Anchor by Competition as it does not violate Equal Consideration for Equal Claims. To see whether it also does better than Anchor by Strength, consider now again Tomlin's second case,

Two-Stage Case B

At stage 1, there is no single claim that is much stronger than any other competing claim, so we ought to do what realizes the best outcome and this depends on n and m. If we now at stage 2 add Ann's death-claim to the finger claim, no matter how large n and m are, we won't now have to switch from satisfying the finger-claims to satisfying the leg-claims. If at stage 1 we ought to satisfy the finger claims, then at stage 2, we will still do so. Unlike Anchor by Strength, Aggregation with Constraints does not violate the Principle of Addition and therefore does better. Adding a claim to one group of claims without at the same time also adding claims to the competing group can never lessen this group's choiceworthiness.

Here is the principled reason why my approach doesn't run into the problems raised by Tomlin. Whereas ARC aggregates some claims (the relevant ones) but not others (the irrelevant ones), my approach either aggregates all claims or none at all. It lacks the discriminating feature in ARC that gives rise to implausible results under both ways of resolving the ambiguity in ARC. My approach is thus not subject to the same objections as both ways of resolving the ambiguity in ARC.

There is, however, a cost to Aggregation with Constraints that some proponents of Limited Aggregation will probably be hesitant to accept. This is the fact that it gives a different verdict from ARC in a well-known case by Frances Kamm.

Within our framework, this case can be represented as follows.

Kamm writes that the benefit of curing Carl's sore throat should be disregarded. It is an “irrelevant utility” and should play no role in our decision. Many proponents of Limited Aggregation agree.Footnote ²¹ According to Kamm's approach, we ought to treat this case just as we would treat the case were Carl not present. That is, we should be indifferent between saving Ann or saving Bob. On my approach, this is not so. Since Ann's claim and Bob's competing claim are equally strong, and therefore obviously close enough, we ought to aggregate and do what realizes the best outcome. This means that we should save Bob and cure Carl's throat, which realizes the best outcome.

While I did share Kamm's intuition when I first encountered her case, I am prepared to give it up on reflection. If we really are indifferent between two alternatives, then any additional reason that speaks in favour of one of the alternatives, but not the other, should make it the case that we are no longer indifferent. On Aggregation with Constraints, this is what happens when we add Carl's very small claim to Bob's side. Before we added Carl's claim, the reasons speaking in favour of saving Ann were exactly equally as strong as the reasons in favour of saving Bob. This no longer is the case once we have added Carl. We now have more reason to save Bob and Carl than we have to save Ann and ought to do so.

As I said, this will come as a significant cost to some proponents of Limited Aggregation who are wedded to Kamm's intuition. Even so, this cost is, I think, much smaller than the costs of accepting the implications of ARC that Tomlin's cases brought out. So, Aggregation with Constraints still has the upper hand over ARC.

Before I conclude my exposition of Aggregation with Constraints, however, there is one more case I want to discuss. Consider the following variation of Sore Throat.

Since in this case, it is not true that one person has much more at stake than any other person whose claim competes with hers, on Aggregation with Constraints, we should still save Bob and cure the sore throats if there are enough of them. This is the case although Dave, also facing death, is now on Ann's side. One could argue that unlike the original Sore Throat Case, Sore Throat 2 is a simple variant of Death vs. Headaches and therefore a stronger objection to Aggregation with Constraints.Footnote ²²

This case does pose a stronger challenge than the original Sore Throat Case. However, as I shall now show, it is similar to the challenge that Two Stage Case C posed and we can give essentially the same answer. We can see this by cashing out Sore Throat 2 as a two-stage case as well.

Two-Stage Sore Throat 2

Stage 1. We can either save Ann from death (A) or cure n different people's sore throat (B).

Stage 2. Dave, whom we can save from death, is added to Ann's side. Bob, whom we can also save from death, is added to the other side.

At stage 1, it is clear that we ought to save Ann. This is just a variant of Death vs. Headaches (or, rather, Death vs. Sore Throats). What we ought to do at stage 2, however, now depends on how large n is. This is because, after adding Bob's death-claim, it is no longer the case that one person has much more at stake than any other person whose interests conflict with hers. Thus, on Aggregation with Constraints, we now ought to do what is best in the aggregate. If n is large enough, we might have to overturn our judgement from stage 1.

This case has the same structure as Two Stage Case C, only that the added claims are of an even stronger kind, so by definition, they are of the kind that is relevant in this situation. As I write above, these sorts of claims have the power to change the normative situation we are in. When we refuse to let a number of weaker claims outweigh one very strong claim, we do so because the person with the strong claim has so much more at stake than any person with a competing claim. In this case, after we added Bob's claim opposing Ann's, this is no longer true. There is no one on either side who has much more at stake than any person with a competing claim. There are death-claims on both sides. This fundamentally changes the kind of justification we can give to the person whom we are not going to save. In a strictly pairwise comparison of the strongest claims, there would now be a tie where there wasn't one before. Therefore, the view that we should always satisfy the single strongest individual claim, which is the view that best embodies the non-aggregation sentiment, judges the two stages of Two-Stage Sore Throat 2 differently. Aggregation with Constraints does so as well.

This concludes my initial exposition of Aggregation with Constraints. We have seen that it does better with regard to some of the objections that have been levelled against other accounts of Limited Aggregation. There are, however, possible objections to Limited Aggregation that also apply to the view I proposed. I will take these up in the remaining two sections.

IV. Separability and Horton's Dilemma

Joe Horton has recently brought forward a new challenge to Limited Aggregation.Footnote ²³ He proposes a dilemma that, in his view, is faced by all views endorsing Limited Aggregation. His challenge is important. This is because both horns of his dilemma include a violation of a plausible separability condition, which he doesn't explicitly recognize. Separability says, roughly, that our choice between two distributions should not depend on elements that are the same in both distributions.

In this section, I show that Limited Aggregation does indeed violate this condition. This, however, is not actually surprising, but at the bedrock of the view's commitments. It brings to the surface the following: there is a tight connection between Limited Aggregation and relational factors as expressed by a violation of separability.

This section proceeds as follows. First, I will introduce the notion of separability as it applies here and show that it is tightly connected to aggregation. Second, I explain how Horton's objection amounts to the claim that Limited Aggregation violates separability. Third, I argue that this violation is not actually an objection to, but rather a central feature of, Limited Aggregation.

In general, separability is the idea that when evaluatively comparing two objects, we can ignore the objects’ elements which are the same in both. It is a prima facie attractive principle and a powerful tool in ethics and decision theory. In decision theory, separability of states of nature is an important axiom of expected utility theory. It holds that when two alternative courses of action have the same outcome in some state of nature, then our choice should not depend on what happens in that state of nature.Footnote ²⁴

In ethics, we are more often concerned with the separability of people. This principle says that when comparing two distributions, we can ignore those people who fare the same in both distributions. In other words, our choice should be independent of the unaffected, where an individual i is unaffected just in case i has the same well-being level in both distributions. Separability is a practically convenient property, as it implies that when choosing between, say, two different policies, we can focus on those people actually affected by the policies. Here is a general formulation that serves our purpose.

To illustrate the attractiveness of this principle, take the following case (Table 1), where the numbers represent the well-being levels of three people, i₁, i₂, and i₃, and we have to choose between two distributions, D1 and D2.

Table 1.

D1 vs. D2

Our choice between D1 and D2 might be affected by many considerations, for example by the fact that total well-being is greater in D2 than in D1. But what our choice should not be affected by, separability holds, is how well i₃ is doing. Her well-being is the same either way and should not affect our choice. This means that on separability, D1 ought to be chosen rather than D2 if and only if (2,2) ought to be chosen rather than (4,1); and D2 ought to be chosen rather than D1 if and only if (4,1) ought to be chosen rather than (2,2).Footnote ²⁶

Separability plays an important role in some of the best positive arguments for utilitarianism and other fully aggregative theories, such as prioritarianism.Footnote ²⁷ In fact, it can be proven that if a theory fulfils a number of prima facie plausible conditions and is separable, then it is fully aggregative, and vice versa.Footnote ²⁸ So, there is a tight, well-known, connection between separability and aggregation. As I show below, Horton's argument exploits that connection.

However, before applying separability to the kinds of cases we are dealing with here, we need to take care of a slight complication. Since i₃ would not have a claim by the lights of our theory, she would not occur in the choices we have been considering thus far. In these choices we only considered the well-being of people who have claims on us. And since, by definition, only people whose well-being we can affect can have claims, i₃ would not be considered. We thus need a slightly stronger principle to show how Limited Aggregation violates separability. We need a principle that tells us that neither i₃ nor i₄ should affect our choice in a case like the following (Table 2).

Table 2.

D3 vs. D4

Such a principle is a little harder to come by, as both i₃ and i₄ have a stake in our decision. i₃ has a claim to D3 and i₄ has a claim to D4. So, in some sense, we should not ignore them. However, it is clear that in the following case (Table 3), in which the decision only includes i₃ and i₄, we should be indifferent between D5 and D6.

Table 3.

D5 vs. D6

We have no reason to favour i₃ over i₄ or vice versa and both have exactly the same stake in our decision, so it is clear that we should be indifferent between both options.Footnote ²⁹ We might then think that attaching such indifferences to our choices should not change anything. In other words, when choosing between two distributions, our decision is unaffected by those elements between which we are indifferent. Call this

This principle implies that the presence of i₃ and i₄ should not affect our choice between D3 and D4 above. Prima facie, this principle will be attractive to many people. Like other separability principles, however, it is violated by some theories in distributive ethics, most notably egalitarian theories.Footnote ³⁰ This is because, for these theories, it is important how individuals fare relative to others. It is also violated by Limited Aggregation and much for the same reason.

For example, to see that Aggregation with Constraints violates Separability*, we can just take another look at

Two-Stage Case C

A and B are equally choiceworthy inasmuch they both include the same number of people with leg-claims, m. Between these two subdistributions we are indifferent. According to Separability* this should then not affect our choice. However, we have seen that under Aggregation with Constraints, adding these claims does affect what we should do.

Now, Horton's dilemma is the following. The first horn: he shows that views of Limited Aggregation either have implications like my Aggregation with Constraints has in Two-Stage Case C above. So, a violation of Separability* is what gives rise to the first horn of Horton's dilemma. The second horn is the following: we can construct cases where, according to Limited Aggregation, we ought to save group A over B, and group C over D, but in a choice between the union of groups A and C and the union of groups B and D, we ought to choose B and D.Footnote ³¹ In other words, we are violating a principle we can call

The specifics of Horton's case need not concern us here, but he is correct that views of Limited Aggregation can violate this principle. Underlying this implication, however, is, again, a violation of separability. This is not recognized by Horton.

Horton's principle follows from Separability* and transitivity. If we ought to choose A over B, then by Separability* we ought to choose (A&C) over (B&C). And if we ought to choose C over D, then by Separability* we ought to choose (B&C) over (B&D). If the transitivity of “ought to be chosen over” holds in this case,Footnote ³³ it follows that we ought to choose (A&C) over (B&D), which is Horton's principle.Footnote ³⁴ So, what gives rise to both horns of Horton's dilemma is the same thing: a violation of a separability. This means that Horton's objection amounts to the claim that all views of Limited Aggregation violate separability.

How surprising is this violation though? To see that it is not very surprising and actually a central part of the view, we just need to look at some of the different formulations of Limited Aggregation. According to Voorhoeve's ARC, “[a] claim is relevant if and only if it is sufficiently strong relative to the strongest competing claim.”Footnote ³⁵ And according to my Aggregation with Constraints, “if one person has much more at stake than any person whose interests conflict with hers, we ought to do whatever serves the interests of that person.” On views of Limited Aggregation, what we should do depends in part on how people fare relative to others. That these views then violate separability is not a surprise or an unwelcome implication, but rather a central building block of the theory. Limited Aggregation is a relational view.

It is important to re-emphasize, however, that this does not imply anything about the underlying axiological structure. Aggregation with Constraints is a deontic view and the separability violations I have outlined occur only at that level. In fact, I am sceptical that any plausible ethical view could have this structure at the level of axiology.Footnote ³⁶

This concludes my discussion of Horton's objection to Limited Aggregation. It wasn't my aim to invalidate the objection. Rather I wanted to show that it is not so much an objection to Limited Aggregation as a restatement of a central part of the view. I argue elsewhere in more detail that there is a tight connection between this violation of separability and scepticism about aggregation. There I argue that aggregation sceptics should be egalitarians, thus violating separability.Footnote ³⁷ For the remainder of this article, however, I now want to turn to a different set of objections that I need to address before Aggregation with Constraints can stand as a contender for the most plausible version of Limited Aggregation.

V. Cases with three or more alternatives

This objection is brought out when we apply my principle not to a binary choice but to a choice between three (or more) alternatives. Given the apparently intransitive structure of Limited Aggregation and the violation of plausible axioms of rational choice that follow from this structure, it is not surprising that Limited Aggregation faces problems when extended beyond binary choices. For Limited Aggregation, the relation “ought to be chosen over” is not always transitive. After all, as we have seen in many examples, it is the very idea of Limited Aggregation, that, if the numbers are right, we ought to save n people from losing an arm rather than Ann from death, and m people from losing a finger rather than n people from losing an arm, but we should not save m people from losing a finger rather than Ann from dying. This structure violates transitivity and in some cases in turn gives rise to a violation of other plausible principles of rational choice. Here, I do not want to add to these well-rehearsed general criticisms and responses to them.Footnote ³⁸ Rather I want to focus on one case of this kind which seems to pose a challenge specifically for Aggregation with Constraints.

Death and Finger vs. Death vs. Fingers. We can either save Ann from a terminal illness and Bob from losing a finger, Annie from the same terminal illness, or n people from losing a finger.Footnote ³⁹

In this case, Aggregation with Constraints seems to imply that we ought to choose C and save the people from losing a finger if there are enough of them. This is because, unlike, for example, Death vs. Headaches, it is no longer the case that “one person has much more at stake than any person whose interests conflict with hers” and so according to Aggregation with Constraints we ought to realize the best outcome, which by assumption means to satisfy the finger-claims, if n is large enough. One could plausibly argue that this is just as implausible as the claim that there is some n such that we ought to satisfy the finger-claims when the only other option is to save Ann, which would be equivalent to giving up our guiding intuition from Death vs. Headaches.

Another problem with a case like this is the following. If we consider pairwise comparisons of the three alternatives, we get a clear winner: A. Aggregation with Constraints would clearly choose A over C. It would also choose A over B. It is odd that the same view would then choose B out of all three options.

As I wrote above, because of their structural properties, such problems are bound to arise for any view of Limited Aggregation. However, if we are unsettled by this particular case, we could amend Aggregation with Constraints in a way that takes care of this case and that is in line with the rationale of the view.

This is first to eliminate all alternatives that do not include a claim that is close enough to the strongest claim. For Death and Finger vs. Death vs. Fingers this would mean eliminating C, and choosing A out of the remaining two alternatives. That is, we exclude all options that could not be chosen in a pairwise comparison with the alternative that includes the strongest claim(s), irrespective of the numbers involved. Only then do we apply Aggregation with Constraints as outlined above. This procedure is in line with the central idea that the strongest claim overall has special significance. The justification of our choice must first and foremost be directed at the person who has the greatest stakes in our decision.