WHO IS AFRAID OF NUMBERS?
Forthcoming in Utilitas
© S. MATTHEW LIAO
Philosophy, Oxford University, Littlegate House, 16/17 St. Ebbes St., Oxford
OX1 1PT, UK; e-mail:
11 December 2006
Who Is Afraid of Numbers?
In recent years, many nonconsequentialists such as Frances Kamm and Thomas Scanlon have been puzzling over what has come to be known as the Number Problem, which is how to show that the greater number in a rescue situation should be saved without aggregating the claims of the many, a typical kind of consequentialist move that seems to violate the separateness of persons. In this paper, I argue that these nonconsequentialists may be making the task more difficult than necessary, because allowing aggregation does not prevent one from being a nonconsequentialist. I shall explain how a nonconsequentialist can still respect the separateness of persons while allowing for aggregation.
Who Is Afraid of Numbers?
I. The Number Problem
Many nonconsequentialists have been puzzling over what has come to be known as the Number Problem, which is how to show that the greater number in a rescue situation should be saved without aggregating the claims of the many, a typical kind of consequentialist move that seems to violate the separateness of persons. That is, suppose there are two islands, one with one person, A, and the other with two people, B&C. There is a tsunami and both islands will soon be immersed in water, killing whoever is on the island. You only have time to go to one of the islands to rescue the people on it. Other things being equal, e.g., assume that there is no morally relevant difference (e.g. special relationship or prior agreements) between these individuals. To which island should you go?
For consequentialists, the answer seems simple: Save the greater number because consequentialism aims to produce the best state of affairs and, other things being equal, more aggregate lives saved may be a better state of affairs than fewer lives saved. Anti-number advocates such as John Taurek argue, on the other hand, that there is no reason to save the greater number per se, because none of the individuals on either island can claim that it is any worse for him to die than it is for the others. As Taurek explains, “Five individuals each losing his life does not add up to anyone’s experiencing a loss five times greater than the loss suffered by any of the five.” As a possible solution, Taurek suggests that perhaps one could flip a coin to decide whether to save the one or the larger group.
Nonconsequentialists who do not want to aggregate the claims of the many, because it seems to violate the separateness of persons – and who, at the same time, do not accept, as Taurek does, that numbers do not matter – have in recent years advanced several novel solutions to the Number Problem. Let us call these nonconsequentialists “pro-number nonconsequentialists.” Others have however argued that the solutions advanced by the pro-number nonconsequentialists do not actually solve the Number Problem, because these solutions covertly involve combining the claims of the greater number.
In this paper, I shall argue that pro-number nonconsequentialists may be making the task more difficult than necessary and that there may be a simpler nonconsequentialist solution to the Number Problem. In particular, I shall argue that a nonconsequentialist can permit aggregation and still respect the separateness of persons.
II. The Kamm-Scanlon Argument and the Weighted Lottery Argument
I begin by giving an overview of two of the best known pro-number nonconsequentialist solutions: the Kamm-Scanlon Argument and the Weighted Lottery Argument.
Developed by Frances Kamm and incorporated into a contractualist framework by Thomas Scanlon, Kamm’s Balancing Argument and Scanlon’s Tiebreaking Argument, or the Kamm-Scanlon Argument, for short, says that in cases of unequal numbers, the ‘unbalanced’ individual must decide the outcome or serve as a tie-breaker. For example, suppose one group contains A, and the other group contains B&C, and other things being equal, the Kamm-Scanlon Argument says that one should begin by balancing off equal and opposing individual claims or needs, e.g. A versus B. Once this is done, C’s claim then serves as a tiebreaker. The result is that we should save the greater number, B&C instead of A.
According to its advocates, the Kamm-Scanlon Argument solves the Number Problem because each person’s claim is taken seriously and is not dismissed just because an individual happens to be a member of a comparatively smaller group. As Scanlon explains, the claims of each person who could be saved have “the same moral force,” and “any nonrejectable principle must direct an agent to recognize a positive reason for saving each person.” Moreover, since B&C are saved, because C’s claim is unbalanced rather than because there is a greater number on the side of B&C, therefore, so the argument goes, this approach does not involve aggregation. As Kamm makes clear, “no one of those in the larger group whose presence is balanced by individuals in the smaller group has a complaint if we do not save the greater number. Only an individual who is not balanced out has a complaint. Those balanced out on his side are, I would say, the beneficiaries of his successful complaint.”
A number of writers have however argued that the Kamm-Scanlon Argument covertly involves combining claims. In particular, Michael Otsuka has argued that in order for C’s claim to count in favor of B&C against A, B must be present. Using a balancing scale metaphor, Otsuka notes that the Kamm-Scanlon Argument requires that one first place A on one of the scales and B on the other scale, at which point the scales balance evenly. One then tip the scales to B’s side by placing C on B’s scale. This suggests that the Kamm-Scanlon Argument still involves covertly combining the claims of B and C. As Otsuka explains, if B were not present, then A and C would balance each other out, and C would not be unbalanced.
Kamm’s response to Otsuka’s challenge is that the Kamm-Scanlon Argument need not involve combining claims in the manner Otsuka has suggested. In particular, Kamm argues that the Kamm-Scanlon Argument can rely on a reasoning process similar to the Argument from Best Outcome, which says the following: Suppose you can save A or B&C. On the basis of Pareto Optimality, it seems worse if both B and C die than if only B dies. At the same time, a world in which A survives and B dies seems just as bad as a world in which A dies and B survives. Given this, it seems that we can substitute A for B on one side of the equation. If so, we obtain the outcome that it is worse if B and C die than if only A dies. The Argument from Best Outcome can be represented as follows:
1. You can save A or B&C.
2. According to Pareto Optimality, saving B&C is better than saving B alone.
3. Saving A is equivalent to saving B.
4. From 3, one can substitute A with B.
5. Therefore, saving B&C is better than saving A alone.
This argument seems then to show that the Kamm-Scanlon Argument need not involve combining claims; it only requires the condition of Pareto Optimality, which is not an aggregative condition.
Kamm anticipates an objection to the Argument from Best Outcome, what she calls the Sore Throat Case. Again, suppose you can save A or B&C. This time, however, while A’s and B’s lives are at stake, C only has a sore throat. As before, saving B&C is better than saving B alone, given Pareto Optimality. Saving A seems equivalent to saving B. If we substitute A with B, it seems that we would save B&C instead of A. Yet, in this case, it is not clear that we should save B&C and give A no chance at all of being saved, given that C’s sore throat seems comparatively trivial. Kamm’s answer to the Sore Throat Case, based on her idea of an “irrelevant utility,” is that the balancing method is appropriate only when the nature of the need of any additional person on one side is serious enough relative to the needs of those in a tie. If the need of the additional person is not serious enough, that is, if the need is irrelevant relative to the needs of those in a tie, e.g. C’s throat versus A’s life, then, according to Kamm, we should treat the case as if only the individuals in a tie are involved. In such a case, we would toss a coin to give each an equal chance.
On its most well-known manifestation, the Weighted Lottery Argument says that if everyone has an equally strong claim to being saved, but if we cannot satisfy the claims equally, then we should give each a 1/(the number of people) chance to be saved. For example, in the case of saving either A or B&C, we cannot save all three. Given this, we should, according to the Weighted Lottery Argument, give each a 1/3 chance of being saved. According to some of its advocates, the weighted lottery then solves the Number Problem if one accepts that if and when B is selected, then having reached B, one should also save C. Or, if and when C is selected, then having reached C, one should also save B. In other words, the Number Problem is allegedly solved on this view, because each has been given a 1/(the number of people) chance, and because those in the group with the greater number will stand a better chance of benefiting from the good luck of someone in that group. As one of its advocates says, “depending on the numbers, it is – more, much more, vastly more – likely that the many will be saved.”
It might be argued that the Weighted Lottery Argument gives the individual in the lesser group too much weight. Suppose there are one million people on one side and one individual on the other. And suppose a one-million-and-one sided dice has been cast in favor of the one individual. One would be required to save the one individual on this approach. But this seems counterintuitive. Still, advocates of the weighted lottery could just bite the bullet and assert that this is what is required to solve the Number Problem.
It might also be pointed out that the fact that the weighted lottery makes it more likely than not – perhaps even overwhelmingly likely – that the greater number will be saved is insufficient to establish that the Weighted Lottery Argument solves the Number Problem, given that the Number Problem is that of explaining why we should always save the greater number in these cases. Again, advocates of the weighted lottery might bite the bullet and argue that this is the best available solution to the Number Problem.
There may be other problems with the Weighted Lottery Argument, but I shall not explore them here. For, the Kamm-Scanlon Argument and the Weighted Lottery Argument also face what might be called the Separateness of Persons Objection.
III. A Separateness of Persons Objection
It might seem ironic that the Kamm-Scanlon Argument and the Weighted Lottery Argument would have problems with the Separateness of Persons Objection. After all, following Rawls’s criticism that utilitarian aggregation, which seems to presuppose a notion of an overall good, is defective because it does not respect the distinctiveness of persons, and Nagel’s argument that an impartial concern for persons is best grounded in pairwise comparison rather than in the aggregation of the claims of separate individuals, pro-number nonconsequentialists believe that what is distinctive about their approach is that they are committed to taking each individual seriously. For example, Kamm states that “If the presence of each additional person would make no difference, this seems to deny the equal significance of each person.”
However, consider the Argument from Best Outcome, the reasoning process of which is similar to the Kamm-Scanlon Argument. It could be argued that someone who believes in the separateness of persons would not allow Premise 4, that one can substitute A with B. After all, on a certain view of the separateness of persons, each person is incommensurable; by incommensurability, I mean something like incomparability and something different from “rough equality” – to borrow a phrase from Joseph Raz. For example, Taurek might be interpreted as holding this view when he says that he would not trade one person's arm for another persons' life. Certainly, outside the literature on saving the greater number, many people also hold that persons are incommensurable. If A and B are both incommensurable, how can one substitute A with B? It is true that A’s and B’s death will both be bad. But unlike ordinary objects which can be substituted and replaced and can be “roughly equal,” arguably A’s and B’s lives are not interchangeable or replaceable, given that A and B are incommensurable.
The Separateness of Persons Objection can also be applied to the Kamm-Scanlon Argument itself. If A and B are both incommensurable, it seems that A’s life could not be ‘balanced’ against B and vice versa, because ‘balancing’ implies that the lives of A and B are commensurable.
In fact, the Separateness of Persons Objection also applies to Nagel’s method of Pairwise Comparison, the method on which the Kamm-Scanlon Argument is based. The Kamm-Scanlon Argument uses the method of Pairwise Comparison to give each person’s life equal and positive weight by engaging in a series of pairwise comparisons of the claims of members of opposing groups. For example, suppose there is A on one side and B&C on the other. A’s claim to being saved would first be compared with B’s claim. Given that the two are equally weighty, the Kamm-Scanlon Argument then says that since C is unbalanced, therefore one should save B&C. However, the method of Pairwise Comparison also faces the Separateness of Persons Objection, because if persons are incommensurable, then surely one could not perform pairwise comparisons. Indeed, if A and B are both incommensurable, one could not compare the claim of one with the claim of the other, as the two claims would simply be incommensurable.
It is worth noting that this point further applies to those who might seek to use the method of Pairwise Comparison to defend the opposite claim, that is, the Taurekean claim that numbers do not matter. For example, suppose there is A on one side and B&C on the other. It might be argued that after comparing A’s claim to B’s, one should not stop; instead one should continue the pairwise comparison and compare A’s claim to C’s. Once one does this, it would seem to follow that everyone’s claim is equally weighty. If so, one might draw the conclusion that one should, following Taurek, flip a coin to give each of A, B and C a ½ chance to be saved. Again, however, if A, B and C are all incommensurable, such a method of pairwise comparison could not be applicable, as the claims of A, B and C would simply be incommensurable.
Finally, the Weighted Lottery Argument also faces the Separateness of Persons Objection. Indeed, if each person is incommensurable, how can each be divided so as to receive only 1/(the number of people) chance to be saved? Arguably, someone who holds the view that persons are incommensurable could argue that incommensurable values simply cannot be divided and proportioned.
Once we see that if persons are incommensurable, then one cannot substitute or balance one life and another or give one individual only 1/(the number of people) chance to be saved, a particular interpretation of Taurek’s anti-number solution becomes quite plausible. For, when one tosses a coin as Taurek suggests, one need not thereby claim that persons are commensurable and can be substituted or balanced or be given some proportional chance. Instead, one could merely be making a decision to save someone, given that everyone is incommensurable, so that saving none at all is the only thing one should not do. In fact, someone who holds the view that persons are incommensurable may not even need to toss a coin. She may simply choose to save someone. Indeed, as noted earlier, Taurek does not say that one must toss a coin. He says that “Perhaps I could flip a coin.” If this is right, then present pro-number solutions such as the Kamm-Scanlon Argument and the Weighted Lottery Argument seem to stand on shaky grounds.
IV. The Arbitrariness Objection
Fortunately, pro-number nonconsequentialists can avoid the Separateness of Persons Objection by rejecting the particular view of separateness of persons that underlies this objection. However, I shall argue that their alternative view of separateness of persons, which refuses to allow aggregation, faces the problem of arbitrariness.
The particular view of the separateness of persons that underlies the Separateness of Persons Objection is the view that persons are incommensurable. Call this the Persons Are Incommensurable View (PAI). While PAI has some plausibility, it has the following counterintuitive implication. Suppose A is in danger of breaking his finger and B is in danger of losing her life. Other things being equal, it seems that we should save B instead of A. However, if, as PAI says, all persons are incommensurable, then there seems to be no way of comparing A and B. If so, then on PAI, it seems that one would be permitted to save either A or B; the only thing one should not do is to do nothing. This seems counterintuitive though, since it seems that we should help B given that B’s life is at stake but A only stands to break a finger. Therefore, it seems that one should reject PAI since it has this implication. If so, and if the Separateness of Persons Objection relies on PAI, then one could also dismiss the objection on this basis. Call this the Broken Finger Objection.
The rejection of PAI opens up an opportunity for pro-number nonconsequentialists to offer a different view of the separateness of persons, one in which pairwise interpersonal comparison, substitution, balancing and division would be permitted. In fact, though not often noted, pro-number nonconsequentialists do have a different view of the separateness of persons. They hold what might be called a Persons as Claimants View (PAC). On PAC, each person’s uniqueness gives each a separate equal claim. Indeed, recall that Scanlon takes it to be the case that “the claims of each person who could be saved as having the same moral force.” Unlike PAI, PAC further holds that these claims are commensurable in certain ways, in particular, they can be compared, balanced out, substituted, and divided. Given this assumption, PAC could handle the Broken Finger Objection, since an individual’s life is arguably weightier than another individual’s finger. Therefore, PAC would require that we save the individual who stands to lose her life instead of the individual who only stands to lose his finger.
The problem with PAC though is that its stopping point seems arbitrary. While it permits comparison, balancing, substitution and division, it refuses to permit aggregation. One might ask, why not? If equal claims can be interpersonally substituted and divided, why can they not be aggregated? They would still be equal claims that are aggregated.
Of course, pro-number nonconsequentialists have shied away from aggregation because they think that it violates the separateness of persons. But it seems that pairwise interpersonal comparison, balancing, substitution, and division may also do the same. Echoing Rawls's discussion of this matter, Robert Nozick explains the problem of making a person undergo some sacrifice for some 'overbalancing' good:
Individually, we each sometimes choose to undergo some pain or sacrifice for a greater benefit or to avoid a greater harm: we go to the dentist to avoid worse suffering later; we do some unpleasant work for its results . . . In each case, some cost is borne for the sake of the greater overall good. Why not, similarly, hold that some persons have to bear some costs that benefit other persons more, for the sake of the overall social good? . . . There are only individual people, different individual people, with their own individual lives. Using one of these people for the benefit of others, uses him and benefits the others. Nothing more . . . To use a person in this way does not sufficiently respect and take account of the fact that he is a separate person, that his is the only life he has. He does not get some overbalancing good from his sacrifice, and no one is entitled to force this upon him. . . .
To give just one example, notice how easily one can transform this passage into an objection to substituting the equivalent good of one individual for another:
Individually, we each sometimes choose to undergo some pain or sacrifice for an equivalent benefit or to avoid an equivalent harm: we do some unpleasant work now so that we would not have to do it later . . . some cost is borne for the sake of an equivalent good. Why not, similarly, hold that some person has to bear some costs that benefit another person, as long as the good of one individual is equivalent to that of another? . . . There are only individual people, different individual people, with their own individual lives. Using one of these people for the benefit of another, uses him and benefits the other. Nothing more . . . To use a person in this way does not sufficiently respect and take account of the fact that he is a separate person, that his is the only life he has. He does not get some substituted, equivalent good from his sacrifice, and no one is entitled to force this upon him. . . .
As far as I can tell, it will be difficult for pro-number nonconsequentialists to show how pairwise interpersonal comparison, balancing, substitution and division are any more or less respectful of the separateness of persons than aggregation is. Note that I am not arguing that nonconsequentialists are wrong to criticize aggregation from the perspective of the separateness of persons. Rather, I am claiming that from this perspective, it is hard to see why substitution and the like would be permitted, but not aggregation. If I am right, PAC seems to avoid the implausibility of the PAI but only at the cost of being arbitrary. Call this the Arbitrariness Objection.
V. A Simple Solution to the Number Problem
At this point, pro-number nonconsequentialists might believe that they are in a dilemma: Given the Arbitrariness Objection, it might seem that if they still reject aggregation, then they must also reject pairwise interpersonal comparison, substitution, and the like. If so, they would in effect be embracing the Taurekean position that numbers do not count. Or, it might seem that they must embrace aggregation and thereby whole-sale consequentialism. I shall now argue that this dilemma may be more apparent than real. There may be an easy way out for pro-number nonconsequentialists, namely, non-consequentialists can accept aggregation and still respect the separateness of persons. How is this possible?
Intuitively, it seems that what distinguishes nonconsequentialism from consequentialism (of the simpler sort at least) is not that states of affairs do not matter at all, but that there are other considerations that also matter such as an agent’s intentions, justice, and so on. Indeed, as Rawls himself says, “All ethical doctrines worth our attention take consequences into account in judging rightness.” For example, consider the Riot Case in which a thousand people will die from a riot unless one prosecutes an innocent individual. A nonconsequentialist need not deny that the number of people who will die may be one consideration in determining what one ought to do. However, unlike a consequentialist (of the simpler sort at least) where the only relevant consideration may be the number of lives at stake, a nonconsequentialist may argue that numbers are not the only relevant consideration; one also needs to consider whether, for example, it is just to prosecute an innocent individual. Once justice is taken into account, a nonconsequentialist may conclude that in this case, the consideration of justice should override the consideration based solely on numbers, and therefore we should not prosecute the innocent individual.
Or, take the classic Transplant Case, in which to save five people from various organ failures, we must intentionally kill a healthy individual and extract his organs. Again, that there are five lives versus one is one consideration in this matter. But for a nonconsequentialist, it will not be the only consideration; the intentional killing of an innocent individual may provide a further consideration in the deliberative process regarding what we ought to do. In such a case, a nonconsequentialist may conclude that intentionally harming an innocent individual is something one should not do, even if this means letting five people die. Call this the Standard Picture of nonconsequentialism.
From the Standard Picture, we have a straightforward solution for the Number Problem. Numbers are one consideration that must be taken into account in our moral deliberation. If there were no other relevant considerations, then numbers would determine what one ought to do. For example, in the case of saving A or B&C, other things being equal, we should save B&C, because numbers are the only relevant factor in this case.
The Standard Picture solves the Number Problem because it holds a plausible account of the separateness of persons. In particular, it holds what might be called the Persons as Agents View (PAA) of the separateness of persons. PAA says that what is distinctive about persons is not that they are incommensurable or that they each embody an equal claim that cannot be aggregated but can be substituted, compared and so on; but that they are moral agents capable of deliberating and being persuaded by moral reasons. Therefore, on PAA, we respect individuals as separate persons when we treat them not just as claimants to be balanced and weighed, but as rational, moral agents who can respond to reasons. This means that even if aggregation presupposes a notion of an overall good, and aggregation is taken into consideration in our moral deliberation, since individual moral agents remain the sources of normativity on PAA, the employment of aggregation must ultimately be justifiable to each moral agent from her point of view.
For example, in the Riot Case, although a great number of people would die if we did not prosecute the innocent individual, PAA holds that moral agents – even the ones who might die as a result of our not prosecuting the innocent individual – could recognize that the consideration of justice may be a legitimate moral reason not to prosecute the innocent individual. Similarly, PAA holds that moral agents could recognize that if numbers are the only relevant factor at issue, then that is a reason in favor of saving the greater number. PAA is more plausible than PAI or PAC because in accepting that aggregation is one important input in our moral deliberation, PAA avoids both the Arbitrariness Objection and the Broken Finger Objection.
At the time, the Standard Picture is not consequentialism because, as the Riot Case and the Transplant Case show, if there are other considerations such as justice and an agent’s intentions, then these other considerations could outweigh the consideration based on numbers. In other words, while consequentialism (of the simpler sort at least) may take into account only the number of lives at stake, the Standard Picture would take into account other considerations such as an agent’s intention, justice, and so on. Hence, numbers play a role in the Standard Picture only as one input among many in the deliberative process of a moral agent. As such, unlike consequentialism, numbers are not the only thing that matters on the Standard Picture.
Some might accept that the Standard Picture is different from consequentialism of the simpler sort, but they might ask how it is different from consequentialism of the more sophisticated sort, which holds that values such as an agent’s intention, justice, and so on, can be, and in fact are already, quantified and incorporated into a broader, consequentialist framework. A response that advocates of the Standard Picture could make is to deny that the further quantification of these values into a broader consequentialist framework is possible. For example, it might be said that there is just no metric to weigh numbers against considerations of justice. This response of course raises the question of how advocates of the Standard Picture would prioritize numbers versus these other values. It is beyond the scope of this paper to discuss this point in full, but advocates of the Standard Picture might be able to employ some sort of prioritarian or sufficientarian principle of prioritization, or some method of reflective equilibrium, none of which is necessarily consequentialist. In any case, I suspect that herein lies one real difference between consequentialism and nonconsequentialism. That is, the problem of the prioritizing numbers versus other values, and not of whether aggregation is permissible, is one factor in determining whether one is a consequentialist or not. If this response is adequate, the Standard Picture is also different from consequentialism of the more sophisticated sort.
Others might argue that there are other reasons to be concerned about aggregation, even if aggregation is no more disrespectful of the separateness of persons than substitution and the like. For example, it has been said that aggregation could lead to a large number of small harms adding up to outweigh a smaller number of large harms, or to the repugnant conclusion. Indeed, Scanlon’s contractualist alternative to consequentialism, which excludes the combined claims of different individuals, is motivated by these concerns. For example, Scanlon’s Television Studio Case, where we are confronted with a choice between interrupting the viewing pleasures of a billion World Cup viewers in order to save Jones who had an accident in the transmitter room and is suffering serious pain from electrical shock, and allowing Jones to be pain until the match is over, is intended to illustrate Scanlon’s worry that aggregation would allow a large number of small harms adding up to outweigh a smaller number of large harms.
But, setting aside the fact that some philosophers actually embrace these implications of aggregation, which suggests that these implications are not obviously wrong, it is not necessary for nonconsequentialists to reject aggregation in order to avoid these implications. For example, to avoid the implication that a large number of small harms can add up to outweigh a smaller number of large harms, nonconsequentialists could employ (and have employed) something like the Principle of Triviality to constrain aggregation without rejecting it. The Principle of Triviality, which is another way of expressing Kamm’s idea of an irrelevant utility or Parfit’s Triviality Principle, says that if some benefits, A, are too trivial when compared to others benefits, B, then any amount of B should outweigh any amount of A. Hence, on this Principle, if the small harms are too trivial when compared with the large harms, then no amount of the small harms can add up to outweigh a smaller number of large harms, thereby constraining aggregation. So, in Scanlon’s Television Studio Case, the trivial pleasures of a billion World Cup viewers would not outweigh the need to alleviate Jones from suffering serious pain. At the same time, the Principle of Triviality does not reject aggregation because if the harms were not trivial, then aggregation would still be permitted.
Lest one thinks that the Principle of Triviality is available only to those who accept the Kamm-Scanlon Balancing/Tiebreaking Approach, it is worth noting how this Principle is independent of such an approach. Recall Kamm’s Sore Throat Case, where one can save A’s life or save B’s life and cure C’s sore throat. On the Balancing/Tiebreaking Approach alone, it should be the case that the additional utility of curing C’s sore throat would break the tie in favor of B&C. Kamm argues though that we should flip a coin in such a case, because curing C’s sore throat is an “irrelevant utility,” given that the alternative would deprive A of her 50 percent chance of being saved. The employment of the idea of an irrelevant utility, which leads to a different recommendation than employing the Kamm-Scanlon Balancing/Tiebreaking Approach alone, shows that the former is independent of the latter. Hence, to deal with some of the implications of aggregation, arguably nonconsequentialists have resources to constrain aggregation without rejecting it.
Pro-number nonconsequentialists have been puzzling over how to show that the greater number in a rescue situation should be saved without aggregating the claims of the many, a typical kind of consequentialist move that seems to violate the separateness of persons. In this paper, I argued that pro-number nonconsequentialists may be making the tasks more difficult than necessary because on the Standard Picture of nonconsequentialism, a nonconsequentialist can allow aggregation and still respect the separateness of persons. In particular, I argued that on the Standard Picture, what distinguishes nonconsequentialists from consequentialists is not that numbers and aggregation do not matter, but that they matter only as one input among many in a deliberative, practical reasoning process about what a moral agent ought to do. For a nonconsequentialist, other considerations such as an agent’s intentions, justice, and so on, could also be relevant. Given this, nonconsequentialist may not have to shy away from aggregation when numbers are the only relevant factor at issue. If this is right, the Standard Picture provides a much simpler solution to the Number Problem for pro-number nonconsequentialists.
 For an excellent overview of this topic, see Wasserman, D., and A. Strudler. "Can a Nonconsequentialist Count Lives?" Philosophy and Public Affairs 31, no. 1 (2003): 71-94.
 Roger Crisp has noted though that even consequentialism may not always require one to save the greater number in the tsunami case, as such a requirement assumes that the population before the tsunami is optimal, but we may just have no clue regarding whether saving more lives will produce the best state of affairs.
 Taurek, J. "Should the Numbers Count?" Philosophy and Public Affairs 6 (1977): 293-316.
 Taurek, "Should the Numbers Count?," p. 307.
 Taurek, "Should the Numbers Count?," p. 303.
 Kamm, F. Morality, Mortality Vol. I:
Death and Whom to Save from It.
 See, e.g., Otsuka, M. "Scanlon and the Claims of the Many Versus the One." Analysis 60 (2000): 288-93.
 Kamm, Morality, Mortality, p. 101; Scanlon, What We Owe to Each Other, p. 232.
 Scanlon, What We Owe to Each Other, p. 232.
 Kamm, F. "Owning, Justifying, and Rejecting." Mind 111 (2002): 323-54.
 Otsuka, "Scanlon and the Claims of the Many Versus the One," op. cit.
 Kamm, F. "Aggregation and Two
Moral Methods." Utilitas 17, no. 1 (2005): 1-23, pp. 11-12. See also Hirose,
 Michael Otsuka has helpfully noted that the conclusion of this argument is not that one should save the greater number. Instead, the conclusion is that saving the greater number is better. According to Otsuka, one needs an additional premise – such as the consequentialist premise that one should bring about the best outcome – to get to the conclusion that one should save the greater number.
See, e.g., Hirose,
 Kamm, Aggregation and Two Moral Methods, pp. 12-15.
 Kamm, Aggregation and Two Moral Methods, p. 13; Kamm, F. Morality, Mortality Vol. I, p. 103.
 Kamm, Aggregation and Two Moral Methods, p. 15.
 John Broome first considered the
weighted lottery, though he did not endorse it.
See Broome, J. "Selecting People Randomly." Ethics 95 (1984):
38-55. See also Timmermann, "The
Individualist Lottery," op. cit.; Brock, D. "Ethical Issues in Recipient
Selection for Organ Transplantation." In Organ Substitution Technology:
Ethical, Legal, and Public Policy Issues, edited by Deborah Mathieu, 86-99.
 Timmermann, "The Individualist Lottery," pp. 110-111; Kumar, R. "Contractualism on Saving the Many." Analysis 61 (2001): 165-70.
 Timmermann, "The Individualist Lottery," p. 111.
 I thank an anonymous referee for this point.
 See Hirose, "Weighted Lotteries in Life and Death Cases," op. cit. for other arguments against the Weighted Lottery Argument.
 Rawls, A Theory of Justice,
 Kamm, F. "Precis of Morality, Mortality? Vol. I: Death and Whom to Save from It." Philosophy and Phenomenological Research 58 (1998): 939-45, pp. 940-41.
 See, e.g., Raz, J. The Morality of
 Taurek, "Should the Numbers Count?," p. 302. It should be noted that Taurek goes on to say that "There may well come a point, however, at which the difference between what B stands to lose and C stands to lose is such that I would spare C his loss. But in just these situations I am inclined to think that even if the choice were Bs he too should prefer that C be spared his loss" (Taurek, p. 302). This further remark may spare Taurek from holding the view that persons are incommensurable, but his position may become confused. In particular, why could someone not say that the point at which one should spare C his loss is precisely when C is in a larger group than B, and that even if the choice were Bs he too should prefer that C be spare his loss, that is, the greater number should be saved? Michael Otsuka has suggested though that Taurek can reject this line of thought by drawing a distinction between pairwise comparisons (which do not involve any appeal to groups) and those comparisons that involve appeals to groups. Let me preface by noting that I do not think that the existence of the view that each person is incommensurable necessarily depends on Taurek’s having held this view. Indeed, as I proceed to point out in the main text, other people have also held this view. This said, let me express some reservations regarding this interpretation of Taurek. In particular, elsewhere, Otsuka has argued that the anti-number position leads to a choice-defeating intransitivity as a result of endorsing the principle of nonaggregation and affirming pairwise comparisons ("Skepticism about Saving the Greater Number." Philosophy and Public Affairs 32: 413-26, 2004). Given this, interpreting Taurek as holding instead the view that persons are incommensurable may enable Taurek to reach his anti-number position while avoiding Otsuka’s charge of choice-defeating intransitivity. See, however, Meyer, K. (2006). "How to be Consistent without Saving the Greater Number." Philosophy and Public Affairs 34(2): 136-146, for the argument that Taurek’s position does not lead to choice-defeating intransitivity even if it affirms pairwise comparisons. As I shall shortly argue though, from the perspective of the view that persons are incommensurable, the method of Pairwise Comparison is itself also problematic.
 See, e.g. Murphy, M. Natural Law in
Jurisprudence and Politics.
 Nagel, T. "Equality," op. cit., pp. 122-27; Kamm, Morality, Mortality, p. 87.
 See Otsuka, "Scanlon and the Claims of the Many Versus the One," pp. 290-91, for this interpretation of the anti-number position.
 Taurek, "Should the Numbers Count?," p. 303.
 Scanlon, What We Owe to Each Other, p. 232.
R. Anarchy, State and Utopia.
 Rawls, A Theory of Justice, p. 30.
 For the proposal of using reflective equilibrium in this context, see Otsuka, M. "Saving Lives, Moral Theory, and the Claims of Individuals." Philosophy and Public Affairs 34, no. 2 (2006): 109-35. Note that one difference between my proposal of the Standard Picture and Otsuka’s proposal of reflective equilibrium is that the Standard Picture explicitly permits aggregation, whereas it is not clear whether reflective equilibrium would permit aggregation.
 Another difference between consequentialism and nonconsequentialism may be whether the theory has a maximizing structure or not. See, e.g., Broome, John. Weighing Goods: Equality, Uncertainty and Time: Blackwell, 1991.
 See, e.g., Parfit, D. Reasons and
 Scanlon, What We Owe to Each Other, p. 235.
 This Principle is stated in a way
that B has lexical priority over A. If
this is too strong, one could state it in a discontinuity form, which would say
that if some benefits, A, are too trivial when compared to others benefits, B,
then enough of B should outweigh any
amount of A. See
Similarly, nonconsequentialists arguably have resources to address the
repugnant conclusion without rejecting aggregation. For example, they could employ something like
would like to thank Michael Otsuka, David Wasserman, Roger Crisp,