² Nagel, Thomas, ‘Equality’, Mortal Questions (Cambridge, 1979), p. 126Google Scholar. First delivered as the Tanner Lecture at Stanford University in 1977.

³ Nagel, Thomas, The Possibility of Altruism (Princeton, 1970)Google Scholar.

⁴ Wattles, Jeffrey, The Golden Rule (Oxford, 1996)Google Scholar.

⁵ Joe South and the Believers, ‘Walk a Mile in My Shoes’ (1970).

⁶ Consequentialists, too, appeal to something like the universalizability or hypothetical consent. See, for example, R. M. Hare, ‘Ethical Theory and Utilitarianism’, and Harsanyi, John, ‘Morality and the Theory of Rational Behaviour’, Utilitarianism and Beyond, ed. Sen, A. and Williams, B. (Cambridge, 1982)Google Scholar.

⁷ Nagel, ‘Equality’, p. 125.

⁸ Parfit, Derek, Equality or Priority? The Lindley Lecture (University of Kansas, 1991)Google Scholar.

⁹ Kamm, Frances, Morality, Mortality: Vol. I (New York, 1983), ch. 6Google Scholar; Kamm, Frances, ‘Precis of Morality, Mortality Vol I: Death and Whom to Save From It’, Philosophy and Phenomenological Research 58 (1998), 939–45CrossRefGoogle Scholar; Rahul Kumar, ‘Unanimity and Aggregation’ (Unpublished MS); Kumar, Rahul, ‘Contractualism on Saving the Many’, Analysis 61 (2001), 165–70CrossRefGoogle Scholar; Scanlon, T. M., What We Owe to Each Other (Cambridge, 1998), chs. 5–9Google Scholar. See also Kumar, Rahul, ‘Defending the Moral Moderate: Contractualism and Common Sense’, Philosophy & Public Affairs 28 (1999), 275–309Google Scholar.

¹⁰ Otsuka, Michael, ‘Skepticism about Saving the Greater Number’, Philosophy and Public Affairs 32 (2004), 413–26CrossRefGoogle Scholar. Otsuka attributes numbers skepticism to Taurek, John, ‘Should the Numbers Count?’, Philosophy and Public Affairs 6 (1977), 293–316Google ScholarPubMed, and Anscombe, Elizabeth, ‘Who Is Wronged?’, The Oxford Review 5 (1967), 16–17Google Scholar.

¹¹ Otsuka, ‘Skepticism’, p. 421. Otsuka includes T. M. Scanlon, for example, in this larger group.

¹² Kamm, Frances, ‘Aggregation and Two Moral Methods’, Utilitas 17 (2005), p. 23CrossRefGoogle Scholar.

¹³ Nagel, ‘Equality’, pp. 123–5.

¹⁴ Nagel ‘Equality’, p. 123.

¹⁵ Some might insist that to take account of an individual's point of view, one must do something more, or different, than assess his well-being. Admittedly, on different conceptions of acceptability and well-being, the individual for whom an alternative is most unacceptable may not be its worst-off individual. But such divergence should not arise in ‘pure numbers’ cases, where differences in preferences, longevity, dependants, health, desert, etc. are stipulated away. In pure numbers cases, as discussed in section III, the individuals are assumed to be the same in all morally relevant respects. Some might insist that even if we use well-being as a proxy for acceptability, different conceptions of well-being may yield different rankings, depending in part on how subjective a conception of well-being is adopted. Again, though, we do not think that this problem complicates the resolution of pure numbers cases. Furthermore, in our discussion of Otsuka's own hypothetical, which is not a pure numbers case, we will follow him in treating the number of limbs restored as a measure of both well-being and acceptability.

¹⁶ Taurek, ‘Should the Numbers Count?’.

¹⁷ Taurek, ‘Should the Numbers Count?’ p. 307. At places in that article, Taurek appears to find an incommensurability between individual losses that would be inconsistent with pairwise comparison. But that strikes us as a distinct strain in his argument.

¹⁸ Anscombe puts forward a similar argument in holding that one is not required to save the larger group for the reason that one would thereby save a greater number of people in ‘Who is Wronged?’. However, in contrast to Taurek, Anscombe does not regard it as impermissible to act upon that reason to save the larger group.

¹⁹ Otsuka, ‘Skepticism’, p. 422.

²⁰ Otsuka notes that the numbers skeptic has additional reason to prefer (ii) to (iii) and (iii) to (iv). By restoring the use of two limbs in a person who has the use of no limbs rather than restoring the use of an additional limb in a person who already has the use of a limb, he aids the person who would otherwise be worse off in absolute terms. See Otsuka, ‘Skepticism’, p. 422.

²¹ Kamm, ‘Aggregation’.

²² Kamm, ‘Aggregation’, p. 22. Kamm also argues that the numbers skeptic has reason to reject consideration of benefit size. She writes, ‘pairwise comparison of degree of benefit depends upon a hidden form of tying that involves balancing and substitution of morally equivalent benefits that anti-numbers people reject’ (‘Aggregation’, p. 21). By ‘balancing and substitution of morally equivalent benefits’ Kamm has in mind the view that ‘we morally get everything we would have got if we had saved A if we instead save B’ – a view the numbers skeptic rejects (‘Aggregation’, p. 19).

²³ Kamm, ‘Aggregation’, p. 18.

²⁴ Kamm, ‘Aggregation’, pp. 22–3.

²⁵ Kamm writes, ‘when Otsuka imagines in the second scenario the anti-numbers theorist preferring to help someone who will have three limbs rather than help each of three people have two limbs, making there be a three-limbed person in existence could involve giving him only one pill, the same number any one of the three could get. This is because, for example, his system employs the pill more efficiently’ (‘Aggregation’, p. 22).

²⁶ Another question we do not resolve is whether someone could complain about an alternative in which he is not the worst-off individual when there is another alternative in which he would be better off and no one would be made worse off than she is in the initial situation, even if someone would remain worse off than the complainant is in the initial situation. (Could complain even if someone would be made worse off in the alternative than in the initial situation, but still not as badly off as he, the complainant, would be in that alternative despite his improvement?) We suspect that much depends on the absolute position of the complainant. If he is sufficiently well off, he won't have a complaint, even if he would be better off in an alternative in which no one would be worse off. This is a contentious issue, which we fortunately need not resolve. In Otsuka's hypothetical, the only individual who is not worst off and could be made better off (Fourth Person in (ii) and (iii)) could not be made better off without someone else being made worst off.

We also do not need to address the non-identity issues raised by the possibility that an individual present in some alternatives is absent in others. For our purposes, we can assume the same population in each – an assumption both Nagel and Scanlon appear to make in discussing the appraisal of results or principles.

²⁷ We use ‘becoming’ in an atemporal sense, consistent with the dynamic interpretation of pairwise comparison we have adopted: an individual ‘becomes’ worse off in one alternative if there is another in which he would have been better off.

²⁸ Fred's complaint about (iv) arguably would be weaker than Ed's, since Ned would have to lose two limbs for Fred to gain two, while Ned would only have to lose one for Ed to gain two. But that difference shouldn't matter to the numbers skeptic, since Fred's complaint about (iii) would be as strong as Ed's about (iv).

²⁹ Things may be more complicated if each of the trio doesn't get to keep his gains, e.g. if Ed gets two limbs in (iii) but Fred and Ted get them in (ii) (or, what amounts to the same thing, if ‘Ed’ etc. aren't rigid designators, but definite descriptions, which may fit different individuals in the four alternatives). But in that case, there will still be complaints about (iii) and (iv), since there will still be one individual in each who could have gained limbs in (iii), and one in (ii), without anyone else becoming limbless. And there will still be no complaints about (i) and (ii), since whoever is limbless in those alternatives could not have gained limbs without someone else becoming limbless.

³⁰ We could make a similar point using Gregory Kavka's argument that the numbers skeptic has intransitive preferences with respect to the three options of (i) saving one person who needs all the available supply of a drug; (ii) saving five others with the drug; and (iii) saving four of those five while letting the fifth die. Kavka claims that the numbers skeptic would be indifferent between (i) and (iii), generating an intransitivity (since the skeptic would be indifferent between (i) and (ii) and prefer (ii) to (iii)). See Gregory Kavka, ‘The Numbers Should Count’, Philosophical Studies 36 (1979), pp. 285–94. David Wasserman and Alan Strudler argue that the numbers skeptic might well prefer (i) to (iii), because (iii) ‘involves the gratuitous waste of a life’. See ‘Can a Nonconsequentialist Count Lives?’, Philosophy and Public Affairs 31 (2003), p. 74. Another way of putting this is in terms of pairwise comparison and the availability of a complaint. No one would have a complaint in (i), since none of the five could be made better off without the one being made as badly off as they are in (i). The fifth would have a complaint in (iii), since he could be made better off without anyone else being made as badly off as he is in (iii) (anyone, that is, who was not already as badly off as he is in (iii)). But if, per Otsuka, we could make only binary comparisons, the fifth could not make this complaint.

³¹ Arguably, Ned would have a complaint about (i) (though not as we've defined complaint), based on the fact that he could get a greater benefit by moving from (i) to (iv) than anyone else could get by any move. Ned's complaint would be that he, the worst off individual, could have received a greater benefit (by one limb) than anyone else without making anyone worse off than he, Ned, is now. But unless the worst off are to be accorded no priority at all, Ned's complaint would surely be far weaker than the complaint from a worst-off individual that he could have been made better off without making anyone as badly off as he, the complainant, is now. It is irrelevant that the limb-additions are stipulated to be of additive value. Going from two limbs to none – the cost Ed, Fred and Ted would each have to pay to give Ned three limbs – is morally more important than Ned's gain of one more limb than anyone else could acquire. If the numbers skeptic recognizes Ned's complaint in (i), he would give the alternatives the perfectly transitive ranking of (ii) > (i) > (iii) = (iv) < (ii). There still is no complaint in (ii), and Ned's complaint in (i) is weaker than Fred's in (iii) and Ed's or Fred's in (iv).

³² Kagan, Shelley, ‘The Additive Fallacy’, Ethics 99 (1998), pp. 23–4Google Scholar.

³³ It is striking that Otsuka employs another transport argument in the first part of his article. The argument in section II of his article characterizes the obligation to save lives as a disjunctive obligation. In the example, each of three people (A, B or C) will die if she does not receive some drug. There is enough of the drug to save one person. The rescuer then discovers a herb that, when mixed with the drug, will allow her to save two people. Otsuka writes that the numbers skeptic could affirm an obligation to save either A&B, or A&C, or B&C even though she need not commit herself to an obligation to save B&C rather than A alone, which is an obligation that Otsuka takes the numbers skeptic to deny (p. 417). Suppose it becomes impossible to save either A&B or A&C. Otsuka points out that some may find it surprising that fulfilling the initial obligation does not require one to save B&C over A. He writes, ‘why does a disjunctive obligation to save any two in the case under discussion – i.e., to save A&B, or A&C, or B&C – not reduce to an obligation to save B&C when one cannot save A&B or A&C’ (p. 420)? Otsuka must be assuming that the evaluation of alternatives is independent of the context of choice, but the obligation that one has when confronted with a choice of saving A&B, or A&C, or B&C, need not be the same as the obligation that one has when confronted with a choice of saving B&C or A. Otsuka's underlying assumption is made clear in n. 17 in which he points out that the numbers-skeptic violates the principle that ‘if x is to be preferred to y when they are elements of the feasible set S, then x must be preferred to y when they are elements of the feasible set T which is a subset of S’. This principle is similar to the axiom of ‘basic contraction consistency’ as described by Amartya Sen, ‘Internal Consistency of Choice’, Econometrica 61 (1993), p. 500. The plausibility of the axiom rests on the assumption that the ranking of an alternative does not change in the light of the feasible alternatives – an assumption we have argued against.