1 By ‘pure case’ I mean a tragic choice situation in which only one factor (in my example, the probability of saving) differentiates the various choices.
2 The ‘all other things being equal’ clause must be understood to include any other feature of the people in each group. In this case, it includes the number of persons being saved, the length of the life after being saved, the quality of life of the saved persons, and so on.
3 This way of presenting the Numbers Case resembles Tooley's example in ‘An Irrelevant Consideration: Killing Versus Letting Die’, Killing and Letting Die. Second Edition, ed. B. Steinbock and A. Norcross (New York, 1994), p. 106. The primary difference is that each of your options is a positive action in the Numbers Case, whereas Tooley presents his example in order to discuss the symmetry (or asymmetry) between actions and omissions.
4 For a definition of ‘aggregation’, see Hirose I., ‘Aggregation and Numbers’, Utilitas 16.1 (2004), p. 66 (‘the combination of separate people's goods, happiness, losses, well-beings, and so on, into an objective value’).
5 Common-sense morality is not consequentialist on many moral issues. However, concerning Numbers Cases, it is clearly compatible with it.
6 Utilitarisnism, as I understand it, is a consequentialist theory, which also is aggregative and welfarist.
7 See Hirose I., ‘Saving the Greater Number without Combining Claims’, Analysis 61.4 (2001), and Hirose, ‘Aggregation and Numbers’, pp. 68–9. This solution is admittedly inspired by what Kamm calls the ‘aggregation argument’ (Kamm F. M., Morality, Mortality, vol. 1 (Oxford, 1993), pp. 85–7).
8 This does not mean that utilitarianism and common-sense morality will agree in every Probability Case. It is not clear, for example, that common-sense morality would require you to press button 2 if pressing button 1 would give A a 98 percent chance of survival instead of a 50 percent chance.
9 See Taurek J. M., ‘Should the Numbers Count?’, Philosophy and Public Affairs 5.4 (1977), pp. 303–4.
10 Admittedly, this result may not be much more counterintuitive than the idea that one should flip a coin when (in a Numbers Case) one has to choose between saving one person and saving one hundred. Still, in this case you are at least sure that one person will be saved.
11 Kamm, Morality, Mortality, pp. 116–17; Scanlon T. M., What We Owe to Each Other (Cambridge, Mass., 1998), p. 232.
12 Note that, when B's chances increase from 50 percent to 100 percent and you accordingly change your decision from flipping the coin to pressing button 2, you are sacrificing A completely in exchange for an increase of B's chances of survival. In Numbers Machine, on the other hand, when you change from flipping the coin to pressing button 2, you are sacrificing A completely in exchange for giving C some consideration. All this is of course controversial. One might plausibly argue that it is not true that C is not taken into account by flipping the coin (in fact, you would be giving him the same chance of survival as A and B). The only reason to press button 2 has to be that C is together with B. This would imply that the argument is, after all, aggregative. See Otsuka M., ‘Scanlon and the Claims of the Many versus the One’, Analysis 60.3 (2000) for this line of argument, and Kumar R., ‘Contractualism on Saving the Many’, Analysis 61.2 (2001) and Hirose, ‘Aggregation and Numbers’, pp. 72–3 for the opposite view. For my purposes, I do not need to push in this direction.
13 Timmermann J., ‘The Individualist Lottery: How People Count, But Not Their Numbers’, Analysis 64.4 (2004).
14 See Broome J., ‘Selecting People Randomly’, Ethics 95.1 (1984), p. 55. The idea is also discussed in Kamm, Morality, Mortality. Volume 1, pp. 128–9 and Scanlon, What We Owe to Each Other, pp. 233–4.
15 As Timmermann rightly points out, this solution is pragmatically identical to the individualist lottery. They are, however, theoretically different because the individualist lottery is (or at least attempts to be) non-aggregative.
16 See Taurek, ‘Should the Numbers Count?’, p. 302.
17 See Otsuka M., ‘Skepticism about Saving the Greater Number’, Philosophy & Public Affairs 32.4 (2004), pp. 421–3. Otsuka's example does not deal with different probabilities but with different amounts of harm, concretely, different numbers of limbs that can be restored.
18 I presented an earlier version of this article as a paper at the symposium ‘Current Problems in Moral and Legal Theory’, Universidad Torcuato Di Tella (June 2007). I am grateful to the audience for many helpful remarks. I also want to thank Marcelo Ferrante, Joshua Gert, Nora Muler, Michael Otsuka, the editor and an anonymous reviewer for their comments and advice.