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Playing Dice with Morality: Weighted Lotteries and the Number Problem


In this article I criticize the non-consequentialist Weighted Lottery (WL) solution to the choice between saving a smaller or a larger group of people. WL aims to avoid what non-consequentialists see as consequentialism's unfair aggregation by giving equal consideration to each individual's claim to be rescued. In so doing, I argue, WL runs into another common objection to consequentialism: it is excessively demanding. WL links the right action with the outcome of a fairly weighted lottery, which means that an agent can only act rightly if s/he has actually run the lottery. In many actual cases, this involves epistemic demands that can be almost impossible to meet. I argue that plausible moral principles cannot make such extreme epistemic demands.

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1 Taurek, John, ‘Should the Numbers Count?’, Philosophy and Public Affairs 6 (1977), pp. 293316, at 307.

2 See, for example, Hooker, Brad, Ideal Code, Real World: A Rule Consequentialist Theory of Morality (Oxford, 2000).

3 See, for example, Rogers, Jason, ‘In Defense of a Version of Satisficing Consequentialism’, Utilitas 22 (2010), pp. 198221. The issue of demandingness will return in section V, below.

4 For example, Douglas Portmore's act-consequentialism includes both agent-relative constraints and agent-relative options. See e.g. Commonsense Consequentialism: Wherein Morality Meets Rationality (New York, 2011).

5 This solution was first proposed in by Frances Kamm in Morality, Mortality, vol. 1 (Oxford, 1993), pp. 129–41. It was given a contractualist interpretation in Scanlon, T. M., What We Owe to Each Other (Cambridge, Mass., 1998), pp. 114–21, and refined in Kumar, Rahul, ‘Contractualism on Saving the Many’, Analysis 61 (2001), pp. 165–70.

6 Hsieh, Nien-hê, Strudler, Alan and Wasserman, David, ‘The Numbers Problem’, Philosophy and Public Affairs 34 (2006), pp. 352–72.

7 One subtext to this debate is the extent to which non-consequentialist moral theory should accommodate the judgments of what Kumar calls ‘common-sense morality’ (‘Contractualism on Saving the Many’, p. 165).

8 As Timmerman, Jens puts it, ‘it seems unfair simply to leave the minority without any chance to be saved’ (‘The Individualist Lottery: How People Count, but Not Their Numbers’, Analysis 64 (2004), pp. 106–12, at 106).

9 This objection is raised in Otsuka, Michael, ‘Scanlon and the Claims of the Many versus the One’, Analysis 60 (2001), pp. 288–93; and in Wasserman, David and Strudler, Alan, ‘Can a Non-consequentialist Count Lives?’, Philosophy and Public Affairs 31 (2003), pp. 7194, though Wasserman and Strudler are also in favour of automatically saving the many.

10 John Broome agrees with Taurek that fairness requires tossing a coin, although he argues that the rescuer nevertheless ought to save the many, since in this case fairness is outweighed by the value of saving extra lives. ‘Kamm on Fairness’, Philosophy and Phenomenological Research 58 (1998), pp. 955–61, at 957.

11 This objection is raised by Scanlon, What We Owe to Each Other, p. 232, and Hirose, Iwao, ‘Weighted Lotteries in Life and Death Cases’, Ratio 20 (2007) pp. 4556, at 47.

12 Kamm calls this procedure ‘proportional chances’ (Morality, Mortality, pp. 128–42). She favours it in some cases, though not in those like the rescue case. Jens Timmermann (‘The Individualist Lottery’) calls it the ‘‘individualist lottery’,’ and defends its use in the rescue case.

13 Timmermann, ‘The Individualist Lottery’, p. 111.

14 Timmerman, ‘The Individualist Lottery’, p. 108.

15 Broome, John, ‘Fairness’, Proceedings of the Aristotelian Society 91 (1990), pp. 87101, at 96.

16 Thanks to an anonymous referee for Utilitas for suggesting this method.

17 Taurek, ‘Should the Numbers Count?’, p. 310.

18 Boats reappear in Kumar, ‘Contractualism on Saving the Many’; Wasserman and Strudler, ‘Can a Non-consequentialist Count Lives?’; Hsieh et al., ‘The Numbers Problem’; Timmerman, ‘The Individualist Lottery’; Hirose, ‘Weighted Lotteries in Life and Death Cases’; Liao, Matthew, ‘Who is Afraid of Numbers?’, Utilitas 20 (2007), pp. 447–61; Brooks, Thom, ‘Saving the Greatest Number’, Logique et Analyse 45 (2002), pp. 55–9; and Peterson, Martin, ‘The Mixed Solution to the Number Problem’, The Journal of Moral Philosophy 6 (2009), pp. 166–77.

19 An example from Foot, Philippa, ‘The Problem of Abortion and the Doctrine of Double Effect’, Virtues and Vices and Other Essays in Moral Philosophy (Oxford, 1977), pp. 1932, at 23.

20 A similar example appears in Hsieh et al., ‘The Numbers Problem’, p. 357.

21 Thanks to two anonymous referees for Utilitas for urging me to consider this objection.

22 Timmerman, ‘The Individualist Lottery’, p. 110.

23 This two-stage process – first, select an individual to be saved, second, save everyone else on his/her island – also appears in Kumar's ‘Contractualism on Saving the Many.’

24 Timmerman, ‘The Individualist Lottery’, p. 111.

25 Thanks to an anonymous reviewer for Utilitas for suggesting this line of argument.

26 Bernard Williams frequently argues that this kind of demandingness counts against utilitarianism. See, for example, ‘A Critique of Utilitarianism’, in Smart, J. J. C. and Williams, Bernard, Utilitarianism: For and Against (Cambridge, 1973).

27 Garrett Cullity argues that rights-based approaches are equally susceptible to such an objection. See The Moral Demands of Affluence (Oxford, 2004). Ashford, Elizabeth has made a similar argument with respect to contractualism in ‘The Demandingness of Scanlon's Contractualism’, Ethics 113 (2003), pp. 273304.

28 Several such approaches are gathered in The Problem of Moral Demandingness, ed. Timothy Chappell (Basingstoke, 2009).

29 Onora O'Neill distinguishes between motivational and practical demands in ‘Demandingness and Rules’, in Chappell, pp. 59–69, at 59. The account of practical demands she outlines is broader than the narrowly epistemic demands emphasized below. O'Neill does emphasize that such demands can be excessive even for agents who are motivated to follow them.

30 Peterson considers, and ultimately rejects, the use of a Gini-index at ‘The Mixed Solution to the Number Problem’, pp. 172–3.

31 David Sobel, for example, argues that the demandingness objection ‘cannot itself provide good reason to break with Consequentialism since it must presuppose the truth of prior and independent breaks with Consequentialism’ (Sobel, David, ‘The Impotence of the Demandingness Objection’, The Philosopher's Imprint 7 (2007), pp. 117, at 1). Kant is even less persuaded: ‘actions of which the world has perhaps so far given no example, and whose very practicability might be much doubted by one who bases everything on experience, are still inflexibly commanded by reason’ (Kant, Immanuel, Groundwork of the Metaphysics of Morals, trans. Mary Gregor (Cambridge, 1997), 4:408). And of course one common target of the demandingness objection, Singer, Peter, is of the view that morality does in fact make extremely stringent demands, perhaps most famously in ‘Famine, Affluence, and Morality’, Philosophy and Public Affairs 1 (1972), pp. 229–43.

32 Lenman, James presents a particularly forceful version of this objection in ‘Consequentialism and Cluelessness’, Philosophy and Public Affairs 29 (2000), pp. 342–70.

33 This distinction can be traced back to J. S. Mill's claim that an agent's motive ‘has nothing to do with the morality of the action’ even if it bears on ‘the worth of the agent’ (Utilitarianism (Indianapolis, 2001), p. 18). For more recent defences of this strategy, see Bales, R. Eugene, ‘Act-utilitarianism: Account of Right Making Characteristics or Decision-making Procedure?’, American Philosophical Quarterly 8 (1971), pp. 257–65; and Railton, Peter, ‘Alienation, Consequentialism, and the Demands of Morality’, Philosophy and Public Affairs 13 (1984), pp. 134–71.

34 Timmerman, ‘The Individualist Lottery’, p. 108 (emphasis in original).

35 Though perhaps not: see Broome, John, ‘Selecting People Randomly’, Ethics 95 (1984), pp. 3855; Kamm, Morality, Mortality, pp. 136–40; and Hirose, ‘Weighted Lotteries in Life and Death Cases’.

36 Thanks to Rahul Kumar, Gwen Bradford, and two anonymous reviewers at Utilitas for helpful comments on an earlier version of this article.

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