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Mixed Feelings About Mixed Solutions

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The numbers problem concerns the question of what is the right thing to do in trade-off cases where one can save different non-overlapping groups of persons, but not everyone. Proponents of mixed solutions argue that both saving the many and holding a lottery to determine whom to save can each be morally right in such cases, depending on the relative sizes of the groups involved. In his book The Dimensions of Consequentialism, Martin Peterson presents an ingenious version of such an approach that avoids a commitment to interpersonal value aggregation, which is highly controversial and rejected by many philosophers for a number of reasons. I criticise Peterson’s proposal by first arguing that it cannot account for the idea that holding a lottery is morally wrong if differences in numbers are very large, and by second pointing out that it relies on implausible assumptions about what is good for an individual. Given the shortcomings of Peterson’s non-aggregationist version of a mixed solution, I next address the issue of how problematic a commitment to interpersonal aggregation really is. To this end, I present an aggregationist version of a mixed solution that is reason-based and bypasses most standard objections to interpersonal value aggregation. I conclude that although there is reason to be optimistic, it remains to be seen whether a mixed solution can be worked out in a fully satisfying way.

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  1. In honour of John Taurek’s seminal 1977 paper.

  2. Such a randomised procedure might be a weighted lottery (Kamm 1985; Saunders 2009), the individualist lottery defended in Timmermann 2004 or a method giving each group the same chance to be saved.

  3. Such accounts have been defended by e.g. Broome (1990) and Hirose (2004).

  4. Peterson defends his version of a mixed solution against the background of what he calls “multidimensional consequentialism”. I will abstract from those parts of Peterson’s framework that play no essential role in his defence of a mixed solution.

  5. It is also better for everyone than the option of saving no one, since the outcome will be 〈0,0〉 for everyone in either group in this case.

  6. Strictly speaking, we need to understand the relevant numbers slightly differently than Peterson suggests. For a lottery to be Pareto better than all alternatives in the 1000 vs. 1001 case, it must be true that the outcome represented by 〈1, u〉 in Table 1 is worse for each member of group B than the outcome represented by 〈0.5, 0〉, and Peterson seems to assume that we can meaningfully subtract u from the number representing the respective individual’s expected well-being, or perform any other mathematical operation such that 〈1, u〉 represents a lower degree of what is good for each member of group B than 〈0.5, 0〉 does (2013, 91). For this to make sense, we must understand u as not (merely) representing the degree of unfairness of the situation, but rather as (also) representing the extent to which this unfairness is bad for the person under consideration. Consequently, we must also take the first elements of the ordered pairs to not represent well-being itself – since well-being cannot be one factor contributing to what is good for a person (i.e. to this person’s well-being) – but rather to represent the impact that some other factor (such as happiness or desire satisfaction) has on the person’s well-being.

  7. If u is taken to be larger than 0.5 in the 1000 vs. 1001 case, it can plausibly be assumed that its value is smaller than 0.5 in the 1 vs. 1000 case. Note that Peterson himself considers the 1 vs. 2 instead of the 1 vs. 1000 case, but the point at issue is more easily illustrated with regard to the latter case.

  8. In this case, we have to drop the assumption that unfairness is an independent aspect of well-being.

  9. Things might be different if you have strong preferences regarding the chances of members of the other group, but this is irrelevant for the present discussion, since Peterson’s version of a mixed solution is meant to cover all types of Taurek cases.

  10. The account presented in what follows differs from Peterson’s, since it does not assume that members of the larger group give the agent reasons that count in favour of holding a lottery in the cases under consideration. It is, however, consistent with Peterson’s proposal to measure unfairness in Taurek cases with the help of the Gini index.

  11. See e.g. Parfit (2011, 32-33); Dancy (2004, 15-17) and Schroeder (2007, 125-129) for different formulations of this idea. I also take this assumption to be essential for Scanlon’s tie-breaking argument for must save many (1998, 232).

  12. Assuming that reasons allow for aggregation does not commit one to any particular view of what reasons there are and what they count in favour of. Therefore, it does not imply the implausible view that the reasons one has to read today’s issue of one’s favourite newspaper five times are stronger than the reasons one has to read it four times (to use a helpful example suggested to me by an anonymous reviewer). The latter assumption is indeed very implausible, but this is due to facts that have nothing to do with whether or not the relevant reasons allow for aggregation. For those facts that give one reasons to read today’s edition of one’s favourite newspaper (such as the fact that it contains useful information and the fact that reading it will be fun etc.) most likely do not also give one reasons to read the same newspaper five times.

  13. Strictly speaking, only reasons that are independent from each other (such that neither derives from the other or a common source) can be suitable for aggregation (cf. Schroeder 2007, 125), but this condition is clearly satisfied in Taurek cases, since all relevant reasons derive from claims of separate individuals.

  14. There is some evidence that this is Taurek’s own view (1977, 303–304), which has been labelled Taureks no worse claim by Lübbe, who also supports it (2008, 75).

  15. This view can also be ascribed to Taurek (1977, 306-307), and it is endorsed e.g. by Timmermann (2004).

  16. For example, the third and fourth criticism seem to miss their target, since it is coherent to assume that whereas fairness and well-being cannot be aggregated, the negative value of unfairness and the positive value of well-being can. With regard to well-being, this point has been made by Parfit (1978, 293).

  17. This could be done by restricting the range of reasons that allow for aggregation to those above a (contextually specified) threshold, which is one way of making sense of the idea that there are, to use a phrase coined by Kamm, irrelevant utilities (1993, 146, see also Scanlon 1998, 239–40). An attractive suggestion for how to restrict the aggregation of claims (which is easily transferred to reasons) is presented in Voorhoeve 2014. However, it is highly controversial whether the idea of restricted aggregation is ultimately tenable (see the discussion of objections in Voorhoeve 2014 and Kelleher 2014). – Temkin has forcefully argued that there is no way to accommodate intuitions such as those that motivate the problem of small losses with regard to comparative value judgements while also accepting the transitivity and independence axiom for the better than relation (2012, 475–476). We should expect similar results for practical reasons as well, since the stronger than relation with regard to reasons and the better than relation seem to share important structural properties (such as being asymmetric, irreflexive and transitive). Given these similarities, one might also try to ground facts about one of these relations in facts about the other. For the purpose of defending a reason-based version of the mixed solution, however, no particular view on this matter is required.

  18. Peterson mentions this as one fundamental objection to aggregative accounts as well (2013, 61).

  19. This point generalises an observation made by Liao (2008).


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I am grateful to Andreas Müller, Thomas Schmidt, Attila Tanyi, Martin van Hees and an anonymous reviewer for very helpful comments on earlier versions of this paper. I would also like to thank Martin Peterson and all participants of the workshop „The Dimensions of Consequentialism“ (16-17 November 2013, University of Konstanz) for a very fruitful discussion.

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Correspondence to Jan Gertken.

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Gertken, J. Mixed Feelings About Mixed Solutions. Ethic Theory Moral Prac 19, 59–69 (2016).

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