¹ Fleurbaey, Marc and Tungodden, Bertil, “The Tyranny of Non-Aggregation Versus the Tyranny of Aggregation in Social Choices: A Real Dilemma” (unpublished, 2007)Google Scholar. In order to streamline the presentation of our later results, we introduced the following modifications of one of Fleurbaey and Tungodden's results. We strengthened both the priority condition and the aggregation condition by requiring the modified benefit distribution to be better than the original—and not merely at least as good. This strengthening permits us to invoke Acyclicity rather than the stronger Transitivity and to drop Weak Pareto for certain results.

² Scanlon, T. M., What We Owe to Each Other (Cambridge, MA.: Belknap Press of Harvard University Press, 1998), 235Google Scholar. For further discussion of the relevance of aggregation of benefits (e.g., the small losses of many adding up to a large total loss), see Brink, David, “The Separateness of Persons, Distributive Norms, and Moral Theory,” in Frey, R. G. and Morris, C. W., eds., Value, Welfare, and Morality (Cambridge: Cambridge University Press, 1993), 252–89CrossRefGoogle Scholar; Kamm, Frances, Morality, Mortality I (New York: Oxford University Press, 1993), chaps. 5–6Google Scholar; Kumar, Rahul, “Contractualism on Saving the Many,” Analysis 61 (2001): 165–70CrossRefGoogle Scholar; Norcross, Alastair, “Comparing Harms: Headaches and Human Lives,” Philosophy and Public Affairs 26 (1997): 135–67CrossRefGoogle Scholar; Alastair Norcross, “Two Dogmas of Deontology: Aggregation, Rights, and the Separateness of Persons,”elsewhere in this volume; Otsuka, Michael, “Scanlon and the Claims of the Many Versus the One,” Analysis 60 (2000): 288–93CrossRefGoogle Scholar; Otsuka, Michael, “Skepticism about Saving the Greater Number,” Philosophy and Public Affairs 32 (2004): 413–26CrossRefGoogle Scholar; Otsuka, Michael, “Saving Lives, Moral Theory, and the Claims of Individuals,” Philosophy and Public Affairs 34 (2006): 109–35CrossRefGoogle Scholar; Parfit, Derek, “Innumerate Ethics,” Philosophy and Public Affairs 7 (1978): 285–301Google Scholar; Parfit, Derek, “Justifiability to Each Person,” Ratio 16 (2003): 368–90CrossRefGoogle Scholar; Scanlon, T. M., “Replies,” Ratio 16 (2003): 424–39CrossRefGoogle Scholar; Taurek, John, “Should the Numbers Count?” Philosophy and Public Affairs 6 (1977): 293–316Google ScholarPubMed; and Thomson, Judith Jarvis, The Realm of Rights (Cambridge, MA: Harvard University Press, 1990), chap. 6Google Scholar.

³ Fleurbaey and Tungodden (“The Tyranny of Non-Aggregation”) also establish an impossibility result which relies on a version of Minimal Aggregation that focuses on the cases where there is a uniquely-worst-off individual who remains the worst off even after he receives the benefit gain, and where Replication Invariance is replaced with a condition called Reinforcement. Reinforcement states that if x is better than y, then x′ is better than y′ if the only difference between x and x′ and y and y′ is that a person who is better off in y than in x is removed in y′ and in x′.

⁴ See Fleurbaey and Tungodden, “The Tyranny of Non-Aggregation Versus the Tyranny of Aggregation.”

⁵ A third way that one might think that the priority condition is too strong is that it requires priority for just one worst-off person. One might agree that priority should be given to benefits to the worst-off individuals when there are enough of them but reject the idea that a single worst-off person should be given such priority. However, given Replication Invariance, this is equivalent to the condition with only one person gaining. Thus, we do not introduce this weakening of the priority condition.

⁶ The proofs of this and all other results are posted online at http://mora.rente.nhh.no/projects/EqualityExchange/.

⁷ This is a standard prioritarian condition from the social choice literature and is named after Arthur Pigou and Hugh Dalton.

⁸ The following relation satisfies all the conditions except for Pigou-Dalton. Consider a sequence of benefit thresholds, g, 2g, 3g,…, where g is the minimum gain of the worst off appealed to in Super Ultra Minimal Nonaggregative Priority. The relation judges that x is morally at least as good as y if and only if (1) for some benefit threshold in the sequence, x has fewer people below that threshold than y does but has the same number of people below each of the lower thresholds, or (2) for every benefit threshold, the number of people below that threshold is the same in x as in y, and the weighted total prioritarian benefits are at least as great in x as in y. This violates Pigou-Dalton because a transfer from a person above a threshold to a person below the threshold may push the better-off person below the threshold, which makes things worse according to this relation.

⁹ The axiological (moral betterness) and deontic (moral permissibility) approaches have been exhaustively studied in the social choice literature, where they are known as the social ordering and social choice function approaches. See, for example, Sen, Amartya, “Social Choice Theory: A Re-Examination,” Econometrica 45 (1977): 53–89CrossRefGoogle Scholar, reprinted in Sen, Amartya, Choice, Welfare, and Measurement (Oxford: Basil Blackwell, 1982), 158–200Google Scholar.

¹⁰ It is important to note that the choice function approach of social choice theory has No Prohibition Dilemmas implicitly built into it. This is because the concept of a choice function—a function that, for any given feasible set, selects the subset consisting of all and only the permissible alternatives relative to that feasible set—is defined to always select a nonempty set. We here posit No Prohibition Dilemmas for practical moral permissibility, but for an argument that prohibition dilemmas are conceptually possible (e.g., not ruled out by deontic logic as such), see Vallentyne, Peter, “Prohibition Dilemmas and Deontic Logic,” Logique et Analyse 18 (1987): 113–22Google Scholar; and Vallentyne, Peter, “Two Types of Moral Dilemmas,” Erkenntnis 30 (1989): 301–18CrossRefGoogle Scholar.

¹¹ Nagel, Thomas, Mortal Questions (Cambridge: Cambridge University Press, 1979), chap. 8Google Scholar; Scanlon, What We Owe to Each Other, chap. 5.

¹² See, for example, Voorhoeve, Alex, “Should Losses Count? A Critique of the Complaint Model,” LSE Choice Group Working Papers, vol. 2 (2006)Google Scholar.

¹³ Rawls, John, A Theory of Justice (Cambridge, MA: Harvard University Press, 1971), 98Google Scholar.

¹⁴ Super Ultra Minimal Aggregation avoids the impossibility result because the condition is applicable only when all the gainers are below some threshold (e.g., none are superaffluent). Another (similar) way of weakening the aggregation condition to avoid an impossibility result is to restrict its application to cases in which the one loser is above some threshold (e.g., a poverty threshold). The idea is that aggregation is required among well-off people, but, below the threshold, absolute priority to the worst-off may prevail. When such a weakening of the aggregation condition is introduced, a possibility result obtains with a modified total affluence relation, which deems x to be at least as good as y if and only if (1) the leximin criterion restricted to positions below the threshold prefers x, or (2) x and y are leximin indifferent with respect to positions below the threshold and x has finitely weighted total prioritarian benefits (for some specified set of weights) that are at least as great as those of y.

¹⁵ Tungodden, Bertil and Vallentyne, Peter, “On the Possibility of Paretian Egalitarianism,” Journal of Philosophy 102 (2005): 126–54CrossRefGoogle Scholar.

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¹⁷ Sugden, Robert, “Why Be Consistent? A Critical Analysis of Consistency Requirements in Choice Theory,” Economica, new series, vol. 52 (1985): 167–83CrossRefGoogle Scholar; Sen, Amartya, “Internal Consistency of Choice,” Econometrica 61 (1993): 495–521CrossRefGoogle Scholar, reprinted in Sen, Amartya, Rationality and Freedom (Cambridge, MA: Harvard University Press, 2002), 121–57Google Scholar; McClennen, Edward F., Rationality and Dynamic Choice (Cambridge: Cambridge University Press, 1990), 182–84, 240, 251CrossRefGoogle Scholar; Kamm, Frances, Morality, Mortality II (New York: Oxford University Press, 1996), 343–44Google Scholar; Kamm, Frances, Intricate Ethics (Oxford: Oxford University Press, 2007), 169–73, 222 n. 16, 298, 484–487CrossRefGoogle Scholar.