¹ If you think that paraplegia is worse than death, then simply swap one for the other.

² For a recent systematic account of Limited Aggregation that sparked a lively debate see Voorhoeve, Alex, How Should We Aggregate Competing Claims?, Ethics, 125.1 (2014), 64–87CrossRefGoogle Scholar. For accounts in support of the Limited Aggregation intuitions preceding Voorhoeve's see e.g. Dorsey, Dale, Headaches, Lives and Value, Utilitas, 21.1 (2009), 36–58CrossRefGoogle Scholar, or Kamm, F. M., Nonconsequentialism, in The Blackwell Guide to Ethical Theory, ed. by LaFollette, Hugh and Persson, Ingmar, 2nd edn (Oxford: Blackwell, 2013), pp. 261–86Google Scholar. For rare examples of strictly non-aggregative views, see e.g. Anscombe, G. E. M., Who Is Wronged? Philippa Foot on Double Effect: One Point, in Human Life, Action and Ethics: Essays (Exeter: Imprint Academic, 2005), pp. 249–51Google Scholar. and Taurek, John M., Should the Numbers Count?, Philosophy & Public Affairs, 6.4 (1977), 293–316Google ScholarPubMed. For fully aggregative views see e.g. Norcross, Alastair, Comparing Harms: Headaches and Human Lives, Philosophy & Public Affairs, 26.2 (1997), 135–67CrossRefGoogle Scholar, and Parfit, Derek, Justifiability to Each Person, Ratio (New Series), 16.4 (2003), 368–90Google Scholar.

³ This is not to say that there could not exist any aggregative view that could resist letting Ann die in Death vs. Headaches. We could, for example have a view with a fully aggregative, but lexicographic axiology, according to which well-being consists of, say, two elements, where no amount of one can ever outweigh any amount of the other. Such views shall not concern me here, however.

⁴ Scanlon, T. M., What We Owe to Each Other (Cambridge, MA: London: Belknap Press of Harvard University Press, 1998), pp. 239–40Google Scholar. It is in fact very hard to allow for even Limited Aggregation within Scanlon's Contractualism. See e.g. Otsuka, Michael, Saving Lives, Moral Theory, and the Claims of Individuals, Philosophy & Public Affairs, 34.2 (2006), 109–35CrossRefGoogle Scholar. This, however, will not be my topic here.

⁵ Parfit, p. 278.

⁶ See Voorhoeve, How Should We Aggregate Competing Claims?

⁷ Voorhoeve, How Should We Aggregate Competing Claims?, p. 66. I have amended the view to reflect a simplifying restriction that the strength of a person's claim only turns on the size of a given benefit and not the absolute well-being level of a person.

⁸ Voorhoeve himself writes that ARC will sometimes “have costs in terms of impersonal goodness”. See Voorhoeve, How Should We Aggregate Competing Claims?, p. 85 (my emphasis).

⁹ Of course, some might disagree about this. See e.g. Temkin, Larry S., Rethinking the Good: Moral Ideals and the Nature of Practical Reasoning (New York: Oxford University Press, 2015)Google Scholar.

¹⁰ Tomlin, Patrick, On Limited Aggregation, Philosophy & Public Affairs, 45.3 (2017), 232–60CrossRefGoogle Scholar.

¹¹ I should note that Voorhoeve himself explicitly places such “diverse group cases” outside the scope of his original paper. However, it is clear that Voorhoeve doesn't think that ARC shouldn't apply beyond the “homogenous group cases” to which he restricts his discussion and none of his arguments would support such a restriction.

¹² See Tomlin, p. 240.

¹³ Tomlin, p. 241.

¹⁴ See Tomlin, p. 245.

¹⁵ Tomlin, p. 245

¹⁶ I should note, however, that there is a phenomenon, familiar from social choice theory, which would also violate the Principle of Addition. This is the so-called “No Show Paradox”. Many otherwise attractive voting procedures have the implication that a voter could be better off “not showing up”, rather than casting the ballot for her favourite candidate in the sense that her voting for her favourite candidate will actually lead to this candidate not being elected when she would have been elected had the voter abstained. See e.g. Moulin, Hervé, Condorcet's Principle Implies the No Show Paradox, Journal of Economic Theory, 45.1 (1988), 53–64CrossRefGoogle Scholar.

¹⁷ See Tomlin, pp. 247–50.

¹⁸ I thank Theron Pummer for prompting me to discuss such a case.

¹⁹ For a similar line of thought concerning this point see Tadros, Victor, Localized Restricted Aggregation, Oxford Studies in Political Philosophy, 5 (2019), 171–204Google Scholar. Tadros proposes an otherwise very different view of Limited Aggregation.

²⁰ See Kamm, F. M., Morality, Mortality Volume I: Death and Whom to Save From It (Oxford: Oxford University Press, 1998), p. 146CrossRefGoogle Scholar.

²¹ See e.g. Voorhoeve, How Should We Aggregate Competing Claims? and Tadros.

²² I thank an anonymous referee for pressing me to discuss this case.

²³ Horton, Joe, Always Aggregate, Philosophy & Public Affairs, 46.2 (2018), 160–74CrossRefGoogle Scholar.

²⁴ In decision theory, this is known as the “Sure Thing Principle”. See e.g. Savage, Leonhard, The Foundations of Statistics, 2nd edn (New York: Dover, 1972)Google Scholar.

²⁵ For a precise formulation and discussion, see Broome, John, Weighing Goods (Cambridge, MA: Basil Blackwell, 1991), ch. 4Google Scholar.

²⁶ I am indebted to an anonymous referee for pressing me to make this point and for improving the following discussion.

²⁷ For an argument for utilitarianism based on separability, see e.g. Broome, Weighing Goods. For an argument for prioritarianism, see Adler, Matthew D., Well-Being and Fair Distribution (Oxford: Oxford University Press, 2012)Google Scholar.

²⁸ See Broome, Weighing Goods, ch. 4. Broome proves that what he calls strong separability for value orderings implies their additivity, and vice versa. In other words, iff an ordering is separable then it is fully aggregative. This is Broome's “First Separability Theorem.” Thus, rejecting full aggregation is equivalent to rejecting separability. However, we cannot readily rely on Broome's result here, since on Limited Aggregation, the “ought to be chosen over” relation is not transitive, and thus not an ordering. The “better than” relation with which Broome is concerned, on the other hand, is an ordering (pace e.g. Temkin). For Broome's method and results, see also Gorman, W. M., The Structure of Utility functions, Review of Economic Studies, 35 (1968), 367–90CrossRefGoogle Scholar, and David, Kranz H. and others, Foundations of Measurement: Volume 1 (Academic Press, 1971)Google Scholar. However, there are positive arguments for a similar result that can largely do without transitivity. See e.g. Kacper Kowalzcyk, Yet Another Argument Against Anti-Aggregation (unpublished manuscript).

²⁹ This is also given by the following principle.

I don't appeal to this general principle here, as it is likely to be rejected by many aggregation sceptics. However, even these sceptics will accept the indifference in this restricted case.

³⁰ In fact, a violation of separability is what distinguishes “true” egalitarian theories from other distribution sensitive theories. See Broome, John, Equality versus Priority: A Useful Distinction, Economics and Philosophy, 31.02 (2015), 219–28CrossRefGoogle Scholar. I agree with Broome on this.

³¹ See Horton, p. 11.

³² Horton does not explicitly mention this principle, but it is clear from his discussion that this is what he has in mind.

³³ I say “in this case”, because I don't want to assume that the transitivity of this relation holds universally. See the next section.

³⁴ I am indebted to Kacper Kowalzcyk for discussion of this point in particular and the issues in this section more general.

³⁵ Voorhoeve, How Should We Aggregate Competing Claims?, p. 66 (my emphasis).

³⁶ I tend to agree with Broome, who argues that, in his terminology, “good is coherent”. See e.g. Broome, Weighing Goods, ch. 6.

³⁷ See Korbinian Rüger, Aggregation and Equality (unpublished manuscript).

³⁸ For some of the criticisms see e.g. Halstead, John, The Numbers Always Count, Ethics, 126.3 (2016), 789–802CrossRefGoogle Scholar; Norcross, Alastair, Intransitivity and the Person-Affecting Principle, Philosophy and Phenomenological Research, 59.3 (1999), pp. 769–76, at 769CrossRefGoogle Scholar. For responses see Kamm, F. M., Intricate Ethics (Oxford University Press, 2007). pp. 297–98, 484–87CrossRefGoogle Scholar; Voorhoeve, How Should We Aggregate Competing Claims?, pp. 76-79; Alex Voorhoeve, Why One Should Count Only Claims With Which One Can Sympathize, Public Health Ethics, 2016, 1–9; Tomlin, p. 236 (fn. 11).

³⁹ Variations of this case were presented to me by Theron Pummer and Bastian Steuwer.

⁴⁰ For comments and discussion, I would like to thank Ralf Bader, Jessica Fischer, Hilary Greaves, Joe Horton, Kacper Kowalzcyk, Michal Masny, Jeff McMahan, Theron Pummer, Tom Sinclair, Bastian Steuwer, Patrick Tomlin, Alex Voorhoeve, two anonymous referees of this journal, and audiences at University College London and at the University of St Andrews.