Arrhenius, Gustaf, 2005, “Superiority in Value”, Philosophical Studies 123, pp. 97-114.
Arrhenius, Gustaf, 2011, “The Impossibility of a Satisfactory Population Ethics”, in E. Dzhafarov and L. Perry (eds.), Descriptive and Normative Approaches to Human Behavior, World Scientific, pp. 51-66.
Arrhenius, Gustaf, forthcoming, Population Ethics: The Challenge of Future Generations, Oxford University Press. (Draft dated September 2012.)
Blackorby, Charles, and D. Donaldson, 1991, “Normative Population Theory: A Comment”, Social Choice and Welfare 8, pp. 261-267.
Carlson, Erik, 1998, “Mere Addition and Two Trilemmas of Population Ethics”, Economics and Philosophy 14, pp. 283-306.
Carlson, Erik, 2010, “Generalized Extensive Measurement for Lexicographic Orders”, JournalofMathematicalPsychology 54, pp. 345-351.
Griffin, James, 1986, Well-Being: Its Meaning, Measurement, and Moral Importance, Clarendon Press.
Holtug, Nils, 2004, “Person-Affecting Moralities”, in J. Ryberg and T. Tännsjö (eds.), The Repugnant Conclusion: Essays on Population Ethics, Kluwer Academic Publishers, pp. 129- 161.
Huemer, Michael, 2008, “In Defence of Repugnance,” Mind 117, pp. 899–933.
Krantz, David, R. D. Luce, P. Suppes, and A. Tversky, 2007, Foundations of Measurement, vol. I: Additive and Polynomial Representations, Dover Publications.
Luce, R. Duncan, D. H. Krantz, P. Suppes, and A. Tversky, Foundations of Measurement, vol. III: Representation, Axiomatization, and Invariance, Dover Publications.
Ng, Yew-Kwang, 1989, “What Should We Do About Future Generations?”, Economics and Philosophy 5, pp. 235-251.
Parfit, Derek, 1984, Reasons and Persons, Oxford University Press.
Parfit, Derek, 1986, “Overpopulation and the Quality of Life”, in P. Singer (ed.), Applied Ethics, Oxford University Press, pp. 145-164.
Roberts, Fred, 2009, Measurement Theory: With Applications to Decisionmaking, Utility, and the Social Sciences, Cambridge University Press.
Tännsjö, Torbjörn, 2002, “Why We Ought to Accept the Repugnant Conclusion”, Utilitas 14, pp. 339–359.
Thomas, Teru, n.d., “Some Possibilities in Population Axiology”, unpublished manuscript.
1 Parfit, 1984, chapter 19.
2 See, e.g., Ng, 1989, Blackorby and Donaldson, 1991, Carlson, 1998.
3 For recent statements of Arrhenius’s results, and references to his earlier work, see Arrhenius, 2011, and forthcoming. (Page references to the latter work are to a draft dated September 2012.)
4 Arrhenius, 2011, p. 23, forthcoming, p. 378.
5 Or at least to all the theorems presented in Arrhenius, forthcoming.
6 Arrhenius, 2011, pp. 7-9, forthcoming, pp. 388-391. The purpose of the “other things being equal”-clauses is to leave room for the possibility that other factors than welfare are relevant to the value of a population.
7 Arrhenius, 2011, p. 9, forthcoming, p. 339. The label the “sixth” axiological impossibility theorem is from the latter work. The proof is found in Arrhenius, 2011, pp. 9-22, forthcoming, pp. 339-346.
8 Arrhenius, 2011, pp. 4-6, forthcoming, pp. 289-291. I have slightly reformulated some of the assumptions. The General Non-Extreme Priority condition seems to require that there is no highest welfare level, and hence that L and L are at least countably infinite.
9 Arrhenius, 2011, p. 5, forthcoming, p. 291. I have simplified Arrhenius’s formulation.
10 Arrhenius, 2011, p. 6, forthcoming, p. 291. To accommodate the possibility that (L, ≿) is incomplete, denseness should be defined as requiring only that there is a welfare level between any pair of distinct and comparable welfare levels. The existence of a welfare level between any pair of distinct levels obviously implies completeness.
11 Arrhenius, 2011, pp. 5-6, forthcoming, p. 291.
12 Arrhenius does not explicitly make this comprehensiveness assumption, but it is implicit in his discussion, as well as necessary for his proof.
13 If (L, ≿) is itself fine-grained and A-discrete, the Crucial Assumption follows trivially.
14 Moreover, it must hold that f(X) > 0 iff X is positive, f(X) < 0 iff X is negative, and f(X) = 0 iff X is neutral.
15 Arrhenius, 2011, p. 6, forthcoming, p. 292. Notation slightly altered.
16 Although a dense order of welfare levels can have holes, it cannot have gaps, if a gap is defined as a (non-maximal) level lacking a merely slightly higher level. While having no gaps seems sufficient for an A-discrete order to be fine-grained, it is not sufficient in the case of dense orders.
17 See, e.g., Luce et al., 2007, p. 49f.
18 It is perhaps not very plausible to suggest that Dedekind completeness is a necessary condition for fine-grainedness, as regards dense orders. Suppose that (L, ≿) is isomorphic to the rational numbers, naturally ordered, and that differences between welfare levels are ordered in accordance with differences between the representing numbers. In this case, (L, ≿) intuitively seems fine-grained, although it is not Dedekind complete.
19 Cf. Luce et al., 2007, p. 50.
20 If (L, ≿) is dense and Dedekind complete, what is required for it to fulfil the Crucial Assumption? A sufficient condition, I conjecture, is that it has a countable suborder (L*, ≿*) which is “order dense” in (L, ≿). Order denseness means that for every X and Y in (L, ≿), such that X≻Y, there is a Z in (L*, ≿*), such that X≿Z≿Y. The real numbers have such a countable order dense subset, viz., the rational numbers, while the lexicographic order of Re × Re lacks a countable order dense subset. (See Krantz et al., 2007, p. 38ff, Roberts, 2009, p. 111f.)
21 Arrhenius, 2011, p. 6. Arrhenius, forthcoming, p. 291f, contains a nearly identical passage.
22 I take it that this notion, rather than A-discreteness, corresponds to the standard mathematical meaning of “discrete”.
23 Note that S-discreteness and denseness are not jointly exhaustive possibilities. Suppose, for example, that (L, ≿) is isomorphic to the following infinite order of rational numbers: -1, -1/2, -1/3, -1/4, …, 0, …, 1/4, 1/3, 1/2, 1. This order is obviously not dense. Nor is it S-discrete, since 0 has no next greater (or next smaller) number in the order.
24 Furthermore, Arrhenius does not argue for the assumption that (L, ≿) is fine-grained. Some adherents of non-Archimedean theories of welfare (see the next section) may want to deny fine-grainedness, and hence the Crucial Assumption. They may argue, for example, that (L, ≿) is dense but contains holes.
25 This may imply, of course, that “merely slight” is used in a way that differs from how it is used in ordinary speech.
26 This assumption introduces a certain degree of “cardinality” into the framework, by presupposing that differences between welfare levels are to some extent comparable. (The term “roughly equal” is meant to allow, though, for some degree of indeterminacy or incommensurability.) However, this presupposition is implicit already in Arrhenius’s assumption that some differences between levels are “merely slight”. Hence, he is not entirely correct in claiming that his “conditions and theorems only presuppose that lives are quasi-ordered by the relation ‘has at least as high welfare as’”. (Arrhenius, 2011, p. 24.)
27 Archimedeanness is very similar to a standard Archimedean axiom in the theory of difference measurement, stating that every “strictly bounded standard sequence” is finite. See Krantz et al., 2007, p. 147, Roberts, 2009, p. 137.
28 Parfit, 1986, p. 161; italics in the original.
29 I interpret Parfit as claiming that the century of ecstasy contains more welfare than the drab eternity. In other words, the welfare level of the former life is higher than the level of the latter life. Parfit could perhaps also be interpreted as holding that the drab eternity contains more welfare than the century of ecstasy, but that the latter life is nevertheless better for the person living it. On this interpretation, his view need not violate Archimedeanness with respect to welfare levels. However, it does not matter for our purposes whether or not Parfit, in particular, holds a non-Archimedean view. The important point is that such views are possible and at least prima facie plausible.
30 Griffin, 1986, p. 85.
31 Griffin, 1986, p. 86.
32 Indeed, he cites Parfit and Griffin, as well as several other authors (Arrhenius, 2005, p. 98, forthcoming, p. 130), and he even refers to these theories as “non-Archimedean”, albeit in a sense that does not correspond exactly to the denial of Archimedeanness, as stated above.
33 Like Arrhenius, I insert an “other things equal”-clause in order not to rule out that other things than welfare can affect the value of a population.
34 Arrhenius (2005, p. 106) labels this kind of trumping “strong superiority” between welfare components. Further, he defines “weak superiority” as, roughly, the view that some amount of the superior welfare components trumps any amount of the inferior components. He is inclined to reject both forms of superiority, since they have implications he finds counterintuitive. However, these implications follow only under the assumption that the relevant welfare components, or composites of such components, can be ordered in a finite sequence, such that the difference in contributive value to a person’s welfare, between any two adjacent members of the sequence, is “marginal”. (Arrhenius, 2005, p. 107, forthcoming, p. 400.) Arrhenius finds this assumption plausible, but it threatens to beg the question to presuppose it in an argument against superiority, since it is exactly the kind of Archimedean principle that believers in superiority will deny. (Note its similarity to the Crucial Assumption.)
35 According to some superiority views, there may be three or more kinds of welfare components, such that the first kind is superior to the second, the second is superior to the third, and so on. Such a view can be represented in terms of ordered n-tuples of numbers, instead of ordered pairs. For a general development of this idea, not restricted to the measurement of welfare, see Carlson, 2010.
36 In other words, the claim that X*≻Xi, for any i, is compatible with Archimedeanness only if the marginal value of seconds of mild pleasure “converges to a finite limit”, in the sense that, for any difference d in (L, ≿), there is a positive integer j, such that, for any k > j, the difference between Xk and Xj is smaller than d.
37 Arrhenius, forthcoming, chapter 11.
38 Arrhenius, forthcoming, p. 299f.
39 This is pointed out by Teru Thomas (n.d.), p. 7.
40 Arrhenius, forthcoming, p. 146f.
41 Arrhenius, forthcoming, p. 150.
42 The proof is on pp. 311-314 in Arrhenius, forthcoming.
43 Arrhenius, forthcoming, pp. 366-376.
44 Arrhenius, forthcoming, p. 368.
45 Some philosophers have responded to population-ethical impossibility results by accepting various versions of Parfit’s “repugnant conclusion” (1984, p. 388). In the case of Arrhenius’s sixth theorem, this would correspond to rejecting Weak Quality Addition. In my opinion, this is a very implausible response. Sometimes, the rejection of adequacy conditions excluding versions of the repugnant conclusion is defended on the grounds that accepting these conditions would force us to deny some other, at least as compelling adequacy condition. (Holtug, 2004, Huemer, 2008, Tännsjö, 2002.) As we have seen, accepting a non-Archimedean theory of welfare allows us to escape this dilemma, at least as it pertains to Arrhenius’s theorems.
46 When this draft was almost completed, John Broome sent me an unpublished paper by Teru Thomas (n.d.), containing criticism of Arrhenius’s impossibility theorems that overlaps significantly with mine. In particular, Thomas points out that “lexic utilitarianism”, a population axiology structurally similar to non-Archimedean totalism, violates A-discreteness of (L, ≿). He also states, without proof, that lexic utilitarianism satisfies all the adequacy conditions of Arrhenius’s first, fourth, fifth, and sixth theorems. Further, Thomas remarks that Arrhenius’s proof that Non-Elitism implies Inequality Aversion presupposes A-discreteness. (This is not quite accurate, since the proof merely requires the logically weaker Crucial Assumption.)