¹ Parfit, Derek, ‘Can We Avoid the Repugnant Conclusion’, Theoria 82 (2016), pp. 100–27CrossRefGoogle Scholar; Derek Parfit, ‘Material from Towards Theory X’, unpublished manuscript (Oxford, 2016), pp. 1–36.

² Parfit, ‘Can We Avoid’, p. 110.

³ Parfit, ‘Can We Avoid’, p. 115. There is another statement at page 112, but it must contain an error; it says ‘non-existence’ instead of ‘existence’. Moreover, on p. 112, Parfit calls it the Imprecise Lexical View. But imprecision only enters through further assumptions, see below.

⁴ See the discussion in Arrhenius, G. and Rabinowicz, W., ‘Millian Superiorities’, Utilitas 17 (2) (2005), pp. 127–46CrossRefGoogle Scholar; Arrhenius, G. and Rabinowicz, W., ‘Value Superiority’, The Oxford Handbook of Value Theory, ed. Hirose, I. and Olson, J. (Oxford 2015), pp. 225–48Google Scholar and Jensen, K. K., ‘Millian Superiorities and the Repugnant Conclusion’, Utilitas 20 (3) (2008), pp. 279–300CrossRefGoogle Scholar.

⁵ Parfit, ‘Can We Avoid’, p. 112.

⁶ Parfit, ‘Can We Avoid’, p. 112.

⁷ Parfit, ‘Material’, pp. 2–3.

⁸ Parfit, ‘Can We Avoid’, pp. 116–17.

⁹ Parfit, ‘Can We Avoid’, pp. 113–16; Parfit, ‘Material’, p. 3, note 3. Broome, John, Weighing Lives (Oxford 2006) pp. 21–22Google Scholar.

¹⁰ Parfit, ‘Can We Avoid’, pp. 116–16.

¹¹ This line of thought is found in Arrhenius and Rabinowicz, ‘Millian Superiorities’ and Jensen, ‘Millian Superiorities’.

¹² Handfield, T. and Rabinowicz, W., ‘Incommensurability and Vagueness in Spectrum Arguments: Options for Saving Transitivity of Betterness’, Philosophical Studies 175 (2018), pp. 2373–87CrossRefGoogle Scholar. W. Rabinowicz, ‘Can Parfit's Appeal to Incommensurabilities Block the Continuum Argument for the Repugnant Conclusion?’, manuscript (2018) applies their result directly to Parfit's attempt to refute Continuum Arguments for the Repugnant Conclusion. See Handfield and Rabinowicz, ‘Incommensurability’ for references to Temkin and Rachels.

¹³ Arrhenius and Rabinowicz, ‘Millian Superiorities’ and Jensen, ‘Millian Superiorities’.

¹⁴ A binary relation R is a weak order, if and only if, R is transitive and complete. R is transitive, if and only if, for all x, y, z: if xRy and yRz, then xRz. R is complete, if and only if, for all x, y: either xRy or yRx.

¹⁵ A relation R is antisymmetric, if and only if, for all x, y: if xRy and yRx, then x and y are identical.

¹⁶ A binary operation ◦ is associative, if and only if, for all x and y, x◦y is equivalent with y◦x; i.e. the order of objects does not matter for the concatenation operation. A set is closed on an operation, if and only if, any object formed by performing the operation on two members of the set is also a member of that set. Thus, concatenating lives always result in a population.

¹⁷ Handfield and Rabinowicz, ‘Incommensurability’ and Rabinowicz, ‘Can Parfit's Appeal’. The fact that they use a worseness relation rather than a betterness relation is insignificant from a formal point of view.

¹⁸ A relation R is asymmetric, if and only if, for all x, y: if xRy, then not yRx.

¹⁹ Arrhenius and Rabinowicz, ‘Millian Superiorities’ and Jensen, ‘Millian Superiorities’.

²⁰ I find the account of Parfit's notion of imprecise equality in Chang, R., ‘Parity, Imprecise Comparability and the Repugnant Conclusion’, Theoria 82 (2016), pp. 182–214CrossRefGoogle Scholar, at 200, somewhat misleading. See Appendix 1.

²¹ A binary relation R is reflexive, if for all x: xRx. R is symmetric, if for all x and y: if xRy, then yRx.

²² See Fishburn, P. C., Utility Theory for Decision Making (New York, 1970), p. 15CrossRefGoogle Scholar.

²³ A real-valued representation of a relation ‘– is better than –’ means that, for all x and y, there are real numbers, such that if x is better than y, then the number assigned to x is larger than the number assigned to y. If the representation is ordinal, only the order of the numbers has significance. The minimal requirement for a cardinal representation is that ratios between differences (intervals) are unique. This allows for a measure of value differences.

²⁴ In the framework, which takes weak betterness as the primitive notion, the analogue of Condition 1 is the assumption that weak betterness is complete; if weak betterness as primitive is not complete, it is not necessarily transitive.

²⁵ The term ‘lexically better’ is Parfit's. I shall use it in the present article rather than ‘weakly superior’, which was used in both Utilitas articles (see note 11). I believe Griffin, James, Well-Being: Its Meaning, Measurement and Moral Importance (Oxford, 1986), p. 85Google Scholar, was the first to introduce the weaker version, which he called ‘discontinuity’; he distinguished it from the stronger version, which he called ‘trumping’. Presumably, the weaker version is thought to be less implausible than the stronger version. Parfit simply adopts the weaker version without any further explanation.

²⁶ Arrhenius and Rabinowicz, ‘Millian Superiorities’ and Jensen, ‘Millian Superiorities’.

²⁷ Krantz, D. H., Luce, R. D., Suppes, P. and Tversky, A., Foundations of Measurement. Volume 1: Additive and Polynomial Representations (San Diego and London, 1971), pp. 4, 25Google Scholar.

²⁸ Arrhenius and Rabinowicz, ‘Millian Superiorities’ and Jensen, ‘Millian Superiorities’.

²⁹ Krantz et al., Foundations, p. 25.

³⁰ Note that this does not exclude numerical representation as such, only representation by a cardinal, real-valued scale. Under Condition 1, if the domain is a countable set, betterness could be represented by an ordinal real-valued scale (Krantz et al., Foundations, p. 39 (Theorem 2.1)).

³¹ Arrhenius and Rabinowicz, ‘Millian Superiorities’, pp. 136–37. This is their Observation 3, based on a transitive and complete weak betterness relation and stated with Parfit's term ‘lexically better’.

³² I follow the presentation of this argument in Handfield and Rabinowicz, ‘Incommensurability’, pp. 2377–78. They present the corollary in note 6. See also Arrhenius, Gustaf, ‘Population Ethics and Different-Number-Based Imprecision’, Theoria 82 (2016), pp 166–81CrossRefGoogle Scholar, at 171–74.

³³ It is also easy to show that, given that betterness is transitive, lexical betterness is also transitive.

³⁴ I owe the statement of this premise, and the point that it is insufficient to support the conclusion, to John Broome, ‘Lessons from Economics’, manuscript (2018), pp. 18, at 13–14, and personal communication. Broome has been complaining for many years that Continuum Arguments only are valid given the strong assumption about a discrete domain.

³⁵ This is a refinement of Observation 1 in Jensen, ‘Millian Superiorities’.

³⁶ Handfield and Rabinowicz, ‘Incommensurability’.

³⁷ I adopt this name from Arrhenius and Rabinowicz, ‘Millian Superiorities’, p. 137. They define this relation that allows for a not necessarily complete weak betterness relation and call it minimal superiority.

³⁸ I adopt this definition from Handfield and Rabinowicz, ‘Incommensurability’, p. 2380. They call it ‘radical incommensurability’, whereas Rabinowicz, ‘Can Parfit's Appeal’ calls it ‘strictly persistent incommensurability’.

³⁹ Handfield and Rabinowicz, ‘Incommensurability’, pp. 2379–80.

⁴⁰ Chang, R., ‘The Possibility of Parity’, Ethics 112 (2002), pp. 659–88CrossRefGoogle Scholar.

⁴¹ Handfield and Rabinowicz, ‘Incommensurability’, pp. 2386–87.

⁴² Handfield and Rabinowicz, ‘Incommensurability’, pp. 2381–82. The point was first made in , Handfield, ‘Rational Choice and the Transitivity of Betterness’, Philosophy and Phenomenological Research 89 (2014), pp. 584–604CrossRefGoogle Scholar, at 599, note 18. Rabinowicz quotes it in ‘Can Parfit's Appeal’, pp. 6–7. Strictly speaking, they prove something slightly different, namely that in a cyclical sequence of objects, where each object is exchangeable with its successor, but the first is better than the last, one case of radical imprecise equality is not sufficient. But it can be proved, see Observation 10 in Appendix 2.

⁴³ Parfit, ’Material’, pp. 9–12.

⁴⁴ Or maybe not. After having stumbled over Jensen, ‘Millian Superiorities’, he approached me for comments to his work, maybe because he sensed a problem here.

⁴⁵ Parfit, ‘Material’, p. 9 considers lexical betterness between qualitatively very similar objects almost a reductio on lexical betterness.

⁴⁶ Parfit, ‘Can We Avoid’, p. 120.

⁴⁷ Arrhenius, Gustaf, ‘The Impossibility of a Satisfactory Population Ethics’, Descriptive and Normative Approaches to Human Behavior, Advanced Series on Mathematical Psychology, ed. Colonius, H. and N., E. Dzhafarov (Singapore, 2011), pp. 1–26Google Scholar.

⁴⁸ Gustaf Arrhenius, ‘Population Ethics’ at p. 174, note 18; cf. Arrhenius, Gustaf, Future Generations: A Challenge for Moral Theory (Uppsala, 2000)Google Scholar, ch. 6.

⁴⁹ Arrhenius, ‘The Impossibility’, p. 10.

⁵⁰ Arrhenius, ‘The Impossibility’, p. 8.

⁵¹ Arrhenius, ’Future Generations’, ch. 6.3.

⁵² Arrhenius, ‘The Impossibility’, at p. 6; cf. Arrhenius Future Generations, at pp. 154–55.

⁵³ Rabinowicz, ‘Can Parfit's Appeal’ struggles with this problem.

⁵⁴ See Jensen, ‘Millian Superiorities’, pp. 297–98.

⁵⁵ Arrhenius, Future Generations, p. 98.

⁵⁶ The major part of this article was written during my stay at Umeå University. I wish to express my gratitude to the Department of Historical, Philosophical and Religious Studies for providing a very fruitful philosophical environment. I am also very grateful for helpful comments from and communication with John Broome and Wlodek Rabinowicz. Finally, I wish to thank one referee for very helpful suggestions, which have improved the overall clarity of the article considerably.