¹ Broome, John, Weighing Lives (Oxford, 2004), p. 50CrossRefGoogle Scholar. Broome writes as if there is a unique comparative relation of F. Yet there are two: one for superiority in Fness, formed by ‘more F than’; and one for inferiority in Fness, formed by ‘less F than’ – see Huddleston, Rodney and Pullum, Geoffrey K., The Cambridge Grammar of the English Language (Cambridge, 2002), p. 1100CrossRefGoogle Scholar. While we shall focus on superiority comparatives, the discussion will also apply to inferiority comparatives, changing what needs to be changed. Perhaps Broome meant that there's only one relation R here, designated by both ‘more F’ and ‘less F’ so that R holds between items x and y if and only if ‘x is more F than y’ is true if and only if ‘y is less F than x’ is true. (I thank Krister Bykvist for this suggestion.)

² This is Broome's ‘special version’ (‘Is Incommensurability Vagueness?’, Incommensurability, Incomparability, and Practical Reason, ed. R. Chang (Cambridge, MA, 1997), pp. 67–89, at 74); his ‘general version’ is stated in terms of degrees of truth (‘Incommensurability’, p. 77). Elsewhere, he states somewhat different versions of the principle; see Broome, , Weighing Lives, p. 174, and ‘Reply to Rabinowicz’, Philosophical Issues 19 (2009), pp. 412–17Google Scholar, at 416. The differences between these versions, however, won't be crucial for our discussion. Broome, ‘Incommensurability’, pp. 74–5, also provides some arguments for the Collapsing Principle; see, however, Gustafsson, Johan E., ‘Indeterminacy and the Small-Improvement Argument’, Utilitas 25 (2013), pp. 433–45, at 437–8CrossRefGoogle Scholar, for some objections to these arguments.

³ Broome, ’Incommensurability’, pp. 73–4.

⁴ Elson, Luke, ‘Borderline Cases and the Collapsing Principle’, Utilitas 26 (2014), pp. 51–60, at 55.CrossRefGoogle Scholar

⁵ Elson, ‘Borderline Cases’, p. 55. This conclusion seems to conflict with the original assumption that Tallish is merely a borderline tall man. If we already have a contradiction, then the second half of the example is superfluous. It makes no difference for my objections whether we adopt this shorter version of the example or the longer one, because my objections apply to the first half of the example.

⁶ Elson, ‘Borderline Cases’, p. 55.

⁷ Elson, ‘Borderline Cases’, p. 56.

⁸ Compare Moore's objection to Brentano's fitting-attitude analysis of ‘good’ and of ‘better’ in Moore, G. E., ‘Review of Franz Brentano, The Origin of the Knowledge of Right and Wrong’, The International Journal of Ethics 14 (1903), pp. 115–23, at 118CrossRefGoogle Scholar: ‘His first suggestion is that since “good” means “worthy to be loved,” “better” must mean “worthy of more love”. . . . It does not seem to have occurred to him that it must mean “more worthy of love,” . . .’ If ‘better than’ is a comparative (which it seems to be), then the same objection should also rule out contemporary versions of Brentano's approach, such as those by Gert, Joshua, ‘Value and Parity’, Ethics 114 (2004), pp. 492–510, at 505CrossRefGoogle Scholar, or Rabinowicz, Wlodek, ‘Value Relations’, Theoria 74 (2008), pp. 18–49, at 38CrossRefGoogle Scholar. They both define that ‘x is better than y’ as ‘it is rationally required that x is preferred to y’. Their definiendum is a comparative but their definiens lacks the structure of a comparative.

⁹ This point also applies to Chang's alleged counter-example based on the relation ‘much heavier than’, which is of the form ‘much Fer than’ rather than ‘Fer than’. A comparative ‘Fer than’ holds if the first relatum has a higher degree of Fness than the second relatum; Chang, Ruth, Making Comparisons Count (London, 2002), p. 166Google Scholar. The relation ‘much Fer than’, on the other hand, does not have this kind of structure; it holds when the first relatum has a much higher degree of Fness than the second relatum. In ‘much Fer than’, ‘much’ modifies the comparative ‘Fer than’; it is not itself part of a comparative. To see this, note that comparatives in English can be modified by ‘much’, ‘far’, ‘somewhat’, ‘slightly’, and other modifiers; see Huddleston and Pullum, The Cambridge Grammar, p. 1131. For example, Smith is somewhat heavier than Jones is grammatical; but *Smith is somewhat much heavier than Jones is not. Hence it should be clear that ‘much heavier than’ is not a comparative. So Chang's alleged counter-example doesn't work against the Collapsing Principle.

¹⁰ Broome, Weighing Lives, p. 50.

¹¹ Broome, Weighing Lives, pp. 50–63, provides an extended defence of comparative relations’ being necessarily transitive against several alleged counter-examples.

¹² I thank Erik Carlson for suggesting this revision.

¹³ Elson, ‘Borderline Cases’, pp. 56–7.

¹⁴ Elson, ‘Borderline Cases’, p. 57.

¹⁵ Elson, ‘Borderline Cases’, p. 57.

¹⁶ Johan E. Gustafsson, Preference and Choice (PhD dissertation, Royal Institute of Technology, 2011), p. 26, and ‘Indeterminacy’, p. 436. Henrik Andersson, How It All Relates: Exploring the Space of Value Comparisons (PhD dissertation, Lund University, 2017), p. 92, points out the similarity between Elson's second example and the Balding Cavalier. The Balding Cavalier is a variation of the following kind of example by Carlson, Erik, ‘Broome's Argument against Value Incomparability’, Utilitas 16 (2004), pp. 220–4, at 224CrossRefGoogle Scholar: ‘suppose that A and B are two identical alarm clocks, except that A is waterproof, and B is not. Is A a better alarm clock than B? There may be no definite answer, since it may be indeterminate whether water resistance is a good-making characteristic of artefacts that are not very likely to come into contact with water. It is clear, however, that B is not better than A, since A’s being waterproof definitely does not detract from its goodness as an alarm clock.’ Carlson's example, however, relies on its being indeterminate which features are good making. Broome, ‘Reply to Rabinowicz’, p. 417, objects that it couldn't be indeterminate whether a certain feature contributes to the value of an item. This objection does not apply to the Balding Cavalier.

¹⁷ Carlson, Erik, ‘Vagueness, Incomparability, and the Collapsing Principle’, Ethical Theory and Moral Practice 16 (2013), pp. 449–63, at 454CrossRefGoogle Scholar.

¹⁸ Here, I follow the terminology of Huddleston and Pullum, The Cambridge Grammar, p. 1162 n. 39; plain degree has traditionally been called ‘positive degree’.

¹⁹ Carlson, ‘Vagueness’, pp. 454–5.

²⁰ Andersson, Henrik, ‘Propping Up the Collapsing Principle’, Ethical Theory and Moral Practice 18 (2014), pp. 475–86, at 482–3.CrossRefGoogle Scholar

²¹ Andersson, ‘Propping Up’, p. 483.

²² I thank Erik Carlson for this point.

²³ I wish to thank Henrik Andersson, Krister Bykvist, Erik Carlson, Luke Elson, Christopher Jay, Cristian Piller, Mozaffar Qizilbash, and two anonymous referees for valuable comments.