¹ This assumption – though plausible given classic ‘Mozart/Michelangelo’ cases – is not undeniable. For an excellent overview, see Chang, Ruth, ‘Introduction’, Incommensurability, Incomparability, and Practical Reason, ed. Chang, Ruth (Cambridge, MA, 1997), pp. 1–34Google Scholar.

² Broome, John, ‘Is Incommensurability Vagueness?’, Incommensurability, Incomparability, and Practical Reason, ed. Chang, Ruth (Cambridge, MA, 1997), pp. 67–89, at 74Google Scholar.

³ For Carlson's counterexamples, see Carlson, Erik, ‘Broome's Argument against Value Incomparability’, Utilitas 16 (2004), pp. 220–4Google Scholar. Broome's response is neatly summarized in Broome, John, ‘Reply to Rabinowicz’, Philosophical Issues 19 (2009), pp. 412–17Google Scholar, at 416–17.

⁴ Constantinescu, Cristian, ‘Value Incomparability and Indeterminacy’, Ethical Theory and Moral Practice 15 (2012), pp. 57–70Google Scholar.

⁵ See Raz, Joseph, ‘Value Incommensurability: Some Preliminaries’, Proceedings of the Aristotelian Society, New Series 86 (1985–6), pp. 117–34Google Scholar, and Chang, Ruth, ‘The Possibility of Parity’, Ethics 112 (2002), pp. 659–88Google Scholar.

⁶ For the canonical presentation, see Broome, ‘Is Incommensurability Vagueness?’.

⁷ See Luke Elson, ‘Heaps and Chains: Could the Chaining Argument for Parity be a Sorites?’, Ethics (forthcoming), for the distinction between fuzzy and sharp hard incomparabilism. The present discussion assumes that vagueness entails the presence of borderline cases (at least that it does in the cases under consideration), and that some non-epistemic account of vagueness is correct. Neither assumption is problematic in this dialectic. The core of the incommensurability-as-vagueness view is that incommensurate options are comparative borderline cases. It's hard to see how this could be sustained without the notion of a borderline case. And if epistemicism is right, then the collapsing principle as presently formulated is empty or trivial.

⁸ Carlson, ‘Broome's Argument against Value Incomparability’.

⁹ Carlson, ‘Broome's Argument against Value Incomparability’, p. 223.

¹⁰ Broome, ‘Reply to Rabinowicz’, p. 417.

¹¹ Constantinescu, ‘Value Incomparability and Indeterminacy’, pp. 67ff.

¹² Again, I am assuming here that there could not be vagueness without borderline cases.

¹³ I am grateful to an anonymous reviewer for this journal, for catching a crucial mistake here.

¹⁴ In terms I discuss in Elson, ‘Vague Projects and Practical Sorites’ (unpublished manuscript), the desire or project involved is wholly binary.

¹⁵ See Fine, Kit, ‘Vagueness, Truth and Logic’, Synthese 30 (1975), pp. 265–300CrossRefGoogle Scholar, and Keefe, Rosanna, Theories of Vagueness (Cambridge, 2000)Google Scholar.

¹⁶ I am grateful to Walter Sinnott-Armstrong for suggesting this way of putting it.

¹⁷ I owe this point to John Broome.

¹⁸ I try to undertake that task in Elson, ‘Incommensurability as Comparative Borderlineness’ (unpublished manuscript). I am indebted to Henrik Andersson, Simon Blackburn, John Broome, Erik Carlson, Patrick Connolly, Ram Neta, Wlodek Rabinowicz, Geoffrey Sayre-McCord, Keith Simmons, Walter Sinnott-Armstrong, Susan Wolf, and participants in the UNC dissertation seminar for discussion of this material and its ancestors.