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Indeterminacy and the Small-Improvement Argument



In this article, I argue that the small-improvement fails since some of the comparisons involved in the argument might be indeterminate. I defend this view from two objections by Ruth Chang, namely the argument from phenomenology and the argument from perplexity. There are some other objections to the small-improvement argument that also hinge on claims about indeterminacy. John Broome argues that alleged cases of value incomparability are merely examples of indeterminacy in the betterness relation. The main premise of his argument is the much-discussed collapsing principle. I offer a new counterexample to this principle and argue that Broome's defence of the principle is not cogent. On the other hand, Nicolas Espinoza argues that the small-improvement argument fails as a result of the mere possibility of evaluative indeterminacy. I argue that his objection is unsuccessful.



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1 Broome, J., ‘Is Incommensurability Vagueness?’, Incommensurability, Incomparability, and Practical Reason, ed. Chang, R. (Cambridge, Mass., 1997), pp. 74–7.

2 Espinoza, N., ‘The Small Improvement Argument’, Synthese 165 (2008), pp. 127–39.

3 de Sousa, R., ‘The Good and the True’, Mind 83 (1974), pp. 534–51, at 544–5.

4 Chang, R., ‘The Possibility of Parity’, Ethics 112 (2002), pp. 659–88, at 669.

5 We employ the following transitivity principle: ∀xyz((xByyEz) → xBz).

6 Broome, ‘Incommensurability’, p. 74. Broome, J., Weighing Lives (Oxford, 2004), p. 174, states the principle in terms of what one can deny:

Collapsing principle. For any predicate F and any things A and B, if we can deny that B is Fer than A, but we cannot deny that A is Fer than B, then A is Fer than B.

7 Carlson, E., ‘Broome's Argument against Value Incomparability’, Utilitas 16 (2004), pp. 220–4, at 224.

8 Broome, J., ‘Reply to Rabinowicz’, Philosophical Issues 19 (2009), pp. 412–17, at 417. Cf. Broome, Weighing Lives, pp. 185–6, where he at least admits the examples as a strong objection.

9 Constantinescu, C., ‘Value Incomparability and Indeterminacy’, Ethical Theory and Moral Practice 15 (2012), pp. 5770, at 68–9.

10 Note that I am not denying that A and B differ in baldness. I just claim that it is indeterminate whether they differ in baldness. One might object, however, that if it is false that A is bald and not false that B is bald, then B is balder than A and hence A and B differ in baldness. This reasoning seems to rely on the following principle posited by Carlson, E., ‘Vagueness, Incomparability, and the Collapsing Principle’, Ethical Theory and Moral Practice 16 (2013), pp. 449–63, at 454:

The monadic collapsing principle. For any x and y, if it is false that y is F, and not false that x is F, then it is true that x is Fer than y.

But this principle is open to counterexamples that are very similar to those offered against the original collapsing principle. Carlson, ‘Vagueness’, pp. 454–5, offers the following:

Let us slightly modify Gustafsson's cavalier case, and assume that B is definitely bald, whereas A is a borderline case of baldness. In all other relevant respects, the two cavaliers are identical. Suppose also that, given their other properties, not being bald is necessary and sufficient for A or B to qualify as a good cavalier. It is thus false that B is good, and indeterminate whether A is good. The monadic collapsing principle then implies that A is definitely better than B. But this seems false, since it is indeterminate whether A lacks the property, viz. baldness, whose absence would constitute the only relevant difference, as compared to B.

Hence it seems question-begging to rely on the monadic collapsing principle in a defence of the original collapsing principle from counterexamples of this type. One might object that, instead of relying on the monadic collapsing principle, one could reason as follows: if it is false that A is bald and not false that B is bald, A must have more hair than B; and if so, B must be balder than A. Yet a problem with this objection is that to be balder is not just to have less hair – the proportion of the scalp covered by hair, for example, also matters. And the relative weights these two factors have in contributing to baldness might be indeterminate. Suppose, for instance, that A has less hair than B but, since it is evenly distributed over his scalp, it is false that A is bald. Furthermore, while B has more hair than A, it is unevenly distributed so some parts of his scalp have little hair, which makes it not false that B is bald. But since each of A and B beats the other in one factor that contributes to baldness and the relative weights of these factors are indeterminate, it is indeterminate whether B is balder than A. A referee for this journal suggests another reply, which is to concede that B is balder than A, but to deny that this difference is relevant to which is the better cavalier. That is, one might deny that being less bald is a better-making relation even though not being bald is good making.

11 Broome, ‘Incommensurability’, p. 74.

12 Broome, ‘Incommensurability’, pp. 74–5.

13 Broome, ‘Incommensurability’, p. 75.

14 The same reply can, mutatis mutandis, be given to the similar example with Sartre's student in Broome, Weighing Lives, pp. 172–4.

15 Espinoza, ‘Argument’, p. 131.

16 Espinoza, ‘Argument’, p. 131. Espinoza has informed me that the ‘Refs.’ in his paper are typos. Formulas (1), (2), (3) and (4) are premises. The argument would make more sense if (5) were replaced by

(5*) ¬D(xEy) ∨ ¬D(x +Bx).

17 Espinoza, ‘Argument’, p. 135.

18 Espinoza, ‘Argument’, p. 137.

19 Espinoza, ‘Argument’, p. 137.

20 Rabinowicz, W., ‘Incommensurability and Vagueness’, Aristotelian Society Supplementary Volume 83 (2009), pp. 7194, at 74. As we shall see in section V, these three claims and axiological completeness are also jointly compatible with the transitivity of ‘better’ and ‘equally good’, which blocks the small-improvement argument.

21 Chang, ‘Parity’, p. 680.

22 Chang, ‘Parity’, p. 682.

23 De Sousa, ‘The Good’, p. 545.

24 Chang, ‘Parity’, p. 684.

25 Chang, ‘Parity’, p. 685.

26 The Oxford English Dictionary, 2nd edn., vol. 11, p. 233, s.v. ‘parity’. Webster's Third New International Dictionary, p. 1642, s.v. ‘parity’. The second part of W3's definition, however, seems to suggest a different analysis, along the lines of the following:

x is axiologically on a par with y if and only if the difference between the value of x and the value of y is small.

x is preferentially on a par with y if and only if the difference between the strength of preference for x and the strength of preference for y is small.

27 Thanks to Gustaf Arrhenius, Campbell Brown, John Cantwell, Erik Carlson, Nicolas Espinoza, Sven Ove Hansson, Martin Peterson, Wlodek Rabinowicz, and an anonymous referee for valuable comments. Financial support from Riksbankens Jubileumsfond and Fondation Maison des sciences de l'homme is gratefully acknowledged.

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