Abstract
John Broome has argued that incomparability and vagueness cannot coexist in a given betterness order. His argument essentially hinges on an assumption he calls the ‘collapsing principle’. In an earlier article I criticized this principle, but Broome has recently expressed doubts about the cogency of my criticism. Moreover, Cristian Constantinescu has defended Broome’s view from my objection. In this paper, I present further arguments against the collapsing principle, and try to show that Constantinescu’s defence of Broome’s position fails.
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Notes
To avoid confusion, it should be pointed out that Chang’s use of ‘incomparable’ differs from the one adopted here. She acknowledges cases of incomparability in the sense I have defined, but is skeptical about incomparability in her own, stronger sense.
The transitivity of’better than’ has been questioned by some philosophers. For the purposes of this paper, we may ignore these doubts.
I shall assume that ‘equally F’ can be defined in terms of ‘Fer than’, as follows: x and y are equally F iff, for all possible items z and w, z is Fer than x iff z is Fer than y, and x is Fer than w iff y is Fer than w. Broome (1997, p. 72) adopts an equivalent definition.
The assumption of a third possibility, besides precise comparability and incomparability, presupposes that we reject ‘epistemicism’ about vagueness. Epistemicists claim that a vague sentence has a definite truth-value, although we cannot know which. (See Williamson 1994.)
In his (2004), Broome formulates the supervaluation theory in terms of what we may assert, rather than in terms of what is true. This is in response to a problem discovered by Timothy Williamson (1994). For the sake of simplicity, I shall ignore this problem. Furthermore, I will move freely between statements in the object-language, such as ‘It is indeterminate that …’, and meta-language statements, such as ‘The sentence … is indeterminate’.
If Harry is a borderline case of baldness, neither ‘Harry is bald’ nor ‘Harry is not bald’ is true, since neither statement is true under all admissible sharpenings of ‘bald’. But the disjunction ‘Harry is bald or not bald’ is true under all admissible sharpenings of ‘bald’, and hence true.
If there is only one item between those that are Fer than the standard and those that are less F, Broome (1997, p. 71f) is inclined to believe that this item must be equally F as the standard.
Broome uses the terms ‘hard indeterminacy’ and ‘soft indeterminacy’ to denote what I call ‘incomparability’ and ‘semicomparability’, respectively.
Broome’s argument assumes that if there is incomparability, then there is some particular pair of items that are incomparable. Supervaluationism allows, however, that an existential quantification is true without having any true instance. Thus, the statement ‘There are incomparable objects’ could be true although there is no pair of items x and y, such that x and y are incomparable. This would be the case if every admissible sharpening of ‘Fer than’ contains an incomparable pair, but no pair is incomparable under every admissible sharpening. Broome (1997, p. 82), gives an additional argument which, if correct, shows that, given the collapsing principle, vagueness excludes also this weak form of incomparability. This argument is somewhat curious, since Broome mentions a possibility that would invalidate it, but without explanation dismisses this possibility as ‘uninteresting’.
This limitation of Broome’s argument is pointed out by Wlodek Rabinowicz (2009a, p. 80, n. 16).
Gustafsson (2011, pp. 26–27), gives a similar criticism of Broome’s argument.
Cf. Gustafsson (2011, p. 27).
It is not even entirely obvious that you are allowed simply to give the prize to x. Perhaps the right thing to do would be to toss a coin, and let the result determine whether to give the prize to x, or to toss a coin again, to determine who to give the prize.
These cases are perhaps more naturally described as involving determinately relevant but indeterminately possessed properties, rather than indeterminately relevant ones. (Jens Johansson pointed this out to me.) Thus, a glacier would be determinately relevant to the height of a mountain if it were definitely a part of the mountain. Inland waters and disputed areas would be determinately relevant to the size of a country if they were definitely geographical parts of the country. A pacemaker would be determinately relevant to a person’s weight if it were definitely a part of her body. Finally, a claim would be determinately relevant to a person’s wealth, if it were definitely a part of her possessions.
This argument assumes the following ‘extended transitivity’ principle: ‘If a is at least as F as b, and b is Fer than c, then a is Fer than c’. Given transitivity of ‘Fer than’, this principle follows from the definition of ‘equally F’ adopted in note 3.
Since spelling out his example would require rather elaborate stage-setting, I refer the reader to Rabinowicz’s article.
Third-order vagueness can be defined in terms of vagueness of “admissible sharpening of ‘admissible sharpening’”. And so on, for higher orders.
This ‘definitely’ operator may seem redundant, since ‘x is definitely Fer than y’ is true iff ‘x is Fer than y’ is true. Note, however, that ‘x is definitely Fer than y’ can be false although ‘x is Fer than y’ is neither true nor false.
This implication follows from the ‘general version’ of the collapsing principle: “For any x and y, if it is more true that x is Fer than y than that y is Fer than x, then x is Fer than y.” (Broome 1997, p. 77) Broome endorses also this version of the principle.
By ‘the contributory values of x and y’ I understand Constantinescu to mean the factors contributing to the Fness of x and y, respectively.
Constantinescu goes on to characterize genuine comparative vagueness as ‘extensional’, and derivative vagueness as ‘intensional’. I find his distinction between extensional and intensional vagueness difficult to grasp, however. (Fine 1975, makes a distinction by means of the same terms, but this is clearly not the distinction Constantinescu has in mind.)
It may be objected that I am here relying on a ‘tolerance principle’, leading to a sorites paradox. If a minuscule subtraction of rigour cannot tip the balance, so that Wittgenstein is definitely better, it seems to follow that he is not definitely better even than a philosopher with no rigour at all. To avoid this conclusion, it may be argued, we must assume a sharp cut-off point, separating philosophers only semicomparable to Wittgenstein from philosophers definitely less good than him, in any chain starting with Quine and containing successively less rigorous philosophers.
The assumption of a sharp cut-off point can be avoided by assuming second-order vagueness. Anyway, this does not matter for the purposes of my argument. If there must indeed be a sharp cut-off point, vis-à-vis Wittgenstein, and if this point happens to be located just below Quine, we can assume, instead, that Puine is slightly more rigorous than Quine. If Wittgenstein is not precisely comparable to Quine, he cannot be better than any philosopher less good than Quine, and less good than any philosopher better than Quine.
Rabinowicz claims to find some support for his reconstruction in the last paragraph of Section 4 of Constantinescu’s paper, but adds that he is not at all sure that this is what Constantinescu has in mind.
See Hirose (2007) for discussion and references.
It might be suggested that the only determinately relevant property of a as well as of b is that of saving at least one life, and that a’s relational property of saving more lives than b is indeterminately relevant. On this construal, the vagueness is derivative. It is very implausible, however, to deny that the property of saving a certain number of lives is a determinately value-relevant property of an option. Consider a case in which one option would save lives, while another option would realize some other value, such as the preservation of natural beauty. If the only determinately relevant property of the former option is that of saving at least one life, it would be difficult to make sense of the reasonable view that the relative values of the two options depend on how many lives that would be saved by choosing the life-saving option.
I am grateful to the following people for valuable comments on drafts of this paper: Wlodek Rabinowicz, Jens Johansson, Karl Pettersson, Folke Tersman, Sven Danielsson, Traugott Schiebe, Kent Hurtig, Melinda Roberts, Karin Enflo, Per Algander, Magnus Jedenheim-Edling, Frans Svensson, and an anonymous referee for this journal.
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Carlson, E. Vagueness, Incomparability, and the Collapsing Principle. Ethic Theory Moral Prac 16, 449–463 (2013). https://doi.org/10.1007/s10677-012-9352-9
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DOI: https://doi.org/10.1007/s10677-012-9352-9