Sunday, March 12, 2006


What Price Bivalence?

(Cross-posted at Arche)

In a talk I've been giving various places recently, I've argued that Delia Graff's theory of vagueness, like other theories which hold onto bivalence, runs into trouble with the forced-march Sorites. Here's the worry in brief. Graff explains our ignorance of where the boundary in a given series lies with the following similarity constraint:

'..if two things are saliently similar [in the relevant respect], then it cannot be that one is in the extension of a vague predicate, or in its anti-extension, while the other is not.' (2000: 57)

The idea is that whenever one is evaluting a pair of items in a suitably constructed Sorites series, both will be saliently similar, and so either both are in the extension of the predicate in question or both are in the anti-extension; the boundary is never where we are looking for it. This also gives an explanation of why we are liable to find Sorites reasoning compelling despite its invalidity (something critics have complained is entirely missing from Williamson-brand Epistemicism, though it's a central concern in Sorensen's Vagueness and Contradiction).

But now imagine a subject shown the items in a Sorites series in order, and asked of each whether it is in the extension of the relevant predicate or the anti-extension. Given the way the series has been constructed, she can neither competently judge all of the items to be in the extension, nor all of the items to be in the anti-extension. So at some point she must 'jump'; that is, judge some item n to be in the extension, yet judge item n + 1 (differing only marginally in the relevant respects from n) to be in the anti-extension. But by the similarity constraint, the boundary in known not to lie between n and n + 1, so how can her judgements be viewed as competent?

As Derek Ball, Geoff Georgi and Crispin Wright all rightly pointed out to me in various discussions, Graff has an answer to that question ready. It's essentially that of Diana Raffman in 'Vagueness without Paradox' (1994); when the subject judges n to be in the extension, n + 1 is too, but when the subject jumps, the boundary shifts so that now both n and n + 1 are in the anti-extension. So the subject's judgements can be seen as competent, yet the similarity constraint is never violated. As Raffman writes (1994: 57):

'There are shifts - events that occur - but these are best viewed as Gestalt like changes of “perspective” or “anchor,” not as boundary crossings - at least not if by ‘boundary’ you mean something that installs a simultaneous and/or fixed category difference between adjacent patches.'

It's clear that my original objection fails to engage with this kind of response, but I've been wondering how it's actually meant to work, given Graff's committment to bivalence - I've reached the conclusion that it doesn't. In presenting the forced-march above, I assumed that the subject only had two available responses. But we should be able to drop this assumption and allow the subject to respond as she likes (see for example Raffman 1994: 45-6). Suppose the subject judges n to be in the extension; then n + 1 is also in the extension. Now suppose when confronted with n + 1, the subjects responds 'I don't know', or 'There's no fact of the matter', or 'It's neither in the extension nor the anti-extension'. How does a shift in the location of the boundary such that both items are now in the anti-extension help to ensure that the subject says something competent? The problem is this; the boundary is supposed to shift so as to accomodate the speakers judgements - that's how we can see the subject who jumps in the face of the forced-march as doing something competent - but while there are a number of judgements a subject might return, the endorser of bivalence only recognises binary options for accomodating them.

(In a recent Analysis note ('How to understand contextualism about vagueness', 2005) Raffman explicitly considers what would happen on her earlier view when a subject switches from judging items to lie in the extension of the relevant predicate to judging them to be borderline. But the issues I raise above don't get into sight, simply because in a parenthetical remark she assumes '[f]or the sake of argument only' that we're working in a three-valued logic (246). With the assumption in place the subject's judgements can be accomodated, but that doesn't help us see how the trick is to be pulled whilst retaining bivalence.)

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I'm still not entirely sure if I understand the issues at stake here, but why can't we accomodate the multiplicity of responses within something like what it sounds like people are saying Graff's account is? When the subject jumps ordinarily and says that person n+1 is not bald, she then also says that person n is not bald, even though a moment ago she said he was bald. Similarly, when the subject jumps to saying there's no fact of the matter about person n+1, then she'll also have to say that there's no fact of the matter about person n.

Instead of just talking about an extention and an anti-extension, we can model the situation with a partition of the space into the extension, the anti-extension, the "no fact of the matter"-extension, the "I don't know"-extension, and whatever all the possible responses are. These extensions shift in a Gestalt-like manner at some point in the series, but at any point, they always either include both or neither of the salient pair of adjacent cases.
We can model things that way, provided we don't restrict ourselves to a bipartite partition of the relevant domain into an extension and an anti-extension. I was suggesting that that's why Raffman avails herself of a three-valued logic when she considers these cases. But I still don't see how this observation is supposed to help the epistemicist, who doesn't recognise any such further divisions in the series. (Well, this isn't quite right. But close enough for present purposes.)

To clarify. My point wasn't that we can't model these situations; as you point out, and as in fact I noted in my closing comment on Raffman, we can. My point is precisely that in doing so, we need to hold that the extension and anti-extension of a vague predicate don't exhaust the series; what's distinctive of the views I've been trying to criticise is their committment to resisting that thought. On this view, there is no 'no fact of the matter' extension, and no 'neither in the extension or the anti-extension' extension, so how are these responses to be accomodated?
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