On Sharp Boundaries for Vague Terms

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<ul><li><p>R. WEINTRAUB</p><p>ON SHARP BOUNDARIES FOR VAGUE TERMS</p><p>1.</p><p>The postulation by the epistemic theory of vagueness (Cargile 1969;Sorensen 1988; Williamson 1994) of a cut-off point between heaps andnon-heaps has made it seem incredible. Thus, Sainsbury (1991, 168, my it-alics) finds crazy . . . the idea that the predicate child divides the universeinto a set and its complement within the universe set.</p><p>I shall argue that an objection of a similar kind can be levelled againstmost theories of vagueness (Sections 2, 3). The only two which avoid itare untenable (Section 4). The objection is less compelling than it ini-tially seems (Section 5). However, even when this obstacle is removed,the epistemic theory is not yet vindicated (Section 6).</p><p>2.</p><p>The epistemic theory is explicitly committed to the claim that vague termsclassify sharply, inducing a cut-off point in the Sorites sequence. But asimilar (and seemingly implausible) commitment to the existence of a cut-off point can be discerned in most other theories of vagueness. Considerthe series:</p><p>p0 0 grains do not make a heapp1 1 grain does not make a heap</p><p>p1,000,000 1,000,000 grains do not make a heap</p><p>For the epistemicist there is a first n such that the nth statement is false,all previous ones being true.1 But similarly, for supervaluationists (Fine</p><p>Synthese 138: 233245, 2004. 2004 Kluwer Academic Publishers. Printed in the Netherlands.</p></li><li><p>234 R. WEINTRAUB</p><p>1975) there is a first n for which it is permissible both to classify n grainsas a heap and as a non-heap. Different, and equally legitimate, ways ofsharpening the concept heap coincide up to n, and diverge beyond it,converging again beyond some value, m. Here, then, there are two cut-offpoints, marking the range within which the predicate leaves matters open.</p><p>Those who model vagueness in terms of degrees of truth (Sanford 1975;Edgington 1992) admit the existence of a first n such that n grains do notconstitute a heap has a degree of truth smaller than 1, whereas those forwhom vagueness involves a third truth-value (Korner 1966) will have torecognise the existence of a first statement whose truth-value is neutral. Forothers, the transition from truth to falsity along the sequence will involvethe existence of a first sentence which is meaningless (Jeffrey 1967, 7) orillegitimate (Black 1963).</p><p>Paraconsistent responses to the paradox countenance a limited numberof contradictions which are either true (Priest and Routley 1989; Hyde1997) or false but analytically correct, constitutive, that is, of the meaningof the vague term (Eklund 2002). In both versions there will be two cut-offpoints, marking the range within which the predicate sanctions both thestatement in the Sorites sequence and its negation.</p><p>Now the epistemic theory seems to be doing quite well: all of itsrivals seem to be countenancing the existence of more than one transition!Sainsbury characterises the epistemic theory as the claim that vague ex-pressions draw sharp boundaries (1995, 589), irremediable ignorance[being] . . . the essence of vagueness . . . [whereas for its opponents] in bor-derline cases there is nothing to know or fail to know (1995, 591). But thiscan now be seen to be very misleading. There are many theories which ad-mit the existence of unknown facts (of some kind or another) pertaining toborderline cases. In fact, the (embarrassing) commitment to the existenceof a sharp transition is hard to avoid. All that is required to force it on us isthe least number principle, according to which if some natural number hasa property, then there is a least number which has it. If the first sentence istrue and the last one isnt, we can apply this principle to argue, there willbe a cut-off point: a first sentence which is not true.</p><p>3.</p><p>Can the transition from truth to falsity be blurred? According to Sainsbury(1990, 14), [s]cepticism about whether boundaryless classification is pos-sible can be set to rest . . . by contemplating the colour spectrum . . . [andnoting that] we can discern no boundaries between the different colours. . . There are bands, but no bounds. But we shouldnt be seduced by the</p></li><li><p>ON SHARP BOUNDARIES FOR VAGUE TERMS 235</p><p>suggestive metaphor just yet. Perhaps we can discern no boundaries in thecolour spectrum, but the question is whether they are genuinely absent.</p><p>We cannot avoid positing a cut-off point by refusing to be drawn intoanswering questions about every member of a sorites sequence (Tye 1994,206, original italics). We may already be committed to definite answers(and sharp transitions) by dint of other things we believe the least numberprinciple, for instance. Our silence cannot cancel this commitment. Butmight there not be a more principled way of blurring the transition fromtruth to falsity so as to reflect the thought that the transition . . . takes placeat no particular point but still takes place (Wright 1987, 256)?</p><p>We were forced to acknowledge the existence of a sharp transition byapplying the least number principle. This is clearly acceptable for thosewho think that the meta-language in which we reason about vagueness isprecise; that each sentence in the sequence can be semantically classified:it is either meaningful or meaningless; it is either true or false or indeterm-inate; it is either within the range in which the predicate leaves us leewayor it isnt. But what if the meta-language is itself vague? Might not thefact that this classification itself has borderline cases open the possibilityof replacing sharp boundaries with (blurred) shadows?</p><p>Two claims may be cited in support of a negative reply. First, ini-tial appearances to the contrary, a vague meta-language doesnt enableus to avoid ignorance about transitions. Second, at least in some cases,the vagueness of the meta-language will prevent us from successfullycontending with the Sorites paradox. I will argue for both claims in turn.</p><p>The vagueness of the meta-language must be construed in accordancewith the semantics under question. For instance, if the semantic predicatehas degree of truth x is itself vague, there will be sentences of the form pnhas degree of truth x which will themselves have an intermediate degreeof truth. And, of course, the ascription of this degree of truth might itselffail to be categorical, and so on. So, similarly, for other theories. Theremight, for instance, be sentences of the form pn is meaningless which arethemselves meaningless.</p><p>We gain nothing by replacing the classification of each element in theSorites sequence in terms of a single value with one involving a (pos-sibly infinite) hierarchy. It will still be possible to apply the least numberprinciple to the series of these hierarchies. There will, again, be a pointin which a semantic transition occurs: an element whose semantic status(given in terms of a hierarchy) differs from that of the first one. For degree-theorists, for instance, the hierarchy of the first element is 1,1,1. . . ., andthat of the millionth is 0,1,1. . . . We havent avoided the postulation of(unknown) sharp boundaries.2</p></li><li><p>236 R. WEINTRAUB</p><p>Here is the second reason for thinking that a vague meta-language wonthelp us out of our quandary. Some philosophers deny that we can theor-ise in a vague language. Our models, Goguen (1969, 327) claims, aretypically purely exact constructions, and we use ordinary exact logic andset theory freely in their development. This amounts to assuming we canhave at least certain kinds of exact knowledge of inexact concepts . . . . Itis hard to see how we can study our subject at all rigorously without suchassumptions.</p><p>I think Goguens scepticism about the possibility of rigorously theor-ising about vagueness in a vague language is too sweepingly formulated.To see why this is so, we have to make more precise the difficulty a vaguemeta-language might pose a theorist. At least in some cases, I will nowshow, it will prevent him from blocking the paradoxical reasoning.</p><p>To reject the Sorites argument we need (categorical) truths about thesemantics of the object language. For instance, the degree-theorist needssome pn to have (categorically) a degree of truth larger than that of pn+1if he is to claim that the Sorites reasoning isnt sound. So, similarly, forWright (1987), there must be two consecutive statements the first of whichis true and the second of which is indeterminate in truth-status, render-ing some conditional premise, pn&gt;pn+1, false or indeterminate, and theSorites reasoning unsound. But these are precisely the truths these the-orists forfeit once they claim (so as to avoid sharp transitions) that themeta-language is vague!</p><p>But the difficulty isnt, pace Goguen, endemic. For an epistemicist, thevagueness of the meta-language is, like its object-language counterpart,merely epistemic. The reasoning is unsound, he claims, since one of theconditionals, pn&gt;pn+1, is false. Of course, we do not know which oneit is. But it exists nonetheless, the vagueness of the predicates true andfalse notwithstanding.</p><p>The supervaluationist, too, avoids the difficulty. The categorical truththat he needs in order to block the paradoxical reasoning isnt one whichhe forfeits in assuming that the meta-language is vague. The argument isntsound, he claims, since one of the conditionals, pn&gt;pn+1, isnt super-true:there is a sharpening of the predicate which renders it false. This, to besure, means that there is a sharp transition in the Sorites sequence of thesharpened predicate, but not in that of the vague one, since there are othersharpenings which draw the line elsewhere.</p></li><li><p>ON SHARP BOUNDARIES FOR VAGUE TERMS 237</p><p>4.</p><p>The idea of blurry transitions having proven illusory, the only way ofavoiding cut-off points is to deny the existence of transitions altogether.This entails the supposition that vague terms are tolerant (Wright 1975),insensitive to marginal changes in the features that matter for their applic-ation. They apply, that is, to adjacent elements in the Sorites sequence tothe same degree. But this means that the term heap will apply to everycollection of grains to the same degree: the relation equally applicableis transitive. Thus, according to Unger (1979), there are no heaps; indeed,there can be no heaps. Alternatively, and still in keeping with the tolerantspirit, any number of grains constitutes a heap. There are, of course, othertolerant positions: it might, for instance, be indeterminate for every nwhether or not n grains constitute a heap. These bring nothing new to thediscussion, so I will focus on the two cited above. Both, I shall argue, areill-motivated and extremely implausible.</p><p>Arguments for tolerance invoke considerations pertaining to the ap-plication, acquisition and function of vague terms, and to our linguisticintuitions about them. All of these arguments can be rebutted.</p><p>Observational terms, it is claimed, must be tolerant. They are, accord-ing to Wright (1975, 335, original italics), coarse, applicable by casualobservation, rough and ready judgement. If a term is observational, heargues, applicable, that is, on the basis of information of one or moresenses, one must be able to tell whether it applies to an object just bylooking at it (1975, 338). And this means that it must be equally applicableto adjacent elements in the sequence, because they are indistinguishable:there is no visual basis for treating them differently.</p><p>The reasoning is specious. Given this intuitive understanding of obser-vationality, it is simply not true that an observational term must be tolerant.Even if theoretical knowledge (about the internal structure of the grains,say) is not allowed to affect the correct application of the term heap toa given collection of grains, the requirement of observationality does notpreclude us from invoking a visual comparison between each of the twoindistinguishable collections and a third one; whether or not two objectsdiffer with respect to their discriminability from a third is itself an obser-vational matter (Wright 1976; Burns 1986). And given the intransitivityof indiscriminability, such comparisons might sanction the application ofheap to one collection of grains and withholding it from another which itmatches.</p><p>But observational terms must be tolerant, it is urged, because they arereliably applied on the basis of observation. If there were adjacent ele-</p></li><li><p>238 R. WEINTRAUB</p><p>ments in the Sorites sequence only one of which was a heap, Dummettargues (1975, 320321), we couldnt correctly apply the term merely bythe employment of our sense-organs. But reliable doesnt mean infallible:we must allow for error, due to bad light, fatigue, etc., even in clear cases.And if observation is merely required typically to be sufficient for correctapplication, the requirement is quite compatible with the existence of asharp line dividing heaps from non-heaps. Around this cut-off point ourapplication of the term heap will not be reliable (even if we are allowedto invoke comparisons). But away from it it will. And the problem-casesare relatively few. Intolerant terms, I conclude, can be reliably applied onthe basis of observation.3</p><p>According to Wright, terms which are acquired ostensively must be tol-erant. To master the sense of a predicate is . . . to learn to differentiate casesto which it is right to apply it from cases of any other sort. If such masterycan be bestowed ostensively, a comparison of two cases must always reveala difference which sense-experience can detect (1975, 342). But sense-experience, we have already seen, can detect a difference between twoitems that match one another: it can invoke comparisons with a third case.Ostensive learning, we must conclude, does not require tolerance.</p><p>Next, consider an argument for tolerance that invokes the putative func-tion of vague terms. It would be absurd to force the question of theexecution of the command, Pour out a heap of sand here, to turn on acount of the grains, Wright (1975, 335) claims. Would it? The claim, pre-sumably, is that visually indiscriminable collections of grains are equallysuitable for the purpose of being a heap, so a cut-off point between twosuch collections would not mark a functionally significant line. Now, it isnot, strictly speaking, true that a tiny, even imperceptible, difference cannotmatter practically: we might be distributing grains of sand among a class ofstudents so that each can study one under the microscope. Still, typically,the tiny difference engendered by crossing the line will be insignificant,and the distinction drawn by the concept seems to subvert its purpose. Butit doesnt really. To be sure, everything else being equal, a concept wouldbe more useful if it didnt distinguish between adjacent elements in theSorites sequence. But everything else isnt equal. A tolerant concept wouldapply indiscriminately, whereas an intolerant one enables us to distinguishbetween clear cases falling on the two different sides of the line. Obviously,the latter, the (somewhat) arbitrary line it draws notwithstanding, is moreuseful....</p></li></ul>


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