On Sharp Boundaries for Vague Terms

R. WEINTRAUB

ON SHARP BOUNDARIES FOR VAGUE TERMS

1.

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”.

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).

2.

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:

p0 0 grains do not make a heap

p1 1 grain does not make a heap

···

p1,000,000 1,000,000 grains do not make a heap

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

Synthese 138: 233–245, 2004.© 2004 Kluwer Academic Publishers. Printed in the Netherlands.

234 R. WEINTRAUB

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.

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) or“illegitimate” (Black 1963).

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.

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 isn’t, we can apply this principle to argue, there willbe a cut-off point: a first sentence which is not true.

3.

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 shouldn’t be seduced by the

ON SHARP BOUNDARIES FOR VAGUE TERMS 235

suggestive metaphor just yet. Perhaps we can discern no boundaries in thecolour spectrum, but the question is whether they are genuinely absent.

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)?

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 isn’t. 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?

Two claims may be cited in support of a negative reply. First, ini-tial appearances to the contrary, a vague meta-language doesn’t 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.

The vagueness of the meta-language must be construed in accordancewith the semantics under question. For instance, if the semantic predicate‘has degree of truth x’ is itself vague, there will be sentences of the form ‘pn

has 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.

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 haven’t avoided the postulation of(unknown) sharp boundaries.2

236 R. WEINTRAUB

Here is the second reason for thinking that a vague meta-language won’thelp 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”.

I think Goguen’s 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.

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+1

if he is to claim that the Sorites reasoning isn’t 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−>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!

But the difficulty isn’t, 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−>pn+1, is false. Of course, we do not know which oneit is. But it exists nonetheless, the vagueness of the predicates ‘true’ and‘false’ notwithstanding.

The supervaluationist, too, avoids the difficulty. The categorical truththat he needs in order to block the paradoxical reasoning isn’t one whichhe forfeits in assuming that the meta-language is vague. The argument isn’tsound, he claims, since one of the conditionals, pn−>pn+1, isn’t 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.


4.

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 applicable’is transitive. Thus, according to Unger (1979), there are no heaps; indeed,there can be no heaps. Alternatively, and still in keeping with the “tolerant”spirit, any number of grains constitutes a heap. There are, of course, other“tolerant” 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.

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.

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.

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 of‘heap’ to one collection of grains and withholding it from another which itmatches.

But observational terms must be tolerant, it is urged, because they arereliably applied on the basis of observation. If there were adjacent ele-

238 R. WEINTRAUB

ments in the Sorites sequence only one of which was a heap, Dummettargues (1975, 320–321), we couldn’t correctly apply the term “merely bythe employment of our sense-organs”. But reliable doesn’t 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

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.

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 doesn’t really. To be sure, everything else being equal, a concept wouldbe more useful if it didn’t distinguish between adjacent elements in theSorites sequence. But everything else isn’t 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.

Even this (limited) arbitrariness is eliminable if the concept ‘heap’is construed as graded rather than categorical (Sanford 1975; Edging-ton 1992), and its ascription true to a degree (between 0 and 1). The


tiny, imperceptible, difference between adjacent items will be reflected inthe (small) difference between the degrees to which they fall under theconcept. The less functionally distinct they are, the smaller it will be.

The proponents of tolerance invoke a similar case taken from ethics.The vague term ‘mature’, they argue, has moral repercussions, and it wouldbe absurd to tie these to an intolerant concept: to attach moral blame to oneperson and not to another the only relevant difference between whom wasan age gap of two seconds (Wright 1976, 231). An intolerant predicatewould “draw lines in places which seem to have no moral significance”(Sainsbury 1991, 180). The absurdity is not removed when we note thatmaturity does not supervene on age. It still seems absurd to suppose thata person would be held morally responsible, whereas another, only veryslightly less emotionally mature, wouldn’t.

The antidote to the sense of absurdity is, again, to remember that despitethe (somewhat) arbitrary line the concept ‘mature’ draws, it enables us todistinguish between clear positive and negative cases, something which atolerant concept cannot do. True, it also forces on us invidious discrimin-ations, but a flawed tool is better than none. And because we are ignorantabout borderline cases, the distinctions we will in fact draw will by andlarge reflect substantial differences. This is a case where an imperfect useof an imperfect tool yields beneficial results.4

Intolerant terms, I conclude, do serve a purpose. The usefulness of toler-ant ones, on the other hand, is very restricted. Terms which are necessarilyempty (’largest prime number’) do have a use (Williamson 1994, 168).‘There is no largest prime number’ is an informative statement. But suchterms cannot be used to make contingent distinctions: to classify some butnot all collections of grains as heaps. And this use is paradigmatic of vagueterms. It is in the empirical realm that they come into their own.

We can now reply to Horgan (1994) and Eklund (2002), who invokeour intuitions in support of tolerance. True, it seems intuitively plausibleto suppose that a single hair cannot make a difference with respect tobaldness. But intuitions are fallible. And we have a reason for thinking thatthey are here mistaken. There is, we have seen, a good reason for having anarbitrary cut-off point. But it is far from obvious. Before we reflect aboutthe issue, it does seem implausible that a predicate should stop applying asa result of a tiny change.

The second form intolerance can take, vague terms applying indiscrim-inately, is even less plausible. Since both ‘heap’ and ‘non-heap’ are vague,they will, on this view, both be applicable to every collection of grains.This suggestion is obviously absurd if the law of contradiction is taken tofail comprehensively with respect to such collections, but it is unacceptable

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even if the contradiction is restricted to our language, keeping the worlditself consistent. To see this, consider two ways of construing this (latter)suggestion.

First, we can suppose that while vague sentences may be assertible,the concepts of truth and falsity do not apply to them (Rolf 1984). Butthis (absurdly) precludes the use of vague sentences to convey informationabout the world.

The second way of “insulating” the world from a (massively) inconsist-ent language is to suppose that the inconsistent meaning-postulates whichsanction the application of the terms ‘heap’ and ‘non-heap’ (say) to everycollection of grains engender truth-conditions which render consistent theset of true sentences. This suggestion, I think, is simply preposterous.

We have canvassed different versions of the “tolerant” position, andfound them all extremely implausible. No wonder Williamson (1994)labels it “nihilist”!

5.

There must be cut-off points (of some kind or another) along the Soritessequence. If their existence seems implausible, this can no longer be usedas a (seemingly very powerful) argument against epistemicists. It is nowseen to be everybody’s problem. But why does it seem implausible, almostincredible?

We could quite easily accept the existence of cut-off points if theyweren’t unknown, perhaps even unknowable. But it is almost universallyacknowledged “that we cannot know – do not know what it would be liketo know – where the cut-off comes . . . we have no conception of what itwould be like to be able to justify any particular belief about where itcomes” (Wright 1994, 150). “We are ineluctably ignorant of the allegedthresholds of vague expressions” (Wright 1994, 153). Sainsbury (1991,181) concurs: “A sharp ‘red’ predicate would draw lines in places wherewe could not detect them”. Wright characterise the view that “speakersdo not know where the sharp boundaries lie” as “bizarre-seeming” (1994,134). Why?

Several putative reasons for rejecting unknown cut-off points have beenadequately rebutted (Williamson 1994). But there is one worry which isyet to be addressed. Sorensen (1988, 245–246) diagnoses qualms aboutthe unknowability of cut-off points as deriving from allegiance to veri-ficationism, a dogma which dies hard. This explanation is unsatisfactory.It is not only verificationists who find objectionable the thought that anunknowable line exists between heaps and non-heaps. Even those who take


in their stride our inability to know the precise momentum and position ofelementary particles at any one time and find undisturbing the idea thatGoldbach’s conjecture may be undecidable balk at the thought that there isan unknowable moment at which one stops being a child. Why?

If there was a “least red thing of which ‘red’ is unqualifiedly . . . true”,Sainsbury suggests (1991, 169), “its identification would be an essentialprerequisite for the full and perfect command of the language”. If wedo not know the cut-off point, he claims, we do not have a completeunderstanding of the vague term, and this is certainly more absurd thanour failure to know empirical truths. Of course, individual speakers neednot fully understand the meaning of every word: young children start bynot understanding the meaning of any. But that is different from supposingthat a concept could be imperfectly grasped by all the speakers in the com-munity. It isn’t verificationism which is driving this objection, but, rather,a very plausible, almost truistic assumption about linguistic competence.

Statements about cut-off points, Sainsbury is (implicitly) claiming, areconceptual. So a speaker who is to be credited with an understanding oftheir constituent concepts must know their truth-values.5 This suggests twoways of making ignorance about their truth-values palatable. The first is toconcede the claim, and argue that there may be unknown conceptual truths.The second response is to deny the claim. I will consider them in turn.

We are not omniscient about meaning. Witness disagreements about themeaningfulness of the positivists’ metaphysical bogeys such as Bradley’s‘The absolute enters into, but is itself incapable of, evolution and progress’.Remember, too, the disputes about the correctness of philosophical ana-lyses of some terms. It is not just that we cannot come up with an analysis:that is not surprising. We find it difficult to determine whether a given ana-lysis is correct. Does ‘knowledge’ mean ‘true and justified belief’? Doesthe meaning of the terms ‘cause’ and ‘effect’ preclude an effect from pre-ceding its cause? Do ethical propositions report the objective obtaining ofethical facts, or does the statement ‘Murder is wrong’ mean ‘I disapproveof murder’? Some of these questions seem irresolvable; none are trivial.

Of course, to point to these related phenomena is not yet to remove thebafflement they (and unknown cut-off points) engender. Indeed, Moore’sparadox of analysis concerns precisely the possibility of an illuminatingphilosophical analysis. “If someone grasps the meaning of two expres-sions”, Dummett writes (1991, 17), “must he not thereby know whetherthey have the same meaning?”. If so, a correct philosophical analysiswould be obvious!

But we have made some progress by apprehending an affinity betweentwo paradoxes, the paradox of analysis and the Sorites. We now have an

242 R. WEINTRAUB

independent reason for countenancing conceptual ignorance: even thosewho find repugnant the supposition that vague terms induce unknown cut-off points will admit the informativeness of philosophical analyses. Whatseems like an implausible philosophical speculation in one case is data forexplanation in the other.

Have we disposed of the seemingly powerful objection we have beenconsidering to theories of vagueness which postulate conceptual ignor-ance? Not yet. It is one thing to admit the existence of unknown conceptualtruths and quite another to classify statements identifying cut-off points ofvague predicates as instances. Can the gap be straddled?

The paradox of analysis shows that we must distinguish between con-ceptual truths recognition of which is essential for understanding and thoseignorance about which does not impugn a speaker’s mastery of the relevantconcepts. An (uncontentious) example of the former sort is ‘Bachelors areunmarried’.6 Of the second kind, it is, in the nature of the case, difficult toadduce instances, but they exist if there are correct philosophical analyses.

Now the question is how Sorites sentences should be classified. If theyare akin to ‘Bachelors are unmarried’, the detour through the paradox ofanalysis will have been useless. And isn’t this the case? Surely someonewho doesn’t know that a hairless pate makes for a bald person doesn’tknow the meaning of the word ‘bald’! Yes, but this needn’t apply to allvague predicates. That there may be informative conceptual truths in someSorites sequences is attested by Hart’s (persuasive) proof that four is thethreshold for heaps. It “is the least number of grains that will form a looselybound but stable cone or pyramid” (1992, 3). The conclusion is reachedthrough a conceptual analysis, but it is surprising. We understand the term‘heap’ without knowing it.

Here is the second way of making unknown cut-off points palatable. Inthe case of some predicates, the statements in the Sorites sequence aren’tconceptual. An example is provided by the word ‘tall’. A tall man, it maybe argued, is one who is above average height (Hanfling 2001). This is aconceptual truth. But the cut-off point for the term ‘tall’ depends also onempirical facts (pertaining to the height-distribution of the population).7

6.

We have been forced to recognise that concepts draw sharp boundaries.But this does not by itself (pace Sorensen 1988, 247) vindicate the epi-stemic theory; a dispute remains about their nature. Is the (sharp) transitionfrom truth to falsity, meaninglessness, indeterminacy or partial truth? Ad-ditional arguments must be brought to bear in adjudicating which logic


and semantics are appropriate for vague discourse. Is the denial of bi-valence incoherent (Williamson 1994, 7.2)? Isn’t it unintuitive to suppose– as bivalence forces us to do – that there is an abrupt transition fromheaps to non-heaps, rather than a graded shift through different degrees ofheaphood (Edgington 1992)? The incredulity engendered by the epistemictheory is due not just to the (unknown) cut-off points it postulates, but tothe dramatic transitions they mark. “The dropping of one grain of sand”,Dummett (1975, 315) writes, “could not make the difference between whatwas not and what was a heap”. “It is . . . absurd”, he says (1975, pp. 308–309, italics mine), “to suppose that there exists a number which is smallbut whose successor is not”.

These are serious issues, which must be addressed. One formidableobjection to the epistemic theory has been rebutted. But the dispute hasnot thereby been settled.

NOTES

1 This is, of course, an oversimplification, since what constitutes a heap doesn’t dependonly on the number of grains. But we can suppose the other factors (density, shape, etc.) tobe fixed.2 Indeed, on some accounts, some sentences in the Sorites reasoning will be assignedmore than one hierarchy; an infinity if we are modelling vagueness in terms of degrees oftruth. If we cannot categorically say that pn has a degree of truth 0.6 (say), we will have tosuppose, instead, that to each possible value corresponds a different degree of truth, withits own degree of truth, and so on ad infinitum. The mind boggles at this profusion. Theepistemic theory’s commitment to a single unknown cut-off point seems very modest incomparison!3 Wright himself has subsequently (1987) recanted. The observationality of a predicateonly requires, he now thinks, the ability to discriminate between cases in which it definitelyapplies and those in which it definitely fails to apply. Judgements about borderline casesare typically uncertain and unstable.4 Wright is no longer daunted by the seeming arbitrariness which sharp distinctions in-volve. “If we troubled to have an absolutely sharp distinction between adolescence andadulthood, it would still be true that most adolescents would differ sufficiently from mostadults to justify the sort of differential treatment and expectations visited upon them”(1987, 269).5 In the case of predicates like ‘red’, there are no conceptual statements of the requisitekind. There is nothing analogous to the number of grains in the case of the term ‘heap’, orheight in that of ‘tall’. Instead, understanding the predicate, it is plausible to suppose here,requires the ability to apply it correctly even in borderline cases.6 This is why Fumerton’s (1983) solution to the paradox of analysis is inadequate. Lin-guistic understanding, he suggests, is a practical capacity. One can follow a semantic rulewithout being able to formulate it, just as one who is able to swim or ride a bicycle needn’tknow how he does it. To understand an expression one must be disposed to “regard it ascorrect” in various possible situations. And one might have such identical dispositions with

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respect to two expressions without realising this. One would then fail to grasp their syn-onymy. But the analogy between swimming and linguistic competence is misleading. Onemight be able to swim without having any propositional knowledge about swimming. Here,“knowledge how” and “knowledge that” are perfectly distinct. But linguistic knowledge isnot like that. Someone who doesn’t hold true the proposition ‘Bachelors are unmarried’cannot be credited with an understanding of the concept ‘bachelor’ even if he is perfectlydisposed to apply it only when he is disposed to apply the concept ‘unmarried’.7 The reference to the population explains why the term is – as has often been noted –context-dependent.

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On Sharp Boundaries for Vague Terms