Reply to McGee and McLaughlin

TIMOTHY WILLIAMSON

REPLY TO MCGEE AND MCLAUGHLIN

Vagueness (hereafter ‘V’) defends an epistemic view of vagueness, onwhich there is an unknowable fact of the matter in borderline cases.1 Intheir thoughtful and fairminded review (hereafter ‘MM’), Vann McGee andBrian McLaughlin defend a modified supervaluationism against V’s argu-ments.2 MM’s arguments face several problems. Section 1 below arguesthat MM’s discussion of bivalence goes wrong by misconstruing the statusof two fundamental principles about truth and falsity. Section 2 illustratesthe difficulties for MM’s attempt to combine a form of bivalence withdisquotational principles about truth and falsity by showing how MM’s ex-planation of the crucial notion of definiteness is in danger of incoherence.Section 3 addresses more technical issues about supervaluationist logic.Section 4 corrects a misleading aspect of MM’s description of V’s epistein-icism. At the end, V’s case for epistemicism and against supervaluationismis unscathed.3

1.

V’s discussion of bivalence is based on two fundamental principlesgoverning the notions of truth and falsity respectively (188):

If an utterance u means that P , then u is true if and only if P .(T)

If an utterance u means that P , then u is false if and only if notP .

(F)

One could gloss “means” in (T) and (F) as “makes the statement” or “ex-presses the proposition”. In that sense, the linguistic meaning of a sentence

1 Williamson, T. 1994, Vagueness, Routledge, London.2 McGee, V. and B. McLaughlin, 1997, ‘Review of Vagueness’, Linguistics and

Philosophy 21, 221–235.3 MM 233–234 discusses the matter of incomplete stipulations raised at V 213–214.

Rather than comment here, I refer the reader to a fuller discussion in T. Williamson, 1997,‘Imagination, Stipulation and Vagueness’, in E. Villanueva (ed.), Philosophical Issues 8:Truth, Ridgeview, Atascadero, CA, pp. 215–228, and 222–227.

Linguistics and Philosophy 27: 113–122, 2004.© 2004 Kluwer Academic Publishers. Printed in the Netherlands.

114 TIMOTHY WILLIAMSON

type s does not fix what an utterance of s means when s is indexical: whenrivals for the throne say “I am the rightful king”, each means somethingdifferent.

From (T) and (F), one can derive a principle of bivalence for utterancesby classical logic that MM accepts:

If an utterance u means that P , then either u is true or u is false.(B)

Quantifying into sentence position, we could paraphrase (B) thus: any ut-terance that means something is true or false. (B) does not imply that everyutterance is true or false. We should not expect a principle of bivalence forutterances to imply that; unless an utterance means something, nothing inthe intuitive rationale for bivalence requires it to be true or false. Thus(B) alone does not imply that utterances of vague sentences are true orfalse in borderline cases. For that, we must add the further assumptionthat utterances of vague sentences mean something in borderline cases;for example, an utterance of “Harry is bald” means that Harry is bald. Vdefends that further assumption (196–197); MM leaves it uncontested.

Suppose that one accepted a disquotational principle about meaning:

An utterance of �P� means that P .(Md)

One could then combine (Md) with (T) and (F) to derive disquotationalprinciples about truth and falsity:

An utterance of �P� is true if and only if P .(Td)

An utterance of �P� is false if and only if not P .(Fd)

From (Td) and (Fd) one could in turn derive by classical logic anunconditional principle of bivalence for utterances:

An utterance of �P� is either true or false.(Bd)

But although (Md) has many true instances, it is not a valid schema. Forexample, your utterance of “I am not TW” does not mean that I am not TW;it means that you are not TW. (Td) and (Fd) are correspondingly invalid;your utterance is not true if and only if I am not TW and false if andonly if I am TW; it is true if and only if you are not TW and false if andonly if you are TW. By contrast, (T) and (F) face no such problem. Theyassert (Td) and (Fd) conditionally on (Md); when the antecedent is false,the conditionals are vacuously true. In such cases, one derives the correcttruth- and falsity-conditions by making substitutions for ‘P ’ in (T) and (F)

REPLY TO MCGEE AND MCLAUGHLIN 115

nonhomophonic with u. The fundamental principles about truth and falsityare (T) and (F), not (Td) and (Fd).

Surprisingly, MM confuses the status of (T) and (F) with that of (Td)and (Fd). The first sign of this is the claim “the schemata [(T) and (F)] aregoing to have to be restricted anyway, because of the liar paradox” (227).The liar paradox provides good reason to restrict (Td) and (Fd), but asV emphasized (197) that reason does not extend to (T) and (F), becausewe can deny the corresponding instances of (Md); (T) and (F) would holdvacuously. MM shows no awareness of the greater flexibility (T) and (F)provide over (Td) and (Fd) in handling semantic paradoxes.

The confusion becomes more serious when MM associates a corres-pondence conception of truth (as opposed to a disquotational one) withthe claim that (T) and (F) are “contingent theses about how languageworks” and can therefore be denied without incoherence (227). Someonewho holds a correspondence conception of truth might indeed regard (Td)and (Fd) as contingent theses about how language works, because (Md)states a contingent fact about what utterances of the sentence type �P�mean, and there is no reason why (Td) and (Fd) should hold when (Md)does not. But these considerations do not threaten (T) and (F). Even ifan utterance could have meant something different, that would not makethe corresponding instances of (T) and (F) contingent; it would merelymake their consequents contingent. If S is a parent, it is still contingentthat S is a mother if and only if S is female, for it is contingent that S is aparent; but it is not contingent that if S is a parent then S is a mother if andonly if S is female. Someone who takes a robust correspondence view oftruth can regard (T) and (F) as analytic, in virtue of conceptual connectionsbetween meaning and correspondence. This removes an essential premisefrom MM’s subsequent argument that denial of (T) and (F) is incoherentonly on a disquotational conception of truth (227).

MM goes on to argue that if (T) and (F) are analytic, they “have no sub-stantive implications whatever about human thought and human language”(227). Certainly we cannot deduce from (T) and (F) any difference betweenhuman thought and language and nonhuman thought and language, butthat was never V’s aim. Epistemicism is motivated by general philosoph-ical and logical considerations; if correct, it is correct for all thought andlanguage, human and nonhuman alike. The analyticity of (T) and (F) nomore voids them of philosophically significant consequences than does theanalyticity of Leibniz’s Law. Furthermore, when (T) and (F) are applied toa particular utterance of a vague sentence in a borderline case, they arecombined with the synthetic assumption that the utterance means some-


thing. Thus MM’s attempt to trivialize the argument from (T) and (F) tobivalence in borderline cases fails.

2.

MM proposes a disquotational account of truth and falsity on whichborderline cases do not exhibit failures of bivalence. To characterize bor-derline cases, MM introduces a ‘definitely’ operator. For example, if a isborderline for ‘bald’, one will say:

a is neither definitely bald nor definitely not bald.(1)

The epistemicist may use the same words, but will explain ‘definitely’ inepistemic terms. If MM’s account is not to collapse into epistemicism, itmust explain ‘definitely’ in non-epistemic terms. V conjectured that thetask cannot be carried out (164); elsewhere I have given reasons in moredetail.4 MM candidly acknowledges the difficulties. They can be illustratedwith MM’s tentative explanation:

To say that an object a is definitely an F means that the thoughtsand practices of the speakers of the language determine con-ditions of application for the word F and the facts about a

determine that these conditions are met.5

(Def)

MM adds “With a suitably restricted understanding of the notion of ‘fact’,this characterization will do the job, or so we think; however, with amore relaxed conception of ‘fact’ it collapses completely” (230). The suit-ably restricted understanding countenances only precise facts; the morerelaxed conception countenances vague facts too. “But we despair ofdistinguishing vague from precise facts” (MM 231).

Let us prescind from the difficulty by concentrating on an example inwhich, at least for purposes of argument, we can delimit the intended factswithout circularity. We shall pretend that whether one is bald depends onlyon the number and configuration of one’s hairs; in other words:

For all N and C, necessarily, if one is bald and the numberand configuration of one’s hairs are N and C respectively, thennecessarily if the number and configuration of one’s hairs are Nand C respectively then one is bald.

(2)

4 Williamson, T. 1995, ‘Definiteness and Knowability’, Southern Journal of Philosophy33, Supplement, 171–191.


(2) is a standard form of supervenience principle. (Def) here presumablycomes to something like this:

To say that a is definitely bald means that the thoughts andpractices of the speakers of English determine conditions ofapplication for ‘bald’ and the number and configuration of a’shairs determine that these conditions are met.

(3)

(3) cannot be quite right as it stands, since one can say that someone isdefinitely bald in languages other than English without meaning anythingabout the English word ‘bald’; for simplicity let us ignore this problem.

What kind of conditions of application do the thoughts and practices ofthe speakers of English determine for ‘bald’? They must determine somesuch conditions, otherwise (Def) would not count paradigmatically baldmen as definitely bald. Surely MM does not postulate that they determineprecise conditions of application, for that would entail a conception oflanguage of just the kind MM rejects. The point is not that they cannotnoncircularly distinguish precise conditions from vague ones (contrast thecase of facts); it is that, even if they could, the conditions of application for‘bald’ determined by the thoughts and practices of speakers of Englishwould fall on the vague side of the distinction, on MM’s view. Let ustherefore suppose that the thoughts and practices of speakers of Englishdetermine vague conditions of application for ‘bald’. Now the vague con-dition of application preeminently associated with ‘bald’ by the thoughtsand practices of speakers of English is simply that one is bald; the idea thatthey determine some vague condition of application for ‘bald’ other thanthat one is bald is quite unappealing. But then (3) yields:

a is definitely bald if and only if the number and configurationof a’s hairs determine that a meets the condition that one isbald.

(4)

What is it to meet the condition that one is bald? Presumably it is just tobe bald; an answer involving ‘definitely’ would make (Def) circular. Thus(4) becomes:

a is definitely bald if and only if the number and configurationof a’s hairs determine that a is bald.

(5)

What is it for the number and configuration of a’s hairs to determinesomething? Determination is not a linguistic relation, for (4) is not a meta-linguistic statement. Rather, to say that the number and configuration of


a’s hairs determine that P is to say that necessarily if a has that numberand configuration of hairs then P . Thus (4) becomes something like:

a is definitely bald if and only if for some N and C, the numberand configuration of a’s hairs are N and C respectively and ne-cessarily if the number and configuration of a’s hairs are N andC respectively then a is bald.

(6)

Since a’s hairs must have some number and configuration, (6) and (2)easily yield:

If a is bald then a is definitely bald.(7)

By a parallel argument:

If a is not bald then a is definitely not bald.(8)

Given (7) and (8), (1) becomes inconsistent by classical reasoning accept-able to MM. For a is both not definitely bald and not definitely not baldby (1); if a is not definitely bald then a is not bald by (7); if a is notdefinitely not bald then a is bald by (8); hence a is both not bald andbald, a contradiction. Thus the attempt to characterize borderline casesusing ‘definitely’ collapses, on the most natural understanding of MM’sexplanation of ‘definitely’. Perhaps something else was intended; if so, itis not clear what.

By contrast, the epistemicist can explain what it is for a to be definitelyF as the absence of obstacles of a special kind to knowing that a is F ;V explains what kind (230–234). Since the absence of obstacles of thatkind to knowing does not entail the absence of all obstacles to knowing,the epistemicist explanation does not collapse ‘definitely’ into ‘knowably’.The explanation is vague: rightly so, because the notion of definitenessis vague. The vagueness is quite different from the threatened collapseof MM’s account. The suggestion that MM’s difficulties in explaining‘definitely’ are no worse than V’s (MM 230 n17) is false.

3.

MM raises some objections to V’s treatment of supervaluationist logic.Although they are largely anticipated in V itself, it may be useful to discussthem here.

Traditionally, supervaluationists identified genuine truth and genu-ine falsity with something like definite disquotational truth and definite


disquotational falsity respectively, and validity with the preservation ofgenuine truth. On this view (unlike MM’s), bivalence fails in borderlinecases, and various standard rules in natural deduction systems involvingthe discharge of premises no longer preserve validity: conditional proof(the deduction theorem), or-elimination (argument by cases) and reductioad absurdum (V 151–152). For suppose that the object-language containsquotation marks �� and a truth predicate T (here and below in this sec-tion ‘truth’ and ‘falsity’ denote the supervaluationist’s genuine truth andgenuine falsity). If the inference from φ to T(�φ�) is valid, it does notfollow that the unpremised conclusion φ ⊃ T(�φ�) is valid. For if φ

is truly neither true nor false, then the conclusion φ ⊃ T(�φ�) is itselfneither true nor false, whereas the fact that the inference from φ to T(�φ�)here leads from an untrue and unfalse premise to a false conclusion doesnot show that it ever leads from a true premise to an untrue conclusion.Thus conditional proof is not guaranteed to preserve validity. The examplecomes from Kit Fine’s classic presentation of a supervaluationist approachto vagueness.6 Nevertheless, MM objects that, on the envisaged nonbi-valent form of supervaluationism, higher-order vagueness invalidates theinference from φ to T(�φ�); for to say that it preserves truth is to assertT(�φ�) ⊃ T(�T(�φ�)�); but just as φ ⊃ T(�φ�) is untrue when φ is trulyneither true nor false, so T(�φ�) ⊃ T(�(T�φ�)�) is untrue when T(�φ�)is truly neither true nor false (224). V makes exactly the same point in afootnote (297 n32, uncited in MM). V uses Fine’s example only for ex-pository reasons, before the complications of higher-order vagueness havebeen introduced.

The same footnote gives an example to show that conditional proof failson traditional supervaluationism in a language with the truth-predicate Teven if, owing to higher-order vagueness, the inference from φ to T(�φ�)is rejected as invalid. For consider the inference from φ & ¬T(�φ�) to φ &¬φ. To say that it preserves truth is to assert T(�φ & ¬T(�φ�)�) ⊃ T(�φ &¬φ�). Now both of the principles T(�φ�) ⊃ φ and T(�φ & ��) ⊃ T(�φ�)& T(��) are easily shown to be valid on the natural supervaluationist se-mantics for higher-order vagueness. MM describes as ‘unproblematic’ thehalf of (T) with only the left-to-right conditional in the consequent (227),which implies the validity of T(�φ�) ⊃ φ given an utterance u that meansthat φ is true; evidently a true conjunction has true conjuncts. Granted thetwo principles both T(�φ & ¬T(�φ�)�) ⊃ (φ & ¬T(�φ�)) and T(�φ &¬T(�φ�)�) ⊃ (T(�φ�) & T(�¬T(�φ�)�)) are valid; since ¬T(�φ�) contra-

6 Fine, K. 1975, ‘Vagueness, Truth and Logic’, Synthese 30, 265–300, cited at MM 223;reprinted in R. Keefe and P. Smith (eds.), Vagueness: A Reader, MIT Press, Cambridge,MA and London, 1996, pp. 119–150, where the relevant passage is at 143.


dicts T(�φ�), ¬T(�φ & ¬T(�φ�)�) is valid, so T(�φ & ¬T(�φ�)�) ⊃ T(�φ& ¬φ�) is valid. Thus even with higher-order vagueness the inference fromφ & ¬T(�φ�) to φ & ¬φ is valid. But the supervaluationist cannot applyconditional proof and assert the unpremised conclusion (φ & ¬T(�φ�))⊃ (φ & ¬φ), for it is truth-functionally equivalent to φ ⊃ T(�φ�), whichthe supervaluationist rejects. So the supervaluationist denies that condi-tional proof preserves validity, and MM’s objection to the original examplemakes no difference.

Perhaps because higher-order vagueness is wrongly held to undermineV’s account of supervaluationist logic, MM misconstrues V’s introductionof the operator “definitely∗” (“definitely and definitely definitely and . . . ”)as “designed to block the escape to higher-order vagueness” (229). In fact,“definitely∗” is introduced in the course of an extended argument whichconcludes “Supervaluationism cannot eliminate higher-order vagueness. Itmust conduct its business in a vague meta-language” (V 161). The discus-sion of “definitely∗” is designed to show that the most promising attempt toescape from higher-order vagueness fails. MM notes that the definition of“definitely∗” does not guarantee its precision because, although it validatesthe principle “definitely∗ φ ⊃ definitely definitely∗ φ”, it need not validatethe principle “¬ definitely∗ φ ⊃ definitely ¬ definitely∗ φ”. V implicitlymakes this point in a footnote (297 n33, uncited in MM) that also givesreasons for doubting that the failure of the latter principle is at the root ofthe ineliminability of higher order vagueness.

MM also contests V’s description of conditional proof and reductio adabsurdum as principles of classical logic.7 By MM’s lights, one acceptsclassical logic by holding “that every classically valid sentence is true, andthat every classically valid inference from true premises leads to a trueconclusion”, even if one rejects conditional proof, reductio ad absurdum,argument by cases and contraposition (225). The issue is partly termino-logical: what shall we count as the extension of ‘classical logic’? But ifwe think of classical logic as comprising those forms of logical inferencetried and tested in mainstream mathematics and other branches of science,then classical logic certainly includes conditional proof, reductio ad ab-surdum, argument by cases and contraposition. What follows is not thatthese forms of inference are sacrosanct, just that the methodological costof rejecting them is similar (perhaps not identical) to the cost of rejectingthose principles that MM would include in classical logic.

7 MM does not mention that V qualifies “a violation of classical propositional logic”by “in a sense” (151).


4.

MM depicts the epistemicist as awaiting vindication from the future ofscience:

The epistemic theory makes a prediction that, when our linguistic understanding finally ad-vances to the point where we understand how words get their meanings, we shall find subtlehidden features of our usage that determine the semantically correct way to adjudicateborderline cases. (MM 235; see also 232–233)

V makes no such prediction, for it does not predict that our linguisticunderstanding ever will advance to the point where we understand ex-actly how words get their meanings. The notion of utterance meaning isvague and recalcitrant to systematic theorizing; the notions of truth andfalsity inherit the same vagueness and recalcitrance via (T) and (F). Wecannot expect ever to be able to adjudicate all borderline cases. Althoughsubtle hidden features of our usage surely will be discovered, they may notgreatly facilitate adjudication. As a first approximation, we can assume thatall relevant features of usage are already in view. What we do not know ishow these features determine meaning; the particular supervenience con-ditionals linking usage to the truth- and falsity-conditions of utterancesremain unknown.

We should not assume that in each case one specific feature is crucial,we know not which. For that would imply that hidden general laws linkusage to meaning. The supervenience of meaning on usage does not entitleus to expect such laws. It implies that absolutely specific configurations ofusage are sufficient for absolutely specific configurations of meaning, butnot that we can generalize across those supervenience conditionals whilestill conceptualizing the usage nonsemantically. Few common patterns arelikely to be discernible at that level. Naturally, this is not to deny that therewill be progress in understanding thought and language. It is to deny thatthe progress is likely to be of a kind that will issue in general formulasexpressing meaning as a function of usage.

The situation is not unique to epistemicism. MM too accepts thatwhatever semantic facts there are supervene on nonsemantic facts. Al-though MM differs from V over what semantic facts there are, MM facessimilar questions about the extent to which we shall ever be able to gen-eralize across the supervenience conditionals. MM provides no reason tosuppose that V is worse off in this respect.

We already have illuminating but imprecise accounts of meaning andreference, which we can reasonably hope to refine short of perfect pre-cision. Any inconsistency between one of those and an epistemic viewof vagueness would cast at least some doubt on the latter. But no one


has ever identified such an inconsistency. I conjecture that any generalaccount of meaning and reference inconsistent with epistemicism also hasconsequences that are implausible on grounds independent of vagueness.

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