THE IMPOSSIBILITY OF VAGUENESS

Abstract

I wish to present a proof that vagueness is impossible. Of course, vagueness is possible; and so there must be something wrong with the proof. But it is far from clear where the error lies and, indeed, all of the assumptions upon which the proof depends are ones that have commonly been accepted. This suggests that we may have to radically alter our current conception of vagueness if we are to make proper sense of what it is. The present investigation was largely motivated by an interest in what one might call the 'global' aspect of vagueness. We may distinguish between the indeterminacy of a predicate in its application to a single case (the local aspect) and in its application to a range of cases (the global aspect). In the first case, it is indeterminate how a predicate, such as a bald, applies in a given case; and, in the second case, it is indeterminate how a predicate applies across a range of cases. Given such a distinction, the question arises as to whether one might understand the indeterminacy of a predicate in its application to a range of cases in terms of its indeterminacy in application to a single case; and considered from this point of view, the result can be seen to show that there is no reasonable way in which this might be done. But the result can also be seen to arise from an interest in higher-order vagueness. It has often been observed that there is a difficulty in conceiving of indeterminacy in the presence of higher-order vagueness. For it cannot consist in these cases being borderline and those other cases not being borderline, for that would be compatible with there being a sharp line between the borderline cases and the non-borderline cases, contrary to the existence of higher-order vagueness; and, for similar reasons, it cannot be taken to consist in these cases being borderline borderline and those other cases not being borderline borderline or in something else of this sort. But it is hard to pin this difficulty down and, certainly, the failure of one particular attempt to characterize indeterminacy in the presence of higher order vagueness does nothing to establish a failure in principle. Considered from this point of view, the present result can be seen to provide a vindication of those (such as Graff-Fara [2003], Sainsbury [1991], Wright ([1987], [1992])) who have suggested that the existence of higher order vagueness does indeed stand in the way of having a reasonable conception of what indeterminacy might be. I begin by giving an informal presentation of the result and its proof and I then consider the various responses that might be made to the alleged impossibility. Most of these are found wanting; and my own view, which I hint at rather than argue for, is that it is only by giving up on the notion of single-case indeterminacy, as it is usually conceived, and by modifying the principles of classical logic that one can evade the result and thereby account for the possibility of vagueness. There are two appendices, one providing a formal presentation and proof of the impossibility theorem and the other giving a counter-example to the theorem under a certain relaxation of its assumptions. The mathematics is not difficult but those solely interested in the philosophical implications of the results should be able to get by without it. The general line of argument goes back to Wright [1987] and further discussion and developments are to be found in Sainsbury[1990, 1991], Wright [1992], Heck [1993], Edgington [1993], GomezTorrente [1997, 2002], Graff-Fara[2002, 2004], and Williamson [1997, 2002]. It would be a nice question to discuss how these various arguments relate to one another and to the argument in this paper. I shall not go into this question, but let me observe that my own approach is in a number of ways more general. It relies, for the most part, on weaker assumptions concerning the underlying logic and the logic of definitely and on weaker constraints concerning the behavior of vague terms; and it also provides a more flexible framework within which to develop arguments of this sort. Suppose we are presented with a sorites series—a series of men a0, a1, … , an+1, for example, which ranges through gradual increments from the first a0, who is hairless, to the last an+1, who is very hairy. Let p0, p1, … , pn+1 be the corresponding propositions that a0 is bald, that a1 is bald, … , and that an+1 is bald. There are then three things that we would be correct—and, indeed, utterly confident—in asserting. One is p0, that the first man a0 is bald, another is not-pn+1, that the last man an+1 is not bald, and the third is that the predicate 'bald' is not completely determinate in its application to the members a0, a1, … , an+1 of the series. Call the last of these claims 'the indeterminacy claim'. It is not altogether clear what is involved in making such a claim. But it does seem clear that its assertion should not be compatible with a complete bipolar resolution of the cases. Suppose that one is presented with a 'forced march'—one is successively asked 'Is a0 bald?', 'Is a1 bald?', … , 'Is an+1 bald?'; and suppose that, upon being presented with a forced march, one gives either the positive answer 'Yes' or the negative answer 'No' to each of the questions. Where there are 25 men, for example, one might respond 'Yes' to the first 12 questions and 'No' to the remaining 13 or perhaps 'Yes' to the first 13 questions and 'No' to the remaining 12. In such a case, there would surely be some kind of incompatibility or incoherence in giving these answers and yet going on to assert that the predicate 'bald' was indeterminate in its application to the various men. It will not be important, in what follows, to insist that the relevant notion of incompatibility should be logical incompatibility. Given an indeterminacy claim, there may perhaps be no logical incompatibility with a bipolar resolution of each case. But it does seem very plausible that there will be an incompatibility in a broadly conceptual sense—that the assertion of indeterminacy, by virtue of its very content and perhaps also by virtue of its being an assertion of that content, will exclude a complete bipolar resolution of the cases. This notion of incompatibility, whether logical or not, is naturally taken to be aligned to a corresponding notion of consequence or commitment by means of the following principle: (*) the assertion of various propositions are jointly incompatible iff their assertion commits one to a contradiction, i.e. to a given proposition and its negation. Thus the assertion of the indeterminacy claim will be incompatible with the complete bipolar resolution of the cases since the claim that there is a bipolar resolution of the cases will commit one to the predicate 'bald'not being indeterminate in its application to the various cases, in contradiction to the indeterminacy claim. Although this principle is very plausible, it is not, in fact, essential to our argument—which could have been stated more directly in terms of compatibility, without regard to its connection with a corresponding notion of commitment. But in order to preserve a sense of familiarity and to avoid certain irrelevant issues, I have found it preferable to state the argument in its present form. The notion of consequence or commitment is normally taken to conform to the principle of reductio ad absurdum: (**) if the assertion of the propositions P1, P2, … along with the proposition Q commit one to a contradiction, then the assertion of the propositions P1, P2, … alone commit one to not-Q. But as has often been observed, this principle is not at all plausible in the context of vagueness. For suppose that one is willing to talk, in such a context, of a proposition's being definitely the case or of its not being definitely the case. The assertion that a given proposition is not definitely the case is then presumably incompatible with asserting that it is the case, since their joint assertion would commit one to the contradiction that it is definitely the case and not definitely the case. So by reductio, the assertion that the proposition is not definitely the case will commit one to the conclusion that it is not the case; and yet surely this is a conclusion we would wish to avoid. Indeed, if it followed, then the assertion that a proposition was borderline (not definitely the case and not definitely not the case) would be self-contradictory, since it would commit one to the proposition's both being the case and not being the case. One might plausibly, and familiarly, explain these counter-examples to reductio along the following lines. In asserting some propositions P1, P2, … , one is committed to more than their actual content, one is also committed to their being definitely the case, definitely definitely the case, definitely definitely definitely the case, and so on. We might say that a proposition is super-definitely the case if it is the case, definitely the case, definitely definitely the case, and so on ad infinitum. Then in asserting some propositions, one is also committed to their being super-definitely the case and it is because of this further content that one cannot infer from the inconsistency of Q with some other propositions that Q alone (apart from its further content) is not the case. In the case of the assertion of not definitely P and P, for example, what explains their joint incompatibility is the straightforward conflict between P's being definitely the case and its not being definitely the case; and no inference from not definitely P to not-P is therefore justified. This suggests that there is an underlying notion of consequence which will conform to reductio and for which the commitment of P1, P2, … to Q will amount to Q being a consequence of P1, P2, … along with their supplementary content P1*, P2*, … . In the case of vagueness, there is no reason to think that any further content beyond the super-definiteness of the propositions in question will be relevant to whether there is a commitment. We are therefore led to the following principle: (***) the assertion of P1, P2, … commits one to Q iff Q is a consequence of its being super-definite that P1, super-definite that P2, … .1 I have used the terms 'commitment' and 'consequence' to mark the distinction between the two kinds of entailment. There is a related distinction between 'compatibility' and 'consistency', where the compatibility of certain propositions is their failure to commit one to a contradiction (as in (*) above) and the consistency of certain propositions is their failure to have a contradiction as a consequence. Certain propositions will then be compatible just in case their being super-definitely the case is consistent. Let us now return to the question of how one might respond to a forced march. We have already remarked that asserting the indeterminacy of the predicate 'bald' in application to the men of the sorites series should not be compatible with giving a positive or negative answer to each of the questions within a forced march. But something more general would also appear to hold. For suppose one were to respond to a forced march by saying that each of the first nine men, say, were definitely bald, that each of the next three men were borderline bald, i.e. neither definitely bald nor definitely not bald, and that each of the remaining men were definitely not bald. Then this would presumably also be incompatible with an indeterminacy claim. For a sharp line is still being drawn, not now between the men who are bald and the men who are not bald, but between the men who are definitely bald and the men who are borderline bald and, in addition, between the men who are borderline bald and the men who are definitely not bald. And the existence of sharp lines at this 'higher' level would appear to be as much in conflict with a claim of indeterminacy, as it might naturally be understood, as the existence of a sharp line at the 'lower' level. What goes for sharp lines at this higher level would appear to extend to sharp lines at higher levels still. It would not do, for example, to respond to each question of a forced march either with the response that the man is definitely definitely bald or with the response that he is not definitely definitely bald and not definitely definitely not bald or with the response that he is definitely definitely not bald (cf. Sainsbury [1991], 168–9 and Hyde [1994], 36). The more general point would appear to be this. Consider any series of responses to a forced march–such as 'Yes, … , Yes, No, … , No' or 'Definitely Yes, … , Definitely Yes, Borderline, … , Borderline, Definitely No, … , Definitely No'. Call such a series of responses sharp if (a) not all of the responses are the same and (b) any two responses that are not the same are inconsistent with one another (should both be given as a response to a single question). Then a claim of indeterminacy should exclude a sharp response to a forced march; it should not be possible to make the indeterminacy claim compatibly with giving a sharp response. We have so far formulated what are, in effect, two requirements on a satisfactory statement of indeterminacy. The first of these, which we may call the Incompatibility Requirement, is that the indeterminacy claim should be incompatible, in the intended sense, with a sharp response to a forced march. The second, which we may call the Compatibility Requirement, is that the indeterminacy claim should be compatible with the 'extremal' responses to a forced march, i.e. with a positive response to the first of the questions and a negative response to the last. For, as we have observed, it will be correct to make an indeterminacy claim in regard to a sorites series and also correct to give a positive response to the first question and a negative responses to the last; and, if it is correct to make the claim and to give these responses, then it will be certainly be compatible with making the claim that one give these responses. We can now state the impossibility result: Impossibility (Version A): No proposition (and hence no putative claim of indeterminacy) satisfies the compatibility and incompatibility requirements. In other words, there is no proposition that is both compatible with a positive and a negative response to the extremal cases of a sorites-like series and yet incompatible with any sharp response. Vagueness will therefore be impossible in so far as there is nothing that can meet the demands upon which its existence would appear to depend. A precise formulation and proof of this result is given in the appendix (under theorem 1), but let me sketch the idea behind the proof. Suppose that the propositions under consideration are p0, p1, … , p4 and that I (the putative indeterminacy claim) is compatible with p0 and not-p4. Consider now the proposition p1 and ask whether it or its negation is compatible with the propositions I, p0 and not-p4. If p1 is compatible, then add it to the propositions; if not- p1 is compatible, then add it to the propositions; and otherwise, add nothing. Now consider the proposition p2 and ask whether it or its negation is compatible with the resulting set of propositions, expanding the set with either if compatible with the set and otherwise leaving the set alone; and similarly in regard to the propositions p3 and p4. Suppose that the outcome of this procedure is a set of propositions consisting of I, p0, not-p1, p3 and not-p4. Then it may be shown that I is compatible with the following sharp response: superdefinitely p0, superdefinitely not-p1, neither superdefinitely p2 nor superdefinitely not-p2, superdefinitely p3, and superdefinitely not-p4. For the superdefinitely responses will be compatible by supposition and the neither-nor responses will be compatible since they are a consequence of the other responses. Thus under the supposition of compatibility, the proof will actually construct a sharp response in which each individual response is of the form super-definitely p, super-definitely not-p, or neither super-definitely p nor super-definitely not-p. The proof rests upon two principal assumptions. The first of these is that the notions of commitment and consequence should conform to (***) above, i.e. that P1, P2, … commit one to Q iff Q is a consequence of P1, P2, … being super-definite. The second is that the notion of consequence should conform to (**) above, i.e. that not-Q should be a consequence of P1, P2, … if P1, P2, … and Q are inconsistent. There are some ancillary assumptions upon which the proof depends but which are much less open to doubt: Consequence is subject to the usual structural rules (such as that P is a consequence of P and that R is a consequence of Q, P1, P2, … if it is a consequence of P1, P2, … alone); Conjunction is subject to the usual introduction and elimination rules (a conjunction is a consequence of its conjuncts and each conjunct is a consequence of a conjunction); The definitely operator is subject to the principles of the modal logic T (what is definitely the case is the case and if Q is a consequence of P1, P2, … then Definitely-Q is a consequence of Definitely-P1, Definitely-P2, …). The scope and interest of the impossibility result is broader than our informal exposition of it might lead one to expect. Although I have stated the result in application to a sorites series, there is nothing in the result itself which requires that this be so. Indeed, the result would appear to exclude a satisfactory formulation of indeterminacy in regard to any collection of propositions, as long as one of them is taken to be true and another to be false. There is no need, in particular, to suppose that the truth of a non-initial member pi+1 of the series implies the truth of its predecessor pi or the falsehood of a non-terminal member pi of the series implies the falsehood of its successor pi+1. Moreover, even when we are dealing with a sorites series, there is no need to suppose that the responses are given in answer to a forced march in which the various questions are asked of each member of the series in turn. The possible contextual interference that arises from the questions being asked in this way (with one set of answers creating the context for another) can therefore be avoided. The result applies with particular force to the standard supervaluational account of vagueness For given that the definiteness of a proposition is taken to be a form of truth, it is especially hard to see how one might plausibly deny any of the assumptions upon which the the result rests. But the result does not merely constitute a difficulty for the standard form of supervaluationism, for almost any other view will be able to make sense of the result and its assumptions and must therefore provide some account of the how the existence of vagueness is to be reconciled with the result. We should also note that, even if one were to reject an assumption upon which the proof of the result depends, one would still face the problem of saying how a global claim of indeterminacy is to be stated. The result points to a genuine difficulty in formulating indeterminacy claims; and when one examines the usual way of formulating these claims, they can immediately be seen to be wanting. It is common, for example, to take a predicate to be indeterminate in its application to a range of objects just in case one of those objects is a borderline case of the predicate. But the predicate's admitting a borderline case is compatible with the sharp response in which one case is taken to be borderline and the rest are not. Thus granted that an indeterminacy claim should be incompatible with any sharp response, then no characterization of this sort will be adequate. In certain cases, it may even be possible to show by other means that the compatibility and incompatibility requirements cannot be met. Consider a three-valued

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