THE IMPOSSIBILITY OF VAGUENESS

Authors


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.

§1 Informal Presentation of the Theorem

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:

  • (I) 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);
  • (II) 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);
  • (III) 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, …).

§2 The Scope of the Result

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 approach, for example, under which propositions can be either true or false or indefinite. We might understand ‘definitely A’ simply to mean A, subject the connectives to the usual strong or weak truth-tables of Kleene, and define consequence as preservation of truth. Then even though reductio will fail for such a logic and our proof will therefore not go through, the result will still hold—there will be no way to formulate a claim that satisfies both requirements.2

In addition to the result as stated, there are also some variants of the result than can be established; and any adequate response to the original result should also be capable of dealing with these variants. In the first place, we have supposed that P1, P2, … will commit one to Q just in case Q is a consequence of P1, P2, … being super-definite and that, likewise, the supposition of P1, P2, … will be compatible just in case the supposition of their being super-definite is consistent. But we may weaken the link between the two forms of inferential relationship and merely suppose that P1, P2, … will commit one to Q just in case definitely Q is a consequence of P1, P2, … being definite and that the supposition of P1, P2, … will be compatible just in case the supposition of their being definite is consistent.

Let us reformulate the compatibility and incompatibility requirements with this new sense of compatibility in place. Say that a proposition is definite (or definitely the case) to iterative degree n if it is definitely definitely … . definitely the case (with n ‘definitely's). Take the propositions constituting the sorites-type series to be P0, P1, … , Pn+1. We then take the modified compatibility requirement on the putative indeterminacy claim to be the requirement that its being definite to iterative degree n should be compatible with the extremal responses being definite to iterative degree n; and we let the modified incompatibility requirement be as before, but using the new notion of incompatibility. We can then establish the following form of the result (theorem 2):

Impossibility (Version B): No proposition satisfies the modified compatibility and incompatibility requirements.

Under the supposition that a given proposition satisfies the modified compatibility requirement, the proof will actually construct a compatible sharp response in which, for some m, each individual response is of the form ‘it is definite-to-iterative-degree-m that p’, or ‘it is definite-to-iterative-degree-m that not-p’, or ‘it is neither definite-to-iterative-degree-m p nor definite-to-iterative-degree-m that not-p’. Thus the previous response involving arbitrarily long iterations of the definitely-operator can be replaced with a response in which the iterations are bound by the length of the sorites-type series under consideration.

Despite appearances, the modified compatibility requirement is weaker than before since it is only compatibility in a relatively weak sense (without indefinite iterations of the definitely operator) that is required. On the other hand, the incompatibility requirement is stronger than before since it is incompatibility in a relatively strong sense (making use of only one iteration of the definitely operator) that is now required. So what we have in effect done is to trade the strength of the one requirement against the other.

Both versions of the result involve the two inferential relations of consequence and commitment, but since commitment, in either case, can be characterized in terms of consequence, we can also state the result directly in terms of the more straightforward notion of consequence (or consistency) and thereby avoid any question as to how it might be related to the less straightforward notion of commitment (or compatibility). Call a a series of responses superdefinite if each of its individual responses is of the form ‘it is superdefinite that …’. The inconsistency requirement is that the superdefiniteness of I (the putative claim of indeterminacy) should be inconsistent with any sharp superdefinite response; and the consistency requirement is that the superdefiniteness of I should be consistent with the super-definiteness of the extremal responses. The first version of the result now takes the form (corollary 3):

Impossibility (Version A*): No proposition satisfies the consistency and inconsistency requirements.

Or again, let us say that a series of responses is definite if each of its individual responses is of the form ‘it is definite that …’. The modified inconsistency requirement is that the definiteness of I should be inconsistent with any sharp definite response; and the modified consistency requirement is that I being definite to iterative degree n + 1 should be consistent with the extremal response being definite to iterative degree n + 1. The second version of the result then takes the form (corollary 4):

Impossibility (Version B*): No proposition satisfies the modified consistency and inconsistency requirements.

§3 What Has Gone Wrong?

Indeterminacy surely can exist. But how in the light of the impossibility results is this possible? Which of the assumptions or requirements upon which the proofs of the results depend should be given up? And why?

There are a limited number of options. One is to object to the principles governing the definitely operator. These are, in effect, the principles of the modal logic T; and it is hard to see on what basis they might be challenged.3

Another option is to question the ‘structural’ rules for the relation of consequence. These are:

Identity Any proposition is a consequence of itself.

Weakening Consequence hold under the addition of superfluous premisses.

Permutation The order of the premisses is irrelevant.

Contraction Repetition of premisses is irrelevant.

Cut Consequences can be chained with the conclusion of one relationship of consequence serving as the premiss of another.4

Various philosophers in the tradition of relevance logic have objected to some of these rules. They have thought that if Q is to be a consequence of the propositions P1, P2, … then these propositions must somehow be used (and perhaps used exactly once or in the right order) in getting to Q. But some of the structural rules may not hold under this understanding of consequence. Even if P1, P2, … are used in getting to Q, for example, there is no guarantee that P1, P2, … and an additional premiss R can be used in getting to Q; and so there is no guarantee that Weakening will hold.

I suspect that the proof can be reconfigured so as to deal with relevantist scruples of this sort. How exactly this is to be done will depend upon what the scruples are. But the critical point is that the places in the proof where an application of reductio is required would all appear to be ones in which the proposition to be discharged is one that is genuinely used in getting to a contradiction.

One might also question the proposed connection between the relations of commitment and consequence. This connection might take one of three forms. Under the strong connection (which is presupposed in the proof), the propositions P1, P2, … will commit one to Q just in case Q is a consequence of P1, P2, … being super-definitely the case; under the weak connection, P1, P2, … will commit one to Q just in case definitely Q is a consequence of P1, P2, … being definitely the case; and under what one might call the ‘null’ connection, the distinction between the two relations will collapse—P1, P2, … will commit one to Q just in case Q is a consequence of P1, P2, ….

I myself find it entirely plausible that the relevant notion of commitment should be subject to the strong connection. The principal question is whether commitment conforms to the rule of ‘D-introduction’: in asserting (or in being prepared to assert) a proposition P, am I thereby committed to its being definitely the case? Surely I am. For the relevant notion of definiteness is one in which it is cognate with the notion of a borderline case. To say that x is definitely F in the relevant sense is to say that it is F and not a borderline case of F. But now the assertion that a man is bald, let us say, will surely commit one to his not being a borderline case of a bald man. For how could one sensibly assert that a given man is bald and yet not thereby be willing to deny that he is a borderline case of a bald man? Given that this is so, it will then follow directly from the above equivalence that the man is definitely bald; and the rule of D-Introduction will have been vindicated.

However, there are various sceptical positions that would challenge the connection in its strong form. It might be acknowledged, for example, that there is an incoherence in asserting both that a certain man is bald and that he is a borderline case of bald, but it might be argued that the weak connection is sufficient to account for the incoherence. For the proposition that the man is definitely bald will be inconsistent with the proposition that he is definitely a borderline case of bald, since the latter proposition will imply that he is is borderline case of bald, which will be inconsistent with his being definitely bald. Thus the incoherence of the assertion can be explained in terms of the inconsistency of its content being definite; and nothing more than the weak connection is required.5 Or again, it might be thought that that there is no genuine incoherence in asserting that a certain man is bald and that he is a borderline case of bald, but only in definitely asserting that he is bald and that he is a borderline case of bald.6

I would prefer not to get embroiled in such issues and, to this end, it will sometimes be convenient to appeal to the formulations of the impossibility results that are stated entirely in terms of the notion of consequence (and the associated notion of consistency). The general question of the connection between commitment and consequence does not then arise; and questions of commitment are only relevant in so far as they are required to justified the corresponding forms of the consistency and inconsistency requirements.

So consider the inconsistency requirement first. There is the original (weaker) form of the requirement under which the superdefiniteness of the indeterminacy claim should be inconsistent with any sharp superdefinite response; and there is the modified (stronger) form of the requirement under which the definiteness of the indeterminacy claim should be inconsistent with any sharp definite response. Is there any reasonable basis upon which either form of the requirement might be challenged?

One possible objection to the weaker form of the requirement is that the existence of a borderline case should have as a consequence that the indeterminacy claim holds; and so the definite existence of a borderline case should have as a consequence that the indeterminacy claim definitely holds. But the sharp definite division of cases into those that are definitely borderline and those that are definitely not borderline will then be consistent with the indeterminacy claim as long as the definite division is itself consistent.

Similarly in regard to the stronger form of the requirement. The existence of a borderline case should have as a consequence that the indeterminacy claim holds; and so the superdefinite existence of a borderline case should have as a consequence that the indeterminacy claim superdefinitely holds. But the sharp superdefinite division of cases into those that are superdefinitely borderline and those that are superdefinitely not borderline will then be consistent with the indeterminacy claim as long as the superdefinite division is itself consistent.

It might be doubted whether in a case of an actual vague predicate such as ‘bald’ it would ever be correct to give sharp responses of this sort. For one might well think that whatever it was that prevented one from correctly making a division of the cases into those that were definitely bald and those that were definitely not bald would also prevent one from correctly making a division of the cases into those that were definitely borderline and those that were definitely not borderline or into those that were superdefinitely borderline and those that were superdefinitely not borderline.

But let it be allowed that a case of this sort might arise and that there was even a sense of ‘indeterminate’ in which it might then be correct to say that the predicate was indeterminate. Still, one would have to admit that there was a more robust, and more usual, way in which a predicate was capable of being indeterminate and which was inconsistent with a sharp definite or superdefinite response; and it is this more robust way of being indeterminate that is here in question.7

Another possible objection reverts to our characterization of the requirements in terms of compatibility. It will be granted that the indeterminacy claim should be incompatible with any sharp response, but it will be questioned whether the consistency of certain propositions being superdefinite is sufficient for them to be compatible.

But what more might reasonably be required? It will do no good to insist that not only should the propositions be superdefinite but that they should also be definitely superdefinite, definitely definitely superdefinite, super-superdefinite, and so on. For given the standard logic of the definitely operator, these supposed strengthenings are not strengthenings at all. The reason is that the claim that P is superdefinite is equivalent, by definition, to the claim that (P & definitely P & definitely definitely P & definitely definitely definitely P & …). A trivial consequence of this claim is that (definitely P & definitely definitely P & definitely definitely definitely P & …). Now it is generally true that (definitely P1 & definitely P2 & definitely P3 & …) has as a consequence that definitely (P1 & P2 & P3 &…); and so, in particular, (definitely P & definitely definitely P & definitely definitely definitely P…) will have as a consequence that definitely(P & definitely P & definitely definitely P & …), which is just to say that it is definitely superdefinite that P. Thus nothing is to be gained by piling on further applications of the definitely operator.

One might, of course, insist that the propositions should also be hyperdefinite, where this is something stronger than being superdefinite, or that it should be hyperdefinite to the ‘second order’, where this is something stronger still than being hyperdefinite or hyper-hyperdefinite or hyper-definite to any finite iterative degree n. If all of the orders of definiteness can be wrapped up into a single ‘transcendental’ order of definiteness, then the result can still be reinstated using the transcendental order of definiteness in place of the original low-level notion. But one might have the view that, even though one can make a compendious assertion of definiteness by way of a ‘scheme’ that runs through all the possible ways to understand the notion, the content of this compendious assertion is not itself capable of being denied (similar positions have been adopted in connection with the semantic and set-theoretic paradoxes). The proof of the result cannot then be reinstated since it requires the application of reductio to the content of the compendious assertion, which must therefore be something that can be denied.

Intriguing as this suggestion may be, I doubt that it can be used to get round the essential difficulty. For it is plausible that the claim of indeterminacy will be formulated using notions of hyper-definiteness up to a certain order (and this would appear to be especially true if it is the kind of claim that can be denied and is not merely a ‘scheme’). But higher orders of hyper-definiteness will then be irrelevant to the existence of an incompatibility between the indeterminacy claim and a sharp response. For the result will tell us that there is a sharp response compatible with the indeterminacy claim as long as the iterations of the hyper-definitely operator are confined to the given order; and this compatibility will remain even when arbitrarily high order iterations of the hyper-definitely operator are allowed.8

Might there be other strengthenings that are relevant to compatibility but do not make use of a definitely operator? Possibly, though it is hard to see what they might be if they are to be peculiarly relevant to the use of a vague language and to a possible conflict with the indeterminacy claim.

With these objections failing, it is hard to see how the inconsistency requirement might plausibily be denied. It may, of course, be allowed that the indeterminacy claim should be consistent with a sharp response. Indeed, it simply follows through repeated applications of reductio that if the indeterminacy claim is itself consistent then it will be consistent with one of the combinations: P0, P1, P2; P0, P1, not-P2; P0, not-P1, P2; P0, not-P1, not-P2; not-P0, P1, P2; not-P0, P1, not-P2; not-P0, not-P1, P2; not-P0, not-P1, not-P2 (and similarly when there are more than three propositions). For either P1 is consistent with the indeterminacy claim or the indeterminacy claim will have not-P1 as a consequence by reductio. Add P0 to the indeterminacy claim in the first case and otherwise leave it alone. Now either P1 will be consistent with the resulting set or the set will have not- P1 as a consequence by reductio. Modify the set accordingly. Continuing in this way, we will end up with a consistent set which, for one of the combinations, has each of its constituents as a member or a consequence and which will therefore be consistent with the combination.9

Thus the mere acceptance of reductio should leave one to expect that an indeterminacy claim (or any consistent claim whatever) will be consistent with a sharp division of cases. And such a position is, of course, entirely consonant with the supervaluationist's or epistemist's point of view. For as long as the indeterminacy claim is consistent, it will be true under some appropriate ‘sharpening’ or ‘epistemic alternative’; and under that sharpening or epistemic alternative one of the eight combinations must be true.

But the matter is entirely different when the division is required to be definite. For surely we do not want a claim of indeterminacy to be consistent with a complete division of the cases into those that are definitely bald and those that are definitely not bald, let us say. And what goes for a straightforward division of this sort should surely extend, under a robust conception of indeterminacy, to a less straightforward division in which the status of the cases as indefinite or borderline may also be in question. It would not do, for example, to suppose that a claim of indeterminacy might be consistent with a complete division of the cases into those that were definitely borderline and those that were not definitely borderline.

If this is right, then even the strong inconsistency requirement should be taken to hold: the indeterminacy claim definitely being the case (or, indeed, its simply being the case) should be inconsistent with any sharp definite response.

What then of the consistency requirement? Again, there are two forms (where the weaker form of the one requirement is linked with the stronger form of the other). Under the original (stronger) form, the superdefiniteness of the indeterminacy claim should be consistent with superdefiniteness of the extremal responses. Under the modified (weaker) form, the indeterminacy claim being definite to iterative degree n + 1 should be consistent with the extremal responses being definite to iterative degree n + 1.

We might attempt to approach the question of consistency by way of truth. For if we can get our opponent to agree to the various propositions whose consistency is in question, then he must surely grant that they are consistent. So consider a typical sorites series involving ‘bald’. Surely we would wish to assert that the first, completely bald, man in the series is bald. And surely we would wish to assert that he is not a borderline case of someone who is bald. And surely we would wish to assert that he is not a borderline case of someone who is bald and not a borderline case of bald. And so on ad infinitum. Whether or not we subscribe to a general principle that allows us to pass from the assertion that x is F to the assertion that x is not a borderline case of F, each of these various assertions seem to be in perfect order; and we certainly have no sense that the extent to which we should be willing to make these further claims is somehow bound by the length of the series.

What goes for the first, completely bald, man in the series also goes for the last, very hairy, man in the series. We will wish to assert that he is not bald, that he is not a borderline case of someone who is not bald, that he is not a borderline case of someone who is bald and not a borderline case of someone who is not bald, and so no ad infinitum. And what goes for the two extreme cases also goes for the indeterminacy claim itself. Why should we hesitate in asserting that the indeterminacy claim is something that definitely holds, i.e. is not a borderline case of something that holds, that it is not a borderline case of something that definitely holds, and so on ad infinitum?

So it looks as if we should accept all of the propositions whose consistency is in question and should therefore grant that they are consistent. We can perhaps strengthen the argument by slightly modifying the predicate in question. Suppose, to simplify, that being bald is a matter of how many hairs one has on one's head and that it is a completely precise matter how many hairs a man has on his head. Define bald* in terms of bald as follows: a man is bald* if he either has 0 hairs on his head or he is bald and does not have 1010 hairs on his head. Let the first member a0 of the soritic series be a man with 0 hairs on his head and its last member an+1 a man with 1010 hairs on his head. Then it is presumably super-definite that a0 has 0 hairs on his head and hence super-definite that he is bald*, and it is presumably super-definite that an+1 has 1010 hairs on his head and hence superdefinite that he is not bald*. But surely the switch from ‘bald’ to ‘bald*’ in no way reduces our confidence in the robust indeterminacy of the predicate. Thus all of the propositions in question should again be accepted and the consistency requirement will be met.

Nor does it help to go epistemic. One can go through the above considerations, using ‘known’ or ‘knowable’ in place of ‘definite’, and it seems that the very same conclusions will hold. Thus if asked whether I know that the man a0 with 0 hairs on his head is bald, then surely I should say ‘Yes’; and if asked whether I know that I know, then surely I should again say ‘Yes’, and so on ad infinitum (cf. GomezTorrente [1997], 245). But it does not really matter whether the conclusions are the same. For vagueness most directly concerns what is definite or borderline. The epistemicist merely provides an epistemic interpretation of these notions; and if our intuitions concerning the one do not square with our intuitions concerning the other, then so much the worse for the epistemic interpretation.10

I should mention, in the interests of full disclosure, that the consistency and inconsistency requirements can both be met as long as the consistency requirement is taken to hold in a suitably weakened form (theorem 3). The inconsistency requirement is as before: the definiteness of the indeterminacy claim should not be consistent with a sharp definite response (or, equivalently, the claim should not be weakly compatible with a sharp response). But for the purposes of the consistency requirement, it is only required that the definiteness of the indeterminacy claim should be consistent with the first case being definitely positive and the last case being definitely negative (or, equivalently, the claim should be weakly compatible with the first case being positive and the last case being negative); it is not also required that the iterative definiteness (to degree n + 1) of the indeterminacy claim should be consistent with the iterative definiteness of the extremal cases being respectively positive and negative.

There is perhaps some comfort to be derived from making such discriminations in the iterative degree of definiteness. But it is hard not to have the sense that the discriminations provide a merely ‘formal’ solution to the impossibility results and have no basis in our intuitive understanding of what it is for a given case to be borderline or definite.

The only remaining way out is to question reductio. We must allow that not-Q may not be a consequence of certain propositions even though those propositions with Q imply a contradiction. But on what basis are we to question reductio?

We have already observed that reductio may fail for the relation of commitment. For the assertion that P and not-definitely P might be taken to commit one to a contradiction even though the assertion that not-definitely P does not commit one to not-P. And the reason for this failure is that in asserting P we are implicitly committing ourselves to definitely-P.

However, in the present case, no such explanation is available to us. For our concern is with the relation of consequence rather than commitment. The actual inferences we employ in the proof in working out the consequences of a given set of propositions are straightforward (they do not involve anything like D-introduction) and so there is no reason to think that they will not lend themselves to the application of reductio. Indeed, if one examines the details of the proof (in its first version), the principle of reductio is applied to propositions of the form ‘it is superdefinite that P’. But it is superdefinite that P implies that it is definite that it is superdefinite that P; and so any explanation of the failure of reductio that works off an implicit strengthening of the premiss to be discharged can have no application in the present case.

We are without an adequate response to the result; and this suggests that there is something deeply misguided about the assumptions upon which it is based. My own view, which I shall not argue for here, is that we lack a proper conception of how global claims of indeterminacy (as embodied in the indeterminacy claim) should relate to local claims on indeterminacy (as embodied in the application of the definitely operator to individual propositions). For it has been presupposed that there is no simple conflict between an indeterminacy claim and a complete bipolar resolution of the propositions under consideration (into those—say P0, P1, P2—that are the case and those—say P3, P4, … , Pn+1—that are not the case) but that the conflict can only be explained in terms of the resolution being taken to have the further status of holding definitely (so that it is implicitly taken to be definite that P0, P1, and P2 and definite that not-P3, not-P4, … , and not-Pn+1). I believe that this presupposition is mistaken and that it is only once we see how we can do without it that we can see how the difficulties raised by the impossibility result might be resolved.

Appendix

Formal Appendix

We first provide a formal statement and proof of the impossibility theorem. We presuppose, by way of background, an infinitary sentential language L(D). The symbols ofL(D) are:

  • (i) the sentence-letters p1, p2, … ;
  • (ii) the negation operator ¬;
  • (iii) the conjunctive operator ⁁; and
  • (iv) the definitely operator D.

The formulas ofL(D) are defined according to the following rules:

  • (i) each sentence letter is a formula;
  • (ii) if A is a formula, then so is ¬A;
  • (iii) if Δ is a countable (i.e. either a finite or countably infinite) set of formulas, then ⁁Δ is a formula;
  • (iv) if A is formula, then so is DA.

It is largely for convenience that we take the conjunction operator ⁁ to apply to a set rather than a sequence of formulas. Formally, we might think of ⁁Δ as the ordered pair <⁁,Δ> but, when Δ={A1, A2…}, we may write ⁁Δ more intuitively as A1∧ A2∧… .

We use the following abbreviations:

  • (B ∧ C) for ⁁{B, C};

  • IA for ¬DA ∧¬D¬A.

  • D0A for A, and Dn+1A for DDnA, n = 0, 1, … ;

  • DA for ⁁{DnA: n = 0, 1, 2, …};

  • InA for ¬DnA ∧¬Dn¬A and IA for ¬DA ∧¬D¬A.

Thus DnA is the formula:

inline image,

while DA is the formula:

A ∧ D1A ∧ D2A ∧… .

Note that InA is not the formula:

I … IA.

Given a set Δ, we let DΔ be {DA: A ∈Δ} and, similarly, we let DnΔ be {DnA: A∈Δ} and DΔ be {DA: A∈ Δ }. We say that the set of formulas Δ is D-closed if DΔ⊆Δ, i.e. if DA∈ Δ whenever A∈ Δ; and we let ΔD be the smallest D-closed set to contain Δ. We sometimes use AD for {A}D, i.e. for {DnA: n = 0, 1, 2, …}. Evidently, ΔD=∪{AD: A ∈ Δ}={DnA: for A ∈ Δ and n = 0, 1, 2, …} and (ΔD)DD.

We take the formulas of language L(D) to be governed by a relation of consequence |- that holds between a countable set of formulas Δ and a single formula A (though there would be no problem in extending the relation to uncountable sets of formulas). We say Δ |- Γ if Δ |- A for each A in Γ and we use lists of formulas and sets to the right or left of |- in an obvious way.

The consequence relation is governed by the following three groups of rules:

  • (I) Structural Rules

Identity: A |- A;

Weakening: if Δ |- A and Δ ⊆ Γ then Γ|- A;

Cut: If Δ |- Γ and Γ, Θ |- A then Δ, Θ |- A.

  • (II) Rules for ¬ and ⁁

¬-Introduction: (Reductio) if Δ, A |- B, ¬B then Δ |- ¬A.

⁁-Introduction: if Δ |- Γ then Δ |- ⁁Γ

⁁-Elimination: ⁁Δ |- A if A ∈Δ

  • (III) Rules for D

D-Elimination DA |- A

D-Distribution if Δ |- A then DΔ |- DA.

Note that Δ |- Δ by Identity and Weakening and so Δ |- ⁁Δ by ⁁-Introduction. Use of the structural rules will often be implicit.

We define definite consequence or commitment by:

Δ |-D A if ΔD |- AD.

Definite consequence only requires D-strengthening on the left (giving us the characterization under (***) in the main text above):

Lemma 1Δ |-D A  iff ΔD |- A.

Proof The left-to-right direction is trivial. For the right-to-left direction, assume ΔD |- A. By D-Distribution, D(ΔD) |- DA. But D(ΔD) ⊆ΔD. So by Weakening, ΔD |- DA. Iterating the argument, ΔD |- DnA for n = 2, 3, … . But then ΔD |- DnA for n = 0, 1, 2, … ; and so ΔD |- DA by ⁁-Introduction.

Lemma 2 |-D conforms to the structural rules of Identity, Weakening and Cut.

Proof We consider each rule in turn.

Identity AD |- A by Identity and Weakening for |-; and so A |-D A, by lemma 1.

Weakening Suppose Δ |-D A and Δ⊆Γ. Then ΔD |- A by lemma 1 and ΔD⊆ΓD. So by Weakening for |-, ΓD |- A; and hence Γ |-D A (again by lemma 1).

Cut Suppose Δ |-DΓ and Γ, Θ |-D A. Then ΔD |- ΓD and ΓD, ΘD |- AD. By Cut for |-, ΓD, ΘD |- AD; and consequently, Δ, Θ |-D A.

It is evident that definite consequence conforms to the rule of D-Introduction, i.e. that A |-D DA, since AD |- DA by Identity and Weakening. I note without proof that |-D is the smallest relation to contain |- and to conform to the structural rules and D-introduction.

Reductio holds in the following modified form for definite consequence:

Lemma 3 If Δ, A |-D B, ¬B then Δ |-D¬DA.

Proof Suppose Δ, A |-D B, ¬B. Then ΔD, AD |- B, ¬B. By ⁁-Elimination, DA |- AD and so, by Cut, ΔD, DA |- B, ¬B. But then by reductio for |-, ΔD |- ¬DA; and consequently,Δ |-D¬DA.

To state the impossibility result, we need some further terminology. A set of formulas Δ is said to be inconsistent if, for some formula B, Δ |- B and Δ |- ¬B and Δ is otherwise said to be consistent. Likewise, Δ is said to be incompatible if, for some formula B, Δ |-D B and Δ |-D¬B and Δ is otherwise said to be compatible. Δ is said to be inconsistent (or incompatible) with the set Γ if Δ∪Γ is inconsistent (or incompatible).

Let p be an any sentence-letter, fixed for the purposes of the following discussion. Then an individual response is a formula A(p) whose sole sentence-letter is p; and the formula A is a response to the question of B if it is of the form A(B), where A(p) is an individual response. A collective response is a sequence A1(p), A2(p), … , An(p) of individual responses; and A1, A2, … , An is said to be a collective response to B1, B2, … , Bn if A1, A2, … , An are respectively of the forms A1(B1), A2(B2), … , An(Bn), where A1(p), A2(p), … , An(p) is a collective response.

We say that the collective response A1, A2, … , An is sharp if:

  • (i) Ai≠ Aj for some i, j ≤ n;
  • (ii) Ai is inconsistent with Aj whenever Ai≠ Aj for 1 ≤ i < j ≤ n.

In a sharp response, we give any two questions the same answer or inconsistent answers, with at least two of the answers not being the same (and similarly in regard to a sharp response to B1, B2, … , Bn).

We call {B0, ¬Bn+1} the extremal response to the formulas B0, B1, … , Bn+1, n ≥ 0, With this terminology in place, we are now in a position to state and prove the result:

Theorem 1 Take any formulas B0, B1, … , Bn+1, n ≥ 0. Then there is no set of formulas Δ0 which is compatible with the extremal response and yet incompatible with any sharp response to B0, B1, … , Bn+1.

Proof The proof is somewhat reminiscent of the proof of Lindenbaum's Lemma. Take any formulas B0, B1, … , Bn+1 and any set of formulas Δ0 compatible with B0 and ¬Bn+1. We show that Δ0 is compatible with a sharp response to B0, B1, … , Bn+1.

To this end, we ‘blow up’Δ0 to a set Δn+1 from which a compatible sharp response can be more readily discerned. We let Δ10∪{B0, ¬Bn+1} and, for k = 1, 2, … , n, we let:

Δk+1k∪{Bk} if Δk is compatible with Bk,

   =Δk∪{¬Bk} if Δk is compatible with ¬Bk

   =Δk−1 otherwise.

It is evident from the construction that:

  • 1Δk is compatible for k = 0, 1, … , n + 1, and
  • 2Δk⊆Δl for 0 ≤ k < l ≤ n +1.

Using Δn+1, we define a collective response A0(p), A1(p), … , An+1(p) (and a corresponding collective response A0(B1), A1(B2), … , An+1(Bn+1) to B0, B1, … , Bn+1). Where k = 0, 1, … , n +1:

  • (a) Ak(p) = D(p) if Bk∈Δn,
  • (b) Ak(p) = D(¬p) if ¬Bk∈Δn, and
  • (c) Ak(p) = I(p) otherwise.

The collective response is well-defined since if Bk∈Δn+1 and ¬Bk∈Δn+1 for some k = 0, 1, … , n +1, Δn+1 would not be compatible by Identity and Weakening for |-D, contrary to (1) above.

We now show:

  • 3Δn+1 |-D Ak(Bk) for k = 0, 1, … , n +1.

Pf. There are three cases:

Bk∈Δn+1 In this case, Ak(Bk) is the formula D (Bk). Bk∈Δn+1, and so Δn+1 |-D Bk. But Bk |-D BkD and BkD |-D⁁(BkD) = D(Bk) by ⁁-Introduction. So by the structural rules for |-D, Δn+1 |- D (Bk).

¬Bk∈Δn+1 Similar to the previous case but with ¬Bk in place of Bk.

Bk, ¬Bk∉Δn+1 In this case, Ak(Bk) is the formula I(Bk). Since Bk, ¬Bk∉Δn+1, it is clear from the construction that neither Bk nor ¬Bk is compatible with Δk. So Δk, Bk |-D C, ¬C for some formula C and Δk, ¬Bk |-D C′, ¬C′ for some formula C′. By the version of Reductio for |-D, Δk |-D¬D(Bk) and Δk |-D¬D(¬Bk); so by ⁁-Introduction, Δk |-D I(Bk). But Δk⊆Δn+1; and so by Weakening for |-D, Δn+1 |-D I(Bk).

Since Δn+1 is compatible by (1) and Δn+1 |-D Ak(Bk) for k = 0, 1, … , n + 1 by (3), it follows that Δn+1 is compatible with the response A0(B0), A1(B1), … , An+1(Bn+1) to B0, B1, … , Bn+1 and hence so is the subset Δ0 of Δn+1. It remains to show that the response A0(p), A1(p), … , An+1(p) is sharp. Since B0∈Δ1, A0(p) = Dp; and since ¬Bn+1∈Δ1, An+1(p) = D¬p. This establishes the first condition for being a sharp response, viz. that two of the individual responses should not be the same. Now the responses A0(p), A1(p), … , An+1(p) are of one of the following three forms: Dp, D¬p, and Ip. The first two are truth-functionally inconsistent with the third (since Ip is the formula ¬Dp ∧¬D¬p) and the first two formulas are inconsistent with one another (since Dp |- p and D¬p |- ¬p). This establishes the second condition for being a sharp response and we are done.

The collective response yielded by the proof of theorem contains three distinct individual responses – Dp, D¬p, and Ip. But we may readily obtain a collective response that contains only two distinct individual responses by replacing the responses Dp or D¬p, wherever they occur, with ¬Ip. Call a response A0(p), A1(p), … , An+1(p) bipartite if there are exactly two formulas in the set {A0(p), A1(p), … , An+1(p)}. We then have:

Corollary 1 Take any formulas B0, B1, … , Bn+1, n ≥ 0. Then there is no set of formulas Δ0 compatible with the extremal response and yet incompatible with any sharp bipartite response to B1, B2, … , Bn+1.

We may state the result without any appeal to the notion of compatibility by making use of the definition of compatibility in terms of consistency and definiteness. A collective response A1(p), A2(p), … , An(p) is said to be superdefinite if each individual response Ai(p), i = 1, 2, … n, is of the form DB(p); and {DB0, D¬Bn+1} is said to be the superdefinite extremal response to the formulas B0, B1, … , Bn+1, n ≥ 0.

Corollary 2 Take any formulas B0, B1, … , Bn+1, n ≥ 0. Then there is no set of formulas Δ0 which is consistent with the superdefinite extremal response and yet inconsistent with any sharp super-definite response to B0, B1, … , Bn+1.

In defining Δ |-D A, we have allowed ourselves to strengthen the premisses of Δ with arbitrary iterations of D's. But we might only allow ourselves to strengthen the premisses of Δ with a single iteration. Accordingly, let Δ |-1 A hold if DΔ |- DA; and let us say that Δ is weakly compatible if there is no formula B for which Δ |-1 B and Δ |-1¬B.

The following version of the result can now be shown to hold, with weak compatibility in place of compatibility:

Theorem 2 Take any formulas B0, B1, … , Bn+1, n ≥ 0. Then there is no set of formulas Δ0 such that DnΔ0 is weakly compatible with {DnB0, Dn¬Bn+1} and Δ0 is not weakly compatible with any sharp response to B0, B1, … , Bn+1.

Proof We shall find it convenient to adopt a slightly different method of proof. Let ±B be the formula B or its negation ¬B (we take the ‘value’ of ±Bi below to be the same in all contexts). Say that a collective response A1(p), A2(p), … , An(p) is a D≤n-response if, for some m ≤ n, each individual response Ai(p), for i = 1, 2, … , n, is of the form Dmp or Dm¬p or Imp. Where B1, B2, … , Bm+1 and C1, C2, … , Cn are any two sequences of formulas, we show:

(*) Suppose that DnΔ0 is 1-compatible with {DnB0, Dn±B1, Dn±B2, … , Dn±Bm, Dn¬Bm+1}. Then Δ0 is 1-compatible with a sharp D≤n-response to B0, B1, … , Bm+1, C1, C2, … , Cn.

The proof is by induction on n. The result is trivial when n = 0 since, by hypothesis, D0Δ0 is 1-compatible with {D0B0, D0±B1, D0±B2, … , D0±Bm, D0¬Bm+1}={B0, ±B1, ±B2, … , ±Bm, ¬Bm+1}, which is a sharp D≤0-response.

Suppose now that n = k + 1. We distinguish two cases:

  • (a) For some i = 1, 2, … , k + 1, DkΔ0 is 1-compatible with {DkB0, Dk±B1, Dk±B2, … , Dk±Bm, Dk¬Bm+1, DkCi} or with {DkB0, Dk±B1, Dk±B2, … , Dk±Bm, Dk¬Bm+1, Dk¬Ci}. Without loss of generality, suppose the former and that i = 1. Then by IH, Δ0 is 1-compatible with a sharp D≤k-response to B0, B1, B2, … , Bm+1, C1, C2, … , Ck+1; and so Δ0 is 1-compatible with a sharp D≤k+1-response to B0, B1, B2, … , Bm+1, C1, C2, … , Ck+1, as required.
  • (b) For each i = 1, 2, … , k + 1, DkΔ0 is 1-incompatible with {DkB0, Dk±B1, Dk±B2, … , Dk±Bm, Dk¬Bm+1, DkCi} and with {DkB0, Dk±B1, Dk±B2, … , Dk±Bm, Dk¬Bm+1, Dk¬Ci}. So:

DkΔ0, DkB0, Dk±B1, Dk±B2, … , Dk±Bm, Dk¬Bm+1, DkCi |-1 F, ¬F for some formula F, and:

DkΔ0, DkB0, Dk±B1, Dk±B2, … , Dk±Bm, Dk¬Bm+1, Dk¬Ci |-1 G, ¬G for some formula G. Therefore:

Dk+1Δ0, Dk+1B0, Dk+1±B1, Dk+1±B2, … , Dk+1±Bm, Dk+1¬Bm+1, Dk+1Ci |- DF, D¬F |- F, ¬F; and

Dk+1Δ0, Dk+1B0, Dk+1±B1, Dk+1±B2, … , Dk+1±Bm, Dk+1¬Bm+1, Dk+1¬Ci |- DG, D¬G |- G, ¬G.

By reductio,

Dk+1Δ0, Dk+1B0, Dk+1±B1, Dk+1±B2, … , Dk+1±Bm, Dk+1¬Bm+1 |- ¬ Dk+1Ci, ¬Dk+1¬Ci |- ¬Dk+1Ci∧¬Dk+1¬Ci (= Ik+1Ci);
and so by D-Distribution,

  (**) DDk+1Δ0, DDk+1B0, DDk+1±B1, DDk+1±B2, … , DDk+1±Bm, DDk+1¬Bm+1 |- DIk+1Ci for i = 1, 2, … , k +1.

Given that Dk+1Δ0 is 1-compatible with {Dk+1B0, Dk+1±B1, Dk+1±B2, … , Dk+1±Bm, Dk+1¬Bm+1}, the set {DDk+1Δ0, DDk+1B0, DDk+1±B1, DDk+1±B2, … , DDk+1±Bm, DDk+1¬Bm+1} is consistent and hence, by (**), so is the set {DDk+1Δ0, DDk+1B0, DDk+1±B1, DDk+1±B2, … , DDk+1±Bm, DDk+1¬Bm+1, DIk+1C1, DIk+1C2, … , DIk+1Cn}. But then {Δ0, Dk+1B0, Dk+1±B1, Dk+1±B2, … , Dk+1±Bm, Dk+1¬Bm+1,Ik+1C1, Ik+1C2, … , Ik+1Cn} is 1-compatible, where Dk+1B0, Dk+1±B1, Dk+1±B3, … , Dk+1±Bm, Dk+1¬Bm+1, Ik+1C1, Ik+1C2, … , Ik+1Cn constitutes a sharp D≤k+1-response.

The theorem follows from (*) upon letting m = 0.

We see that a slight strengthening of the compatibility condition serves to secure the result for the case of 1-compatibility. Note that the sharp response delivered by the proof will be one in which each individual response is, for some fixed m ≤ n, of the form Dmp or Dm¬p or Imp. And, as before, the result can be stated without appeal to the notion of compatibility. For say that a collective response A1(p), A2(p), … , An(p) is definite if each individual response Ai(p), i = 1, 2, … n, is of the form DB(p). Then:

Corollary 4 Take any formulas B0, B1, … , Bn+1, n ≥ 0. Then there is no set of formulas Δ0 which is such that Dn+1Δ0 is consistent with {Dn+1B0, Dn+1¬Bn+1} and yet DΔ0 is inconsistent with any sharp definite response to B0, B1, … , Bn+1.

Somewhat surprisingly, theorem 2 fails when the compatibility condition is not strengthened.11

Theorem 3 Consider the sentence-letters p0, p1, … , pn+1, n > 0. Then there is a set of formulas Δ0 which is weakly compatible with {p0 and ¬pn+1} and yet not weakly compatible with any sharp response to p1, p2, … , pn.

Proof Sketch We focus on the case n = 1, the proof for the general case being similar. Let |- be the appropriate notion of consequence for the modal logic T. To construct Δ0, we define four types of formula in the sentence-letter p (I have found it easier on the eye to revert to the standard notation for modal logic, with □ in place of D):

Necessary Truth   NT(p) for □p;

Necessary Falsehood NF(p) for □¬p;

Contingent Truth  CT(p) for p ∧¬□p;

Contingent Falsehood CF(p) for ¬p ∧¬□¬p.

We might, in the present context, call either of NT, NF, CT and CF a modality; and we say that the modality ψ is consonant with the modality ϕ (or that ψ(p) is consonant withϕ(p)) if ψ≠ϕ and either ϕ= NT and ψ= CT or ϕ= NF and ψ= CF or ϕ= CT or ϕ= CF. Intuitively, ψ(p) is consonant with ϕ(p), for ψ≠ϕ, if the possibility of ψ(p) is consistent with the truth of ϕ(p).

We now let:

Γ0={□n(NT(p0) ∧ NF(p2) ⊃ CT(p1) ∨ CF(p1)): n = 1, 2, …}∪{□n(ϕ(pk) ⊃◊ψ(pk)): ψ is consonant with ϕ, k = 0, 1, 2, and n = 1, 2, …}; and

Δ0={p0, ¬p2}∪Γ0.

0 has been so constructed that the failure to be □-closed arises solely from the presence of p0 and ¬p2.)

We first show that Δ0 is 1-compatible, i.e. that there is a T-model M and world w of M such that □Δ0 is true at w in M. The model will be a tree-model in which the points are sequences of elements which get extended by one element in moving from a given point to an accessible point. Each element will be a triple of formulas (E0, E1, E2), where Ek, for k = 0, 1, 2 is one of the formulas NT(pk), NF(pk), CT(pk) or CF(pk) and it is not the case that: E0= NT(p0), E2= NF(p2) and either E1= NT(p1) or E1= NF(p1). Intuitively, the triple (E0, E1, E2) is used to indicate that the formulas E0, E1, and E2 are to be true at the given point. The element (F0, F1, F2) is said to be consonant with the element (E0, E1, E2) if, for k = 0, 1, 2, Fk is consonant with Ek. We take a point w to be a sequence e1, e2, … , en of elements, n > 0, for which ei+1 is consonant with ei for i = 1, 2, … , n−1; and we use wL for the last element of the point w. The model M0= (W0, R0, υ0) is then defined by:

W0={w: w is a point};

R0={(w, v): w, v ∈ W and v = w or v result from appending the element e to w};

υ0={(w, p): w ∈ W, p is pk for some k = 0, 1, 2, and, for wL= (E0, E1, E2), Ek is either NT(p) or CT(p)}.

Let w0= (NT(p0), CT(p1), NF(p2)). It is then readily verified that:

  • 1□Δ0 is true at w0 in M0.

We now deal with the incompatibility condition. Given any model M = (W, R, υ) and world w of M, say that the type of p at w in M – Type(p, w, M) – is the modality ϕ for which ϕ(p) is true at w in M. Clearly, each world has exactly one type in a given sentence-letter.

(2) Suppose that M and N are generated models with respective base points w0 and v0, that Γ0 is true at w0 in M and at v0 in N, that type(pk, w, M) = type (pl, v, N), for 0 ≤ k, l ≤ 2, and that A(p) is a formula whose sole sentence letter is p. Then A(pk) is true at w in M iff A(pl) is true at v in N.

Proof By a straightforward formula induction.

Finally, suppose for reductio that A0(p0), A1(p1), A2(p2) is a sharp response in p0, p1, p2 and that it is 1-compatible with Δ0. □Δ0∪{□A0(p0), □A1(p1), □A2(p2)} is then true at some point w in a model M = (W, R, υ). We may show that □A1(p0) is true at w. For take any v for which wRv. Since NT(p0) and NF(p2) are true at w, it follows that CT(p1) or CF(p1) is true at w. Without loss of generality, suppose that CT(p1) is true at w. It should then be clear that for some u, wRu and type(p1, u, M) = type(p0, v, M). Since □A1(p1) is true at w, A1(p1) is true at u and so, by (2) above, A1(p0) is true at v. But v was arbitrary, and so □A1(p0) is true at w. But then A0(p), and A1(p) are not inconsistent (indeed, not even incompatible) and so A0(p) = A1(p). Similarly, A2(p) = A1(p) and the response is not sharp after all.12

Notes

  • 1

    The argument given against such reductions towards the end of Graff-Fara [2003] is not relevant to the present case since she assumes, somewhat strangely, that the supplementary content will always be the same regardless of which propositions P1, P2, … are in question. Indeed, her semantic characterization of the consequence relation for the supervaluationist directly supports the kind of reduction I have in mind.

  • 2

    This is because the truth-tables for the connectives are monotonic and so, given that the indeterminacy claim is true under some assignment of truth-values, it will remain true under a classical extension of that assignment. I conjecture that the ‘neighborhood’ semantics of Field [2008], 264–6) is also unable to evade the result. Cases such as these suggest that there should be a more general formulation of the result, though it is not clear to me how it should go.

  • 3

    Though I should mention that Field ([2008], 270–71) is one of the few authors to reject the modal logic T for the definitely operator.

  • 4

    I say ‘in effect’ since Permutation and Contraction are not stated as such but are built into the assumption that consequence is a relation that holds between a set of premisses and a given conclusion.

  • 5

    Williamson [1994] might be taken to endorse a view of this sort.

  • 6

    David Barnett has suggested such a position to me.

  • 7

    See Wright ([1976], p. 226) and Sainsbury ([1991], 179) for related considerations. It is conceivable that one might think that this more robust form of indeterminacy only has application when the series of propositions is sufficiently long.

  • 8

    The technical reason is this. Write the hyper-definitely operator of order α as D<α> and, where I0 is the indeterminacy claim, let α0 be an upper bound for all the α for which D<α> occurs in I0. The proof produces a sharp response R (also bound by α0) which is compatible with I0 under arbitrary iterations of D<α> for α≤α0. But if R were to be incompatible with I0 under arbitrary iterations of the D<α>, for anyα, then it should remain incompatible under the ‘identification’ of D<α> with D<α0> for all α > α0.

  • 9

    This argument is essentially just a reworking of the proof of theorem 1 for the special case in which DA is taken to coincide with A.

  • 10

    Some related issues concerning the epistemic interpretation are discussed in Gomez-Torrente [2002], Graff [2002] and Williamson [2002].

  • 11

    A related construction is given by Williamson ([2002], 147–8), though he is not there interested in formulating an indeterminacy claim.

  • 12

    I presented earlier versions of the paper at an Arche seminar, at a graduate discussion group at NYU, and at talks to the philosophy departments at the University of Nebraska and Stanford University; and I should like to thank the participants at these various events for their many helpful comments. I am extremely grateful to David Barnett, who carefully read through an earlier draft of the paper and made many valuable points and criticisms; and I owe a special debt to my colleagues Hartry Field, Stephen Schiffer and Crispin Wright, who over the years have helped awaken me from my supervaluational slumbers.

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