INTUITIONISM AND THE MODAL LOGIC OF VAGUENESS

Rumfitt Abstract An intuitionistic solution to the Sorites Paradox is defended by relating it to the sentential modal logic of Wright Some philosophers have proposed intuitionistic logic as the one best suited to provide the foundation for a theory of vagueness.\ 3 / As we remind readers in §1, that logic provides an elegant solution to the Sorites Paradox which avoids the implausible sharp cut-offs in classically based epistemicist theories. Its acceptance has been hampered by two factors. First, the lack of an accepted semantics for languages containing vague terms has led even philosophers sympathetic to intuitionism to complain that no explanation has been given of why intuitionistic logic is the correct logic for such languages (see §2 below). Second, switching from classical to intuitionistic logic, while it may help with the Sorites , does not appear to offer any advantages in dealing with the so-called paradoxes of higher-order vagueness. In this paper we offer a proposal that makes strides on both issues:


Wright's argument for intuitionism as the logic of vague statements
In some recent papers, Crispin Wright has drawn attention to a version of the Sorites Paradox that is especially stark and difficult to solve (see [38], [39], [41] and especially [40]). Let a1,…, a100 be a sequence of a hundred transparent tubes of paint with the following properties: tube a1 is clearly red; tube a100 is clearly orange and hence clearly not red; but for each n, tube an+1 is only marginally more orange (and hence only marginally less red) than its predecessor an. Indeed, let us suppose that an observer with good eyesight, when viewing any pair of adjacent tubes an and an+1 together in white light but without comparing them with other tubes, is unable to perceive any difference in colour between them. That is, we suppose that any two adjacent members of the sequence are indiscriminable in colour.
Let L be an interpreted first-order language which contains: the connectives '', '', and ''; the quantifier 'n' ranging over the natural numbers from 1 to 100; numerals standing for each of those numbers; a one-place functional expression ' + 1' with the expected sense; a one-place predicate 'R' with the sense of ' is red'; and complex singular terms in the form 'an' with the sense of 'the n th tube of paint in the sequence'. In the situation described, the tube a1 is clearly red, so we may assert in L (1) Ra1.
The tube a100, by contrast, is clearly not red, so we may also assert as a premiss (2) Ra100.
Let us also consider-not as a premiss, but merely as an assumption, to see where it leads-the formula (3) n(Ran  Ran+1).
Let us now consider, also as an assumption, (4) Ra99.
(8) is the contradictory of (1) and, since assumption (3) has led to a contradiction, we may now discharge it, and infer (9) n(Ran  Ran+1), with (1) and (2) as the only premisses. In classical logic (9) yields (10) n(Ran  Ran+1) paradox commits him, then, to the thesis that the intuitionistic sentential calculus, IPC, is the correct sentential logic to use when reasoning with vague statements. A logic for vagueness must eventually cover the quantifiers too, and below we shall say a little about how our analysis could be extended to them. Our focus, however, will be on the thesis that IPC is the correct sentential logic of vagueness.

Dummett's challenge
What should we make of this argument for intuitionism? In his reply to [40], Michael Dummett put on record his 'admiration for the beautiful solution of the Sorites advocated by Crispin Wright, clouded by a persistent doubt whether it is correct' ( [12] p.453). Let us call a statement vague if a vague expression is used in it. Then what unsettles Dummett is that Wright's suggestion that the logic of vague statements is intuitionistic is not underpinned by any detailed semantic analysis. Wright points to broad analogies between the intuitionists' attitude to mathematical statements and the attitude that he recommends taking towards vague statements: in each case, he says, statements' 'truth and falsity have to be thought of as determined by our very practice, rather than by principles which notionally underlie' that practice ([40] p.441). However, those broad analogies fall well short of a developed semantic theory against which the soundness of a given logic could be assessed. 'It is not enough to show that the Sorites paradox can be evaded by the use of intuitionistic logic; what is needed is a theory of meaning, or at least a semantics, for sentences containing vague expressions that shows why intuitionistic logic is appropriate for them rather than any other logic' ( [12] p.453).
We agree with Dummett that it would be nice to have a semantic theory for languages which contain vague expressions. At this point in the philosophical debate about vagueness, though, there is no semantic theory that we can appeal to either as empirically well-confirmed or as widely accepted by contributors to the debate. To the contrary, even the outlines of such a theory are a hotly contested matter. There is, then, some interest in seeking another sort of argument for the appropriateness of intuitionistic logic in the present context, one which does not take as a premiss the correctness of any semantic theory for vague expressions.  39).\ 8 / That is to say, one can interpret the intuitionistic sentential calculus IPC in a system of classical logic enriched with an additional unary operator 'ℬ' if 'ℬ' is understood to mean 'it is provable that' and is assumed to conform to the axioms and rules of the modal propositional logic S4.
Here, 'provable' means 'provable by some correct method', not 'provable in a given formal system S'.
However, if ℬ means 'is provable in S', then by Gödel's own Second Incompleteness Theorem Gödel's paper is rather telegraphic; what it shows may be spelled out as follows. Let t be a mapping from the well-formed formulae of IPC to well-formed formulae of the language of  which meets the following constraints: For any sentence letter p, t(p) = ℬp Gödel demonstrated that whenever IPC├ A, ├ t(A). In modern terms, he showed that if IPC├ A, where S4 is the modal system with 'ℬ' serving as the necessity operator. He also conjectured the converse, which was eventually proved by McKinsey and Tarski [19]: Rasiowa and Sikorski [25] extended these results to intuitionistic predicate logic, IQC. They first extended Gödel's mapping t to quantified formulae as follows: 8 Gödel's use of different symbols for classical and intuitionistic connectives suggests that he held that the negation sign (e.g.) used by a classical logician eo ipso has a different sense from that used by an intuitionist. See [28] §7.5 for an argument against this thesis. If this sort of argument is to help explain deviations from classical logic, then the translation t must provide a plausible account of the extra-mental conditions for asserting the relevant sentences or formulae, as they are used by the relevant speakers. We do not ourselves believe that translation t is a particularly faithful rendering of these conditions: for one thing, it omits the important point that an intuitionist is prepared to assert an existentially quantified formula only when she has a method for constructing a verifying instance. That is one reason why Gödel's interpretation of the language of intuitionistic mathematics has been superseded by semantic theories which capture the intended meanings of the connectives and quantifiers more accurately. All the same, the failure of one application of this method of interpreting a language modally does not mean that others might not succeed. One goal of this paper is to propose a non-mathematical application of the method.
In considering whether a Gödel-type modal translation might explain why intuitionistic logic is the right logic to use when reasoning with vague predicates, it helps to bear two formal facts in mind. First, Gödel's own scheme of translation is not the only scheme that yields comparable results.
In fact, McKinsey and Tarski used an alternative mapping,  below, from the language of IPC into the language of S4 (whose necessity operator we will henceforth write as ''): (i) For any sentence letter p, (p) =  p It is easy to show that S4├ (A)  t(A), so we also have that IPC├ A if and only if S4├ (A).
Because it is simpler, the McKinsey-Tarski mapping has largely superseded Gödel's original, and  will prove to be a more convenient translation for our purposes than t. Note that A1 Then any modal logic in the closed interval whose weakest member is S4 and whose strongest member is S4 + Grz is a modal companion of IPC.\ 9 / There are similar results for IQC and indeed for the entire class of 'superintuitionistic' logics, i.e. those which are intermediate in strength between the intuitionistic and classical systems. In applying Gödel's method to try to make sense of why intuitionistic logic might be the right one to use in a given context, one is not confined to S4. Any modal logic between S4 and S4 + Grz will serve. Since any such logic includes the rule of necessitation, we shall again have, for any modal companion ML of IPC, the result that In the results just cited, the underlying logic of the various modal companions is taken to be classical. In pursuing our subsidiary project of explaining to a classical logician why Wright's philosophical assumtions lead precisely to IPC as the appropriate non-modal sentential logic for a language with vague terms, we shall adopt this assumption until the end of §4. In §5, however, we shall ask what might be the appropriate companion extension of IPC, in which the intuitionistic sentential logic is taken to underlie the modal system.

The modal logic of vagueness
How might  vindicate the claim that IPC is the correct logic to use when reasoning with vague terms? In outline, our argument will run as follows. Let us interpret the symbol '' of the modal language as meaning 'it is clear that' or 'it is definite that' (terms which we use interchangeably). We first contend that, on Wright's philosophical assumptions, and where A is a sentence or formula that contains a vague term, (A) gives the extra-mental condition for A to be assertible. Thus, applying this principle to the language L of §1, the extra-mental condition for 'Ra50' to be assertible (as a formula of L) will be that it is clear that tube a50 is red; the corresponding condition for 'Ra50' is that it is clear that it is not clear that tube a50 is red; and so forth. As in the mathematical case, whether a particular language user is entitled to assert such a statement will depend on her epistemic situation: a blind speaker may not be entitled to assert 'Ra1' even though tube a1 is clearly red. All the same, we hypothesize, (A) spells out the factor to which speakers must stand in the appropriate epistemic relationship if they are to be warranted in asserting A. Support for this hypothesis, which we will lay out in §4, comes from the sense it makes of the assertive behaviour of competent users of L. Second, we argue that the correct modal logic of the operator 'it is clear that' is one of the modal companions of IPC. Putting these things together, we conclude that IPC is the correct logic for a language such as L. In this way, we answer Dummett's challenge to 'show why intuitionistic logic is appropriate for [vague terms] rather than any other logic'. The argument rests on philosophical premisses which adherents of classical logic will resist. Given those premisses, though, it vindicates the precise choice of IPC as the appropriate logic to use when reasoning with vague predicates.
It is natural to use the notion of clarity in spelling out the extra-mental conditions for the assertibility of vague statements. However, a distinctive feature of our approach is that clarity-and definiteness, and determinacy-are to be explained in terms of being a borderline case of a property.
We take this last notion to be philosophically basic. There are distinctive marks of borderlineness: competent speakers hedge, or otherwise manifest uncertainty, when called upon to apply a predicate that signifies a given property to one of its borderline cases. They might allow that different speakers may reasonably have different views about whether the predicate is correctly applicable to such a case. We express no view on the reasons for this hedging behaviour. Rather, we take the hedging per se to be constitutive of something's being borderline.
In constructing the modal language in which we spell out the extra-mental conditions for assertibility of vague sentences, we take as primitive a sentence-forming operator '' with the sense of 'it is borderline whether'. (We choose the symbol '', pronounced 'nabla', because 'it is borderline whether' stands to 'it is clearly the case that' as 'it is contingent whether' stands to 'it is necessarily the case that'.\ 10 /) What axioms and rules govern ? First, it is highly plausible to maintain that, where A is a theorem of a satisfactory logic of borderlineness, the truth of A is not a borderline matter; we expect truths of logic to be determinately true. We thus have the following rule of proof: Second, we understand the notion of its being borderline whether such-and-such is the case so as to be neutral between affirmative and negative verdicts. Thus we have the schema 1 A  A.
Third, we postulate The justification for schema 2 is as follows. An immediate consequence of these definitions is that A  A is equivalent to A. To establish the equivalence from right to left, suppose A. Since A is defined as A  A, A follows by -introduction. Since  is defined as A  A,  implies A by -elimination, whence Should we also assume that the notion of clarity conforms to 4? Otherwise put, if it is clearly the case that P, is it clearly the case that it is clearly the case that P?\ 12 / Given our definition of '', the question reduces to the correctnesss of the following schema: Otherwise put, if it is borderline whether it is borderline whether P, is it borderline whether P?
The matter is delicate but, since one of us has argued at length for an affirmative answer (see [1] ), we here give only a rough sketch of an argument for 4 as it applies to sentences of the most pertinent form, Fa. Recall that we interpret Fa as 'it is borderline whether Fa' and take 'borderline whether Fa' to be constituted by the range of hedging behaviour of speakers cognitively and perceptually fully competent with regard to F and a (cf §3). Assuming compositionality of , then, Fx can be paraphrased as 'x is such that speakers competent with F hedge over applying F to x' 12 Wright is committed to accepting the 4 principle for 'it is clear that'. In his analysis of the socalled paradoxes of higher-order vagueness [37], he postulates a rule of proof, DEF, which in our terms may be formulated as follows: DEF Given a derivation of B from A1, … , An, construct a derivation of B from A1, … , An.
Since there is a derivation of A from A, DEF licenses the construction of a derivation of A from A, whence 4 follows by →-introduction.
DeVidi [8], on the basis of a sketch of a semantic theory, motivates and defends Wright's solution to the Paradox of Sharp Boundaries by relating it to an intuitionistic modal logic. Unlike Wright, however, DeVidi rejects the 4 principle for 'it is clear that', so his logic is an intuitionistic version of KT. The resulting theory is consequently very different from that to be developed here. 13 Sketch of proof. To show that 4 implies 4, assume 4, and suppose A, i.e. A  A. We need to deduce A, i.e. A  A. The first conjunct is trivial. For the second, suppose (for a contradiction) that A, i.e. (A  A). By 1 this yields (A  A) and hence A  A by 3. By 1, the first disjunct implies A, as does the second disjunct, by 4. Given A, however, we have A. The resulting contradiction establishes A, thereby completing the deduction of A.
To show conversely that 4 implies 4, assume 4 and suppose that A. Then, by the equivalence between A and A  A, we have (A  A)  (A  A). By -elimination, this yields (A  A). Since (B  C) implies B, we may infer A, which combines with 4 to give A. Since (B  C) implies C, we may also infer A, i.e. A which, given 4, implies A. By introduction, then, we reach A  A, which is A, as required. and Fx can be paraphrased as 'x is such that speakers competent with F hedge over applying F to x'.
We maintain that if relevantly competent speakers hedge over whether to hedge when called upon to apply F to a, then one expects relevantly competent speakers also to hedge when called upon to apply F to a. For, since any relevant cognitive or perceptual incompetence on the side of the speakers has been ruled out, there remain no reasons why, if relevant speakers do not hedge over Fa, relevant speakers would hedge over whether they hedge over Fa.
The modal logic of clarity is, then, at least as strong as S4.\ 14 / Is it stronger, though? We shall soon contend that it is, but we first note that any normal modal system which contains every instance of the modal principle 5,

5
A → A, whether as an axiom or as a theorem, is too strong to serve as an adequate logic of clarity. (In particular, then, that logic cannot be the system S5 itself.) This is because, given the definitions of '' and '', and assuming our postulates for , 5 implies (in fact, it is equivalent to) A. (For a proof of this equivalence, see [20] Theorem 13.) That is, 5 implies that that it is never borderline whether such-and-such is borderline. We do not think that this thesis is tenable. Wright's example of the sequence of tubes of paint shows why. Given the thesis, and assuming that if tube an is clearly red then am is also clearly red for any m  n, and if tube an is clearly not red then am is also clearly not red for any m  n,\ 15 / the sequence will contain precisely two sharp boundaries, one between the clear cases of red and the clear borderline cases, and the other between the clear borderline cases and the 14 A referee asks for reassurance that adopting 4 as part of the logic of clarity does not collide with well-known arguments against 'luminosity'. The relationship between borderlineness, luminosity, and 'margin for error' principles is one we have broached elsewhere (in [1] §8). For now, it suffices to remark that we are treating its being clearly the case that a is red simply as the extra-mental condition for asserting 'a is red'. Adopting 4, then, in no way implies that if it is assertible that a is red then it is assertible that it is assertible that a is red, let alone any version of the KK principle. 15 This assumption-dubbed 'monotonicity of clarity' in [2]-is weaker than Wright's assumption (cf. n.6 above) of monotonicity in respect of redness.
cases which are clearly not red. Since 5 precludes us from introducing borderline borderline cases, the sharpness of the transition from clear cases to borderline cases cannot be dulled. Having two such sharp boundaries is no more acceptable than the original single sharp cut-off point between the red and the non-red cases. Therefore any logic with 5 as a theorem is too strong to serve as an adequate modal logic of clarity.
Our proposed additional principle says that it is never clearly the case that it is borderline whether P. In symbols, A. We showed earlier that A is equivalent to A  A, so we may reformulate this principle as the schema The reader may be surprised at our labelling (A  A) as 'M'. In modal logic, that letter is usually reserved for the axiom A → A. Quite so, but (A  A) and A → A are classically equivalent in all normal modal logics. As is well known, (A  B) is equivalent in K to A  B (for a proof see e.g. [18] p.31). Thus (A  A) is equivalent in K to A  A and hence (A  A) is equivalent in K to (A  A), which is equivalent in the classical sentential calculus to A → A, i.e. A → A. It will be convenient to take (A  A) to be definitive of M. The claim that it is never clearly the case that it is borderline whether P has more obvious implications for the logic of vagueness than the more traditional formulation of M.
Whatever the fate of M on other assumptions, there is a compelling reason why anyone who adopts an intuitionistic treatment of vagueness should accept it. The intuitionist has just as little time as the classical logician for any alethic value other than true and false. Although she refrains from asserting Bivalence Every statement is either true or false, she asserts the principle that Dummett labelled As the flipside of its rendering borderlines elusive in this sense, M ensures that we do not have to worry about the perplexities that come with substantial hierarchies of higher-order vagueness, i.e. orders of vagueness that are not logically guaranteed to be co-extensive (see e.g. [15], [29], [32], [33] and [37]

Consequence for vague statements
With this in place, we return to Wright's thesis that IPC is the correct sentential logic to use when reasoning in a language which contains vague terms.\ 18 / A logic provides conditions for when a conclusion follows from some premisses. Where L is a sentential language which contains at least one vague expression, and where A1, … , An, B are formulae of L, let us write A1, … , An╞V B to mean that conclusion B follows from premisses A1, … , An. A logic of vagueness ought to provide conditions for╞V. (For present purposes, we need concern ourselves only with the case where a conclusion follows from a finite number of premisses.) Now if there were an agreed semantic theory for languages containing vague terms, it would be a routine matter to parlay that theory into a necessary and sufficient condition for╞V. As remarked in §2, however, there is no such theory, so we have to proceed less directly.
To see how to proceed, it helps to revert to the explanation touted in §2 of why Brouwer and others employed intuitionistic logic when reasoning about mathematics. This rested on two postulates: first, that the Gödel translation spells out the extra-mental conditions in which a mathematical sentence or formula A is assertible; second, that a formula A is logically true when, and only when, there is a logical guarantee that the extra-mental condition for its assertibility (the existence of a proof of A) obtains. This provided the needed explanation, for, when considering the translation, a classical logician will agree that there is no logical guarantee that the extra-mental condition for the assertibility of every instance of Excluded Middle is met. Rather, the logical guarantee extends only to cover the validities of the intuitionistic calculus.
Similarly, we propose, classical logicians can understand why Wright accepts only the rules of intuitionistic logic when reasoning with vague statements, if they adopt the following two postulates.
The first connects the translation with the extra-mental condition for the assertibility of vague 18 See again the end of §1 for why the crucial thesis concerns sentential logic. sentences or formulae: (A) specifies this condition for a vague sentence or formula, A, so long as '' in (A) is read as meaning 'it is clearly the case that'.
Since some of the relevant claims (such as Bivalence and Tertium Non Datur) involve quantifiers, we need to extend  so that it covers these operators. We duly propose The second postulate says that a sentence or formula in a vague language is logically true when, and only when, there is a logical guarantee that the extra-mental condition for its assertibility obtains.
More generally, we postulate that an argument in a vague language is logically valid when, and only when, there is a logical guarantee that the extra-mental condition for the assertibility of its conclusion obtains whenever the corresponding conditions for the assertibility of all its premisses obtain.\ 19 / Why are these two postulates a good way of interpreting Wright? A touchstone for interpreting any intuitionist is to make sense of the fact that she asserts Tertium Non Datur while refraining from asserting Bivalence. When read straight, a classical logician can make no sense of this, because the two statements are classically equivalent, and obviously so. Now, the interpretation we recommend makes this central aspect of the intuitionist's position intelligible, indeed, sensible. 19 If one further assumes that the property which is logically guaranteed to be preserved in a valid argument is truth, then the second postulate implies that a vague statement is true just in case what it says is clearly the case. Not everyone will accept this implication; indeed, rejecting it is one natural way to resist the conclusion that IPC is the correct sentential logic for vague languages. The implication, however, is not so outré as to mar our explanation of why certain philosophical assumptions lead to that conclusion. Dummett [9,256] accepts the implication.
When formalized using the obvious notation, and with the domain of quantification restricted to statements, Bivalence is which maps under  to x(Tx  Fx).
The classical logician will agree that the condition expressed by the latter formula is not met. The condition in question is that every statement is either clearly true or clearly false. This condition fails if the relevant language contains any vague statements.

Tertium Non Datur, by contrast, is
which maps under  to x(Tx  Fx), which is classically equivalent to x(Tx  Fx).
The condition expressed by this last formula is that it is clearly the case that every statement is not clearly neither clearly true nor clearly false. The condition, in other words, is that it is clearly the case that every statement is not clearly borderline. As explained in §3, philosophers like Wright, who reject the idea that vague statements possess any third alethic value, hold that this condition is met.
Moreover, its being met does not entail that every statement is either clearly true or clearly false, even in classical logic. So the proposed scheme of interpretation provides us with our corollary: it enables the classical logician to understand how Wright can consistently-indeed reasonably-assert Tertium Non Datur, while refraining from asserting Bivalence.
It is vital to this explanation that the scheme of interpretation should map an atomic formula such as 'Tx' to 'Tx'. So the case provides some confirmation that clause (i) in the specification of  with a50 supposed to be a borderline case of R. If the scheme of interpretation left the sign for negation unchanged, there would be no explanation of this unwillingness, for the condition for asserting 'Ra50  Ra50' would be: This formula is another instance of Excluded Middle, so an interpretation that leaves negations unchanged would not help the classicist to understand why intuitionists like Wright are unwilling to assert 'Ra50  Ra50'. Clause (v) of the McKinsey-Tarski translation, however, maps  A  to  A  , so the assertibility condition for 'Ra50  Ra50' will be: That is, 'Ra50  Ra50' will be assertible if and only if either it is clearly the case that tube a50 is red or it is clearly the case that it is not clearly red. Since a philosopher who accepts M will refuse to accept that a borderline case is ever clear, the classical logician can understand why such a philosopher is not prepared to assert this instance of Excluded Middle.\ 20 / The proposed interpretation of negation also illuminates the way negation interacts with the quantifiers. When the quantification is over a domain of transparent paint-filled tubes, the classicist hears an intuitionist who utters 'xRx' as saying that every tube is clearly red. When the intuitionist says 'xRx', the classicist hears her as saying that there is a clearly red tube. Now 'There is a clearly red tube' is clearly incompatible with 'Every tube is clearly not red', which explains why the intuitionist accepts the inference from 'xRx' to 'xRx'. However, 'There is a clearly red tube' does not follow classically from 'Clearly not every tube is clearly not red', which explains why the intuitionist does not accept the converse inference from 'xRx' to 'xRx'.
What, finally, of clause (iv), which deals with the conditional? This raises issues of a different sort, for few people would maintain that the conditional operator of either classical or intuitionistic logic corresponds at all closely to the vernacular 'if...then'. Both systems validate 20 In S4M, A is consistent with A. Given our scheme of interpretation, then, 'Ra50' may be asserted in a circumstance where it is borderline whether tube a50 is red, whereas 'Ra50' may not be. Such an asymmetry may disturb classical logicians, but it is built into any scheme for interpreting intuitionists: the conditions for asserting A cannot simply mirror those for asserting A, on pain of there being no explanation of why A is assertible in circumstances where A is not. (We are here indebted to comments from a referee.) In the previous section, we argued that the appropriate modal logic ML is S4M. Hence we reach:  conditions of the formulae of L. By reference to that account, we were then able to show precisely why sentential 'intuitionistic logic…rather than any other <sentential> logic' is appropriate for L. The proffered explanation itself does not presume any antecedent commitment to intuitionistic logic. To the contrary, the explanation is one that a classical modal logician could understand and find explanatory. For, throughout the explanation, the sentential logic that underpins S4M is assumed to be the classical propositional calculus, CPC.\ 21 / 21 A referee asks whether the theory we put forward is to be understood as normative or descriptive.
We are proposing neither that English speakers should nor that most of them do use intuitionistic logic, nor that they should or do use S4M, classical or intuitionistic, when reasoning with vague expressions. Rather, given that certain people use intuitionistic logic in such contexts (including rehearsals of Sorites arguments), we advance an explanation of why they do so. It is true that, on our view, the assertibility of borderline cases cannot be established, but neither can their non-assertibility, since it cannot be established whether they are borderline or non-borderline cases. Speakers with a proclivity to assert borderline cases can keep their discourse consistent by adopting a Stalnakerian conversational score policy along the lines suggested by Shapiro in [33].

An intuitionistic modal logic of clarity
This last point raises a question about the coherence of Wright's position. The analysis of §3 proceeded on the assumption that the sentential logic underpinning the modal logic of clarity would be CPC. Suppose, though, that one takes the plunge and joins Wright in restricting oneself to the weaker system of intuitionistic logic when reasoning with vague statements. Can one then persist in taking the sentential logic underpinning the modal logic to be classical, or must it, too, be weakened to IPC?
There is no formal inconsistency in holding that matters concerning clarity are subject to classical logic, with intuitionistic logic making its appearance only via the McKinsey-Tarski translation into this classical modal language. Indeed, on some philosophical accounts of the nature of clarity, this will be the correct way to go. It seems to us, though, that such a view would not cohere with the philosophical motivations of Wright's position. Remark first that the intuitionist has good reason to introduce notions of clarity, or being borderline, into her language (if they are not already present). As Wright observes, the key to the intuitionistic solution to the Paradox of Sharp Boundaries is that, if she is asked flat out 'Does the sequence contain a tube which is red but whose successor is not red?', the intuitionist can answer neither 'Yes' nor 'No'. Indeed, if she is confined to the simple interpreted language L, she is fated to remain silent if asked whether she accepts or rejects formula (10) of §1. This is, in fact, a pervasive feature of the intuitionistic position. Consider a forced march Sorites-the 'LEM Sorites'-in which an intuitionist is successively asked whether a1 is either red or not, whether a 2 is either red or not, …., and finally whether a 100 is either red or not. The intuitionist will begin by answering 'Yes' (because a 1 is red) and end by also answering 'Yes' (because a 100 is not red), but there is a range of intermediate cases in which she has to remain silent. When asked (as it might be) whether a50 is either red or not, she cannot answer 'Yes', for an intuitionist may affirm a disjunction only when she is in a position to affirm one or other disjunct. Equally, though, she cannot answer 'No'. Answering 'No' to 'Is it the case that P?' is tantamount to affirming 'Not P' but, for the intuitionist as much as for a classicist, the negation of any instance of Excluded Middle yields a contradiction.
There are, then, questions which may be posed in L which an intuitionist cannot answer in L.
But it is not as though she has nothing to say in response to these questions. She could offer that in such cases she does not assert 'Ra50  Ra50' because she may be unable to assert either disjunct. She could explain that she may be unable to assert 'Ra50' because it may not be clear that a50 is red; and that she may be unable to assert 'Ra50' because it may not be clear that it is not clear that a50 is red.
Or she may invoke borderlineness for these cases. Such explanations require a language in which the resources of L are reinforced with sentence-forming operators meaning 'it is clear that' and 'it is borderline whether'.
The question then arises what the underlying sentential logic of this richer language is, and it seems to us that on Wright's premisses it ought to be intuitionistic rather than classical. Consider the sentence 'It is clear that tube a30 is red', or the corresponding formula 'Ra30'. Tube a30, we may suppose, is a borderline case of clear redness just as tube a50 is a borderline case of redness. On the view under consideration, however, we shall be entitled to assert 'Ra30  Ra30' and 'Ra30 → Ra30', even though we may not assert 'Ra50  Ra50' or 'Ra50 → Ra50'. Since, from Wright's perspective, there is no good argument for treating these two kinds of case so differently, we shall explore the hypothesis that the sentential logic underpinning the modal logic of vagueness is IPC, not CPC. We shall then need to check that the key features of Wright's approach to vagueness are retained when the logic underpinning the account of it is weakened in this way.
They are retained. The key feature of the account proposed in §3 was the equivalence between axiom M and the  Principle viz. A  A. We now introduce a plausible intuitionistic modal logic of vagueness and supply derivations that show that this equivalence is retained in it.
In classical modal logic, it is common to select one of the basic modal notions  and  as primitive and to define the others in its terms. What underpins this are two familiar principles of duality: () A  A and () A  A.
The first of these principles, however, undercuts the view we are trying to develop. An intuitionistic consequence of () is A → A. In intuitionistic logic, however, we have B  B for any formula B, so A → A. Thus A → A which yields A → A by () again (see [3] p.3). Even in the context of intuitionistic logic, then, () ensures that Double Negation Elimination may licitly be applied to any formula in the form 'A', whereas we want to ensure that 'Ra30 → Ra30' is not a logical theorem.\ 22 / In constructing an intuitionistic modal logic, the most common way of getting round this problem is to follow Gisèle Fischer Servi [16] and take both '' and '' to be primitive operators. In the system we shall present, '' and '' are not duals. However, neither is primitive. As in the classical case, '' is understood to mean A  A and 'A' to mean A  A. We now assume, however, that the logic of , , and  is intuitionistic, not classical. As we shall see, this enables us to justify a number of the principles which Fischer Servi takes to be axiomatic.
The first point to note is that these definitions ensure that A remains equivalent to A   even when the background sentential logic is intuitionistic. An argument given in §3 (see the text at n.11) already shows that A intuitionistically entails A  . As for the converse, we also showed there that A   intuitionistically entails A. What is more, any formula A is stable, i.e. we always have A → A. This follows directly from the meaning of '': if it cannot be ruled out that something is not borderline, then it is borderline. Given stability, then, we may complete the argument to establish that A   intuitionistically entails A. Even in the context of an intuitionistic sentential logic the equivalence between A and A   stands, and M continues to 'say' that there are no clear borderline cases. 22 Note that our definitions of  and  in terms of  do not have this problematical consequence when the background sentential logic is intuitionistic. While A  A (i.e. ) intuitionistically implies (A  A) (i.e. A), the converse implication does not hold intuitionistically. That converse, however, is needed to complete the proof that A yields A.
Fischer Servi includes the rule of necessitation, N, in her intuitionistic modal system. Given the definition of '', this is a derived rule of our system: the derivation of N from R is intuitionistically acceptable. She lays down four further principles which, she claims, are needed for a normal intuitionistic modal logic. The first two concern ' ': and the latter two concern '':

5 ⊥
This axiom is compelling: since ⊥ is stipulated to be a falsehood, it is certainly not a borderline matter whether it is true. The left-to-right half of K(a) follows directly from 3 and the right-to-left half also follows immediately from the following

K-rule
Given A and a derivation of B from A, infer B.
This rule of proof is also compelling. Given our definition, 'A' may be read as saying 'A cannot be ruled out'. If A cannot be ruled out, and B follows from A, then B cannot be ruled out either.
Fischer Servi similarly postulates two principles corresponding to the classical T axiom: These follow immediately from the definitions of '' and '' in intuitionistically acceptable ways. In the spirit of this paper, we also accept as working hypotheses Fischer Servi's two components of the classical 4 principle: Fischer Servi adds two 'mixing principles' which constrain the interaction of  and : The best justification we know of Mix2 is due to Simpson [34]. He formalizes intuitionistic modal logic as a natural deduction system in which the introduction and elimination rules for every connective are 'harmonious' and 'stable' in Dummett's sense. Both Mix1 and Mix2, along with Fischer Servi's other axioms, are theorems of his system. (Mix 1 may be derived using his rules of elimination, -elimination, and -introduction; Simpson ([34] p.71) already presents derivations of K(a), K(b), and Mix2 as formulae 4, 3, and 5 respectively.) In [11] ch.13, Dummett had justified IPC precisely on the basis that its natural deduction rules have these properties while those of CPC do not. Simpson's formalization, then, is supported by one influential argument for intuitionistic logic. The Dummettian justification of IPC in terms of harmony, however, is remote from the considerations that have inspired Wright and others to adopt it as the logic of vague statements. Mix1 and Mix2 are also theorems of Ewald's system of intuitionistic modal logic [13]: apply his definitions of  and  in terms of tense operators to his tense-logical axioms (G11b) and (G11a) respectively. Ewald, however, advances no philosophical justification for his axioms.
We are unsure if Mix2 may be justified given our definitions of  and  but, since we shall not need it, we leave the matter unresolved. Mix1, however, can be given a justification. It follows intuitionistically from the following principle: In order to justify 6, suppose it is borderline whether This lemma is important, for it guarantees that the first conjunct of A  A excludes the possibility that A is clearly false.
A key result of §3 was the equivalence between M and the thesis A → A, which says that if something is borderline it is borderline whether it is borderline. In fact, this equivalence is retained even in a weak system of intuitionistic modal logic. Let the system IT be that sentential modal logic whose non-modal base is IPC and whose modal axioms are T and T Then the following are correct derivations in IT: In the context of that logic, then, the M principle still suffices to ensure that 'there is no real hierarchy' of higher-order vagueness (to quote [29] once more). That is, this distinctive implication of M is retained even as the sentential calculus that underpins the modal logic is weakened from CPC to IPC.\ 24 / M also continues to ensure that the borderline cases are elusive in the sense that any attempt to give an example of one will produce, at best, a borderline case of an example.  24 What first-order logic should be built on the foundation of the Fischer Servi-style intuitionistic modal logic S4M? It is suggested (in [2]) that, when the underlying propositional logic is classical, S4M should be supplemented with a finality axiom (for which see [7]), so as to preserve at the firstorder level completeness with regard to the same Kripke frames and to extend to the first-order level the desired absence of clear borderline cases. We leave for another day the interesting questions of whether the finality axiom can, or should, play an analogous role in the first-order extension of intuitionistic S4M.
Intuitionists, Williamson supposes, will wish to assert that the Law of Excluded Middle 'does not hold always' in cases of vagueness. He takes this to mean that they will wish to assert x(Fx  Fx).
However, by (W) and contraposition, this yields (Fa  Fa ), the negation of a particular instance of Excluded Middle. As Williamson rightly points out, this is inconsistent in IPC ( [36] p.33).
This argument may be resisted at two places (at least). If an intuitionist says that Excluded Middle 'does not always hold' in cases of vagueness, she is not charitably interpreted as meaning to assert x(Fx  Fx). Rather, she is to be understood as saying that we are not entitled to assert arbitrary instances of Excluded Middle when the language contains vague expressions. When so understood, she will not be committed to asserting the negation of the consequent of (W) and there will be no question of applying contraposition to reach a contradictory conclusion. On the view propounded in this paper, however, Williamson's argument does not even get off the ground. The very formulation of (W) assumes that there are paradigm borderline cases for vague terms. In fact, the argument does not require the assumption of a paradigm borderline case; the weaker assumption of any case that is an example of a borderline case of F would do. Either way, intuitionists who favour the modal company of S4M will reject the assumption: no example of a borderline case of F can be given. So, to them Williamson's first argument is irrelevant.
His second argument against intuitionism as a viable basis for a theory of vagueness ( [36] p.33) fares no better. It revolves around the question 'Was Mars always either dry or not dry?', in a context where it is assumed that Mars was definitely not dry at some point in the past and has reached its current state of total dryness by a gradual process of evaporation. Williamson's discussion is complicated by the fact that the 'always' in his question is naturally heard as quantifying over a continuum, so we prefer to switch to a simpler case. Imagine a sequence of steps in each of which a single grain of sorghum is removed from what is clearly a heap until only one grain remains, and consider the question 'At every step was there either a heap or not a heap?' Williamson argues that the intuitionist is poorly placed to defend a negative answer: 'the denial of the conjunction of any finite number of instances of the law of excluded middle is intuitionistically inconsistent. The denial of the universal generalization of the law over a finite domain is therefore intuitionistically false too' ([36] p.33).
We agree with Williamson that the intuitionist cannot return a negative answer to this question; she will not, of course, give a positive answer either. But we reject his assumption that a good theory of vagueness has to validate one answer or the other. That assumption simply dismisses intuitionistic theories like Wright's. As we saw in §1, it is essential to Wright's position that he gives neither of the expected answers to the question 'Is it the case that some red tube in the sequence is immediately followed by a tube which is not red?' The negative answer leads to a contradiction, and the positive answer is Wright's 'Unpalatable Existential'. Williamson's question involves a universal, rather than an existential, quantifier but the intuitionist's stance towards it should be the same: she should refrain from answering it negatively, on pain of contradiction, but also refrain from answering it positively, on the ground that we have no reason to believe in the sharp cut-offs that a positive answer implies. That is, the intuitionist neither asserts nor denies that there was either a heap or not a heap at every step It is true that intuitionistic logic is a poor basis for a negative answer to Williamson's question. Then again, an intuitionist has no reason to give that answer. Williamson's second argument is just as ineffectual as the first against the intuitionist approach to vagueness that has been defended here.
While more work needs to be done, we hope to have shown that that approach is far more promising than Williamson's discussion of intuitionism would suggest. We argued that the general philosophical principles that underpin the intuitionistic approach-in particular, the thesis that borderline cases present no third alethic value to stand alongside true and false-recommend S4M as the correct modal logic of 'it is clearly (definitely, ...) the case that', and does so independently of whether the ambient logic is classical or itself intuitionistic. S4M vindicates the belief that there is higher-order vagueness, while avoiding the hierarchies of higher orders which generate paradox. We were also able to meet Dummett's challenge and explain why IPC is precisely the right logic to use when reasoning with vague terms at the sentential level. Moreover, we were able to do that without constructing a semantic theory for a language with vague terms: the only premiss we needed was that the McKinsey-Tarski translation specifies the conditions in which an intuitionist (such as Wright) who adopts the aforesaid general principles is prepared to assert vague statements. We do not, of course, claim to have shown that intuitionistic logic provides the best solution to the Sorites Paradox: we have not in this paper compared that solution with any of those which retain classical sentential logic. The connection we have drawn between intuitionistic logic and the modal logic of vagueness should make it easier to compare that solution with classical theories which use modal operators to express notions of determinacy, definiteness, or clarity. In any event, the connection already shows that the intuitionistic approach has a logico-philosophical coherence which ought to lead any theorist of vagueness to take it seriously.