If it’s clear, then it’s clear that it’s clear, or is it? Higherorder vagueness and the S4 axiom ⋆
If it’s clear, then it’s clear that it’s clear, or is it? Higherorder vagueness and the S4 axiom ⋆
Abstract and Keywords
This paper challenges some widespread assumptions about the role of the modal axiom S4 in theories of vagueness. In the context of vagueness, S4 usually appears as the principle ‘If it’s clear (determinate, definite) that A, then it’s clear that it’s clear that A’, or, formally, CA→CCA. We argue first that contrary to common opinion, higherorder vagueness and S4 are perfectly compatible. This is in response to claims such as Williamson’s that if vagueness is defined using a clarity operator that obeys S4, higherorder vagueness disappears. Second, we argue that contrary to common opinion, bivalencepreservers (such as epistemicists) can without contradiction condone S4, and bivalencediscarders (such as opentexture theorists and supervaluationists) can without contradiction reject S4. To this end, we show how in the debate over S4 two different notions of clarity are in play, and elucidate their respective functions in accounts of higherorder vagueness. Third, we rebut several arguments produced by opponents of S4, including those by Williamson.
Keywords: vagueness, higherorder vagueness, S4 axiom, modal logic, clarity, definiteness, epistemicism, opentexture, supervaluationism, KK Principle, Sorites
1. Introduction
In his pioneering essay on the treatment of vague expressions and the Sorites paradox in antiquity, Jonathan Barnes writes: ‘Ifyou don’t know whether a_{i} is clearly F, then it is not clearly clearly F and hence not clearly F’^{1} This sentence contains two conditional claims about vague predicates. The present paper is about the second: if a is not clearly clearly F, then a is not clearly F. Or, contrapositively, if it is clear that a is F, then it is clear that it is clear that a is F. Formally, the principle at issue can be expressed thus:
(CC) CA → CCA.
In analogy with standard modal logic, this principle is sometimes called the S4 axiom. To indicate that the principle is about clarity rather than, say, knowledge, we call it the CC Principle. Instead of ‘clear’ and ‘C’, some use the expressions ‘determinate’ or ‘definite’ and the operators Def DET, Δ, or similar. For the purpose of this paper, these will all be treated alike. (CC) does by no means find universal approval among vagueness theorists. Some take it for granted that it should be rejected. Thus Richard Heck writes: ‘Surely, anyone who takes higherorder vagueness seriously is going to want to deny Def(P) → Def(Def(P))’.^{2} In fact, it seems to be the prevalent view, shared by bivalencepreservers (like Williamson) and bivalencediscarders (like Dummett), that (CC) is to be rejected, since it is incompatible with higherorder vagueness.^{3} Yet, (p.190) in recent years, (CC) has found several supporters of rank among bivalencediscarders.^{4} But it has not, so far, been shown how exactly the presence or absence of (CC) manifests itself within the various mainstream theories of vagueness.
The purpose of this paper is to challenge some widespread assumptions about (CC). First, we argue that, contrary to common opinion, higherorder vagueness and (CC) are perfectly compatible (Section 4). This is in response to claims like the one by Timothy Williamson that, if vagueness is defined with the help of a clarity operator that obeys (CC), higherorder vagueness disappears. Second, we argue that contrary to common opinion, (i) bivalencepreservers (such as epistemicists) can without contradiction condone (CC); and (ii) bivalencediscarders (such as opentexture theorists, supervaluationists) can without contradiction reject (CC) (Sections 5–7). To this end, we show how in the debate over (CC) two different notions of clarity are in play and what their respective functions are in accounts of higherorder vagueness (Section 3). Third, we rebut a number of arguments that have been produced by opponents of (CC)—in particular, by Williamson (Section 8). Since discussants of (CC) have employed rather heterogeneous nomenclatures, we introduce diagrams to facilitate comparisons between the various theories.
2. Higherorder borderline vagueness: some preliminary remarks
(i) This paper is concerned only with Soritesvague predicates—that is, predicates that give rise to Sorites paradoxes. Every Sorites paradox runs on what we call its dimension D. Thus, for a paradox built on ‘tall’, the dimension D is height.^{5} Multidimensional Soritesvague predicates are considered only insofar as they give rise to onedimensional Sorites paradoxes. The most basic case of (CC) is then
CFa → CCFa,
with F for a simple, ordinary language, Soritesvague predicate and a for a designator. An example would be ‘if it is clear that Curly is bald, then it is clear that it is clear that Curly is bald.’ Given the context sensitivity of Soritesvague predicates, the semantic value of Fa is always assumed to be relative to a context C.
(ii) We consider in the first instance those theories of higherorder vagueness that intend to capture higherorder borderline cases; that is, borderline borderline cases, borderline borderline borderline cases, and so on. This type of higherorder vague ness—which we also call higherorder borderline vagueness—is the one most frequently (p.191) discussed.^{6} Here is an illustration. People may think that Curly is clearly bald, and that Moe is clearly borderline bald, but that with respect to baldness, a third man, Larry, falls somewhere between Curly and Moe; more precisely, somewhere on the border of clear baldness and borderline baldness. In that case, Larry is a borderline borderline case of bald. And we could imagine a fourth man, Shemp, who is a borderline borderline borderline case of bald.
(iii) It has become customary to define ‘borderline case’ with the help of a clarity (definiteness, determinateness) operator. Often, first a clarity operator is introduced, and then an account of ‘borderline case’ or ‘borderline utterance’ or similar^{8} is given in terms of the absence of such clarity. Sometimes an unclarity operator is defined in terms of the respective clarity operator, for instance (with ‘U’ for ‘it is unclear whether’, ‘it is indeterminate whether’, and so on):
UA =_{df} ¬CA & ¬C¬A.
That is, ‘it is unclear whether A iff it is neither clear that A nor clear that not A’. The most basic case is ‘It is unclear whether Fa iff it is neither clear that Fa nor clear that not (p.192) Fa.’ The unclarity of Fa is then explicitly or implicitly equated with a being a borderline case of F. As the basic semantics of ‘unclear whether’ suggests, when it is unclear whether A, it is also unclear whether ¬A:
(U) UA ↔ ¬A^{9}.
Higherorder borderline cases can be described by saying that it is unclear whether it is unclear whether A, and so on, and formalized as UUA or U^{2} A, U^{3} A, … U^{n} A.
Diagram 1a represents the distribution on a dimension D in a context C of first order borderline cases (B) with the respective—uninterpreted¬Clarity and unclarity operators indicated underneath.
Diagram 1b illustrates the distribution on a dimension D in a context C of second order borderline cases (B^{2}) with the respective combinations of—uninterpreted¬Clarity and unclarity operators indicated underneath:
In fact, the diagrams make visually explicit several assumptions that seem to hold of any Soritesvague predicate F relative to a dimension D. First, two assumptions about penumbral connections: if an object a on D is n times clearly F, then all objects on D to the left of a are n times clearly F; and if an object a on D is n times clearly not F, then all objects on D to the right of a are n times clearly not F, with n 〉 0. These two assumptions can be expressed formally as:
(PC_{1}) 
C^{n}Fa _{m} → C^{n}Fa _{m–1} 
(PC_{2}) 
C^{n}¬Fa _{m} → C^{n}¬Fa _{m+1} 
(LA) 
∀x [C^{n}Fx ∨ U^{n}Fx ∨ CU^{n–1}Fx ∨ C^{2}U^{n–2}Fx ∨ C^{3}U^{n–3}Fx ∨… ∨ C^{n–1}UFx ∨ C^{n}¬Fx]. 
Higherorder borderline vagueness differs from what we call simple higherorder clarity. The latter has (HOC) instead of (LA):
(HOC) 
C^{m} Φa _{n}→ C^{m} Φa _{n–1}, 
(PC_{1}), (PC_{2}), (C_{∃}), (LA), and (HOC) are not themselves part of the logical systems that we are considering—those which may or may not contain (CC). Rather, these assumptions allow us to check what consequences such a logical system would have, if we restrict our consideration to a set of objects on a dimension that form a Sorites series.
3. Selfrevealing versus concealable clarity
With these preliminaries in place, we return to (CC). The dispute over (CC) concerns the logical relations that hold between the different orders of clarity. In terms of borderline cases, it can be rephrased thus: are we to allow for the possibility that something a is a secondorder borderline case of F but not a firstorder borderline case, but rather a firstorder clear (definite, determinate) case of F? Epistemicists generally allow for this possibility, whereas bivalencediscarders often do not. Intuition has been invoked in defense of either view, and this fact can be used to direct us towards two distinct notions of clarity with which the discordant parties appear to operate: self revealing and concealable clarity. We consider them in turn.
What we call selfrevealing clarity is a clarity that cannot be concealed but always reveals itself. It shines through all higherorder levels—if you want.^{14} If something is selfrevealingly clear, it cannot be unclear whether it is clear. The fact that many people judge that, considered in a single context, paired sentences such as the following contradict each other, exemplifies how intuition supports this notion:

(p.195) • ‘It is clear that Tallulah is tall’—‘It is unclear whether it is clear that Tallulah is tall.’

• ‘This is undoubtedly sterling silver’—‘It is doubtful whether this is undoubtedly sterling silver.’

• ‘It is somewhat indefinite whether this colour patch is definitely purple’—‘This patch is definitely purple.’
And exploiting the fact that we are considering higherorder borderline vagueness, we can add cases such as:

• ‘Baldwin is borderline borderline bald’—‘Baldwin is not borderline bald but clearly bald.’

• ‘Selma is definitely slim’—‘Selma is a borderline borderline case of slim.’^{15}
In terms of borderline cases, selfrevealing clarity leads to the motto: what smacks of being borderline is borderline.^{16} For, if something is in any way tinged by unclarity, it cannot be selfrevealingly clear. The utility of this notion of selfrevealing clarity lies in the fact that it enables us to separate the clear cases from any cases that may have any lack of clarity to them—even if it is just a hint of a trace of a smidgeon of unclarity. It is a notion of clarity for those who want to be able to express that some fact is surefire and utterly unquestionable.
By contrast, concealable clarity is a clarity that is not automatically detectable through higherorder levels of clarity or unclarity. In this case, something can be clear while it is unclear whether it is clear. Here, plausibility may be gained from intuitions about the semantics of ‘unclear whether’. If it is unclear whether she is in the bedroom or the bathroom, then surely she could be in the bedroom. Now, if on a dimension D something a is on the border of clear cases and unclear cases of F (as, for example, Larry in Diagram 0), we may express this by saying that it is unclear whether it is clear that Fa or whether it is unclear whether Fa. Hence, analogously to the bedroom/bathroom example, if (i) it is unclear whether it is clear that Fa, or unclear whether Fa, then surely it could be the case (ii) that it is clear that Fa. But then (i) and (ii) are compatible; and (p.196) since (i) entails (iii), that it is not clear that it is clear that Fa, (ii) and (iii) are also compatible. Hence, with concealable clarity, (ii) and (iii) are not contradictories, as the notion of selfrevealing clarity had suggested. This notion of concealable clarity is useful as follows: with it we can express for higher orders of vagueness that, if something is in the border area of a vague expression Φx (Fx, CFx, C^{2}Fx…), it may after all be Φ, even if this is not apparent.
4. Concealable and selfrevealing clarity and modal logic
Before we consider how selfrevealing and concealable clarity correlate with some of the more popular theories of vagueness, one prevalent prejudice against (CC) needs to be laid to rest. Many proponents of higherorder vagueness have remarked on the partial similarity between the logic of clarity (determinateness, definiteness) and modal logic, and in particular between necessity and clarity^{17} One theorem that is generally granted in modal logic,
has an analogue in logics of clarity that is equally generally granted:(T) □A → A
that is, like necessity, clarity is considered to be veridical. The converse,(T_{c}) CA → A;^{18}
A → CA
is assumed neither for concealable nor for selfrevealing clarity. Similarly, most modal logics do not have a theorem
A → □A.^{19}
(p.197) Where selfrevealing and concealable clarity come apart is
(CC) 
CA → CCA. 
(C4) 
CA ↔ CCA, 
CA ↔ CCA
holds, does not have this effect at all^{24} In S4, not only is saying that A is necessarily necessary a longwinded way of saying that A is necessary, but also is saying that A is possibly possible just a longwinded way of saying that A is possible^{25} However, in the case of higherorder vagueness, the relevant two notions of clarity and unclarity (or nonborderlinehood and borderlinehood) are related in a different way. ‘It is unclear whether A’ is not synonymous to ‘it is not (the case that it is) clear that not A’; that is, we do not have
UA ↔ ¬C¬A.
Nor, by the by, is ‘It is unclear whether A’ synonymous with ‘it is not (the case that it is) clear that A’; that is, we do not have
UA ↔ ¬CA
either. Rather, as we stated in Section 2, the relation between clarity and unclarity is
UA ↔¬CA ∧ ¬C¬A.
(p.198) This difference to the necessity/possibility relation is marked by the fact that we typically say ‘clear that A’ but ‘unclear whether A’. The logic that governs unclarity is thus analogous to the logic of contingency,^{26} not of possibility. Acknowledging this, we can introduce an operator parallel to the noncontingency operator. We simply define ‘it is clear whether A’ or C_{w} A by ¬UA. Using the definition of UA, it follows that
(C/C_{w}) CA → C_{w} A^{27}
We have seen above that it is UFa—and not CFa—that is used directly in the account of ‘borderline case’: a is a borderline case of F iff UFa. Now it becomes apparent that with a notion of selfrevealing clarity, and (CC) granted, saying that A is clearly clear is just a uselessly longwinded way of saying that A is clear^{28} Yet, saying that it is unclear whether it is unclear whether A is not a uselessly longwinded way of saying that it is unclear whether A. For¬At least with the theorems introduced so far—it is not the case that
UUA ↔ UA.
It may hold that
(UU/U) 
UUA → UA 
But, or so most vagueness theorists assume, it certainly does not hold that
UA → UUA.
For it seems possible for there to be a and F such that a is a clear borderline case of F, or, in other words, such that it is possible that it is both unclear whether Fa and clear that it is unclear whether Fa; that is
UFa ∧ CUFa^{29}
By equivalence transformation we obtain
¬[UFa → ¬CUFa]
(p.199) from which, by (C/C_{w}), it follows that
¬[UFa → ¬C_{w}UFa]
which, by the definition of C_{w} is equivalent to
¬[UFa → UUFa].
We can infer that
UA → UUA
does not hold. Hence (CC) does not make higherorder vagueness disappear.^{30} Diagram 1b in Section 2 illustrates that this is the result we would expect for Sorites series on D. At level 2, there are sections for clearly clear cases, and for two kinds of borderline cases: U^{2} cases and CU cases. They are distributed on the dimension D as follows:
C^{2}Fx 
U^{2}Fx 
CUFx 
U^{2}Fx 
C^{2}¬Fx. 
C^{3}Fx 
U^{3}Fx 
CU^{2}Fx 
U^{3}Fx 
C^{2}UFx 
U^{3}Fx 
CU^{2}Fx 
U^{3}Fx 
C^{3}¬Fx. 
The center sections on D, CUFx from level 2 and C^{2}UFx from level 3, would be equivalent. This is, in fact, just an instance of
CA ↔ CCA
but neither U^{3}Fx nor CU^{2}Fx is equivalent with CUFx, nor are they equivalent to each other. And although U^{3}Fx entails U^{2}Fx, the converse does not hold. At higher levels we have increasingly more different kinds of borderline cases, with an increase of one kind per level. It follows that it is simply wrong to assume that just because a logic of selfrevealing clarity adopts the characteristic axiom of S4, higherorder borderline vagueness disappears, or that all statements of higherorder unclarity collapse into those of firstorder unclarity. Rather, at each level we obtain just as many different types of higherorder borderline cases as we expect.^{31} Thus the structural identity of (p.200) (CC) with the S4 axiom alone provides no reason for dismissing a logic of higher order borderline vagueness that includes (CC)^{32}
5. Comparison between concealable and selfrevealing clarity
There is thus no difference between concealable and selfrevealing clarity with regard to the possible number of types of higherorder borderline cases per order. The difference between logics of vagueness with and without (CC) is more subtle. As mentioned above, it concerns the relation between different orders. We have just seen that both with and without (CC) in place, the extensions (or quasiextensions)^{33} of the borderline cases, borderline borderline cases, and so on, do not necessarily coincide. For example, something can be a borderline case without being a borderline borderline case. But with (CC), the extensions (or quasiextensions) of clear cases and clearly clear cases, and so forth, do coincide, even though the boundaries between the clear cases and the unclear cases need not be sharp. By contrast, without (CC), the extensions (or quasiextensions) of clear cases and clearly clear cases need not coincide. We can show the impact of (CC) on higherorder borderlinehood by considering Shemp in Diagram 0. There, Shemp was borderline borderline borderline bald. Without (CC), for all we know, Shemp could be neither firstorder nor second order borderline bald, but instead both clearly and clearly clearly bald, despite the fact that he is thirdorder borderline bald. By contrast, if (CC) is taken to hold, Shemp would be a borderline case of bald, though only borderline borderline so; and he would be a borderline borderline case, though only borderline so. It is the coextensionality of clear and clearly clear cases by which (CC) endears itself to some supervaluationists and opentexture theorists, but proves unpopular with epistemicists¬As will be seen presently.
(p.201) 6. Bivalencediscarders and concealable clarity
Among bivalencediscarders, those who prefer systems with selfrevealing clarity either hold that
(C4) 
CA ↔ CCA 
However, it is by no means necessary for bivalencediscarders to accept (CC). In particular, those who wish to allow for an epistemic element in their theory may find a notion of concealable clarity useful for expressing this. To give an example, they may define firstorder borderline cases as those where it is unclear whether Fa in the sense that one can’t tell whether Fa,^{37} and hold that among those cases there are some a_{m} – a _{n} which make Fx true, some a ^{p} – a_{q} which leave Fx indeterminate,^{38} and some a_{s} – a_{t} which make Fx false, as depicted in Diagram 2a. (The curly brackets ‘{’ indicate the absence of a sharp border.)
7. Bivalencepreservers and selfrevealing clarity
Epistemicists—the most notorious of bivalencepreservers—tend to maintain that the only workable notion of clarity is concealable clarity. For example, Timothy Williamson suggests ‘knowably A’ as an interpretation of ‘it is clear that A’ or ‘definitely A’.^{40} (CC) then becomes ‘if it can be known that A, then it can be known that it can be known that A’, and is consequently rejected.
With regard to a dimension D, at the first level, the epistemicist assumption is that all firstorder unclarity utterances are de facto either true or false; we just cannot know which one. That is, there are on D some a _{m} – a _{n} which make Fx true and some a _{p} – a _{q} which make Fx false, but we cannot know that they do so. At the second level, there are assumed to be on D some a _{m} – a _{n} which make CFx true and some a _{p} – a _{q} which make CFx false, but of which, again, we cannot know that they do so. For these a it is unclear whether they are clearly F. Higher orders work accordingly. (At each level, it is possible for there to be cases which make C^{n}Fx true but C^{n+1}Fx false.) Thus we have a theory with concealed clarity, as illustrated in Diagram 3. (p.203)
Still, bivalencepreservers, including epistemicists, need not postulate concealable clarity to keep their theory consistent for higher orders of unclarity. The following scenario shows how this works. We retain the assumptions (i) that classical logic and semantics hold for vague predicates; (ii) that there are sharp boundaries between the clear and the unclear cases at every order of clarity; and (iii) that the sharp boundaries are unknowable. With regard to any dimension D, we keep the requirements for simple higherorder clarity, as introduced in Section 2 (and accepted, for example, by Williamson). Now we simply add (CC), thereby trading concealable clarity for selfrevealing clarity. The result is perfectly consistent. The only change of note is that instead of having the higher orders of clarity—possibly—staggered with regard to the lower ones, now the sharp boundaries between the clear and unclear cases of all orders of clarity coincide, as illustrated in Diagram 4.
Ifwe suspend judgment about the question whether, at any level, the boundaries are sharp, we obtain an agnostic position like the one developed by us in detail elsewhere^{41} Either way (agnostic or epistemicist), the theory has an advantage over Williamson’s own in that it can be interpreted as a theory of higherorder borderline vagueness (see Section 2), with the proviso that there are no instances of clear borderline cases at any level. In its agnostic version, it has the attractive advantage over bivalence discarding theories that it preserves radical higherorder vagueness but escapes the various paradoxes of higherorder vagueness.^{42}
In its epistemic version, this theory is open to any bivalencepreserver who is happy to give up at least one of Williamson’s assumptions that force him to reject (CC), as there are:

(i) The Coperator is governed by rule (RN).

(ii) The Coperator is governed by axiom (K).

(iii) Clarity is cashed in directly in terms of a notion of knowability that by definition includes a margin for error, such that for any Sorites series it holds that CFa _{n} →Fa _{n+1}.

(iv) If CFa _{n}→Fa _{n+1}, then also C^{m}Fa _{n}→C^{m1}Fa _{n+1}, for all m〉1.
Since each one of (i) to (iv) can be questioned, there may well be epistemicist takers for the option here presented.
8. Discussion of some arguments against (CC)
We conclude the paper by considering the main arguments that have been marshaled against (CC).
(p.205) 8.1. (CC), (KK), and propositional attitudes
Assume that the KK Principle—that if I know something, then I know that I know it^{43}—concerns propositional attitudes and that it is false. One standard objection to (CC) is that when it is cashed in in epistemic terms, the difficulties encountered in (KK) reoccur for (CC). For example, if Cp is interpreted as ‘it is known (by some person 5) that p’^{44} or ‘a person 5 can tell that p’, we appear to deal with propositional attitudes. CCp then becomes ‘it is known (by 5) that it is known (by 5) that p’ or ‘a person can tell that they/a person can tell that p’, respectively. For any philosopher who rejects (KK) and accepts an interpretation of Cp like the ones given, (CC) would appear to be unacceptable too.
However, first of all it is not at all obvious that in the context of vagueness, clarity must be defined in epistemic terms. And even if it i5 thus defined (as we believe it should be), it does not follow that we are dealing with propositional attitudes. For example, CFx could be defined roughly along the lines of ‘x is such that all relevantly competent and informed speakers of English, if asked, could tell whether Fx’.^{45} CFx would then not express a propositional attitude, but rather a property of items a that involves (but cannot be reduced to) certain human dispositions to react to certain things. In that respect, CFx would be on a par with expressions such as ‘amusing’, ‘amazing’, ‘understandable’, ‘comprehensible’, ‘instructive’, ‘funny’, ‘gloomy’, ‘readable’, ‘(il)legible’ (said of a story, for example), ‘(un) illuminating’, ‘(un) intelligible’, ‘boring’, ‘tedious’, ‘uninspired’ (said of a performance, for example). Just as ‘I find it amusing that p’ and ‘he finds it boring that p’ express propositional attitudes, but ‘it is amusing that p’ and ‘it is boring that p’ often do not,^{46} so ‘I cannot tell whetherp’ and ‘I can tell that p’ may express propositional attitudes, but ‘all relevantly competent and informed speakers of English, if asked, could tell whether p’ need not. Hence, whenever CA and UA are defined in epistemic terms, but do not express propositional attitudes, the standard objections against (KK) fail when transferred to (CC).
8.2. Williamson’s argument from intuition
In the Appendix of his book Vaguene55, Timothy Williamson bases his rejection of (CC) on intuition:^{47} He argues as follows: ‘Intuitively, any formula [any propositional (p.206) formula with or without clarity operator in the logic of clarity that Williamson is presenting] permits a margin in which it is true but not clearly true, unless it takes up all or no conceptual space.’^{48} This recourse to intuition is unconvincing. Let us admit for the sake of argument that any firstorder vague expression (say, Fx) permits a margin in which it is true but not clearly true; that is, that there are a such that Fa is true but not clearly true. This may have some plausibility. Still, it is not at all intuitively true that a secondorder vague expression (say, CFx) permits a margin in which it is true but not clearly true; that is, that there are a such that CFa is true but not clearly true. This becomes obvious when we reformulate this case on the objectlevel: it is not at all intuitively true that any secondorder vague expression ‘it is clear that Fx’ permits a margin for which it holds that ‘it is clear that Fx but it is not clear that it is clear that Fx’. In fact, here many people experience a strong intuition to the contrary^{49} Williamson’s presumed ‘intuition’ will be shared only by those who, like himself, already think of clarity in terms of concealable clarity. But whether this is how we should think of clarity in the context of vagueness is exactly the point under dispute. Intuition does not settle the dispute. There are some philosophical frameworks in which a notion of concealable clarity is suitable, and others in which a notion of self revealing clarity is suitable. Accordingly, when confronted with the question whether (CC) holds, we can summon intuitions either way.
In his ‘On the structure of higherorder vagueness’,^{50} Williamson uses (a model that corresponds to) standard modal logic to analyze the notion of definiteness relevant to vagueness, claims to provide a system that clarifies structural issues regarding this notion ‘without addressing deep questions about the nature of vagueness’ (128),^{51} and suggests that his theory covers supervaluationist theories as well as epistemicist theories. He writes that ‘for the supervaluationist, definiteness is truth under all… admissible sharpenings’ and ‘for the epistemicist, definiteness is truth under all sharp interpretations of the language indiscriminable from the right one’ (128). About the S4 axiom he says that ‘to deny 4 is to deny that accessibility is transitive; intuitively, there is a close connection between the nontransitivity of indiscriminability and higherorder vagueness’ (134) and that ‘the addition of 4 alone to KTB… gives S5’ and then suggests—correctly, in our view—that S5 is not compatible with second order vagueness (134)^{52} This notwithstanding, Williamson’s arguments are, again, not compelling. First, the ‘close connection between the nontransitivity of indis criminability and higherorder vagueness’ that he mentions exists only if the (p.207) indiscriminability is based on his own marginforerror principle.^{53} However, non epistemicists need not accept this principle (see Section 8.3 below). But then, contrary to what Williamson suggests, his argument is not compelling for supervaluationists. Second, the fact that ‘the addition of 4 alone to KTB… gives S5’ is irrelevant. For, Williamson himself suggests, four pages later, that axiom B should be abandoned (138);^{54} and axioms K and T together with the S4 axiom do not give S5.^{55}
8.3. Williamson’s argument from his marginforerror principle
In both his (1992) and his (1994), Williamson argues that (CC) does not hold for vague sentences, since it is incompatible with his marginforerror principle.^{56} Here we present a condensed form of his proof. Since Williamson cashes out clarity in terms of knowledge, he uses K rather than C as operator. K denotes ‘it is known that’.^{57} His analogue to the CC Principle—the KK Principle—can then be expressed as:
(KK) 
KFa _{n} — KKFa _{n}, 
where a _{n} and a_{n–1} mark adjacent objects of a Sorites series. Roughly, the marginfor error principle is justified by the fact that it precludes that Fa could be false in circumstances very similar to those in which it is known that Fa. Williamson then ‘substitutes’ KFx for Fx in (ME):(ME) KFa _{n}→Fa _{n+1},^{58}
(ME_{K}) 
KKFa _{n}→KFa _{n–1} 
(ME _{Kcontra}) ¬KFa _{n–1}→ ¬KKFa _{n}.
Next, he introduces two suppositions: (S1) ‘KFa _{n} is true’, and (S2) ‘KFa _{n–1} is false’. Applying modusponens to (S2) and (ME_{Kcontra}), he derives ¬KKFa _{n}. He concludes that there is a possible situation in which we have both KFa _{n} (by (S1)) and ¬KKFa _{n} (by derivation); and that hence, given his marginforerror principle, (KK) does not hold in all cases.
At first blush, this argument may seem plausible for Soritesvague predicates. However, neither epistemicists nor bivalence discarders are bound by it. Epistemicists can question both (ME) and the step from (ME) to (ME_{K}). The purpose of (ME) was to introduce a margin for error for knowledgability by precluding that Fa could be false in circumstances very similar to those in which it is known that Fa. But the truth of (ME) requires more than the existence of a margin for error for knowledgability. It additionally requires that the size of the margin for error that comes with the knowledge of Fa _{n} always equals or exceeds the distance on dimension D between two adjacent a _{n} and a_{n+1} of a Sorites series. Yet for all we know, this second requirement may be false in any number of cases.^{59} There are obvious examples—such as cases where the number of objects in the Sorites series or in the borderline zone is very small (‘a couple of us stayed for dinner’ stated by one of a group of five afternoon tea visitors). But even in commonorgarden Sorites arguments there seem to be no compelling reasons why this second requirement should always be satisfied.^{60} Second, the step from (ME) to (p.209) (ME_{K}) is not a logical step. It may seem like some kind of logical substitution, but it is not.^{61} Rather, the step relies on a philosophical assumption which Williamson never makes explicit. This is the assumption that with the addition of the two Koperators in (ME_{K}) the margin for error never decrea5e5 to a size less than the distance between a _{n} and a _{n}+_{1} on D. This assumption, too, can be challenged. It follows that even epistemi cists are not forced to accept Williamson’s conclusion that (CC) does not hold for vague sentences. This result is in line with our argument in Section 7 above.
Next, we show that bivalencedi5carder5 have no reason to grant that Williamson’s marginforerror principle (ME) is relevant to vagueness in the way he suggests. Take any bivalencediscarding theory that accepts the following assumptions (A1) and (A2):

(A1) If in a context C a is borderline F, then with regard to C (an utterance of) the sentence Fa either has a semantic value other than Truth or Falsehood, or has no semantic value at all.

(A2) For any Sorites series of F, there is always at least one a _{n} in the borderline zone of F.
In most theories ofvagueness, including Williamson’s own, (A2) is trivially true. (A1) is accepted by most bivalence discarders, including supervaluationists and opentexture theorists. Given (A1), Fx has a morethantwovalued semantics. Consequently, the formalization of the marginforerror principle needs modification, if Williamson’s argument is to fly at all. The least intrusive and most plausible change may be to
meaning roughly ‘if (in C) it is clear that Fa _{n}, then it is not the case that (in C the utterance of) the sentence Fa _{n–1} is false. In fact, (ME)^{7} follows from the conjunction of (A1) and (A2). Thus there is de facto no need for invoking any idea of a knowledge margin for error for the acceptance of (ME)^{7}. The step analogous to Williamson’s step from (ME) to (ME_{K}) would then be from (ME)′ to(ME)′ CFa _{n}→ —[Fa _{n–1} is false] (with ‘–’ for exclusion negation),^{62}
(ME_{K})′ 
CCFa _{n}→ —[CFa _{n–1} is false]. 
Acknowledgments
Thanks are due to Ruth Barcan Marcus, Zoltan Gendler Szabo, Timothy Williamson, David Kaplan, Tyler Burge, Jennifer Hornsby, the audience at UCLA, where an earlier version of the paper was presented in 2006, the participants in my graduate classes on vagueness, and especially to Stewart Shapiro, who read the paper for Oxford University Press. I also wish to acknowledge support from the Institute for Advanced Studies, Princeton, the National Endowment for the Humanities, and the Mellon Foundation.
References
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Notes:
(^{⋆}) It is my pleasure to dedicate this paper to Jonathan Barnes, from whom I have learned more than from anyone else, and from whose kindness and generosity I have benefited more than I can ever hope to return.
(^{3}) For example, Dummett (1975), 311; Wright (1987) and (1992); Williamson (1994, pp. 159, 271–2 and (1999); Greenough (2003); cf. also Garrett (1991) 347.
(^{4}) In addition to Barnes (1982), these include Field (2000) and Soames (2003). It is considered by Shapiro in his (2006), for example, ch. 5.6, and by Fine (1975). Fine does not accept (CC) outright, but he ponders the possibility that ‘definitely A’ is simply elliptical for ‘“A” is true’.
(^{5}) For ‘bald’, typically the dimension is number of hairs, increasing by one; for ‘red’ it may be a series of color patches from red to orange, and so on.
(^{6}) For example, in Sainsbury (1991), Wright (1992), Fara (2003), Shapiro (2005) and (2006) ch. 5, and Greenough (2005). We believe that the examples usually given, including our own, are best analysed not in terms of higher orders but as borderline nestings. However, in this paper we consider mostly how the mainstream theories of vagueness and their notions of higherorder vagueness relate to the S4 axiom, and for that reason retain the description in terms of higherorder vagueness.
(^{7}) A quite different problem of higherorder vagueness, which we disregard in this paper, is the question of the vagueness of the predicate ‘vague’. This is discussed, for example, in Sorensen (1985), Varzi (2003), and Hyde (2003).
(^{8}) An utterance of Fa is a borderline utterance of Fa iff a is a borderline case of F. A sentence Fa is a borderline sentence with regard to context C iff with regard to C a is a borderline case of F.
(^{9})
UA 
↔ ¬C¬A Λ ¬C¬¬A 
definition of UA, substitution of ¬A for A 
UA 
↔ ¬C¬A Λ ¬CA 
double negation 
(^{10}) Diagram 1b shows that U^{2}Fa is equivalent to ¬C^{2}Fa Λ ¬CUFa Λ ¬C^{2}¬Fa. Generally, U^{n}Fa is equivalent to ¬C^{n}Fa λ ¬C^{n(n1)}U^{(n1)}Fa ∧ ¬C^{n(n2)}U^{(n2)}Fa ∧…∧ ¬C^{n(nm)}U^{(nm)}Fa ∧ ¬C^{n}¬Fa, with n+1 conjuncts and m = (n–1).
(^{11}) (LA) does not state that at every level every a clearly falls into one of the categories. Rather, for higher order borderline vagueness, each higher level can be thought of as improving on the previous one. The lower levels may be considered as incomplete (inviting some precisification), or as provisional (inviting some revisionary specification). Either way, level n would be superseded and replaced by level n+1. Of course, given (T_{c}), the lower levels are all in some sense preserved in the higher ones. (LA) does, however, throw out theories which introduce more than two possible semantic values for (utterances of) sentences governed by the clarity operator (for example, true, false, indeterminate), rather than expressing lack of clarity, as clarity that is unclear. (Thus, for example, the option considered by Edgington (1993), 193–200, is precluded by (LA).)
(^{12}) It has been suggested that this type of higherorder vagueness gives rise to the socalled ‘higherorder vagueness paradoxes’ if it is taken to be ‘radical’—if it allows for an infinite number of higher orders (cf. Wright (1987) §5, Wright (1992), Fara (2003), Greenough (2005), and Shapiro (2006), ch. 5). Several philosophers (such as Heck (1993) and Edgington (1993)) have proposed ways of defusing these paradoxes. For a theory of radical higherorder borderline vagueness that avoids the paradoxes, see Bobzien (2010). In any event, no paradox ensues for theories that assume relatively small and finite numbers of higher orders.
(^{13}) See Williamson (1994 chs. 7–8, Williamson (1999), and the discussion of Williamson’s view of higher order vagueness in Sections 8.1 and 8.2 below.
(^{14}) Alternatively, you could imagine that at any point on the dimension, higherorder unclarity casts, as it were, a shadow of unclarity down on all lower orders. This notion of selfrevelation is similar to Williamson’s notion of luminosity; cf., for example, Williamson (2000), ch. 4.
(^{15}) Objection: In the examples we do not have true contradictions, but only some kind of pragmatic conflict, similar to the one we observe in Moore’s paradox (‘p. I don’t believe that p.’). Reply: In Moore’s paradox, the two sentences are in conflict only because they are (assumed to be) uttered by the same person. By contrast, ‘it is clear that p’ and ‘it is not clear that it is clear that p\ etc., neither contain an indexical (like ‘I’) nor do they have to be (assumed to be) uttered by the same person, or be in any way relative to a speaker’s perception, view, or similar, for them to appear contradictory. (See also Section 8.1 below).
Since vague predicates are, as a rule, context sensitive, there may, of course, be no appearance of contradiction, if the contexts of the two assertions differ. Moreover, even in the same context, if we have what one may call negotiations regarding a temporary stipulation of the extension of a vague predicate, there need be no contradiction. For example, if in a conversational context (along the lines of the theories of conversation in Stalnaker (1979) and Lewis (1979)) the conversationists try to settle the relevant borderlines with regard to that context, tentative proposals may be made by different speakers (cf. Shapiro (2003) and (2006)). The appearance of contradiction is relative to the assumption that the utterance(s) are considered as assertions of how things are, regardless of whether the utterance(s) are made.
(^{16}) This motto differs from Kit Fine’s suggestion that anything that smacks of being borderline is treated as a clear borderline case; cf. Fine (1975), 297.
(^{17}) For example, Fine (1975), Dummett (1975), Heck (1993), Williamson, (1994), (1999), McGee and McLaughlin (1995), Field (2000) and (2003), Shapiro (2006), and Wright (2010).
(^{18}) Our observations should hold for all systems of higherorder borderline vagueness and of simple higherorder clarity. To provide a general idea of what an axiomatic system might look like, here is a set of rules and axioms which could be used to introduce a basic logic of borderline cases, or more generally, of the kind of clarity relevant to vagueness:
(PC)^{⋆} 
If A is a truthfunctional tautology, then – A 
(MP) 
If –A → B and – A, then –B 
(∧I) 
If — A and – B, then – A∧B 
(T_{c}) 
– CA → A 
(C∧) 
– [CA ∧ CB] → C[A ∧ B] 
(^{⋆}) (PC) works for supervaluationists and epistemicists. For those who maintain that classical logic may apply in the borderline zone, the rule can be replaced by one that makes truthfunctional tautolo dependent on the clarity of their component formulae (cf. Bobzien (MS)).
(^{19}) Most standard modal logics do have a metarule that if A is provable (from no undischarged premi then so is □A. The equivalent in a logic of borderline cases would be that if you can prove A, you prove that A is not borderline, or, in short, that nothing provable is borderline. Here is not the place to di this point.
(^{20}) We get CCA → CA by substituting CA for A in (T_{c}). From CCA → CA and (CC) we get (C4) by using the rule (∧I) and the definiens of ↔, [A→B]∧[B→A].
(^{21}) We say ‘resemble’, since nothing has been said so far about whether a logic of clarity must include counterparts for the axiom (K) and rule (RN); or for the rule that if A is derivable from premises, all of which begin with ‘□’, then □A is derivable from those very premises. The equivalent for the latter in a logic of clarity would be what Crispin Wright christened ‘rule DEF’ (Wright (1992), 131), and the validity of which is questioned by Edgington (1993) and Heck (1993).
(^{24}) The only author we are aware of who has commented on this fact is Hartry Field, in Field (2000), §V
(^{25}) ◊A =_{df} ¬□¬A
◊◊A ↔□¬◊A ↔□¬¬□¬A ↔ ¬□□¬A ↔ ¬□¬A ↔◊A:
(^{27})
UA ↔ ¬CA ∧ ¬C¬A 
definition UA 
UA ↔ ¬CA 
by [A ↔ B ∧ C) → [A→B) 
CA → ¬UA 
by contraposition, double negation 
CA ¬C_{w}A 
by definition of C_{w} A. 
(^{28}) This seems intuitively plausible. When someone says ‘it is clearly clear that Simone is small’, it is hard to detect in it any substantial addition to ‘it is clear that Simone is small’. But intuitions vary, and those at work here are obviously those that support selfrevealing clarity.
(^{29}) When this paper was originally conceived (in 2006), it was almost universally assumed by vagueness theorists that if there is higherorder borderline vagueness, then there are clear borderline cases. This assumption has been challenged in Bobzien (2010). Those who doubt the very existence of true higher order vagueness (for example, Sainsbury (1991), Wright (1992), (2003 implied) and (2010), Fara (2003), and Shapiro (2006)—to which, in response, Bobzien (2009)—Smith (2009), and Raffman (2010)) are invited to consider the argument in the main text ad hominem.
(^{30}) For a more complex argument that shows that even if there are no clear borderline cases, (CC) does not make higherorder vagueness disappear, see Bobzien (2010).
(^{31}) Åkerman and Greenough (2010a), p. 287, n. 37, observe that if we use the logic of S4 (KT4), ‘then there is only a finite number of modalities (in fact at most fourteen distinct modalities…). Consequently, there cannot be borderline cases to borderline cases ad infinitum.’ However, this seems to be just another instance of the mistaken analogy of borderlinehood with possibility rather than with contingency. We saw in Section 2 that the modal operator for borderlinehood (the unclarity operator U) is defined as ¬CAa¬C¬A. Iterations of the operator U (together with its combination with the clarity operator C) thus do not collapse into the fourteen distinct basic modalities of the standard modal system S4.
(^{32}) Objection: a logical system of the kind suggested that contains (CC) is inconsistent:
(1) 
Assume 
UUFa 
assumption. 
(2) 
Hence 
UFa 
(1), by (UU/U), assuming this holds. 
(3) 
Hence 
CUFa 
since surely anything that can be inferred must be clear in the sense defined. 
(4) 
Hence 
C_{w}UFa 
(3), (C/C_{w}). 
(5) 
Hence 
¬UUFa 
(4), definition of C_{w}. 
But (5) contradicts (1). Hence the system is inconsistent. Reply: step (3) is problematic. It assumes that the clarity of inference is identical with or entails the clarity of nonborderline utterances. Or, expressed differently, it assumes that the clarity operator C is governed by a rule that allows one to infer CA from A in natural deduction modal logics. However, we have no reason to believe that this is the case.
(^{33}) A quasiextension is for a theory of predicates without sharp boundaries what an extension is for a theory of predicates with sharp boundaries; cf. for example, Soames (2003).
(^{34}) For example, Field (2000), Soames (2003), Shapiro (2005), Shapiro (2006) ch. 5, and considered by Fine in his (1975), §5.
(^{35}) For example, Fine (1975), 296, with ‘definitely’ instead of ‘clearly’. Fine points out two problems with this ‘ellipsis’ view.
(^{37}) Here we leave aside the question of who exactly it is who cannot tell. In Section 8 we show that the interpretation of UFx as inability to tell whether Fx does not commit one to the KK Principle.
(^{38}) We here use ‘indeterminate’ as a substitute for whatever alternative semantic status a bivalence discarder may wish to assign to these cases.
(^{40}) Williamson (1994), pp. 164 and 195. We noted in Section 2 that Williamson’s logic of clarity is not a logic of higherorder borderline vagueness, but of simple higherorder clarity, and that it precludes clear borderline cases (cf. 1c). With classical logic and standard modal logic, it seems impossible to provide a satisfactory model for higherorder borderline vagueness that permits clear borderline cases.
(^{42}) Cf. Bobzien (2010), Section 11. Elsewhere we provide an explanation of the common assumption that there are clear borderline cases.
(^{43}) This is Williamson’s version of the principle (Williamson (1994), p. 223. For a more formal version see, for example, Greenough (2003), 275.
(^{45}) For example, Shapiro takes this line in his (2003) and (2006), but gives up compositionality for C (definiteness, in his terminology). In Bobzien (2009) we show that compositionality can be preserved for this general type of clarity.
(^{46}) In the simplest case, such expressions can be analyzed along the lines of standard secondary quality predicates; for example, ‘is boring’ as ‘apt to cause, in a normal way, boredom in normal subjects under normal conditions’; more subtle alternatives that involve normative or relativist elements are also available.
(^{47}) Williamson (1994), pp. 271–2. Williamson suggests (in his (1994), p. 274) that he has presented ‘the logic of clarity’ (italics ours). We do not believe that there is precisely one correct logic of clarity, even in the restricted sense in which the notion of clarity is introduced to illuminate what borderline cases are. Rather, as the present paper illustrates, depending on to what features of borderline cases one wishes to draw attention, different logical systems can be used.
(^{49}) Cf. above, Section 3.
(^{51}) Here we ignore the fact that the question of whether classical logic is the appropriate logic for vagueness is itself a ‘deep question about vagueness’, and that Williamson simply assumes that it is.
(^{52}) So also Shapiro, who in his (2005) and (2006) bites the bullet and denies the existence of true higher order vagueness, if at the cost of giving up compositionality. For details, see Bobzien (2009).
(^{53}) See Williamson (1999), 137, his reference to his (1994), where he argues that the acceptance of the marginforerror principle leads to the rejection of KK. For details, see Section 8.3 below.
(^{54}) In the same vein, in his (1994), p. 272, Williamson states that axiom B ‘of KTB is unobvious’, and suggests a model with KT instead of KTB.
(^{55}) The only other support that Williamson provides for his rejection of (CC) is in footnote 5 of his (1999), where he states that ‘Schema 4 [(CC)] plays a critical role in the supposed paradox of higherorder vagueness of Wright (1987) and (1992); it is likely to be rejected in any plausible account of higherorder vagueness.’ However, Edgington (1993) and Heck (1993) have plausibly argued that the relevant critical role is played by Wright’s ‘DEF’ rule rather than by (CC). Moreover, it is possible to both keep (CC) and avoid Wright’s paradox, as is shown in Bobzien (2010), Section 11. It is also worth noting that none of the authors whom Williamson mentions in the footnote actually provides an argument against (CC).
(^{56}) Cf. Williamson (1992), Sections 1, 2, and 5, and Williamson (1994), ch. 8. We say his marginforerror principle, since a marginforerror constraint can be introduced into a theory of vagueness by different principles; see, for example, Glanzberg (2003), Section 4.3, and Bobzien (2010), Section 4.
(^{58}) Williamson uses n1 instead of n+1, but the difference is insignificant. What matters is that the a in the consequent of (ME) is closer to the borderline zone than the a in the antecedent, as is clear from his assumptions (S1) and (S2); see below.
(^{59}) Thus, for Williamson’s theory to succeed in solving the Sorites paradox for some F, it requires the additional constraint on simple higherorder clarity that the number of orders of clarity is smaller or equal to the number of clear cases of F on D. Since standard Sorites paradoxes have a finite number of members of the Sorites series, the numbers of orders will be finite (and hence cannot converge towards some point on D). As Williamson himself describes the nonclear (nondefinite) cases of the various orders as borderline cases (for example, Williamson (1999), 132), this leads to the awkward result that for ‘bald’, for example, there would be an nth order borderline case of being bald who has zero hairs (with n finite). When Williamson in his Theoria interview says that ‘epistemicism can easily handle higherorder vagueness’ (Williamson and Chen (2011), 9), this may be true. However, his simple and elegant theory of higherorder vagueness has no explanatory power when it comes to borderline cases of vague predicates as they are commonly understood.
(^{60}) Could we not restrict our attention to Sorites series in which the difference between adjacent items is (much) smaller than the margin for error? We could. But since Williamson’s solution to the Sorites is tied to the size of the margin for error, this would have the unhappy consequence that his solution only works for a fraction of Sorites paradoxes. And if we would not want this fraction to be determined arbitrarily, the answer to the question of how small ‘(much) smaller’ would have to be is: so small that the distance on D between adjacent a _{n} and a _{n+1} of the Sorites series is not larger than the size of the margin for error that comes with the knowledge of Fa _{n}—which is just the point that we made in the main text.
In Williamson’s theory, (ME) is needed to explain why people—mistakenly—believe that the inductive premise of the Sorites is true. In simplified terms, they overlook the difference between the true (ME) and the false inductive premise. However, contextualist theories, which have become the predominant theories of vagueness, offer a sheaf of alternative explanations for why people mistakenly accept the inductive premise, so that Williamson’s (ME) is not required. (In addition to the groundbreaking work by Diana Raffman (1994), see Soames (1999), Fara (2000), Shapiro (2003), (2005), and (2006), and the more recent clarifications and defenses in Akerman and Greenough (2010a), Akerman and Greenough (2010b), and Akerman (forthcoming)).
(^{61}) Cf. the following examples: [KFa→Fb] → [KKFa→KFb]; [[Clearly p] → q] → [[Clearly clearly p] → [clearly q]]; [[Clearly 2+2=4] → [This is red]] → [[Clearly clearly 2+2=4] → [Clearly, this is red]]; [[I know that 2+2=4] → [4+4=8]] → [[I know that I know that 2+2=4] → [I know that 4+4=8]]. Of course, Williamson does not claim that the step is a logical step.
(^{62}) The choice of exclusion negation (‘S is true if and only if S is not true’) is motivated by the goal to cover as many theories as possible. It allows for theories in which, if in C a_{n–1} is borderlineF, then an utterance of the sentence Fa _{n–1} in C is the kind of utterance of which a truthvalue cannot be meaningfully predicated. (Cf., for example, Glanzberg (2003), Bobzien (2010), Sections 5 and 9, Rayo (2010), pp. 42–3, and Iacona (2010) for consideration of this option.)
(^{63}) Their dispensability was noted above.
(^{64}) See Section 2 and Diagram 1c. To preclude this possibility, we would need something along the lines of strong simple higherorder clarity (for example, Dummett (1975), p. 311), which differs from weak simple higherorder clarity by the added constraint that for any n 〉 1, 3x[C→x λ ¬C^{n+}→x]. Rules (PC→ and (PC2) would guarantee that the cases singled out by this constraint are in the right place on D to make (MEK)’ true.
(^{65}) A different argument against (CC) on the basis of an epistemic interpretation of ‘It is clear that’ comes from Greenough ((2003), 274–8). He uses the phrase ‘It is known that p (to a speaker s)’ to characterize borderline cases, and consequently considers (KK) relevant. He suggests that if (KK) fails for all orders n, then there must be radical higherorder vagueness (276). He reconfigures Crispin Wright’s socalled paradox of higher–order vagueness in epistemic terms, and subsequently develops a complex quattrolemma from which he concludes that (KK) fails for all orders n (276–8). His argumentation is not persuasive, for two reasons. First, in his proposal, ‘borderline case’ is relativized to a speaker, which is not how we usually understand the term. Second, his adaptation of Wright’s argument against (KK) works only because it lumps together two distinct parameters: it does not distinguish between very similar objects an, am and very similar contexts, C1, C2, but only between very similar ‘normal cases of judgment conditions for the speaker s’ α, β (260; 276–7). Greenough’s argument deserves detailed discussion, but this goes beyond the scope of this paper.