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)’ is equivalent to ‘p’; so A(ii) is equivalent to ‘he’s either old or not old’, which is strictly weaker than A(i); so when conjoined with A(i) one just gets A(i), i.e. ‘he’s old’. With (B), adding (ii) does produce a genuine strengthening. But given that ‘appropriate’ is obviously vague, there’s still no reason to think that ‘determinately old’ has sharp boundaries. The situation with (C) is similar to that of (B): it’s unclear how exactly to explain ‘fuzzy’, but it seems like however one explains it, it’s bound to itself be vague. Still, there’s a substantial worry: that we could produce a sharp border by iterating a ‘non-fuzziness’ operator into the transfinite. That is, why doesn’t the sequence old; old and not fuzzy whether old; old, not fuzzy whether old, and not fuzzy whether fuzzy whether old; and so forth
collapse to a bivalent predicate by level ω or by some higher transfinite level γ ? If this were to happen—and it does happen in many standard proposals for nonclassical logics for vagueness, e.g. the Lukasiewicz continuum-valued logic—then that would be a disaster. For then there would be a number N such that Russell was determinatelyγ old at nanosecond N but not determinatelyγ old a nanosecond before; we’d have a sharp boundary for ‘determinatelyγ old’, so why not just take this as the sharp boundary for ‘old’?
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If the determinately operator collapses to bivalence in this way, nothing would be gained by going non-classical. And it is a delicate matter to get a non-classical logic of vagueness in which such a collapse is avoided. Still, it can be done: there are reasonable logics of vagueness in which such a collapse never occurs.² Admittedly, a non-classical logic appropriate to vagueness is somewhat complicated. Given that almost every term is somewhat vague, wouldn’t the non-classical approach make proper reasoning about ordinary subjects difficult? I think this worry is exaggerated. It might be useful to compare the case to geometric reasoning. We all know that space is not quite Euclidean, and indeed fails to be Euclidean in a quite complicated way; nonetheless, we are safe in using Euclidean reasoning except in special contexts, because the error involved in doing so is so slight. That is the policy I recommend for logic: reason classically, except for those situations where there is reason to think that the errors induced by such reasoning are significant. Situations where we derive boundaries for vague terms look like just the sort of situation to worry about! 11.4
B ROA D E N I N G T H E R A N G E O F C O N S I D E R AT I O N S
How do we decide between a classical logic approach to vagueness, which must postulate sharp borders, and a non-classical approach that avoids this but complicates the logic? It’s a matter of weighing costs and benefits. I haven’t tried to argue that the weight of the benefits is on the non-classical side: that would be a big task. Rather, I’ve just tried to argue that the non-classical approach is not without motivation (especially if it avoids the danger of collapsing determinately operators). I’d like to conclude by mentioning an additional item on the nonclassical side of the ledger: Berry’s paradox. Say that a 1-place formula F (x) of English is uniquely true of an object c if it is true of c and of nothing else. Let an S-formula be a formula of English with less than a thousand symbols. Then 1. There are only finitely many S-formulas; since there are infinitely many natural numbers, there must be natural numbers that no S-formula is uniquely true of. So, by the least number principle, 2. There is a smallest natural number M such that no S-formula is uniquely true of it. But ‘x is the smallest natural number such that no formula of English with less than a thousand symbols is uniquely true of it’ is an S-formula. So 3. ‘x is the smallest natural number such that no formula of English with less than a thousand symbols is uniquely true of it’ isn’t uniquely true of M ; that is, it isn’t uniquely true of the smallest natural number such that no formula of English with fewer than 1000 symbols is uniquely true of it. In other words, ² The logic advocated in Field 2008 is one such.
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3∗ . ‘x is the smallest natural number such that no formula of English with fewer than 1000 symbols is uniquely true of it’ either (i) isn’t true of the smallest natural number such that no formula of English with fewer than 1000 symbols is uniquely true of it, or (ii) is true of things other than the smallest natural number such that no formula of English with fewer than 1000 symbols is uniquely true of it. Either option is thoroughly counterintuitive, and a gross violation of the schema ( T ) ‘F (x)’ is true of c if and only if F (c). Horwich often emphasizes the centrality and importance of the truth-of schema, and for good reason. But we see that the unrestricted least number principle forces a violation of that schema. This is a substantial consideration in favor of restricting the least number principle in the context of vagueness, and hence in favor of restricting the law of excluded middle which underlies it. Horwich takes the opposite stance, of restricting not excluded middle but the truth-of schema. But that has a high cost. Let’s look at the point of the notions of truth and truth-of. Sticking to truth for simplicity: suppose I forget the details of what a doomsayer said yesterday, but remember the gist well enough to conclude: If everything he said yesterday is true, then we’re in trouble. On the assumption that what he said was p1 , . . . , pn , this had better be equivalent to If p1 and . . . and pn , then we’re in trouble. This requires the intersubstitutivity of True(
) with p in extensional contexts. Given the very minimal law p ↔ p, this yields the truth schema True(
)↔ p. ( The situation with ‘true of ’ is similar.) Restricting intersubstitutivity restricts the ability to generalize in a reasonable way, leading to extreme pathologies in theories of truth that reject intersubstitutivity or the truth schema. But we can keep the intersubstituvity principle and the truth schema unrestricted if we weaken excluded middle (and more or less equivalently, the least number principle); similarly for truth-of. We can do this in a way that allows for fully classical reasoning when no ‘ungrounded’ uses of ‘true’ are present (and when vagueness isn’t at issue): for instance, we can accept classical reasoning within mathematics without restriction. And the logics that keep the truth schema (and the intersubstitutivity of True(
) with p) seem to be fully suited to deal with vague and indeterminate concepts in the way sketched earlier. I think this is no accident: there’s a strong intuitive connection between the Sorites example (Russell is old) and the Berry paradox example. Moreover, the obstacles that must be overcome in getting a logic that adequately handles vagueness and the semantic paradoxes are pretty much the same in
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both cases: for instance, in both cases we must make sure that no determinacy operator collapses to bivalence when iterated. ( This is why the Lukasiewicz logic fails both as a logic for vagueness and as a logic for the semantic paradoxes.) That there is a connection here is of no surprise. Vague concepts and ‘true’ seem species of indeterminate concepts. ‘True’ initially seems determinate, because it seems that the truth schema True(
) ↔ p settles its extension. But once we reflect on ‘ungrounded’ sentences (such as Truthteller sentences, which assert their own truth; and Liar sentences, which assert their own untruth), we see that this is an illusion. This connection makes it natural to use the same logic for such ‘ungrounded’ applications of ‘true’ as for vague predicates. That there is a link between the semantic paradoxes and the paradoxes of vagueness is perhaps further suggested by another paradox of the same ilk, which seems to have ties to both. (I think I first heard of it many years ago in Martin Gardner’s Scientific American column.) Some natural numbers aren’t very interesting. So there must be a smallest one that isn’t very interesting. The smallest one that isn’t very interesting! What an interesting number! Contradiction. (In case anyone is tempted to regard this as a proof that every natural number is very interesting, it’s worth remarking that an analogous proof using the classically correct least ordinal principle yields that every ordinal number is very interesting. Since for any cardinal number c, there are more than c ordinal numbers, it seems quite surprising that interest can extend so far!) Another kind of paradox that suggests a connection is what Sorenson calls a ‘nono’ paradox: Person A asserts that what person B is saying is not true, at the same time that person B says that what person A says isn’t true. Classically, either what A says is true and what B says isn’t, or vice versa; and yet A and B seem symmetrically placed. (We might even imagine that A and B are Doppelgangers in a completely symmetric universe; in which case we have a failure of truth to supervene on non-semantic facts.) Intuitively this is a kind of underdetermination reminiscent of vagueness, and the paradox arises only from the supposition of excluded middle. To summarize, I think there is considerable pressure in the vagueness case to slightly weaken the logic so as to avoid postulating counterintuitive boundaries, and even more pressure in the semantic paradox case to weaken the logic in the same way to enable us to keep the truth and truth-of schemas. These two pressures to weaken the logic are, I think, mutually reinforcing, and succumbing to this joint pressure is not the desperate measure that Horwich suggests it is. Re f e re n c e s Field, Hartry (2008), Saving Truth from Paradox, Oxford University Press, Oxford. Horwich, Paul (2005), Reflections on Meaning, Oxford University Press, Oxford.
12 Perceptual Indiscriminability and the Concept of a Color Shade Leon Horsten
12.1
I N D I S C R I M I N A B I L I T Y A N D C O LO R S H A D E S
We shall be concerned with visual indiscriminability, and more specifically on perceptual indiscriminability of colors. But, as usual, the discussion is intended to carry some more generality. It intends to contain some lessons for the philosophical theory of perceptual qualia in general. According to the received view, perceptual indiscriminability is a nontransitive relation. Until about a decade ago, there was a nigh consensus in the philosophical literature that the received view is correct. Today, this consensus has dissolved. This has prompted me to make an attempt at reevaluating the question of the transitivity of perceptual indiscriminability. In the first sections of this chapter, recent challenges to the thesis of the nontransitivity of perceptual indiscriminability will be critically examined. Such challenges can take different forms. Some of them concentrate on philosophical arguments for the nontransitivity thesis; some concentrate on alleged empirical evidence for the claim that indiscriminability is nontransitive. Fara’s challenge belongs to the first kind. She has criticized Wright’s influential philosophical argument that was intended to establish that indiscriminability is nontransitive. We shall investigate to what extent Wright’s argumentation can be upheld in the face of Fara’s reply. Raffman has emphasized that the question of the transitivity of the relation of perceptual indiscriminability is ultimately empirical in nature. And she believes that the alleged empirical evidence in favor of the nontransitivity thesis is far from I am indebted to the participants in the Arch´e Vagueness Conference (St. Andrews, 2007) for insightful comments and helpful suggestions. My thanks go especially to Paul Egr´e (who commented on this chapter), Diana Raffman, Patrick Greenough, Crispin Wright, Nathan Salmon, and Anthony Everett. Section 12.3 and section 12.6 are based on earlier joint work with Rafael De Clercq, to whom I am also grateful for discussions specifically relating to the problems with which the present chapter is concerned. It should not be assumed that Rafael De Clercq agrees with the overall position that is articulated and defended in this chapter.
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conclusive. She argues that a contextualist interpretation of the data can be given which leaves ample room for the thesis that perceptual indiscriminability is a transitive relation after all. We shall see that whereas her arguments are not conclusive, there is much to be learned from her considerations. If perceptual indiscriminability is transitive, then at least a criterion of identity for color shades is readily obtained by Frege’s familiar method of abstraction. But if perceptual indiscriminability is a nontransitive relation after all, then we cannot rely on the method of abstraction to yield an identity criterion for color shades—at least not in any straightforward manner. The question then becomes acute whether the concept of a color shade is coherent in the first place, and, if so, which form it could take. It shall be argued that in this situation, it must be conceded that the concept of a color shade is to some extent theoretical in nature. But since the concept of a color shade is invoked in the first place to make sense of our color discrimination judgements, it must respect these as much as possible. We shall therefore inquire how a concept of a color shade can be obtained which is, in the face of nontransitivity of indiscriminability, maximally faithful to our color indiscriminability judgements. But faithfulness to indiscriminability judgements is not the only requirement. The resulting concept of a color shade should also be well in agreement with the way in which the concept of a color shade is used in natural language. So we must also ask ourselves whether this requirement, too, is satisfied. In what follows, I have in mind the notion of indiscriminability in color for a given agent which is kept fixed throughout the discussion. This agent is assumed to have capacities for discriminating between colors of objects that are or fairly average for a human being. Two objects are said to be indiscriminable in color by the agent if she cannot discriminate between them with respect to color on the basis of a direct color comparison. Thus the relation of discriminability with which we are operating coincides with what Goodman called the matching relation. Indiscriminability with respect to color does not exclude that there are other ways in which the agent can distinguish the colors of objects. Suppose, for instance, that there is a nontransitive triad of objects x, y, and z. On the basis of a direct color comparison, x cannot be discriminated from y, y cannot be discriminated from z, but x can be discriminated from z. Then our agent can reason that since x is discriminable from z whereas y is not, x and y must have a slightly different color. Still, in our sense of the word, x and y remain indiscriminable. In other words, the concept of indiscriminability as it will function in our discussion is assumed to be phenomenal. Indiscriminability will also be taken to be a judgemental relation. For two objects to be indiscriminable in color for our agent, she has to be able to judge them to be indiscriminable when she visually compares them. Likewise, for these objects to be discriminable for her, she has to be able to judge them to be discriminable on the basis of a visual comparison.
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W R I G H T ’ S N O N T R A N S I T I V I T Y A RG U M E N T
Perceptual indiscriminability is nontransitive if it is possible to have three items x, y, z such that x is perceptually indiscriminable from y, y is perceptually indiscriminable from z, and x is perceptually discriminable from z. This entails that precisely one of the following two theses must hold: The Transitivity Thesis Perceptual indiscriminability is a transitive relation. The Nontransitivity Thesis Perceptual indiscriminability is a nontransitive relation. Until fairly recently, it was common philosophical practice to accept the nontransitivity thesis (and thus to reject the transitivity thesis) on the strength that we can imagine a process of gradual change in which a series of unnoticeably small changes finally add up to a noticeable change (in respect of a given quality). Several authors have sought to show that perceptual indiscriminability is nontransitive in this way. Here we discuss an influential philosophical argument by Crispin Wright (Wright 1975, 345–7). Wright presents his proof as a reductio showing that the nontransitivity thesis follows from the possibility of phenomenal continua. The argument can be paraphrased as follows. Suppose that indiscriminability is transitive. Then consider a process of change in respect of some observable property (think of it as a determinable such as color, position or pitch). The process is composed of stages between which there is no seemingly abrupt transition, and is non-recurrent in that for two distinct stages x and y, with x preceding y, there is no later stage z such that z is more like x (in respect of the observable property) than y is. Take any two stages Di and Dj such that Dj is discriminable from Di and yet close enough to it to guarantee that all stages lying in between are either indiscriminable from Di or indiscriminable from Dj . In other words, the intermediate stages will appear to have the same determinate of the determinable as either of the two surrounding stages (e.g. the same shade of color). They cannot be indiscriminable from both Di and Dj since being-indiscriminable-from is supposed to be a transitive relation. As a result, the region between Di and Dj will divide into two adjacent sub-regions, one consisting of stages indiscriminable from Di , the other consisting of stages indiscriminable from Dj . Since indiscriminability is supposed to be transitive and since Di is discriminable from Dj any stage belonging to the first sub-region will likewise be discriminable from any stage belonging the second sub-region. However, if this is true, then, contrary to what we have been assuming, a seemingly abrupt change must occur between Di and Dj . In recent years, the nontransitivity thesis has been called into doubt. These challenges tend not to consist of arguments that directly support the thesis of the transitivity of indiscriminability. Rather, they consist in the first place of attempts to undermine the alleged evidence for the nontransitivity thesis. In the next section, I shall scrutinize Fara’s critique of Wright’s philosophical argument. In section 12.4, we shall turn to Raffman’s attempt to deflect the charge that we have conclusive empirical support for the nontransitivity thesis.
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U N N OT I C E D A P PA R E N T C H A N G E S
According to Fara, Wright’s proof relies on two assumptions: the possibility of phenomenal continua and the finiteness of human discriminatory powers (Fara 2001, 931). The first assumption is needed to deny the existence of a seemingly abrupt transition from one stage to another. The second assumption allows for perceptually indiscriminable stages in the process. According to Fara these two assumptions ‘are, taken individually, not implausible [but] they are in so much tension with each other that it is utterly unreasonable to accept them jointly when neither has anything remotely like adequate support’ (Fara 2001, 931). Closer inspection of Wright’s argument reveals that the first assumption, concerning the possibility of phenomenal continua, is not necessary for his argument. In fact, it is not hard to see that Wright’s argument, reduced to its essentials, is exceedingly simple. Aside from a plausible physical assumption, the finiteness of our powers of discrimination is all that is needed for the argument. Let there be given an observable physical quantity Q. Suppose that the value of this quantity can be expressed as a real number. (Thus, the quantity Q can be regarded as a determinable with specific values as determinates). And adopt the physical continuity assumption that the value of Q varies according to some smooth continuous function (in the mathematical sense of the word!) through time. Let ri refer to the value of quantity Q at time i. Now we assume finite discriminability in the sense that (i) there are ra , rb such that the subject can discriminate between them and (ii) there is a d ∈ R such that if ri − rj < d , then a given person is unable to perceptually discriminate between Q at i and Q at j. Now consider a finite chain ra = r0 , r1 , . . . , rn = rb , such that for each ri in the chain, ri+1 − ri < d . The foregoing assumptions entail that such a chain exists. Moreover, finite discriminability entails that a subject perceiving the chain will not notice ‘an abrupt change’, which means that the change in Q will be perceived as continuous in Wright’s (phenomenal) sense. Elementary mathematical considerations show immediately that this chain must contain a violation of transitivity of indiscriminability. After all, since each element in the chain is indiscriminable from the next with respect to Q, transitivity would imply that the first element is indiscriminable from the last. However, by assumption ra is discriminable from rb with respect to Q. So in the final analysis, all rests on the assumption of finite discriminability.¹ Fara does not find this assumption evident. In her discussion of the phenomenon of ‘slow motion’, she writes (Fara 2001, 928): . . . we have two competing explanations of what is going on when the hour-hand of a clock looks to have moved over some long [time] interval, but also seems to have looked still during every sufficiently short sub-interval. The first explanation is that when we judge the hour-hand to look still, say for every twenty-second period, it does in fact look to be in the same position at the end of each period as at the start. The alternative explanation is that when we judge the hour-hand to look still, although there is at least one twenty-second period for which it does ¹ One may wonder at this point why Fara believes that the phenomenal continuity assumption is in tension with the finite discriminability assumption. A diagnosis is offered in De Clercq and Horsten (2004, section II).
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not look in the same position at the end as at the start, we do not notice this. Noticing the change in an apparent position requires not only that there be an apparent change, but also that we believe there to be one. [emphasis in the original]
In other words, according to one explanation of what happens when the hour-hand of a clock changes unnoticeably, there is no apparent change because there does not appear to be a change: at least at a conscious level, things look exactly the same before and after the change. This explanation seems plausible enough. However, Fara’s sympathy lies with the other explanation: the apparent position of the hour-hand of a clock—the position it appears to have—changes constantly, i.e. even within time intervals that are so short that we are unable to tell (‘notice’) whether there has been a change. Neuro-psychological research has born out that we routinely respond to visual stimuli of which we have no conscious awareness.² In the situation that Fara describes, too, it may be that we are behaviorally able to respond in some way to the change of the position of the hour-hand in a twenty seconds period even though we are not consciously aware of a position change. So, in this sense, it might be said that the position change may not completely escape our attention even if it escapes our conscious attention. Elsewhere in her paper, Fara argues that accepting the nontransitivity of ‘looking the same as’ does insufficient justice to the phenomenal character of looks (Fara 2001, 932). After all, if ‘looking the same as’ is transitive, then looks can simply be taken to be equivalence classes of the relation; and if ‘looking the same as’ is nontransitive, then one must either maintain that there are things which look the same (in some respect) but nevertheless do not have the same look, or that there are things which look different but have the same look, or both. However, if this objection is justified, then it might be argued that Fara stands guilty of a similar charge. Fara’s concept of indiscriminability is in the end a subjudgemental and subconscious relation; it can fail to obtain in a comparison test without the agent being conscious that it fails to obtain. By separating the notion of apparent change from the notion of noticed change, Fara may then be said to deprive the notion of apparent change of its phenomenal nature. At best, the notion of apparent change becomes one that is determined by our (partially unconscious) response-behavior rather than by the contents of our explicit consciousness. Nothing can prevent Fara from abstracting a notion of color shade from such a behavioral relation. But this notion of color shade will fail to qualify as fully phenomenal. 12.4
A P PE A R I N G I N C O N T E X T
Raffman has articulated a contextualist position on the basis of which she criticizes Wright’s argument for the nontransitivity of indiscriminability (Raffman 2000). She ² See Weiskrantz (1986).
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is concerned with the relation of perceptual indiscriminability of objects, in respect of a perceptual property (Raffman 2000, 155). This should be interpreted loosely. It is intended to include situations where someone is asked to discriminate visual images or parts of visual images with respect to color. Raffman’s arguments are not intended to establish but only to make room for the hypothesis that perceptual indiscriminability is a transitive relation (Raffman 2000, 154–5). She says of her her argumentation that much of it is ‘speculative [...] and in need of empirical test’ (Raffman 2000, 155). The contextualist hypothesis can roughly be formulated in the following way: The Contextualist Hypothesis When objects x and y are mentally compared with respect to perceptual property P with the aim of reaching a discrimination judgement, contextual factors can and typically do influence the agent’s perception of x and of y. This hypothesis seems eminently plausible, and is supported by a large body of empirical data, which go under the rubric of ‘contrast effects’. The contextualist hypothesis is used by Raffman to undermine arguments for the nontransitivity of perceptual indiscriminability such as Wright’s as logical fallacies. Suppose first x and y are compared by someone with respect to color; then y and z are compared; and then x and z are compared. And suppose that the person’s discrimination judgements are no, no and yes, respectively, thereby forming a prima facie case for nontransitivity of the relation of perceptual indiscriminability. Then the contextualist will point out that, to a first approximation, in the first comparison, x constitutes the context of y, and in the second comparison, z constitutes the context of y. If the contextual hypothesis is correct, this causes the person’s perception of y in the first comparison to differ from the person’s perception of y in the second comparison. So one is not allowed to ‘carry over the middle term’ in the argument for nontransitivity (Raffman 2000, 161).³ In a contextualist framework, the transitivity thesis takes the following form. If objects are compared in view of a perceptual property P in such a way that there is no contextual disturbance of perception, indiscriminability is a transitive relation. According to Raffman, this hypothesis is not excluded by what is presently known. And we have seen that Raffman expresses the hope that empirical evidence might be brought to bear on it. It is clear that even in experiments set up to determine whether one participant can discriminate between the color of stimuli, her answers will display a statistical distribution. Hardin emphasized the importance of this phenomenon for the philosophical discussion about the (non-)transitivity of indiscriminability.⁴ It will happen that even for a single pair of color patches, she will sometimes answer that she can discriminate them, whereas at other times she will judge them to be indiscriminable in color. So it must be conceded at the outset that our assumption that an agent always makes ³ Schroer (2002) develops a similar, but less fully articulated, line of reasoning. ⁴ See Hardin (1988).
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the same discriminability judgement makes us guilty of grossly oversimplifying the situation. We shall have to come back to this: in section 12.6, the implications of retracting this assumption will be (all too briefly) discussed. Raffman considers possible counter-evidence to the contextualist version of the transitivity thesis. She considers circles, presented against a uniform background color, and divided into 3, 4, 5 or more equally large sectors, where adjacent sectors receive very similar colors. In particular, she considers such a circle divided into 3 sectors (Raffman 2000, 163ff). Participants are asked to compare the three sectors two by two with respect to color. The advantage of such a set-up is that the ‘context’ of y, for example, when compared to x is the same as when compared to z, for in each act of comparison the whole circle remains firmly in view of the participant.⁵ Raffman contends that for circles divided in three sectors, solid empirical evidence pointing in the direction of nontransitivity has not been forthcoming. It appears to be impossible to construct a circle divided into three sectors x, y and z of slightly different colors in such a way that the agent systematically judges x and y as well as y and z indiscriminable, but also systematically judges x and z discriminable. Here the statistical nature of the distribution of discriminability answers may play a substantial role. The statistical distribution is presumably just not sufficiently sharply peaked to come to a reliable conclusion in such a triad situation. This point is not without importance. It entails that reports such as the following just cannot be accepted without further ado: Suppose I focus on just patches 1–3 and claim sincerely to attend simultaneously to the colorappearances of all three patches. ([. . .] I can attest from my own case—as you can probably attest from yours—that we can [focus on three patches at a time].) At a given instant, I claim, patches 1 and 2 looks the same to me, as do patches 2 and 3, but patch 1 and 3 look different. (Mills 2002, 395)
At this point, it is important that a background condition for success is firmly kept in mind. This condition was highlighted by J. A. Burgess (Burgess 1990, 209), but it has not received much attention in the literature. For the experiment to succeed, x, y and z have to be perceived as uniformly colored. This forces us to design the experiment carefully. A first question is whether the three sectors are separated from each others by lines of a uniform color which clearly differ from the colors of x, y and z. And a second question is whether the three sectors really touch each other or, alternatively, are ‘pulled apart’ some distance against a uniform background. If the answer to both these questions is no, then it can legitimately be questioned whether the background condition is satisfied. When one tries to construct a pie consisting of three contiguous sectors that forms a counterexample to the transitivity hypothesis, the participants in the experiment have some tendency to see it as a phenomenal continuum. To the extent that that is the case, the participants will be hesitant to affirm that each of the three sectors are uniformly colored. This problem can be mitigated by pulling the sectors apart against a uniform background color. Schroer notes that the sectors should ⁵ Raffman admits that she is ‘helping [her]self to the notion of a visual context more or less unexamined’ (Raffman 2000, 159), but let us grant her that.
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not be pulled apart far (Schroer 2002, 265–7). If the sectors occupy parts of the visual field of the participant that are not very close to each other, then the participant cannot compare them directly. So on account of physiological limitations she will have to shift her focal visual attention from one sector to the other. And this entails that in her act of comparison, she has to rely on her memory, which is much less precise than occurrent appearances are. Let us assume in the sequel that the experiments are set up in such a way that the participants perceive the monochromatic colors as uniformly colored. Then one may ask: how could empirical evidence bear out that the nontransitivity hypothesis is false? In any actual mental comparison of two objects x and y with respect to P, representations of x and y are necessarily present in a special way. When a person compares x and y, she is focused on x and y in a manner that she is not focused on z. Suppose, perhaps contrary to the facts, that it were possible to construct a three-sector circle in such a way that it would reliably generate prima facie evidence for the nontransitivity thesis. Then the contextualist is free to conjecture that in experiments of this sort, concentrating on x and y (in a comparison of x and y) versus concentrating on y and z (in a comparison of y and z) makes a difference in the perception of y. Thus the prima facie evidence for nontransitivity would on closer inspection disqualify as genuine evidence for nontransitivity of the indiscriminability relation. Just this move is executed by Raffman when she considers variants of this experiment with circles divided in at least five sections (Raffman 2000, section III). A knee-jerk reaction to the difficulty of constructing a three-sector circle which generates prima facie evidence for the nontransitivity hypothesis is to divide the circle into more sectors—as many sectors as it takes! One might, for instance, divide the circle into five sectors such that sector 1 is very close in color to sector 2, which is in turn very close in color to sector 3, and so on, until one arrives at sector 5, which is not very close in color to its neighboring sector 1. Raffman thinks that on closer inspection, even prima facie evidence of this kind does not refute the transitivity hypothesis (Raffman 2000, 169): . . . it may be that in any series whose adjacent members are indiscriminable (would be judged the same in a same / difference comparison), [. . .] at any given time at least one member looks different in its two hypothetical comparisons with adjacent items, even in the case the entire series is viewed simultaneously.
But there is no reason why a contextualist could not take this line even concerning circles divided into three sectors. In fact, this kind of move is always open to the contextualist. It even provides a scheme for reacting to diachronic comparison experiments. Suppose we modify the experiment so that at a given time, the whole visual field is occupied by exactly one monochromatic paint chip, and the agent is asked whether she can discriminate the color presently in her field of view from the color she saw two seconds ago. Even though the color sample is now not presented against a visual background, it is presented against a ‘visual memory context’. Presumably visual memory contexts can influence the content of a color perception in a way similar to the way in which visual contexts influence colors perception of objects.
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Raffman’s attempt to give hard empirical content to the contextualist defense against the prima facie five-sector counterevidence against the transitivity hypothesis is unsuccessful. She suggests the following diachronic comparison experiment (Raffman 2000, 167): Suppose you are given a new task, involving just the pairs #3 / #4 and #4 / #5: you are to attend first to #3 and #4, then to #4 and #5, this time with the aim of judging whether #4 looks the same or different in the two pairings—in the two attendings, one might say. In other words, rather than making two comparisons, of #3 to #4 and #4 to #5 as in the serial scenario, your new task is to perform a single, cross-contextual comparison. As I will put it, your task is to compare #4 viewed in an act of focused attention to #3 and #4, with #4 viewed in an act of focused attention to #4 and #5. First you attend to the pair #3 / #4, then you shift your attention to #4 / #5, keeping #4 in view continuously, and you make a same / difference judgement of #4. Isn’t it possible that #4 should look different?
It is not clear that Raffman’s diachronic comparison experiment is well-conceived in the first place. The memory-image of #4 may not be sharp enough for an average participant to compare with the occurrent experience of #4 (Wright 1975, 336), (Schroer 2002, 267). But even if memory-images are sufficiently crisp for the task, there is no reason why the contextualist should feel cramped by consistent responses that in the two comparison acts #4 looks the same, if such responses were forthcoming. She is free to conjecture that after the second comparison task, the memory of #4 in the first comparison task is influenced by the subsequent focus on #5. This is after all, for all we know, possible. And if it is the case, then ‘no difference’-responses to the diachronic task have little import for the transitivity hypothesis. At this point it becomes evident that conclusively refuting the contextualist defense of the transitivity hypothesis on the basis of empirical evidence, is impossible. This is just a consequence of the fact that was noted earlier, namely that in an explicit color comparison between two items x and y, these items are present in a special manner; they are present in a way that is unlike the way in which features of the context are present in consciousness. This always leave the contextualist free to deny that it is permissible to ‘carry over the middle term’. One might say that the situation with the transitivity hypothesis is no different from that with scientific hypotheses in general. One may hold as fast to them as one likes. But there is a point at which rescue missions begin to look somewhat ad hoc. It seems that if empirical circumstances would force Raffman to ascribe contextual effects to focusing acts, i.e. when empirical evidence would force her to question whether in the five-sector circle experiment #4 looks the same when compared to #3 as when compared to #5, this point has been reached. For all that has been said, the transitivity hypothesis may still be correct. It may be correct even in the face of the sort of hypothetical prima facie empirical counterevidence that we have discussed. Still, as Raffman has emphasized, it remains at least in part an empirical matter. For it may well be that the empirical data look much more favorably upon the contextualist theory than in the pessimistic scenarios that we have been considering.
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Raffman was right to insist that it is at least to some extent an empirical question whether perceptual indiscriminability is a transitive notion. But we have seen that it is not as easy to test the transitivity hypothesis as it appears at first sight. If we want to empirically determine color shades, how should we proceed? What are the possible outcomes of such an investigation? And what is their significance? As mentioned before, color shades are supposed to make sense of our sense experience. But the discussion of Fara’s position entails that we ought to be more precise. We must decide whether we want color shades to make sense only of what we are consciously aware of, or whether it should also make sense of what we are unconsciously or behaviorally aware of. This is purely a matter of decision. But it appears best in line with the historical development of this enterprise if we restrict ourselves to conscious sense experience, which the agent is able to make explicit in discriminability judgements. From a contextualist point of view, the situation then looks as follows. Contextual factors typically influence indiscriminability judgements. So when indiscriminability judgements are combined in arguments, the context must be kept as uniform as possible. Raffman advises us to keep the context constant (as much as possible) throughout the indiscriminability experiment by making sure that all the stimuli remain clearly within the visual field of the participant throughout the successive comparison tasks. For concreteness, let us suppose that the participants are presented with sectors of a circle which are slightly differently colored. To ensure that the sectors are perceived as uniformly colored, they are pulled apart a bit. And the sectors are presented against a uniformly colored background. To conclude, let us suppose that we are working with a large but finite number of color stimuli (‘paint chips’). In this way, we obtain a total indiscriminability graph in a fairly straightforward way. But on closer inspection even this much is far from clear, for the following reason. Surely there will be a maximum on the number of sectors that can be presented in one experiment (Raffman 2000, 170). And it seems likely that this number will be less than the number of color stimuli that are of a slightly different color from which we want to abstract the color shades. Hence not all distinct stimuli can be present in one run of the experiment. Instead, many runs will have to be done with many sextuples (say) of colored sectors. Even if the transitivity hypothesis can explain each run of the experiment individually, the question remains how the results of the different runs of the experiment should be patched together. It will not be sufficient to assume without further ado that if one stimulus is present as a sector of the circle in different runs of the experiment, the appearance in one run of it is the same as the appearance of it in another run. After all, the same stimulus was offered in different contexts, and the contextualist hypothesis will predict that contextual differences typically change the appearance of the stimulus. In sum, it is not at all clear how the results of the sequence of experiments can be taken to give rise to one global graph on the class of stimuli. It cannot be excluded
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that the simultaneous focal presence of one color may influence the appearance of the other—this is unavoidable. And this would mean that the contextualist may still cry foul: it may be that it is impermissible to combine the individual discrimination judgements into a discriminability graph. But from an operational point of view, this appears the best we can do. We shall not pursue this problem further. Instead we shall for the sake of argument assume that it can be dealt with so that in the end, an indiscriminability graph is obtained. So let us suppose that an experiment is set up and carried out in this way. Then there are in general two possible outcomes. We shall examine them in turn. First, it may turn out that the empirical data can be explained very well by the transitivity hypothesis in the framework of a version of contextualism which does not appear ad hoc. Then, it seems, color shades can be abstracted in the familiar way from explicit indiscriminability judgements that are corrected as much as possible for contextual effects. Abstraction does not yet tell us what the nature of color shades consist in. But it does give us a criterion of identity for color shades: The Innocent Criterion The color shade of paint chip x is numerically the same as the color shade of paint chip y if and only if x and y are perceptually indiscriminable with respect to color. In these circumstances, and given the supposition that the transitivity hypothesis can be upheld in the light of the data, there will be sharp cutoff points between color shades. A marked advantage of the innocent criterion is that it ensures that color shades harmonize perfectly with the indiscriminability relation. Color shades are admitted into our ontology in the first place in order to make sense of our visual experiences. So if we can best make sense of the discrimination experiment on the basis of a global transitive graph on the stimuli, then we are in the fortunate situation that the abstracted collection of color shades will be completely faithful to our (corrected) discrimination judgements. One may wonder whether a theoretical element has nevertheless entered into the contextualist defense of the transitivity hypothesis. After all, on this account there are no ‘pure’ indiscriminability judgements; it is still the case that all we have are objects that are indiscriminable in a context. But this appears unproblematic. It just seems hard to deny that context is a parameter that plays a role in the discriminability of objects with respect to color—even though until recently philosophers have hardly recognized it. A second possible outcome is that empirical evidence makes it hard even for a contextualist to maintain the transitivity hypothesis without resorting to seemingly ad hoc maneuvers. Then it is impossible to abstract color shades from indiscriminability judgements in the familiar way. Either some patches that cannot be distinguished in color will be judged to have different color shades, or some patches that are discriminable in color will be judged to have the same color shade, or both.
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This is what Fara finds objectionable (Fara 2001, 909). In his earlier writings, Wright also deemed this unacceptable (Wright 1975, 352).⁶ Color shades (‘looks’) were supposed to make sense of our sense experience first and foremost. If the concept of a color shade does not fully respect our indiscriminability judgements, then it is not a phenomenal concept. Wright went so far as to say that in such a situation we have a transcendental concept (Wright 1975, 357). Yet what are our options, in the situation under consideration? We have access to color shades only via color comparisons. In this sense, indiscriminability judgements are prior to the color shades themselves. First, one can deny that color shades (or sense data in general) exist. This is not an uncommon stance. Armstrong takes this position (Armstrong 1961). Fara does not explore this option; she merely mentions in passing the possibility to give up on ‘looks’ altogether (Fara 2001, 916, fn 13). Raffman does not embrace this position either, although she is careful not to rely anywhere in her argumentation on the assumption that color shades exist (Raffman 2000, 160–1). Perhaps from a contextualist point of view denying the existence of shades should appear as an attractive option. For would it not be natural to say, from a contextualist perspective, that color shades (or ‘looks’) are essentially relative to a context? This would mean that it makes sense to say that in a given visual context in which two objects x and y are present, the color shade of x is the same as the color shade of y. But it does not make sense to ask whether the color shade of an object z presented in one visual context is the same as the color shade of u presented in another visual context. (As far as I am aware, the prospects of this position have not yet been fully explored in the literature.) A second option consists in conceding that the concept of a color shade is to some extent a theoretical concept. This line was taken by J. A. Burgess (Burgess 1990, 218–19):⁷ . . . theories that respect the phenomenology of perception are just that: theories. This means that they might not only be required to postulate properties that are (in some sense) not presented in experience in order to do justice to some facts of experience [. . .]; it also means that they might need to discard some apparent data as illusory.
The concept of a color shade is not clearly a philosophers’ concept, but it is also not clearly a pre-theoretical concept that is as pure as driven snow. It appears to be a low level theoretical concept. This should not be taken to contradict the fact that it is intimately tied to experience. Indeed, it can and should be upheld that its aim remains first and foremost to make sense of our indiscriminability judgements. Whether it is fruitful to develop a theoretical concept of color shade depends on what use it can be put to. In ordinary language, color shades play an important communicative and judgemental role. We routinely make judgements about color shades, for instance when we say ‘this shade is the same as that shade’ while pointing at parts of the surfaces of two objects. If we develop a theoretical concept of color shade that respects indiscriminability judgements as much as possible, we may hope to validate ⁶ He has since then sought to qualify his position on this issue. See below, section 12.7. ⁷ Linsky defends a similar position (Linsky 1984).
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many of the judgements of this kind. In communicating our visual experiences to others, we use the concept of a color shade. We can construct a theoretical model of what such color shades could be. In the quotation above, Burgess appears to take a realist view of color shades. Such a view is certainly tenable, but it is not forced upon us. One can also take color shades to be useful theoretical fictions. Which of these two stances is more appropriate is, as in all forms that the realism debate takes, difficult to adjudicate. 12.6
A P P ROX I M AT I N G B Y E QU I VA L E N C E R E L AT I O N S
Let us proceed on the assumption that we want to recognize the existence of color shades in our ontology. More specifically, let us assume that we want to find a criterion of identity for color shades in the face of apparent nontransitivity of the indiscriminability relation. Then we are faced with two requirements that are pulling in opposite directions. On the one hand, we want color shades to respect the indiscriminability relation as much as possible. On the other hand, we need an equivalence relation to base our abstraction on. In this predicament, we should try to strike a good compromise. Being phenomenal or observational is a matter of degree in this context, and should not be taken to be a matter of principle. We should look for an equivalence relation that is somehow as close as possible to the nontransitive indiscriminability relation. From such an equivalence relation, a criterion of identity for color shades can be obtained. The advocates of color shades in the light of nontransitivity have been aware of this task. But some of them have underestimated the subtlety of the problem. A first proposal was made by Nelson Goodman (Goodman 1966, ch. IX). He thinks that the innocent criterion should be replaced by (roughly) the following criterion:⁸ Goodman’s Criterion The color shade of paint chip x is numerically identical with the color shade of paint chip y if and only if for every paint chip z, x is perceptually indiscriminable with respect to color from z if and only if y is perceptually indiscriminable with respect to color from z. Even if indiscriminability is nontransitive, the right-hand-side of Goodman’s identity criterion is an equivalence relation. If Goodman’s criterion is correct then discriminability of paint chips x and y is a sufficient, but not a necessary condition for the paint chips to be of a different color shade. So the effect of Goodman’s criterion is to loosen the tie of color shades with indiscriminability in exchange for having an equivalence relation. Goodman’s criterion will result in what Wright calls Goodman Shades. Wright has observed that under fairly general circumstances, Goodman Shades will be very finely grained (Wright 1975, 354). Indeed, in many circumstances, each pair of paint chips ⁸ Burgess (1990) also defends a version of Goodman’s criterion. Linsky (1984) shows sympathy for it, but he has reservations and does not quite endorse it. See Linsky (1984, 379).
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will be judged by the criterion to be of different color shades. This restricts the usefulness of Goodman’s criterion. For surely the color shades that we attempt to communicate and quantify over in ordinary language are not so fine grained. Williamson has proposed methods for finding equivalence relations which are much closer to the nontransitive indiscriminability relation (Williamson 1986, 1990). We are given an indiscriminability relation G which is reflexive and symmetrical but nontransitive. Our task is to approximate G by an equivalence relation. We can approximate G from above by taking the transitive closure of G. Or we can approximate G from below by considering maximal equivalence relations G − ⊂ G. Either way, we obtain equivalence relations that are in a qualitative sense close to G: we call these qualitatively best equivalence-approximations. In the present context, the approximation from above does not make sense. For if the indiscriminability graph is connected (as in most cases it will be) then the transitive closure is the total graph on the underlying collection of paint chips. And that would mean that we have only one color shade. In other words, compared to Goodman’s criterion we would have landed in the other extreme. Qualitatively best equivalence-approximations from below seem more promising. They are of the same kind as Goodman Shades: they ensure that discriminability is a sufficient but not in general a necessary condition for having different color shades. The difference is that qualitatively best equivalence-approximations are more coarsely grained than Goodman Shades. Consequently, qualitatively best equivalence-approximations from below are in general more faithful to the indiscriminability relation than Goodman Shades. One drawback of qualitatively best equivalence-approximations from below is that they are in general not unique. For all nontransitive indiscriminability graphs, there exist more than one best equivalence-approximations from below. This problem can be mitigated by choosing the most coarse grained best equivalence-approximation from below. But even imposing this as an extra requirement does not always ensure that there is a unique best approximation. This entails that a conventional element is inherent in the proposal of choosing a qualitatively best equivalenceapproximation as the basis for an identity criterion for color shades. Sometimes all the best equivalence-approximations are equivalent to each other up to a simple transformation. In such cases, the conventional element does not play a deep role in the theory of color shades. But there are indiscriminability graphs for which the equivalence-approximations are not equivalent up to a simple scaling factor. In such situations, the conventional component plays a deeper role. Williamson’s methods can be somewhat improved upon by taking a quantitative point of view (De Clercq and Horsten 2005).⁹ Our task again is to approximate the indiscriminability graph as closely as possible by a transitive graph. Taking a quantitative view of the matter, we consider those transitive graphs that result from G by cutting and/or pasting a minimum number of edges from/in G. Equivalence ⁹ The method that we are about to discuss can be fairly straightforwardly extended to infinite domains. See again De Clercq and Horsten (2005).
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approximations in this sense are quantitative approximations. It is easy to show that often the quantitatively closest approximation to G results from cutting and pasting edges. In other words, often a closest equivalence approximation is one that partially overlaps with G. It is also clear from this that quantitatively best approximations that only cut edges are qualitatively best approximations from below (in Williamson’s sense). Nontransitive indiscriminability graphs are more likely to have a unique quantitatively best equivalence-approximation from below than to have a unique qualitatively best equivalence-approximation from below. But uniqueness can still not be guaranteed in all circumstances. So the conventional element will play a smaller role, but it cannot be guaranteed that it plays no role whatsoever. But in general, quantitatively best equivalence-approximation will be more faithful to the indiscriminability relation than qualitatively best equivalence-approximations. So quantitatively best equivalence-approximations can be said to be more ‘phenomenal’ than Williamson’s qualitative approximations. This may be an appropriate place to illustrate the differences between the different strategies for abstracting color shades from nontransitive relations. As a basis for this illustration, we use the indiscriminability graph G = V , E = {e1 , e2 , e3 , e4 }, {e1 e2 , e2 e3 , e3 e4 } . It scarcely needs to be mentioned that this is a highly simplified indiscriminability graph. In an actual experiment, we shall want a much larger domain of paint chips. And it is very unlikely that the associated indiscriminability graph will be serial in the way that G is. If we follow Goodman’s criterion, then there is a one-to-one correspondence between the paint chips and the shades, for the transitive graph E g to which it gives rise is the totally unconnected graph. Since G is connected, the qualitatively best equivalence-approximation from above E + is the total graph. There are two qualitatively best equivalence-approximations E1− , E2− from below, of which the edges are, respectively: e2 e3 ; e1 e2 , e3 e4 . We see that according to E1− there are three color shades ({e1 }, {e2 , e3 }, {e4 }) whereas according to E2− there are only two ({e1 , e2 }, {e3 , e4 }). So if we want the most course grained qualitative approximation from below, we must choose E2− . Also, we see that from a quantitative point of view there is only one best approximation: E2− . The latter makes only one ‘mistake’ against G, whereas E1− makes two mistakes and E g and E + contain even more discrepancies with G. One advantage of the innocent criterion and of Goodman’s criterion is that they specify a rule for deciding, at least in principle, whether two paint chips are of the same color shade. On finite domains, best equivalence-approximations (both in the qualitative and in the quantitative sense) are of course decidable in principle. But
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quantitatively best equivalence-relations even on a decidable infinite nontransitive graph need not be decidable at all. And even in the finite situations to which we confine our discussion, best equivalence-approximations do not give us a rule in any natural sense of the word for deciding equality of color shade for paint chips. Also, and related to this, most discriminability graphs are such that their best equivalenceapproximations cannot be succinctly expressed in natural language. For some indiscriminability graphs, the quantitatively best equivalence-approximation both cuts and pastes edges.¹⁰ In other words, sometimes the quantitatively best equivalence-approximation is an equivalence relation that partially overlaps the original graph. When closeness to the agent’s discriminability judgements is our primary consideration, it seems that one should in such a situation opt for such an overlapping approximation. But this would entail that we count some patches that are discriminable in color as having the same shade of color. Surely many participants in the debate will find this objectionable. But is it really? Are we not sometimes willing to count certain objects as having the same shade even if we can discriminate them in color? Don’t we sometimes say ‘yes, I see a slight difference between them, but they are the same shade’? So if fitting common language usage is an important consideration, then it is not clear that our concept of color shade is not permitted to function in this way. Admittedly such considerations fall short of being decisive. Perhaps in such cases what is meant is that the relevant objects are of roughly the same shade. So we shall in the sequel refrain from challenging the thesis that it is part of the meaning of the concept of a color shade that discriminability entails difference in color shade. As mentioned before, indiscriminability with respect to color is in practice not an all or nothing affair. Participants are typically more sure of some of their indiscriminability judgements than of others. This can be captured by assigning weights to the edges in the discriminability graph (De Clercq and Horsten 2005, 388). At the very least we should allow participants to opt out of some indiscriminability judgements. This would be captured by setting the set of weights equal to {0, 0.5, 1}, where an edge with weight 0.5 corresponds to a pair of paint chips for which the participant is unsure whether they are indiscriminable with respect to color. Assigning weights to edges of course limits the scope of the non-uniqueness problem even further. But it will still not dissolve the problem completely. So what should we do when we have a weighted indiscriminability graph for which there exist two or more quantitatively best equivalence-approximations? As the reader will expect, familiar maneuvers present themselves. For instance, it is possible to maintain that two paint chips are only of the same (a different) color shade if the corresponding edge belongs to all (no) quantitatively best equivalence-approximation. Where some equivalence-approximations disagree, we could say that there is no matter of fact whether the relevant paint chips are of the same color shade. Suppose that we have a global nontransitive indiscriminability graph of which there are more than one quantitatively best equivalence-approximations. Then how do we individuate color shades? ¹⁰ A simple example is given in De Clercq and Horsten (2005, 377).
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One option is to take the stance that color shades are low-level theoretical constructs seriously and simply abstract them from one particular quantitatively best equivalence-approximation—perhaps from one that is particularly easy to describe. Another option would be to argue that the ‘belonging to’ relation between color stimuli and color shades is to some extent vague. A supervaluation idea could be applied in the following way. Take any two color stimuli a and b from the global indiscriminability graph. Say that a and b (determinately) belong to the same color shade if a and b are indiscriminable according to all quantitatively best equivalence-approximations. a and b (determinately) do not belong to the same color shade if a and b are discriminable according to all quantitatively best equivalenceapproximations. And in the remaining case, it is indeterminate whether a and b belong to the same color shade. If one wants to remain as close as possible to the way in which the concept of a color concept functions in natural language, then the latter might look like the best option. But this position posits vagueness in the world, and arguments have been formulated which purport to show that the only vagueness that can exist is linguistic vagueness (Evans 1978). But in the present situation the vagueness involved—if there is any—cannot be easily shifted to the meaning of expressions. The reason is that natural languages do not contain simple names for all color shades. Perhaps natural languages do not even for every color shade contain a definite description that singles it out.¹¹ So if one is persuaded by Evans’ arguments against worldly vagueness, then it is not easy to see in which way an appeal to vagueness can solve the non-uniqueness problem. 12.7
C O LO R S H A D E S A N D N AT U R A L L A N G UAG E SEMANTICS
From the foregoing we may conclude that the question whether indiscriminability relation is a transitive relation, is at least to some extent an empirical one. But to some extent, it is also a theoretical one: the answer depends to some extent on theoretical interpretations of the evidence. Of course, it is often that way in empirical science. If in the final analysis indiscriminability turns out to be transitive, then a concept of a color shade can be defined that is completely faithful to the indiscriminability judgements. If indiscriminability is not a transitive relation, then it is futile to look for a (non-contextual) concept of color shade that completely respects the relation of indiscriminability. But in these circumstances it is still open to us to define a concept of color shade that is as faithful as possible to our indiscriminability judgements. This would result in a concept of color shades that falls short of being completely phenomenal, even though it will be as phenomenal as possible. As adumbrated before, for Wright an important theoretical function of (in his view phenomenal) concepts such as that of a color shade consists in assisting us in ¹¹ Cf. the next section.
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constructing a semantics for natural language. A concept of color shades abstracted from a best equivalence-approximation to our indiscriminability judgements could fit this bill. Hardin doubts that color shades modeled on the basis of the indiscriminability relation are of much direct relevance for natural language semantics. The reason is that natural languages have names for only a few of them (Hardin 1988, 226–7). But considerations such as these should not be taken as decisive objections against Wright’s project. Aside from color predicates, English contains the concept of a color shade. And this concept is not so easily dispensable. Many statements containing the term ‘color shade’ can be paraphrased by statements that contain the concept of indiscriminability instead. Thus, instead of saying These two walls are painted in the same shade of orange, we can equivalently say: These two orange walls are indiscriminable with respect to color. But the paraphrases of many statements containing the expression ‘color shade’ sound awkward. And for some statements containing the expression ‘color shade’ it is difficult to see how they can be paraphrased in terms of the concept of indiscriminability at all. An example of such a statement may be: There are more shades of colors than we are able to produce samples of in our laboratory.
So it is not excluded that a concept of a color shade developed along the lines that have been outlined in this article may be of value for natural language semantics. Such a concept of color shades might count certain paint chips that are indiscriminable with respect to color as being of a different color shade. In this sense, the concept of a color shade coheres imperfectly with indiscriminability judgements. But this is only problematic if indiscriminability somehow gives the rule by which identity judgements of color shades are made. Already in his (Wright 1975), Wright thought that that this is precisely the assumption that should be given up anyway. Our semantic competence does not consist exhaustively in using expressions according to implicit rules. In his more recent publications, Wright has connected this with a softening of his stance on the observability of color predicates (Wright 1987, 246): ‘How could the appropriate kind of sensitivity operate selectively among indiscriminabilia?’ [. . .] it should now seem as if this question has rather disappeared. ‘Looks red’ ought certainly to qualify as observational on this count. But the kind of sensitivity to appearance which someone who understands ‘looks red’ [. . .] must have does operate selectively among items which, in respect to apparent color, cannot be told apart. [. . .] The suggestion that there is some kind of tension between such selectivity of response and its being the sort of response appropriate in the case of an observable predicate, depends on the thought that it cannot then be purely in response to appearance—either it is unprincipled or it is a principled response to more than the appearances. But ‘unprincipled’ here just means: not guided by rules correlating responses with appearances. So we should embrace the first alternative: such responses are indeed unprincipled, and no less appropriate or less purely ‘to’ appearances on that account.
These remarks are in perfect agreement with the theory that proposes identity criteria for color shades in terms of best equivalence-approximations to the indiscriminability relation. An identity criterion for color shades expressed in terms of equivalenceapproximations points away from the picture of indiscriminability furnishing the
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ingredients of a compact rule that language users are implicitly guided by when forming judgements of identity and difference of color shades. Indeed, we do not even have succinct expressions in natural language for most of these equivalenceapproximations. But in case indiscriminability is nontransitive, best equivalenceapproximations give us identity criteria for entities that cohere best, albeit not perfectly, with our indiscriminability judgements. Re f e re n c e s Armstrong, D. (1961), A Materialist Theory of Mind, Routledge and Kegan Paul. Burgess, J. A. (1990), ‘Phenomenal qualities and the nontransitivity of matching,’ Australasian Journal of Philosophy 68, 206–20. De Clercq, R. and Horsten, L. (2004), ‘Perceptual indiscriminability: In defense of Wright’s proof,’ Philosophical Quarterly 54, 439–44. (2005), ‘Closer,’ Synthese 146(2005), 371–93. Evans, G. (1978), ‘Can there be vague objects?’ Analysis 38, 208. Fara, D. G. (2001), ‘Phenomenal continua and the sorites,’ Mind 110, 905–35. Goodman, N. (1966), The Structure of Appearance, 2nd edition, Bobbs-Merill, 1966. Hardin, C. L. (1988), ‘Phenomenal colors and sorites,’ Noˆus 22, 213–34. Linsky, B. (1984), ‘Phenomenal qualities and the identity of indiscernibles,’ Synthese 59, 363–80. Mills, E. (2002), ‘Fallibility and the phenomenal sorites,’ Noˆus 36, 384–407. Raffman, D. (2000), ‘Is perceptual indiscriminability nontransitive?’ Philosophical Topics 28, Vagueness, ed. Christopher Hill, 153–75. Schroer, R. (2002), ‘Matching sensible qualities: A skeleton in the closet for representationalism,’ Philosophical Studies 107, 259–73. Weiskrantz, L. (1986), Blindsight: A Case Study and its Implications, Oxford University Press. Williamson, T. (1986), ‘Criteria of identity and the axiom of choice,’ Journal of Philosophy 83, 380–94. (1990), Identity and Discrimination, Basil Blackwell. Wright, C. (1975), ‘On the coherence of vague predicates,’ Synthese 30, 325–65. (1987), ‘Further reflections on the sorites paradox,’ reprinted in R. Keefe and P. Smith (eds.), Vagueness: A Reader, MIT Press, 1997, 204–50.
13 The Sorites, Linguistic Preconceptions, and the Dual Picture of Vagueness Mario G´omez-Torrente
Following roughly a formulation of Delia Graff Fara, we may say that we have an instance of the sorites paradox when, in a particular occasion of use, we are confronted¹ with a group of sentences of the following form,² each of which seems highly compelling in that occasion of use: (A) (∃x1 ) . . . (∃xn )([Kx1 & Kx2 & . . . & Kxn &]Rax1 & Rx1 x2 & . . . & Rxn−1 xn & Rxn z); (B) [Ka &] Fa; (C) (∀x)(∀y)([Kx & Ky ⊃](Fx & Rxy ⊃ Fy)); (D) [Kz &] ∼ Fz (cf. Fara 2000, 49f ). The brackets indicate that the bracketed parts will appear in some instances of the sorites paradox (those involving a comparison class; see below) but will not appear in others. Here ‘F’ is to be replaced with the sorites susceptible predicate, ‘a’ with a name of a case of application of the predicate which is intuitively clear in the occasion of use, ‘z’ with a name of a case of negative application which is intuitively clear in the occasion of use, and ‘R’ with a name of some binary relation. The occasion of use is understood as containing some factors that supply at least a universe of discourse (and range for the quantifiers) which includes the things named by ‘a’ and ‘z’. In the case of at least some sorites susceptible predicates (such as ‘is small’ or ‘is expensive’), interpretation seems to require the occasion of use to supply a comparison class, which may coincide with or be properly included in the universe of discourse. We may view the comparison class as the interpretation of ‘K’ in the bracketed parts (even though sometimes there may not be a predicate naming the I am indebted to many people for their comments. Richard Dietz, Manuel Garc´ıa-Carpintero, Sven Rosenkranz, Timothy Williamson, and audiences at the UNAM, the University of Barcelona, the University of Saint Andrews, the University of Lisbon, and the Institut Jean Nicod in Paris deserve special gratitude. Research was supported by DGAPA (IN 401909–3) and by the Spanish MICINN (FF 12008–04263). ¹ The sentences need not be physically uttered for the paradox to arise, but at least a mental utterance of some sort will occur if the paradox is to be considered by a thinker at all. ² There are other versions of the sorites; the basic considerations of this chapter will apply to them without substantive changes.
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comparison class, which may be left implicit). In these cases interpretation also seems to require the occasion of use to provide further standards of some sort for the application of ‘F’ and its relationship with ‘R’, which are partly responsible for the fact that (B), (C) and (D) are compelling in the occasion of use. And there may be other things that full interpretation requires the occasion of use to provide. The paradox is that every set of fully interpreted utterances of sentences of the form of (A), (B), (C) and (D) (whether we include the bracketed parts or not) is inconsistent according to classical semantics and logic, and yet many such sets are highly compelling in their corresponding occasions of use.³ For example, ‘F’ may be replaced with ‘is small’, ‘R’ with ‘has a population of 1 inhabitant less than’, ‘a’ with ‘Smalltown’, the name of a town with just 100 inhabitants, and ‘z’ with ‘Nonsmalltown’, the name of a town with 49,900 inhabitants; the relevant comparison class (the interpretation of ‘K’) may be taken to be the set of towns in the world that at present have 50,000 inhabitants or less, and the universe of discourse may be any set that includes that set of towns. Then (Bsmall ), (Csmall ) and (Dsmall ) are all highly compelling under normal standards,⁴ and we may also suppose for the sake of the example that we know (Asmall ) to be true as a matter of fact: (Asmall ) (∃x1 ) . . . (∃x49,799 )(Kx1 & . . . & Kx49,799 & Smalltown has a population of 1 inhabitant less than x1 & x1 has a population of 1 inhabitant less than x2 & . . . & x49,798 has a population of 1 inhabitant less than x49,799 & x49,799 has a population of 1 inhabitant less than Nonsmalltown); (Bsmall ) K(Smalltown) & Smalltown is small; (Csmall ) (∀x)(∀y)(Kx & Ky ⊃ (x is small & x has a population of 1 inhabitant less than y ⊃ y is small)); (Dsmall ) K(Nonsmalltown) & ∼ Nonsmalltown is small. Assuming that the truth of (Asmall ) is not in dispute, a solution of the paradox must convince us that one or more of (Bsmall ), (Csmall ), and (Dsmall ) is not true, or that classical logic or semantics do not apply. The same holds, of course, of every highly compelling set of fully interpreted utterances of sentences of the form of (A), (B), (C), and (D). Not all predicates that have been held to be sorites susceptible seem sorites susceptible for the same reasons. Many (though not all) sorites susceptible degree adjectival predicates (such as ‘is small’ or ‘is expensive’) give rise to very compelling (A)–(D) sets, as in the example above. Many predicates whose predicative element is a scalar noun (such as ‘is a heap’ or ‘is a youngster’)⁵ do not seem substantively different in ³ The use of (semi-)formalized language is not essential to the formulation and existence of the paradox; it just helps make somewhat clearer its formulation and its relevance for classical semantics and logic. ⁴ But not under all standards. For example, it is imaginable that we can work with a standard under which even Smalltown counts as not small, and only some very small towns of under fifty inhabitants count as small. (See related remarks in Fara 2000, 65; see also section 13.2 below.) ⁵ I understand a scalar noun as one that has an analytically associated dimension of comparison, usually also analytically connected with a degree adjective—a heap, on the acceptation that I take to be relevant to the sorites discussion, is a big pile of suitable things lying one on another.
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this respect. However, other predicates, and in particular some whose predicative element is a non-scalar noun, e.g. a noun for a typical natural kind (such as ‘is a dog’) do not seem to give rise to (A)–(D) sets that are compelling for the same reasons. And the situation with other predicates may not be subsumable under any of these two usually discussed types. In this chapter I will sketch a picture of the workings of sorites susceptible predicates in English and similar natural languages—including an outline of a solution of the sorites paradox—that is especially applicable to many adjectives and scalar nouns. The picture is applicable also to other predicative words and phrases, but lack of space will prevent extended discussion of this topic here (see footnote 28 and surrounding text). The picture is a dual picture, because it is based on a division of occasions of use of a sorites susceptible predicate into regular and irregular, according as to whether the predicate has a reference (extension) in the occasion of use or not. It is also based on two distinct sub-pictures of how language, and in particular the mechanisms for the fixing of reference (and, more generally, of intension), work in regular occasions of use and fail to work in irregular occasions of use. On the picture, the meaning of a typical word is pretty meager, though it comes together with a number of firmly accepted sentences containing the word, its associated ‘preconceptions’, which are not part of its meaning but are somehow designed to help fix its extension (and intension) in particular occasions of use. Typically, some of these preconceptions intuitively state that a certain predicate has some paradigmatic cases of application and negative application, while other preconceptions intuitively postulate generic principles for the expansion of the extension of the predicate beyond its paradigmatic range. These ideas are explained in section 13.2. According to the dual picture, in occasions of use where the preconceptions and the facts of the matter about a typical degree adjective (or scalar noun) give rise to an instance of the sorites paradox, the occasion of use is of the irregular kind: the adjective (or scalar noun) lacks an extension (and an intension) and as a result the utterances of sorites-relevant sentences containing it don’t have truth conditions. (But sorites-paradoxicality is only one source of irregularity.) In particular, for example, (Bsmall ), (Csmall ), and (Dsmall ), as uttered in the occasion of use described above, are neither true nor false. This thesis about sorites-paradoxical occasions of use is related to treatments that postulate reference or truth value gaps to deal with other paradoxes and inconsistencies, but it is radically different from standard theories of vagueness that postulate truth value gaps exclusively for alleged so-called ‘borderline cases’ of sorites susceptible predicates. The latter theories postulate that a sorites susceptible predicate has some kind of non-classical extension, and that the logical expressions operate on that extension through some suitably ad hoc non-classical semantics or logic. On the dual picture, a typical sorites susceptible degree adjective (or scalar noun)⁶ has no extension at all in those occasions of use. But the picture as developed here will also postulate (in section 13.4) that this is psychologically obscured by the fact that the default mechanism designed to fix a classical extension for these words often succeeds in ⁶ Henceforth I will omit the parenthetical addition of ‘and/or scalar noun(s)’ to ‘typical degree adjective(s)’ in many cases in which it should be tacitly understood.
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doing so, even in closely connected regular (thus non-paradoxical) occasions of use. Section 13.4 also compares the ‘extension gap’ thesis about sorites-paradoxical occasions of use with an analogous and independently plausible thesis about extension (and intension) gaps arising from certain linguistic conflicts among paradigm and generic preconceptions for other adjectives and other nouns. This comparison makes plausible the extension gap thesis for typical sorites susceptible degree adjectives in sorites-paradoxical contexts. It will also help me defend some further theses about the psychological reasons why an appearance is created, even after exposure to the sorites paradox, that utterances of sentences of the forms (B), (C), and (D) have truth conditions even in paradoxical occasions of use, and in particular of why people exposed to the sorites paradox tend to give a preference to their intuitions about the truth value of the (B) and (D) sentences over their intuitions about the truth value of the (C) sentence. These theses will appeal to plausible conjectures about the psychology of any reference-fixing mechanism that relies heavily on paradigms. It is plausible to assume that if a certain mechanism of reference-fixing exists (and persists), it must be successful in at least a vast number of occasions. The dual picture postulates that in a vast number of occasions of use of a typical sorites susceptible degree adjective, the default reference-fixing mechanism of preconceptions works successfully and turns the occasions of use into regular ones in which even (B), (C), and (D) sentences have truth conditions. This happens when the paradigm and generic preconceptions and the facts of the matter about the adjective determine an extension and an anti-extension which are mutually exclusive and jointly exhaustive with respect to the universe of discourse in the occasion of use; in these cases no compelling (A) sentence is in sight. The hypothesis that these occasions of use are very numerous is a purely empirical one, and cannot be fully decided on the basis of a priori linguistic reflection on the semantics of sorites susceptible predicates. Some considerations that favor it are offered in section 13.3, where the picture’s semantic treatment of regular occasions of use is sketched as well. It may also be tempting to assume, additionally, that successful communication with grammatically declarative sentences must nearly always use utterances with truth conditions. But I don’t take this to be a compulsory thesis. What I do take as a very reasonable thesis is that communication with declarative utterances generally occurs under the tacit assumption that these utterances have truth conditions. Plausibly, it also involves some understanding, however implicit and inchoate, of how those truth conditions should be determined if they in fact exist, as well as of how the referents of particular classes of words should be determined if they in fact exist. I take it that semantic theory often appeals to this implicit understanding as evidence in the construction of theories. So I take it as a reasonable burden on semantic theory to describe reference-fixing mechanisms for classes of words that plausibly underlie our tacit understanding of the referential properties of those words. But this is compatible with the possibility that there may be frequent instances of successful communication by means of utterances without truth conditions. In the presence of an adequate theory of the mechanisms of reference-fixing, successful communication by means of such utterances is explained by the fact that speakers can elicit from hearers all sorts of desired responses under the tacit common understanding of how the truth conditions
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of those utterances would be determined (if they had in fact existed). To take a simple example, a mother can say to her child ‘Santa Claus will bring you presents tomorrow’ and get the child to expect happily the presents from an unknown man that at some point someone has named ‘Santa Claus’. This particular instance of communication is not prevented by the lack of a referent for ‘Santa Claus’ or by the presumable lack of truth conditions for the mother’s sentence. The search for a convincing, or even promising, idea for a solution of the sorites paradox(es) has proved to be very elusive, perhaps surprisingly so. Standard attempts are predictably affected by some sophisticated problem or other, but more importantly, they are also generally unpersuasive even when taken as rough pictures of the workings of the sorites susceptible predicates and their interaction with the logical expressions. I will start (in section 13.1) with a brief survey of what I take to be some fundamental (as opposed to sophisticated) problems afflicting the most familiar standard theories. The survey is based on a tripartite classification of standard solutions of the sorites as ‘paradigmatist’, ‘genericist’, or ‘strongly nihilist’, which I think is illuminating in its own right. The dual picture is a neutral picture, in that it is intermediate between all these standard (and extreme) positions. Later it will be useful to have made explicit the fundamental problems of standard theories, for much of the support I wish to marshal for the neutral dual picture will consist in noting that it does not suffer from those problems. The picture is also, I hope, not unpersuasive at first sight. It is of course not free from potential objections, only some of which can I try to describe and defuse here. Nevertheless, I hope this initial exposition can convince some that it is a step in the right direction. I also hope that it can be expanded and refined in future work. 13.1
A C L A S S I F I C AT I O N O F FA M I L I A R T H E O R I E S , A N D T H E I R P RO B L E M S
The theories of the sorites paradox that we may call optimistic claim that, despite appearances, (C) utterances, regardless of the occasion of use, must be false, and do not postulate a semantics or logic for the logical expressions distinct from the classical. For example, they propose that, for any one of the sorites series t1 , t2 , . . ., t49,798 , t49,799 making true (the matrix of ) (Asmall ) that can be drawn from the comparison class above, the negation ‘it is not the case that for all ti and ti+1 , if ti is small then ti+1 is small’ is true; or, what is equivalent given classical semantics and logic, the existential quantification ‘there is a town with i inhabitants that is small and a town with i + 1 inhabitants that is not small’ is true. Optimism is often (but need not be) accompanied by epistemicism, the additional claim that we cannot know which number i is. (See e.g. Cargile 1969, Williamson 1994.) If epistemicism is true, it provides a certain kind of explanation of the natural repugnance we feel for optimism: we cannot accept the existential quantification according to which there are such towns because we cannot find out which number i is. There are a number of sophisticated problems with epistemicist theories (see e.g. G´omez-Torrente 2002), but the basic problem would seem to be that, no matter how sophisticated the defenses of optimism get, it
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is very hard to believe that some fact about the actual semantics of ‘small’ makes those existential quantifications true, at least in occasions of use such as the one above.⁷ And indeed, we have no plausible semantic model of how they could be true.⁸ One appealing feature of optimism, however, is that it does not postulate a non-classical semantics or logic for the standard logical expressions. Supervaluationists (Fine 1975 is the prime source) also say that those existential quantifications must be true. But unlike optimists, they claim that the existential quantifier does not have its classical semantics, at least when it interacts with sorites susceptible predicates. They claim that ‘(∃x)(∃y)(Kx & Ky & x is small & x has 1 inhabitant less than y & ∼y is small)’ may be true without there being some specific towns that make true ‘(x is small & x has 1 inhabitant less than y & ∼y is small)’. This is supposed to calm our worries about how the existential quantification could be true. But it sounds very implausible: assuming, as seems reasonable, that classical semantics is the semantics that we intuitively ascribe to the existential quantifier, why should we believe that it adopts an ad hoc semantics in certain cases? Despite occasional timid claims to the contrary, I see no evidence that we intuitively occasionally ascribe this semantics to the existential quantifier. Other theories of the sorites are based on even more radical departures from classical semantics. For example, typical degree theorists (such as Machina 1976) claim that the existential quantifier works in such a way that the existential quantification in question comes to have some alternative degree-theoretic truth value—it is (approximately) ‘half-true’. For what we might call primitivists, sorites predicates also determine a sui generis semantics for the logical expressions, but they claim that we need not know (or even that we may be unable to know) what this semantics is, at most that it is different from the classical. ( This is the way I read Sainsbury 1990.) As with the supervaluationist, the fundamental problem for the degree theorist and the primitivist is that postulating a (possibly unknowable) non-classical semantics for the logical expressions merely because they interact with sorites susceptible predicates is counterintuitive and ad hoc. Like optimists, I find the thesis that the logical expressions are in all essential respects governed by classical semantics and logic more than compelling. We may call all of these views paradigmatist, since they stick to the truth of the (B) and (D) sentences in each particular occasion of use, affirming the application and ⁷ Some optimists have theories compatible with anti-epistemicism that appeal to a special kind of contextualism to explain why (C) sentences seem true to us. Fara (2000) claims that some factor that context contributes to interpretation makes it the case that the sharp cut-off point between the extension and the anti-extension of ‘small’ in the context is never ‘where we look’, which explains why we believe of any pair of towns ti and ti+1 that it is not the case that ti is small and ti+1 is not small. But even if this were true, it would give us no reason to believe that the cut-off point is in some place where we don’t look (see e.g. Heck 2003, 120). Soames (1999), ch. 7, uses a similar strategy, although he is not strictly an optimist, since he postulates that there is no sharp cut-off point between the small and the non-small towns. But he postulates an equally implausible semantically determined sharp cut-off point between the small towns and towns which are supposed to be neither in the extension of ‘small’ in the context nor in its anti-extension (the ‘borderline’ towns). ⁸ As emphasized e.g. by Schiffer (1999) and Wright (2003).
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negative application of the sorites predicate in contextually paradigmatic cases, and they claim that the (C) sentence is somehow not true. An altogether different kind of option is to claim that somehow the (C) sentence is made true and either the (B) or the (D) sentence, or both, are made false by the relevant semantic mechanisms. For this reason we may call these views genericist. In the case above, and assuming classical semantics and logic, there are three suboptions: (a) to claim that ‘Smalltown is small’ is true and ‘∼Nonsmalltown is small’ is false—and presumably that all towns are in fact small; (b) to claim that ‘Smalltown is small’ is false and ‘∼Nonsmalltown is small’ is true—and presumably that no town is in fact small;⁹ (c) to claim that ‘Smalltown is small’ and ‘∼Nonsmalltown is small’ are both false. (c) is absurd and can safely be discarded. One general problem with (a) and (b) is that they give a preference to the intuitions about (C) utterances over the intuitions about (B) and (D) utterances. However, to me they seem more or less equally strong intuitions, and if I were forced to choose, I would say that the intuitions about the truth value of (B) and (D) utterances feel somewhat stronger after exposure to the sorites paradox. Also, the idea that, if sorites susceptible predicates do have extensions, then these are non-trivial extensions that affect semantically real distinctions between objects seems very plausible.¹⁰ But for the genericist, the extensions of sorites predicates never manage to do that. Williamson (1994, 165 ff ) has placed the (b) suboption together with other views on which the (logically atomic) sorites susceptible predicate also lacks application, although in this case because the predicate is nonsensical or in some milder way semantically defective, but in any case lacks an extension altogether. All these views on which the sorites susceptible predicate lacks application he calls nihilist. The views on which the sorites predicate is in some way semantically defective we might call (in order to distinguish them from the (b) suboption above) strongly nihilist (Dummett 1975 is an (imperfect) example; Eklund 2002 is closer to the idea). Strong nihilists are certainly not paradigmatists, but they are not genericists either, since the sorites predicate is just as defective in the (C) sentence as it is in the (B) and (D) sentences. Something that seems to me to be a problem for strong nihilists (or at least for representative strong nihilists) is that they seem to think that all of the (B), (C) and (D) sentences are compelling because the semantic rules for the sorites susceptible predicate in some way dictate that they must be accepted as true, if one is to abide by the meaning or broadly speaking the semantics of the predicate. (More exactly, what they claim is that, e.g. given the meaning of ‘small’ and uncontroversial facts about the population of Smalltown, (Bsmall ) must be accepted.) But I doubt that (Bsmall ), (Csmall ), and (Dsmall ) seem compelling because of that. If they did, the paradigmatist and genericist positions alike would seem to us to be excluded ⁹ Unger (1979) embraces this view for logically atomic copular predicates whose predicative element is a count noun applying to what he calls ‘ordinary things’: predicates like ‘is a table’, ‘is a house’, etc. have an empty extension. Elsewhere he held the same view for natural kind predicates, but he changed his view about these in Unger (1984). I am unsure of what he would say of predicates like ‘is small’. ¹⁰ As I will note in footnote 23, however, options like (a) or (b) are clearly the right options in some special non-paradoxical occasions of use of sorites predicates.
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on purely semantic grounds; we would have the feeling that abandoning either of (Bsmall ), (Csmall ), or (Dsmall ) would inevitably amount to changing the meaning or the semantics of ‘small’. But I don’t think we have that intuition. I think that when in theoretical discussion we are exposed to optimism or to a (b)-type genericism we just see them as weird, not analytically excluded speculations that might conceivably uncover the concealed semantics of the sorites susceptible predicates. (We don’t see them the way we would see a theory which claimed that being unmarried is not an analytically necessary condition of bachelors.) In any case, a problem for all varieties of strong nihilism is that it is incompatible with the assumption (made at the beginning of this chapter) that the default reference-fixing mechanism for typical sorites susceptible degree adjectives must work successfully in a vast number of occasions. In fact, for the strong nihilist, just like for the genericist, we can never effect semantically real distinctions between objects by means of sorites susceptible degree adjectives. A final problematic consequence of strong nihilism is that it’s self-referentially ‘instable’: if it were true, it could not be stated with truth, given that, in all probability, it would have to use sorites susceptible predicates in its own statement. My purpose in what follows is to sketch a basis for a picture of vagueness that is not affected by the fundamental problems of the theories that we have reviewed in this section. How the picture avoids each of the problems will be pointed out along the way.
13.2
LINGUISTIC PRECONCEPTIONS
The picture, or at least my defense of it, relies on a broadly Kripkean view of language as evolving through the appearance and modification or abandonment of what we might call preconceptions. (Kripke has used the word ‘prejudices’ for this or a closely related concept.¹¹ I prefer ‘preconception’ both because it indicates that there may be differences between the two concepts and because it seems less negatively charged than ‘prejudice’.) In general, these preconceptions are sentences which are very resolutely assented to by minimally sophisticated normal people at relatively pretheoretical levels of use, sentences that are very difficult or even nearly impossible to abandon without exposure to relatively extensive reflection or empirical research. Preconceptions need not be a priori, necessary, analytic or even in any sense dictated to be of obligatory acceptance by the semantics of an expression; they just have to be very hard to give up. Nevertheless, among preconceptions some (perhaps all) are what we might call ‘linguistic’: they have a bearing on the extensions (and the intensions) that we assign to words, and in particular on the extensions that we assign to predicates. But not even linguistic preconceptions are invariably or even usually analytic or dictated to be of obligatory acceptance by the semantics of an expression. The Kripkean picture I have in mind postulates that we may view most predicates (and most words) as having a pretty meager meaning that usually does not suffice ¹¹ For exposition of the Kripkean notion of a prejudice see G´omez-Torrente (forthcoming).
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to fix their extension (or their intension),¹² but also as ‘introduced’ in some way together with a number of linguistic preconceptions involving them, preconceptions that are somehow designed to help fix their extensions, possibly with respect to particular occasions of use. Now a further idea, less definitely Kripkean, but certainly suggested by Kripke’s presentations, is that an ‘initial’ set of preconceptions {(P), (P), X(P), . . .} about the extension of a predicate ‘P’ is ‘introduced’ together with a tacit conditional instruction. We may also view this as a linguistic preconception (perhaps one that in some sense is a priori or dictated to be of obligatory acceptance by the semantics of the predicate), having a form similar to ‘If there is exactly one set Q such that (Q), (Q), X(Q), . . ., then the extension of ‘‘P’’ is that set’. These instructions help fix the extension of ‘P’ when their antecedents are satisfied. But when their antecedents are not satisfied, i.e. when the preconceptions in the initial set are not jointly uniquely satisfied, it may often be unclear that an extension is determined for the predicate. There are presumably other general, but less definite or less compelling linguistic preconceptions that may help in some cases in which those antecedents are not satisfied. For example, there may be general preconceptions exhorting us to try to assign extensions to predicates by abandoning those of the preconceptions in the initial set that intervene less in the use of the predicate, or by abandoning those whose abandonment provides for the simplest way of obtaining an extension (if any), etc. But even after (implicit) attempts to apply these further preconceptions, conflicts of unsatisfiability or other problems may often remain unresolved. The mentioned conditional instructions are similar to Kripkean conditionals by means of which in some way an explicit or implicit attempt is made to fix the reference of some proper names and general terms for natural kinds, substances or phenomena. One example is ‘If there is exactly one planet causing the perturbations in the orbit of Uranus, then ‘‘Neptune’’ refers to that planet’, which successfully fixes the extension (reference) of ‘Neptune’. (As well as its intension, which on account of its rigidity is simply the function assigning Neptune to each possible world.)¹³ Here the sentence ‘There is exactly one planet causing the perturbations in the orbit of Uranus’ cannot be called a preconception in the strict sense above, as it is certainly not resolutely accepted by normal people at relatively pretheoretical levels of use, though it is clearly not analytic or a priori, and it has a role in fixing the reference of ‘Neptune’. But stricter examples are provided by general terms for natural kinds, substances and phenomena. In the case of ‘dog’, it is natural to assume the existence of some such conditional instruction as ‘If there is exactly one set of which (most of ) a, b, c,. . . are members and such that the things that are in it are exactly the instances of a certain natural kind, then the extension of ‘‘is a dog’’ is that set’ (where ‘a’, ‘b’, ‘c’, . . . are names of things which are taken as paradigms of dogs). In this case ‘(Most of ) a, b, c,. . . are dogs’ and ‘The things that are dogs are exactly the instances of a certain natural kind’ are plausibly viewed as initial linguistic preconceptions about ‘dog’. ¹² Henceforth I will omit the parenthetical addition of ‘intension(s)’, ‘its(their) intension(s)’, etc. in many cases in which it should be tacitly understood. ¹³ Assuming that ‘Neptune’ is ‘obstinately rigid’ in the sense of Nathan Salmon (1982).
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They have an apparently successful (implicit) role in fixing both the extension and the intension of ‘is a dog’.¹⁴ (On account of the rigidity of ‘dog’, the latter is simply the constant function assigning the set of possible dogs to each possible world.¹⁵) One distinction between two important kinds of preconceptions stands out, and it is already illustrated in the case of ‘dog’. It’s the distinction between paradigm preconceptions and generic preconceptions. Paradigm preconceptions are relatively simple preconceptions whose intuitive content is either that a certain predicate applies to a certain specific object or objects (positive paradigm preconceptions) or that it negatively applies to a certain specific object or objects (negative paradigm preconceptions). (One example of (positive) paradigm preconception would be ‘(Most of) a, b, c,. . . are dogs’.) Generic preconceptions, on the other hand, are simply preconceptions which are not paradigm preconceptions. Often they intuitively state necessary, sufficient or other sorts of general conditions for the application or negative application of a predicate. They are designed to guide us in the expansion of our use of the predicate beyond its paradigmatic range of use. (One example would be ‘The things that are dogs are exactly the instances of a certain natural kind’.) One specific proposal of the picture in this chapter is that typical sorites susceptible degree adjectives (and scalar nouns) provide yet another example of predicative words associated with a set of preconceptions that includes paradigm and generic preconceptions designed to help fix the extension of those words.¹⁶ (B) and (D) sentences in occasions of use where they are compelling are examples of paradigm preconceptions; (C) sentences in occasions of use where they are compelling are examples of generic preconceptions. But there is one important difference one must emphasize with respect to ‘is a dog’. The intuitive truth value of paradigm and generic sentences for typical degree adjectives varies extremely with the occasion of use, unlike in the case of ‘dog’ and related words. It is natural to postulate the existence of abstract linguistic preconceptions associated with typical sorites susceptible degree adjectives that are not intended to help fix an absolute extension but one relative to an occasion of use, or in other words, regulatory principles for the adoption of concrete paradigm and generic preconceptions relative to particular occasions of use.¹⁷ An abstract preconception regulating the acceptability of specific paradigm preconceptions relative to an occasion of use for ‘small’ presumably takes a form similar to this: In an occasion of use, that already provides a universe of discourse U and a comparison class K included in U , one may take members r1 , r2 , etc. of K as cases of small things and/or ¹⁴ Successful at least over usual universes of discourse, one of which presumably constitutes the intended domain of quantification in these preconceptions. See the text surrounding footnote 28 for more on this qualification. ¹⁵ Assuming that a natural kind predicate is rigid because its designation in all possible worlds is the same set of possible objects. (Cf. the notion of ‘obstinate essentiality’ in G´omez-Torrente 2006.) ¹⁶ The importance of paradigms for our understanding of sorites susceptible predicates is emphasized by Sainsbury (1990). ¹⁷ This is not to imply that preconceptions associated with ‘dog’ and related nouns do, when successful, fix an absolute extension for them, even if in a way they may be so intended. See again the text surrounding footnote 28.
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members s1 , s2 , etc. of K as negative cases of small things, just as long as the relevant sizes of r1 , r2 , etc. are smaller than the relevant sizes of s1 , s2 , etc.
This would account for the fact that the intuitive truth value of paradigm sentences for ‘is small’ varies extremely with the standards in operation in the occasion of use. In one occasion of use (e.g. a conversation between wealthy people looking for an apartment) 100 square meters is a positive paradigm of a small size for an apartment; in another (a conversation between people with very modest incomes) 100 square meters is a negative paradigm of a small size. On the other hand, even the intuitive truth value of a concrete generic preconception for ‘is small’ in the occasion of use involving the wealthy (or even the not so wealthy) people, such as ‘(∀x)(∀y)(Kx & Ky ⊃ (x is small & x has 1 square meter less than y ⊃ y is small))’, can vary with the occasion of use. In a conversation between people in Hong Kong looking to buy a micro-apartment where they can fit all their furniture, 1 square meter may make all the difference between smallness and non-smallness. So an abstract preconception regulating the acceptability of concrete generic preconceptions for ‘is small’ presumably takes a form similar to this: In an occasion of use, that already provides a universe of discourse U and a comparison class K included in U , and that may provide members r1 , r2 , etc. of K as cases of small things, and/or members s1 , s2 , etc. of K as negative cases of small things, one may take ‘(∀x)(∀y)(Kx & Ky ⊃ x is small & x has a size inferior by 1 u to y ⊃ y is small))’, where u is a relevant size unit, as a generic principle holding in the occasion of use, provided just that the difference between the ri with the greatest size and the si with the smallest size is greater than 1 u.
Note that ‘(∀x)(∀y)(Kx & Ky ⊃ (x is small & x has a size inferior by 1 u to y ⊃ y is small))’ is logically equivalent to ‘(∀x)(∀y)(Kx & Ky ⊃ (∼ y is small & x has a size inferior by 1 u to y ⊃ ∼x is small))’, which will thus be a concrete generic preconception in play in those occasions of use where its equivalent is in play. It is also natural to postulate that ‘small’ has associated with it an abstract preconception regulating the acceptability of concrete conditional instructions for the fixing of extension/anti-extension pairs relative to particular occasions of use. It would be something like this: In an occasion of use, that already provides a universe of discourse U and a comparison class K included in U , that provides members r1 , r2 , etc. of K as cases of small things, and/or members s1 , s2 , etc. of K as negative cases of small things, and that provides some general principle(s) ‘(∀x)(∀y)(Kx & Ky ⊃ (x is small & x has a size inferior by 1 u1 to y ⊃ y is small))’, ‘(∀x)(∀y)(Kx & Ky ⊃ (x is small & x has a size inferior by 1 u2 to y ⊃ y is small))’, etc., a principle of this form is acceptable: If there is a unique pair <E, A> of subsets of U which are mutually exclusive and jointly exhaustive over U , and are such that • r1 , r2 , etc. are in E, • everything in K that has a size inferior to something in E is in E, • everything in K that has a size superior by 1 u1 to something in E is in E, • everything in K that has a size superior by 1 u2 to something in E is in E, etc. • s1 , s2 , etc. are in A, • everything in K that has a size superior to something in A is in A,
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• everything in K that has a size inferior by 1 u1 to something in A is in A, • everything in K that has a size inferior by 1 u2 to something in A is in A, etc. • U -K is included in A,¹⁸
then <E, A> is the extension/anti-extension pair of ‘is small’ relative to the occasion of use.¹⁹
If something like this principle underlies the fixing of a reference for ‘small’ relative to an occasion of use, then in occasions of use where a unique pair <E, A> satisfies the antecedent of the concrete conditional instruction in operation, a reference for ‘small’ (in the sense of an extension/anti-extension pair) gets fixed;²⁰ in other occasions of use, a reference may not get fixed. Williamson (1999) has argued that, if some mechanism fixes the extension of a predicate F, then the same mechanism, by default, fixes an anti-extension for F: the set of things that (are in the universe of discourse and) are not in the extension of F; as Williamson puts it, fixing the extension and the anti-extension of F are not ‘independent achievements’ (509). This is a reasonable idea, and is not contradicted by the just postulated mechanism for the fixing of a reference for ‘small’ relative to an occasion of use. Note that no proposal is made that there is a set of preconceptions giving intuitively jointly necessary and sufficient conditions for membership in the extension of ‘small’ (unlike what happened in the case of ‘dog’); and no proposal is made that there is an independent set of preconceptions giving intuitively jointly necessary and sufficient conditions for membership in its anti-extension. The extension/anti-extension pair is fixed (when it is) in a ‘coordinated’ fashion, i.e. when the positive and negative paradigm preconceptions and the generic preconceptions are jointly satisfied by a pair of classes. If someone wished to use Williamson’s remark as the basis for an objection to the mechanism postulated here, he might try to argue that there is some independent reason to think that the extension of a predicate must in successful cases be fixed by a ‘non-coordinated’ mechanism, and that its anti-extension must only then be fixed by default. However, I see no reason why this should be so in general, and the apparent possibility of the mechanism just postulated in the text goes against this radical thesis. Furthermore, there are special reasons to think that the thesis is false for actual typical degree adjectives. As emphasized by Sainsbury (1990), a typical degree adjective generally comes together with an antonym (e.g. ‘big’ in the case of ‘small’), and the antonym is analytically connected with a sufficient condition for membership in the anti-extension of the original adjective (e.g. ‘If something is big, it’s not small’). Assuming only that positive paradigm preconceptions play a role in fixing the extension of each lexically different degree adjective by giving sufficient conditions for ¹⁸ I assume (somewhat artificially) that everything that is not in the comparison class is not small in the sense relevant to the occasion of use, and thus that it is in the anti-extension of ‘small’. ¹⁹ There are surely preconceptions about ‘small’ (and as we will see, also about ‘dog’) other than the paradigm and generic preconceptions postulated in the text, and thus the hypothesized conditional instructions are simplifications. ²⁰ As noted above, reference-fixing conditional instructions may often be analytic or a priori. In particular, the preceding abstract principle and its concrete instances may well be analytic or a priori.
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membership in it, it follows that the positive paradigm preconceptions corresponding to two antonyms F and G must be ‘coordinated’ in some way if the predicates are to possess suitable extension/anti-extension pairs, for there should be no overlap of the set of positive paradigms of F (which should be in the extension of F) with the set of positive paradigms of G (which should be in the anti-extension of F). It can already be seen from what has been said so far that the present picture implies that the semantic rules for ‘is small’ do not per se (or even in conjunction with uncontroversial facts about the populations of Smalltown and Nonsmalltown) dictate that (Bsmall ), (Csmall ), or (Dsmall ) must be accepted as true. (Bsmall ), (Csmall ), and (Dsmall ) are merely preconceptions, analogous to the non-analytic paradigm and generic principles that have a role in (implicitly) fixing the reference of natural kind predicates. This is consistent with the intuition, mentioned in section 13.1, that (Bsmall ), (Csmall ), and (Dsmall ) are not really analytic or dictated as of obligatory acceptance by the semantics of ‘is small’.
13.3
THE FIXING OF REFERENCE IN REGULAR OCCASIONS OF USE
The classical logic and semantics of the logical expressions, in particular of the quantifiers, requires essentially one thing of the interpretation of a predicate: that the interpretation fix an extension and an anti-extension for the predicate which are mutually exclusive and jointly exhaustive over the previously given universe of discourse. This will not be sufficient by itself for the predicate to be endowed with an intension, but in some cases the fixing of an extension may determine the fixing of an intension if some additional factors are in play. In many occasions of use, the (concrete) paradigm and generic preconceptions and conditional instructions for a sorites susceptible degree adjective or scalar noun provide a classical extension/anti-extension pair for them, and perhaps they are also enough to fix an intension. Let’s consider the following example. A couple of modest income are looking to buy an apartment, and they are having a conversation in which they will try to decide which one to buy. There aren’t that many options. Their choice is reduced to four apartments, with sizes of 65, 70, 100, and 105 square meters; call them ‘A65’, ‘A70’, ‘A100’, and ‘A105’. We may take this set K of apartments as the relevant comparison class for ‘small’ in the conversation. Given their standards in the situation, A65 counts as small for them, and A105 as not small. Also given their standards, they take it that 5 square meters don’t make a difference as to whether an apartment is small or not. We may also postulate that the following concrete conditional instruction (licensed in this occasion of use by the last abstract preconception of section 13.2) is in operation: If there is a unique pair <E, A> of subsets of the universe of discourse U which are mutually exclusive and jointly exhaustive over U , and are such that • A65 is in E, • everything in K that has 5 square meters more than something in E is in E, • A105 is in A,
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• everything in K that has 5 square meters less than something in A is in A, • U -K is included in A,
then <E, A> is the extension/anti-extension pair of ‘is small’ relative to the occasion of use.
Under all these assumptions, the conditional instruction and the paradigm and generic preconceptions in play fix a classical extension/anti-extension pair for the predicate ‘is small’: its extension in the occasion of use is {A65, A70}, and its antiextension is the set containing A100 and A105 plus any other thing that is not in K . Extension and anti-extension are mutually exclusive and jointly exhaustive over the universe of discourse relevant in the conversation. Perhaps also a (classical) intension for ‘is small’ is fixed in the mentioned occasion of use with the help of the mentioned principles and others that might be plausibly postulated. Given that a classical extension/anti-extension pair has been fixed, this pair in turn may induce in the obvious way a pair of scales of associated numbers (measuring sizes in square meters, in this case), which we may take simply as a pair of sets; in the example, the pair of scales would be <{65, 70}, {100, 105}>. Then an intension for ‘is small’ in the mentioned occasion of use might be computed with the help of this principle: the extension of ‘is small’ over the previously given universe of discourse U in a world w contains an element a of K just in case a’s size in w is less than or equal to one of the sizes in the first scale, {65, 70}; and the anti-extension of ‘is small’ over U in a world w contains an element a of K just in case a’s size in w is greater than or equal to one of the sizes in the second scale, {100,105} (and it also contains everything that is not in K ). The resulting intension puts A100 in the extension of ‘is small’ in worlds in which, say, its builder changed the architect’s plans and gave it a size of just 68 square meters.²¹ An occasion of use of a degree adjective or scalar noun may presuppose a comparison class that is even more reduced than the comparison class in the apartment example, and that does not create any obstacle to the fixing of a classical extension/anti-extension pair. If we are talking about a figure such as the following
in most occasions of use we will be able to say felicitously and truly things like ‘The small circle is to the left’, ‘There is a small circle to the left’, ‘The circle to the left is small’, ‘The non-small circle is to the right’, etc.²² In occasions of use like these, ²¹ This mechanism is of course very sketchy and leaves questions unanswered. In worlds w where one of the apartments of the given 4-element set has a size in the open interval (70,100), the mechanism doesn’t assign a classical extension/anti-extension pair to ‘is small’ in w. This seems tolerable and not incompatible with usual possible worlds semantics, which contemplates intensions which are partial functions. The description in the text is not meant as a complete one, but only as indicative of the direction a more complete description might take. ²² These two-element comparison classes (and figures similar to the one in the text) are considered in Klein (1980). See also Kennedy (2007).
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it is clear that the extension of ‘small’ relative to the universe of discourse will have been taken to be the set consisting of the circle to the left, and its anti-extension will have been taken to contain the circle to the right. In these cases, the paradigm preconceptions in play suffice to fix the extension/anti-extension pair, given that no appropriate generic preconception will enter into conflict with them. In general, mutatis mutandis the same can be said of cases in which, as in the apartment case or the two circles case, the comparison class is clearly divided into two mutually exclusive and jointly exhaustive subsets consisting, respectively, of positive paradigms and/or individuals that can be ‘reached’ from the positive paradigms by generic preconceptions, and of negative paradigms and/or individuals that can be ‘reached’ from the negative paradigms by generic preconceptions. A great number of uses of typical degree adjectives do not seem to presuppose large, sorites-prone, or even not clearly divided comparison classes. Many of them, on the contrary, seem to be what we might call contrastive uses: they seem to presuppose precisely a comparison class consisting of two clearly separated sets of objects, not infrequently sets of just one element, that need to be forcefully contrasted for the conversational purposes of the situation. Consider the italicized degree adjectives in the following passages, all taken from the first page of a widely used reference work: abbess. (. . .) In the Middle Ages wide powers were claimed by some abbesses, but the Council of Trent put an end to most special prerogatives. Abbot, George (1562–1633), Archbishop of Canterbury from 1611. (. . .) he won James I’s favour by his mission to Scotland (1608) (. . .). As archbishop he was severe on Roman Catholics and partial to Calvinists at home and abroad. (. . .) The strong line which he took over the Essex nullity suit (1613) won him respect and a temporary popularity. In 1621 he accidentally shot a gamekeeper and his position was considered to have become irregular; James decided in his favour and he resumed his duties. He crowned Charles I but had little influence in his reign. (Livingstone (2006), 1)
It is clearly forced to view the use of ‘wide’ as applied to the powers claimed by some abbesses as presupposing a large comparison class consisting of (classes of) powers that people have claimed, or even of powers that the heads of monasteries have claimed; certainly, no knowledge of such comparison class is required of the reader for the understanding of the sentence. Its use simply seems to presuppose a contrast with the powers of the other abbesses who did not claim those same powers. This is made somewhat clearer by the next sentence, in which these powers are called ‘special’, in a use that does not even seem to admit of comparatives. The use of ‘severe’ as applied to Abbot on the Catholics could hardly presuppose an extended class of men or of acts of severity; presumably Abbot’s actions against the Catholics do not rank especially high in the universal classification of acts of severity. The author apparently means to contrast Abbot’s attitude toward the Catholics with his attitude toward the Calvinists, as made clear by the next clause. The use of ‘partial’ presumably should be understood along the same lines. The use of ‘strong’ as applied to Abbot’s line over the Essex nullity suit is nearly impossible to understand as presupposing a comparison class of several ‘lines’ ordered by strength; the author just means to contrast Abbot’s adverse attitude with the favorable attitude of the other side. The use of ‘irregular’ (if
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it’s a use that admits comparatives) seems again not based on a comparison with several positions men (or archbishops) could have, but simply with the position Abbot enjoyed before the shooting incident as an archbishop who had not been involved in any strange circumstances. Finally, the use of ‘little’ as applied to his influence in Charles I’s reign is not meant to locate that influence at a low point in a ranking of ‘influences’ of people or Canterbury archbishops, but simply to contrast that influence with the influence he enjoyed under James I, from whose favor he had benefited. In contrastive uses like these (and the examples could be multiplied at will), there is no obstacle to the fixing of a classical extension/anti-extension pair over the contextual universe of discourse by means of the mechanism of preconceptions sketched above, given that the relevant comparison class is smallish (or even a two-element one) and clearly divided.²³ The ease with which these numerous and useful uses are accommodated without abandoning the presuppositions of classical semantics suggests that, even though other uses are problematic, the linguistic practice involving the employment of sorites susceptible adjectives is sustainable in the face of paradox. If speakers using typical degree adjectives were constantly faced with uses which did seem to create problems for classical semantics or logic, that linguistic practice would probably be hard to sustain. The above examples of uses of adjectives, together with the preconceptions picture of how they obtain classical extensions (and thus of how the utterances in which they appear obtain classical truth conditions), vindicates the plausible idea that an often successful mechanism for the fixing of reference underlies our use of degree adjectives. On the present picture, then, justice is done to the convincing idea that we manage to effect semantically real distinctions between objects with the help of typical sorites susceptible degree adjectives, and even that we do so in a vast number of occasions. Furthermore, the picture does this without postulating an ad hoc non-classical semantics for the logical expressions when they interact with those adjectives. On the picture the intuition is preserved that when we deal with typical sorites susceptible adjectives, we use the classical semantics for the logical ²³ As advanced in footnote 10, in yet other non-problematic cases the adjective will have either an empty or a universal extension over the comparison class; i.e. either an (a) or a (b) genericist suboption (in the sense of section 13.1 above) will be the right option in some special occasions of use. Suppose the comparison class and the universe of discourse are the same, the set of natural numbers; suppose that we take the first ten numbers as (positive) paradigms of small numbers, but we abstain from taking any number as a negative paradigm of smallness; and suppose that we accept the principle ‘(∀x)(∀y)(x is small & x + 1 = y ⊃ y is small)’. The mechanism of preconceptions postulated above then generates as the extension of ‘small’ the whole set of natural numbers: every number is small under the exacting standards in the situation. (Not unreasonable standards, if we reflect that every number is only greater than finitely many numbers but smaller than infinitely many.) This is a (b) case; analogous (a) cases are also easily imagined. And similar cases can be created with many adjectives for other universes of discourse if we suppose the comparison class to be greatly unrestricted, e.g. when it contains many merely possible objects. It seems that there could have been towns of all finite numbers of inhabitants. Suppose then that our universe of discourse contains all such possible towns, and that the comparison class is the set of all possible towns. In a perfectly acceptable occasion of use with these features the extension of ‘is small’ will be universal with respect to the comparison class: every town will count as small. By analogous arguments, one could argue that in some occasions of use in which the comparison class is greatly unrestricted every man counts as bald, no man counts as tall, every man counts as poor, etc. (Perhaps occasions of use of this kind have motivated the proposal of genericist theories; but I’m unaware that they have.)
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expressions. We are simply working under the assumption, however tacit or inchoate, that our linguistic preconceptions fix a classical extension/anti-extension pair for the adjectives, and thus no theoretical hypothesis that the logical expressions operate in a non-classical way on a non-classical extension is called for.
1 3 . 4 T H E FA I LU R E O F R E F E R E N C E I N I R R E G U L A R , E S PE C I A L LY PA R A D OX I C A L , O C C A S I O N S O F U S E In the case of names and general terms for natural kinds, substances and phenomena, the Kripkean mechanism of preconceptions described at the beginning of section 13.2 plausibly fails to generate extensions (and intensions) in some cases. These include cases in which the preconceptions enter directly into contradiction with relevant truths that are not preconceived (and they may include non-conflictive cases in which they are nevertheless insufficiently specific to generate a unique extension). Similarly, in the case of typical sorites susceptible degree adjectives and scalar nouns the mechanism of preconceptions described later in section 13.2 fails to generate extensions in cases in which the preconceptions are in conflict with a truth of the form of (A), with the content that a sorites series can be drawn from the comparison class; and the preconceptions also fail to generate extensions in some cases in which no true sentence of the form of (A) is in sight, but they are nevertheless insufficiently specific to divide the comparison class (and hence the universe of discourse) uniquely into two mutually exclusive and jointly exhaustive subclasses. I will begin this section explaining and illustrating these failures. I take it to be fairly uncontroversial that in some cases the descriptive identifications and conditional instructions by means of which an attempt is made to fix a reference for certain names fail to do so. A well-known example is ‘If there is exactly one planet causing the perturbations in the orbit of Mercury, then ‘‘Vulcan’’ refers to that planet’, which fails to fix a reference for ‘Vulcan’. It seems also most reasonable to think that, even though in a vast number (or even a majority) of cases of terms for natural kinds, substances and phenomena, the initial linguistic preconceptions and conditional instructions about a predicate successfully fix an extension (and an intension) for it, in at least some cases they fail to do so, just as in the proper name case.²⁴ Consider ‘If there is exactly one set of which most of a, b, c,. . . are members and such that the things that are in it are exactly the instances of a certain disease, then the extension of ‘‘is an instance of madness’’ is that set’. Here ‘Most of a, b, c,. . . are instances of madness’ and ‘The things that are instances of madness are exactly the instances of a certain disease’ can be viewed as initial linguistic preconceptions about ‘is an instance of madness’. If the former mentions paradigms of all traditional kinds, the two preconceptions are not jointly satisfied, for it has turned out that there are many kinds of equally frequent traditional paradigms, which are instances of diseases ²⁴ It is natural to conjecture, as we did with degree adjectives, that the vast number of cases in which the mechanism of preconceptions fixes a reference for terms for natural kinds makes the linguistic practice involving them sustainable even in the presence of problematic cases.
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or other phenomena that don’t have anything to do one with the other—epilepsy, tetanus, dementia praecox, delirium tremens, all kinds of so called neuroses and psychoses, etc. Now, of course normal people at a relatively pretheoretical level of use do not sense any problem preventing sentences containing ‘madness’ or ‘mad’ to have truth values. And it seems to me that, after exposure to the theoretically varied nature of the diseases or other phenomena that prompt attributions of madness, most people tend to reject the generic preconception ‘The things that are instances of madness are exactly the instances of a certain disease’ and stick to the preconception to the effect that at least a majority of paradigm cases must fall under the extension of ‘madness’ and ‘mad’ (Charles Manson, Dr Samuel Johnson,²⁵ my extremely agoraphobic neighbor, and so on). However, on what seems to me to be the most reasonable view after reflection, there is actually no fact of the matter as to whether, e.g. Manson, Dr Johnson, my neighbor, etc. are instances of madness, or there is no such thing as an instance of madness among them; thus no extension is fixed for ‘is an instance of madness’, as there isn’t even a fact of the matter whether its extension should contain them or not. Of course we may use ‘madness’ with a definite extension if either we stick to the preconception that most people in our paradigms list are instances of madness but accept that madness is not a (common) disease; or if we stick to the preconception that the instances of madness are precisely the instances of a certain disease but accept that it’s not a disease exemplified by most people in the list, in which case it’s false that most of them are instances of madness and ‘is an instance of madness’ has a reduced extension, possibly empty, consisting of the instances of a single disease, possibly an imaginary one. But regardless of any initial inclination we may have, reflection suggests that these options ultimately require arbitrary decisions not justified by preexisting usage. In particular, if we decide to stick to most of our paradigm preconceptions, it is unclear that we can appeal to any principle determining exactly which majoritarian subset of these preconceptions we should stick to. And even if there is such a principle (e.g. if for some reason ‘the’ principle is to stick to all of our paradigm preconceptions), once we abandon the idea that most instances of madness must be instances of a common disease, it becomes unclear how to evaluate new cases for membership in the extension of ‘madness’, and thus how to obtain a determinate extension merely from the paradigms. Is ‘the’ general principle to include in the extension all the new cases which exhibit the same descriptive symptoms (assuming we can specify these) as the initial paradigms? Is ‘the’ principle to include all the new cases which fall under one of the diseases or other phenomena exemplified by the initial paradigms? Is it something else? Of course, we may revert to sticking to the generic preconception to the effect that the instances of madness ought to be precisely the instances of a certain disease, and then probably to considering ‘madness’ as naming some sort of imaginary disease having no real instances. But this seems no more compulsory than any of the paradigmatist options. The most reasonable stand seems to be to acknowledge that ‘madness’ is in some way defective, and that the reason is that its preexisting semantics ²⁵ A well-known case of a personality with extremely obsessive-compulsive habits.
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together with its preexisting associated preconceptions fail to fix an extension for it in the presence of a conflict they were not designed to cope with. A crucial proposal of this chapter is that, provided we accept that typical sorites susceptible degree adjectives and scalar nouns are governed by something like the mechanism of preconceptions of section 13.2, they fail to have a reference (a classical extension/anti-extension pair) in occasions of use in which the preconceptions are in conflict with a truth of the form of (A), stating that a sorites series can be drawn from the comparison class. Consequently, utterances of usual sentences containing typical sorites susceptible degree adjectives and scalar nouns in such occasions of use will lack truth conditions;²⁶ these are paradoxical occasions of use. The sorites reasoning makes explicit the existence of a conflict between, e.g. the truth (Asmall ), on the one hand, and the paradigm preconceptions (Bsmall ) and (Dsmall ), and the generic preconception (Csmall ) on the other, as uttered or considered in a paradoxical occasion of use such as the one described at the beginning of this chapter.²⁷ We can ask again: does preexisting usage determine that some of these sentences are true while at the same time the others are false? Is it the generic preconception that is false, or is it some of the paradigm preconceptions? (Or is it the case that we have been under some illusion that classical semantics and logic govern our use of ‘small’?) Needless to say, the conflict is unresolved, as reflected in the existence of genericist and strongly nihilist theories, even though, as noted earlier, there is some initial intuitive pressure for paradigmatism. The most reasonable view seems again to be that ‘small’, as used in paradoxical occasions of use, is defective, as its semantics and associated preconceptions are not enough to get an extension for it in the presence of the unexpected sorites conflict. Despite the by now predictable initial intuitive pressure for paradigmatism, the sorites case is one in which it is particularly clear that paradigmatism is not the right option. It is not only that, as in the case of ‘madness’, no paradigmatist option for obtaining a full extension from the paradigms seems singled out by preexisting usage, but also that all standard paradigmatist options seem clearly false after some reflection (as noted in section 13.1). Abandoning (Csmall ) means either postulating and ad hoc semantics or logic for the logical expressions, or else accepting the negation of (Csmall ) as classically understood, and hence the truth of the corresponding optimistic, and so hard to believe, existential quantification. One virtue of the present picture is that it explains in a simple way why this existential quantification is so hard to believe. In all probability there is no further preconception that provides for the determination of the sharp cut-off point that is needed in this case; hence abandoning (Csmall ) does not provide any way of assigning an extension to ‘is small’. The presumable scarcity of preconceptions, and especially the inexistence of a preconception providing for the determination of needed sharp cut-off points, explains our natural ²⁶ Perhaps some utterances of sentences containing sorites susceptible predicates in some irregular occasions of use have truth conditions, e.g. some where the predicates are in the scope of locutions of propositional attitude. ²⁷ In speaking of (Bsmall )–(Dsmall ) as sentences in conflict, or as elements of reasoning, etc., I am of course not implying that they do after all have a truth value. They have those properties roughly in the same sense that schemata can be inconsistent or are usable in schematic reasonings.
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repugnance for optimism, and even implies that it is false as a matter of fact, at least if we further accept that sharp cut-off points could only be determined by some feature of either the meaning or the non-analytic linguistic preconceptions about a predicate. The present proposal thus satisfies in a very strong sense the desideratum that a picture of the sorites phenomenon should not be optimistic. More generally, the picture has the welcome implication that paradigmatist options as a group are not really semantically superior to genericism, despite initial appearances. Many other kinds of predicates besides degree adjectival predicates have been claimed to be sorites susceptible, including natural kind predicates. For example, the following has been claimed to be a compelling (A)–(D) set, where the quantifiers range over a greatly unrestricted universe of discourse containing billions of particle aggregates, n is some huge number, ‘Rigo’ is the name of a dog, and ‘Molly’ the name of a single molecule of some sort: (Adog ) (∃x1 ) . . . (∃xn )(x1 results from Rigo by the removal of a single molecule & x2 results from x1 by the removal of a single molecule & . . . & xn results from xn−1 by the removal of a single molecule & Molly results from xn by the removal of a single molecule); (Bdog ) Rigo is a dog; (Cdog ) (∀x)(∀y)(x is a dog & y results from x by the removal of a single molecule ⊃ y isa dog); (Ddog ) ∼ Molly is a dog. If these are real paradoxes, the present picture suggests that the solution for them may lie in acknowledging that, while the more natural preconceptions associated with a natural kind predicate (mentioned in section 13.2) are in some sense designed to fix an absolute extension for it, they may only manage to fix one over the tame universes of discourse which are presumably quantified over in those preconceptions. If (Bdog ), (Cdog ), and (Ddog ), for example, are further preconceptions associated with ‘dog’, then, in occasions of use involving a universe of discourse containing billions of suitably weird aggregates of particles, the preconceptions associated with ‘dog’ will not be jointly satisfied, ‘dog’ will not get an extension and (Bdog ), (Cdog ), and (Ddog ) will all lack a truth value. Nevertheless, the natural preconceptions for ‘dog’ mentioned in section 13.2 surely fix an extension for it if they quantify, as they presumably do, over more usual universes of discourse, that contain only normal objects and don’t contain billions of weird aggregates of particles. Even if sorites conflicts create obstacles to the fixing of an extension in weird occasions of use involving greatly unrestricted universes of discourse, a vast majority of occasions of use involving ‘dog’, and other non-scalar nouns for natural kinds, artifacts, etc. will be non-problematic occasions of use (similarly for non-degree adjectives like ‘canine’). Space limitations prevent detailed examination of these predicates here.²⁸ ²⁸ Another case that I can only mention cursorily is that of ‘appearance’ predicates like ‘looks red (to John)’. Such a predicate is often thought to give rise to a sorites paradox when the relation in the relevant (A) and (C) sentences is ‘looks the same in color (to John)’. Again the present picture suggests the possibility that ‘looks red (to John)’ fails to get an extension over universes of discourse
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There are also reference failures in some cases in which no true (A) sentence is in view, now due simply to insufficient specificity of the paradigm and generic preconceptions in play. These will be non-paradoxical but still irregular occasions of use. Think of this variant of the apartment example. A different couple (with a less modest income) must choose an apartment from a set K containing 65, 70, 100, 130, and 135 square meters apartments (A65, A70, A100, A130, and A135). A65 and A135 count clearly as small and not small for them, respectively, and 5 square meters don’t make a difference for them as to whether an apartment is small or not. The relevant preconceptions then imply that A70 is small, and that A130 is not small, but fail to imply that A100 is either in the extension of ‘small’ or in its anti-extension. In a case like this, even though there is no paradox, it is most reasonable to think that the predicate ‘is small’ fails to have a uniquely determined reference, for there seems to be no preconception determining that A100 should be either in its extension or in its antiextension.²⁹ In both irregular paradoxical and irregular non-paradoxical occasions of use, an impression is created that some objects far away from the positive and negative paradigms along the relevant dimension of comparison are ‘borderline cases’, objects that fall outside the extension and anti-extension of the adjective in question. The explanation of this impression according to the present picture is that these objects, besides being neither positive nor negative cases of application of the adjective, are not even preconceived as paradigms, and are psychologically far away from them along the relevant dimension of comparison.³⁰ The objects in question are not ‘borderline cases’ in the sense that they fall outside the extension and anti-extension of the adjective while the paradigms and objects easily reached from them by the generic that contain suitable sorites series of color patches, while it gets an extension in tamer universes. However, it is also quite possible that this case is in fact like other related cases which sometimes are thought to be sorites paradoxical but are not really so. We could train a pigeon to peck at big heaps of seed and to refrain from pecking at small heaps. Substitute ‘is pecked at (by the pigeon)’ for ‘F’ in (B)–(D) and ‘is indiscriminable for pecking purposes (by the pigeon)’ for ‘R’ in (A) and (C), and think of n as some suitably large number. Here the relevant (A) sentence seems simply false (while ‘is pecked at (by the pigeon)’ does get an extension). For any sorites series h1 , h2 , . . ., hn there will be some number i of seeds for which the pigeon will eventually fail to peck at hi+1 , after having pecked at hi , and thus it will after all discriminate in some way between the sizes of hi and hi+1 . In the same way, the camel’s back will break with a number j+1 of straws even though it did not break with j straws. (These ‘cut-off’ numbers will vary from circumstance to circumstance, but this doesn’t show that the predicates involved have any interesting semantic peculiarity; it only shows that the constitutions of the pigeon and the camel suffer minute changes from circumstance to circumstance.) ²⁹ There may also be cases in which the preconceptions about a term for a natural kind are insufficiently specific to generate a unique extension, even if they don’t enter into conflict with any truths. ³⁰ A speaker who considers successively the items in a sorites series will presumably reach a point where, e.g. though he is (already baffled but) ready to count as small a certain town ti , he is baffled and not ready to count ti+1 as small, and he may perhaps in some cases be (baffled and) ready to count ti+1 as not small. Contextualist theorists (see footnote 7) may explain these baffled shifts as arising from subtle context changes. The present theory postulates that the judgments in question lack a truth value, and that presumably the shifts (and the bafflement) of the speaker are to be explained by the truth value gaps rather than by any concealed context change.
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preconceptions fall inside. There is of course no extension/anti-extension pair in the problematic cases, and so no ‘borderline cases’ in the mentioned semantic sense.³¹ Nor are there ‘borderline cases’ in an epistemic sense, i.e. objects that fall either in the extension or in the anti-extension but are not known to fall in any of the two places. There are ‘borderline cases’ in a purely psychological sense. Normal speakers at relatively pretheoretical levels of use tend strongly to believe that utterances containing degree adjectives, of both paradigm and generic sentences, have truth values even in irregular occasions of use; the thought of an extension gap and accompanying truth value gaps is very hard to elicit from them. This fact was of course to be expected in speakers not exposed to the sorites paradox and to sufficient theoretical reflection on it, and constitutes no problem at all for the dual picture of this chapter. This case is no different from the case of failed natural kind terms, in which the plausible lack of reference and of truth conditions is nevertheless accompanied by a resolute acceptance by normal speakers of both the relevant generic and paradigm preconceptions. It is important, however, to stress the fact that, under the assumptions made earlier, this resolute acceptance would be hard to explain if we did not have at hand an often successful reference-fixing mechanism for natural kind terms that plausibly underlay normal people’s tacit understanding of how these terms come to have a reference (when they do). Analogously, in the case of degree adjectives this is precisely what is provided by the description of the mechanism of preconceptions in section 13.2. A further psychological factor that may contribute to the resoluteness with which normal speakers accept the relevant preconceptions is what we may call the closeness phenomenon. When we are confronted with a compelling (B)–(C)–(D) set in a particular paradoxical occasion of use, and in fact even when we are confronted with a compelling (B)–(C)–(D) set in an irregular but non-paradoxical occasion of use, there are potential very close regular occasions of use in which the same paradigm and generic sentences work as preconceptions, but generate a classical extension/anti-extension pair. The existence of these close occasions of use may even ³¹ The notion of a ‘borderline case’ has been closely associated with attempts to characterize vagueness. It is by now generally accepted that the existence of ‘borderline cases’ in a semantic sense could not characterize what it is for a predicate to be vague, for, assuming that ‘borderline cases’ in the semantic sense were possible, one could define predicates with precise cut-off points separating the positive and negative cases from the ‘borderline’, thus predicates with ‘borderline cases’ but not sorites susceptible. An alternative, and by now apparently popular proposal is to say that a vague predicate is a ‘boundaryless’ predicate (in the sense of Sainsbury 1990), i.e. one for which there is simply no semantically determined precise border separating the positive and the negative cases from the ‘borderline’. But think of the occasion of use in the second apartment example, and imagine that we introduce an adjective ‘small*’ stipulating that the sentences resulting from replacing ‘small’ with ‘small*’ in the paradigm and generic preconceptions of the original example are to be taken as of required acceptance by the semantics of ‘small*’ (and no other principle governs its semantics). Then, in a clear sense, no semantically determined precise border exists separating the positive and negative cases of apartment smallness* from the borderline (we don’t stipulate jointly necessary and sufficient conditions for membership in either the extension or the anti-extension of ‘is small*’). And yet there is no sorites susceptibility, because no sorites series is in sight in the occasion of use. Assuming that sorites susceptibility is a necessary condition on vagueness, it follows that ‘boundarylessness’ does not characterize it.
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divert to some extent our attention from paradox or irregularity in general. Let’s go back to the (Asmall )–(Dsmall ) set of our initial example and its described original occasion of use. A very close occasion of use is one in which the comparison class consists exclusively of Smalltown and/or a few tiny towns ‘reachable’ from it with the help of the generic preconception(s) in play in the original occasion of use, plus Nonsmalltown and/or a few big towns ‘reachable’ from it with the help of that(those) same generic preconception(s). In the new occasion of use the comparison class provides no sorites series, and the paradigm and generic preconceptions in play (the same as in the original occasion of use) suffice to divide the comparison class (and hence the universe of discourse) into two mutually exclusive and jointly exhaustive subclasses. Similarly, in the second apartment example, which was irregular but not paradoxical, a very close occasion of use is one in which the comparison class consists exclusively of the 65 square meters apartment and/or the 70 meters apartment, plus the 135 meters apartment and/or the 130 meters apartment. Here again there is no sorites series, but neither is there any psychological ‘borderline case’, and the paradigm and generic preconceptions in play suffice to divide the comparison class (and hence the universe of discourse) into two mutually exclusive and jointly exhaustive subclasses. In general, the strong tendency to believe that paradigm and generic sentences, and other sentences, are true even in irregular occasions of use, may to some extent be reinforced by the existence of closely similar and simpler occasions of use of the basic contrastive, regular kind described in section 13.3.³² In my view, one main strength of the picture of this chapter is the understanding it can provide of the fact that, even after reflection on paradox, people tend to give a preference to their intuitions on the truth value of paradigm preconceptions over their intuitions on the truth value of generic preconceptions—the phenomenon of the preference for paradigm intuitions, for short. Most theories of the sorites paradox have taken the preference for paradigm intuitions at semantic face value, and thus have assumed that paradigm preconceptions must be true. However, as argued in section 13.1 and recalled a few paragraphs ago, reflection suggests that standard paradigmatist theories of the sorites paradox are all false. I take this as a datum, and I think that what is needed is a theory that, while implying the falsity of paradigmatism, ³² In (2009) Peter Pagin has proposed a much more ambitious contextualist thesis, according to which in most contexts of use of a sorites susceptible predicate, some contextual factor restricts the domain of quantification so that a classical extension/anti-extension pair over the restricted universe is delivered for the predicate, and in such a way that sentences involving the predicate retain their intuitive truth value. However, it seems implausible that any factor determining contextually the domain of quantification works in such fine-tuned coordination with the mechanisms for predicate reference-fixing. For one thing, it is unlikely that any contextual factor determines two unique sharp cut-off points, between the extension of the predicate and the intermediate excluded cases and between these and the anti-extension. Pagin seems to agree with this, and to propose his theory not as a theory of the determination of reference, but as a theory about the determination of a class of extension/anti-extension pairs that might all equally well play the role of referents for a sorites predicate in paradoxical occasions of use. However, I doubt that even such a class can be determined in paradoxical cases: in the (Asmall )–(Dsmall ) example, it is implausible that there is a biggest town that will not appear in the extension of any of the extension/anti-extension pairs that might all equally well play the role of referents of ‘small’. (For discussion of Pagin’s theory I am indebted to Sven Rosenkranz.)
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can at the same time explain its appeal. The picture in this chapter does precisely that, when it is supplemented with a number of allied plausible conjectures about the psychology of paradigm beliefs. The preference for paradigm preconceptions has several plausible sources in the reliance on paradigms for reference-fixing and its associated psychology. Probably paradigm preconceptions are psychologically more basic than generic preconceptions in many respects. For example, it is well known that the inclination to classify under a common predicate certain paradigms or prototypes develops earlier in children than any implicit generic idea as to how one should expand the extension of the predicate starting from the paradigms. This inclination is also probably of more adaptive or practical value at pretheoretical levels of use than the development of any generic idea; one can classify and contrast particular objects by means of paradigm preconceptions (even if these turn out to contain semantically defective predicates), thus getting a means to influence and react to one’s hearer’s responses to particular objects, but one cannot do that merely with generic preconceptions. At a more theoretical level of use, and especially after exposure to paradox or conflict, other factors may contribute psychologically to the preference for paradigm preconceptions. It is clear that stipulatively rejecting the paradigm preconceptions would involve a more radical departure from established usage than stipulatively rejecting the generic preconceptions. We might, for example, fix a generic principle determining the extension of ‘madness’ by stipulation, but we could not stipulate paradigm preconceptions about ‘madness’ to be false without suppressing our ability to effect distinctions with the help of ‘madness’; similarly, we might fix cut-off points for ‘small’ by stipulation in irregular occasions of use, but we could not stipulate paradigm sentences about ‘small’ to be false without suppressing our ability to effect distinctions with the help of ‘small’. There is also the presumable fact that occasionally, after the emergence of paradox or conflict, and without the help of explicit stipulations, linguistic practice settles on some generic principle that is compatible with the original paradigm preconceptions. The psychologically evident possibility of stipulations or implicit choices of generic principles compatible with the original paradigm beliefs may well cause to some extent the preference for paradigm preconceptions. But of course, this possibility does not imply that preexisting usage does, or even can, single out non-arbitrarily any generic principle that fixes the extension of a problematic predicate. These psychological factors are presumably in operation whenever a referencefixing mechanism relies heavily on paradigm preconceptions. Other plausible psychological factors contributing to the preference for paradigm preconceptions are specific to the use of degree adjectives. Paradigm preconceptions about these adjectives are often less variable than generic ones, even with respect to the same comparison class. Presumably the 65 meters apartment will count as small under any standard with respect to all (or most) comparison classes in which it is the smallest apartment; but a generic preconception with the intuitive content that five meters don’t make a difference as to smallness will vary widely in perceived truth value even with respect to a fixed comparison class where the 65 meters apartment is the smallest apartment. Other paradigm preconceptions are stable even across all occasions of use of a
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predicate; for example, a man with no hairs counts as intuitively bald in all occasions of use. It seems to me that together, all these probable psychological factors provide considerable support for the thesis that the preference for paradigm preconceptions does not have a semantic root. If we take it as a datum that paradigmatist semantic theories of the sorites paradox are not determined to be correct by preexisting usage, the existence of these psychological explanations goes quite a bit of the way toward eliminating the paradigmatist inclination suggested by the preference for paradigm preconceptions. Finally, I should stress that, unlike strong nihilist theories, the dual picture is not necessarily self-referentially instable. Its proponent says ‘All sorites susceptible predicates, in paradoxical occasions of use, lack an extension, and all sorites susceptible predicates, in occasions of use in which the mechanism of preconceptions works, have an extension’. In order for his utterance to be true, the occasion of use in which he makes it must be one in which the predicates appearing in that sentence have an extension. Do they have an extension in the relevant occasion of use? Assuming that predicates in general have their extensions determined (when they do) by mechanisms of preconceptions related to the ones postulated for degree adjectival predicates and natural kind predicates, the problem is basically the problem of what is the typical universe of discourse presupposed in that occasion of use or similar ones, and of whether the preconceptions associated with the predicates appearing in the sentence fix classical extension/anti-extension pairs for them over that universe. This in turn reduces to the question whether the proponent of the dual picture needs to quantify over things which, in the relevant occasion of use, are neither clear predicates nor clear non-predicates, or neither clear occasions of use nor clear non-occasions of use, etc. I conjecture that he doesn’t need so to quantify. Metaphysical theories often quantify over large universes of discourse. But it’s unclear that an appropriate theory of the basic linguistic phenomena surrounding the sorites must be a metaphysical theory. It may be a linguistic theory that doesn’t need to quantify over universes of discourse containing inordinately large numbers of things. For example, it might be claimed that some types of sounds emitted by humans in some counterfactual, imaginable or even real cases are neither clear cases of predicates nor clear non-cases, perhaps because they are neither clear cases of words nor clear cases of non-words. But the proponent of the dual picture doesn’t need to consider the properties of those sounds, just as a syntactician doesn’t typically theorize about sounds or expressions that are not clear words. The dual picture is intended only for things of the type we find in dictionaries, in English and similar languages. In the universe of things it quantifies over, its proponent can assume a clear division between words and non-words, and presumably between predicates and non-predicates. Related remarks hold for the picture’s use of ‘occasion of use’ and other predicates that appear in its formulation. Unfortunately, space limitations again prevent further discussion of this issue in this preliminary presentation.
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Re f e re n c e s Cargile, J. (1969), ‘The sorites paradox’, British Journal for the Philosophy of Science 20, 193–202. Dummett, M. (1975), ‘Wang’s paradox’, Synthese 30, 301–24. Eklund, M. (2002), ‘Inconsistent languages’, Philosophy and Phenomenological Research 64, 251–75. Fara, D. G. (2000), ‘Shifting sands: An interest-relative theory of vagueness’, Philosophical Topics 28, 45–81. (Originally published under the name Delia Graff.) Fine, K. (1975), ‘Vagueness, truth and logic’, Synthese 30, 265–300. G´omez-Torrente, M. (2002), ‘Vagueness and margin for error principles’, Philosophy and Phenomenological Research 64, 107–25. (2006), ‘Rigidity and essentiality’, Mind 115, 227–59. (forthcoming), ‘Kripke on color words and the primary/secondary quality distinction’ in A. Berger (ed.), Saul Kripke, Cambridge University Press, Cambridge, forthcoming. Heck, R. G. (2003), ‘Semantic accounts of vagueness’ in Jc Beall (ed.), Liars and Heaps, Oxford University Press, New York, 106–27. Kennedy, C. (2007), ‘Vagueness and grammar: The semantics of relative and absolute gradable predicates’, Linguistics and Philosophy 30, 1–45. Klein, E. (1980), ‘A semantics for positive and comparative adjectives’, Linguistics and Philosophy 4, 1–45. Livingstone, E. A. (2006), The Concise Oxford Dictionary of the Christian Church, 2nd rev. edn., Oxford University Press, Oxford. Machina, K. (1976), ‘Truth, belief and vagueness’, Journal of Philosophical Logic 5, 47–78. Pagin, P. (2009), ‘Vagueness and Central Gaps,’ in this volume. Sainsbury, M. (1990), ‘Concepts without boundaries’, reprinted in R. Keefe and P. Smith (eds.), Vagueness. A Reader, MIT Press, Cambridge, MA, 1996, 251–64. Salmon, N. (1982), Reference and Essence, Blackwell, Oxford. Schiffer, S. (1999), ‘The epistemic theory of vagueness’, Philosophical Perspectives 13, 481–503. Soames, S. (1999), Understanding Truth, Oxford University Press, New York. Unger, P. (1979), ‘There are no ordinary things’, Synthese 41, 117–54. (1984), Philosophical Relativity, University of Minnesota Press, Minneapolis. Williamson, T. (1994), Vagueness, Routledge, London. (1999), ‘Schiffer on the epistemic theory of vagueness’, Philosophical Perspectives 13, 505–17. Wright, C. (2003), ‘Vagueness: A fifth column approach’ in Jc Beall (ed.), Liars and Heaps, Oxford University Press, New York, 84–105.
14 Vagueness and Central Gaps Peter Pagin
Ordinary intuitions that vague predicates are tolerant, or cannot have sharp boundaries, can be formalized in first-order logic in at least two non-equivalent ways, a stronger and a weaker. The stronger turns out to be false in domains that have a significant central gap for the predicate in question, i.e. where a sufficiently large middle segment of the ordering relation (such as taller for ‘tall’) is uninstantiated. The weaker principle is true in such domains, but does not in those domains induce the sorites conclusion. This fact can be used for interpreting ordinary uses of vague expressions by means of a new kind of contextual quantifier domain restriction. A central segment is cut from the domain, if consistent with speaker intentions. As long as this is possible, tolerance, bivalence and consistency can all be retained. This chapter focuses on the basic semantic properties in a model-theoretic setting. The natural language application is sketched and the nature of the approach briefly discussed.
14.1
TO L E R A N C E P R I N C I P L E S
A sorites argument in the inductive format is normally taken to have the following form: (1) 1 F (k1 ) 2 ∀i(F (ki ) → F (ki+1 )) 3 F (kn ) This chapter was presented as a paper at the 6th Arch´e vagueness workshop in St Andrews, in March 2006, at the Logic and Language conference in Birmingham, in April 2006, and at the Stockholm Logic and Language seminar in September 2006. I am grateful to the audiences on those occasions, in particular to Herman Cappelen, Manuel Garc´ıa-Carpintero, Richard Dietz, Patrick Greenough, Patrick Grim, Jeff King, Andrew McGonigal, Augustin Rayo, Tim Williamson, and Crispin Wright. I owe especially much to Sven Rosenkranz who was my commentator in Birmingham, and who also provided me with further helpful comments on a later version. I also owe much to Kathrin Gl¨uer for discussions of these ideas over several years. The work on this paper was funded by a research grant from The Swedish Research Council.
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The terms ‘k1 , . . . kn ’ are taken to denote objects a1 , . . . an in a sequence A along some ordering relation, such as taller than. We have an instance of the sorites paradox if the premises are apparently true while the conclusion is apparently false. We get a typical example by choosing ‘is tall for a man’ for ‘F ’, and a1 , . . . an as a sequence of men so that a1 is 200 cm tall, and each ai+1 is 1 mm shorter than his predecessor, while an is 150 cm tall. In such a case both premises do seem true, while the conclusion also does seem false. The argument is inductive, and the second premise is the inductive premise. The apparent acceptability of the inductive premise derives from a basic intuition about typically vague terms: that they do not have sharp boundaries. In the case imagined, that intuition would typically be expressed by means of (2) 1 mm cannot make the difference between being tall and not being tall. Put in a more inductive format, we would say, in this case: (3) If a man of n + 1 mm is tall, then a man of n mm is tall. When put in the format of (3), the formulation is apt for expressing a version of the basic intuition: that the predicate in question is insensitive to small differences. The property of being insensitive to small differences is what Crispin Wright has called ‘tolerance’ (Wright 1976, 156). Wright says What is involved in treating these examples as genuinely paradoxical is a certain tolerance in the concepts which they respectively involve, a notion of a degree of change too small to make any difference, as it were. . . . Then F is tolerant with respect to φ if there is also some positive degree of change in respect of φ insufficient ever to affect the justice with which F applies to a particular case.
For reasons that are briefly given later (section 14.5), I prefer to speak of linguistic expressions as being vague or tolerant, rather than concepts, but that is for now less significant.¹ Wright’s introduction of the notion of tolerance, over and above the notion of lacking sharp boundaries, was important, not just because a term may in fact lack sharp boundaries for one reason or other,² but because the tolerance of a vague term intuitively explains why it lacks sharp boundaries. It is conceivable that a term lack sharp boundaries in all worlds where it has the same meaning, even though there is a nomological or metaphysical explanation of why this is so that has nothing to do with vagueness.³ For present purposes I shall refer to principles like (3) as tolerance principles, even though, for the reasons just stated, they can be true because of other factors than ¹ I also find they claim, that the justice with which the term applies isn’t affected, unnecessarily strong. It would be enough for purposes of an account of vagueness, it seems to me, if the degree of justification drops by approximating a limit value that is high enough. ² This could be, for instance, because by nomic regularity, animal species cannot be very similar. As long as we treat vagueness extensionally, it can also be because the predicate has an empty extension. ³ We cannot explain in general how the distinction between being true in virtue of meaning properties and being true in virtue of properties of what is denoted is to be applied (there does not seem to be an analytic/synthetic dichotomy in this sense), but we may be able to do somewhat better when it comes to tolerance.
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insensitivity. A tolerance principle is an inductive principle that involves a tolerant predicate—‘tall’ in the case of (3)—and a tolerance level —1 mm in the case of (3). A tolerance level, as I use the term, then, does not depend on any particular sequence of objects, but makes explicit what the difference is to be in a sequence of objects that is appropriate for an inductive premise like the one in 1. Since tolerance principles are themselves inductive principles we can as well state a sorites argument with their help, without relying on any particular sequence. In order to so, we must make the logical form precise. It quickly turns out, however, that the informal statement (3) admits of two different formalizations. The form of antecedent and consequent, (4) A man of x mm is tall as it occurs in (3), has a generic interpretation, which, I assume, is equivalent to (5) ∀y(Man(y) & H (y) ≥ x → Tall(y)) with ‘H ’ for ‘the height of . . . in mm’.⁴ Accordingly, given that (3) itself is of conditional form, the natural predicate logic rendering would be (T1) ∀y(Man(y) & H (y) ≥ n + 1 → Tall(y)) → ∀y(Man(y) & H (y) ≥ n → Tall(y)) On the other hand, we would also take (3) to have a reading like this: (6) For any two men, if the one is (not more than) n + 1 mm tall and the other is (at least) n mm tall, then if the former is tall, so is the latter. And, the natural formalization of (6) is (T2) ∀x∀y((Man(x) & Man(y) & H (x) ≤ n + 1 & H (y) ≥ n) → (Tall(x) → Tall(y))) But although we seem to get (T1) as well as (T2) out of the intuitive formulation (3), (T1) and (T2) are not equivalent. The difference is made more perspicuous by noting that (T2) is equivalent with (T2 ) ∃y(Man(y) & H (y) ≤ n + 1 & Tall(y)) → ∀y(Man(y) & H (y) ≥ n → Tall(y)) I shall call tolerance principles of the form (T1) strong tolerance principles, and those of the form (T2)/(T2 ) weak tolerance principles.⁵ Weak and strong tolerance principles work differently. With respect to some cases they induce the same result, but not in all. Consider the original setup, with the sequence A = a1 , . . . , an , and Let G(n) =def ∀y(H (y) ≥ n → Tall(y)) ⁴ The generic reading of (4) involves an element of nomicity that is not captured by the quantification in (5), but I shall not concern myself with that aspect. ⁵ I shall henceforth take quantification to be restricted to a domain of men, and so the ‘Man’ conjunct will be dropped.
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We then get with (T1) a modified sorites argument: (7)
1 2 3 4 5 6
G(2000) H (kn ) = 1500 & ¬ Tall(kn ) ∀i(G(i + 1) → G(i)) G(1500) Tall(kn ) ⊥
assump. assump. (T1) 1, 3 2, 4 2, 5
Note that the absurdity conclusion requires the assumption (2), for by itself, (4), i.e. ∀y(H (y) = 1500 → Tall(y)) is true if there are no men of 150 cm height in the universe of discourse. Similarly, 1 is true if there are no men of height 200 cm in the universe of discourse. We could add a dependence on actual instantiation and drop the assumption 1. We then replace it with assumption about k1 , together with a number uniformity principle, stating that anyone is tall whose height is at least equal to that of someone who is tall:⁶ (U) ∀x, y(H (y) ≥ H (x) → (Tall(x) → Tall(y))) We then have the simple derivation (8)
1 H (k1 ) = 2000 & Tall(k1 ) 2 G(2000)
assump. 1, (U)
which can replace the first assumption in (7). In that case the contradiction depends on the (U), (T1), and the facts about the heights and tallness attributes of k1 and kn . In either case, at least one singular fact is needed. If instead we use the weaker tolerance principle (T2), the derivation will be rather different. First, another abbreviation: Let E(n) =def ∃x(H (x) ≤ n & Tall(x)) With respect to the same domain of men, we will then have the sorites argument: (9)
1 2 3 4 5 6 7
H (k1 ) = 2000 & Tall(k1 ) H (kn ) = 1500 & ¬Tall(kn ) ∀i(E(i + 1) → G(i)) E(2000) G(1999) H (k2 ) = 1999 → Tall(k2 ) H (k2 ) = 1999
assump. assump. (T2) 1 (T2), 4 5 assump.
⁶ This is of course a simplification, since a man of 200 cm height and 250 cm width would not count as tall, but I shall disregard shape. Note that (U) implies both the claim that men of the same height have the same tallness attribute, and the so-called penumbral principle that anyone taller than someone who is tall, is tall. The penumbral principle follows on the assumption that x is taller than y iff H (x) > H (y).
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Tall(k2 ) H (k2 ) = 1999 & Tall(k2 ) E(1999) G(1998) .. .
12 13 14 15 16
G(1500) H (kn ) ≥ 1500 → Tall(kn ) Tall(kn ) ¬Tall(kn ) ⊥
6, 7 7, 8 9 10, (T2)
12 2, 13 2 14, 15,
For deriving the contradiction with (T2), we need as premises both the heights and tallness attributes of a positive and a negative specimen (i.e. premises 1 and 2 of sorites argument (9)), but also each of the height instantiations (including 7). That is, the sorites argument with (T2) requires that there be an unbroken sequence of individuals from a positive instance to a negative instance, where adjacent members differ by at most the tolerance level (in the example, 1 mm). Initially, this may seem to mean nothing more than that a simple and elegant sorites argument can be made complex and cumbersome. However, the difference between (T1) and (T2) gains importance in relation to a domain where the chain in question has a significant central gap. Informally, a significant central gap is a gap in the chain which is bigger than any admissible tolerance level, and such that there are only positive instances on the one side of the gap and only negative instances on the other side. With respect to our domain of men of different heights, we would have such a gap if, say, all members of heights between 190 cm and 160 cm were removed. Thus, suppose that from the vocabulary, the terms k102 , . . . , k400 , denoting men shorter than 190 cm and taller than 160 cm, are removed. What happens with the sorites arguments? As regards (T1), the argument (7) goes through as before, since (T1) only requires one negative instance (and in the expanded version, one positive and one negative instance). As regards (T2), however, the argument (9) clearly does not go through: in order to derive the intermediate conclusion that all men of a height of 1898 mm are tall, we need as a premise that there is a man of at most 1899 mm who is tall, but since this time there are no men in the domain of heights between 1900 mm and 1898 mm, the derivation is broken at this step. The instantiation picks up again at the term k401 , denoting a man of 160 cm height, but that man is already clearly short, and so the premise Tall(k401 ) cannot be justifiably added. Therefore, the derivation cannot be resumed at this step, and because of that the conclusion Tall(kn ) cannot be reached. That is, because the tolerance principle that would be needed to bridge the gap postulates an unacceptably high tolerance level—30 cm—the conclusion cannot be reached, and the sorites argument fails.
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That the (T2) argument fails is not of great significance if the (T1) argument, which does not rely on an unbroken chain of instantiations goes through anyway. However, it doesn’t. Interpreted in a domain with a significant central gap, the (T1) principle is straightforwardly false. With respect to our example, consider the instance of (T1) at the lower end of the gap: (10) G(1601) → G(1600) i.e. (10) ∀y(H (y) ≥ 1601 → Tall(y)) → ∀y(H (y) ≥ 1600 → Tall(y)) To make things explicit, we add as an extra premise (11) ¬ Tall(k401 ) where the term ‘k401 ’ denotes the man of 1600 mm height. Now, the antecedent of (10) is true: anyone in the domain that is 1601 mm or higher is 1900 mm or higher, and so tall. The consequence, however, is false: there is an individual of 1600 mm, the referent of ‘k401 ’, and that individual is not tall, by the truth of (11). Hence, the instance (10) of (T1) is false, and so (T1) itself is false. Therefore, in a gappy domain, the falsity of (T1) does not entail the existence of a sharp boundary. By contrast, (T2) is true at the lower end of the gap, for (12) ∃y(H (y) ≤ 1601 & Tall(y)) → ∀y(H (y) ≥ 1600 → Tall(y)) is true: the antecedent is false, since any domain member shorter than or equal to 1601 mm is shorter than or equal to 1600 mm, and therefore not tall. But since the antecedent is false, we can’t detach the consequent, and don’t have a contradiction. Similarly, (T2) is true at the upper end of the gap. (13) ∃y(H (y) ≤ 1900 & Tall(y)) → ∀y(H (y) ≥ 1899 → Tall(y)) The antecedent is true, but so is the consequent, for any domain member with a height of at least 1899 mm has a height of at least 1900 mm, and so is tall. So with respect to a domain with a significant central gap, the strong tolerance principle (T1) is false, but the falsity does not entail that there is a sharp boundary. The weak tolerance (T2) is true, but does not lead to inconsistency. Only the existence of a sharp boundary makes (T2) false (see next section). Again, these facts might seem to be of marginal interest, since the difficult problems concerning vagueness relate to non-gappy domains. I shall, however, make use of these facts for a semantics of natural language sentences where the existence of significant central gaps is not presupposed. In the next section, the (T1) and (T2) principles, and their semantic relation, will be characterized model-theoretically. It will turn out that under natural conditions, (T2) is a consequence of (T1). The non-technical reasoning, and the application to vagueness, will be resumed in section 14.3.
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M O D E L S O F TO L E R A N C E A N D G A P S
To characterize the (T1) and (T2) principles formally, we define a class of models: Definition 2.1. A V–model M is a classical model D, I, T , H for a first-order language L, where D is a domain of individuals and I an interpretation function, T a function that for each predicate letter F in L as argument assigns a real number TF ≥ 0, and H a function that for each predicate letter F in L assigns a total function HF from D to R, such that (+) for any a, b ∈ D (HF (a) ≥ HF (b) → (b ∈ I(F ) → a ∈ I(F ))) By a classical model I mean a model where the truth definition gives the standard classical clauses for the first-order logical constants. These clauses will be assumed below. The function TF will be used below for assigning a tolerance level to the predicate F in the model. Definition 2.2. A V-model M has a central F-gap iff there are no objects a, b ∈ DM such that a ∈ I(F ), b ∈ I(F ) and 0 ≤ HF (a) − HF (b) ≤ TF Remark 2.3. It would be natural to identify a central gap with a pair i, j of real numbers such that i is the least upper bound and j the greatest lower bound of the gap. But although this can be done, there is no simple and uniform definition of the pair. For if we allow dense and continuous domains of objects, there need not be any smallest member of the set (for instance, for every long period of time in the domain, there may be a shorter period of time in the domain that is still long, even though not every period is), nor any largest number that is not the measure of a member (e.g. for every real number u that is not the length of a long period in the domain, there is greater number v that is also not the length of a long period in the domain, even though there are long periods). We are only guaranteed by the Least Upper Bound Axiom of real analysis that one of the two must exist. But then to identify a gap with a pair of numbers leads to a considerable increase in the complexity of definitions and proofs. Secondly, as the definition is stated, a domain has a central F -gap even in case all objects are F :s or no object is. Again, it simplifies the definition and the reasoning with it not to require both positive and negative instances in the domain. But if we don’t, there need not even be a definite pair i, j to identify with the gap. However, for the natural language semantics to be presented in section 14.4, it is more convenient to identify gaps with definite pairs of numbers. End of remark. Now we can characterize the two tolerance principles model theoretically: Definition 2.4. A (T1)-model M for a predicate F of L is a V-model D, I, T , H for L such that TF > 0, and (ti) For all k (if (for all a ∈ DM (if HF (a) ≥ k + TF , then a ∈ I(F )), then (for all a ∈ DM (if HF (a) ≥ k, then a ∈ I(F )))
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Definition 2.5. A (T2)-model M for a predicate F of L is a V-model D, I, T , H for L such that TF > 0, and (tii) For all k, (if (there is a ∈ DM (HF (a) ≤ k + TF , and a ∈ I(F )), then for all a ∈ DM (if HF (a) ≥ k, then a ∈ I(F ))) We shall now prove some elementary properties of the (T1) and (T2) models. Definition 2.6. A V-model M is (a) (b) (c) (d) (e)
HF -unbounded iff for all k ∈ R there is a ∈ DM such that HF (a) ≥ k empty iff DM = ∅ F -full iff DM ∩ I(F ) = DM F -empty iff DM ∩ I(F ) = ∅ F -free iff M has a central F -gap
Fact 2.7. For all V-models M, M is a (T1)-model for a predicate F of L iff M is empty, or M is non-empty and F -full, or M is non-empty and F -empty and HF unbounded. Proof. Left to right. Assume for reductio that M is a) non-empty, and b) not F -full, and c) not both F -empty and HF -unbounded. Because of a) and b) there is a non-F , i.e. an object b− ∈ DM − I(F ). Because of a) and c), either—case A—there is an F , i.e. an object b+ ∈ DM ∩ I(F ), or—case B—M is HF -bounded, i.e. there a k ∈ R such that for all a ∈ DM , HF (a) < k. Consider first case A. By clause (+) of the definition of a V-model, it holds that (i) for all a ∈ DM (if HF (a) ≥ HF (b+ ), then a ∈ I(F )) Then by repeated applications of (ti), from (i) we finally get (ii) for all a ∈ DM (if HF (a) ≥ HF (b− ), then a ∈ I(F )) Since HF (b− ) ≥ HF (b− ), by (ii) we can conclude that b− ∈ I(F ), contradicting the assumption. Hence, case A cannot hold. Consider then case B. Let kj > HF (a) for all a ∈ DM . Then (iii) for all a ∈ DM (if HF (a) ≥ kj , then a ∈ I(F )) Then we can again apply (ti) from (iii) repeatedly, until we have derived (ii), again concluding that b− ∈ I(F ), contradicting the assumption. Hence, case B cannot hold either. Right to left. Three cases: A) M is empty, B) M is non-empty and F -full, C) M is non-empty and F -empty and HF -unbounded. Consider case A. Since by assumption DM = ∅ it holds for any k ∈ R that (iv) for all a ∈ DM (if HF (a) ≥ k, then a ∈ I(F ))
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Therefore, it also holds for any k ∈ R that (v) if for all a ∈ DM (if HF (a) ≥ kj + TF , then a ∈ I(F )), then for all a ∈ DM (if HF (a) ≥ kj , then a ∈ I(F )) and then (ti) holds as well. Case B. Since by assumption it holds for all a ∈ DM that a ∈ I(F ), (iv) will again hold for any k ∈ R, and so we have the same conclusion as in case A. Case C. Assume for reductio that (ti) is false, and hence there is a kj such that (vi) for all a ∈ DM (if HF (a) ≥ kj + TF , then a ∈ I(F )) is true, while (vii) for all a ∈ DM (if HF (a) ≥ kj , then a ∈ I(F )) is false. Since by the first C-case assumption it holds for any a ∈ DM that a ∈ I(F ), (vi) is true only if it holds that (viii) for all a ∈ DM (HF (a) < kj + TF ) But since by the second C-case assumption M is HF -unbounded, this does not hold. Hence (ti) is true. In all three cases, (ti) true, and therefore M is a (T1) model. Fact 2.8. For all V-models M, M is a (T2)-model for a for a predicate F of L iff M is empty, or M is non-empty and F -full, or M is non-empty and F -empty, or M is non-empty and F -free. Proof. Left to right. Assume for reductio that a) M is non-empty, b) not F -full, c) not F -empty, and d) not F -free. Because of a) and b), there is a non-F , i.e. an object b− ∈ DM − I(F ). Because of c), there is an F , i.e. an object b+ ∈ DM ∩ I(F ). Since M is not F -free, there are two objects a, b ∈ DM such that a ∈ I(F ), b ∈ I(F ) and 0 ≤ HF (a) − HF (b) ≤ TF . Hence (i) HF (a) ≤ HF (b) + TF Because a ∈ I(F ), we have (ii) there is c ∈ DM (HF (c) ≤ HF (b) + TF , and c ∈ I(F )) Applying (tii) we can conclude from (ii) (iii) for all c ∈ DM (if HF (c) ≥ HF (b), then c ∈ I(F ))) Instantiating, we have the conclusion that b ∈ I(F ), contrary to assumption. Therefore, M cannot be simultaneously non-empty, not F -full, not F -empty, not F -free. Right to left. We have four cases: A) M is empty, B) M is non-empty and F -full, C) M is non-empty and F -empty, and D) M is non-empty and F -free. Case A. By assumption, M is empty, and so (iv) there is a ∈ DM (HF (a) ≤ k + TF , and a ∈ I(F ))
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is false for all k ∈ R, and hence the conditional (v) if (there is a ∈ DM (HF (a) ≤ k + TF , and a ∈ I(F )), then (for all a ∈ DM (if HF (a) ≥ k, then a ∈ I(F )) is true for all k ∈ R, and so (tii) follows. Case B. By assumption, M is non-empty and F -full. Then, all a ∈ DM are also in I(F ), and so (vi) for all a ∈ DM (if HF (a) ≥ k, then a ∈ I(F )) is true for all k ∈ R. Hence, (tii) is again true for all k ∈ R. Case C. By assumption, M is non-empty and F -empty. As in case A, (iv) is false for all k ∈ R, and the rest follows. Case D. By assumption, M is non-empty and F -free. If M is F -empty or F -full, (tii) follows, so assume M is neither. In order for (v) to be false for a particular k ∈ R, there must a pair b+ , b− ∈ M such that b+ ∈ I(F ), b− ∈ I(F ), and (vii) HF (b+ ) ≤ HF (b− ) + TF Then (v) is false for k = HF (b− ). However, if there is such a pair b+ , b− ∈ M, then either HF (b− ) > HF (b+ ), or 0 ≤ HF (b+ ) − HF (b− ) ≤ TF . The first disjunct is ruled out by condition (+) of the definition of V-models. If the second disjunct is true, then by definition 2.2, there is no central F -gap in M. Since by assumption there is a central F -gap, the second disjunct is false as well. Then, (v) cannot be false for any k, and hence (tii) holds. In all four cases, (tii) holds, and therefore M is a (T2) model. Fact 2.9. (T 1) |V (T 2) but (T 2) |V (T 1) Proof. It follows from Facts 2.7 and 2.8 that the class of (T1)-models is a proper subclass of the class of (T2)-models. (T1) will be false in (T2)-models M that are nonempty, not F -full, not F -empty and F -free, as well as in (T2)-models M that are non-empty, not F -full, F -empty but not HF -unbounded.
14.3
M E T H O D O LO G I C A L I N T E R LU D E
The standard alternatives for coping with the sorites paradox are (a) (b) (c) (d)
reject the validity of the argument question or reject the (strict) truth of the inductive premise reject the truth of the minor premise or the falsity of the conclusion accept the whole reasoning and conclude that the vague predicate is incoherent.⁷ ⁷ Cf. Keefe and Smith 1996b, 10.
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I think it is fair to say alternative b) has been the most popular one in the literature of recent decades. Epistemicists such as Timothy Williamson (1994) take the inductive premise to be straightforwardly false, although it is not knowable where the boundary is. Supervaluationists, such as Kit Fine (1975) or Rosanna Keefe (2000), take the inductive premise to be (super-)false, since false in every classical evaluation. Degree theorists, such as Kenton F. Machina (1976), take the inductive premise to be almost completely false, even though each of its instances is almost completely true. Contextualists, like Diana Raffman (1994, 1996), Scott Soames (1999), Delia Graff Fara (2000), and Stewart Shapiro (2003, 2006) take the inductive premise to be false in each context, although the boundary shifts between contexts. I shall propose that we accept the inductive premise, i.e. in the form of the surviving tolerance principle, exemplified by (T2), or rather, a certain revised version of it. Without going deeply into polemics with the dominant trend, I take it to be intuitively part of the semantics of vague predicates to be insensitive to small differences. Our inability to locate any sharp boundary of vague predicates, and even more the intuitive rejection by ordinary speakers of the very idea of a sharp boundary, suggests that it is part of the meaning of vague expression as used by ordinary speakers not to have them. Although it is well known that all the theories of vagueness that involve rejecting the inductive premise come at a high cost, rejection in any of the proposed forms may still seem to involve smaller costs than does acceptance. Since drastic revisions of logic are required to treat the standard sorites argument as invalid, and since it is implausible to deny the existence of tall men, or of nontall men, the only remaining option seems to be that the use of vague vocabulary is incoherent, or even inconsistent. This position has been advocated e.g. by Michael Dummett (1975). Dummett says Wang’s paradox merely reflects this inconsistency. What is in error is not the principles of reasoning involved, nor, as on our earlier diagnosis, the induction step. The induction step is correct, according to the rules of use governing vague predicates such as ‘small’: but these rules are themselves inconsistent, and hence the paradox. Our earlier model for the logic of vague expressions thus becomes useless: there can be no coherent such logic. (1975, 265)
A little before that, Dummett provides a gloss on ‘consistent’: ‘Consistent’ here means that it would be impossible to force someone, by appeal to rules that he acknowledged as correct, to contradict himself over whether the predicate applied to a given object. (1975, 264)
That the use of vague vocabulary is inconsistent may be seen as rendered plausible from considerations of so-called forced march sorites, which ‘is designed to force us, one step at a time, into a separate verdict on each successive pair of adjacent items in a sorites sequence’ (Horgan 1994, 173). Our inclination to respect initial intuitions about clear cases while not accepting any sharp boundary, naturally leads us into trouble.
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On the other hand, there is a frequent use of vague vocabulary that by common sense standards fulfill its communicative function pretty well. In fact, most applications of vague predicates appear unproblematic. This observation has been used in an objection against Dummett by Crispin Wright: [ . . . ] what is actually responsible, on this view, for the large degree of coherence and communicative success which our use of color vocabulary enjoys? Indeed, what is the justification for continuing to think of the use of such expressions as governed by rule? Knowledge of appropriate rules was supposed to constitute linguistic competence. But it cannot do so if competent usage essentially has a coherence which, in Dummett’s view, the rules lack. Dummett’s response needs supplementing with an explanation of our communicative success with such vocabulary in which the idea of knowledge of inconsistent rules has an ineliminable part to play. For either such knowledge is still to be a basic ingredient in competence or we should drop the idea. (Wright 1987, 212)
Directly following this, Wright makes a related point: That brings us to [ . . . ] a decisive objection to Dummett’s response. I do not see how we can rest content with the idea that certain implicitly known semantic rules are incoherent when nobody’s reaction, on being presented with the purported demonstration of the inconsistency, i.e. the paradox—even if they can find no fault with it—is to lose confidence in the unique propriety of the response—e.g. ‘That’s orange’—which the demonstration seems to confound. (Wright 1987, 213)
I agree with Wright. We should try to account for the apparent communicative success of the use of vague vocabulary. Most of natural language lexical items are vague, and to dismiss the use as governed by inconsistent rules is bad theory. On the other hand, it seems to me absurd to try to rescue every single sequence of applications of vague predicates. Speakers do contradict themselves. It cannot be the goal of semantic theory to represent natural language as a foolproof means of making good sense (after all, people do paint themselves into corners and cut off the branches they are sitting on). These considerations suggest that we try to find a semantic account of ordinary applications of vague predicates that does not reject all tolerance principles as unacceptable. For a forced-march sorites is not a sequence of ordinary applications. I shall therefore propose a combination of two strategies. For ordinary contexts of use, I propose that we opt for strategy a): reject the validity of the argument. For certain extreme contexts, on the other hand, I propose alternative d): the use of the vague predicate is incoherent. In the next section, I shall sketch such an account, based on the observations of the first section. It is a contextual account, but unlike the mainstream of current contextual accounts, the most important feature will not be the shift between contexts, but an extra element in the context itself.
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CENTRAL GAP SEMANTICS
I think that the following correctly describes ordinary speaker psychology of ordinary applications of vague predicates: A predicate, like ‘tall’, is applied to a clear case of tallness or non-tallness, and although the speaker does not think that a sufficiently small difference, like 1 mm, can decide between being the one or the other, the difficult intermediate cases are simply ‘forgotten’, or ‘ignored’. It is enough that the case considered is clear, and that there are clear contrastive cases. There is no need to consider intermediate cases. We could say that intermediate cases are dismissed. It is this dismissal of intermediate cases that can be modeled by means of interpretation that introduces significant central gaps in the domain of discourse in those cases where it is needed. It is widely agreed that for interpreting (normal) utterances of sentences like (14) Everyone went to bed at midnight a contextually induced restriction on the quantifier domain is needed. The present proposal is that we extend the tool of quantifier domain restriction to give a context semantics for vague expressions that respects tolerance, bivalence as well as consistency for normal use. I shall propose that for a vague predicate F , like ‘. . . is tall’, a speaker that accepts it as tolerant with respect to a particular dimension, uses it with a tacit assumption that there is a restriction on the domain of discourse, such that the domain in question has a significant central F -gap. In this application, it is convenient to think of a central gap for a predicate F in a context c, relative to a dimension of variation, as a pair of real numbers (i, j) with respect to a measurement scale and a measure function HF . With an initially given domain D of individuals, the gap determines a proper subset of D, the set Fc = {a ∈ D : j ≤ HF (a) ≤ i} of individuals in D whose F measures are in the gap. The gap then forces a cut in the domain, consisting in subtracting Fc from D. That is, the restricted domain is D − Fc .⁸ The type of semantics that I suggest is a context semantics.⁹ It involves the assumption that in each context c, for each vague predicate F that is used in c, a tolerance level and a central gap is determined for F . It assumes the existence of a general gap function G that maps contexts of utterance on central gaps. Since the pair of number selected is arbitrary within limits, G must be a choice function with restrictions on the values it can give. It must be required that in each context a full semantics is given for the full fragment of a language that is used in the context, but not for linguistic material outside that fragment. This means in particular that any referring singular term that is used in c must have referent in the quantifier domain of c. For maintaining bivalence, the ⁸ Some predicates are associated with more than one dimension of variation, e.g. predicates formed from simple predicate by means of connectives. In those cases we will need n-tuples of simple real number pairs, one for each dimension of variation. ⁹ The account is more fully worked out in my Vagueness and Domain Restriction, to appear in a volume on vagueness and language use edited by Paul Egr´e and Nathan Klinedinst.
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relevant measure of that term for the predicate F cannot fall in the central gap for F at c. Hence, the central gap is required not to include the measure of that referent. So the location of the gap depends on the terms used in the context. This heads the list of restrictions, ordered according to importance. (GAP)
(i) The central gap must be selected so that the full fragment of language used in the context, including pragmatically determined contextual updates, is taken account of. (ii) The size of the central gap must be at least equal to the tolerance level. (iii) The location of the central gap must be selected so as to ensure consistency. (iv) The location of the central gap must be selected so that what the speaker says comes out as true, to the extent this is possible and reasonable.
(GAPi) cannot be compromised, but this has the consequence that the other three may not be jointly satisfiable, given collateral facts. When they are not, it is not always clear e.g. that consistency of the speaker should take precedence of the truth of individual judgments. But this is a matter of further investigation. That the gap must be at least equal to the tolerance level is necessary to preserve consistency in normal situations. If we make the further decision to let the size of the central gap for a predicate in a context be equal to the tolerance level for that predicate in the context, then the central gap can be regarded as determined by two well-known contextual standards: a standard of comparison and standard of precision.¹⁰ We can simply identify the standard of precision with the tolerance level. If we further let the standard of comparison correspond to the center of the gap, then the gap is determined as a function of the standard of comparison and the standard of precision: where i is the value of the standard of comparison and k is the standard of precision, the gap is simply (i + k/2, i − k/2). There are then two basic ideas for the semantics: The first is that for each predicate F for which a gap is introduced, the extension of F consists of the individuals a in D − Fc with measures above the gap (HF (a) > i) and the anti-extension of individuals b with measure below that gap (HF (b) < j). The second idea is that quantifiers are domain-restricted by means of the cut. We can then verify that a tolerance principle, stated with binary quantifiers, such as (15) Some x(man(x) & height(x) ≤ n + k mm, tall(x)) All x(man(x) & height(x) ≥ n mm), tall(x)
→
is true in any context c where the tolerance level for ‘tall’ is k mm or greater. We assume that the relevant measure function maps men on their heights in mm. Let’s assume here that ‘man’ is non-vague. Then the antecedent of (15) is true just in case some individual a in the restricted domain is a man and has a height above the upper edge of the ‘tall’ gap in c. Then any individual b in the c-restricted domain that is a man has a height at least that of the height of a minus k mm itself has a height above ¹⁰ Cf Lewis 1979, 244–6.
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the upper edge of the gap, for there is no individual in the restricted domain that has a height in the gap, and any individual c in the domain with a height below the gap is more than k mm shorter than a. So the the consequent of (15) is true.¹¹
14.5
MEANING AND CONTENT
You might think that no adult human being, at least these days, who is 150 cm in height, could reasonably be counted as tall. That is, it would be part of the linguistic meaning of ‘tall’ that no interpretation function with an associated upper gap boundary below 150 cm is admissible. A consequence of this view is that if there is no sharp boundary between admissible and non-admissible gaps, there will be no sharp boundary between admissible and non-admissible gap functions, and thus it would be unclear how a speaker could be interpreted. This is not, I think, a severe problem. We cannot anyway reasonably hope that our entire meta-language vocabulary is precise, and there is no good prior reason, from the present perspective, to believe that the domain of admissible gap functions, in some particular context of utterance, is sharply delimited. A leading idea of the present approach is that it is enough for adequate interpretation that there is a least one clearly admissible gap function (see below). A more unwelcome result, however, is that two speakers A and B who disagree about how short an adult human can be and still count as tall, by such a view would be using ‘tall’ with different linguistic meanings. That is, they would not disagree substantially about the lower boundary of tallness, but would be speaking different languages, with phonetically and orthographically identical but semantically distinct predicates ‘. . . is tall’.¹² There would then be neither agreement nor disagreement between them on matters of tallness. I find that implausible. Rather, with one proviso, if two speakers agree on the number uniformity principle (U) (stating so-called penumbral connections) of the predicate in question, then they share the concept. If they agree on the uniformity principle, then they agree on the dimension of comparison, and they agree on the direction relevant for the predicate (e.g. if taller, then more disposed to be counted as ¹¹ The present account has some similarity to the account in Manor 2006, although the two were developed independently. Manor provides a non-standard semantics where the usual inductive premise fails if there is a suitable gap in the sequence of measures. According to Manor, the existence of a unique gap in the contextual domain provides a natural demarcation of the extension of a vague predicate in that context. To effect such a demarcation, the gap must be unique and well placed, and on Manor’s account vague terms are used with the presupposition that there is such a gap. The present account differs from Manor’s inter alia in that gaps are provided as part of interpretation, rather than declaring a speech act as failed when the presupposition isn’t met. ¹² For an epistemicist, like Williamson (1994, 205–12), there is a sharp boundary of tallness determined for the language of the speech community, and from this perspective at least one of the two speakers would be mistaken about what the lowest admissible boundary is. Moreover, on this view, both speaker would be mistaken in thinking that there is a range of admissible alternatives. As many other non-epistemicists, I find such a determination of sharp boundaries from non-uniform use implausible.
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tall). The proviso is that for something to have a property as depending on its position in some order, then it must be possible that something has a lower (or, depending on the predicate, higher) position in that order. So, an object x is not tall unless it is possible that some other object y is shorter, and is not non-bald unless it is possible that some other object y has less hair. This means that for agreeing on the meaning of the predicate, two speakers must also agree on such absolute limits. Accordingly, I take it e.g. to be part of the meaning of ‘bald’ that a person with no hairs on his scalp is bald, but for no number greater than zero is it part of the meaning of ‘bald’ that a person with that number of hairs is bald. Accordingly, I take two speakers who assign the same meaning in this respect to a tolerant predicate to agree on the general concept, such as the concept of tallness, or baldness or heapness. However, such a concept is not individuated directly by application conditions. That is, the concept of being tall, for instance, is not individuated by a set of conditions C such that if an object x satisfies the conditions in C, x is tall and if it does not, it is not. The sorites paradox itself is a reason against this view, provided the premise of tolerance is accepted. For then, if there is such a concept of tallness, it is tolerant, and hence a tolerance principle is valid for that concept, which together with facts about the distribution of heights among adults of the world, leads to a sorites-type contradiction. We cannot avoid such a contradiction by changing the semantics of the concept, for concepts have their semantics built-in. Only for linguistic expressions, or other sign-like entities, can we devise alternative semantic theories. The conceptual semantic paradox can be avoided only if such a concept of tallness does not exist. Rather, the concept of tallness, on the present view, is to be seen as a function from standards of application to extensions. And, on the present approach, admissible standards of application involve a contextual significant central gap. A central gap together with a measure function and the condition of belonging to the extension just in case one’s measure is at least as great as the upper boundary, does fix the extension (and correspondingly for the lower boundary and the anti-extension). For the issue of utterance and belief content, we must switch to an intensional framework. Within a possible-worlds framework, it is natural to regard the concept associated with a tolerant predicate as a function from possible worlds and standards of application to extensions. If we take the standard of application as the first argument, the value of the function is an ordinary intension, i.e. a function from possible worlds to extensions. For instance, an upper gap boundary for tallness of 180 cm determines an intension that for any possible world w fixes an extension consisting of the set of (adult male) humans in w that have a height of at least 180 cm. It is not reasonable, however, to attribute such a precise belief content to the normal speaker. More plausibly, for each speaker and context there is a range of standards that the speaker is prepared to count as admissible. And plausibly, that range is not itself sharply delimited. That is, we have an unsharply bounded set of propositions that are admissible as intensions of the sentence in the context of utterance. This provides one dimension along which the expression ‘believes that’ is itself tolerant, and the present suggestion is, accordingly, that central gap domain restriction is to be applied to sentences containing it.
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It is in line with this suggestion to take two speakers A and B to agree on a particular statement in a context c, not just conceptually but also doxastically, just in case there is a standard of application that is admissible to both A and B. This entails that for each tolerant predicate F involved in c, there is a pair (i, j) which is admissible for both GFA and for GFB . As a consequence, doxastic agreement in a context is not transitive: there may be a standard that is admissible to both A and B, and a standard that is admissible to both B and C but still no standard that is admissible to both A and C. But note that if two values are both admissible, a switch between them will not affect the truth value distribution over sentences. Hence, if A and B, and B and C, respectively, agree doxastically, they agree on truth values, and hence so do A and C, for the given linguistic fragment and the given domain. The non-agreement between A and C is conceptual, not a doxastic dis-agreement.
14.6
P RO B L E M S
The basic strategy of the current proposal is to account for ordinary non-paradoxical use of vague expressions in one way, and allow inconsistency in the extreme cases where we have to do with sorites sequences. Accordingly, it is a necessary condition for the viability of the proposal that the account does give intuitively correct results for ordinary utterances, and in particular that ordinary utterances that do seem intuitively true or intuitively false, do not come out as incoherent on the proposed semantics. It is, however, not obvious that this is the case. Potential sources of trouble are the combining of tolerant predicates, and the use of quantifiers. Here, there is not space for a thorough investigation of the matter. I can only discuss a few examples. One thing that can happen is that objects are added consecutively to the domain in a way that eventually eliminates the possibility of a central gap. If we say (16) Julia is tall. So is Georgina, and so is Elsa, and so is Amanda . . . we may in the end populate the domain of quantification so that a sorites sequence results, given reasonable tolerance levels for ‘tall’. I don’t think this is a problem, however. Rather, this is one way in which the use of vague predicates can run into trouble.¹³ This is perfectly in line with the present proposal. The combination of tolerant predicates in a single context of utterance can give rise to the opposite effect. Since we need a gap and hence a domain cut for each vague predicate, the result might be that so much is cumulatively taken out of the domain of discourse that the topic is distorted. This consequence is avoided in case we are allowed context shifts that need not be conservative, in the sense that new cuts can be made and old cuts undone. This is an issue that requires further investigation. But there is a problem even with a small cut. Suppose a basketball coach says (17) Every player in my team is tall ¹³ The idea that verb phrase ellipsis provides problems for other contextualist accounts of vagueness is due to Jason Stanley (2003).
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and it so happens that, given what counts as tall in the context, and given the contextual tolerance level, there is a sorites sequence in the team between the tall and the non-tall. Then some domain cut is needed, let’s say with the effect that two players must go out of the contextually restricted domain. Have we thereby not misrepresented the content of the utterance of the coach? After all, the coach seems to be saying that every player of his team is tall, not that every member in a set consisting of all except two of the players in his team are tall. The content ascribed to the utterance is not identical to the intuitive content, a content which would be incoherently ascribed. Rather, the content ascribed only approximates the intuitive content. The general idea of imposing contextual domain restrictions on utterance interpretation is that of getting the utterance content right. Isn’t it used here with the opposite effect? Let’s note, first, that the intuitive truth value assignment is not affected. Intuitively, the utterance is false, since the team contains at least one non-tall player (or otherwise there would be no sorites sequence). But with a central gap, the cut in the domain is such as to leave anti-extension members in the domain. Hence, even with respect to the restricted domain the utterance comes out as false. Had all members been tall by the given standard, then no cut had been needed in the first place. So, either way, the intuitive truth value is preserved.¹⁴ Second: We have assumed that the utterance of the coach in itself makes sense. What doesn’t make sense is to interpret it with the contextual tolerance level and the initially determined domain, i.e. the entire team. To bring out what is intuitively right, or in this case intuitively wrong, with the utterance, we need to deviate from the intuitive content ascription. This could be done in other ways, e.g. by assigning greater precision, i.e. a lower tolerance or even a zero tolerance level, going beyond any discrimination that the speaker himself would be prepared to make. This too would, on the current assumption, amount to distorting the intuitive content. Applying it across the board would have the effect of making the tolerance principle (15) come out false, despite being held true by the speaker, and thus force an error theory about the speaker, and about natural speakers in general to the extent that such tolerance principles are generally affirmed. In order to assign content in a consistent way, some approximation has to be made. The type of approximation suggested here generally saves the intuitively assigned truth values and avoids the need of adopting an error theory about tolerance itself. Re f e re n c e s Dummett, M. (1975), ‘Wang’s paradox’, Synthese 30: 301–24. Reprinted in Dummett 1978. Page reference to the reprint. (1978), Truth and Other Enigmas, Harvard University Press, Cambridge, MA. Fara, D. G. (2000), ‘Shifting sands: an interest-relative theory of vagueness’, Philosophical Topics 28: 45–81. ¹⁴ We can indeed get more complicated cases with other quantifiers or determiners, like ‘most’, but there is not enough space here to discuss these cases. Some of them are more problematic, but then again problematic also on all standard accounts of vagueness.
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Fine, K. (1975), ‘Vagueness, truth and logic’, Synthese 30: 265–300. Horgan, T. (1994), ‘Robust vagueness and the forced-march sorites paradox’ in J. E. Tombelin (ed.), Logic and Language, vol. 8 of Philosophical Perspectives, Ridgeview, Atascadero, CA. Keefe, R. (2000), Theories of Vagueness, Cambridge University Press, Cambridge. Keefe, R. and Smith, P. (eds.) (1996a), Vagueness: A Reader, MIT Press, Cambridge, MA. 1996b, ‘Introduction: theories of vagueness’, in R. Keefe and P. Smith, eds., Vagueness: A Reader, 1–57, MIT Press. Lewis, D. (1979), ‘Scorekeeping in a language game’, Journal of Philosophical Logic 8: 339–59. Reprinted in Lewis 1983. Page reference to the reprint. 1983, Philosophical Papers. Volume I , Oxford University Press, Oxford. Machina, K. F. (1976), ‘Truth, belief, and vagueness’, Journal of Philosophical Logic 5: 47–78. Manor, R. (2006), ‘Solving the heap’, Synthese 153: 171–86. Raffman, D. (1994), ‘Vagueness without paradox’, Philosophical Review 103: 41–74. (1996), ‘Vagueness and context-relativity’, Philosophical Studies 81: 175–92. Shapiro, S. (2003), ‘Vagueness and conversation’ in Jc Beall, ed., Liars and Heaps, 39–72, Oxford University Press, Oxford. (2006), Vagueness in Context, Oxford University Press, Oxford. Soames, S. (1999), Understanding Truth, Oxford University Press, Oxford. Stanley, J. (2003), ‘Context, interest relativity and the sorites’, Analysis 63: 269–80. Williamson, T. (1994), Vagueness, Routledge, London. Wright, C. (1976), ‘Language-mastery and the sorites paradox’ in G. Evans and J. McDowell, eds., Truth and Meaning, 223–47, Clarendon Press, Oxford. (1987), ‘Further reflections on the Sorites paradox’, Philosophical Topics 15: 227–90. Reprinted in Keefe and Smith 1996a (with omission of section 5), 204–50. Page references to the reprint.
IV Vagueness in Context
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15 Hold the Context Fixed—Vagueness Still Remains ˚ Jonas Akerman and Patrick Greenough
Contextualism about vagueness (hereafter ‘Contextualism’) is the view that vagueness consists in a particular species of context-sensitivity and that properly accommodating this fact into our semantic theory will yield a plausible solution to the sorites paradox.¹, ² But Contextualism, as many commentators have noted, faces the following immediate objection: if we hold the context fixed, vagueness still remains, therefore vagueness is not a species of context-sensitivity. Call this ‘the simple objection’.³ Absent a convincing reply to the simple objection, Contextualism is in very bad shape. Oddly enough, defenders of Contextualism have said very little in reply. Proponents of the objection have tended to assume that this is because no reply is in the offing—the simple objection is taken to be unassailable. In this paper, we sketch two replies to the simple objection which result in two very different kinds Parts of this were jointly presented at Seventh Arch´e Vagueness Workshop in November 2006 and at the Arch´e Audit in June 2007. Thanks to the following folk for very useful feedback (on either or both of those occasions): Elizabeth Barnes, Maria Cerezo, Richard Dietz, Dan L´opez de Sa, Aidan McGlynn, Sebastiano Moruzzi, Peter Pagin, Graham Priest, Diana Raffman, Sven Rosenkranz, Mark Sainsbury, Stewart Shapiro, Paula Sweeney, Jordi Valor, and Crispin Wright. Thanks also to Sven Rosenkranz and Elia Zardini for particularly valuable comments on the penultimate draft. This paper was completed while one of the authors (Greenough) was a Postdoctoral Fellow in the Epistemic Warrant Project at ANU (2007–8). Thanks go to the many philosophers at ANU for their hospitality—philosophical and otherwise. ¹ According to a generic version of Contextualism, the vagueness of the predicate type ‘is tall (for a Ugandan Pygmy)’, for example, consists, in part, in the fact that relative to different contexts of utterance (where these contexts of utterance differ only in respect of certain designated parameters), the extension of this predicate can differ (even though the heights of all people in Uganda remain fixed). For Fara (2000) (originally published under Delia Graff), the designated contextual parameters are the interests and purposes of the speaker (and their conversational participants). For Raffman (1994, 1996) the designated parameters concern the psychological states and dispositions of the speaker. For Lewis (1979), Soames (1999, 216–17), Shapiro (2003, 2006), the designated parameters concern ˚ the operative standards of precision. See Akerman and Greenough (2009) for a critical discussion of the various ways in which vagueness may consist in a particular species of context-sensitivity. ² There are two broad kinds of contextualist solutions to the sorites paradox (see Section 15.2). ³ The objections raised against Contextualism in Stanley (2003) and in Keefe (2007) are strictly ˚ independent of the simple objection discussed here. See Akerman and Greenough (2009) for a critical discussion of some of Keefe’s objections.
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of Contextualism: Epistemicist Contextualism and Radical Contextualism. With these two theories in hand, the simple objection loses much, if not most, of its force. 15.1
C O N T E X T UA L I S M A N D W E A K TO L E R A N C E
All extant forms of Contextualism are committed to something like the following principle of weak tolerance: (WT) It is not the case that: there is a context of utterance C and there is an x such that x and x are considered together as a pair by a single subject in C and ‘is F ’ (as used in C) is true of x and ‘is F ’ (as used in C) is false of x , (where x is adjacent to x in the sorites series running from F to not-F ).⁴ Roughly, WT says that, when considered pairwise, adjacent members of the series are never category different.⁵ WT is a principle of weak tolerance since it permits that (a) there can be a context C and a context C such that ‘is F ’ (as used in C) is true of x and ‘is F ’ (as used in C ) is false of x , and that (b) there can be a sharp boundary within C if x and x are not considered together as a pair in C. One of the characteristic symptoms of vagueness is that vague predicates draw no known boundary across their associated dimension of comparison.⁶ WT can explain how this symptom of vagueness arises: as we inspect each pair of adjacent items in the sorites series, WT ensures that the members of each adjacent pair cannot be category different. Given the factivity of knowledge, it follows that there is no context of utterance C such that there are two adjacent items x and x , which are considered together in C, such that a subject knows that ‘x is F and x is not-F ’ is true. Roughly, no (context in which there is a) boundary between saliently similar objects in the series entails no (context in which there is a) known boundary between those objects. (We shall encounter two further symptoms of vagueness in section 15.3.) But do vague predicates draw sharp boundaries or not? WT is compatible with either view. On this score, there is an important (and generally overlooked) distinction between what may be termed Boundary-Shifting Contextualism (BSC) and Extension-Shifting Contextualism (ESC).⁷ ⁴ It’s a further question whether WT holds in all contexts (see Shapiro 2003, 44, fn. 1, for some relevant remarks). ⁵ WT is cognate to both Raffman’s principle IP∗ which, with respect to ‘is red’, says that ‘for any n, if patch #n is red then patch #(n + 1) is red, relative to a pairwise presentational context’ (1994, 68) and Fara’s salient-similarity constraint which says that ‘if two things are saliently similar, then it cannot be that one is in the extension of the predicate, or in its anti-extension, while the other is not’ (Fara 2000, 57). Cf. Soames (1999, 214–16) and Shapiro (2003, 42–3). ⁶ In Greenough (2003), this symptom is called ‘epistemic tolerance’. ⁷ See Greenough (2005, 178–9) for more on this distinction. Raffman (1994) and Shapiro (2003, 2006) both defend forms of ESC, while Fara (2000) defends a form of BSC. Soames (1999, 216–17) appears to defend a form of BSC whereby there is a shifting boundary between the extension/anti-extension of a predicate and the undefined cases in the borderline area—cases for which there is a truth value gaps of sorts. Thus, while all forms of BSC are committed to sharp (variant) cut-offs, not all forms are committed to classical logic. Stanley (2003), Heck (2003), Priest (2003), and Keefe (2007) simply assume that Contextualism is exhausted by BSC.
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1 5 . 2 B O U N D A RY- S H I F T I N G C O N T E X T UA L I S M A N D E X T E N S I O N - S H I F T I N G C O N T E X T UA L I S M BSC says that in every context there is a cut-off. That is, across a sorites series for ‘is F ’, for every context of utterance C, there is an x such that ‘is F ’ (as used in C) is true of x and ‘is not-F ’ (as used in C) is true of x .⁸ Thus, BSC is a form of epistemicism in that vague predicates draw sharp, bivalent, boundaries. Unlike the epistemicism of Sorensen (1988) and Williamson (1994), however, it is constitutive of vagueness that the boundary can shift as a function of changes in the context of utterance (see fn.1). Thus, the following principle is invalid: there is an x such that, for every context of utterance C, ‘is F ’ (as used in C) is true of x and ‘is not-F ’ (as used in C) is true of x . This latter principle amounts to the claim that there is a cut-off such that it obtains in every context. Furthermore, as we should expect, BSC plus WT entails that the cut-off drawn by a vague predicate is not only unknown but unknowable—at least via the method of inspecting adjacent items. What does BSC say about the standard sorites paradox? With respect to a typical sorites series for the predicate ‘is red’, it is given that the first colour patch in the series is red and the last colour patch is not red. The major premise of the standard version of the paradox says that, for all colour patches x in the series, if patch x is red then patch x is red. Given mathematical induction, it follows that all patches in the series are red. But that contradicts the fact that the last member is not red. In order to resolve the paradox, BSC—just like standard epistemicism—holds the major premise to be outright false. But if the major premise is false why did we find it so plausible (and so believe it) in the first place? Importantly enough, BSC and standard forms of epistemicism differ with respect to this key question. Standard epistemicism can offer something like the following ‘confusion’ diagnosis: in confronting the paradox we systematically confuse the (true and plausible) claim that there is no known boundary across a sorites series with the (false) claim that there is no sharp boundary. Such a confusion confers plausibility onto the stronger claim—explaining why we come to believe the stronger claim.⁹ BSC is able to offer a related, but distinct, ‘confusion’ diagnosis: in confronting the paradox we systematically confuse the (true and plausible) weak principle of tolerance WT (and kindred principles) with the following (false) strong principle of tolerance (and kindred principles): (ST) It is not the case that: there is a context of utterance C and there is an x such that ‘is F ’ (as used in C) is true of x and ‘is F ’ (as used in C) is false of x , (where x is adjacent to x in the sorites series running from F to not-F ).¹⁰ ⁸ Such a formulation assumes that we can never set standards so low or so high such that either everything or nothing counts as an F . ⁹ In fact this diagnosis is available to any theory which takes the major premise of the sorites to be false—such as a supervaluational or intuitionistic conception of vagueness (see Greenough 2003, 272–4 for further discussion). ¹⁰ ST entails that for all contexts of utterance C and for all x, if ‘is F ’ (as used in C) is true of x then ‘is F ’ (as used in C) is true of x . In other words, the predicate ‘is F ’ is tolerant in all contexts.
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Very roughly, we confuse the (true and plausible) claim that there is never a boundary between any two adjacent items considered together as a pair with the stronger (and false) claim that there is a never a boundary between adjacent items. Again, such a confusion confers plausibility onto the stronger claim—explaining why we come to believe the stronger claim.¹¹ (We shall return to these diagnoses in section 15.4.) ESC represents a radically different form of contextualism. Given ESC, in no context of utterance is there a cut-off.¹² For ESC there can only be ‘quasiboundaries’—boundaries which hold, as it were, across, but not within contexts.¹³ With respect to the standard sorites, the paradox is not to be resolved by taking the major premise to be unequivocal and false as in the case of BSC. Rather, the sorites is taken to exhibit a fallacy of equivocation.¹⁴ There is a true reading of the major premise: for all colour patches x in the series, if patch x is red then patch x is red relative to a pairwise presentational context whereby x and x are presented together as a pair to a competent judge. And there is a false reading: for all colour patches x in the series, if patch x is red relative to a singular presentational context then patch x is red relative to a singular presentational context, whereby the context in which x is presented to a competent judge may differ from the context in which x is presented to a judge.¹⁵ It follows that, in the present context, for all x, if x is F then x is F . That is, the major premise of the standard sorites follows from ST. Given classical logic, and the fact that the first member of the series is F and the last member of the series if not-F , then the major premise is outright false and so ST is outright false. ¹¹ Why does such a confusion take place? The thought is that subjects are typically (pretheoretically) unaware of the effect that context has in the determination of the extension of a predicate. ¹² The alert reader will have noticed that this is just to assert ST. But ST classically entails the major premise of the standard sorites. As it turns out, ESC can retain ST without fear of paradox because the classical consequence relation is restricted within contexts given ESC—in particular, the classical least number principle is not valid (see main text below). For the special case of the sorites paradox under which one uses the negation of ST to derive a contradiction, the solution given by ESC is as follows: the major premise ST is not equivocal at all but simply true, however the paradox does not arise because classical logic fails. ¹³ With respect to the forced march sorites, Raffman (1994, 46–7, passim) and Shapiro (2003, 51–3) allege that a (competent) subject will always ‘jump’ in the forced march—thus delivering a differential verdict with respect to adjacent items in the series. But this jump does not mark a boundary (within a context) but rather a shift in context. ¹⁴ See Raffman (1994, 68–9). Shapiro defends a form of ESC but, oddly, takes the major premise to be false (see Shapiro 2003, 53). In Greenough (2005, 178) it is argued that Shapiro should posit a fallacy of equivocation. ¹⁵ As it turns out, BSC can offer an alternative (and incompatible) explanation of the seductiveness of the major premise by also positing a fallacy of equivocation: the major premise equivocates between a strong (and false and implausible) reading (via ST and cognate principles) and a weak (and true and plausible) reading (via WT and cognate principles). We have resisted this way of presenting matters because extant defenders of BSC (e.g. Fara) represent themselves as taking the major premise to be false and so this premise is not, for Fara at least, equivocal. A further point of note is that it is not possible for ESC to co-opt the solution to the sorites posited by BSC under which ST and the major premise are taken to be false. The reason for this is that ESC takes ST to be true—see fn. 12. So, there is an asymmetry between BSC and ESC: both can offer a diagnosis under which the major premise equivocates between a true reading and false reading, while only BSC can offer a diagnosis under which the major premise is both false and yet taken to be true/plausible because a
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Which of these two species of Contextualism is the better view? Here is a quick argument in favour of BSC over ESC: According to ESC, in no context of utterance is there a cut-off. It follows that within a context of utterance, whereby the first member of the series is F and the last member is not-F , the classical least number principle is invalid—otherwise we could derive that there is a cut-off between the F ’s and not-F ’s in that very context. Thus, classical logic fails given ESC. Given that BSC preserves classical logic, and ESC does not, then BSC is the more plausible view.¹⁶ The reason is simple: the contextualist has no need to both posit context-sensitivity and give up on classical logic in order to resolve the sorites paradox. This argument provides a pretty strong reason to prefer BSC over ESC. So, in what follows we shall only defend BSC against the simple objection.¹⁷ (From now on, by ‘Contextualism’, we shall mean BSC.) 15.3
THE SIMPLE OBJECTION
Some prominent exemplars of the simple objection are as follows: Vagueness remains even when the context is fixed. (Williamson 1994, 215) we should distinguish vagueness from paradigm context-dependence (i.e. having a different extension in different contexts) even though a term may have both features (e.g. ‘tall’). Fix on a context which can be made as definite as you like (in particular choose a specific comparison class): ‘tall’ will remain vague, with borderline cases, and fuzzy boundaries, and the sorites paradox will retain its force. This indicates that we are unlikely to understand vagueness or solve the [sorites] paradox by concentrating on context-dependence. (Keefe and Smith 1997, 6, see also Keefe 2000, page 10) the first blush response that almost everyone seems to have [towards Contextualism] is: OK, fix the context; the extension of ‘red’ in that context is still vague [. . .] The sorites reasoning is just as appealing when one nails the extension down as it is when one allows it to vary. (Heck 2003, 120)¹⁸
If we follow Keefe’s particular example and assume that the context-sensitivity which is constitutive of vagueness is exhausted by the sensitivity to a comparison class then the objection is persuasive. However, no extant or sensible form of Contextualism invokes that kind of context-sensitivity to make sense of vagueness.¹⁹ Even so, the subject when first confronting the paradox confuses it with a true and plausible principle of weak tolerance. ¹⁶ See Greenough (2005, 178–9). ¹⁷ We do not mean to imply that ESC is any worse off than BSC when it comes to the simple objection. However, ESC must offer a rather different range of responses to the simple objection than the range of responses that are available to BSC. ¹⁸ A form of the simple objection also appears in an unpublished paper ‘A problem for contextualism about vagueness’ by Max K¨olbel, 2007. ¹⁹ See fn. 1. In her (2007, 276), Keefe recognizes that vagueness-related context-sensitivity is independent of sensitivity to shifts in comparison class.
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objection has a more general form: suppose we hold all aspects of the context of utterance fixed (e.g. speaker, world, time, place, orientation, conversational partners, contextually salient comparison class, the operative standards of precision, the psychological states of the conversationalists, and so on) then the extension of ‘is red’ in that context will still exhibit all the symptoms of vagueness and will thus count as vague. Since, by hypothesis, the predicate ‘is red’ cannot vary its extension within the fixed context in hand, and since this predicate remains vague, then vagueness is not a species of context-sensitivity.²⁰ We’ve encountered one (epistemic) symptom of vagueness already: vague predicates draw no known boundary across their respective dimension of comparison. Two other symptoms are important. The second symptom is also epistemic: vague predicates give rise to borderline cases, cases such that we do not know whether or not the predicate applies.²¹ The third symptom is quasi-psychological in nature: vague predicates are sorites-susceptible—they are such that (pre-theoretically) we are seduced into accepting the major premise of the sorites paradox.²² For the purposes of this paper we will assume that these symptoms are individually necessary and jointly sufficient for the presence of vagueness.²³ WT as we have already seen can be used to explain why there is no known boundary across the series: when adjacent items in a sorites series are considered together as a pair, those items are never category different and so there is no known boundary between them. This means that when we employ the (very natural) method of inspecting adjacent members of the series in order to discover the whereabouts of the boundary we cannot locate the boundary since WT ensures that the boundary can never be where we are looking. Furthermore, the contextual factors which (in part) go to determine the extension cannot be held fixed through a complete inspection of the series using this method since successively considering adjacent items as pairs inevitably entails a change in those very factors.²⁴ Thus, WT ensures that there are ²⁰ Raffman’s distinction between internal (‘psychological’) contexts and external contexts (which concern the relevant comparison class, operative standards, and so forth) is of no help in resolving this more general form of the simple objection (for the distinction see Raffman 1994, 64–6; cf. Shapiro 2003, 60–1). ²¹ While extant forms of ESC (as given by e.g. Raffman 1994, Shapiro 2003, 2006) allow that first symptom of vagueness is a genuine symptom, these theories nonetheless permit a subject to know whether or not a predicate applies across the borderline area—and so the second symptom of vagueness is not a genuine symptom. This feature of these views issues from the fact that, in borderline cases, whether or not a predicate applies is taken to be a response-dependent matter such that what a (competent) subject judges to be the case determines what is the case (where such a judgment also puts the speaker in a position to know what is the case). Strictly speaking, such a response-dependent conception is not an essential feature of ESC. ²² We use the expression ‘quasi-psychological’ because in giving an explanation as to what gives rise to this third symptom of vagueness we not only need to give some psychological explanation as to why we come to believe that vague predicates are strongly tolerant, but we also need to establish why the claim that vague predicates are tolerant is so pre-theoretically intuitive. The two parts of this explanation are, of course, connected. ²³ Arguably, they are also individually sufficient, though substantiating that fact lies outside the scope of this paper. In Greenough (2003, 265–72) two proofs are given which show that the first two symptoms are equivalent given some pretty plausible background assumptions. ²⁴ It follows from WT that a subject cannot simultaneously bring all pairs in the series to salience.
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certain conditions under which we cannot hold the context fixed. Under those conditions, the simple objection cannot arise. Even so, this only helps defuse a certain version of the simple objection. Even if the relevant contextual factors cannot be held fixed in the required way, it seems that we can introduce a new predicate via stipulation which is intuitively just as vague as the original one but is apparently not sensitive to differences in the context. Heck has a version of this objection as follows: Suppose I say, [in context C0 ]: Some of the patches are red; call them the reddies. I might ask which is the last of the reddies. [. . .] The question is why we cannot locate the last of the reddies. Maybe the extension of the word ‘red’ as we would then be using it would indeed shift, but the point does not seem relevant. There is no such shift in the extension of ‘the reddies’. (Heck 2003, 118–19)²⁵
Heck’s stipulation licenses the following double biconditional: (S) ‘is a reddie’ is true of x if and only if ‘is red in context C0 ’ is true of x if and only if ‘is red’ (as used in C0 ) is true of x. Heck assumes that the predicate ‘is a reddie’ cannot shift in extension (as a function of which pairs in the series we happen to be considering). Given (S), this assumption entails that the predicate-context pair ‘is red’ (as used in C0 ), and the predicate ‘is red in context C0 ’ likewise cannot shift in extension.²⁶ The general form of the puzzle then becomes: absent such shiftiness, what explains (a) why we don’t know the cutoff drawn by these predicates, (b) why these predicates give rise to borderline cases, and (c) why these predicates are sorites-susceptible? However, if this is the nub of the simple objection, then a further issue emerges: it’s not immediately obvious that the predicate ‘is red in context C0 ’ is genuinely soritessusceptible.²⁷ Here the immediate thought is that this predicate is a theoretical predicate of sorts—and we simply lack the requisite intuitions in natural language to say with conviction that this predicate exhibits the symptom of sorites-susceptibility. But ²⁵ Williamson also has a version of this objection (see Mills 2004, 640). ²⁶ Elia Zardini has suggested to us that if ‘is red’ is context-sensitive then the predicate ‘is red in context C0 ’ is a ‘monstrous’ predicate (in the sense of ‘monstrous’ given by Kaplan 1989, 510–11). If that is right then the open sentence ‘It is true in context C0 that x is red’ is also monstrous. However, that would only seem to be so under the assumption that the context-sensitivity of ‘is red’ is indexical context-sensitivity. Indexical context-sensitivity demands that the operator ‘It is true in context C0 that’ cannot operate upon character—because indexicals (in English at least) are such as to always take wide scope. In other words, if ‘is red’ is an indexical then this predicate, as used in a context in which the sentence ‘It is true in context C0 that x is red’ is uttered, determines an extension (relative to a circumstance of evaluation) given some value which is supplied from the context of utterance rather than from C0 itself. However, on a non-indexical model of context-sensitivity that need not be so. On such a model, the operator ‘It is true in context C0 that’ is akin to the modal ˚ operator ‘It is true at world W0 that’. See Akerman and Greenough (2009) for several arguments in favour of non-indexical over indexical contextualism. ²⁷ Cf. Stanley (2003, 279, fn. 13) who assumes without scruple, following Williamson, that a predicate such as ‘is tall at time t’ is sorites-susceptible. Presumably Stanley and Williamson would say the same concerning the predicate ‘is red in context C0 ’.
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if there is some doubt that ‘is red in context C0 ’ is genuinely sorites-susceptible and we then reflect on (S), then that doubt may spread to the predicate ‘is a reddie’ and, in turn, to the predicate-context pair ‘is red’ (as used at C0 ). Given that sorites-susceptibility is a necessary condition of the presence of vagueness then the simple objection lapses since vagueness is no longer present once we hold the context fixed. Perhaps all this shows is that the notion of sorites-susceptibility is too elusive to rely on as a reliable indicator of vagueness. After all, once one has been exposed to enough theory then it’s often hard to be drawn to think that vague predicates are strongly tolerant or think that the major premise of the standard sorites paradox simply must be true. In any case, it turns out that one can defuse the simple objection even if all the predicates in (S) are taken to be sorites-susceptible and so, for the purposes of argument, we shall assume that these predicates exhibit all three symptoms of vagueness. (To simplify matters, however, in much of what follows we shall focus on the predicate-context pair ‘is red’ as used in C0 .) What replies to the simple objection are in the offing? 15.4
R E P LY O N E : E PI S T E M I C I S T C O N T E X T UA L I S M
In brief, this reply runs as follows: Let it be granted that the predicate-context pair ‘is red’ (as used in C0 ) has a sharp and invariant extension. Let it also be granted that this predicate-context pair exhibits the first symptom of vagueness such that there is no known boundary between the extension of this predicate and its anti-extension. However, let the explanation for this ignorance be a purely epistemological explanation. One can flesh-out the required epistemological explanation by invoking something like a safety-based account of knowledge to explain our ignorance of the cut-off. On such an account, a belief that p is safe just in case there are no nearby worlds where I form the false belief that p on the same basis (see Williamson 1994, ch. 8, Williamson 2000 chs. 5, 7). The basic idea is that even if a subject formed a true belief, on a basis B, that the boundary for ‘is red’ (as used in C0 ) lies between a certain pair, this belief cannot constitute knowledge since the subject could easily have formed a false belief about the whereabouts of the cut-off on the same basis. Here the thought is that the extension of the predicate-context pair could easily have been different since the boundaries drawn by such predicates are unstable—even relative to a fixed context (see below). Such a story can also serve to explain why the second symptom of vagueness arises.²⁸ Suppose that a subject forms a true belief, on a basis B, that a certain item in the series belongs to the extension of the predicate-context pair ‘is red’ (as used in C0 ). Suppose also that this item lies near to the boundary drawn by the predicate-context pair. The subject’s belief fails to constitute knowledge because this belief could easily have been false. Again, the thought is that the extension of the predicate-context pair is unstable (relative to a fixed context) and so it could easily have been the case that the item failed to belong to the extension of the predicate (see below). ²⁸ And indeed the story can be used to explain why ‘is a reddie’ and ‘is red in context C0 ’ also exhibit the first two symptoms of vagueness.
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A hybrid theory of vagueness is thus called for. A form of epistemicism is required to explain why we lack knowledge of the invariant cut-off for ‘is red’ (as used in C0 ), while a contextualist explanation, drawing on WT, would explain why we can’t know the cut-off for ‘is red’ relative to a fixed context where we are considering adjacent items together. Call this hybrid theory Epistemicist Contextualism. Is this reply ad hoc? Hybrid theories of vagueness are not uncommon. Ironically, Heck (2003, 124–5) himself sponsors a hybrid conception of vagueness under which first-order vagueness is taken to be semantic, but the boundary between the borderline area and the non-borderline regions is taken to be sharp (and unknowable). Heck says: ‘there is nothing ad hoc about the refusal to go epistemic at one point but not the other’ (ibid., 124). But then Heck can have no principled complaint with the reply in hand to the simple objection.²⁹ Even so, those who accept standard forms of epistemicism (e.g. Sorensen and Williamson) are likely to be unmoved by this reply on the grounds that considerations of simplicity and uniformity dictate that a non-hybrid theory of vagueness is called for.³⁰ This counter-reply can itself be resisted. The most well-worked out form of epistemicism—Williamson’s—is an impure form of epistemicism in that Williamson posits that the sharp boundaries drawn by vague predicates are themselves ‘unstable’ (1994, 231) such that ‘the extension of ‘‘thin’’ as used in a given context could very easily have been slightly different’ (ibid., 230). This (modal) instability in extension arises because the pattern of usage of ‘thin’ (even with respect to a fixed context) is itself unstable. Even though such usage may be invariant from context to context in the actual world, nonetheless, such usage could easily have been different. For Williamson, this instability in extension plays a key role in explaining why I cannot, for example, know the truth value of the sentence ‘Everyone with exact physical measurements x, y, z, is thin’. Suppose this sentence is true and I believe it to be so, why does my belief fail to constitute knowledge? If true, this sentence expresses a necessary truth (Williamson 1994, 204, 230). But since there are no worlds in which the proposition expressed by this sentence is false, then a fortiori there are no nearby worlds in which the proposition expressed by this sentence is false. Hence, my belief that the sentence is true is guaranteed to be safe. It thus seems a safety-based account of knowledge cannot explain the requisite kind of ignorance. However, if the sentence could easily have expressed a different, and indeed false, proposition (relative to a fixed context) then my belief that the sentence is true could easily have been false and so cannot constitute knowledge. It is for this reason that Williamson posits unstable cut-offs for vague predicates to fully explain the ignorance which may arise because of vagueness. A pure form of epistemicism, in contrast, posits only an epistemological explanation for our ignorance of cut-offs. Impure forms of epistemicism are hybrid theories because they posit a special vagueness-relevant semantic (or metaphysical) feature of vague predicates and invoke an epistemological story from there. For this reason, an ²⁹ Koons (1994, 447) sponsors a similar hybrid view. Goguen (1969) also seems to defend a hybrid of fuzzy logic and epistemicism, whereby the borderline area is also sharply-bounded. See also Simons (1992). ³⁰ See also Keefe and Smith for this objection (1997, 47).
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epistemicist form of Contextualism and Williamson’s impure epistemicism are simply on a par with respect to the desiderata of simplicity and uniformity.³¹ The preceding considerations show that Contextualism can not only allow, but even predicts, that the first two (epistemic) symptoms of vagueness arise even when the context is held fixed. But what about the third symptom of vagueness? Why are vague predicates sorites-susceptible? Recall from above that BSC offers the following ‘confusion’ diagnosis as to why we find the major premise of the standard sorites paradox so plausible: we confuse the following two principles (and their respective kin): (WT) It is not the case that: there is a context of utterance C and there is an x such that x and x are considered together as a pair in C and ‘is F ’ (as used in C) is true of x and ‘is F ’ (as used in C) is false of x , (where x is adjacent to x in the sorites series running from F to not-F ). (ST) It is not the case that: there is a context of utterance C and there is an x such that ‘is F ’ (as used in C) is true of x and ‘is F ’ (as used in C) is false of x , (where x is adjacent to x in the sorites series running from F to not-F ). The question then arises: does the diagnosis mooted by Contextualism above as to why we find the major premise of the sorites so compelling retain its force when the context is held fixed? According to the diagnosis in hand, ST derives its plausibility from being confused with WT. For the purposes of argument, let that part of the diagnosis stand. It is also the case that ST, as applied to ‘is red’ entails: for all x and for all contexts C, the predicate ‘is red’ (as used in C) is not true of x and false of x . In other words, take any context you like, the predicate ‘is red’ (as used in that context) does not draw a boundary. So, take the context C0 . It follows that ‘is red’ as used in C0 does not draw a boundary. So, if we are confused into accepting ST, then we are confused into accepting that the predicate-context pair ‘is red’ as used at C 0 draws no boundary. On that basis, we accept the major premise of the standard sorites as applied to the predicate-context pair ‘is F ’ (as used at C0 ). In other words, this predicate-context pair is sorites-susceptible even though it draws a sharp and invariant boundary across the dimension of comparison. Thus, not only can Contextualism allow that sorites-susceptibility remains even when the context has been held fixed, it predicts that such sorites-susceptibility will remain. The simple objection simply does not get a grip when it comes to the third symptom of vagueness. Even if one resists the details of the diagnosis just given, epistemicist forms of Contextualism have a fallback diagnosis. Recall that the standard epistemicist diagnosis as to why the major premise of the standard sorites is so plausible also posits a confusion. But this confusion is more humdrum: we confuse the (true and plausible) claim that vague predicates do not draw a known boundary with the (false) claim that they do not draw a (sharp) boundary.³² Given Epistemicist Contextualism, the predicates ‘is a reddie’, ‘is red in C0 ’, and the predicate-context pair ‘is red’ (as used at C0 ), all exhibit ³¹ In his most recent defence of epistemicism, Sorensen (2001) is also committed to a hybrid view of sorts since he posits a metaphysical explanation for the unknowability of the sharp cut-offs drawn by vague terms in terms of what he calls ‘truthmaker gaps’. ³² Where to lack a sharp boundary is to lack a boundary.
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the first symptom of vagueness—they all draw no known boundary across the sorites series for ‘is red’. Given the ‘confusion’ diagnosis just posited, this first symptom is easily confused, when first thinking about the paradox, with the claim that they draw no (sharp) boundary. If we are confused into believing that these predicates draw no sharp boundary then we are confused into believing that the major premise of the sorites is valid. Hence, these predicates are sorites-susceptible. There are various ways in which one can finesse such a diagnosis.³³ However for our purposes it doesn’t matter whether such a diagnosis is compelling. What matters is that the simple objection posits no special objection to Contextualism since Contextualism can also draw on epistemicist resources to explain why the sorites-susceptibility of a predicate remains even when the context has been held fixed. The overall upshot, then, is that (an epistemicist form of) Contextualism can allow, and even predicts, that each of the three symptoms of vagueness arise when one holds the context fixed. The simple objection is no objection to Contextualism. The trouble with this reply is that it is committed to a form of epistemicism and so is unlikely to persuade everybody. Is there a viable alternative? 15.5
R E P LY T WO : R A D I C A L C O N T E X T UA L I S M
In brief, this reply runs as follows: Let it be granted that all three predicates in (S) give rise to our three symptoms of vagueness. So, ‘is a reddie’, ‘is red in context C0 ’, and the predicate-context pair ‘is red’ (as used in context C0 ) are all vague. But note that ‘is red’ (as used in C0 ) is true of x if and only if x satisfies ‘is red’ in context C0 . Given (S), this means that the vagueness of the object-language predicates ‘is a reddie’, ‘is red in context C0 ’, and the predicate-context pair ‘is red’ (as used in C0 ) will co-vary with the vagueness of the meta-linguistic predicate ‘x satisfies ‘‘is red’’ in context C0 ’. Meta-linguistic vagueness represents a kind of higher-order vagueness.³⁴ Thus, to ask whether the predicate-context pair ‘is red’ (as used at C0 ) is vague is a way of asking whether the predicate type ‘is red’ is higher-order vague (in the requisite sense of ‘higher-order vague’). The first-order vagueness of this predicate type consists in the fact that, relative to different contexts of utterance, this predicate type can differ in extension (relative to a given world). The second-order vagueness of this predicate type consists in the fact that a meta-linguistic predicate such as ‘x satisfies ‘‘is red’’ in context C0 ’ can itself differ in extension relative to different contexts of utterance. The vagueness of this metalinguistic predicate ‘x satisfies ‘‘is red’’ in context C0 ’ may have one of two sources. Either it is vague what the singular term ‘context C0 ’ refers to, or it is vague what the quotation name ‘ ‘‘is red’’ ’ refers to. We shall simply focus on the former source. In Heck’s statement of the simple objection we are ³³ One way to finesse the diagnosis would be to argue that this confusion itself arises from an internalist conception of meaning and understanding which licenses the transparency claim that for all n, if it is true that patch n in the series is red then one is in a position to know this (cf. Williamson 1994, 205–12). ³⁴ On this score, we agree with Keefe and Smith (1996, 15–16).
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supposed to be able to nail down a sharp and invariant extension for the reddies by saying, at a particular time T0 : ‘Some of the patches are red; call them the reddies.’ However, such a saying has temporal spread because all speech acts take place over time. It is thus unclear, and indeed vague, at what exact time the saying—the stipulation—has taken place. That is, it is vague just what time is picked by the name ‘T0 ’, and so vague just what context is picked out by the name ‘C0 ’. It follows that there are some objects such that it is vague whether or not these objects fall under the predicate ‘is a reddie’. On a contextualist model of vagueness, according to certain strict standards, the saying may be deemed to have taken place in a very narrow interval of time, while relative to more lax standards, the saying may be deemed to have taken place at a broader interval. Thus the predicate ‘x satisfies ‘‘is red’’ in context C0 ’ is itself subject to contextual variation. There is thus no difference in kind between the vagueness of the predicate type ‘is red’ and the vagueness of ‘is a reddie’. The status of the simple objection now ought to be clear: it amounts to the claim that Contextualism cannot allow for (a certain type of) higher-order vagueness. Epistemicist Contextualism can be seen as an attempt to offer a semantic model of first-order vagueness and an epistemic model of higher-order vagueness. Radical Contextualism can be seen as an attempt to offer a uniform characterization of all orders of vagueness. Is Radical Contextualism defensible? For our purposes it doesn’t matter. What matters is whether it is co-defensible with what the leading non-epistemic (non-contextualist) theories of vagueness say concerning higher-order vagueness. One way of making sense of higher-order vagueness given Radical Contextualism is to offer a type-theoretical model. According to such a suggestion, contexts of utterance should be typed to a level: level-1 contexts of utterance, level-2 contexts of utterance, and so on. Semantic closure is thus to be rejected and a hierarchy of increasingly expressive meta-languages is called for. So, for example, context C0 , is a level-1 context of utterance, whereas the context of utterance in which the predicate ‘x satisfies ‘‘is red’’ in context C0 ’ determines an extension is a level-2 context of utterance. This kind of radical model defuses the simple objection as follows: when it is said ‘hold all the features of the context fixed, vagueness still remains, therefore vagueness is not context-sensitivity’ this should simply be read as ‘hold all the features of the level-1 context fixed, vagueness still remains, therefore not all vagueness is level-1 contextsensitivity’. On this score, it is notable that perhaps the most sophisticated response to issue from the non-epistemic camp concerning the various puzzles of higher-order vagueness is given by Keefe (2000, ch. 8).³⁵ It turns out that Keefe can have no principled objection to the broad type-theoretic strategy just mooted with respect to the simple objection. That’s because she alleges that a Tarskian style hierarchy of increasingly expressive meta-languages is required if we are to address a central puzzle of higherorder vagueness given by Williamson. Williamson’s puzzle can be given as follows: Suppose we define a notion of absolute definiteness as follows: It is absolutely definite that A =df A and it is definite that A and it is definite that it is definite that A ³⁵ Every other non-epistemic theory of vagueness has notably failed to address all of the pressing puzzles concerning higher-order vagueness.
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and . . . The notion of absolute definiteness intuitively ought to be vague. But it also follows from the definition, given some simple logic, that an S4 reduction schema for absolute definiteness is valid: If it is absolutely definite that A then it is absolutely definite that it is absolutely definite that A.³⁶ But if this schema holds then absolute definiteness cannot exhibit genuine higher-order vagueness.³⁷ If that is so, it is not genuinely vague at all. Contradiction.³⁸ Keefe concedes that absolute definiteness is vague but that it’s vagueness cannot be expressed within the meta-language. A richer metalanguage is needed. But within this richer meta-language we can define a new notion of absolute definiteness (‘absolute definiteness∗ ’), which itself is vague. Yet this new notion cannot be used to express the fact that it is vague without contradiction and so the problem re-occurs. A richer meta-meta-language is needed to express the vagueness of this new notion. To fully resolve the problem the hierarchy of meta-languages is non-terminating. To this she adds: If there is no general objection to the claim that the sequence of metalanguages for metalanguages is potentially infinite, then what is the difficulty with adding ‘and each of these languages is vague’? [. . .] There is no vicious infinite regress forced upon us. It is just that the vague is not reducible to the non-vague. (2000, 208)
Is Keefe’s model of higher-order vagueness defensible? Again, for our purposes it doesn’t matter.³⁹ What is clear is that it is broadly co-defensible with what Radical Contextualism is committed to with respect to higher-order vagueness in order to address the simple objection. The upshot is that what Radical Contextualism says in response to the simple objection yields a set of commitments which, broadly, are no more implausible than the commitments incurred by the most promising non-epistemic (noncontextualist) theories of vagueness with respect to higher-order vagueness. Likewise, what Epistemicist Contextualism says in response to the simple objection yields a ³⁶ We freely assume the closure of the D-operator here. That is not uncontroversial of course. ³⁷ If S4 (i.e. KT4) is the logic for absolute definiteness then there is only a finite number of modalities (in fact at most fourteen distinct modalities, see Chellas 1980, 149). Consequently, there cannot be borderline cases to borderline cases ad infinitum. ³⁸ See Williamson (1994, 160–1) for the puzzle and for the anticipation of Keefe’s reply. ³⁹ A further issue concerns the possibility of quantifying over all levels. If that is possible then a strengthened version of the simple objection can be formulated thus: hold all features of all contexts of whatever level fixed, the vagueness of the predicate relative to that (infinite) sequence of contexts/levels still remains, therefore vagueness is not context-sensitivity. Is there a reply? In the first place, note that Keefe’s model of higher-order vagueness also suffers from a strengthened version of Williamson’s puzzle of higher-order vagueness if we are allowed to quantify over all meta-languages. So, again, (Radical) Contextualism is no worse off than its most sophisticated competitors. Secondly, as Elia Zardini has pointed out to us, to make this strengthened version of the challenge stick we would need to make sense of the infinite embedding ‘. . . satisfies ‘‘satisfies ‘‘satisfies . . . ‘‘satisfies ‘‘red’’ in C1 ’’ in C2 ’’ in C3 ’’, . . .’. But it is far from clear that such a sentence can be understood. Its length can be arbitrarily large (well beyond omega), and already at omega (and then at any limit ordinal) the string is not going to be well founded (it has no starting point as can be seen from the initial dots).
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set of commitments which are, broadly, no more implausible than the commitments incurred by the leading epistemic theories with respect to vagueness and higher-order vagueness. Either way, the simple objection to Contextualism loses much, if not most, of its force. Re f e re n c e s ˚ Akerman, J. and Greenough, P. (2009), ‘Vagueness and non-indexical contextualism’ in New Waves in the Philosophy of Language, ed. Sarah Sawyer, Basingstoke, Palgrave Macmillan. Beall, Jc (2003), Liars and Heaps: New Essays on Paradox, New York, Oxford University Press. Chellas, B. F. (1980), Modal Logic, Cambridge, Cambridge University Press. Fara, D. G. (2000), ‘Shifting sands: an interest-relative theory of vagueness’, Philosophical Topics 28, 45–81. (Originally published under the name ‘‘Delia Graff’’). Goguen, J. A. (1969), ‘The logic of inexact concepts’, Synthese 19, 325–73. Greenough, P. (2003), ‘Vagueness: a minimal theory’, Mind 112, 235–81. (2005), ‘Contextualism about vagueness and higher-order vagueness’, Proceedings of the Aristotelian Society, Supplementary Volume, 79, 167–90. Heck, R. (2003), ‘Semantic accounts of vagueness’ in Liars and Heaps, ed. Jc Beall, 106–27, New York, Oxford University Press. Kamp, H. (1981), ‘The paradox of the heap’ in Aspects of Philosophical Logic, ed. U. M¨onnich, 225–77, Dordrecht, Reidel. Kaplan, D. (1989), ‘Demonstratives’ in Almog, Perry, and Wettstein (eds.), Themes from Kaplan, Oxford, Oxford University Press. Keefe, R. (2000), Theories of Vagueness, Cambridge, Cambridge University Press. (2007), ‘Vagueness without context change’, Mind 116, 275–92. Keefe, R., and Smith, P., eds. (1996), Vagueness: A Reader, Cambridge, MA, MIT Press. K¨olbel, M. (2007), ‘A problem for contextualism about vagueness’, Paper presented at the Joint Session of the Mind Association and Aristotelian Society, July 2007. Koons, R. (1994), ‘A new solution to the sorites problem’, Mind 103, 439–49. Lewis, D. (1979), ‘Scorekeeping in a language game’, Journal of Philosophical Logic 8, 339–59. Mills, E. (2004), ‘Williamson on vagueness and context-dependence’, Philosophy and Phenomenological Research, 68, 635–41. Priest, G. (2003), ‘A site for sorites’ in Liars and Heaps, ed. Jc Beall, 9–23, New York, Oxford University Press. Raffman, D. (1994), ‘Vagueness without paradox’, Philosophical Review, 103, 41–74. (1996), ‘Vagueness and context relativity’, Philosophical Studies 81, 175–92. Shapiro, S. (2003), ‘Vagueness and conversation’ in Liars and Heaps, ed. Jc Beall, 39–72, New York, Oxford University Press. (2006), Vagueness in Context, Oxford, Oxford University Press. Simons, P. (1992), ‘Vagueness and ignorance’, Aristotelian Society, suppl. 66, 163–77. Soames, S. (1999), Understanding Truth, Oxford, Oxford University Press. Sorensen, R. (1988), Blindspots, Oxford, Clarendon Press. (2001), Vagueness and Contradiction, Oxford, Oxford University Press. Stanley, J. (2003), ‘Context, interest-relativity, and the sorites’, Analysis 63, 269–80. Williamson, T. (1994), Vagueness, London, Routledge. (2000), Knowledge and Its Limits, New York, Oxford University Press.
16 Saying More (or Less) Than One Thing Andrea Iacona
In a paper called Definiteness and Knowability, Tim Williamson addresses the question whether one must accept that vagueness is an epistemic phenomenon if one adopts classical logic and a disquotational principle for truth. Some have suggested that one must not, hence that classical logic and the disquotational principle may be preserved without endorsing epistemicism.¹ Williamson’s paper, however, finds ‘no plausible way of substantiating that possibility’. Its moral is that ‘either classical logic fails, or the disquotational principle does, or vagueness is an epistemic phenomenon’.² The moral of this chapter, on the contrary, is that there is a plausible way of substantiating that possibility. The option it contemplates looks like a view that Williamson dismisses at the beginning of his paper, and that others regard as unworthy of serious consideration.
16.1 A couple of preliminary clarifications. The first concerns the expression ‘borderline case’. It is widely accepted as a matter of definition that a vague word is a word that admits of borderline cases. Typically, a vague predicate is a predicate that admits of borderline cases. Yet this leaves unsettled what ‘borderline case’ means exactly. According to one reading of the expression, a borderline case is a situation that involves actual problems of evaluation. We are ‘in’ a borderline case so understood when a speaker assertively utters a sentence and we don’t know whether the assertion is correct or incorrect. For example, it may happen that a speaker assertively utters ‘A is tall’ and we don’t know how to evaluate the assertion, because we don’t know whether it is correct to say that A falls into the extension of ‘tall’. According to another reading of the expression, a borderline case is an object that neither clearly belongs to the extension of a predicate neither clearly does not belong This chapter has a long history, as it went through a considerable number of changes and revisions before taking its present shape. In the course of this history, several people have helped me with it in various ways. Richard Dietz, Max K¨olbel and Tim Williamson are definitely among them. But there are many others. ¹ See Horwich 1998, 78–84 and Field 1994, 409–22. ² Williamson 1995, 171.
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to it. This reading hides an ambiguity. An object may be a borderline case for a predicate in the sense that it is unclear simpliciter whether it belongs to the extension of the predicate. That is, an object may neither paradigmatically have a property nor paradigmatically not have it. For example, if the height of A is 1.75, A is neither paradigmatically tall nor paradigmatically not tall. Thus A may be called a borderline case for ‘tall’. Alternatively, an object may be a borderline case for a predicate relative to a certain context in the sense that it is unclear in that context whether it belongs to the extension of the predicate. For example, A is a borderline case for ‘tall’ relative to a certain context if it is unclear in that context whether ‘tall’ applies to A. The distinction between borderline situations and borderline objects doesn’t really matter as far as borderline objects in the second sense are concerned. Since the characterization of such an object is parasitic on the notion of a borderline situation, it turns out to be trivial that borderline situations arise just in case borderline objects are involved. However, it is important to avoid confusion between borderline situations and borderline objects in the first sense, because there is no reason to think that they must always go together. It is plausible to say that—given the due qualifications—whenever a borderline situation arises, a borderline object in the first sense is involved. But the converse does not hold. Borderline objects in that sense do not necessarily make borderline situations. Suppose that A is 1.75. This does not entail that whenever ‘A is tall’ is assertively uttered, problems of evaluations arise. It is easy to imagine cases in which the sentence is assertively uttered, yet no unclarity affects the evaluation of the assertion. If A is teacher of a class of kids, and one of them assertively utters ‘A is tall’ in order to explain to another why A is able to write on a part of the board they can’t reach, the assertion may be taken to be correct. If instead ‘A is tall’ is assertively uttered by someone who is seriously considering A’s chances to join a basketball team, the assertion may be taken to be incorrect. Here only the first reading of ‘borderline case’ will be adopted. To avoid confusion, the expression will not be used to refer to objects. In the literature the two readings are often mixed, and the ambiguity of the second is often neglected. This is why sometimes philosophers talk as if whenever (and just because) a borderline object in the first sense is involved, a borderline case should arise. And given that borderline objects are as common as paradigmatic objects, this amounts to talking as if borderline cases were as common as unproblematic cases. But such talk is just a theoretical habit that takes us very far from ordinary linguistic practice. In reality, it almost never happens that a speaker assertively utters a sentence and we don’t know how to evaluate the assertion. We normally take for granted some way of understanding the sentence according to which the assertion turns out clearly correct or clearly incorrect. Borderline cases almost never arise. As a matter of fact, borderline objects in the first sense are correctly or incorrectly described this or that way depending on the occasion. The second clarification concerns the expression ‘what is said’. Roughly, what is said by uttering a sentence on a certain occasion depends on how the words occurring in the sentence are understood on that occasion, where understanding a word involves grasping its linguistic meaning and, possibly, specifying its reference in the context of utterance. This leaves room for at least two distinct notions. One is that,
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when a sentence is assertively uttered by a speaker, there is something that the speaker has in mind. For example, a speaker may use the sentence ‘He is there’ to convey the information that A is in St. Andrews, while another speaker may use it to convey the information that A is in Mexico City. A natural way to express this difference is to say that the two speakers have different things in mind when they utter the sentence. So there is a reading of ‘what is said’—call it intentional reading—according to which what is said by uttering a sentence on a certain occasion is a matter of what understanding of the sentence can rightfully be ascribed to the speaker on that occasion. The other notion is that, when a sentence is assertively uttered by a speaker, there is something to which truth or falsity can be ascribed. The something in question is naturally understood as a specification of the reference of the words occurring in the sentence such that, according to it, either the sentence describes things as they are or it describes things as they are not. Suppose that A is in Mexico City. Then ‘He is there’ turns out true if ‘he’ refers to A and ‘there’ refers to Mexico City, while it turns out false if ‘he’ refers to A and ‘there’ refers to St. Andrews. So there is a reading of ‘what is said’—call it truth-conditional reading—according to which what is said by uttering a sentence on a certain occasion is a matter of what understanding makes the sentence evaluable as true or false on that occasion. It is a naive temptation to put together the two notions. This amounts to thinking that, when a sentence is assertively uttered by a speaker, there is one understanding of the sentence that can rightfully be ascribed to the speaker and involves a specification of the reference of the words occurring in it that makes it evaluable as true or false. Contrary to this temptation, here it will be assumed that the two readings of ‘what is said’ do not coincide. It may happen that a speaker assertively utters a sentence and has something in mind, yet no specification of the reference of the words in the sentence that makes it evaluable as true or false can rightfully be ascribed to the speaker.³
16.2 The thought entertained in this chapter is that borderline cases are cases in which there is no such thing as what is said in the truth-conditional sense. An example of borderline case may help illustrate. Suppose that the grandmother is in the kitchen and the cat is in the living room in such a position that half of its body lies on the mat and the other half lies on the floor. The grandmother makes a guess and assertively utters the sentence (1) The cat is on the mat Apparently, it is hard to tell whether the grandmother’s assertion is correct or incorrect, because it is hard to tell whether the word ‘on’ that occurs in (1) applies to the cat ³ Iacona (2006) deals with cases in which the two readings seem not to coincide.
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and the mat. This is not simply due to the fact that the cat is in the position described. It is easy to imagine situations in which (1) is uttered and the cat is in the very same position, yet no problem of evaluation arises. Imagine that the cat got muddy in the garden, and the mat is brand new and very expensive. The grandmother could be concerned about the mat, and wonder whether at least part of the body of the cat lies on it. Or imagine that it is winter and the floor is cold. The grandmother could be concerned about the health of the cat, and wonder whether its whole body lies on the mat. On the assumption that there are cases in which the two readings of ‘what is said’ do not coincide, this can be regarded as one of them. The grandmother has something in mind. That is, something can rightfully be ascribed to her as the actual understanding of (1). But that understanding does not amount to one specification of the reference of the words occurring in (1) that makes (1) evaluable as true or false. Whether (1) is true or false depends on whether or not the pair formed by the cat and the mat belongs to the extension of ‘on’. But the extension of ‘on’ can be specified both in such a way as to include the pair and in such a way as not to include it. As the two alternative situations imagined show, neither of the two possibilities is ruled out by the linguistic meaning of ‘on’. This is to say that there are ways of understanding ‘on’ according to which (1) is true and ways of understanding ‘on’ according to which (1) is false. The point is that the actual understanding of (1) does not decide between the former and the latter. What is said in the intentional sense does not uniquely determine something said in the truth-conditional sense. The case of the grandmother may be described as a case of underspecification, that is, as a case in which the actual understanding of a sentence is not sufficiently specific for the purpose of ascribing truth or falsity to the sentence. For there are at least two ways of understanding (1) that go beyond its actual understanding, in the sense of not being uniquely determined by it. A case of underspecification is not a normal case, as normally the actual understanding of a sentence is sufficiently specific. The account may be phrased in a more rigorous way by using familiar terminology. Let a valuation be an assignment of semantic properties to the sentences of a language that determines definite truth conditions for them. For example, in the case of (1) a valuation will assign an object to ‘the cat’, another object to ‘the mat’, and a set of pairs to ‘on’. Valuations amount to what supervaluationists call ‘precisifications’, that is, ways of making language precise.⁴ A valuation is admissible when it corresponds to a legitimate way of making language precise, that is, when it respects the constraints imposed by the linguistic meaning of its expressions. For example, a valuation that assigns a dog to ‘the cat’ will not be admissible. The actual understanding of a sentence—what is said in the intentional sense—may be described as a set of admissible valuations, namely, the set of all the admissible valuations that are not ruled out by that understanding.⁵ Truth in a valuation is defined in the standard way. In the case of a simple sentence, it depends on whether the predicate applies to the objects denoted by the terms. Thus (1) will be true in a valuation v just in case the pair formed by the objects that v ⁴ As in Fine (1975).
⁵ Admissibility is again as in Fine (1975).
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assigns to ‘the cat’ and ‘the mat’ belongs to the set that v assigns to ‘on’. The truth of a complex sentence depends on that of its constituents, in accordance with the usual compositional rules. For example, if (1) is true in v and ‘Snow is white’ is true in v, then also ‘The cat is on the mat and snow is white’ will be true in v. Let us stipulate that two valuations overlap on a sentence s when they are alike as far as the truth or falsity of s is concerned. For example, two valuations overlap on (1) if they assign to ‘the cat’ and ‘the mat’ the same objects a and b, and they assign to ‘on’ two sets that differ only in that one includes a pair
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false in that interpretation. The case of the grandmother belongs to neither of these two categories, as it is a case in which there is no unique thing said. One way to make sense of this is to allow that more than one thing is said, in that more than one interpretation is compatible with the actual understanding. Another way is to hold that nothing is said, in that no interpretation can be attributed to the speaker. The latter seems preferable. First of all, it is quite implausible to hold that a speaker says both something true and something false at the same time. Secondly, although it is sensible to assume that any thing said in the truth-conditional sense is an interpretation, there seems to be no reason in addition to grant that any interpretation is something said in the truth-conditional sense. For example, any set containing one admissible valuation is by definition an interpretation of (1). But we wouldn’t describe any such set as something said by the grandmother. A case of underspecification differs from a normal case in that it involves a conflict of interpretations. The problem that characterizes it concerns the attribution of interpretations, it is not primarily a problem of truth and falsity. There is an obvious sense in which underspecification is compatible with the principle of bivalence, according to which truth and falsity are mutually exclusive and jointly exhaustive values. It is the sense in which bivalence holds in any interpretation. By definition, for any interpretation α of a sentence s, either all the valuations that belong to α make s true or they make s false. This entails two claims. One is that either s is true in α or s is false in α: truth and falsity are jointly exhaustive. The other is that it cannot be the case that s is true in α and s is false in α: truth and falsity are mutually exclusive. 16.3 So far it has been suggested that the case of the grandmother is a case of underspecification. Since the case of the grandmother is clearly a borderline case, this entails that at least some borderline cases are cases of underspecification. However, it does not entail that all borderline cases are cases of underspecification. The stronger claim may be justified as follows. Let C be a case in which S assertively utters s, and let α be the actual understanding of s manifested by S. Now suppose that α is an interpretation. Given that bivalence holds relative to interpretations, s is either true or false in α. If s is true in α, then an interpretation in which s is true can be attributed to S. This means that no problem of evaluation arises in C. Similarly, if s is false in α, then an interpretation in which s is false can be attributed to S. This means that no problem of evaluation arises in C. Therefore, if α is an interpretation then no problem of evaluation arises in C. Since C is a borderline case only if some problem of evaluation arises in C, we get that if α is an interpretation then C is not a borderline case. It follows that if C is a borderline case then α is not an interpretation. That is, underspecification obtains in all borderline cases. We saw that if α is not an interpretation, there are at least two interpretations compatible with α that yield opposite truth values for s. Therefore, the claim that underspecification obtains in all borderline cases entails that if C is a borderline case, there are at least two interpretations compatible with α that yield opposite truth values for
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s. A fortiori, it entails that if C is a borderline case, there are admissible valuations that yield opposite truth values for s. This amounts to a presupposition that is at the core of semantic theories of vagueness, namely, that borderline cases are cases in which the linguistic meaning of the words occurring in the sentence does not settle the question of whether the assertion is correct or incorrect. For example, a case in which a speaker assertively utters ‘A is tall’ and we don’t know how to evaluate the assertion is a case in which the meaning of ‘tall’ rules out neither the possibility that the assertion is correct nor the possibility that it is incorrect. A general tenet of semantic theories of vagueness is that the kind of unclarity that is characteristic of borderline cases depends on the linguistic meaning of the words occurring in the sentences we utter. The argument above shows that this tenet holds, in that it is a necessary condition of borderline cases that the linguistic meaning of the words occurring in the sentence uttered does not determine whether the assertion is correct or incorrect. On the other hand, however, underdetermination in this sense is not a sufficient condition of borderline cases. The linguistic meaning of ‘on’ and the fact that the cat is in such a position that half of its body lies on the mat and the other half lies on the floor do not suffice to make the case borderline. The same goes for the linguistic meaning of ‘tall’ and the fact that the height of A is 1.75. In order to have a borderline case we need three ingredients: the linguistic meaning of the words occurring in the sentence, the state of affairs, and the actual understanding. The first two do not suffice. This point may be phased in terms of the customary distinction between speaker’s meaning and semantic meaning, that is, between what the speaker means by uttering certain words, and what those words mean. Since borderline cases are effects of the actual understanding of sentences, they are features of speaker’s meaning. Therefore, there is a sense in which vagueness concerns speaker’s meaning rather than semantic meaning. Semantic theories of vagueness usually do not recognize this sense. Yet there is also a sense in which vagueness concerns semantic meaning, that in which it amounts to underdetermination of linguistic meaning. The semantic meaning of a vague sentence may be identified with a set of valuations some of which are in conflict. In this sense it is correct to say that borderline cases are grounded on linguistic meaning. Asking which of these two senses comes first, or which is more fundamental, is like asking whether the chicken or the egg comes first. On the one hand, vagueness in speaker’s meaning depends on vagueness in semantic meaning, in that the latter is a condition of the possibility of the former. On the other, vagueness in semantic meaning depends on vagueness in speaker’s meaning, in that words have linguistic meaning as a result of the way speakers use them. 16.4 The account of borderline cases in terms of underspecification contradicts Williamson’s disjunctive moral. In the first place, borderline cases pose no threat to classical logic. For they involve no violation of bivalence. In the second place, the disquotational principle is preserved. Assuming that truth and falsity apply to
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sentences relative to interpretations, there are two ways of stating the principle. One is to phrase it as a schema that holds for sentences as they are actually understood and involves ascription of truth relative to interpretations. That is, for any interpretation α, the following biconditional is true relative to α: ( T1) ‘p’ is true if and only if p Here ‘p’ is a substitutional variable, and ‘true’ is a predicate whose extension varies with α. Borderline cases do not affect the principle, in that the only trouble they give concern the choice of α. The other way is to phrase the principle as a schema that holds for interpreted sentences. For any α, let ‘pα ’ stand for ‘p’ as it is interpreted according to α. The following biconditional is true simpliciter: ( T2) ‘pα ’ is true if and only if pα In this case, to see that borderline cases do not affect the principle it must be acknowledged that ( T2) follows from a more general schema: ( T3) If ‘p’ says that pα , then ‘p’ is true if and only if pα A borderline case is a case in which ‘p’ is uttered, but there is no α such that ‘p’ says that pα . This means that the antecedent of ( T3) is not satisfied, hence that ( T3) is vacuously true.⁶ In the third place, the account does not entail that vagueness is an epistemic phenomenon. According to epistemicism, the meaning of the words occurring in a sentence determines a truth value for the sentence, in borderline cases just as in normal cases. This is to say that in the case of the grandmother the extension of ‘on’ is such as to make (1) true or false. The idea is that words have sharp boundaries, but we don’t know exactly where these boundaries lie. That is, given any cat and any mat, either the pair formed by them belongs to the extension of ‘on’ or it doesn’t. The fact is that we don’t know exactly the borders of that extension, hence it may happen that we don’t know, of a certain cat and a certain mat, whether or not the pair formed by them belongs to it. The account in terms of underspecification entails nothing like that. Truth and falsity apply to (1) relative to interpretations, so only relative to this or that interpretation the extension of ‘on’ is such as to make (1) true or false. Since borderline cases are cases in which there is no such thing as ‘the’ correct interpretation, there is no such thing as the interpretation that makes (1) true or false. So there is nothing to be ignorant of. ⁶ ( T3) provides an argument against the hypothesis—whose plausibility is called into question in §2—that borderline cases are cases in which more than one thing is said. As it is shown in Andjelkovic and Williamson 2000, 225–6, ( T3) entails a principle of uniformity to the effect that everything said by a sentence on a certain occasion has the same truth value: if ‘p’ says that pα and ‘p’ says that pβ , then pα if and only if pβ . For suppose that there are two interpretations α and β such that ‘p’ says both that pα and that pβ . By instances of ( T3), this yields that ‘p’ is true if and only if pα and that ‘p’ is true if and only if pβ . It follows that pα if and only if pβ . The principle of uniformity rules out the hypothesis that there are cases in which ‘p’ is uttered, and there are two interpretations α and β such that ‘p’ says both that pα and that pβ but α and β yield opposite truth values.
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This provides a response to an argument that is a mainstay in Williamson’s defence of epistemicism. The argument goes as follows. Suppose that ‘A is tall’ is uttered and we are in a borderline case. If one accepts a plausible unknowability principle for borderline cases, one gets that we don’t know whether A is tall. But if one assumes classical logic and the disquotational principle, one also gets that ‘A is tall’ is either true or false. This means that there is something to be known: ‘the speaker’s ignorance has an object’.⁷ The response is that it must not be taken for granted that classical logic and the disquotational principle entail that ‘A is tall’ is either true or false simpliciter. On the assumption that truth and falsity apply to sentences relative to interpretations, bivalence can be shown to hold relative to interpretations, in accordance with classical logic and the disquotational principle. Therefore, what one gets is that, in any interpretation, ‘A is tall’ is either true or false. This is consistent with the hypothesis that the speaker’s ignorance has no object in the case considered. For in that case there is no such thing as the correct interpretation of ‘A is tall’.⁸ 16.5 Showing that there is a coherent view that preserves classical logic and the disquotational principle without being epistemicist wouldn’t be enough if the view were implausible. For mere coherence does not justify acceptance. This is what Williamson seems to think of a view that looks pretty much like that outlined in the previous sections: At least one view does combine classical logic and ( T+) with the denial that vagueness is an epistemic phenomenon in a way that seems at any rate not formally inconsistent. This is the view that vague sentences do not say that anything is the case, in borderline cases if not elsewhere. Thus, if ‘b’ is a borderline case for ‘bald’, to say ‘b is bald’ or ‘b is not bald’ is to make an utterance without propositional content. One says nothing by uttering either sentence, so neither is true. Thus, the corresponding antecedents of ( T+), in ‘in c ‘‘b is bald’’ says that P’ and ‘in c ‘‘b is not bald’’ says that P’, fail whatever legitimate substitution is made for ‘P’, and ( T+) holds vacuously. For the same reason, ( T ) would have no relevant instance. Thus, there would be no relevant true sentence, and therefore nothing for speakers of the language to be ignorant of. [. . .] No attempt will be made here to argue against that extreme view. Many of the philosophers who wish to accept orthodoxy while denying that vagueness is an epistemic phenomenon are willing to instantiate ( T ) with vague sentences even in borderline cases. They allow that ‘b is bald’ is true if and only if b is bald; ‘b is bald’ says that b is bald, even if b is borderline for ‘bald’.⁹
Here ( T+) is a principle equivalent to ( T3), ( T ) is the disquotation schema, and ‘orthodoxy’ stands for the combination of classical logic with ( T ). The account outlined in the previous sections seems to fit the description. There is a sense in which ⁷ Williamson 1995, 174–5. ⁸ Obviously, here the assumption is that Williamson’s use of ‘borderline case’ does not differ from ours. ⁹ Williamson 1995, 173.
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one says nothing by uttering a sentence in a borderline case, namely, the truthconditional sense. In this sense it is correct to say that a principle equivalent to ( T+) holds vacuously, in that the antecedent of ( T3) is not satisfied. Why should the view be ‘extreme’? The alleged motivation is that an utterance of ‘b is bald’ in a borderline case would have no content. Consequently, the disquotation schema wouldn’t be instantiated, and we couldn’t say that ‘b is bald’ is true if and only if b is bald. But that isn’t so. It is plausible that an utterance of ‘b is bald’ in a borderline case has content. The obvious sense in which it has content is that some understanding of the sentence can rightfully be ascribed to the speaker. In other words, such a case differs from one in which, say, ‘He is there’ is written on the board to make a point of grammar. However, the view does not entail that ‘b is bald’ lacks content in that sense. For that sense is the intentional sense, and the view does not deny that something is said in the intentional sense.¹⁰ Similarly, it is plausible that ( T ) is instantiated with sentences uttered in borderline cases, as it is quite natural to say that ‘b is bald’ is true if and only if b is bald. But again, the view does not deny it. The disquotational principle can be phrased in terms of a schema—( T1)—that holds for sentences as they are actually understood, and these obviously include sentences uttered in borderline cases. So the view is not extreme, and perhaps some attempt should be made to argue against it. To appreciate its plausibility, it is important to distinguish it from another view that is indeed extreme, namely, that according to which only precise expressions have meaning, so whenever we utter a sentence containing vague expressions we say nothing at all.¹¹ What the account of borderline cases outlined requires is simply that that there are cases in which we say nothing sufficiently precise. This is compatible with there being cases—the normal ones—in which we say sufficiently precise things. For example, even if there are cases in which ‘on’ is understood in such a way as to say nothing sufficiently precise by uttering (1), most of the time ‘on’ is understood in ways that are sufficiently precise for the descriptive goals that guide our use of the sentences containing it. This is not to say that most of the time ‘on’ is understood in a completely precise way. The use of a predicate almost never involves complete specification of its extension. Rather, it involves a partial specification of it, in that the only part of extension that matters on each occasion concerns the objects that are salient on that occasion. In other words, the use of a predicate on a given occasion determines a set of valuations, the set of all the valuations that share the part specified. Thus if one uses (1) to describe a cat a and a mat b, one takes the extension of ‘on’ to include the pair
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of ‘on’ includes
16.6 The foregoing sections show how borderline cases can be accounted for in accordance with the thesis that truth and falsity apply to sentences relative to interpretations. This section shows that the thesis provides a straightforward solution to the paradox of the sorites. In its original form, the sorites says that if the removal of one grain from a heap always leaves a heap, the successive removal of every grain still leaves a heap. The argument goes as follows: (2) 1000 grains make a heap (3) For every n, if n grains make a heap then n − 1 grains make a heap (4) 0 grains make a heap The inference from (2) and (3) to (4) seems legitimate, (2) and (3) seem acceptable, but (4) seems unacceptable. Therefore, it is natural to think that there is something wrong with the argument. To provide a solution to the paradox is to say what exactly is wrong. The thesis that truth and falsity apply to sentences relative to interpretations points to the following definitions of validity and soundness. An argument is valid if and only if, necessarily, every interpretation that makes its premises true also makes its conclusion true. The criterion involved is the classical one, that according to which validity is necessary truth preservation. But since the premises and conclusion of an argument can be true only relative to interpretations, truth can be necessarily preserved only relative to interpretations. Similarly, an argument is sound in an interpretation if and only if it is valid and all its premises are true in that interpretation. Again, the criterion involved is the classical one, that according to which a sound argument is a valid argument whose premises are true. But since the premises of an argument can be true only relative to interpretations, the argument can be sound only relative to interpretations. These two definitions tell us what is wrong with the sorites. Although the argument is valid, there is no interpretation that makes it sound. For there is no interpretation in which (3) is true. An interpretation of (3) is a way of understanding (3) relative to which truth or falsity can be ascribed to it. Since (3) is equivalent to a list of conditionals, the interpretation must allow ascription of truth or falsity to each of the conditionals in the list, hence it must allow ascription of truth or falsity to the antecedent and to the consequent of each of the conditionals in the list. This entails that, for each of the collections of grains featuring in the series that goes from 1000 to 0 grains,
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it has to be specified whether or not it is a heap. Therefore, an interpretation of (3) involves a way of understanding ‘heap’ according to which there is a cut-off point in the series, that is, there is a number n such that a collection of n grains belongs to the extension of ‘heap’ while a collection of n − 1 grains does not belong to it. The sorites draws its appeal from the fact that we normally use ‘heap’ without specifying its extension as required by (3). Thus, (2) seems acceptable in that ‘heap’ is normally understood in such a way that a collection of 1000 grains belongs to its extension. Similarly, (4) seems unacceptable in that ‘heap’ is normally understood in such a way that a collection of 0 grains does not belong to its extension. But such ways of understanding ‘heap’ do not involve complete specification of its extension. In particular, they do not involve a delimitation of that extension sensitive to differences of one grain. Normally, when we understand ‘heap’ in such a way that a collection of n grains belongs to its extension, we do not have in mind a specification which prescribes that a collection of n − 1 grains does not belong to it. Therefore, we are apt to exclude that a collection of n grains is a heap but a collection of n − 1 grains is not a heap. This is why (3) seems acceptable. In other words, what makes the existence of a cut-off point for ‘heap’ unwelcome is that we normally do not specify such a point. Whenever we use the word to describe a certain object, we take for granted that no cut-off point lurks in the vicinity of that object, namely, that relevantly similar objects may equally be described in the same way.¹²
16.7 The account of vagueness outlined in this chapter—call it the underspecification view —may be contrasted with two similar accounts that are well known. The first is standard supervaluationism. The underspecification view substantively differs from standard supervaluationism, as the latter does not contradict Williamson’s moral. The basic idea of supervaluationism is that the vagueness of natural language consists in its capacity in principle to be made precise in more than one way. Following this idea, the method of supervaluations is adopted in order to deal with sets of precisifications of the language. A supervaluation is an assignment of truth values based on a quantification over assignments of truth values relative to precisifications. A sentence is ‘supertrue’ if it is true on all precisifications, ‘superfalse’ if it is false on all precisifications, and neither otherwise. Supervaluationism identifies truth with supertruth. This is why it obeys Williamson’s moral. In the first place, supertruth does not conform to the disquotational principle. If a sentence is true on some precisifications and false on others, the biconditional obtained by plugging the sentence in the disquotation schema is not supertrue. In the second place, supertruth violates bivalence, hence ¹² Assuming that the use of ‘heap’ on a given occasion is guided by what is psychologically or conversationally salient on that occasion, one may say that in normal circumstances no cut-off point is located within the area of salience. This, however, is not quite the same thing as to say that a cut-off point exists but is located somewhere outside that area, as suggested in Fara 2000.
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classical logic. Moreover, as it has been emphasized by its critics, supervaluationism fails to preserve some classical principles about consequence and rules of inference. This is due to the definition of validity as necessary preservation of supertruth, that is, the definition according to which an argument is valid just in case necessarily, if its premises are supertrue then its conclusion is also supertrue.¹³ The underspecification view shares with supervaluationism its basic idea. It is plausible to say that a vague language is a language that in principle can be made precise in more than one way. This is couched by the assumption that a sentence allows different admissible valuations. However, the underspecification view differs from supervaluationism in that it does not identify truth with supertruth. Truth in an interpretation conforms to the disquotational principle and does not violate bivalence.¹⁴ Moreover, the underspecification view has no problem with consequence and rules of inference, in that it does not involve a supervaluational definition of validity. The latter says that an argument is valid just in case, necessarily, if its premises are true on all precisifications then its conclusion is true on all precisifications, while the definition given in §6 says that an argument is valid just in case, necessarily, in all interpretations, if its premises are true then its conclusion is true. The difference between the two definitions may be seen as a difference in the scope of the quantification over precisifications or interpretations.¹⁵ The second account to be considered is the non-standard version of supervaluationism advocated by Van McGee and Brian McLaughlin, and then adopted by Cian Dorr. The version at issue is non-standard in that it does not identify truth with supertruth. McGee and McLaughlin claim that a distinction must be drawn between ‘truth’ and ‘definite truth’, where the former is defined in accordance with the disquotational principle and classical logic, while the latter is characterized by using a supervaluational model-theoretic apparatus. Given that this characterization of definite truth is not epistemic, their view is like the underspecification view—and unlike standard supervaluationism—in that it contradicts Williamson’s moral.¹⁶ But some significant differences remain. In the first place, it is not clear whether the actual understanding of a sentence plays some role in the determination of definite truth. McGee and McLaughlin say at a certain point that ‘the thoughts and practices of the speakers of the language, together with the non-linguistic facts, pick out a set of sentences as definitely true’.¹⁷ However, it is not clear whether this entails that what a speaker has in mind on a certain occasion may contribute to determine what ¹³ See Williamson 1994, 146–53. The problem concerns supervaluationism in its standard version. But other definitions may be adopted. See Keefe 2000 and Varzi 2007. ¹⁴ The simple fact that truth in an interpretation is defined in terms of a quantification over valuations does not mean that it is a form of supertruth, at least not in the sense that matters here. For the definition does not allow for a third status between truth and falsity. ¹⁵ Williamson 1994, 147–8, Keefe 2000, §3, and Varzi 2007 spell out this distinction. The two kinds of definitions are called ‘global’ and ‘local’ respectively. ¹⁶ McGee and McLaughlin 1995, Dorr 2003. Williamson 1997, however, questions that the characterization of definite truth in McGee and McLaughlin 1995 is not epistemic. ¹⁷ McGee and McLaughlin 1995, 227.
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is definitely true on that occasion. If it doesn’t, then in the case of the grandmother the view must be that the unclarity that affects the evaluation of (1) is independent of what the grandmother has in mind, although it may depend on the thoughts and behaviour of the linguistic community as a whole. This is to say that the unclarity at issue is due entirely to the fact that the linguistic meaning of ‘on’ and the position of the cat are as they are, contrary to what is assumed here.¹⁸ In the second place, even supposing that McGee and McLaughlin do take into account the actual understanding of a sentence, or that their view may be adjusted in such a way as to take it into account, a difference remains. When McGee and McLaughlin talk of truth and definite truth, they seem to have in mind what is said in the intentional sense, whereas here the bearer of truth and falsity is what is said in the truth-conditional sense. This is in part a matter of focus. The underspecification view could as well be phrased in terms of the intentional reading of ‘what is said’. Borderline cases could be described as cases in which a speaker asserts a unique thing that is not evaluable. This would not essentially differ from saying that borderline cases are cases in which the thing asserted is neither definitely true nor definitely false, given that ‘evaluable’ would mean ‘evaluable as true or false’. However, there is one crucial respect in which the difference is not simply a matter of focus. McGee and McLaughlin ascribe truth simpliciter to what is said in the intentional sense. This entails that the things we say in borderline cases are true or false simpliciter. By contrast, even if the underspecification view were phrased in terms of the intentional reading of ‘what is said’, such ascription would be ruled out. For its main thesis would be that the things we say are evaluable as true or false only relative to interpretations. Re f e re n c e s Andjelkovic M. and Williamson, T. (2000), ‘Truth, falsity, and borderline cases’, Philosophical Topics 28, 211–44. Dorr, C. (2003), ‘Vagueness without ignorance’, in Hawthorne, J. and Zimmerman, D., eds., Philosophical Perspectives 17, Blackwell, 83–114. Fara, D. G. (2000), ‘Shifting sands: an interest-relative theory of vagueness’, Philosophical Topics 28, 45–81. (Originally published under the name ‘‘Delia Graff ’’). Field, H. (1994), ‘Disquotational truth and factually defective discourse’, Philosophical Review 103, 405–52. Fine, K. (1975), ‘Vagueness, truth and logic’, Synthese 30, 265–300. Garc´ıa-Carpintero, M. (2007), ‘Bivalence and what is said’, Dialectica 61, 167–90. Horwich, P. (1998), Truth, Oxford University Press, Oxford. Iacona, A. (2006), ‘True in a sense’, Grazer Philosophische Studien 72, 141–54. Keefe, R. (2000), ‘Supervaluationism and validity’, Philosophical Topics 28, 93–106. McGee, V. and McLaughlin, B. (1995), ‘Distinctions without a difference’, The Southern Journal of Philosophy 33 (Supplement), 203–51. Varzi, A. (2007), ‘Supervaluationism and its logics’, Mind 116, 633–75. ¹⁸ This is what they seem to think when they say things such as ‘Harry is bald’ is definitely true if Harry is ‘definitely bald’, 210. Similar considerations hold for Dorr 2003.
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Williamson, T. (1994), Vagueness, Routledge, London. (1995), ‘Definiteness and knowability’, The Southern Journal of Philosophy 33 (Supplement), 171–91. (1997), ‘Imagination, stipulation and vagueness’ in Villanueva, E. (ed.), Philosophical Issues 8, Ridgeview, Atascadero (CA), 214–28.
17 Vagueness as Semantic Max K¨olbel
I shall argue that vagueness, understood as a semantic phenomenon, can be accommodated within standard semantics by assimilating it to contingency in standard modal semantics and suitably modifying the pragmatics. I claim that vagueness in natural language is not a defect and that accommodating it is therefore obligatory for semantic frameworks for natural languages. In section 17.2, I interpret the claim that vagueness is a semantic phenomenon as involving at least the claim that vague predicates do not determine an extension. I then outline three ways in which standard semantics can account for the failure of an expression to determine an extension, namely ambiguity, indexicality and relativity to circumstances of evaluation (e.g. contingency). I point out some problems with treating vagueness as a form of ambiguity or as a form of indexicality. Then I explain the view that vagueness is a form of relativity to circumstances of evaluation, and why such a view needs to provide an account of the normative significance of truth for assertion and belief. I show how this normative role is constrained by the two desiderata that we explain the seductiveness of sorites arguments and give an account of borderline cases. Finally I briefly consider higher-order vagueness and conclude by comparing the account given with other views of vagueness.
17.1
VAG U E N E S S I S N OT A D E F E C T
Many natural language predicates are vague in the sense that they seem subject to tolerance constraints and therefore generate sorites paradoxes. For example, the predicate ‘is rich’ is vague because it seems to be subject to the constraint that if someone is not rich then receiving a small amount of money such as one cent will not make that person rich. Thus a sorites paradox can be formulated as follows: (A) A person with possessions worth 0 Euros is not rich. This chapter was first presented at the Fifth Arch´e Vagueness Workshop, 18–19 November 2005, and subsequently at a few other occasions. I would like to thank the participants for their comments, especially Hartry Field, Manuel Garc´ıa-Carpintero, Mark Sainsbury and Achille Varzi.
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(B) If a person with possessions worth n Euros is not rich, then a person with possessions worth n + 0.01 Euros is not rich either.¹ (C) A person with possessions worth 100 million Euros is not rich. (C) seems to follow from (A) and (B), but while (A) and (B) seem clearly true, (C) seems clearly not true. This phenomenon is widespread. Countless natural language predicates are vague in this sense. The vagueness of these predicates does not seem to be an impediment to their usefulness in communication. Similarly, the concepts expressed by vague predicates do not seem to create any problems for our thought. The vagueness of natural language predicates and the concepts they express is therefore not some deficiency, shortfall or malfunction. Vagueness is perfectly normal. If vagueness is normal, then semantic frameworks for natural languages, ought to be able to accommodate it. If standard frameworks cannot accommodate vagueness, then they need to be abandoned in favour of new or modified frameworks that do accommodate vagueness. In the interest of continuity, it is therefore desirable to explore if and how standard semantics can make room for vagueness. As we shall see, there are several ways in which room can be made for vagueness within standard semantics, some better than others.
17.2
VAG U E N E S S A S E X T E N S I O N A L I N D E T E R M I N AC Y
There is a minority of philosophers, the epistemicists, who hold that vagueness is not a semantic problem, but rather reflects our inability to know the exact borderlines of the extensions of the predicates (and concepts) we use. On this view, premise (B) in the above sorites is simply false. There is a truth of the form (D) A person with possessions worth n Euros is not rich and a person with possessions worth n + 0.01 Euros is rich. But we cannot know that truth because of general principles concerning knowledge.² It is this fact that explains why (B), despite its falsity, is so attractive. According to epistemicism, then, vagueness is an epistemic, and not a semantic phenomenon. The meaning of vague as well as non-vague predicates determines for each object whether it is in that predicate’s extension or not. The majority, however, finds the epistemic view incredible, in large part because it remains mysterious how the precise extensions of vague predicates are determined. The majority instead believes that vagueness is a semantic phenomenon, i.e. that the meanings of vague predicates fail to determine exact extensions. I will not provide any reasons to favour semantic views over epistemicism. I shall merely assume that ¹ There are different ways of formalizing ‘a person’ in (B), see Pagin 2009. These differences will not matter for the current discussion. ² See Williamson 1994 and Sorensen 1988.
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the semantic view is correct. Starting from that assumption, I will make a case for a certain treatment of vagueness, understood as semantic, within standard semantic frameworks. Thus the conclusion of this chapter is a conditional one. Some of those who believe that vagueness is a semantic phenomenon may think that the failure of vague predicates to determine an extension is a kind of imperfection that it would be better to reform away. I have already argued above that vagueness should not be treated as an imperfection because it is a normal, widespread and unproblematic aspect of language use. Others may infer from the extensional indeterminacy of vague predicates that we must modify the semantic framework, e.g. by allowing three truth values. But before going down that path, we ought to examine the resources of the existing standard framework to accommodate the kind of extensional indeterminacy characteristic of vague predicates.
17.3
S TA N D A R D S E M A N T I C S F O R C O N T E X T- S E N S I T I V E L A N G UAG E S W I T H I N T E N S I O N A L O PE R ATO R S
The semantic framework I shall be using can be called ‘double-index semantics for context-sensitive languages’.³ According to this framework, the meanings of natural language sentences determine characters, and these are functions from contexts of use to contents (propositions). Contents in turn are (or determine) functions from circumstances of evaluation to truth values (see diagram below). In the language of intension and extension: the meaning of each expression determines, in a context of use, an intension, and an intension determines, in each circumstance of evaluation, an extension, as shown in the diagram. For example, the English sentence ‘I am hungry now.’ has a character that determines different contents in different contexts of utterance. Utterances of the sentence can express propositions about different people and different times, depending on who utters it when. Now, consider one of these propositional contents, about some person John and some time t. This content, i.e. the proposition that John is hungry at t, has a truth value. Which truth value it has will depend on how things are with John at t, or, in other words, it will depend on the circumstances of evaluation. If John goes long enough without eating before t and is otherwise normal, then the proposition is true. If John has a large breakfast just before t, then it is not true. Thus the proposition is true in some possible worlds and not in others. The extension of the concept is hungry at t varies from one possible world to another. To summarize: the determination of the truth value of an utterance is generally a matter of two stages: the meaning or character of the expression determines, ³ This type of framework is familiar from Kaplan (1977). Lewis (1980) argues against Kaplan’s two-stage approach and proposes a competing one-stage theory, which does not postulate contents expressed by sentences in contexts. I shall assume that Kaplan is right to introduce double-indexing, but that otherwise the difference between Lewis’s index theory and Kaplan’s two-stage theory is not directly relevant to present purposes. For discussion see Recanati 2007.
Vagueness as Semantic Sentence (type)
Meaning (character)
Intension (content, proposition)
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for each context of utterance, a content, and then that content determines, for each circumstance of evaluation, an extension. Such variation in extension with different circumstances is a feature of contents that is exploited in the standard semantics of modal operators. It is worth adding that natural language sentences, in addition to their characters, also exhibit force indicators. These force indicators indicate the communicative function of utterances of the sentence.⁴ Thus, the proposition that John is hungry at t could be expressed, for example, with assertoric force, or as a question. Even though semanticists are traditionally more concerned with the truth conditional content of utterances, their theories must ultimately connect up with the theory of speech acts. One particular link that will play a role below is the normative role the extension of an utterance has for assertoric speech acts. Usually it is thought that assertion in some sense aims at truth, so that asserting an untrue proposition constitutes some kind of mistake. A semantic theory for a language in some sense⁵ represents part of the competence of users of that language. One aspect of linguistic competence, however, is usually treated as pre-semantic: the ability to resolve ambiguities. The input from which a semantic theory can be used to derive intensions (and illocutionary forces) of utterances, is thought of as unambiguous syntactic forms. Which unambiguous syntactic form is expressed by the utterance of an ambiguous sentence (such as ‘The bill was huge.’) is something that will again be resolved by recourse to the linguistic and nonlinguistic context of the utterance. Despite some similarities, there is a difference between disambiguation and assignment of content to indexicals.⁶ Disambiguation is usually treated as pre-semantic, while assignment of contents to indexical elements is treated as part of semantics. This is why the diagram above does not represent the determination of character or meaning of an expression type as yet another semantic function, in addition to the functions from context to content and from circumstance to truth value. ⁴ See K¨olbel forthcoming for more detailed reflections on force indicators and assertoric force. ⁵ In what sense is a notoriously difficult question which I shall not broach here. ⁶ It is not easy to justify the relegation of disambiguation to the pre-semantic realm, or the strict separation of the two phenomena. There are great similarities between some of the phenomena that are standardly treated as cases of ambiguity and those that are standardly treated as cases of indexicality. Names, such as ‘John’ are often treated as ambiguous, and such treatment may explain why in some contexts one can say literally and coherently ‘John is home, but John isn’t.’. Some however, will treat this as evidence for the indexical character of personal proper names—compare ‘He is French and he isn’t.’ or ‘Now the lights are on and now they are not.’, when demonstrating different people or times at the moment of uttering the different occurrences of ‘he’ and of ‘now’.
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T H R E E S TA N D A R D S O U RC E S F O R E X T E N S I O N A L I N D E T E R M I N AC Y
As already noted, on the semantic view of vagueness, vague predicates fail to determine for each object whether the predicate applies to it. For short, vague predicates are extensionally indeterminate. Diagnoses of this sort are not at all alien to the semanticist. There are three ways in which a semantically non-deficient predicate (thought of as an expression type) can fail to determine an extension: the predicate may be ambiguous, indexical or its extension may vary with the circumstances of evaluation. Let me briefly review these three sources of extensional indeterminacy. The first source is ambiguity: ambiguous expressions fail to determine an extension. For example, the word ‘coach’ has several distinct and unrelated meanings in English. Each meaning determines a different extension. On one meaning, some people are, some people are not in the extension of ‘coach’. On the other meaning, no person is in the extension.⁷ When ambiguous expressions are used in communication, confusion is avoided because the context of use allows communicators correctly to disambiguate, i.e. to focus on one of the meanings of the ambiguous expression and to ignore the others. For example, if someone utters ‘The coach is waiting.’, then successful communication seems to require correct disambiguation. Correct disambiguation would here seem to involve at least that speaker and audience disambiguate in the same way. The second source is indexicality. A predicate may fail by itself to determine an extension because its character is a non-constant function. For example the predicate ‘is my uncle’ expresses different properties when used by different speakers. When used by you it expresses a (relational) property instantiated by your uncles (if any), and when expressed by me it expresses a property instantiated my uncles. The third source is sensitivity to circumstances of evaluation. A predicate may fail to determine an extension because the content it expresses is a non-constant function from circumstances of evaluation to extensions. The best known and least controversial type of example is that of predicates expressing contingent properties. The extension of the property of being a photographer varies according to what actually happens. Modotti is in the extension the property has in some circumstances of evaluation, including the actual circumstance. But had she met different people in her youth, she would not have become a photographer, and would consequently not have been in the extension of the property. Sensitivity to circumstances of evaluation is best known in the case of contingency, and often circumstances of evaluation are interpreted merely as possible worlds. However, the framework does in principle allow further parameters in the circumstances. For example, one circumstantial parameter that ⁷ Some prefer to use ‘word’ (‘expression’, ‘predicate’) in such a way that by definition each word has only one meaning. Thus instead of having one word with several meanings, we have several words that are phonetically and orthographically indistinguishable. On this terminology, it is not words, but, for example, phonetic types that are ambiguous. This is just a terminological variation.
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has been much discussed is the time parameter that temporalists (including Kaplan himself ) want to add.⁸ The standard framework thus allows for three sources of extensional indeterminacy, and if vagueness is a form of extensional indeterminacy, each of these is a potential source of vagueness. I shall discuss the three sources in turn.
17.5
VAG U E N E S S A S A M B I G U I T Y
Suppose we want to account for the phenomena of vagueness by treating vagueness as a special form of ambiguity. This is how the story goes: a vague predicate, such as ‘rich’ has many meanings, in fact countless meanings, each of which draws a different precise boundary between the rich and the non-rich. This explains why the predicate by itself does not determine an extension. Thus, the semantics is standard, but we have an especially complicated pre-semantics.⁹ There is an immediate worry. When successfully communicating with ambiguous expressions, communicators are generally required to disambiguate, and to do so correctly. This means, at least, that in successfully interpreting an ambiguous utterance, speaker and audience have in mind the same of the candidate meanings. But there does not seem to be an analogous requirement of disambiguation in the case of vague communication. It seems wrong to say that when I hear ‘Anita is rich.’ I need to select one of many precise meanings of ‘rich’, and then to use that meaning (and only that meaning) in interpretation. There are answers to this worry. The ambiguity theorist might argue that communication with ambiguous expressions does not always require disambiguation. Consider an uncontroversially ambiguous sentence: ‘The coach is waiting.’ True, understanding an utterance of this sentence will often require correct disambiguation. However, there may be occasions, when no disambiguation is required. Suppose the coach of a second division football club doubles as the team’s chauffeur. Everyone knows that after the match, the team’s bus is waiting iff the team’s trainer is waiting. For it’s the team’s trainer who conducts the bus and if the trainer is waiting after a match, he is always waiting in the bus, the engine running. Given this background knowledge, neither speaker nor audience may need to disambiguate ‘coach’ in an utterance of ‘The coach is waiting.’. Another example: in order successfully to argue about whether a chemical purifier factory ought to be built down the road, we may not need to disambiguate ‘chemical purifier factory’. The differences ⁸ For an overview of other forms of variation of extension with circumstances of evaluation, see K¨olbel 2008. ⁹ The actual position closest to this view is that defended by Linda Burns in her 1991. Her view in turn is inspired by some remarks in Lewis 1975. Kit Fine’s classic exposition of supervaluationism (Fine 1975) also has some affinity, as Fine calls vagueness ‘ambiguity on a grand and systematic scale’. However, supervaluationism is not usually read as a form of the ambiguity view presented in this section.
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between the various possible meanings may be irrelevant to our communicative purposes.¹⁰ No doubt, the ambiguity theorist of vagueness could devise a systematic account of what is involved in communicating ambiguously without disambiguation, and she could apply this account to the special case of vague predicates. But whatever that account is, it faces some further difficulties. On the standard account sketched above, it is assumed that any non-defective utterance expresses a unique content or proposition. So, what, on the view that vagueness is ambiguity, is the proposition asserted by an utterance of ‘Anita is rich.’? There are only two ways the ambiguity theorist can go. Either she retains the principle that non-defective utterances express a unique proposition, or she does not. Let’s consider the first case first. The predicate ‘rich’ has countless precise meanings. In a given utterance of ‘Anita is rich.’, each of these meanings corresponds to one non-vague proposition concerning Anita. Let’s call these propositions the ‘candidate propositions’. The ambiguity theorist’s account will devise a way that allows us to derive the proposition expressed by an utterance of ‘Anita is rich.’ in some way from the set of candidate propositions. For example, she might say that the context of use determines a certain range of relevant candidate propositions, and that the proposition expressed is a conjunction, or perhaps a disjunction, of the relevant candidate propositions. Whatever the merit of these proposals, it is clear that they are proposals that move away from the idea that ‘rich’ is ambiguous. For what the so-called ‘ambiguity theorist’ is now claiming is that the content expressed by utterances of ‘rich’ (not the meaning of ‘rich’) is determined systematically by the many meanings of ‘rich’ and the context of use. So, while there may be a viable theory in the neighbourhood, it is highly misleading to describe it as a theory according to which vagueness is a form of ambiguity. The resulting theory will belong to the group of views that treat vague predicates as varying in intension with the context of use, i.e. as being indexical at least in a wide sense. These views will be considered in the next section. Now consider the second case. Suppose the ambiguity theorist wants to give up the principle that each non-defective utterance of a declarative sentence expresses a unique proposition. She might say, for example, that in using a sentence like ‘Anita is rich.’, the utterer does not determinately assert any single proposition but indeterminately asserts a range of the candidate propositions. She might abandon the idea that assertion is a propositional act that relates a person to a single proposition and ¹⁰ This example is from Sainsbury 2001. Sainsbury uses it to support the view that certain unspecificities in compound expressions are due to very unspecific meanings, rather than to hidden indexical variables, as claimed by Stanley (2000), or to ambiguity in the pre-semantic sense, as defended by Travis (1985, 1996). Unlike Sainsbury, I am here taking it for granted that ‘chemical purifier factory’ is ambiguous. For whatever one may think about the unspecificity in the compounding operation (i.e. ‘purifier factory’ can be read as a factory that makes purifiers or as a factory that employs purifiers in making something), the phrase also exhibits a classic scope ambiguity (‘chemical’ can qualify ‘purifier’ or ‘factory’). Thus, even if Sainsbury’s view about the first unspecificity were correct, the example can still illustrate my thesis that disambiguation is not always necessary.
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instead think of assertion as an act relating the asserter to a range of propositions.¹¹ I cannot see any principled reason why such proposals could not be made to work. They would need to be complemented with an appropriate theory of assertion and indeterminate assertion, and similarly with an account of the belief states we express by vague utterances (e.g. an account of indeterminately believing a range of propositions, or belief as a relation towards a range of propositions). This may ultimately yield an explanation of the phenomena of vagueness. But any such proposal does require some major modifications to the standard framework. If, on the ambiguity view, utterances of vague sentences express multiple propositions then we are in fact dealing with a phenomenon quite different from ordinary ambiguity. I shall not here examine the ambiguity view further because the approach discussed in section 17.7 below seems to me to involve less of a departure from the standard framework.
17.6
VAG U E N E S S A S I N D E X I C A L I T Y
Let us now consider the view that vagueness is a special form of indexicality. Indexical expressions do not by themselves determine an extension because they determine a content (intension) only in a context of use (their character is not constant). Thus, the predicate ‘is my uncle’ does not have a specific content until it is used by someone in a suitable context of utterance. At that point, it expresses a specific content (namely the property of being that person’s uncle), and determines an extension in any possible world. If vague predicates are indexical, then the content expressed by them similarly varies with context. Which property is expressed by ‘is rich’ will vary from one context of utterance to another. But what are these variable contents, and how does the context of utterance determine which of these variable contents is expressed? Consider ‘rich’. The extension of ‘rich’ clearly varies with a comparison class. In a context where we are talking about the wealth of sub-Saharan refugees, the threshold for membership in the extension of ‘rich’ will be much lower than in a context where we are discussing the comparative wealth of European royalty. Richness for a refugee and richness for a royal are two different properties with different extensions. The same goes for many vague adjectives: ‘tall’, ‘small’, ‘poor’, ‘bald’, ‘young’, etc. However, this form of context-sensitivity, as obvious as it is, is not particularly useful in accounting for the phenomena of vagueness. For the tolerance constraints characteristic of vagueness, and responsible for sorites paradoxes, govern ‘rich for a sub-Saharan refugee’ just as much as they do ‘rich’. ¹¹ Soames 2003, as well as Cappelen and Lepore 2005 and Cappelen 2008 distinguish the proposition semantically expressed from the proposition(s) asserted by an utterance. I will not discuss this complication here for reasons of clarity of exposition. These theories face the challenge of specifying how the proposition semantically expressed constrains the proposition(s) asserted. If semantic properties of expressions are to be determined by their use (and use is in turn to be constrained by semantic properties) then this challenge cannot be ignored, as it is deliberately by Cappelen 2008 (see also Pagin and Pelletier 2007).
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Those who invoke context-relativity in trying to account for vagueness¹² usually appeal to forms of context-relativity that explain why we are unable to find the exact border between the members and the non-members of the extension of a vague predicate. The idea is that the extension of a vague predicate depends on the context in such a way that the border between the members and the non-members is never where we are currently looking. Thus, in a context where we are considering two people whose level of wealth is only marginally different, our very act of considering them ensures that the boundary between the rich and the non-rich (at that context) does not separate these two people.¹³ Thus, the borderline is never where we are looking, and each of the instances of the generalized conditional premise of the sorites ((B) above) will be true at every context. However, it does not follow that the generalized conditional premise is therefore also true at every context. With the right account of a context, it will not be true in any context.¹⁴ Whatever the details of such an account, it will crucially claim that when we make utterances concerning the F -ness of objects located at different parts of a sorites series for F -ness, then the contextrelativity of ‘F ’ will be such that the contexts of these utterances differ significantly, i.e. they differ in a way that triggers a change in ‘F ’ ’s content and therefore extension. Let’s introduce a neutral term for that feature of an utterance context that allegedly determines the extension of vague predicates. Let’s say that each vague predicate has a determinate extension only relative to a standard, and that one aspect of each relevant context of utterance is precisely such a standard. In other words, each context of utterance determines a precise standard of richness, poverty, youth, baldness etc. I shall leave open how exactly an ordinary context of utterance determines one such standard—my observations will be neutral as to the exact implementation of the indexical approach. The indexical approach faces several problems. Like epistemicism, it holds that vague predicates determine a precise extension at every context of use, but that we are ignorant of it.¹⁵ While the epistemicist explains our ignorance of the extension of ¹² Here I have in mind primarily Kamp 1981, Raffman 1994, 1996, Soames 1999, 2002, Fara 2000 and Shapiro 2005. I am not claiming that all these writers regard vague predicates as indexicals, just that they invoke context-dependence of some sort in resolving the paradox. In the last section I will say more about this question. For now I just want to consider the position (no matter whether actually held by anyone) that vague predicates are indexical, and that their indexicality is the source of the characteristic extensional indeterminacy. ¹³ Raffman’s 1994 constraint (IP*) is that two adjacent members of a sorites series that are being judged at a context must be both in the extension at that context or neither. Soames requires that if two objects are sufficiently and relevantly similar, and one of them is salient at a context, then either both or neither are in the predicate’s extension at that context. Fara 2000 speaks of a ‘similarity constraint’, which in her case requires that any two things that are relevantly and sufficiently similar and whose similarity is salient at a context, are either both in the extension at that context or neither is. Kamp’s (1981) treatment is different in that his context-dependence involves the semantics for the conditional, but this is beside the point here. ¹⁴ Kamp 1981 is most thorough on this point, by offering a formal theory of contexts that rules out a context in which an entire sorites series is salient as incoherent. Raffman’s account is reminiscent of Kamp’s but gives a psychological explanation of why usually, before we regard an entire sorites series, our inner context switches. ¹⁵ This means that it shares a problem with epistemicism: if the meaning, and a fortiori the extension, of an expression is ultimately determined by the way we use it, then it seems mysterious how vague predicates in context should have acquired these meanings (extensions).
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vague predicates with certain general limitations on knowledge, the indexical contextualist attempts to explain this with the way in which vague predicates vary in extension with context, which is governed by some similarity constraint. The moment we consider two sufficiently similar objects (or consider their similarity), we are forced to conclude that they are either both inside or both outside the extension (compare Fara 2000, 59). However, this explanation is not satisfactory. Consider another context-dependent predicate: the predicate ‘is an object I am currently not considering’. Clearly, whenever I consider whether some particular object o is in the extension of this predicate, my very act of considering o causes o to be excluded from the extension. This does not mean that the extension of the predicate is empty. The way to convince yourself of that is to think of some particular utterance of ‘There is an object I am currently not considering.’ in some context c. After c, you can retrospectively, consider which objects where in the extension of our predicate in c. Equipped with a sufficiently detailed and reliable introspective memory, you could then determine for any object whether it is in that extension. The same goes for vague predicates, if the indexical contextualist story is right. It should be possible to consider a particular utterance of ‘Bob is rich.’, made in context c1, and retrospectively to consider for any object whether the extension of ‘is rich’ in c1 includes that object. There is no danger that in so considering we change the context, because we are thinking about the extension of the predicate in c1, a context that can no longer be changed. However, it seems utterly mysterious how we should go about it. Holding the context fixed does not make the limits of the extension in any way less elusive.¹⁶ The second and third problem for the indexical approach is that vague predicates do not behave like typical indexicals in certain respects. It is important to be clear from the start that these two points can merely show that vague predicates are not typical indexicals. They leave open whether vague predicates are a special or unusual kind of indexical (where indexicality is understood to be the phenomenon of variation of content with context of use). The second problem concerns speech reports. In general, when reporting indexical speech one must adjust the words used in the report to any relevant changes in the context. For example, if reporting an utterance of ‘You are a fool.’, one can use the same words in the report as originally uttered only if the addressee of the context of the report is the same as the addressee of reported utterance. If Otto addresses Peter saying ‘You are a fool.’, then I can report his utterance with the words ‘Otto said you were a fool.’ only if in making the report I am also addressing Peter. Otherwise I would have to adjust and say something like ‘Otto said Peter was a fool.’ Thus reporting indexical speech follows the following general rule: ¹⁶ A related problem is that the indexical contextualist does not have an appropriate way to characterize borderline cases. For, if she is to remain within the standard classical framework, a vague predicate in a context of use determines a precise concept. So the indexical contextualist will have to say that borderline cases are objects that either are or are not in the extension of the vague concept, but that we simply do not know which. But why this should be so is mysterious. Perhaps the indexical contextualist would at this point show her true colours and adopt one of the epistemicists’ explanations.
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(SR) If a sentence s is indexical in such way that the content expressed by s varies with contextual feature f , and utterance u is an utterance by S of s in context c1 , and context c1 differs from context c2 in feature f in a way that would alter the content of s in c2 as compared to its content in c1 , then an utterance in context c2 of ‘S said that s’ is incorrect. Now, clearly, if vague predicates are indexical in the way described above (i.e. if their extension varies with the context of use in such a way that a similarity constraint is met), then they do not comply with this rule. Consider a speaker who is being ‘force-marched’ from left to right through a sorites series of 50 coloured patches which range from paradigmatic red at the left to paradigmatic orange to the right. Suppose the speaker at some point utters ‘Patch 25 is red.’. Later on, she is forcemarched through the same series from right to left. This time she utters ‘Patch 25 is not red.’. Remembering her earlier utterance, she might add ‘but a while ago I said that it is red’. I believe that this would be a correct report. However, if the indexical contextualist theory of vagueness is correct, then (SR) predicts that this is not a correct report. The conclusion is that if vague predicates are indexical in the suggested way, then they are reported in an exceptional way at relevantly changed contexts. The third problem is related. When we evaluate the correctness of indexical utterances retrospectively, we evaluate them with respect to the original context of utterance. Thus, if I utter ‘I am hungry.’ before lunch and then consider the correctness of my utterance after lunch, I will evaluate what I said before lunch as correct just if I believe that I was hungry then. Thus, we would expect that if the above-mentioned speaker re-evaluates her earlier verdict on patch 25 when considering it in the later context, she should without hesitation evaluate the earlier utterance as correct. However, it would seem decidedly odd, if she said ‘Patch 25 is not red. A while ago I said that it was red, and what I said is true.’. Again, the conclusion is that if vague predicates are indexical in the way proposed then these indexicals behave unexpectedly when utterances of them are evaluated at a relevantly changed context. These three problems do not conclusively refute the indexical approach to vagueness. However, I believe that the first represents a serious challenge, while the second and third show that at the very least we are dealing with indexicals of an exceptional variety. This should be sufficient motivation for exploring the third potential source of the extensional indeterminacy of vague predicates.
17.7
R E L AT I V I T Y TO C I RC U M S TA N C E S O F EVA LUAT I O N
The third potential source of the extensional indeterminacy of vague predicates is a variation of extensions with circumstances of evaluation. In order to illustrate this possibility, I shall first briefly discuss another, better known case in which such variability has been debated, namely the case of tensed sentences.
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17.7.1 Eternalism and temporalism Eternalism and temporalism are two alternative ways of construing the semantics of tensed sentences, such as ‘MK is hungry.’ or ‘The root canal treatment is over.’. Eternalism treats tensed sentences as expressing different propositions with eternal truth value at different times of use. Thus, the indeterminacy of truth value exhibited by tensed sentences is treated as indexical, and as being resolved by placing the sentence in a context of use. Temporalism on the other hand treats tensed sentences (qua tensed sentences) as non-indexical.¹⁷ Tensed sentences (qua tensed sentences) express the same proposition in all contexts of use. However, these propositions are so-called ‘tensed propositions’. Tensed propositions do not have absolute truth values. Like contingent propositions, they vary in truth value with circumstances of evaluation. The difference is that while contingent propositions are sensitive merely to a possible world parameter in the circumstances, tensed propositions are sensitive to a time parameter in the circumstances. Thus the sentence ‘MK is hungry.’ expresses the tensed proposition that MK is hungry, and this proposition changes its truth value regularly. Before lunch, on most days, it is true at the actual world, and after lunch, on most days it is false (when events take their normal course). One advantage of, and motivation for, temporalism is the fact that it can accommodate certain intuitions about propositions conceived of as the objects of belief, assertion, etc. For example, believing the tensed proposition expressed by the sentence ‘My root canal treatment is over.’, will typically cause relief. However, believing a corresponding eternal proposition (the proposition that my root canal is over 12 March 2006 at noon) does not, by itself warrant any relief, for one might believe that proposition truly even before the 12 March at noon, and relief would be out of place then.¹⁸ The disadvantage of temporalism may be that there are also different intuitions regarding the objects of assertion and belief, which it does not accommodate. For example, suppose I sincerely use the sentence ‘Clinton is US president.’ twice, once in 1996 and once in 2000. Clearly, what I have asserted first (the belief I expressed) is true, and what I have asserted on the second occasion false. So it would seem that the objects of assertion cannot be the same on both occasions.¹⁹ Put in this form, the argument can be resisted: for why should it follow from a difference in truth value that the propositions expressed are also different? If tensed propositions have different truth values at different times then we can continue to maintain that both utterances expressed the same tensed proposition. However, there does seem to be a robust intuition that in some sense the object of belief changes. The most sensible reaction would seem to be the ecumenical one of allowing both tensed and eternal propositions, and to say that I expressed the same tensed, but different eternal propositions on the two occasions.²⁰ ¹⁷ See, e.g. Kaplan 1977 and Prior 1967. ¹⁸ See Recanati 2007, Book I, for a detailed defence of temporalism. ¹⁹ Compare Richard 1981. ²⁰ Temporalism and eternalism are also different in their treatment of tenses and other temporal qualifications, such as ‘sometimes’. Thus, temporalists construe tenses and temporal qualifications as operators while eternalists treat them as quantifiers. While the quantifier treatment is clearly more
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Whatever we may think about the debate between temporalists and eternalists, I here merely want to draw attention to the way in which temporalists need to refine the way they think about the normative significance of propositional truth. A tensed proposition, such as the proposition that my root canal treatment is over, does not have an absolute truth value, but varies its truth value not just from possible world to possible world but also from time to time. This raises the question (also asked by Evans 1979): truth at which time is relevant for the correctness of an assertion, or for the correctness of a belief ? Suppose at t1 I assert (and believe) the tensed proposition p, that the root canal is over. At t2 (later than t1) we can ask: is the assertion (and the belief ) correct? The question is unclear. We could look at the truth value of p at t1, we could look at the truth value of p at t2, or we could look at the truth value of p at any other time or even range of times. In principle all these manners of evaluation could be interesting and legitimate. But it is obvious that only one manner of evaluation is relevant if we want to test our semantics against language use: the truth value of p at t1, the time at which the assertion was made (or at which the belief occurred). Temporalism in the semantics of natural language makes sense only on the background of certain assumptions of how truth at a circumstance is relevant for the evaluation of assertions (or beliefs) as correct. The obvious principle expressing this relevance is: ( TP) An assertion (belief ) that p occurring in context c is correct only if the proposition that p is true at the time of c.
17.7.2
Vague propositions
Let us turn to the third potential source of extensional indeterminacy, according to which the extensions of vague predicates vary not with the context of use, but with the circumstances of evaluation. The proposal is to mimic the temporalist by adding another parameter to the circumstances of evaluation, and saying that the propositions expressed by sentences containing vague predicates vary in truth value with this parameter. They vary in this way because the vague predicates used to express these propositions express vague concepts which themselves vary their extensions with this parameter. What are the values of the circumstantial parameter with which the truth values (extensions) of vague propositions (predicates) vary? They are ways of making vague predicates precise consistently with clear cases and with certain a priori principles, i.e. functions that assign to the vague concepts expressed by vague predicates precise extensions. We could call these functions ‘reasonable standards of precisification’. But I will here rely on the terminology familiar from supervaluationism and call them ‘(admissible) sharpenings’. On this view, then, vague predicates express properties that are extensionally sensitive to a sharpening component in the circumstances of evaluation (just as contingent properties are extensionally sensitive to a possible world popular among semanticists, there does not seem to be any compelling reason for this preference (for discussion see King 2003, Recanati 2007).
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component in the circumstances of evaluation). Consequently sentences containing vague predicates will sometimes express vague propositions, i.e. propositions that are sensitive in their truth value to a sharpening parameter in the circumstances of evaluation.²¹ Just as in the case of tensed propositions, in order now to make sense of vague propositions as the objects of assertions or beliefs, we again need to spell out the normative significance of propositional truth. We need to know which sharpening or sharpenings are relevant for the evaluation as correct of an assertion or belief. In the case of tensed propositions, it was plausible to say that for each assertion (or belief ) there was just one time for evaluation, namely the time at which the assertion (the belief ) occurs. In the case of vague propositions, the situation will be more complicated. What we need is an appropriate completion of the following schematic principle: (VP) An assertion (belief ) that p occurring in context c is correct only if the proposition that p is true at . . . I would like to broach this task by first distinguishing in the abstract two dimensions in which completions of (VP) can vary, and then argue for each of these dimensions what our completion should look like. In principle, the completion of (VP) could either (a) privilege a unique sharpening or (b) privilege a range of several sharpenings and it could either (1) privilege the same sharpening(s) in each situation of assertion/belief or (2) privilege a different sharpening (or different sharpenings) in different situations of assertion/belief. In the next two sections I shall explain why I advocate a b-2 completion of (VP). 17.8
L E A R N I N G F RO M C O N T E X T UA L I S TS : T H E S O R I T E S
I believe that contextualists about vagueness teach us how option (2) helps us avoid sorites paradoxes. Let’s consider a non-inductive version of the sorites of section 17.1. Consider a sorites series of people, P0 , P1 , P2 , . . . P1,000,000 , such that P0 has ¤0, P1 has ¤1, P2 , has ¤2 and so on, each Pi having exactly i Euros. ²¹ For simplicity, I am ignoring the indexicality of many vague predicates, such as their sensitivity to a contextually salient comparison class. Thus I am strictly speaking considering only a subclass of vague predicates, namely those whose character is constant—such as, perhaps, ‘is tall for a British male born between 1975 and 1980’. Such predicates are no doubt still vague.
318 (P0) (P1) (C1) (P2) (C2) . . . (P1,000,000) (C1,000,000)
Max K¨olbel P0 is not rich. If P0 is not rich then P1 is not rich. P1 is not rich. If P1 is not rich then P2 is not rich. P2 is not rich.
If P999,999 is not rich then P1,000,000 , is not rich. P1,000,000 , is not rich.
Now, on an indexical contextualist approach, each of the constituent modus ponens arguments is valid in the sense that if its premises are true in a context c, then the conclusion is also true in c. However, there is no context c such that all the nonconditional premises (C1)–(C1,000,000) are true in it, and there is no context such that all conditional premises (P1)–(P1,000,000) are true in it. This is because considering more and more people in the series will accumulatively change the context until at some point the context undergoes a sudden reversal (cf. Raffman’s 1994 ‘gestalt switch’ and Kamp’s 1981 ‘incoherent’ context). If someone were to begin pronouncing the entire argument, then in each of the premise pairs (Pn)/(Cn), ‘rich’ would express a slightly different property, until suddenly it would express a significantly different property. This means that the corresponding generalized conditional premise (GP) For all x, y: if x is not rich and y has only ¤1 more than x, then y is not rich either. [There are no x, y, such that x is not rich, y has only ¤1 more than x and y is rich.] is false in every context. Nevertheless, there is no context in which a counterexample of the form (B) a is not rich, b has only ¤0.01 more than a, and b is rich. could be uttered and be true at that context. This explains (GP)’s appearance of truth. If sentences containing vague predicates express vague propositions in the sense outlined above, and if the normative significance of propositional truth is given by a type (2) completion of (VP), then a structurally analogous response to the sorites is available. On the non-indexical approach, the premises of the sorites argument are not indexical, but express the same propositions in all contexts of use. However, the propositions expressed vary in truth value with the sharpening parameter in the circumstances of evaluation. The question we are now considering concerns the normative significance of these relative truth values, e.g. under what conditions it is correct to assert or believe such a proposition (i.e. how to complete (AP)). According to response (2), the sharpening or sharpenings relevant for evaluating an assertion will vary as a subject is marched along the sorites series. At the beginning of the series,
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when we are asking ourselves whether some Pn is rich, we’ll have to say that she is not, because we have just previously ruled that Pn−1 is not rich. The sharpening(s) relevant for the correctness of an utterance in a context c obey the following constraint: if two individuals x and y are relevantly similar (e.g. they differ only by ¤1), and their similarity is salient²² in c, then the sharpening(s) relevant for judging correct assertability (believability) in c will not classify differently the proposition that x is not rich and the proposition that y is not rich. As the subject moves further and further along the series, however, there will come a point at which the context undergoes a sudden leap (perhaps just because proximity to clear cases of rich is becoming all too obvious). This explains why (GP) is not correctly assertable (believable) in any of the contexts, yet each of its instances is. This in turn explains the deceptive pull exerted by (GP) despite its unacceptability. Despite emulating some aspects of indexical contextualism, the non-indexical approach here proposed clearly differs in other respects. According to the indexical approach, ‘rich’ expresses a different property, and concept, at each stage of the march through the sorites series, whereas on the relativist approach, the property and concept expressed by ‘rich’ typically remain constant as a subject is moving along a sorites series. It is merely the correctness of calling an individual ‘rich’ and the correctness of believing an individual to be rich that varies as we move along the series. A comparison with the more familiar case of contingent properties and propositions may be illuminating: in the actual situation it is correct to call Modotti (or believe her to be) a photographer. In some non-actual situations, it would not have been correct to call her (believe her to be) a photographer. Nevertheless, the property ascribed to her in the different situations is the same: the property of being a photographer. It’s just that that property has an extension that varies from one possible world to another. Similarly, the proposition accepted in each case is the same, it is merely the truth value of that proposition that changes from one world to another. The proposal is that we treat the variability in the range of things to which ‘rich’ can be correctly applied analogously with this variability in the extension of ‘is a photographer’. In summary, when completing (VP), we should make assertability and believability depend on a variable (range of ) sharpening(s): (VP) An assertion (belief ) that p occurring in context c is correct only if the proposition that p is true at S(c). where ‘S’ is some contextual function that will be further described in the next section. 17.9
BORDERLINE CASES
It remains to argue that in completing (VP) we should privilege a range of sharpenings rather than an individual one, and then to superevaluate. The motivation for this ²² I here go with Fara’s (2000) ‘saliently similar’ rather than with Soames’s (1999) ‘similar and salient’.
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comes from our intuitions about borderline cases. There are three obvious options for construing S: (VPa) An assertion (belief ) that p occurring in context c is correct only if the proposition that p is true at the sharpening determined by c. (VPb) An assertion (belief ) that p occurring in context c is correct only if the proposition that p is true at all sharpenings in the range determined by c. (VPc) An assertion (belief ) that p occurring in context c is correct only if the proposition that p is true at some sharpenings in the range determined by c.²³ The consequence of (VPa) would be that vague propositions are, in any context, either correctly assertable or correctly deniable, and never both (where correct deniability of p is equivalent to correct assertability of not-p). This goes against all intuitions: against the intuition that in borderline cases of a predicate one may neither assert nor deny and also against the intuition that in borderline cases one may both assert and deny. Thus, I believe, (VPa) can be discarded. As for the remaining two options: it seems that there are two ways of thinking about borderline cases. According to one view (I believe the majority’s), borderline cases of richness are cases where it is neither correct to affirm nor to deny richness. Thus, for some n, it may neither be correct to call Pn rich, nor to call her not rich, at least in certain contexts (not, for example, when one has just judged Pn−1 to be not rich). Option (VPb) is the way to make room for this intuition.²⁴ Some have argued that borderline cases are cases where both verdicts are permissible (e.g. Wright 2003). According to them, in a borderline case it is both correct to assert and deny the property in question. A theorist supporting this view would naturally opt for (VPc). However, I am persuaded by the more common conception of borderline case. There is a close structural similarity, then, between the characterization of borderline cases adopted here and the supervaluationist position. So it will be worth pointing out the differences. Supervaluationists typically claim that truth is super-truth and that falsity is super-falsity. Thus, supervaluationist semantics involves the claim that some utterances are neither true nor false. The relativist here described, however, does not superevaluate in the semantics: the semantics does not specify super-truth conditions. Rather, the relativist superevaluates at the pragmatic level, when it comes to spelling out the normative significance of the semantic properties of expressions. One of the difficulties of supervaluationism is that it is committed to the truth of the negation of the general premise in the Sorites: (¬GP) For some x, y: x is not rich, y has only ¤1 more than x, and y is rich. ²³ These three options are clearly not exhaustive. For example we might replace ‘all’ in (VPb) with ‘most’, ‘many’, ‘a few’, or even with ‘twenty’. However, I do not see any reason to think that any of these options is promising. ²⁴ It s worth noting that even an epistemicist like Williamson can accept this characterization of borderline cases as cases where it is neither correct to assert (believe) nor to deny (disbelieve). For according to Williamson, correct assertability requires knowledge, and belief that is not knowledge is ‘botched’. See Williamson 2000.
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For (¬GP) is supertrue, i.e. true on each admissible sharpening. This seems to be a problem because there does not seem to be a true instance of (¬GP). It might seem that the current proposal is similarly committed to (¬GP) being correctly assertable. However, this is not so because, as pointed out above, the range of sharpenings which are relevant for assertability vary with the context. A similarity constraint will ensure that (¬GP) is not assertable in any context, while (GP) is assertable in every context. The upshot, then, is that a principle along the lines of (VPb) states the normative significance of propositional truth. This explains the seductiveness of the sorites and makes good sense of borderline cases without in any way departing from standard semantics. What is new is the pragmatics, i.e. the role truth plays in assessing assertions and beliefs for correctness. 17.10
H I G H E R - O R D E R VAG U E N E S S
There are at least two ways in which higher-order vagueness might arise on the current proposal. First, the notion of an (admissible) sharpening might be vague, and secondly, correct assertability (believability) may be vague, due to the contextual determination relation mentioned in (VPb) being vague. I shall discuss these in turn.
17.10.1 Is ‘sharpening’ vague? According to the semantic account of vagueness here proposed vague predicates vary in their extension with a sharpening parameter in the circumstances of evaluation. I likened this parameter to the sharpenings or precisifications familiar from supervaluationism: they are ways in which all concepts could be made precise consistently with clear cases and certain a priori principles. George Soros and Anita Roddick, for example, are clear cases of richness. An example of an a priori principle is the principle that if one person, A, is richer than another, B, then it cannot be that B is rich and A is not. Thus, according to my rough exposition of the relativist semantics, a precisification that does not count Carlos Slim as rich would not qualify as an admissible sharpening in any context, nor would a precisification according to which a non-rich person has more money than some rich person. Higher-order vagueness can arise in connection with the former issue: does a precisification qualify as a sharpening if it counts someone with possessions worth 10 thousand Euros as rich? What about one Euro less? It looks like the border between admissible and inadmissible precisifications in a given context is fuzzy. There are at least two ways of dealing with this. One is to accept that the notion of an admissible sharpening, as it figures in the semantic meta-language, is a vague notion. This, I believe, is in principle unproblematic. However, it is important to notice that admitting this form of higher-order vagueness is not required to make room for the phenomena of higher-order vagueness. For example, it is a phenomenon of higher-order vagueness that there does not seem to be a clear cut-off point between those who, in some context, may be called rich (or believed to be so) and those who
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are on the borderline between the two, i.e. those of whom it is neither correct to say (or believe) that they are rich nor that they are not. This phenomenon, in the current account will be accounted for by the vagueness of the notion of correct assertability (believability), which in turn derives from vagueness in the relation of determination between contexts of utterance and relevant sharpenings. I shall say more about this form of higher-order vagueness in a moment. What then is at stake in the question whether the notion of an admissible sharpening is vague? Consider a proposal according to which it is not vague. We might say that any precisification that respects the relevant ordering principles (e.g. ‘a non-rich person cannot have possessions worth more than some rich person’s’) is an admissible sharpening. In the case of supervaluationism, this would lead to the unwanted consequence that everyone is a borderline case of ‘rich’. However, the current approach characterizes borderline cases at the level of correct assertability or believability, which in turn requires truth in all sharpenings in the range determined by the context. As long as that range is occasionally restricted, we avoid the trivializing result that everything is borderline. What is at stake in the question whether ‘sharpening’ is vague is something quite different. According to the proposal that ‘sharpening’ is not vague, ‘rich’ varies in extension with various sharpenings, and for every person there is a sharpening that classifies him or her as a member of the extension and for every person there is a sharpening that classifies him or her as a non-member. Most of these sharpenings are pragmatically irrelevant because there is no context in which they are determined as relevant. So, even if everyone is classified as ‘rich’ in some sharpening, not everyone can correctly be called ‘rich’ or correctly be believed to be rich. However, a side-effect of this is that the semantic content of ‘rich’ will not differ in the expected way from that of ‘very rich’; ‘small’ not from ‘tiny’, ‘large’ not from ‘huge’ etc. The difference between ‘tiny’ and ‘small’ will not be that the extension of the former is less comprehensive than that of the latter. The differences between these concepts will show up only in the sharpenings that are determined as relevant by context.
17.10.2
Is correct assertability/believability vague?
Higher-order vagueness in the usual sense is, on this account, an entirely pragmatic phenomenon, in the sense that it concerns correct assertability and believability. Typical vague concepts have borderline cases: objects of which it is neither correct to assert (believe) nor to deny (disbelieve) the concept. However, correct assertability (believability) seems itself to be subject to tolerance constraints that lead to vagueness. Just as there seems to be no n such that Pn is not rich and Pn+1 is, there also seems to be no n such that it is correct to assert that Pn is not rich and not correct to assert that Pn+1 is not rich. Given the analysis of correct assertability proposed above, this form of higher-order vagueness could come about in two ways. First, the range of sharpenings determined by a context to be relevant to adjudicating the correctness of an assertion (belief ) might be vague. Thus, in a given context c, there is no n such that Pn is in the extension of ‘rich’ relative to all c-relevant sharpenings and Pn−1 is not. Thus, the
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determination relation that determines for each context of use a range of relevant sharpenings is itself a vague relation. Secondly, it may be that in each context a precise range of sharpenings is determined as relevant for correct assertability. Thus, for any context c, there is an n such that Pn is in the extension of ‘rich’ relative to all c-relevant sharpenings and Pn−1 is not. In that case it may still be true that there is no n such that it is correct to assert that Pn is not rich and not correct to assert that Pn+1 is not rich, i.e. higher-order vagueness may be present in this sense. However, this is only the result of the context changing when different Pn are under discussion. Hold any a context c fixed, and there will be an n such that it is correct to assert in c that Pn is not rich and not correct to assert in c that Pn+1 is not rich. The second view, I believe, faces the challenge of explaining how the predicates in question acquire the pragmatic features that determine a sharp borderline of correct assertability in a context, given that we manifestly have no idea where that borderline is located. This is analogous to the challenge facing epistemicists and indexical contextualists in explaining how vague predicates acquire their precise extensions (in contexts of use). I therefore prefer the first account of higher-order vagueness. 17.11
C O N C LU S I O N
I have shown how a standard semantic framework along the lines of those proposed by Kamp and Lewis for modal indexical languages can accommodate vague predicates, conceived of as extensionally indeterminate. I discussed three ways in which standard semantics makes room for extensional indeterminacy of predicates: ambiguity, indexicality and sensitivity to circumstances of evaluation (e.g. contingency). After discussing the prospects for treating vagueness as a phenomenon of ambiguity or indexicality, I moved on to develop an account of vagueness that assimilates the extensional indeterminacy of vague predicates to that of contingent predicates. Vague predicates, on this view, vary their extension with an additional parameter in the circumstances of evaluation, a ‘sharpening’. On this approach the semantics remains absolutely standard, and it is only the way in which the semantic notion of truth figures in pragmatic norms, the norms of assertion and belief, that requires some modification. The account bears similarities with both contextualism and supervaluationism, so it will be worth once more to point out the differences. The difference between viewing vagueness as a form of indexicality and the relativist view here proposed is clear. Indexical contextualists claim that the extensional indeterminacy of vague predicates is owed to their content being context-sensitive. Vague predicates (qua vague predicates) express different properties and concepts in different contexts of use, and vague sentences (qua vague sentences) express different propositions or contents in different contexts. The non-indexical view I proposed claims that the contents of vague predicates and sentences (qua vague predicates and sentences) are invariant, and that it is merely their extension that varies with circumstances of evaluation.
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This difference is, I believe, sufficiently clear. What is not completely clear is whether well-known contextualists about vagueness should be read as putting forward a version of the indexicality view here described. As far as I can tell, much of what contextualists say is indeterminate between what I have called indexicality and sensitivity to circumstantial parameters. In so far as this is true, the account here outlined should be taken as a contribution to developing further the views of contextualists about vagueness. In so far as this not true, i.e. in so far as contextualists about vagueness subscribe to an indexical view, this chapter should be taken as a proposal for modifying contextualism about vagueness.²⁵ ²⁵ Soames is the only clear case of an indexical contextualist about vagueness because in his 2002 he makes it explicit. Kamp comes at times very close to being an explicit indexicalist, though since the framework treated here as standard was only emerging at that time (with Kamp one of the pioneers), the terminology may well be misleading here. Even though Kamp’s contexts seem clearly intended as contexts of use, Kamp considers only one narrow aspect of the context of use, namely sentences that have previously been uttered. Raffman’s account (1994, 1996) is closely related to Kamp’s. The way she discusses the various aspects of context in her 1994, 64, suggests strongly that she thinks of the context as an utterance context, the quote by Kamp suggests it especially. But literally and strictly, what she says is compatible with both an indexical and a non-indexical reading of contextualism. For she does not usually consider the question whether the property or proposition expressed by a predicate or sentence varies with the context, but only whether the extension thus varies. For this reason, her account seems to be undecided between an indexical and a non-indexical reading. In her 2005 she explicitly distances herself from the indexicality view, though in the context it is not clear whether she here intends ‘indexical’ to mean ‘pure indexical’ (in Kaplan’s sense) or indexical in the wider Kaplanian sense of ‘the content varies with the context of use’. Fara (2000) is a complicated case because she considers many kinds of context-sensitivity of vague predicates. ‘is tall’ means roughly the same as ‘is significantly (x) taller than is typical (y) for (z)’. The extension of ‘tall’ varies with a comparison class (z), a norm of what’s typical by way of tallness for the comparison class (y), and also with standards of significance (x). The latter, she repeatedly says, is interest-relative. All these seem clearly to be intended to be aspects of the context of utterance, and the phenomenon one of indexicality. On this view, the property expressed by a vague predicate changes with the context of utterance. However, she insists on 64 and 75, that at least the interest factor is not to be understood in this way: ‘the property attributed to John by a particular utterance of ‘‘John is tall’’—that is, once all contextual elements are fixed—is still a property the extension of which may vary even as the heights of everything remain stable, since the extension of the property may vary as the interests of the relevant parties vary, that is, as different differences become more or less significant as different similarities become more and less salient.’ (75) This is puzzling without any further explanation. The idea is the property expressed by ‘tall’ in a given context remains invariant but its extension still varies. On this view of a property, differences in extension are not sufficient for differences in property, and therefore what Fara had in mind in her 2000 may well have been something akin to what I have been proposing in this chapter. Fara 2008 makes this more explicit: here she speaks explicitly of interest-relative properties and interest-relative propositions. Stanley 2003 claims that on Fara’s view, vague predicates are not indexicals because they express invariant properties, just as ‘is a US citizen’, which always expresses the same property, but that property’s extension varies with time. Thus, ‘is tall for a British male’ expresses the same property, namely that of being significantly taller than is typical for a British male, but of course that property changes its extension not just with time but also with interests.
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What is the difference between the current relativistic account and supervaluationism? First, the relativist does not accept the ‘supervaluationist’s slogan’ (Keefe 2000, 202) that truth is super-truth.²⁶ The semantic truth-conditions of a sentence (even of a sentence in a context) are relativized truth conditions, not conditions of super-truth. Supervaluation comes at the level of assessing assertions or beliefs for correctness in their context, not at the level of assessing the truth of the contents of such assertions of beliefs. Another difference is that the current relativistic account makes the range of sharpenings over which we superevaluate (when assessing correctness) contextually variable. This is not usually part of supervaluationism. Re f e re n c e s Burns, Linda (1991), Vagueness: An Investigation into Natural Languages and the Sorites Paradox, Dordrecht, Kluwer. Cappelen, Herman (2008), ‘Content relativism and semantic blindness’ in Manuel Garc´ıaCarpintero and Max K¨olbel (eds.), Relative Truth, Oxford, Oxford University Press, 265–86. Cappelen, Herman and Ernie Lepore (2004), Insensitive Semantics, Oxford, Blackwell. Ellis, Jonathan (2004), ‘Context, indexicals and the sorites’, Analysis 64, 362–4. Fara, Delia Graff (2000), ‘Shifting sands: An interest-relative theory of vagueness’, Philosophical Topics 28, 45–81. (2008), ‘Profiling interest relativity’, Analysis 68, 326–35. Evans, Gareth (1979), ‘Does tense logic rest on a mistake?’ in his Collected Papers (1985), 341–63, Oxford, Clarendon Press. Fine, Kit (1975), ‘Vagueness, truth and logic’, Synthese 30, 265–300. Kamp, Hans (1981), ‘The paradox of the heap’ in U. M¨onnich, ed., Aspects of Philosophical Logic, Dordrecht, Reidel, 225–77. Kaplan, David (1977), ‘Demonstratives’ in Almog et al., eds, Themes from Kaplan, Oxford, Clarendon Press 1989. Keefe, Rosanna (2000), Theories of Vagueness, Cambridge, Cambridge University Press. King, Geoffrey (2003), ‘Tense, modality and semantic value’, Philosophical Perspectives 17, 195–245. K¨olbel, Max (2008), ‘Motivations for relativism’ in Manuel Garc´ıa-Carpintero and Max K¨olbel, eds., Relative Truth, Oxford, Oxford University Press, 1–38. (forthcoming), ‘Assertion, intention and convention’ forthcoming in Sarah Sawyer, ed, New Waves in Philosophy of Language, Hampshire, Palgrave Macmillan. Lewis, David (1975), ‘Languages and language’ in Minnesota Studies in the Philosophy of Language 7, 3–35. Reprinted in Lewis (1983). However, Stanley also attributes to Fara the view that one of the effects of this interest-relativity is that it will depend on the interests of an utterer which proposition is expressed by the utterance of a vague sentence. This suggests that on the wider Kaplanian sense of ‘indexical’ (the one I have been using earlier in this chapter), vague sentences are indexical in the sense that they express different propositions in different contexts of use, and that predicates are indexical if the content they contribute to the proposition expressed depends on the context of use. ²⁶ Though McGee and McLaughlin 1995, despite being supervaluationists, do not accept the slogan.
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Lewis, David (1983), Philosophical Papers, vol. 1, Oxford, Oxford University Press. ¨ (1980), ‘Index, context, and content’ in Stig Kanger and Sven Ohman, eds., Philosophy and Grammar, Dordrecht, Reidel. Reprinted in Lewis, Papers in Philosophical Logic, Cambridge, Cambridge University Press 1998. McGee, Vann and Brian McLaughlin (1995), ‘Distinctions without a difference’, Southern Journal of Philosophy 33, 203–51. Pagin, Peter (2009), ‘Central gap domain restriction’. This volume. Pagin, Peter and Jeff Pelletier (2007), ‘Content, context and composition’ in G. Peter and G. Preyer, eds., Content and Context. Essays on Semantics and Pragmatics, Oxford, Oxford University Press, 25–62. Prior, Arthur (1959), ‘Thank goodness that’s over’, Philosophy 34, 12–17. (1967), Past, Present and Future, Oxford, Clarendon Press. Raffman, Diana (1994), ‘Vagueness without paradox’, Philosophical Review 103, 41–74. (1996), ‘Vagueness and context relativity’, Philosophical Studies 81, 175–92. (2005), ‘How to understand contextualism about vagueness: Reply to Stanley’, Analysis 65, 244–8. Recanati, Franc¸ois (2007), Perspectival Thought, Oxford, Oxford University Press. Richard, Mark (1981), ‘Temporalism and Eternalism’, Philosophical Studies 39, 1–13. Sainsbury, Mark (2001), ‘Two ways to smoke a cigarette’, Ratio 14, 386–406. Shapiro, Stewart (2003), ‘Vagueness and conversation’, in Jc Beall, ed., Liars and Heaps, Oxford, Oxford University Press. Soames, Scott (1999), Understanding Truth, Oxford, Oxford University Press. (2002), ‘Replies’, Philosophy and Phenomenological Research 62, 429–52. (2003), Beyond Rigidity, Oxford, Oxford University Press. Sorensen, Roy (1988), Blindspots, Oxford, Clarendon. Stanley, Jason (2000), ‘Context and logical form’, Linguistics and Philosophy 23, 391–434. (2003), ‘Context, interest-relativity and the sorites’, Analysis 63, 269–80. Travis, Charles (1985), ‘On what is strictly speaking true’, Canadian Journal of Philosophy 15, 187–229. (1996), ‘Meaning’s role in truth’, Mind 105, 451–66. Tye, Michael (1989), ‘Supervaluationism and excluded middle’, Analysis 49, 141–3. Williamson, Timothy (1994), Vagueness, London, Routledge. (2000), Knowledge and its Limits, Oxford, Oxford University Press. Wright, Crispin (2003), ‘Vagueness: A fifth column approach’ in Jc Beall, ed., Liars and Heaps, Oxford, Clarendon Press.
18 How to Respond to Borderline Cases Dan L´opez de Sa
It seems that Hannah and her wife Sarah may disagree as to whether Homer Simpson is funny, without either of them being at fault. This is an (almost) uncontroversial case of apparent faultless disagreement. More cases are arguably provided in other philosophically interesting domains: predicates of personal taste, evaluative predicates in general, epistemic modals, and knowledge attributions. With respect to any of these, it is held, it seems that there could be contrasting judgements without fault on the part of any of the participants. Some philosophers seem to think that vagueness should be included in the list above: borderline cases provide further cases of apparent faultless disagreement. My aim here is to argue against such a suggestion. After elaborating briefly on the notion of apparent faultless disagreement, I present the case for my main claim: with respect to borderline cases, people typically do not respond by taking a view—in contrast to what is the case in genuine cases of apparent faultless disagreement (section 18.1). The status of this kind of claim, both descriptive (of paradigm cases, at least) and normative—though familiar in many other domains, such as the theory of meaning, decision theory, or moral psychology—is likely to raise suspicion. The main part of this chapter is devoted to alleviating such suspicion. I argue that my claim is indeed respected and actually accounted for by paradigm cases of semantic and epistemic views on the nature of vagueness (section 18.2). And I also argue that my claim turns out to be, initial appearances notwithstanding, compatible with other claims in the literature—to the effect that, in appropriate circumstances, there are indeed, or there might well be, ‘macho,’ admissible, forced, and hesitant responses to borderline cases (section 18.3). Earlier versions were presented at the 7th Arch´e Vagueness Workshop and the LOGOS Seminar. Thanks to the audiences then, and in particular to Richard Dietz, Manuel Garc´ıa-Carpintero, Max K¨olbel, Sebastiano Moruzzi, Diana Raffman, Sven Rosenkranz, Stewart Shapiro, Crispin Wright, and Elia Zardini, for very helpful objections and suggestions, and to anonymous referees for Oxford University Press. Research partially funded by projects HUM2004-05609-C02-01 and FFI200806153/FISO, and a GenCat-Fulbright Postdoctoral Fellowship. Thanks to Mike Maudsley for his linguistic revision.
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T H E PH E N O M E N O N O F A P PA R E N T FAU LT L E S S D I S AG R E E M E N T
We are quite familiar with situations such as that of Hannah and Sarah regarding whether Homer Simpson is funny: disputes as to whether spinach is tasty or disgusting, or as to whether getting to the party late is cool or lame, or as to whether Brad Pitt or Uma Thurman are really sexy or rather overrated. In cases like these, people often take views on the matter, even strong ones, which sometimes issue in (long) discussions and arguments. Still, it seems that none of the parties need be mistaken with regard to their views, after all. Whether such an appearance of faultless disagreement is to be endorsed—or even whether it could be endorsed—is, of course, a matter of controversy. Following the lead of Crispin Wright 1992, one can conceive of relativism in general as precisely the attempt to so endorse the appearances of faultless disagreement, in the different domains—for such needs to involve, in one way or other, some relativity to contrasting features of the subjects in question. The different sources the relativity might be held to have are what give rise to the different relativisms. Moderate relativism has it that such an endorsement can be done within the general Kaplan–Lewis–Stalnaker two-dimensional framework, in which the basic semantic notion is that of a sentence s being true at a context c at the index i.¹ It may in effect be the case that s is true at c (at its index ic ) but false at c ∗ (at ic∗ )—due to the content of sentence s at c being different from that of s at c ∗ (indexical contextualism); or, even if the content is the same, due to relevant differences in the indices ic and ic∗ determined by c and c ∗ (non-indexical contextualism). Radical relativism, by contrast, claims that appropriately endorsing appearances of faultless disagreement requires departing from the two-dimensional framework, in that s at the very context c can be true from a certain perspective but false from another—where perspectives are to be thought of as the same sort of thing as contexts, but representing a location from where a sentence, as said in a (possibly different) location, could be viewed or assessed.² ¹ The jargon I adopt is from Lewis (1980). A context is a location—time, place, and possible world, or centered world for short—where a sentence could be said. It has countless features, determined by the character of the location. An index is an n-tuple of features of context, but not necessarily features that go together in any possible context. Thus an index might consist of a speaker, a time before his birth, a world where he never lived at all, and so on. The coordinates of an index are features that can be shifted independently, unlike those of a context, and thus serve to represent the contribution of sentences embedded under sentence operators, such as ‘possibly’ or, more controversially, ‘somewhere,’ ‘strictly speaking,’ and so on. Given a context c, however, there is the index of the context, ic : that index having coordinates that match the appropriate features of c. Given this uniqueness, the basic two-dimensional relation can be abbreviated in this special case: sentence s is true at context c iff s is true at context c at index ic . ² I propose to use ‘perspectives’ instead of MacFarlane’s ‘contexts of assessment,’ see his 2003, 2005. I think this terminology helps to avoid confusions with ‘context of use/utterance’ (‘context’ here) and, more importantly, with ‘circumstance/point of evaluation’ (‘index’ here). My taxonomy is greatly indebted to—and some of the labels due to—John MacFarlane. I elaborate on the details in L´opez de Sa (2009b).
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Whether the appearances of faultless disagreement are (can be) endorsed is, as I said, controversial. But that such appearances exist is, I take it, a datum for nonrelativists and relativist alike—appearances that are to be explained away, if they are not endorsed. Hannah may have a judgement she might naturally express in an ordinary context by uttering ‘Homer Simpson is funny’ with its literal meaning; whereas Sarah may have a judgement she might naturally express in an ordinary context by uttering ‘Homer Simpson is not funny.’ And these contrasting judgements need not involve, apparently, any error on the part of Hannah nor Sarah. Similarly in some other philosophically interesting cases—including future contingents, predicates of personal taste, evaluative predicates in general, epistemic modals, and knowledge attributions—there can be contrasting judgements about an issue in the domain that do not seem to involve fault on the part of any of the participants: they all involve cases of (at least) apparent faultless disagreement.³ Some philosophers seem to think that vagueness should be included in the list above: borderline cases provide further cases of apparent faultless disagreement. However, this does not seem to be so. Take Jason and his husband Justin, and consider a borderline green towel.⁴ Typically, I submit, they would not respond to it by taking a view as to whether the towel is green or not. They would simply lack the judgements that they would naturally express in an ordinary context by asserting ‘The towel is green’ or ‘The towel is not green’ with its literal meaning: rather, if questioned about it, they would easily converge in something like that ‘it sort of is and sort of isn’t,’ ‘it’s greenish,’ etc.—and they would be rational in so doing. But then they would lack the building blocks for the appearance of faultless disagreement clearly present in the other cases considered above: the (contrasting) judgements. Hannah and Sarah do typically form polar opinions with respect to issues such as whether Homer Simpson is funny; Jason and Justin typically do not form such verdicts with respect to issues such as whether the towel is green. So this is in essence why I think that vagueness does not provide further cases of apparent faultless disagreement: with respect to borderline cases, people typically do not respond by taking a view—in contrast to what is the case in genuine cases of apparent faultless disagreement. The status of this kind of claim, however, is likely to raise suspicion. I am submitting Jason and Justin as paradigmatic with respect to people’s actual ways of responding to borderline cases. And I am also suggesting the normative view that it is indeed rational for them so to respond. I take it we are familiar with this kind of situation—claims that are submitted as both descriptive (of ³ Some use ‘faultless disagreement’ in a more restricted sense, requiring that there be a single content or proposition which is contrastingly judged, see for instance K¨olbel (2003). According to this more restricted sense, it cannot just be taken as a datum for relativists and non-relativists alike that there are apparent faultless disagreements, nor do all versions of relativism endorse that there are in effect faultless disagreements in the relevant domains. These I take to favor my more liberal usage. ⁴ To provide an adequate characterization of what it is for something to be a borderline case is of course part of what is at stake. For present purposes, however, it suffices to point to cases with respect to which the different views as to how to respond to them differ.
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paradigm cases, at least) and normative—in many other domains, such as the theory of meaning, decision theory, or moral psychology. But this familiarity by itself need not remove suspicion. As I said at the beginning, what follows is my best attempt to alleviate such suspicion in the case at hand. I will argue that my claim is indeed respected and actually accounted for by paradigm cases of semantic and epistemic views on the nature of vagueness. And besides, I will also argue that my claim turns out to be, initial appearances notwithstanding, compatible with other claims in the literature—to the effect that, in appropriate circumstances, there are indeed, or there might well be, ‘macho,’ admissible, forced, and hesitant responses to borderline cases.
18.2
N O R M S O F A S S E RT I O N A N D T H E N AT U R E O F VAG U E N E S S
One of the main views on the nature of vagueness has it that vagueness is a phenomenon of semantic indecision: (roughly) whatever it is that in the thoughts, experiences and practices of language users determines the meaning of expressions, it fails to determine, for vague expressions, any single one from a given range of similarly natural candidate references. Each way of (‘arbitrarily’) fixing what is left semantically indeterminate gives rise to a precisification or sharpening of the original vague expression. Although all such sharpenings are, by essence, arbitrary to a certain extent, not all of them are admissible. In the case of predicates, admissible ones should preserve clear cases, both of application and of non-application—Yul Brynner should count for ‘is bald,’ while Andy Garc´ıa cannot—, and they should also preserve penumbral connections —‘Whoever is bald is bald,’ ‘If someone is bald, then so is anyone who is balder,’ and so on—.⁵ What one says by means of a vague expression is true, according to this view, if it would be true however one (admissibly) precisifies it—or, as I will put it, if it counts as true according to all admissible sharpenings. And it is false if it counts as false according to all admissible sharpenings. Otherwise, if there are admissible ways of precisifying it which give rise to truths, but also admissible ways of precisifying it which give rise to falsehoods, the vague sentence is indeterminate: neither true nor false. That is indeed the situation with respect to borderline cases, as the view has it. Take Harry, a borderline case with respect to ‘is bald,’ having exactly 3,833 hairs on his scalp. Whatever it is that in the thoughts, experiences and practices of language users determines the meaning of expressions, it fails to determine whether someone with this very number of hairs does or does not fall under ‘is bald.’ Thus ‘is bald’ can be admissibly precisified by (let us assume) ‘has at most 3,832 hairs on his scalp,’ but ⁵ Thus sharpenings are, strictly speaking, of the language as a whole, and not of isolated expressions, see Fine (1975). How to characterize in an explicit satisfactory way the notion of admissible constituted by these connections (possibly among others) would of course be crucial for a full defense of the view of vagueness as semantic indecision. Notice that ‘is admissible’ is, of course, itself vague: this is arguably part of what accounts, in this framework, for the phenomenon of ‘higher-order’ vagueness. Complications arising from this will be set aside here.
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also by ‘has at most 3,834 hairs on his scalp.’ Hence, ‘Harry is bald’ fails to be true, given that ‘Harry has at most 3,832 hairs on his scalp’ is false, but it also fails to be false, given that ‘Harry has at most 3,834 hairs on his scalp’ is true.⁶ As most of its critics also acknowledge, the view of vagueness as semantic indecision is certainly—at least initially—intuitively very plausible. But it is at odds with the claim that borderline cases exhibit apparent faultless disagreement: it predicts that people would typically not (and should not) take a view with respect to borderline cases, as the relevant statements would lack a truth value and thus would not be true. In this way, I hold, the view provides further support to my main claim. Let me elaborate. As we have just seen, the phenomenon of faultless disagreement requires that people do typically form judgements on the matter (which may be contrasting while apparently fault-free). Judgements like these are typically manifested by people’s asserting the relevant statements in question, at their respective contexts. Thus, as we saw, Hannah could perfectly well express her judgement by asserting ‘Homer is funny’ at her context, and Sarah by asserting its negation at hers. Assertions are (arguably) acts governed by norms. The weakest sensible norm for assertion, most would agree,⁷ is the truth rule: One must: assert s at c only if s at c (at ic ) is true. The truth rule forbids untrue assertions.⁸ Borderline cases exhibiting the phenomenon of apparent faultless disagreement would require things like (say) Jason forming a view to the effect that the towel is green, and Justin forming a contrasting view to the effect that the towel is not green. But according to the view of vagueness as semantic indecision, ‘The towel is green’ is not true at Jason’s context, nor is it true ‘The towel is not green’ at Justin’s. Thus Jason should not assert ‘The towel is green,’ nor Justin assert its negation. And this is, as we saw, in clear contrast with the case of Hannah and Sarah. To the extent to which it is sensible to assume that people’s actions typically conform to their characteristic norms (at least in paradigmatic instances of the relevant action types), the view of vagueness as semantic indecision also accounts for why people typically do not take a view with respect to borderline cases. I have argued that the view of vagueness as semantic indecision, which is certainly (at least initially) plausible from an intuitive point of view, respects and accounts for my main claim and thus provides support to the contention that borderline cases do not exhibit the phenomenon of apparent faultless disagreement. The same is true, ⁶ Thus the characteristic denial of the principle of bivalence: not everything that says something is either true or false, as borderline cases are indeterminate. Williamson (1994) contains an argument for the incompatibility of this feature with Tarskian views about truth and falsity, which apparently convinced most people in the field. In my view, however, Andjelkovi´c and Williamson (2000) contains the key elements for resisting it: see for elaboration and further discussion L´opez de Sa (2009a). ⁷ As also noticed by defenders of radical relativism, their view is committed to depart from standard ones on this (related) count as well. ⁸ See (Williamson, 2000, ch. 11) for further (critical) discussion. He says: ‘The truth rule forbids false assertions’ 2000, 242, my emphasis, which is just a proper consequence—in the absence of the (independent) principle of bivalence.
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I now claim, about one of the other main rival views on the nature of vagueness: epistemicism as defended by Williamson (1994)—provided that some epistemic norm for assertion holds. Were Jason to form a view to the effect that the towel is green, this would not constitute knowledge. And neither could Justin knowledgeably form a contrasting view to the effect that the towel is not green, for the same reasons. Most people would agree: if it is borderline, then there is no knowing that the towel is green, and no knowing that it is not green. According to defenders of the view of vagueness as semantic indecision, the explanation of this is straightforward (and shows why it would be misleading to label the situation as one of ignorance): there is no knowing because, as noted above, there is no truth there to be known. By contrast, according to epistemicism as defended by Williamson (1994), we may suppose, either ‘The towel is green’ is true at Jason’s context, or it is true ‘The towel is not green’ at Justin’s.⁹ Still, the epistemicist holds, neither of the judgements that Jason or Justin could naturally express in their respective contexts would constitute knowledge.¹⁰ Now, although admittedly more controversial, a case has been made for assertions being acts governed by the (stronger) knowledge rule, see (Williamson, 2000, ch. 8): One must: assert s at c only if one knows p, where p is the content of s at c. But if this holds, we have a corresponding support for my main claim, even according to epistemicism. For again borderline cases exhibiting the phenomenon of apparent faultless disagreement would require things like (say) Jason forming a view to the effect that the towel is green, and Justin forming a contrasting view to the effect that the towel is not green. But according to epistemicism, neither could knowledgably form such judgements. Thus again Jason should not assert ‘The towel is green,’ nor should Justin assert its negation, in clear contrast with the case of Hannah and Sarah. To the extent to which it is sensible to assume that people’s actions typically conform to their characteristic norms (at least in paradigmatic instances of the relevant action types), the epistemicist view of vagueness also accounts for why people typically do not take a view with respect to borderline cases.
18.3
F U RT H E R R E S P O N S E S TO B O R D E R L I N E C A S E S
I have submitted that the case of Jason and Justin regarding whether borderline green towel is green is intuitively very different from that of Hannah and Sarah regarding whether Homer Simpson is funny. The former typically won’t (and shouldn’t) take a view on the matter, thus lacking the judgements that are the building blocks for the phenomenon of apparent faultless disagreement, present in the latter. I have also ⁹ The main positive argument offered by Williamson (1994) in favor of epistemicism is precisely the one in favor of the principle of bivalence mentioned in footnote 6. ¹⁰ The explanation of this given the presence of a truth on the issue, is much more complex: see (Williamson, 1994, ch. 8) and (Williamson, 2000, ch. 5)
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argued that this claim is indeed respected and actually accounted for by paradigm cases of semantic and epistemic views on the nature of vagueness, provided that certain plausible norms of assertion hold. In the remainder of this chapter I will try to show that my claim turns out to be, initial appearances notwithstanding, compatible with other claims in the literature concerning various responses to borderline cases.
18.3.1
‘Macho’ responses
In his discussion of Williamson (1994), Paul Horwich contends that the essence of [the phenomenon of vagueness] is not that borderline predications cannot be known to be correct. The essence of it . . . is that in certain cases we are normally unwilling to apply the predicate, unwilling to deny that it applies, and confident that no further investigation could yield a decision. The problem of knowledge in such circunstantes is a result of this paralysis of judgement. (Horwich, 1997, 931)
I would not say that the issue of how to respond to borderline cases constitutes the essence of vagueness, as opposed to being part of its characteristic manifestation. As implied above, I take the different views of the nature of vagueness to be views such as the view of vagueness as semantic indecision and epistemicism, which account for the manifestation of vagueness in how to respond to borderline cases. As a result, pace Horwich, I do not take the contention that a certain kind of ‘paralysis of judgement’ with respect to borderline cases is a characteristic manifestation of vagueness to be in tension with an epistemic view on the nature of vagueness. On the contrary, as I have just argued, epistemicism seems to account nicely for this, provided that certain epistemic norms for assertion hold. My main claim, however, that with respect to borderline cases people typically don’t and shouldn’t form categorical judgements—in contrast with what is clearly the case in domains that exhibit apparent faultless disagreement—seems clearly in tune with Horwich’s contention that a certain kind of ‘paralysis in judgement’ with respect to borderline cases is indeed a characteristic manifestation of vagueness: Jason and Justin would typically not respond to a borderline green towel by taking a view as to whether it is green or not. It is important to observe that this claim is compatible with the possibility of Jason and Justin being, on occasions, more opinionated and ‘macho’ than one typically is (and should be), and thus with them forming the judgements as to whether the towel is green or not. After all, we have all discovered ourselves, on occasions, in discussions as to whether something is or is not a certain way, just to realize that the case in question was simply borderline. My claim has it, however, that giving such ‘macho’ responses is not the way we typically respond to borderline cases, nor of course the way we should respond to them, and that this is clearly in contrast with situations such as that of Hannah and Sarah regarding whether Homer Simpson is funny. In his intriguing reply to Horwich, however, Williamson says: Horwich notwithstanding, paralysis in judgement is quite unnecessary for vagueness. Consider an opinionated macho community, in which everyone applies the term ‘bald’ or its negation
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confidently and unhesitatingly on the basis of impressions gained from causal observations whenever the issue arises. There is no appeal to precise necessary and sufficient conditions. Speakers accept that the application of ‘bald’ depends only on the exact distribution of hairs on someone’s scalp, but vagaries of mood and perception cause them often to apply the term ‘bald’ when the distribution is exactly the same as in a previous case which they classified as ‘not bald.’ When they disagree, each dogmatically insist that the other is clearly wrong. When inconsistencies are pointed out in a single speaker’s application of the term, they are denied ad hoc (‘I never said that!’, ‘His hair has grown since then!’). On Horwich account, ‘bald’ is not vague in the language of this community, because there are no cases in which speakers ‘are normally unwilling to apply the predicate, unwilling to deny that it applies, and confident that no further investigation could yield a decision.’ But ‘bald’ is vague in the language of this community. (Williamson, 1997, 945–6)
I think there are two ways of conceiving of such a community. If Jason and Justin can on occasions give such ‘macho’ responses, we can conceive of them as always giving them—and we can also conceive of the rest of the population being similarly ‘macho.’ On this way of conceiving the community, it is certainly the case that the relevant expressions are still vague in the language of the community, as Williamson contends. But this being so does not contradict Horwich’s contention about ‘paralysis of judgement,’ at least understood along the lines of my main claim. For it would still be the case that, in the relevant sense, this would be the conceiving of a community in which people typically wouldn’t and shouldn’t give such ‘macho’ responses: on this way of conceiving the community, people are disposed as we are, it is just that we imagine the conditions to be such that they do not manifest their dispositions. Many other domains, such as the theory of meaning, decision theory, or moral psychology—or indeed basic dispositions to judge that the lines in the M¨uller–Lyer illusion are the same length—provide situations that are structurally analogous. If, by contrast, we conceive of a situation in which people simply lack the relevant dispositions to manifest ‘paralysis of judgement’ with respect to borderline cases at all, then I submit we no longer have the intuition that the relevant expressions are, indeed, vague —as ours indisputably are.
18.3.2
Admissible responses
On the face of it, my main claim that, with respect to borderline cases, people typically don’t (and shouldn’t) take a view seems in tension with the idea that, with respect to borderline cases, people can ‘go either way.’ Here is Stewart Shapiro’s recent statement of this idea: Suppose . . . that a is a borderline case of P. I take it as another premise that, in some situations, a speaker is free to assert Pa and free to assert ¬Pa, without offending against the meanings of the terms, or against any other rule of language use. Unsettled entails open. The rules of language use, as they are fixed by what we say and do, allow someone to go either way in the borderline region. Let us call this the open-texture thesis. (Shapiro, 2003, 43)
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However, I do not think that the tension is genuine. For the idea seems to amount to the thought that, in certain conversations and given the presence of certain particular knowledge, purposes, etc., participants are capable of altering the standards of precision prevalent in that conversation, with the effect of producing local (explicit, or more plausibly, implicit) stipulations that give rise precisely to a precisification of the relevant vague expression—at least, for the purpose of the conversation at hand. The presence of the mechanism can be motivated independently, in a straightforward enough way within the framework of the view of vagueness as semantic indecision. I assume that similar moves might be available to the friend of epistemicism, although I will not attempt to adapt the consideration here. In his ‘Scorekeeping in a Language Game,’ David Lewis 1979 famously introduced the figure of a conversational score, whose kinematics—including prominently the rules of accommodation—he precisely illustrated with, among others, the case of vagueness. If Fred is a borderline case of baldness, the sentence ‘Fred is bald’ may have no determinate truth value. Whether it is true depends on where you draw the line. Relative to some perfectly reasonable ways of drawing a precise boundary between bald and non-bald, the sentence is true. Relative to other delineations, no less reasonable, it is false. Nothing in our use of language makes one of these delineations right and all the others wrong. We cannot pick a delineation once and for all (not if we are interested in ordinary language), but must consider the entire range of reasonable delineations. If a sentence is true over the entire range, true no matter how we draw the line, surely we are entitle to treat it simply as true. But we also treat a sentence more or less as if it is simply true, if it is true over a large enough part of the range of delineations of its vagueness. (In short: if it is true enough). . . . When is a sentence true enough? Which are the ‘large enough’ parts of the range of delineations of its vagueness? This is itself a vague matter. More important for our present purposes, it is something that depends on context. What is true enough on one occasion is not true enough on another. The standards of precision in force are different from one conversation to another, and may change in the course of a single conversation. Austin’s ‘France is hexagonal’ is a good example of a sentence that is true enough for many contexts, but not true enough for many others. Under low standards of precision it is acceptable. Rise the standards and it loses its acceptability. (Lewis, 1979, 244–5)
As I suggested, the idea that, with respect to borderline cases, people can ‘go either way,’ as I understand it, can indeed be seen as providing further ways in which accommodation can alter the standards of precisions in force in a given conversation. For consider the following conversation between Jason and Justin: [Jason has just finished having his shower. In their bathroom, there is both the borderline green towel and another, white one.] —Justin, please, pass me the towel, would you? —Which one you want? —I don’t know . . . whichever . . . Just give me the green one, but please hurry up, I’m freezing!
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I submit that, in a conversation such as this, participants would accommodate by relaxing standards of precision dramatically, so as to count sentences as ‘The towel is green’ at the context as true, regardless the fact that the part of the range of reasonable delineations which do so may be not large at all. But all this is, I take it, compatible with my main claim that, with respect to borderline cases, people typically don’t (and shouldn’t) take a view: in the absence of the particular knowledge, purposes, etc., that we naturally imagined partly informing the conversational score, one would regard an unqualified utterance of ‘The towel is green’ by Jason at his context to be simply a ‘macho’ response.
18.3.3 Forced responses Following Diana Raffman 1994, Shapiro considers the responses in a ‘forced march’ scenario—where subject are asked to say (say) ‘yes’ or otherwise to the question ‘Is this towel green?’ concerning items that conform a sorites series—with respect to (among others) borderline cases. In her response, Rosanna Keefe wonders: Is it reasonable to draw any significant conclusions from the response subjects are driven to make when they are marched through a Sorites series and forced to judge each case either one way or the other? . . . [S]uppose you make subjects respond with ‘yes’ or ‘no’ to questions involving unfulfilled presuppositions; e.g. you ask them ‘Have you stopped φ-ing?’ when they’ve never φ-ed. They may be reluctant to answer yes or no—both answers are misleading—but they may nonetheless choose one of those answers when forced. Surely their choice in that situation should not be taken as deeply significant, nor as helping to illuminate the semantics of sentences involving unfulfilled presuppositions. . . . Second analogy: reading too much into the response to forced march paradoxes seems rather like forcing someone to guess the weight of something and then taking that guess to reveal that the subject believes that the weight is exactly that. (Keefe, 2003, 79)
I do not want to assess here whether Keefe is right in these comments, nor how this would affect the tenability of the contextualist proposals of Raffman and Shapiro. For my present purposes, a much weaker and rather uncontroversial remark is pertinent. Whichever way one conceives of the relevance of these forced responses, the fact that in situations like those envisaged people are asked to issue them is compatible with my main claim. For, indeed, it would seem that awareness of the past items in the series, of the likely future one, and their respective similarity in the relevant respects, among other things, gives rise to a peculiar conversational score in which the forced responses in question can be regarded as admissible. But this is so even if, typically, one would not (and should not) issue them.
18.3.4 Hesitant responses Third Possibility can be seen as the generic view that, if sentence s at context c is borderline, then it has some kind of third status incompatible with each of the poles,
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truth and falsity—be it lacking a truth value, possessing a third value, or what have you. The view of vagueness as semantic indecision is a paradigm case of Third Possibility. Call verdict a judgement to the effect that something is F , or that it is not F —where the question of whether something is or is not F might be borderline. Verdict Exclusion says that with respect to borderline cases no such verdict constitutes knowledge. Both the view of vagueness as semantic indecision and epistemicism are paradigm cases of Verdict Exclusion. One consequence of my main claim in this chapter might be a plea for Verdict Exclusion. As Crispin Wright—to whom these labels are due—says: According to Verdict Exclusion, one ought, all things considered, to offer no verdict about a borderline case and to have no opinion which could be expressed in such a verdict. (Wright, 2003, 92)
In effect, I have claimed, intuitively, and in sharp contrast with the case of Hannah and Sarah regarding whether Homer is funny, Jason and Justin would typically not, and should not, offer a verdict as to whether the towel is green, nor have any opinion which could be expressed in such a verdict. This is accounted for by two of the main views about the nature of vagueness, and is compatible with their issuing ‘macho,’ admissible, forced, and, as we are now about to see, with their issuing hesitant responses. As Wright points out, The manifestation of vagueness, in the kinds of case we are concerned with, is not a consensus on certain cases as borderline—not if that is to be a status which undercuts both polar verdicts. Rather, the impression of a case as borderline goes along with a readiness to tolerate other’s taking a positive or negative view—provided, at least, that their view is suitably hesitant and qualified and marked by a respect for one’s unwillingness to advance a verdict. (Wright, 2003, 92–3, my emphasis)
I think, however, that the defender of Third View and Verdict Exclusion can—with a qualification to come—fully appreciate this insight. For what her view excludes, as we have seen, is that people typically offer—non-hesitant, unqualified—verdicts concerning borderline cases. Jason and Justin can indeed give such responses, as in Williamson’s ‘macho’ opinionated community. But they would typically not do so, nor should do so. This does not mean that they should issue no response at all, refusing to form any opinion on the matter whatsoever. They may eventually refuse to do so, but in most contexts it would be more natural (and rational) for them precisely to issue the suitably hesitant and qualified responses: ‘yeah . . . it’s kind of green,’ ‘sort of is’n sort of isn’t,’ ‘ . . . greenish . . . ,’ ‘it’s more green than blue, I guess’—or even ‘it’s green,’ which the appropriate gestural and/or intonational vagueifying markers. So, the defender of Third View and Verdict Exclusion can, it seems, fully appreciate the insight contained in the second part of Wright’s quote. She would probably resist, and this is the qualification announced above, the remark in the first part. For, she might hold, the predicted consensus on certain cases as borderline can indeed take the form of people precisely issuing the suitably hesitant and qualified opinions—not necessarily confining themselves to an aseptic agnostic silence. What I
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am in effect suggesting is that Third View and Verdict Exclusion can indeed turn out to be compatible with the contention that there is a characteristic psychological attitude of the sort advocated by Wright 2003 himself—see also, for a related proposal, Schiffer (2003).¹¹ (Of course, one may hold that vagueness is characteristically manifested by a certain way of responding to borderline cases and still hold that the nature of borderline cases has to do with semantic indecision, irremovable ignorance, and so on.) Substantiating this suggestion of mine is something I am not in a position to do here.¹² Fortunately, defending my main claim does not require it. For my main claim has been simply that people typically do not and should not respond to borderline cases by forming—non-hesitant, unqualified—verdicts concerning them. This is compatible with their forming an opinion—provided they are suitably hesitant and qualified. 18.4
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It seems that Hannah and Sarah may disagree as to whether Homer Simpson is funny, without either of them being at fault. They may typically form (non-hesitant, unqualified) judgements on the matter, and it is not clear at all that they should not: hence the appearance of faultnessness in their disagreement, which most people are, in the case at hand, inclined to endorse. By contrast, Jason and Justin do not typically form (non-hesitant, unqualified) judgements on whether the (borderline green) towel is green or not. That this is so is not only the intuitive view but also indeed respected and actually accounted for by paradigm cases of semantic and epistemic views on the nature of vagueness. And it turns out to be compatible with their issuing ‘macho,’ admissible, forced, and hesitant responses with respect to borderline cases. Thus Jason and Justin just lack the (eventually contrasting) judgements, which are the building blocks of apparent faultless disagreements. Borderline cases do not provide further cases thereof. Re f e re n c e s Andjelkovi´c, M. and Williamson, T. (2000), ‘Truth, falsity, and borderline cases’, Philosophical Topics 28, 211–43. Fine, K. (1975), ‘Vagueness, truth and logic’, Synth`ese 30, 265–300. Horwich, P. (1997), ‘The nature of vagueness’, Philosophy and Phenomenological Research 62, 929–35. ¹¹ Wright explicitly notes that the quoted reflections on the characteristic manifestation of vagueness ‘are, to stress, strictly inconsistent neither with Third Possibility nor, therefore, with Verdict Exclusion.’ 2003, 93 He nonetheless adds: ‘What they are inconsistent with is our knowing that either of those proposals correctly characterizes borderline cases—or better, if someone insist that either is a correct characterization, with there being any definite (known) borderline case in the sense of the characterization.’ 2003, 93. I am suggesting that they can be consistent in the latter case as well. ¹² I hope to attempt this elsewhere.
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Keefe, R. (2003), ‘Context, vagueness, and the sorites’, in Jc Beall, ed., Liars and Heaps, Oxford University Press, Oxford. K¨olbel, M. (2003), ‘Faultless disagreement’, Proceedings of the Aristotelian Society 104, 53–73. Lewis, D. (1979), ‘Scorekeeping in a language game’, Journal of Philosophical Logic 3, 339–59. Reprinted in his Philosophical Papers vol. 1, Oxford University Press, 1983 (q.v.). ¨ (1980), ‘Index, context, and content,’ in S. Kanger and S. Ohman, eds, ‘Philosophy and Grammar’, Reidel, Dordrecht. Reprinted in Papers in Philosophical Logic, Cambridge University Press, 1998 (q.v.). L´opez de Sa, D. (2009a), ‘Can one get bivalence from (Tarskian) truth and falsity?’, Canadian Journal of Philosophy 39, 273–82. (2009b), ‘The many relativisms: Index, contex, and beyond’ in S. D. Hales, ed., The Blackwell Companion to Relativism, Blackwell, forthcoming. MacFarlane, J. (2003), ‘Future contingent and relative truth’, Philosophical Quarterly 53, 321. (2005), ‘Making sense of relative truth’, Proceedings of the Aristotelian Society 105, 321–39. Raffman, D. (1994), ‘Vagueness without paradox’, Philosophical Review 103, 41–74. Shapiro, S. (2003), ‘Vagueness and conversation’ in Jc Beall, ed., Liars and Heaps, Oxford University Press, Oxford. Schiffer, S. (2003), The Things We Mean, Oxford University Press, Oxford. Williamson, T. (1994), Vagueness, Routledge, London. (1997), ‘Reply to commentators’, Philosophy and Phenomenological Research 62, 945–53. (2000), Knowledge and Its Limits, Harvard University Press. Wright, C. (1992), Truth and Objectivity, Harvard University Press, Cambridge. (2003), ‘Vagueness: A fifth column approach’ in Jc Beall, ed., Liars and Heaps, Oxford University Press, Oxford.
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PA RT I I T H E LO G I C O F VAG U E N E S S
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V Supervaluationism
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19 Supervaluationism and the Report of Vague Contents Manuel Garc´ıa-Carpintero
In two recent papers, Schiffer (1998, 196–8; 2000, 246–8) advances an argument against supervaluationist accounts of vagueness, based on reports of vague contents. Suppose that Al tells Bob ‘Ben was there’, pointing to a certain place, and later Bob says, ‘Al said that Ben was there’, pointing in the same direction. According to supervaluationist semantics, Schiffer contends, both Al’s and Bob’s utterances of ‘there’ indeterminately refer to myriad precise regions of space; Al’s utterance is true just in case Ben was in any of those precisely bounded regions of space, and Bob’s is true just in case Al said of each of them that it is where Ben was. However, while the supervaluationist truth-conditions for Al’s utterance might be satisfied, those for Bob’s cannot; for Al didn’t say, of any of those precisely delimited regions of space, that it is where Ben was. From a perspective more congenial to supervaluationism than Schiffer’s, McGee and McLaughlin (2000, at 139–7) pose a related problem about de re ascriptions of propositional attitudes and indirect discourse. The same difficulty is gestured at in this argument: ‘there are additional concerns about the ability of supervaluational proposals to track our intuitions concerning the extension of ‘‘true’’ among statements involving vague vocabulary: ‘‘No one can knowledgeably identify a precise boundary between those who are tall and those who are not’’ is plausibly a true claim which is not true under any admissible way of making ‘‘tall’’ precise’ (Wright 2004, 88). In an earlier version of the material that I will present here (Garc´ıa-Carpintero 2000) I replied to Schiffer’s argument that supervaluationism has an independently well-motivated defense. The response is essentially based on the point that the occurrence of ‘there’ in Bob’s utterance (and of ‘tall’ in Wright’s argument) occurs An earlier version of this chapter was presented at talks at the university of Navarra and Arch´e, St Andrews; I thank the audience for criticisms and suggestions. My work has benefited from comments by Pablo Cobreros, Richard Dietz, Cian Dorr, Dan L´opez de Sa, Josep Maci`a, Daniel Nolan, Manuel P´erez Otero, Timothy Williamson, and Crispin Wright. Thanks also to Michael Maudsley for his grammatical revision. Financial support was provided by the research project HUM2006–08236, funded by the CICYT, Spanish Government, and by a Distinci´o de Recerca de la Generalitat, Investigadors Reconeguts 2002–2008, DURSI, Generalitat de Catalunya.
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in indirect discourse, and supervaluationists may allow that it shifts its referent there.¹ Schiffer’s (2000b) reply to this response shows that it was not made sufficiently clear.² In this chapter I will try to improve on that score. In his more recent reply, Schiffer (2000b, 325) dismisses a proposal like the one I will make, mainly because it ‘undermines . . . a leading virtue of supervaluationism . . . its implication that vagueness is . . . not a feature of the world.’ I will argue that my reply does not undermine the fundamental contentions of the supervaluationist account. Suppose that, in a context where the size of a given rod is being discussed, Alex utters (1) while placing his symmetrically extended hands one opposing the other at a certain distance: (1) The rod was this length. In uttering (1) Alex makes an assertion, the kind of speech act that we routinely classify as true or false and has therefore truth-conditions, which illustrates the sort of data that theories of vagueness attempt to account for. The basic datum, put in a way as neutral as possible among possible potentially conflicting accounts, is this: the facts about the rod that Alex wanted to report might be such that it is indeterminate whether (1) is true, and it is indeterminate whether (1) is false; the size of the rod being discussed in the context might be a borderline case of the type of length that Alex signified with the predicate ‘was this length’. Call this ‘DV’, the datum of vagueness. Supervaluationism is an account of vagueness that upholds certain claims for which DV poses a problem requiring theoretical elucidation. Or, rather, it is not supervaluationism per se that provides the account. Supervaluationism is a mathematical model-theoretic technique, and, as McGee says (1998, 156): ‘It has been thought that the model theory provides a deep explanation of the way we use vague language; specifically, it has been thought to explain the fact that we are able to use classical logic even in the face of semantic indeterminacy. But that can’t be right. Model theory is just mathematics, and, as such, it can’t explain anything about language use.’ The explanation is provided by a philosophical account that applies the model theory. It concerns the nature of vague language, illustrated by (1), and distinguishes itself from others by upholding those intuitions. Following David Lewis, I will refer to the explanatory philosophical theory as vagueness as semantic indecision,‘VSI’. A first claim with which DV is prima facie in conflict is the correspondence claim. Language and thought are representational at their root: some expressions are semantically substantively related to objective, mind- and language-independent objects. Consider (1). As we said, it is used to make an assertion, assessable as true or false, and has therefore a certain truth-condition such that, together with the facts of the actual world, determines (1)’s truth value (and, together with the facts of ¹ After the reference shift, the term still refers indeterminately because of higher-order vagueness. The arguments here discussed differ from objections to supervaluationism based on higher-order vagueness; hence, for the sake of simplicity I will ignore it here. ² While he rejects my answer, he proposes ‘to stay with the topic’ so as ‘to consider a supervaluationist response I was too quick to dismiss’ (Schiffer 2000b, 322); the response is in substance the one I was intending to convey.
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other possible worlds, its truth value across possible worlds). This truth-condition is compositionally determined by (1)’s logical form, its semantically relevant syntactical composition out of lexical units and phrases formed from them. The correspondence claim is that (1)’s truth-condition is such that, if it is met and (1) is true, there is a mind- and language-independent truth-maker in the actual world making it so, on which (1)’s truth would then depend. In particular, ‘this length’ in (1) contributes to (1)’s truth-condition a mind- and language-independent object constituting that truth-maker, a specific length. Secondly, there is the clear-cut world claim; this is the contention that the objective, mind- and language-independent world does not include vague objects, kinds or properties. D. Lewis provides a compelling rationale for it: ‘I doubt that I have any correct conception of a vague object. How, for instance, shall I think of an object that is vague in its spatial extent? The closest I can come is to superimpose three pictures. There is the multiplicity picture, in which the vague object gives way to differences between precisifications, and the vagueness of the object gives way to differences between precisifications. There is the ignorance picture, in which the object has some definite but secret extent. And there is the fadeaway picture, in which the presence of the object admits of degree, in much the way that the presence of a spot of illumination admits of degree, and the degree diminishes as a function of the distance from the region where the object is most intensely present. None of the three pictures is right. Each one in its own way replaces the alleged vagueness of the object by precision. But if I cannot think of a vague object except by juggling these mistaken pictures, I have no correct conception’ (Lewis 1993, 27).³ Finally, we have the claim that the logical validity of our ordinary arguments is to be accounted for ultimately on the basis of the classical, Tarskian model-theoretic validity of arguments, by formalizing them in the languages devised by logicians. Now, relative to our illustrative case (1), we can see how the three claims create a difficulty in the presence of DV. For given the third, a predicate like ‘was this length’ in (1) should signify a subset of a domain of discourse, a class of lengths; given the first and the second, this should be a class containing a precisely delimited length (one thus to which any given length either belongs or does not belong, tertium non datur). This conflicts with DV, unless we could account for it on epistemic grounds; but supervaluationists assume that this is excluded by the notion that semantic properties in general, and the truth-conditions in particular of speech acts and thoughts, depend on their role in rational activities in which conscious, potentially reflective beings like us engage, and that as a result such vagueness as it is illustrated by (1) is not a matter of ignorance.⁴ It is here that VSI, vagueness as semantic indecision, together with the supervaluationist technique, comes to the rescue, reconciling the claims with DV. As Williamson ³ Elaborating on suggestions from Evans, McGee (1998) provides an argument against the view that a term like ‘Kilimanjaro’, intending to refer to a mind- and language-independent mountain, refers to a vague object. ⁴ See, for instance, Horgan (1997) as an expression of this well-known form of skepticism about Williamson’s (1994) epistemic theory of vagueness.
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(1994, 142) aptly puts the idea:⁵ ‘the vagueness of a language consists in its capacity in principle to be made precise in more than one way. Not every substitution of precise meanings for vague ones counts as making the language precise . . . vague meanings are conceived as incomplete specifications of reference. To make the language precise is to complete these specifications without contradicting anything in the original content.’ As required by our three claims, the intended models for our discourse are the sharp models for, say, a logician’s first-order language; they allow for classical, bivalent definitions of truth in a model. Vagueness is due to the fact that ‘our thoughts and practices do not pick out a unique model as the actual model. They pick out a class of models’ (McGee 1998, 154). As McGee puts it, the fundamental hypothesis of VSI is that the semantics of a vague language can be described by singling out an appropriate class of models such that a sentence is determinately true if and only if it is true in every model in the class. According to VSI, there are two notions of truth required to account for DV while validating the claims. There is the fundamental notion involved in stating the truth-conditions of our assertions and judgments, given the representational character of language and thought. This is the fundamental non-bivalent determinate truth or super-truth, which comes handin-hand with a related correspondence notion of reference; the adjustment required by the correspondence claim in view of vague sentences such as (1), according to VSI, is that they do not just represent a unique truth-maker, but a plurality thereof. And there is, in addition, the semantically ancillary notion of truth, the bivalent truth in a model, and the related notion of reference.⁶ Let us consider now the original problem based on indirect discourse posed by Schiffer (1998, 197): ‘Suppose that in uttering ‘‘Harry is bald’’, Renata said that Harry was bald. Then the sentence ‘‘Renata said that Harry was bald’’ is true. But the supervaluationist must say that it wouldn’t be true if the that-clause in [it], ‘‘that Harry was bald’’. . . indeterminately referred . . . to various precise propositions . . . not one of those precisifications will be true, since, even taking into account the vagueness of ‘‘say’’, Renata obviously didn’t say any precise proposition . . . Evidently, then, the supervaluationist must say that [its] that-clause refers to the vague proposition that Harry is bald’. In my reply, I relied on the following theoretical basis: ‘propositional attitude verbs . . . express relations between agents and interpreted logical forms (ILFs). ILFs are annotated constituency graphs or phrase-markers whose nodes pair terminal and non-terminal symbols with a semantic value’ (Larson and Ludlow 1993, 305). Larson and Ludlow’s semantic values are classical semantic values: objects for terms, sets for predicates, truth values for sentences. On an alternative version (Pietroski, 1996), symbols are paired with Fregean senses in ILFs (which, in their turn, determine semantic values). ILFs, under either of those proposals, are the sort of entity that can be vague, in the sense that they admit different precisifications, and admit thereby a supervaluationist treatment. On Larson and Ludlow’s version, vague ILFs can be ⁵ Williamson is far from accepting it, of course. ⁶ This corresponds to the distinction by McGee and McLaughlin (1995) between the senses of truth answering, respectively, a ‘correspondence’ and a ‘disquotational’ conception.
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neither true nor false as a result of the fact that (ignoring higher-order vagueness) at least some terminal node (say, the one corresponding to ‘bald’ in Schiffer’s example) is paired, not with an appropriate semantic value, but with a class of them (its admissible precisifications). On Pietroski’s version, the same obtains if the mode of presentation with which the symbol is paired does not determine a unique semantic value, but a class of admissible ones. Before moving on to the problem posed by de re ascriptions which will be the main focus of the present chapter, let me emphasize that the proposal so far substantially agrees with Schiffer’s diagnosis, quoted two paragraphs back. To put it impressionistically, the supervaluationist agrees in accepting, besides the precise truth-makers indeterminately represented in vague sentences, some ‘vague entities’: i.e. vague contents, modeled along the ILF accounts. But, far from being incompatible with VSI, this is taken to be a crucial aspect of it. What matters is that truth and falsity (in their fundamental, non-ancillary senses linked to the correspondence claim) are ultimately determined relative to the class of precisifications.⁷ To make this more vivid, consider the following Schifferian argument. Supervaluationism treats all vague expressions as indeterminately referring to precise referents. In particular, supervaluationism treats ‘this length’ in (2) as indeterminately referring to precise lengths in a given class. However, none of those lengths is an observable property, if by ‘observable’ we understand something like discriminable by the naked eye. Thus, for any of the lengths to which ‘this length’ indeterminately refers, (2) is false. Hence, (2) should be superfalse, against compelling intuitions: (2) This length is an observable property. To provide an adequate response to this argument, it is enough to characterize a prima facie plausible way to reject it, compatible with VSI. The response could legitimately rely on contentious philosophical views, if they can be defended independently of the present issue. We do not need to go further into the details of a well-argued defense of the proposal; we do not need to defend the contentious philosophical assumptions. For we will have already shown that Schiffer has at most established a conditional: supervaluationism is false, unless such-and-such philosophical view is correct. A response of this kind to the Schifferian argument goes like this. A first premise is that the very same expression (‘this length’) that in a given context (its occurrence in (1)) refers, albeit indeterminately, to the precise lengths constituting the objective world, in a different context (its occurrence in (2)) refers to something else. The second premise is that, in addition to containing precise types of lengths, the world also contains what, in a manner of speaking, can be intelligibly called ‘imprecise lengths’, of which it is not determinate of all lengths in the first group ⁷ Garc´ıa-Carpintero (2007) elaborates on this, on the basis of more detailed considerations on the nature of truth and its relation to what is said. Keefe (2008)—a nice presentation of the main ideas defining supervaluationism—also emphasizes the centrality of quantification over precisifications to the account, and its compatibility with ‘vague entities’ of some such representational sort.
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whether or not they instantiate them. Combining these two premises, we can counter the Schifferian argument by saying that, although the occurrence of ‘this length’ in (1) should be semantically treated following the supervaluationist guidelines (it indeterminately refers to lengths of the precise kind), the one in (2) somehow shifts its reference, determinately denoting instead a length of the imprecise sort.⁸ The second premise might superficially appear to be incompatible with the philosophical motivation we have provided for supervaluationism. However, let us reflect more carefully on it, in order to clarify the qualification ‘in a manner of speaking’. Does it follow from the three claims that VSI tries to accommodate that there are no vague entities in the world at all? It is not just that the answer to this is negative; it should be clear that VSI rather requires that the world include vague entities. VSI only assumes that the objective, mind- and language-independent world does not contain vague entities, and that truth and falsity is ultimately to be accounted for on the basis of supervaluationist quantification over those entities. However, VSI assumes that representational facts create vagueness, and representational facts are, of course, facts (albeit obviously not mind- and language-independent facts). They induce (in a manner of speaking) new properties and kinds instantiated by the precise objects constituting the objective world, which are (in a manner of speaking) in their turn new objects, potential objects of reference, which can in a clear sense be called ‘imprecise’.⁹ The first premise is also in good philosophical standing. It has it that the very same expression that in a context refers indeterminately to entities in the mind- and language-independent world, in a different context might refer (whether determinately or indeterminately, depending on the issue of higher-order vagueness, which we are putting aside here) to the indeterminately instantiated objects induced by the representational fact involving indeterminate reference in the previous context. Fregean theories assert the existence of this kind of systematic ambiguity to account for quotation and direct discourse in general, and for indirect discourse.¹⁰ This is what, on the present suggestion, happens to ‘this length’ in (2). It does not refer indeterminately to a length, but (as it were) to a new kind of ‘lengths’, a length as referred to by a demonstrative expression with the contextual help of a certain way of grasping lengths. This way is constituted by perceptual experiences of the same kind as that on which the speaker is relying, and takes his audience to be relying, in the context of his utterance (2). This way of grasping lengths is distinguished by its not being able to discriminate among a given set of (precise, as there are no others in the ⁸ Remember that, for the sake of simplicity, we are ignoring higher-order vagueness. A more realistic treatment should also use the supervaluationist strategy with respect to ‘this length’ in (2), allowing that it indeterminately refers to a length of the imprecise variety. ⁹ The mechanism though which ‘precise’ and ‘imprecise’ acquire a new sense is the same creating metonymies and other cases of ‘semantic transfer’; this is why I qualify my claims with ‘in a manner of speaking’. All these apparent references to, and quantification over, imprecise ‘objects’ should at a fundamental level be subjected to a fictionalist explanation; see Garc´ıa-Carpintero forthcoming. ¹⁰ As I indicated in my original contribution, we do not need to have recourse to a strictly speaking Fregean theory to justify this; a theory which attributes the shift in reference to the implicit presence in the utterance of a ‘hidden-indexical’ could serve as well, and in fact my own Davidsonian sympathies when it comes to the account of quotation suggest that much.
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mind- and language-independent world) lengths. It induces a mind-dependent sort of (mind-independent, precise) length, an ‘imprecise’ kind of length which is not just one precise length instantiate, but several—a sort that counts thereby as imprecise.¹¹ Thus, for the present purposes, we can take the semantic value of ‘this length’ in (2), in contrast to (1), to be a length of such a sort. These contentions involve no violation of the fundamental assumptions of VSI; on the contrary, they are to be properly justified ultimately on their basis. For these imprecise lengths are in effect representational entities, entities constituted by their role in representational activities; and the supervaluationist apparatus of precisifications is still required to obtain the truth-conditions of utterances—and mental states—whose content they help characterize, such as our original (1). This puts me in agreement with the main claim in Merricks (2001), that VSI is either a form of metaphysical vagueness, or a form of epistemic vagueness, by my embracing the first disjunct. Notice, however, that this is only because, in characterizing metaphysical vagueness, Merricks does not distinguish, as I have done, among entities in general, those responsible for fundamentally accounting for the semantic values of expressions, in particular the truth values of assertions and judgments. Metaphysical vagueness just consists for him in that ‘for some object and some property, there is no determinate fact of the matter whether the object exemplifies the property’ (145); properties are understood here in a fully liberal, ‘abundant’ sense. Merricks then considers a proposal like the one I have made concerning the sentence ‘Bald’ applies to Harry, with ‘Harry’ denoting a borderline case of baldness. Against the perhaps more orthodox supervaluationist line, on which such a sentence signifies many different precise propositions, I have granted that there is a sense in which such a sentence expresses a vague proposition, one ascribing to Harry the vague property λx(‘Bald’ applies to x): think of ‘Bald’, as previously suggested, as referring to a semantically individuated word. However, whether or not an object exemplifies such vague representational property is to be accounted for, at a fundamental level, relative to supervaluationist quantification over precise properties (as there are no others at the fundamental level). Merricks (op. cit, 155–6) is right, however, that supervaluationist arguments against metaphysical vagueness, such as the one by Lewis quoted before, do not mention any distinction between fundamental and non-fundamental entities of the VSI account. But it is not that difficult to insert adverbs such as ‘fundamentally’ at the proper places, and it seems to me more charitable to do so. Thus, what is unintelligible is not that there is a vague ‘object’, with an indeterminate spatial extent—we have already envisaged vague ‘kinds’ (vague sorts of lengths), and presently we will be considering vague ‘‘particulars’’ (vague locations). What is unintelligible is rather that such objects have an explanatory fundamental role in accounting for the truthconditions of our assertions and judgments. And, on the present view, they don’t; those are explained in terms of supervaluationist quantification over precise entities.¹² ¹¹ At the risk of boring the reader, I should insist that I am ignoring higher-order vagueness. ¹² Williams (2008) diagnoses a loophole in the usual semantic ways of dealing with Evans (1978) infamous argument against vague objects, which question the λ–conversion step. That
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Let us now move on to arguments involving de re ascriptions. Schiffer argues that a proposal along the previous lines cannot work in cases involving singular terms, such as (3) below, or ‘there’ in (5), taken as a report of Al’s utterance of (4): (3) (4) (5) (6) (7)
Alex said that the rod was this length. Ben is at that mountain. Al said that Ben was there. ∃x(x is where Al said Ben was). There is where Al said Ben was.
Notice that both ‘this length’ in (3) and ‘there’ in (5) are de re at least in that, say, (5) entails (6) and, (‘by demonstrative specification’), (7). ‘Here . . . the supervaluationist evidently has to take her standard line: in a sentence of the form ‘‘There is such-and-such’’, ‘‘there’’ must be taken to indeterminately (or partially) refer to each member of a set of precise places, the set of places that can be used to give the supervaluationist truth-conditions of the sentence in which the demonstrative occurs’ (op. cit., 198). This is how Schiffer’s argument goes: ‘There was no problem initially in the idea that the that-clause in ‘‘Renata said that Harry was bald’’ referred to a vague proposition, because there was no problem initially in the idea that ‘‘bald’’ in that that-clause expressed a vague property, a property with a penumbra. The problem with (5) comes when we try to make sense of the idea of there being a vague place to which ‘‘there’’ might refer. What could possibly be both a place, a region of space, and fail to have precise boundaries? It might be thought that the supervaluationist could take a vague proposition to be a set of precise propositions, those used to give the supervaluationist truth conditions of the vague proposition. Then the reference of ‘‘there’’ can be taken to be a set of precise places. But I don’t think this will work . . . A set of places is not a place. The problem is that the occurrence of ‘‘there’’ in (5) is de re and thus occurs as a demonstrative seeking to refer to a place’ (op. cit., 198).¹³ step cannot be validly instantiated with referentially indeterminate expressions; but their referential indeterminacy could be the result of ontic vagueness, and not its cause. Thus, the argument does not after all dispose of ontic vagueness, even granting the controversial assumptions it requires. Williams goes on to provide a model for ontic vagueness, based on an ersatzist conception of possible worlds. On such a view, worlds are abstract maximal properties that the one and only Reality could have; ‘the’ actual world is one more abstract property, and not Reality itself. This allows that there is not just one ‘actuality’, if for w to be actualized is for w not to be determinately uninstantiated. On this view, propositions understood as sets of worlds—properties predicated of Reality, in assertions and judgments—are themselves indeterminately instantiated, and thus vague; by Merricks’s lights the view counts as propounding metaphysical vagueness. However, the truth or falsehood of assertions and judgments expressing those vague propositions is ultimately explained on the basis of supervaluationist quantification over precise propositions, and thus, to the extent that I find this view intelligible, it is just a form of VSI. ¹³ The problem that Weatherson (2003, 482) takes to be Schiffer’s, and for which he offers a solution, is that for (4) to be true, Al must have said of every candidate-mountain that Ben was there; but Al ‘could not have said all those things’. But this does not distinguish between the problem posed by predicates, as in Schiffer’s original example with ‘Harry is bald’, previously discussed, and the problem posed by singular terms, as Schiffer does here. The problem Schiffer poses is not that supervaluationism has Al saying too many things, if (4) is to be true, but that none of those
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But Schiffer’s conclusion does not follow. Let us take Kaplan’s (‘Quantifying In’) account of the truth-conditions of de re ascriptions, as in ‘Joan does not know that her best friend betrays her’, which we could formalize (only for the sake of the discussion, for a full account should be more complicated) as (8), where ‘R’ stands for an appropriate representational relation, one (involving acquaintance, or what have you) sustaining correct de re ascriptions between constituents in ILF and their semantic values, ‘VPA’ for any verb ascribing propositional attitudes, and the Greek variables such as ‘α’ range over modes of presentation (in Kaplan’s original presentation), over parts of ILF on the view of attitude ascriptions I am assuming here: (8) ∃α(R(α, τ , S) ∧ S VPA σ (α)) On such a view, we describe the ILF in indefinite terms, existentially quantifying over some of its nodes, by indicating only its semantic value—the omitted complication consists in that typically some additional information about the mode of presentation α is given in de re ascriptions, such as that it is a mode of presentation of a mountain, it is demonstrative, or, indeed, it is (im)precise. Where α is a constituent of a vague ILF, the simplest theory is that an instance of this schema obtains just in the case that τ is one of the several semantic values with which the vague term in α is paired (one of the semantic values determined by its paired sense). Under this interpretation, the ascription (5) may well be true. It is misleading to object, as Schiffer does, that ‘Al didn’t say, of any precise place, that it was where Ben was’; for this rings true only by contextually suggesting that, under the proposal, the truth of (5) requires Al to have expressed a precise thought (one with a precise ILF); the omitted complication would properly deal with this, if it is explicitly specified that α was indexical and vague.¹⁴ The present proposal rejects this claim by McGee: ‘In order for us to have de re beliefs, at least on our usual understanding of them, our thoughts and practices have to pick out one particular thing as the object the belief is about’ (1998, 147); for (5) is things, being precise, are good candidates for reporting what Al said; moving to saying-relations to imprecise contents is OK when we only consider predicates, as in ‘Renata said that Harry was bald’, but de re ascriptions, according to Schiffer, make this move irrelevant. Unlike Weatherson’s proposal, mine properly deals with the problem posed by de re ascriptions which I take Schiffer to be raising here. I will come back later to Weatherson’s views. ¹⁴ In her contribution to this volume, Rosanna Keefe (2009) provides a more orthodox reply, which avoids vague entities by assuming only the penumbral coordination of the precisification of the embedded sentence in an attitude report, and that of the reported sentence or mental item. She discusses an objection: ‘Someone might object to the above solution that ‘‘Renata said that Harry is bald1 ’’ should come out determinately false (where bald1 is a precisification of ‘‘bald’’), whereas on the above treatment, it comes out indeterminate. (Schiffer, 2000, 248, suggests something like this objection.) But, this intuition, if there is one, is far less strong than the intuition that ‘‘Renata said that Harry is bald1 ’’ should not be determinately true.’ On my account, however, we could say more, if we take the ‘suggestion’ that the representational device used by Renata was a precise one, produced by the use of a precise device in the ascription, to go into the truth-conditions of the report; for, if so, the ascription would turn out to be determinately false after all. (Otherwise, we could appeal to a pragmatic explanation of the incorrect impression that the ascription is false.) I take it that the possibility of thus capturing the intuition, even if it is a weak one, is an advantage of my proposal.
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a correct de re report of a de re assertion, but neither the reported asserter, nor the reporting utterer, need in any way to have been in a position to ‘pick out’ one in particular of the several precise mountains that could legitimately be invoked in order to precisify the utterances. The proposal so far does not require that precisifications are coordinated so as to assign the same candidate-mountain to ‘that mountain’ in (4) and ‘there’ in (5), as in Weatherson’s (2003, 482–3) and Keefe’s (2009) more orthodox replies to Schiffer. (4) is (super-) true just in case, for each admissible value for ‘that mountain’, (4) is true; (5) is true just in case, for each admissible value for ‘there’, Al was in the proper R-relation with it through whatever corresponds in the thought he expressed to ‘that mountain’ in his utterance (4). But perhaps the proposal does require such coordination among precisifications implicitly, in the conditions for a candidate-mountain to be an admissible value for ‘there’ in (5). For in specific contexts, it may be part of the intended meaning of those singular terms in de re ascriptions that they are in a sort of anaphoric relation with corresponding ‘singular terms’ in the vehicle for the reported propositional attitude.¹⁵ I will conclude by discussing a different, but related objection, made by McGee and McLaughlin (2000, 145–6). They consider an atom at or around the base of Kilimanjaro, called Sparky, and define Kilimanjaro(+) ‘to be the body of land constituted . . . by the atoms that make up Kilimanjaro together with Sparky [and] Kilimanjaro(−) [to] be the body of land constituted . . . by the atoms that make up Kilimanjaro other than Sparky’ (2000, 129); and they argue as follows, about someone like Al in the previous example: ‘In fact, there isn’t anything, either in his mental state or in his neural state or in his causal relations with his environment that would make one of Kilimanjaro(+) and Kilimanjaro(−), rather than the other, the thing [Al’s assertion] is about. [The thought he expressed] can with equal justice be imagined to be the singular proposition obtained from the propositional function described by the English open sentence ‘‘that it is the snow-capped mountain within sight of the equator where Ben is’’ by supplying Kilimanjaro(+) as argument and the proposition obtained by supplying Kilimanjaro(−) as argument. But exactly one of those propositions is true. The possibility that [Al said] all of the countless billions of singular propositions obtained by supplying Kilimanjaro candidates as arguments of the proposition function can be readily dismissed, for it implies that, no matter how careful and knowledgeable a geographer [Al] may be, his every true [thought] about Kilimanjaro is accompanied by countless billions of false [thoughts]’. In discussing this argument, we need to keep in mind a warning made by McGee himself (‘Kilimanjaro’, 152): ‘Just to make sense of the attachment of the word ¹⁵ As Keefe (2009) points out, Weatherson’s proposal that precisifications should be given wholesale, for every word in the language, is no modification of VSI, for penumbral connections, a fundamental ingredient of the supervaluationist account, are holistic in that way. That precisifications should be given in this holistic way not just for words, but for tokens thereof (or words-in-context), as Weatherson rightly insists, is a consequence of context-dependence in general, such as the long-term discourse anaphoric relations envisaged in the main text in particular. Keefe is nonetheless right that this coordination of token-precisifications raises further problems, which do not depend specifically on issues of vagueness.
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‘‘determinately’’ to an open sentence containing free variables is a bit of a stretch, since we primarily think of determinacy as an attribute of sentences. A sentence is determinately true, determinately false, or unsettled. We need to go beyond this familiar usage if we want to say of an object that it either determinately satisfies, determinately fails to satisfy, or is indeterminate with respect to an open sentence with one free variable’. This is a warning that Williams (2006) ignores, in providing an argument in support of Lewis’s (1993) supervaluationist solution to the problem of the many, a solution which differs from the equally supervaluationist one I prefer; let me elaborate, in order to provide a useful background for the discussion of McGee and McLaughlin’s argument. The supervaluationist solution I prefer has it that on every way of making the language precise, exactly one of the many candidates for being the referent of ‘Kilimanjaro’ will count as a mountain. Williams (2006, 415) argues that, given what he takes to be the ‘standard’ treatment of ‘Definitely’ as applied to open sentences, this solution entails the falsity of (9): (9) ∃x Definitely (x is a mountain) Williams argues that this is bad news for supervaluationists, because it conflicts with their standard ‘confusion’ explanation for our intuitions regarding the mayor premises in sorites arguments. The standard explanation is that we confuse ‘Definitely ∃x . . .’, truly stating that there is a cut-off point in every precisification, with ‘∃x Definitely . . .’, falsely asserting that there is a definite such cut-off; i.e. we read existentially quantified claims in terms of the ‘∃x Definitely . . .’ scope relations, not the other way around. If this account is generally correct, he contends, we should read ‘there are mountains’ as in (9), and therefore (given the proposal to deal with the problem of the many we are assuming) judge it false, against what our intuitions in fact tell us. To preserve the confusion explanation, Williams proposes to adopt instead Lewis’s solution to the Problem of the Many, according to which all mountain-candidates are indeed in the extension of ‘mountain’,¹⁶ and thus (9) turns out to be true, assuming that ‘standard’ interpretation of the interaction of ‘Definitively’ with open sentences. But this will not do, because, generalizing the confusion explanation in the same way, we would read ‘there is exactly one snow-capped mountain within sight of the Equator’ as: ∃!x Definitely (x is a snow-capped mountain within sight of the Equator), and, assuming now the Lewis solution that Williams is arguing for, judge it false— which is not what we in fact do. Instead of arguing on the basis of claims about our intuitions very difficult to uphold in this area, it is in my view preferable to rethink the interpretation of the interaction of ‘Definitely’ with (what intuitively corresponds to) open sentences, attending to McGee’s own warning. Consistently with my account so far, I propose to appeal to a representational relation R (with a contextual parameter C instead of the subject parameter S), generalizing the previous proposal to interpret de re locutions in ascriptions of propositional attitudes. Thus, I propose to ¹⁶ Lewis deals with the problem by appealing to flexible standards for counting.
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analyze sentences involving ‘quantifying in’ the Definitely-operator, such as (9), along these lines: (10) ∃x ∃α(R(α, x, C) ∧ Definitely σ (α)) Thus: there is an ILF-part α, which in the context C represents x, such that the logical form consisting in plugging α in the frame σ (x) is supertrue. Given my preferred, standard supervaluationist solution to the Problem of the Many, in any particular precisification only one of the many Kilimanjaro-candidates will be in the extension of ‘mountain’, and will therefore be an acceptable candidate for being represented in the context C by the relevant instance of α. This proposal is therefore compatible with the confusion explanation of sorites reasoning.¹⁷ This proposal allows us to diagnose the problem with McGee and McLaughlin’s claims about the singular propositions obtained by supplying Kilimanjaro(+) or Kilimanjaro(−) as arguments for the propositional function described by the English open sentence ‘that it is the snow-capped mountain within sight of the equator where Ben is’—in particular, the claim that ‘exactly one of [them] is true’. What is true is only that, assuming the supervaluationist solution to the problem of the many, in each precisification the English open clause ‘that it is the snow-capped mountain within sight of the equator where Ben is’ is made true at most by assigning to ‘‘it’’ as value one of Kilimanjaro(+) or Kilimanjaro(−). But the issue is what follows from this with respect to the correctness of de re ascriptions like (11): (11) Al said of n that it is the snow-capped mountain within sight of the equator where Ben is. Given the previous proposal, what McGee and McLaughlin’s considerations— including the assumption that Al is a careful and knowledgeable geographer—show is only that the conditions on a candidate-mountain to be an admissible value for the referential expression that Al used (the one on which ‘it’ in (11) is ultimately anaphoric), in particular the condition required to deal with the problem of the many that there is at most one snow-capped mountain saliently within sight of the Equator, will extend to the conditions a candidate-mountain should meet to be an acceptable value for ‘n’ in (11) in each precisification. Weatherson (2003, 488) is right in assuming that ‘there is a penumbral connection between the subject of [Al’s assertion] . . . ¹⁷ Williams (op. cit., 415) argues that on the standard supervaluationist solution to the problem of the many ‘the ability of the confusion hypothesis to explain intuitions about the sorites premise is undermined. To illustrate this, let us put the explanatory challenge in the following contrastive form. (a) In the original case presented above, where we have a range of emanations from Kilimanjaro to Glastonbury Tor, decreasing in height by a few metres from one to the next, we have strong ‘‘no cut-off’’ intuitions. (b) Consider a new range, which consists in Kilimanjaro standing next to Glastonbury Tor. In this scenario, we have strong intuitions that there is a cut-off: a mountain standing next to a non-mountain. The datum to be explained is the contrasting intuitions in the two cases (a) and (b).’ The present proposal to understand the interaction of ‘Definitively’ and intuitively open sentences accounts for this datum. In both cases, we read the existential quantifier outside the scope of the definitely-operator; in (a), we get a falsehood (but the narrow-scope reading is in the vicinity, which explains our confusion); in (b), understood as I have proposed, in terms of (10), we get a truth.
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and the word ‘‘mountain’’ ’.¹⁸ Thus, in a sense Al said ‘‘all of the countless billions of singular propositions’’ (the same sense in which, against Schiffer, Al did say, of every relevant (precise) location, that it was where Ben was). But this does not imply at all that ‘‘his every true [thought] about Kilimanjaro is accompanied by countless billions of false [thoughts]’’. Reckoning by the more intuitive counting in terms of vague thoughts, Al just had one true de re thought about Kilimanjaro. Let me summarize. Both in his original paper and in his reply to my original criticism, Schiffer argues against the kind of proposal I have made, by contending (correctly in my view, as I have said) that claims like (3) are de re in that they entail reports like (12): (12) This length is such that Alex said that the rod was it But in view of our discussion about (2), this does not pose new problems. That the report (3) is de re, as shown by the fact that the inference to (12) holds, only requires that ‘this length’ in it—as in (2)—still refers to a type of lengths, an imprecise kind instantiated by particular lengths. This is compatible with its referring to a kind individuated in part by mind-dependent matters (a perceptual way of grasping lengths), which accounts for its being an imprecise kind in accord with the intuitions that VSI tries to support.¹⁹ In his reply, Schiffer (2000b, 322) asked me to characterize ‘the nature of the modes of presentation’ under which Alex said something about a myriad precisely delimited lengths; to say how a set of lengths, which is not a property, can be a property of modes of presentation; and what the truthconditions of statements like (3) are. These requests are well taken, but I think I have met them here. The modes of presentation at stake are in part types of contextually salient perceptual experiences. The relevant property is the property of being a type of perceptual experience presenting any length in the given set to a perceiver experiencing it.²⁰ The truth-conditions can be given (with some licenses, mostly in the metaphoric reference to parts of contents) as follows: (3) is true iff Alex made an assertion whose propositional content ‘consist’ of a ‘part’ contributed by ‘the rod was’ and another ‘part’ signifying the imprecise kind of length determinately referred to by ‘this length’. As mentioned before, we may or may not additionally assume that a contextual indication of the sort of perceptual experience on which Alex contextually relied to refer to a length is part of the full characterization of that propositional content. ¹⁸ But this solution has nothing to do with Weatherson’s previous appeal to naturalness. I find it difficult to understand how, although ‘in reality Kilimanjaro(+) is no more natural than Kilimanjaro(−)’, nevertheless ‘according to any precisification, one of them will be more natural than the other, for precisifications determine content by determining relative naturalness.’ I cannot see how precisifications, which are arbitrary reinterpretations of the language, can determine naturalness; rather, naturalness and other facts about the language such as penumbral connections determine which of them are acceptable. ¹⁹ That is to say, it is compatible with the occurrence of ‘this length’ in (3) being only weakly de re, in terms of the distinction I made in my original reply to Schiffer, Garc´ıa-Carpintero (2000). ²⁰ By referring to sets we can of course refer to the properties determining them, when we are not in a fastidious mood.
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Schiffer (2000b, 325) criticizes accounts of this sort on three counts: ‘First, it requires finding truth-conditional ambiguity’ in forms like (1) and (2), ‘when in fact those forms don’t seem ambiguous in any relevant way. Second, the move looks suspiciously like saying that the supervaluationism is to be limited to those cases that aren’t clear counterexamples to it. Third, it undermines what some will have thought was a leading virtue of supervaluationism—namely, its implication that vagueness is either not a feature of the world at all, but of our ways of describing it, or, failing that, a feature of the world that is wholly reducible to, a construct out of, non-vague features of the world. Evidently, the supervaluationist theory that survives doesn’t have this ‘‘virtue,’’ since it recognizes that vague objects and properties may have features not possessed by the precisifications of those objects and properties.’ As regards the third and main point, as we have seen, properly understood supervaluationism is not only compatible with this consequence, but actually requires it. The relevant claim that VSI tries to validate is only about the mind- and languageindependent world; the leading virtue of supervaluationism lies in its capacity to buttress this claim. The account, however, entails (rather than being incompatible with it) that our representational activities induce imprecise particulars, properties and kinds, possessing distinctive properties of their own; and these induced particulars, properties and kinds are, of course, also part of the wider world.²¹ Against what Schiffer says, this is compatible with their being ‘constructed out of’ the precise objects, at least in a sense which can be precisely explicated in terms of some form of supervenience: no difference in the imprecise objects, without a corresponding difference in precise objects. Schiffer’s main criticism is thus shown to depend on a misleading characterization of supervaluationism’s ‘leading virtue.’ As regards the first and second points, the claim of ambiguity, as I have suggested, can be motivated on independent, Fregean-like considerations. Schiffer only has validly argued for a conditional: if referential expressions never shift their referents in the way suggested by Fregean-like theories, then supervaluationism is wrong. However, the reader should only realize how wide-ranging ‘Fregean-like’ is in the antecedent of this conditional, to appreciate the extent to which its falsity is probable. As I said, even theories that explain the shift of reference attributing it to other expressions implicitly or explicitly present in the utterance (hidden-indexicals, or other expressions in the sentence) count, for present purposes, as Fregean-like. I conclude that Schiffer has not given us a compelling new argument against VSI. Weatherson (2003) and Keefe (2009) offer alternative solutions to Schiffer’s challenge, on which I have made some critical remarks before. The main difference is that they do not countenance vague entities, such as the vague representational items my proposal envisages. I think that in that way they miss what I see as its main virtue, that it allows us to capture the sense in which, as Schiffer insists, Al didn’t say, of any precisely delimited regions of space, that it is where Ben was; or the corresponding sense in which Wright’s claim in the quotation provided in the first paragraph is correct. ²¹ As I said before, I take this reference to imprecise entities to be amenable to a fictionalist treatment; but we do not need to go into this for present purposes.
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Re f e re n c e s Garc´ıa-Carpintero, Manuel (2000), ‘Vagueness and indirect discourse,’ Philosophical Issues 10, E. Villanueva, ed., Boston, Blackwell, 258–70. (2007), ‘Bivalence and what is said’, Dialectica 61, 167–90. (forthcoming), ‘Fictional entities, theoretical models and figurative truth’ in Frigg, R, and Hunter, M., eds., Beyond Mimesis and Convention—Representation in Art and Science, Springer. Horgan, Terence (1997), ‘Deep ignorance, brute supervenience, and the problem of the many’ in Philosophical Issues 8: Truth, E. Villanueva (ed.), Ridgeview, Atascadero, CA, 229–36. Keefe, Rosanna (2008), ‘Vagueness: supervaluationism,’ Philosophy Compass 3 (2), 315–24. (2009), ‘Supervaluationism, indirect speech reports and demonstratives’, this volume. Larson, Richard, and Ludlow, Peter (1993), ‘Interpreted logical forms,’ Synthese 95, 305–55. Lewis, David (1993), ‘Many, but almost one,’ in Ontology, Causality and Mind, J. Bacon, K. Campbell and Ll. Reinhardt, eds., Cambridge, Cambridge University Press, 23–38. McGee, Vann (1998), ‘Kilimanjaro’, Canadian Journal of Philosophy: Meaning and Reference, supp. vol. 23, A. Kazmi, ed., 141–63. McGee, Vann and McLaughlin, Brian (1995), ‘Distinctions without a difference’, Southern Journal of Philosophy, supp. vol. 33, 203–51. (2000), ‘The lessons of the many’, Philosophical Topics 28, 129–51. Merricks, Trenton (2001), ‘Varieties of vagueness,’ Philosophy and Phenomenological Research 62, 145–57. Pietroski, Paul (1996), ‘Fregean innocence,’ Mind and Language 11, 338–70. Schiffer, Stephen (1998), ‘Two issues of vagueness’. The Monist 81, 193–214. (2000a), ‘Vagueness and partial belief,’ Philosophical Issues 10, E. Villanueva, ed., Boston: Blackwell, 220–57. (2000b), ‘Replies,’ Philosophical Issues 10, E. Villanueva, ed., Boston: Blackwell, 321–43. Weatherson, Brian (2003), ‘Many many problems’, Philosophical Quarterly 53, 481–501. Williams, J. Robert (2006), ‘An argument for the many’, Proceedings of the Aristotelian Society 106, 409–17. (2008), ‘Multiple actualities and ontically vague identity’, Philosophical Quarterly 58, 134–54. Williamson, Timothy (1994), Vagueness, London: Routledge. Wright, Crispin (2004), ‘Vagueness: A fifth column approach’ in Jc Beall, Liars and Heaps, Oxford, Oxford University Press, 84–105.
20 Supervaluationism, Indirect Speech Reports, and Demonstratives Rosanna Keefe
According to the supervaluationist theory of vagueness, a vague sentence such as ‘Bob is tall’ is true iff it is true on every way of making it precise. In general, the truthconditions of sentences containing vague terms involve quantification over different ways of making the various vague components of the sentence precise.¹ In this chapter I consider whether, as Stephen Schiffer argues, this popular theory of vagueness is undermined by considerations about indirect speech reports. As a very brief summary of the potential problem, consider the sentence ‘Carla said that Bob is tall’. The worry is that this sentence will be true only if it is true on all ways of making precise its vague terms, including ‘tall’, resulting in the condition that the speech report is true only if Carla said that Bob was over 6.0001 feet tall and said that he was over 6.0002 feet tall etc. But, the objection goes, she clearly didn’t say any, let alone all, of those things. I will argue that the supervaluationist can satisfactorily deal with indirect speech reports in general, and I will offer solutions to the various problems raised by cases such as Schiffer’s. The most interesting cases involve demonstratives, I will argue, but these can be handled by the supervaluationist as well. 20.1
SCHIFFER’S OBJECTIONS
In his 1998, Schiffer sets up the problem as a dilemma.² With a vague sentence such as An earlier draft of this chapter was presented to the Third Navarra Workshop on Vagueness in Granada: I am very grateful to the participants and organizers, especially Elia Zardini, Pablo Cobreros and Maria Cerezo. For comments and advice on other drafts of this chapter, thanks to Jenny Saul, Sebastiano Moruzzi, and Richard Dietz. I am also very grateful to the Arts and Humanities Research Council who funded a period of leave during which this chapter was originally written. ¹ This is subject to penumbral constraints: we consider acceptable ways of making the whole language precise at once, respecting relations between different vague terms, such as the fact that nothing will count as both red and orange on a precisification of those two terms. On the supervaluationist theory of vagueness, see Fine 1975 and Keefe 2000. ² See also his 2000a, 246–8 and 2000b, 321–6, where the problems are presented differently.
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[a] Harry is bald, supervaluationists have to say one of two things about the propositional content expressed by that sentence. They can either say that propositions are precise and it is indeterminate which proposition is expressed by [a], or they can maintain that there are vague propositions and that [a] expresses one of them. If Harry is borderline tall then [a] will turn out neither true nor false on both options, either because of the divergence in truth value of the precise propositions that [a] indeterminately expresses, or because the vague proposition that it (determinately) expresses is itself neither true nor false in the context. The problems Schiffer identifies for both options involve indirect speech reports. Take [b] Renata said that Harry was bald The first option (involving indeterminate reference to many precise propositions) supposedly comes to grief because ‘in order for [b] to be true, according to the supervaluationist, it must be true under every way of precisifying the reference of its that-clause. Yet not one of those precisifications will be true, since, even taking into account the vagueness of ‘‘say’’, Renata obviously didn’t say any precise proposition, let alone all of the precise propositions to which the that-clause partially, or indeterminately referred.’ (Schiffer 1998, 197). In relation to the second option (involving vague propositions), Schiffer focuses on ‘Al said that Ben was there’, where this can also be reported as ‘there is where Al said Ben was’, which, he says, is surely not true for each precisification of ‘there’.³ We would equally expect this problem to arise with ‘Everest is what Al said Ben climbed’: the key feature is that the relevant singular term has wide scope, so I shall call this the de re problem. Before proceeding, I will clarify why the above argument ignores the vagueness of ‘says’. Like most other English expressions, ‘says’ is vague: there will be instances of ‘S says that p’ that intuitively are borderline due to the vagueness of ‘says’; for example, it might not be clear whether or not S has said that p, because he was muttering. Schiffer’s objections, however, can be run on a case where there is seemingly no unclarity about whether, say, Renata said that Harry was bald (for example, she uttered the very words ‘Harry is bald’ in appropriate circumstances). Then, assuming we are right about that intuitive classification of the report, it should be the case that, no matter how we make ‘says’ precise, the indirect speech report comes out true. The construction of the argument as a dilemma looks misplaced, however. For the truth-conditions of [b] turn on the truth values that result when we make precise those components, whatever we decide about the nature of propositions. Even if ‘Harry is bald’ determinately refers to a unique vague proposition, quantification over ³ He also objects that we cannot make sense of a vague place to be the referent of ‘there’, but I think we can pass quickly over this problem. Indeed, Schiffer himself has dropped it by his subsequent presentations of the problem (2000a and 2000b), arguing that the greater Boston metropolitan area could be a vague place. Note that on a Fregean conception of propositions, there would be no need for a vague place as a component of the proposition.
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precisifications is still needed.⁴ Similarly, the de re problem is equally a problem for the supervaluationist who rejects vague propositions. The objection in relation to ‘there is where Al said Ben was’ is that the supervaluationist must maintain that Al said of each of the relevant range of precise places that Ben was there. And this is so whether or not they accept vague propositions. In section 20.3, I will argue that the general de re problem can be solved by the supervaluationist. I will then go on to ask whether there is a particular problem with demonstratives in relation to indirect speech reports (though not specifically de re ones). 20.2
I N D I R E C T S PE E C H R E P O RTS
The heart of Schiffer’s objection involving the sentence ‘Renata said that Harry was bald’, is summarized in the following quotation: ‘Renata obviously didn’t say any precise proposition, let alone all of the precise propositions to which the that-clause partially, or indeterminately, referred.’ (1998, 197). Supervaluationists must indeed say that ‘Renata said that Harry was bald’ is true on each precisification if it is to count as true simpliciter, as intuitively it should. But, as we’ll see, that doesn’t commit them to saying that she said any precise proposition. Consider a precisification, s1 , according to which ‘bald’ means bald1 , for some precisification of bald, and ‘Harry is bald’ says that p1 (for some precise proposition p1 ). When I use ‘bald’ in reporting Renata’s utterance, then according to precisification s1 , I mean bald1 by ‘bald’. Now, Renata uttered the words that, according to s1 , mean p1 , so surely according to that precisification, she did say that p1 and my report to that effect is true on that precisification. More generally, according to precisification si , Renata said that pi (where pi is precise) and according to the same precisification I report her as having said pi . According to different precisifications she said different precise things and is reported as having said different precise things. But it isn’t true (i.e. true simpliciter) that she said p for any precise p, for there is no precise proposition that, according to all precisifications she said. What she said differs according to the precisifications. So, it certainly doesn’t need to be the case that she said all of the precise propositions: that is clearly false on all precisifications, so false simpliciter. If A says ‘a is F’, and B says ‘A said that a is F’, then B’s report is true iff it is true on all precisifications. But, to put it somewhat loosely, whether B’s utterance is true on a precisification depends on what, according to that precisification, A says. To demand, for the truth of the report, that it is true (so true on all precisifications) that A said all of the precise things is, in effect, to recognize the variation between precisifications over what B is reporting A as having said, without acknowledging the corresponding variation over the actual content of A’s utterance according to those precisifications. ⁴ By analogy, compare an unusual supervaluationist who maintained that there were vague properties, and that ‘bald’ determinately referred to one of them, but that something counts as having that vague property iff it has all the precise properties appropriately related to it. Supervaluationist quantification is needed when there is vagueness, whether it is an indeterminacy of reference or determinate reference to a somehow vague entity.
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Analysing the supervaluationist’s truth-conditions of indirect speech reports in further detail may require commitment to some particular account of indirect speech reports (which is typically determined by one’s account of propositional attitudes more generally). Alternative such accounts include sententialist accounts—according to which the truth of the speech report turns on whether the speaker uttered a sentence appropriately related to (e.g. saying the same as), the sentence attributed to them—and accounts of indirect speech reports as relations to propositions, where these may be Fregean, Russellian, sets of possible worlds or various other possibilities. I maintain, though I will not argue it here, that the above solution to the problem is available on any of these accounts.⁵ For example, ‘A said that a is F’ may express a relation between A and a different precise Russellian proposition on different precisifications (differing as to the precise property picked out by ‘F’ on that precisification). But for each of those precisifications, the proposition which is the relata of this relation is also the content of A’s utterance according to that precisification. So the speech report can be true on all precisifications. What the case of indirect speech reports brings out is that sometimes the truth value of a sentence on a precisification depends on the values of other sentences on that precisification. This is a kind of penumbral connection and is unproblematic for the supervaluationist.⁶ Someone might object to the above solution that ‘Renata said that Harry is bald1 ’ should come out determinately false (where bald1 is a precisification of ‘bald’), whereas on the above treatment, it comes out indeterminate. (Schiffer, 2000, 248, suggests something like this objection.) But, this intuition, if there is one, is far less strong than the intuition that ‘Renata said that Harry is bald1 ’ should not be determinately true. If it is indeterminate whether ‘Harry is bald’ means that Harry is bald1 , it is reasonable to maintain that it is indeterminate whether Renata said that Harry is bald1 , when she uttered ‘Harry is bald’. It might then be thought that since according to each precisification, there is some precise p such that Renata says that p, (albeit a different one according to different precisifications), then it will come out true simpliciter (since true on all precisifications) that she says something precise. But this putative consequence does not in fact follow, given the supervaluationist treatment of sentences involving ‘precise’, ‘vague’ etc. A sentence such as ‘ ‘‘bald’’ is precise’ does not come out true on all, or indeed on any precisifications, despite the fact that ‘bald’ receives a precise interpretation on all those precisifications. For ‘ ‘‘bald’’ is precise’ is a metalinguistic claim and whether it is true on a precisification depends on what is true on other precisifications, not ⁵ Garc´ıa-Carpintero (2000) defends supervaluationism against Schiffer’s objection by adopting what he calls a syncretic account of propositions, in which modes of presentation play a role even though the constituents of propositions are entities, as on a Russellian picture. See Schiffer (2000b) for his response, where he argues that adopting this account of propositions does not solve the problem. See Garc´ıa-Carpintero (2009) for further discussion. For a brief argument that Schiffer’s problem does not arise on Davidson’s paratactic account of indirect speech (Davidson 1968), see Keefe (2000, 158). ⁶ Weatherson (2003) tackles the de re problem and similarly appeals to penumbral connections. For more discussion of Weatherson, see below.
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just at the precisification in hand (see Keefe 2000, 186–7; and compare the way that the truth of a modal sentence at a world depends on the truth of sentences at other worlds). Similarly, then, for a sentence such as ‘Renata said something precise’: the differences in what she said on the different precisifications are enough to ensure that this comes out false, just as the differences in the values of p at different worlds makes ‘p is contingent’ come out true. So, Schiffer’s objection fails and the supervaluationist can accept normal, vague speech reports without being committed to the absurd consequences he claims. It needn’t be determinately true that Renata said any of the relevant range of precise propositions for the speech report to be true. As an analogy to Schiffer’s objection, consider an objection to supervaluationism centring on the compelling claim ‘ ‘‘Harry is bald’’ means that Harry is bald’. On each precisification the second ‘Harry is bald’ gets some precise interpretation (e.g. Harry has less than 2003 hairs on his head). We can parallel Schiffer’s objection as follows: ‘Harry is bald’ does not mean any of these precise things, let alone all of them. Again, this objection would be misguided. It needn’t be true simpliciter that ‘Harry is bald’ has any of the relevant precise meanings for the meaning claim to be true. Rather, it is true according to each precisification that it has some such meaning. On p1 , ‘Harry is bald’ means Harry is bald1 , while on p2 , it means Harry is bald2 . So, according to each precisification, ‘Harry is bald’ has some precise meaning, but there is no precise meaning that it is true that this sentence has. Next, consider the situation with propositional attitude reports. Consider ‘Simon believes that Harry is bald’. On a given precisification, ‘bald’ will get a well-defined extension and, putting it loosely, that extension will figure in the content of Simon’s belief according to that precisification. Now, it might seem strange that well-defined extensions get into the content of beliefs, even on precisifications. An opponent might argue as follows: why should the content of Simon’s beliefs depend on how a particular expression is made precise? To say that the belief ascription is true on a given precisification, s1 —where ‘Harry is bald’ means p1 , say—is to claim that it is true that Simon believes that p1 according to s1 (ignoring precisifications of ‘believes’). And, the opponent might continue, Simon’s belief—his state of mind—does not change with change in how expressions are made precise, so it seems as if it should thus also be true on precisification s2 that Simon believes that p1 . This would mean that if it is to be true on all precisifications that he believes that p, then it must be true (i.e. true on all precisifications) that he believes all the precisifications of p. To see how the supervaluationist can reply, consider first what the epistemicist says about Simon’s belief. According to epistemicism, there will be a single precise interpretation of ‘Harry is bald’ and Simon’s belief will have the content given by that interpretation. The contents of our beliefs, according to the theory, depend on the meanings of our words, which are, in turn, determined by the use of those words in the community.⁷ This means that the content of Simon’s belief could have been slightly different if the meaning of ‘bald’ had been different, and Simon may not have noticed this. The situation for the supervaluationist can be similar in relation to each ⁷ See Williamson 1994, ch. 7.
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precisification. What Simon counts as believing on s1 depends on the meaning of ‘bald’ according to s1 , and if ‘Harry is bald’ means that p1 on s1 , then on s1 it is true that Harry believes that p1 and this does not imply that on s2 he believes that p1 . So, the belief report can be true without it being true at all precisifications that he believes each precisification of ‘Harry is bald’. We will return to the comparison between the supervaluationist and epistemicist later.⁸ Next, I turn to the de re problem.
20.3
VAG U E S I N G U L A R T E R M S A N D D E M O N S T R AT I V E S
Recall the de re problem with ‘there is where Al said Ben was’. Since I will argue that there are distinctive issues surrounding demonstratives, I shall start by considering the example ‘Everest is what Al said Ben climbed’. The worry, recall, is that the supervaluationist will have to say that it is true of all precise delimitations of Everest that Al said that Ben climbed them. I will argue that the supervaluationist can solve the alleged problem with this example in the same way that the previous version of the problem was solved. On precisification s1 , ‘Everest’ names O1 , say. Al utters the words ‘Ben climbed Everest’ and on s1 this is true iff Ben climbed O1 . On s1 , then, it is true that O1 is what Al said Ben climbed. On s2 , though, it is true that O2 is what Al said Ben climbed and false that O1 is what Al said Ben climbed. So, the sense in which it is true of all precise delimitations of Everest that Al said that Ben climbed them, is that of each of those objects, according to some precisification, Al said that Ben climbed it. And this does not have the unwanted consequence that Al said something about a huge quantity of precise objects. There is no precise object, Oi , of which it is true that Al said Ben climbed it, for of no such object is this true on all precisifications. The de re presentation of the issue poses no additional problems. This solution seems to turn on the way in which the reference of ‘Everest’ in ‘Everest is what Al said Ben climbed’ is guaranteed to be the same on a precisification as the reference of that name in Al’s report. This guarantee is provided simply by the use of the same expression. Whatever the reference is on a precisification will be the reference for any occurrence of the name. Next, consider a case where Al says ‘Ben climbed the highest mountain in the world’ and, again, I report this with ‘Everest is what Al said Ben climbed’. Vagueness aside, there would be disagreement over whether this will be true—whether this change in manner of picking out the mountain is ⁸ One kind of influential objection to the epistemicist has centred on the question ‘how are the exact extensions to our vague predicates determined’, where the thought is that ‘they are determined by use’ is not sufficiently specific and remains problematic (see e.g. Keefe 2000, 76–83). Now, the opponent might suggest that since the supervaluationist needs to say exactly the same about extensions on particular precisifications, then they face the same objection with respect to each precisification (so they face it many times over!). But, the supervaluationist, unlike the epistemicist, can still maintain that use does not determine a unique well-defined extension to a vague term: for the supervaluationist, each of the precisifications is compatible with use and there is nothing that selects between them. We don’t have to pick out a precise interpretation and what is true at it: no unique one is singled out by our use. So, the supervaluationist, here and below, is not simply appealing to an unattractive feature of epistemicism to solve their problem.
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compatible with the report being accurate. ⁹ But such debates should be independent of one’s theory of vagueness. And the truth and falsity of the report could each be accommodated by the supervaluationist, if other features of one’s views on indirect speech reports allow it. For the reference on a precisification of my use of ‘Everest’ to be guaranteed to match that of Al’s use of ‘the highest mountain in the world’, there must be some kind of penumbral connection. For example, Everest is definitely a mountain, and so any simultaneous precisification of ‘mountain’ and ‘Everest’ must make that true. On a given precisification, there is only one object in the right vicinity for Everest that counts as a mountain, and that is the same object that counts as Everest on that precisification. So, the use of different expressions in the speech report from those used in the reported speech act need not deliver the kind of problem Schiffer highlights. The details of the required penumbral connections have not been drawn out here, but such a story is needed to accommodate the truth of various other compelling sentences such as ‘Everest is the tallest mountain in the world’. Although on different precisifications the statement will be about different precise objects, the penumbral connection guarantees that the two sides of the identity claim refer to the same thing. What about the cases with demonstratives? Again, to account for the truth of a speech report containing a demonstrative, the aim is to establish a connection on each precisification between the referent of the speakers’ demonstrative and the referent of the reporter’s term (or, for the epistemicist, a connection between the actual referent of each). Here there are two kinds of cases. Suppose Al points at Everest and says ‘Ben climbed that’ and the reporter similarly points at Everest and says ‘that is what Al said Ben climbed’. We can reasonably take ‘that’ to mean ‘that mountain’ and the case can be solved. On any precisification, there will be but one object that counts as the mountain in the vicinity, and on that precisification, the same object will count as the mountain in assessing both Al’s assertion and the reporter’s report. So, the report will be true on all precisifications. (And, if your view of indirect speech reports allows the truth of ‘that is what Al said Ben climbed’ when Al has said ‘Ben climbed Everest’, then this can again be accommodated by the supervaluationist by taking into account the penumbral connections between ‘Everest’ and ‘mountain’.) For the first kind of case, then, the demonstrative is coupled (perhaps implicitly) with some sortal that does the job of ensuring a treatment of the above kind will work.¹⁰ The second kind of case involves a bare demonstrative, where Schiffer’s ‘there’ is one of the most forceful examples. Being a bare demonstrative, there is no sortal that ⁹ Cappelen and Lepore (2004), among others, would allow that the report can be true in this case—at least assuming that all parties involved know that Everest is the highest mountain in the world—since it is something a reasonable person might endorse as a correct report. ¹⁰ In fact, this may not work on all accounts of complex demonstratives. On a minimal theory, with ‘this F’ the sortal, F, does not play a semantic role in determining the content of what is said, just a pragmatic role in helping the hearer pick up on the speaker’s referent (see, e.g. Larson and Segal 1995). On such a theory, there may be no significant difference between the cases just discussed and the cases of bare demonstratives discussed below, depending on the exact details of the story.
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could play the role ‘mountain’ played in the previous example in ensuring co-reference across the utterances. Even if ‘there’ means ‘that place’, ‘place’ is not the right kind of sortal to do the above job, since, for example, ‘place’ will not get precisified in the way that ‘mountain’ does such as to ensure that on a given precisification there are no two substantially overlapping places. Appeal to the community’s use of the chosen expression (e.g. ‘there’ or ‘that’) will not help here either, given that such demonstratives can be used to refer to almost anything.¹¹ But appeal to use could still help, if we focus on the individual’s use. Consider the problem in relation to the epistemicist again. What, for the epistemicist, could make it the case that I refer to place P1 with ‘there’ rather than a very similar, but slightly differently delineated precise place, P2 ? As usual, the epistemicist will surely say that this is determined by features of use (in particular the use by the speaker, but perhaps relevant utterances from other people). How this occurs is mysterious, but an epistemicist like Williamson allows that ‘meaning may supervene on use in an unsurveyably chaotic way’ (1994, 209). If you then report my utterance, also using the word ‘there’, then the epistemicist can say that it is features of your use of that expression that determine its reference. A key feature of your use of ‘there’ in that speech report, is that you intend it to have the same reference as my use of the expression. Perhaps that intention is enough (in suitable circumstances) to guarantee that it does have the same reference. If so, your speech report will come out true. Now, the supervaluationist can say exactly the same in relation to a given precisification without having to say that there is some unique precise referent for Al’s term. According to some particular precisification, s1 , when Al says ‘Ben was there’, he refers to a particular precise place. But my use of ‘there’ when I report ‘there is where Al said Ben was’ gets to pick out the same place, due to the key feature of my use (which will hold for all precisifications) that I intend to pick out the same place as Al. My intentions ensure the existence of penumbral connections. This is, then, at least the beginning of a way out of the apparent problem with demonstratives for both the epistemicist and the supervaluationist. But the viability of this solution depends on the treatment of demonstratives offered, and that is questionable. In general, for my use of a demonstrative to refer to the same thing as yours, it is not enough that I intend it to. For, I can have several referential intentions which conflict. For example, suppose I point to John and say ‘you said he was F’ intending to refer to the same person you were talking about when you pointed to Mark and said ‘he is F’. In Kaplan’s terminology, my ‘directing intention’ here picks out John, and it is plausible to contend that this intention trumps my intention to co-refer with you and that I thereby incorrectly report you as having said something about John. ¹¹ The above treatment could, however, be used for a case where the speaker and the reporter each seek to refer with ‘there’ to a vague place which can be picked out independently, for example the greater Boston metropolitan area. Here, the relevant place will be a different precise area on different precisifications, but each speaker’s use of ‘there’ will pick out, on a precisification, whatever is the referent of the associated term on that precisification. Note that the problem with demonstratives here is not dependent on the de re formulation of the reporter’s utterance: it would equally arise if I say ‘Al said Ben was there’.
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Once directing intentions are considered central, the above solution is questionable: in the reporter’s context, ‘there’ will be associated with some set of precise interpretations and in the speaker’s it will be associated with another set and there will be no penumbral connection between them.¹² (Or for the epistemicist, the reporter and reportee will not count as picking out exactly the same place.) In other words, the reporter’s intentions cannot piggyback on the speaker’s to guarantee co-reference, as was assumed in the solution above.¹³ I find this general story about demonstratives rather compelling. But vagueness poses a problem. It isn’t merely supervaluationism that faces it: it may be that no theory of vagueness is any better placed. Consider the Epistemic View. Al’s intentions determine an exact referent of his utterance of ‘there’, but this is by no means guaranteed to coincide with what my intentions determine as the referent of my use of ‘there’ when I report his speech. It is thus highly likely that my utterance of ‘there is where Al said Ben was’ will be false. A theory such as a Degree Theory or other many-valued theory surely offers no new way out of the problem. Allowing degrees of truth for sentences is of no help in guaranteeing the truth of the various reports. Schiffer himself, who advocates a view of vagueness involving vague partial beliefs that come in degrees, does not tackle speech reports involving demonstratives within his own framework, and the way of dealing with them is equally unclear. There is no advantage for the currently popular contextualist theories of vagueness either. The context clearly changes between the reportee’s original utterance and the reporter’s report, and with it the referent of terms like ‘there’ is liable to change. (For wider problems with indirect speech reports facing the contextualist, see Keefe 2007.) It is tempting to say that this is not a problem of vagueness; it is a problem of a reporter matching demonstrative reference with the reportee, or of giving an account of demonstratives that accounts for this. Given the prevalence of vagueness and other necessary features of the example, it is hard to come up with a suitable problematic example in which there is no vagueness, but that doesn’t make it a task for a theory of vagueness. Is there anything other than intentions that could guarantee co-reference between the speaker’s and the reporter’s use of ‘there’, thereby ensuring the truth of the speech report ‘Al said Ben was there’? Weatherson (2003) offers a proposal that would fill the gap. He draws on Lewis’s notion of naturalness (e.g. Lewis 1983). In the case of some terms—natural kind terms, for example—the referent or extension of our term is determined by our use of the term in conjunction with the world. Our term gets ¹² See also Bach (1992), who maintains that you refer to the thing that you intend and expect the audience to recognize as your referent. When I say ‘there is where Al said Ben was’, my intention to refer to whatever Al referred to cannot be the intention by which I intend my audience to pick up on my reference, since they have no independent grasp on that—they are expected to realize what I am referring to from my demonstration. ¹³ Alternatively, the fact that a speaker has conflicting referential intentions, such as in the John/Mark case above could be taken to render the report neither true-nor-false since there is no unique thing being talked about. On that picture, then, the vague demonstrative case is equally messy and we should not be discouraged by a verdict of neither true nor false in such a case, seeing this as a problem which is not due to vagueness.
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to pick out the most natural candidate that is compatible with its use. Now, with a vague predicate such as ‘bald’, there is no most natural candidate property to be its referent. All the various candidate precisifications are equally natural or unnatural and nature does nothing to choose between them. Weatherson’s suggestion is that associated with each precisification is a complete ordering with respect to naturalness, so that there is always an answer to which of two properties is most natural. Then, according to a given precisification, ‘bald’ picks out whatever is the most natural candidate according to the naturalness well-ordering associated with that precisification. And different precisifications will have different well-orderings and so different chosen extensions for ‘bald’. The same, Weatherson assumes, will go for objects or places or whatever are the referents of singular terms, including demonstratives: the naturalness ordering for each precisification selects, for example, some exact area of space as the referent of an utterance of ‘there’. The thought is that this can then explain the penumbral connections—between the reportee’s and reporter’s use of ‘there’, for example—which will guarantee the truth of the kinds of indirect speech reports in question. For, on a given precisification, Al’s use of ‘there’ will pick out what counts as the most natural of the candidate regions according to that precisification. And in reporting Al’s utterance, my use of ‘there’ will pick out that same region, since that will be the most natural of the candidates again. Can Lewisian naturalness really be employed for this purpose? One problem is that naturalness is, for Lewis, a feature of properties, while Weatherson needs it equally to be a feature of whatever entities are the referents of singular terms and other terms, including objects and regions of space etc. For Lewis, the feature of naturalness lines up with qualitative duplication between things sharing the property: perfect qualitative duplicates share all their perfectly natural properties. So, for example, two chairs can be qualitatively the same without sharing the non-natural property of being my favourite chair. Perhaps the naturalness of an object o can be smuggled in by considering the naturalness of the property ‘is identical to o’. But this will not do, partly because all instances of any property of the form ‘is identical to x’ will automatically completely resemble each other in the relevant respect, so all such properties seem to be on a par as regards to naturalness. A second problem for Weatherson’s purposes is that Lewisian naturalness is not a vague notion subject to complete sharpening via imposition of a complete ordering. Even if there is some vagueness that could be resolved in different ways on different precisifications, no precisification should result in a complete ordering. For, there are ties for naturalness, such as ties between all the perfectly natural properties, of which none are more natural than the others. Imposing a complete ordering on candidates for naturalness is not simply resolving vagueness in the idea of naturalness, but revising the idea in more radical ways. Dropping the requirement of a complete ordering and allowing ties within the ordering will not do either. For, then there would be no guarantee that there was just one most natural candidate region to be the referent of Al’s utterance of ‘there’, for example. As long as several candidates can be equally natural, the problems will all re-emerge. Now, perhaps Weatherson can accept the deviation from Lewis’s own notion of naturalness and deny that his precisifications of the language are giving a precisification of ‘natural’ in the ordering they are committed to. We could just stipulate that
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there is a complete ordering of entities on each precisification, which preserves certain relations in the naturalness ordering. But then it isn’t clear what we are being offered in addition to a claim that the required penumbral connections do exist. It is simply a picture according to which on precisifications, truth-conditions are given as if certain objects (or regions of space etc.) are privileged, independently of the intentions of speakers. Certain exact regions of space are singled out above others to be semantic values on a given precisification, for example, and they will be the privileged one among the candidates for various different, apparently independent uses of ‘there’. This will generate penumbral connections where you might think there should be none; e.g. where there are two completely unrelated uses of ‘there’ pointing in roughly the same direction. Consider whether an approach like Weatherson’s would help out the epistemicist, where there would be a unique complete ordering of naturalness among objects, properties and other entities. Nature surely doesn’t pick out a precise privileged area to the referent of Al’s ‘there’, and it is natural to think that it is entirely Al’s intentions that determines such an area if, as the epistemicist maintains, one such is determined. On the Weatherson approach, however, nature does determine the referent, in conjunction with the speakers’ intentions. That seems, at best, very surprising and properties such as ‘big animal’ and ‘zebra’ come out as much more on a par than expected. It was an advantage of the details of Williamson’s epistemicism that he could explain the existence of sharp boundaries to our vague predicates without metaphysical commitment to implausible privileged boundaries in nature; but this advantage is lost on the Weathersonian approach in question. As argued above, the approach cannot merely maintain that it is simply employing a notion of Lewisian naturalness which is sharp but about which we are ignorant, so the epistemicist would seem to be saddled with a highly significant metaphysical commitment to an implausible counterpart to Lewisian naturalness. Is there another approach available to deal with demonstratives in indirect speech reports? When Al says ‘Ben was there’ and I say ‘Al said Ben was there’, on some precisifications, the place Al denotes will diverge from the one I denote, if there is no penumbral connection between them. But on other precisifications, the referents will coincide, and the reporter’s utterance will come out true. Being true on some precisifications and false on others, the utterance will count as neither true nor false overall. Maybe this is an acceptable consequence. We might informally describe the case as one where the original speaker and the reporter have each picked out a rough place, where they were only roughly the same place. There may then be a further pragmatic story to be told about why such indirect speech reports appear true and/or are useful to make—assuming they do and are—but I won’t enter into this in detail here.¹⁴ Of course, a response to the problem that trumpets the fact that the problematic reports will come out neither true nor false, rather than false, is not available to the ¹⁴ According to the standards required for truth on various accounts of indirect speech reports, many such apparently true reports are strictly false (e.g. if the reporter uses a different but coextensive expression). Such accounts will naturally be combined with a pragmatic explanation of such speech reports, and many of the devices used there could be transposed for our purposes here.
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epistemicist. If the reporter’s directing intention is all that is relevant to determining the referent of the demonstrative, then the (unknown) exact referent for the original speaker and the reporter is likely to be different (albeit only slightly). The report would thereby come out false rather than neither true nor false. On the other hand, a pragmatic story may be shared, where for the epistemicist, this would be a matter of explaining why a false—not indeterminate—utterance is useful or compelling. On the vexed issue of demonstratives within reported speech (where vagueness is not identified as the key issue), see Sainsbury 2004 and Altham 2004. One aspect of the problem, as they see it, is that accurately to report speech involving demonstratives, it is not enough to indicate the same thing: it must be indicated in the same way. So, considering a case where Jill says ‘there’s a bird on that post’, Altham writes, ‘Tom cannot properly report what Jill said in indirect speech by saying ‘‘Jill said that there was a bird on that post’’ . . . ‘‘that post’’ . . . refers from Tom’s perspective rather than Jill’s and so, even if it refers to the same post, does not do so as Jill did’ (2004, 237). This suggests that reports of speech involving demonstratives will typically not be true, regardless of vagueness and that vagueness raises no new problems that didn’t already face a theory of demonstratives. To summarize: supervaluationism can allow the truth of most intuitively compelling indirect speech reports, whether they involve vague predicates or vague singular terms, de dicto or de re. For the reports can be true on all precisifications because of penumbral connections with the reportee’s terms. The most challenging cases involve demonstratives, specifically bare demonstratives. The account of these cases turns more on one’s account of demonstratives and indirect speech reports than on one’s theory of vagueness. For example, if one’s account of demonstratives allows a reporter’s reference to be determined by his/her intention to co-refer with the reportee, then the reports come out true. Or if one requires that a speech report involving a demonstrative exactly matches the perspective of the reportee, then the truth of such speech reports will be hard to come by, regardless of vagueness. In the hard cases, a supervaluationist may accept that speech reports are strictly neither true nor false, where apparent truth can be explained. There is, then, a range of options compatible with supervaluationism; and the options are also compatible with other theories of vagueness, for which the issues surrounding demonstratives within speech reports arise as much—or as little—as for supervaluationism. Re f e re n c e s Altham, J. E. (2004), ‘Reporting indexicals’ in Studies in the Philosophy of Logic and Knowledge, T. R. Baldwin and T. J. Smiley, eds., Oxford University Press, Oxford. Bach, K. (1992), ‘Intentions and demonstrations’, Analysis 52. Cappelen, H. and Lepore, E. (2004), Insensitive Semantics: In Defense of Semantic Minimalism and Speech Act Pluralism, Basil Blackwell, Oxford. Davidson, D. (1968), ‘On saying that’, Synthese 19. Fine, K. (1975), ‘Vagueness, truth and logic’, Synthese 30. Reprinted in Vagueness: A Reader, R. Keefe and P. Smith, eds., MIT Press, Cambridge MA. Garc´ıa-Carpintero, M. (2000), ‘Vagueness and indirect discourse’, Philosophical Issues 10.
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Garc´ıa-Carpintero, M. (2009), ‘Supervaluationism and the report of vague terms’, in this volume. Keefe, R. (2000), Theories of Vagueness, Cambridge University Press, Cambridge. (2007), ‘Vagueness without context change’, Mind 116. Larson, R. and Segal, G. (1995), Knowledge of Meaning, MIT Press, Cambridge MA. Lewis, D. (1983), ‘New work for a theory of universals’, Australasian Journal of Philosophy 61. Sainsbury, R. M. (2004), ‘Indexicals and reported speech’ in Studies in the Philosophy of Logic and Knowledge, T. R. Baldwin and T. J. Smiley, eds., Oxford University Press, Oxford. Schiffer, S. (1998), ‘Two issues of vagueness’, Monist 81. (2000a), ‘Vagueness and partial belief ’, Philosophical Issues 10. (2000b), ‘Replies’, Philosophical Issues 10. Weatherson, B. (2003), ‘Many many problems’, Philosophical Quarterly 53. Williamson, T. (1994), Vagueness, Routledge, London.
21 Scope Confusions and Unsatisfiable Disjuncts: Two Problems for Supervaluationism Delia Graff Fara
Supervaluationism as a theory of vagueness has its advantages and its disadvantages. Most of the advantages of supervaluationism are ones that favor it over those of its rivals that also reject bivalence. Since I believe in bivalence, those advantages do not hold much sway over me. I think that the disadvantages of supervaluationism far outweigh its advantages. But I offer no cost-benefit analysis here. Rather, I want to provide some detailed discussion of a couple of the disadvantages. The view discussed here is canonical supervaluationism, which I’ll take to be the view presented by Kit Fine in his 1975 article and defended by Rosanna Keefe in her more recent book (Keefe 2000).¹ On this view, a claim is supervaluationally true (or just true) when it is true-according-to-classical-semantics on all of the different admissible ways of collectively ‘precisifying’ the vague expressions in its language.² The same goes for falsity: a claim is supervaluationally false (or just false) when it is false on all of the different admissible ways of collectively precisifying the vague expressions in its language. Since some claims involving vague expressions are true on some admissible precisifications but false on others, there are claims containing vague expressions that are deemed by the supervaluationist to be neither true nor false. This is what it is to reject bivalence. A predicate is vague on the canonical view only if its extension gap is non-empty, only, that is, if there are objects of which it is neither true nor false. To precisify a vague predicate is to assign all of the objects in its extension gap to one or the other of its extension (the things of which it’s true) or anti-extension (the things of which it’s false). What makes some precisifications of the vague expressions jointly admissible?³ Those precisifications must not alter the truth value of any claim that ¹ But see also Hans Kamp (1975) and Dominic Hyde (1997) for very different applications of the supervaluationist techniques to some problems of vagueness. ² By a claim, I mean an utterance with the following properties: it’s made in a particular language, it’s made in a particular context, and it says something (it ‘expresses a proposition’). When I say that a claim contains a certain expression, I mean that the sentence of which that claim is an utterance contains that expression. ³ The precisifications must be done collectively, since some admissible precisification of ‘small’, for example, will render a number of precisifications of ‘tiny’ inadmissible: everything tiny is small,
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already has a truth value. Some of these claims will relate different vague predicates to one another (‘no one huge is tiny’); others will relate vague predicates to associated relational expressions (‘everyone larger than someone huge is also huge’); still others will relate vague predicates to their clear cases (‘anyone under one meter tall is tiny’).⁴ These claims are said to express ‘penumbral connections’—ones that hold even when the quantifiers in them range over borderline cases of the vague predicates in them: over the things in the predicate’s ‘penumbral region’.⁵ It would be mistaken to describe these constraints by saying that the supervaluationist deems some precisifications admissible whenever they’re compatible with our current usage of vague terms, since on her view a lack of extension gap is not compatible with vagueness. ( This turns out to be the underlying source of the focus of Section 21.2, below) After a brief inventory in Section 21.1 of four advantages and disadvantages of canonical supervaluationism, we will focus on two of the disadvantages: (Section 21.2) canonical supervaluationism allows not merely for true disjunctions with no true disjuncts, but also for true disjunctions with no satisfiable disjuncts; (Section 21.3) supervaluationists have yet to provide us with any convincing answer to the question of how we could ever find the false premise of a sorites argument to be as appealing as we find it to be.
21.1
B E N E F I TS A N D C O S TS
The main advantages of supervaluational treatments of vagueness are these: a. Their account—in terms of truth value gaps—of what it is to be a borderline case of a vague predicate is satisfying to most philosophers not wedded to bivalence (which unfortunately may well be most philosophers). b. They preserve the truth of truisms expressing penumbral connections, such as ‘anyone shorter than a short person is short’ or ‘everything tiny is small’, whereas rival theories that also ditch bivalence do not. c. They account for the fallacy involved in soritical reasoning: in deeming false the inductive premise of a sorites argument (e.g. ‘any man just one nanometer taller than a short man is himself also short’), they declare the argument unsound. d. They preserve classical logic, for the most part anyway.⁶ Not surprisingly, these advantages come with their disadvantages. (No theory is perfect.) The chief ones of these are: and there are some small things that aren’t tiny; in semantic talk, all, but not only, things in the extension of ‘tiny’ on a precisification must be in the extension of ‘small’ on that precisification. ⁴ Which cases are clear cases may vary with context. If ‘tiny’ means ‘tiny for an adult’, then one meter is a clear case; though not if it means ‘tiny for a two-year old’. ⁵ The terminology is adapted from Bertrand Russell (1923). ⁶ See Timothy Williamson (1994) and Delia Graff Fara (2004, §2) for arguments that the preservation of classical logic is incomplete in important ways.
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a. a commitment, despite the rejection of bivalence, to vague predicates’ having sharp boundaries—in the sense of there being a least tall height, one nanometer below which renders a person not tall,⁷ a reddest red, a thinnest fat person, etc; b. a rejection of standard forms of reasoning, such as contraposition of arguments and reductio ad absurdum; c. a failure to provide a satisfactory explanation of why we’re mistakenly tempted by sorites reasoning; and d. an assignment of extremely counter-intuitive truth conditions to the logical particles other than negation.
21.2
T RU E D I S J U N C T I O N S W I T H U N S AT I S F I A B L E D I S J U N C TS
Supervaluation yields truth conditions that are extremely counter-intuitive for the logical particles other than negation. It is often criticized, and justly so, for allowing there to be true disjunctions or true existential generalizations that have no true disjunct or instance; and, correspondingly, false conjunctions or false universal generalizations that have no false conjunct or instance. These failures of the classical truth conditions are alleged to arise due to what Fine called a ‘truth value shift’. In the case of disjunctions and existentials, different disjuncts or instances can be the verifying ones on different admissible precisifications; but as long as there is at least one verifier on any given admissible precisification, the disjunction or the existential will be true simpliciter. The main problem with this as a justification, as I will discuss shortly, is that a sentence can be true on an admissible precisification without its even being possible for the sentence to be true simpliciter. This makes room for there being true disjunctions both of whose disjuncts could not possibly be true. That there could be a shift in truth from disjunct to disjunct when neither disjunct could be true is repellent. I say: ‘Someone in this room is the shortest tall person’. Supervaluationists say: ‘We agree, but there needn’t be a correct answer to the question Who?’ And I say: ‘Either Juan or Carlos is the shortest tall person in this room’. Supervaluationists say: ‘We agree, but there needn’t be a correct answer to the question Which one?’ And supervaluation allows all statements made in the following dialog to be correct: YOUR CHAIR: Don’t be late for the 4:00 meeting. YOU: OK, what’s the latest I can be without being late? CHAIR: A short time after the hour. ⁷ This isn’t exactly accurate; the commitment is rather to there being either a shortest tall height or a tallest non-tall height, and likewise for the other cases mentioned. When variation occurs along a continuous scale, a region can be bounded without itself containing that boundary.
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Delia Graff Fara YOU: I figured that, but what time? CHAIR: I couldn’t truly tell you any particular time.
This leaves me staring incredulously, as it should you. In fact, it’s such a strange position to hold that one often hears this said of supervaluationists: They think that (i) there’s a shortest tall man, even though they don’t think that (ii) someone in particular is such that he is the shortest tall man.
But of course, supervaluationists do believe both of these things; like all of us, they regard these claims as trivially equivalent. What they do believe, which the mistaken attributer is trying to capture, is that there is a shortest tall man, even though no one in particular is such that it’s true that he is the shortest tall man. Supervaluationism is so discordant with the way we actually speak that there are philosophers who understand the view but who aren’t yet fluent in the language we would be speaking if it were correct. But these philosophers are fluent in the language we’re in fact speaking. Whatever prescriptive merits supervaluationism might have, it is not descriptively correct. The following supervaluationist explanation of the anomaly has significant appeal. The reason ‘It’s either pink or red’ can be true even when neither disjunct is true results from the vagueness of ‘pink’ and ‘red’, and in particular from the fact that the object in question is a borderline case of both predicates. We can tell by looking at the thing that it falls somewhere on the spectrum between pink and red, which explains why the disjunction is true. But it’s indeterminate which of pink or red it is, since there are different ways of drawing a boundary between pink and red that count as admissible precisifications of our vague usage of these words, and the object falls on different sides of that boundary on different ones of these ways. And that is why neither disjunct is true simpliciter.
The problem with the explanation is that it does not sit at all well with the following fact about supervaluational semantics. It allows not only for true disjunctions where neither disjunct is true, it allows also for true disjunctions where neither disjunct could be true. I assume that the supervaluationist must say that a sentence could not be true when there is no supervaluational model in which it is true. Let us use s in the following way: s φ is true on a precisification just in case is true in some supervaluational model.⁸ s is the supervaluational satisfiability operator. Then for an appropriately chosen and , supervaluationists allow for the truth of the following: ( ∨ ) ∧ ¬ s ∧ ¬ s . ⁸ The connection with genuine possibility should be apparent. One would think that possibility should amount to truth in some possible world, where on the supervaluationist’s conception each possible world would correspond to some supervaluational model, namely that model that verified and falsified exactly the same sentences as it. Depending on the supervaluationist’s treatment of modality, the diamond used here might not be the diamond of modal logic, since it remains open for the supervaluationist to say that for the modal operator , is true if is true in some possible world, but false only if is false in every possible world. In that case, since our (and too) will be untrue in every supervaluational model, but also neither true nor false in some supervaluational models, would be neither true nor false, while s is false.
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The disjunction of this appropriately chosen and is true because at least one of them is true on every precisification, even though each disjunct is truth valueless, and even though neither disjunct could be true. If we are to accept the claim of the form ‘ ∨ , but it’s indeterminate which’ on the grounds that things could go either way depending on how you drew precise boundaries for the predicates involved, then it should be indeed that things could go either way. But in the case to be presented, it is not. So what’s the example? It’s very simple. Suppose we have something that’s a borderline case of ‘pink’. Let’s represent the claim that it is pink as p. This claim is indeterminate—i.e. neither true nor false according to supervaluationism. We’ll use B to stand for the borderline-case operator. So Bp is true. Supervaluation verifies every instance of excluded middle, even for indeterminate propositions like p. Take the true disjunction p ∨ ¬p; conjoin each of its disjuncts with the true claim Bp; then the result, (Bp ∧ p) ∨ (Bp ∧ ¬p), is true.⁹ But neither of its disjuncts could be true. To see that neither could be true, note that when Bp is true, neither p nor ¬p can be true: we’ll have p true on some admissible precisifications of ‘pink’, ¬p on others. But Bp manages to be true on all of these precisifications, for its truth at a precisification depends on the values p takes over the whole space of precisifications: Bp is true on a precisification when p is true on some admissible precisifications, but false on others of them.¹⁰ This is not something that can vary from precisification to precisification, so Bp is true simpliciter. But Bp is incompatible with p, and also with ¬p. For the truth of Bp requires precisely that neither of these claims be true on every precisification, that each be true on some, false on others. So, we have the supervaluational truth of the following, when Bp is true: ((Bp ∧ p) ∨ (Bp ∧ ¬p)) ∧ ¬ s (Bp ∧ p) ∧ ¬ s (Bp ∧ ¬p). We have substituted (Bp ∧ p) for and (Bp ∧ ¬p) for in the schema given earlier. The underlying source of the difficulty is that supervaluationists (i) think that the existence of borderline cases is not compatible with precision, understood as no gap in truth value; yet they (ii) supply a semantics on which claims of borderline status remain true upon complete precisification of the vague expressions in the language. They are constrained to supply such a semantics by their thoughts that (iii) there are claims of borderline status that are true simpliciter while (iv) truth simpliciter is to be identified with truth-on-every-precisification.¹¹ Putting these views together requires ⁹ Another way to see the commitment is to note that the conjunction of two true claims is true. So Bp ∧ (p ∨ ¬p) is true. By classical distribution laws, (Bp ∧ p) ∨ (Bp ∧ ¬p) is true. ¹⁰ The truth conditions associated with the B operator are structurally like those associated with contingency in modal logic. Those for the B operator involve quantification over admissible precisifications, while those for contingency quantify over possible worlds. ¹¹ Cf. Keefe (2000, 186f). Here Keefe discusses whether supervaluationists face the problem that many of their own theoretical claims (e.g. ‘a predicate is vague only if it has an extension gap’) are not true on any precisification, much less on all of them. She argues that the best supervaluational response is to supply a semantics of the kind described as feature (ii) above. On that point, I agree with her.
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there to be claims of borderline status (ones of the form B) that are true on each precisification, even though it must be that either or its negation is true on that precisification. I take (i–iv) to be at least partly constitutive of of the view I’m here criticizing; to reject any of these in light of my criticism is to concede defeat. In particular, to give up (i), the incompatibility of precision with the existence of borderline cases, is to allow for bivalence in spite of vagueness; while to refuse (ii) is to reject at least one of (iii) and (iv). To reject (iii) is to reject vagueness. To reject (iv), meanwhile, is to reject supervaluationism at its core. I would like to say that if a sentence cannot be true, then that suffices for its being impossible. The supervaluationist might object to this philosophical platitude, but if I were to help myself to it, I could put the main point of this section this way. Supervaluationists think that a single impossibility need not be false, and that the disjunction of two impossibilities can be true. This is because they think that it’s impossible for something to be both borderline pink and pink.¹² Yet, for anything that is a borderline case of pink, the claim that it’s both borderline pink and pink can only be indeterminate, hence not false, according to their neither-true-nor-false account of indeterminacy.
21.3
S C O PE C O N F U S I O N A N D T H E P S YC H O LO G I C A L QU E S T I O N
Supervaluation falsifies the inductive premises of sorites arguments, but does not directly provide an answer to the question of why we’re mistakenly inclined to believe them in the first place. Kamp was prompted to spurn his (1975) supervaluational theory of vagueness for just this reason. Fine, who published an independentlydeveloped supervaluational account of vagueness that same year, did offer some explanation (1975, 286). I have argued elsewhere that that explanation was unsatisfactory (2000, 50–2). This was primarily on the grounds that it implicitly required that we tend to equate truth with non-falsity, something which an opponent of bivalence cannot do without undermining his own theory. Keefe, one of the most prominent current supervaluationists, has offered a supervaluationist answer to this ‘psychological question,’ as I have called it (2000, 50). The psychological question is the question why we are so inclined to believe a sentence with the form of (U) in many cases where it is in fact false. I know of only two places where a supervaluationist answer to this psychological question is offered or defended: in Fine’s article (1975, 286) and in Keefe’s book (2000, 183–6). (U) ∀x(x → x ). Here is some sorites-susceptible predicate and x is the successor of x on some sorites series for that predicate. ¹² This is not the case on the typical view that affirms bivalence, although Raffman’s (2005) pro-bivalence view is a prominent exception.
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By a supervaluationist answer to the question, I mean one that derives from supervaluationism per se, not from supervaluationism as it might be supplemented with some further view better designed to answer the psychological question, such as a contextualist theory or some boundary-shifting theory more generally.¹³ The main flaw in both of Fine’s and Keefe’s explanations is an implicit reliance on our tending to equate truth with non-falsity. This is at best in serious tension with the supervaluationist rejection of bivalence. In the remainder of this section I argue against the success of Keefe’s answer in particular to the psychological question.¹⁴
21.3.1 Keefe’s scope-confusion explanation of our mistake Keefe’s explanation of why we believe (U)—when we do, which as she points out (183f ), may not be always—‘turns on the fact that supervaluationism can distinguish between the falsity of (U) and its having a false instance, and correspondingly between (E)’s being true and its having a true instance’ (184). ((E) is an equivalent of the negation of (U).) (E) ∃x(x ∧ ¬x ), ( TE) true: ∃x(x ∧ ¬x ), (ET ) ∃x true: (x ∧ ¬x ). ‘true:’ here represents the operator ‘it is true that’. We mistakenly confuse (ET) with (TE) (and hence with (E), an equivalent of the latter), so that our correct denial of (ET) mutates into an incorrect denial of (E).
21.3.2 The problem with the explanation The first question to address is how exactly a confusion of (ET) with (E), and a concomitant confusion of the denial of (ET) with the denial of (E), explains our incorrect attitude towards an entirely different sentence, (U). It behooves us to spell out the chain of reasoning. We begin with the correct denial of (ET), which, due to a putative scope confusion, leads to an incorrect denial of (TE). Then a chain of good logical reasoning leads us to the incorrect affirmation of (U). The connecting links are spelled out below. In the ‘Reason’ column, I cite the justification for the judgement made in that row on the basis of the judgement made in the immediately preceding row. In the ‘Correctness’ column I indicate whether the judgement made in that row is good, not whether the reasoning is good. Each step of reasoning is good, except for the scope confusion made at the outset. What’s missing, as I’ve indicated, is an explanation of why we move from step (4) to step (5). Why would we go from denying a certain ¹³ Boundary-shifting theories—whether they be contextualist ((Kamp, 1981), (Raffman, 1994, 1996), (Soames 1999) or invariantist (Fara, 2000, 2008))—are not in competition with any particular account of what it is to be a borderline case of a vague predicate, where these latter accounts include epistemicism, degree theories, supervaluationism, or truth value gap theories more generally. ¹⁴ I have argued against Fine’s elsewhere (Fara, 2000, 50–2).
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Table 21.1.
(ET) (TE) (3) (4) (5) (6)
Sentence
Judgement
Justification
Correct?
∃x TRUE: (Fx ∧ ¬Fx ) TRUE: ∃x(Fx ∧ ¬Fx ) TRUE: ¬∀x(Fx → Fx ) FALSE: ∀x(Fx → Fx ) TRUE: ∀x(Fx → Fx ) ∀x(Fx → Fx )
Deny Deny Deny Deny Affirm Affirm
Good Judgement Scope Confusion Substitution of Equivalents FALSE: ≡ TRUE: ¬ ???? True: ≡
Correct Incorrect Incorrect Incorrect Incorrect Incorrect
falsity ascription to affirming the truth of its embedded clause? For given the supervaluationist rejection of bivalence, there are two ways for a claim to fail to be false. It could be true or it could be indefinite. Two further questions are pressing. First, why is our good judgement directed at (ET) rather than (TE); why don’t we correctly affirm (TE) and then because of scope confusion incorrectly affirm (ET), leading us eventually to the correct conclusion that the inductive premise is false?¹⁵ If anything, our attitude to (TE) should be the dominant one, since of the two sentences, only it (according to the supervaluationist) is equivalent to a simple non-metalinguistic claim. Second, why do we tend toward confusion of the relative scopes of a quantifier and a truth value operator only in the case of ∃ and ‘true:’? If we ignore quantifiers other than the ones typically appearing in formal first-order languages, there are four possible scope confusions to be considered. Table 21.2 represents the only even remotely plausible position the supervaluationist could take on the chance of our making any of these scope confusions. It is telling that the only case in which we’d be at all inclined to confuse the relative scopes of a truth value operator and a quantifier is the one where the supervaluational opinion about equivalence differs from the classical opinion. Table 21.2. Equivalence? ?
∃x TRUE: x ≡ TRUE: ∃xx ?
∃x FALSE: x ≡ FALSE: ∃xx ?
∀x FALSE: x ≡ FALSE: ∀xx ?
∀x TRUE: x ≡ TRUE: ∀xx
Chance for Mistake?
Classical?
Supervaluational?
Yes
Yes
No
No
No
No
No
No
No
No
Yes
Yes
Table 21.2 displays, incidentally, that something like the reasoning in table 21.1 must be involved in our inference to the wrong conclusion in the case of (U)—if that ¹⁵ I have in mind this chain of inference: Correct affirmation of true: ∃x (x ∧ ¬x ) ⇒ Incorrect affirmation of ∃x true: (x ∧ ¬x ) ⇒ incorrect affirmation of ∃x true: ¬(x → x ) ⇒ incorrect affirmation of ∃x false: (x → x ) ⇒ correct affirmation of false: ∀x(x → x ).
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inference is to involve a scope confusion. There is no other illegitimate scope confusion of the relevant kind which we’re in danger of making. Keefe does allege that that we are in danger of confusing the relative scopes of ‘true:’ and ∃ in at least one other case. I don’t find the example convincing. She writes: [The confusion of (E) and (ET)] is thus like a confusion between saying that it is true that someone ought to do X and saying that it is true of someone that they ought to do X : the latter may be false while the former is true. We would run the two together if we thought the only way that ‘someone ought to do X ’ could be true was if there was someone, y who ought to do X . But . . . the former could hold because X being done is a right of z’s and so it ought to be done by someone, though it is no individual’s duty to do it (185).
In my assessment, we should use a deontic operator ‘ought:’ to represent the conflated sentences as follows. (OE) true: ought: ∃yXy, (EO) ∃y true: ought: Xy. There is a scope confusion all right. But it is one between ought: and ∃, not true: and ∃, and can be represented without involving truth at all: (OE*) ought: ∃yXy, (EO*) ∃y ought: Xy. Let me summarize the points made in this section. The scope-confusion explanation of our failure to recognize (U) as false fails for the following reasons: first, the scope explanation succeeds only if we’re apt to confuse non-falsity with truth, which we should not be if supervaluational semantics were correct; second, scope confusion is symmetric, but the explanation on offer requires an unexplained and unlikely asymmetry in the inferential order of our judgements; third, there is no good explanation for why we might make such a scope confusion, since we’re not at all in danger of doing so in any of the relevantly similar cases (table 21.2); while fourth, the only supposedly clear example of the confusion in question is in fact an example of an unrelated sort of scope confusion.
21.4
C O N C LU S I O N
No plausible or satisfying supervaluationist answer to the psychological question has yet been offered. I probably phrased the question somewhat badly at the outset, however. But it is the question so phrased that Keefe, and Fine before her, answered it. The problem with the phrasing is that it demands an explanation of our attitude to a false universal generalization, whereas what is really needed is an explanation of why we’re inclined to believe each instance of the generalization given that we have overwhelming evidence that not all of its instances are true. Without such an explanation, we have no answer to the question why we’re inclined to accept sorites reasoning
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when its premises do not include a universal generalization but rather, in the place of one, a series of its instances—a series of claims about adjacent pairs of minimally differing objects, e.g. ‘these are either both tall or both not’ or ‘this one is tall if that one is’. Combined with this, ideally, would be an explanation of why we’re unable to locate a shift in any kind of status along a sorites series, not only from truth to falsity, but also, e.g. from truth to truth valuelessness, from clear cases to borderline cases, or even from clear cases to cases about which nothing relevant could truly be said (perhaps, not even that). A supervaluationist might remedy the deficit by supplementing her view with some complementary contextualist or boundary-shifting answer to these questions. The considerations in Section 21.2, however, suggest that we should be leery of the prospects for ultimate success. Since supervaluationists allow for true disjunctions with only unsatisfiable disjuncts, they cannot legitimately appeal to ‘truth value shift’ in order to explain away the strangeness of the truth conditions they assign to disjunctions, conjunctions, etc. Re f e re n c e s Fara, Delia Graff (2000), ‘Shifting sands: An interest-relative theory of vagueness’, Philosophical Topics 28(1), 45–81. Published under the name ‘Delia Graff’. (2004), ‘Gap principles, penumbral consequence and infinitely higher-order vagueness’ in Jc. Beall, ed., Liars and Heaps: New Essays on Paradox, Oxford University Press, 195–221. Published under the name ‘Delia Graff’. (2008), ‘Profiling interest relativity’, Analysis 68(300), 326–35. Fine, Kit (1975), ‘Vagueness, truth and logic’, Synthese 30, 265–300. Hyde, Dominic (1997), ‘From heaps and gaps to heaps of gluts’, Mind 106(424), 641–60. James, E., Slater, J. et al., eds. (1983–), The Collected Papers of Bertrand Russell, Allen and Unwin/Unwin Hyman, London. Kamp, Hans (1975), ‘Two theories about adjectives’ in E. L. Keenan, ed., Formal Semantics of Natural Language, Cambridge University Press, Cambridge, 123–55. (1981), ‘The Paradox of the heap’ in U. M¨onnich, ed., Aspects of Philosophical Logic, D. Reidel, Dordrecht, 225–77. Keefe, Rosanna (2000), Theories of Vagueness, Cambridge University Press, Cambridge. Raffman, Diana (1994), ‘Vagueness without paradox’, Philosophical Review 103(1), 41–74. (1996), ‘Vagueness and context-relativity’, Philosophical Studies 81, 175–92. (2005), ‘Borderline cases and bivalence’, Philosophical Review 114(1), 1–31. Russell, Bertrand (1923), ‘Vagueness’, Australasian Journal of Philosophy and Psychology 1, 84–92. Page references are to reprint in E. James, J. Slater et al., eds. 1983–. Soames, Scott (1999), Understanding Truth, Oxford University Press, New York. Williamson, Timothy (1994), Vagueness, Routledge, London.
VI Paraconsistent Logics
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22 The Prospects of a Paraconsistent Response to Vagueness Dominic Hyde
Might the challenge to logic and semantics presented by the sorites paradox, and vagueness more generally, be met with a paraconsistent response? Some have thought so but the majority view has been that a paracomplete response¹—roughly, a truth value gap response—is as radical a departure from classical semantics as is necessary (and, of course, there are those, e.g. epistemicists, who see no need to depart from classical semantics at all). Paraconsistency is often thought to represent a revision of logical theory that is too radical to be defensible. For those theorists convinced that vagueness is a semantic, rather than merely epistemic, phenomenon the current best contender appears to be that paracomplete response known as supervaluationism. The logic of supervaluationism, SpV , is not only taken to be more conservative than a paraconsistent response by virtue of its paracomplete, gappy, semantics but it also is commonly said to ‘preserve classical logic’ in spite of its non-classical semantics and this too is taken to speak in its favour in respect of theory choice. The view then is that SpV is superior to any paraconsistent theory both at the level of semantics and logic. A paraconsistent response will require the abandonment of classical logic and requires a more radical departure from classical semantics. The choice is seen by many then to be clear. I have argued elsewhere, however, that the choice is far from clear.² The paraconsistent logic SbV , or subvaluationism, is no less conservative than SpV nor more so. In defence of SpV , Keefe (2000) responds by suggesting that paraconsistency is objectionable per se and, more interestingly I think, that SbV is indeed less conservative than its rival. The debate throws up a range of interesting issues but in the end, I shall argue, such issues only serve to reinforce both the conservatism of subvaluationism and the radicalism of supervaluationism. In the end both logics offer equally compelling theoretical approaches to vagueness. Each approach is, I shall argue, equally objectionable with neither providing an adequate account of vagueness but this criticism arises from a feature shared by each approach that is independent of their paracompleteness or paraconsistency per se. ¹ More exactly defined below.
² See Hyde (1997), (1999), (2001).
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For all that has been said, a paraconsistent approach and the associated recourse to truth value gluts remains a contender in accounting for vagueness.
22.1
S U PE RVA LUAT I O N I S M
Let us turn, firstly, to briefly describe salient aspects of that revision of classical twovalued logic advocated by supervaluationists in response to vagueness, the logic SpV .³ ( Those familiar with Hyde (1997) can simply skip the next section.)
22.1.1
Describing supervaluationism
The theory begins from the observation that vague predicates are such that their determinate extension and anti-extension are not exhaustive; there are objects which are in neither. And thus, assuming a full complement of names in the language, there are sentences for which the possibility arises of their being neither determinately true nor determinately false. The supervaluationist then equates determinate truth with truth simpliciter, or ‘supertruth’, thus defining a concept of truth for which bivalence fails and in terms of which we may describe borderline cases as giving rise to truth value gaps. On this view, where a is a borderline case of P, the indeterminacy of Pa amounts to its being neither true nor false.⁴ The non-bivalent logic SpV is thus an example of an incomplete logic. Let us say that a logic is complete if and only if, for any valuation or model of any contradictory pair of sentences A and ∼A, one or the other must be true in the valuation or model. i.e. A, ∼A.⁵ Classical logic is a paradigm of a complete logic. A logic will count as incomplete just in case it is not complete—i.e. for some sentence A, neither it nor its negation need be evaluated as true. i.e. A, ∼A. Non-bivalent approaches to the ³ An informal account appears to have first been proposed in Mehlberg (1958). Interestingly, Mehlberg was a former student of the Lvov–Warsaw School of philosophy, itself a well-known centre of logical innovation and the origin of the subvaluationist paraconsistent approach to vagueness proposed a decade earlier by Ja´skowski. Despite its early advocacy by Mehlberg, supervaluationism as applied to the phenomenon of vagueness is generally considered a reinterpretation of the ‘presuppositional languages’ of van Fraassen, formally described in his (1966). It is most extensively described and defended in Keefe (2000). ⁴ Note that the semantics for the precise fragment of natural language is usually taken to be classical—truth and falsity are considered exclusive and exhaustive. This assumption is not essential. A supervaluational model structure could equally well be built upon an underlying semantics that was nonclassical, e.g. intuitionist, relevant, etc. In this sense a supervaluationist approach merely aims to provide a non-bivalent semantic superstructure sensitive to vagueness which collapses to one’s preferred underlying semantics where vagueness does not arise. However, since it is traditionally a development of a non-classical semantics from a classical base, and this tradition has circumscribed the ensuing issues, problems and debate, supervaluationism as it is discussed and debated is now synonymous with this classically oriented theory—classical supervaluationism. We shall continue in this tradition and take classical supervaluationism as our object of focus, referring to it simply as supervaluationism. ⁵ ‘’ represents the generalized, multiple-conclusion consequence relation. Given a set of sentences (the multiple-premise set) and a set (the multiple-conclusion set) we shall say that if and only if whenever all the members of are true then some member of is true.
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problem of vagueness like SpV that postulate truth value gaps exemplify logics which are incomplete since, given the shared assumption that a sentence is false if and only if its negation is true, to admit sentences which are neither true nor false is to admit sentences which fail to be true whilst their negations also fail to be true. Thus: SpV A, ∼A. Now there is obviously a trivial sense in which a logic might be incomplete—namely, if whenever a sentence and its negation fail to be true every sentence fails to be true. This is not the sense of incompleteness to which truth value gap approaches in general, and SpV in particular, are committed. Such approaches countenance quarantined gaps by rejecting the spread-principle, B A, ∼A, according to which if there are truth value gaps anywhere then they are everywhere. That is, they accept that some sentence B can be true whilst not every sentence or its negation is. Gaps do not implode everywhere—the logic is non-implosive. So in addition to incompleteness there is also a commitment to the non-triviality of the incompleteness. i.e. B A, ∼A. Let us say that a logic which is non-trivially incomplete is paracomplete. The cornerstone of mainstream responses to the logical and semantic problems posed by vagueness, and SpV in particular, amounts to the view that vagueness necessitates a paracomplete response at worst. Thus: B SpV A, ∼A. Given this non-classical constraint on the semantics of vague expressions we may wonder to what extent classical logic remains intact. For instance, if a sentence A and its negation ∼A are indeterminate what of their conjunction and disjunction? Are they likewise indeterminate? Are the classical laws of excluded middle and noncontradiction still theorems? Do inferences like modus ponens or proof by cases remain valid in such semantics? What supervaluationists aim to do in developing a logic of vagueness is to admit truth value gaps whilst respecting what they describe as ‘penumbral connections’. To paraphrase Fine (1975, 269f ), suppose that a certain blob is a borderline case of ‘red’ and let S be the sentence ‘the blob is red’. Though we may agree that S is indeterminate as is its negation, ∼S, nonetheless their conjunction should count as false since they are contradictories. The boundary of the one shifts, as it were, with the boundary of the other. Similarly, since S and ∼S are complementary over the given colour range, their disjunction, S ∨ ∼S, is true. Penumbral connection is the possibility that just such logical relations hold among indeterminate sentences. The supervaluationist’s claim then is that penumbral truths must be respected (and, as a consequence, some non-truth-functional approach must be sought). They are insisting, in effect, that classical theorems, in so far as they reflect these supposed penumbral connections, must be respected despite the failure of bivalence in accommodating vagueness. It is this which makes classical supervaluationism especially interesting. Consequently, the paracomplete logic that is sought differs crucially from other paracomplete logics that have been proposed to deal with vagueness, for example Łukasiewicz’s three-valued logic or Kleene’s popular strong three-valued logic, where
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lem fails. The logic that is sought is what, following Arruda (1989), we may describe as a weakly paracomplete logic. That is to say, though it admits of non-trivial incomplete valuations which do not make true either A or ∼A (i.e. B A, ∼A), nonetheless all such non-trivial valuations make true A ∨ ∼A. (By contrast, strongly paracomplete logics do not distinguish between the non-truth of A ∨ ∼A and the non-truth of both disjuncts, and consequently such theories do not contain A ∨ ∼A yet do not implode and may contain some truths, thus being non-trivial.) Thus: B SpV A ∨ ∼A. The supervaluationist model of vagueness attempts to deliver just such a weakly paracomplete semantics. In the now familiar way, supervaluationism defines supertruth as truth in all admissible valuations, identifies supertruth with truth simpliciter, and subsequently defines SpV -consequence in the obvious way: SpV if and only if whenever all the members of are (determinately) true then some member of is (determinately) true. Such an account of consequence will indeed establish all classical (CL) theorems as theorems of SpV , as desired, since in the special case where = ∅ and is a singleton set, it is easily shown that: (I) SpV A if and only if CL A.⁶ This confirms, for example, that though the principle of bivalence is rejected (i.e. despite the failure of the logical theory to be complete) the law of excluded middle remains valid. Thus: lem: SpV A ∨ ∼ A. Where a restriction is placed on the consequence relation to the effect that be a singleton set, the multiple-conclusion consequence relation narrows to the more commonly studied single-conclusion consequence relation. This relation is coextensive with classical consequence and admits as valid all and only those inferences that are classically valid. Thus: (II) SpV A if and only if CL A. It is not hard to show then that rules like modus ponens, contraposition, conditional proof, proof by cases and reductio are all SpV -valid. Notably, as Williamson points out, all these principles fail in a language extended to include a determinacy operator, ‘D’. But even in the unextended language currently under consideration, cracks are already evident in the supposedly conservative logical veneer of SpV . Multipleconclusion consequence already deviates from its classical counterpart. As a result of being paracomplete but only weakly so, SpV fails subjunction: A ∨ B SpV A, B.⁷ ⁶ See Williamson (1994, ch. 5, §3). ⁷ Since, were subjunction to hold, substituting ∼A for B, lem would mandate completeness (i.e. one of A and ∼A would have to be true).
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This non-classical feature of SpV manifests the non-truth-functional account of disjunction required to underwrite a weakly paracomplete theory of vagueness and, as we shall see shortly, is a major source of concern when evaluating the system’s adequacy. How can it make sense to deny the truth of two unacceptable claims considered separately whilst accepting them jointly by accepting their disjunction? The anomaly is evidence of the fact that SpV ’s preservation of classical single-conclusion consequence incurs a correlative cost at the level of classical multiple-conclusion consequence more generally. Classical multiple-conclusion consequence is preserved in SpV only in the following restricted sense: (III) SpV A1 ∨ A2 ∨ . . . ∨ An
22.1.2
if and only if CL A1 , A2 , . . . , An .
Defending supervaluationism
The conservatism of the supervaluationist approach expressed through (I) and (II) above is frequently cited as a major virtue. Retaining classical logic to that extent whilst accommodating vagueness would be a significant achievement if, indeed, it is achievable.⁸ There are, however, problems with such an approach—problems attend the resolution of the sorites paradox and the retention of classical laws. Consider, firstly, how the paradox is resolved. In particular, consider the following version: A man with 1 hair on his head is bald. For any n, if a man with n hairs on his head is bald then a man with n + 1 hairs on his head is bald. ∴
A man with a million hairs on his head is bald.
The conclusion is deemed unacceptable yet the reasoning is valid by classical lights. By (II) above then the SpV theorist accepts the argument as valid, but deems it unsound. Despite the prima facie truth of the premises, supervaluationists deny the truth of the second, universally quantified conditional premise. Of course, the mere non-truth of a premise does not entail its falsity given the failure of bivalence, nonetheless for the SpV theorist the paradox in this form does indeed have a false premise. The universally quantified conditional premise is false and the following is accepted: (1) It is true that there is some n such that a man with n hairs on his head is bald whilst a man with n + 1 hairs on his head is not bald. However, the acceptability of (1) is not to be confused with another, very similar claim which in SpV is nonetheless quite distinct: namely, that there is some n for which it is true that a man with n hairs on his head is bald and a man with n + 1 hairs on his head is not bald. The truth of there being a hair-splitting n no more entails ⁸ More generally, retaining one’s preferred underlying logic—classical or otherwise—to that extent in the face of vagueness would be a virtue if achievable. (Recall that we are considering what I earlier termed classical supervaluationism, thus the default logic is assumed to be classical.)
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there being an n of which it is true that it is hair-splitting than the truth of A ∨ ∼A entails the truth of A or the truth of ∼A. The following is rejected: (2) For some n, it is true that a man with n hairs on his head is bald whilst a man with n + 1 hairs on his head is not bald. In an attempt to chart a course between the acceptance of paradox and the rejection of vagueness, it is recommended that one accept (1) whilst rejecting (2). Acceptance of (1) is considered reasonable (vagueness is not so ‘serious’ as to warrant its rejection) and does not preclude the vagueness of the relevant predicate, whereas acceptance of (2) would do so for it would amount to a rejection of the truth value shift, so one should reject (2). In response to this counterintuitive recommendation Varzi (2001) makes explicit one line of defence that remarks of Fine (1975, 285) gesture at, namely a justification by appeal to an associated metaphysical thesis taken to underlie the formal theory of SpV . Vagueness is a real enough phenomenon to warrant logical reform but is not serious enough to be ontologically grounded—it is purely representational, and more particularly in the case of vague language it is purely semantic. The Varzi thesis that the non-standard analysis of the quantifier is mandated by the purely semantic nature of vagueness deserves closer scrutiny. However, my own view is that even if the anomalous behaviour can be shown to follow from a purely semantic account, such an account should be rejected anyway. Keefe opts for a quite different response. Rather than any metaphysical defence, the choice of theory is made on pragmatic grounds. ‘I advocate the indirect argument that we should accept the phenomenon because of its role in an altogether successful theory of vagueness’ (2000, 182). Agreeing that (2) precludes the vagueness of the predicate involved, it is indeed to be rejected. Nonetheless (1) is true, thus undermining the soundness of the sorites paradox above. This acceptance of (1) is admittedly counterintuitive, however the costs are said to be offset by the overall benefits of the SpV theory—‘any costs [that accrue by virtue of the acceptance of (1)] are easily worth paying given the advantages of the theory’ (2000, 183). Attempts to mitigate the costs though are seriously flawed. Firstly, claims that the major premise of the paradox would not always be assented to, even if plausible, hardly serve to show the plausibility of the truth of its negation, i.e. the truth of (1). Of course, they would, if successful, serve to show that those who object to (1) on the grounds that the major premise is true are on shaky ground; the truth of the premise is not beyond question. But in the absence of the Principle of Bivalence one might agree with Keefe on the non-truth of the major premise and nonetheless object to its being counted false, as (1) demands. Arguably, the most compelling (and frequently cited) reason for objecting to (1) is that already discussed earlier—namely, its apparently untoward consequence in the form of (2). On this point Keefe seeks to redress the balance in favour of (1) by pointing to the fact that in SpV the untoward consequence simply doesn’t follow. ‘The rest of my defence turns on the fact that supervaluationism can distinguish between [(1) and (2)]’ (2000, 184). But this is not itself a non-question-begging defence unless the distinction, whose maintenance
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depends on the failure of the inference from (1) to (2), is itself independently plausible. And it is not, at least not without further argument, for as we have seen it requires an interpretation of existential quantification at odds with our ordinary understanding. The costs of accepting (1) are not mitigated, but rather, the attempted mitigation simply shifts the costs of accepting (1) onto the semantic analysis of quantification. Not surprisingly then, the apparent anomaly surrounding quantification is one which supervaluationists are keen to dispel. They sometimes appeal to facts about the behaviour of the quantifiers in other contexts and Keefe is no different in this respect. The change of scope involved in the inference from (1) to (2) is compared to syntactically similar scope changes involving the existential quantifier. The generally agreed unacceptability of the latter is then offered as evidence for the (required) unacceptability of the former. For example, most, if not all, people would agree that its being true that someone ought to perform action X does not entail there being someone of whom it is true that they ought to perform action X. Similarly, it seems plain that its being true that some seat has been promised to me on my booked flight to destination Y does not entail there being any seat of which it is true that it has been promised to me. In each case, the change in scope of the quantifier is deemed illicit. Obligation and promising do not distribute over ‘there is’. Similarly, claims the supervaluationist, it is to be expected that truth does not distribute over ‘there is’ and objections to the theory based on an assumption to the contrary are misguided. The claims to similarity however do not obtain in the relevant sense. To be sure, there is a syntactic similarity to all three cases considered. But this is irrelevant. The claim underlying the objection is not that truth ought to distribute over the quantifier since any operator ought to do so. Clearly some operators do not. (Examples to do with obligation and promising are cases in point.) The claim is that truth ought to distribute by virtue of what ‘there is’ means. Given our ordinary understanding of the meaning of existential quantification an existential claim is true if and only if it has some true instance. Truth and the existential quantifier interact in this way given a proper semantic understanding of the quantifier. There is no analogous reason for thinking that operators representing obligation and promising should behave similarly, moreover their not doing so reveals nothing salient concerning the meaning of ‘there is’ and so nothing salient concerning the illegitimacy of the change of scope involved in the inference from (1) to (2). We are left with no reason for rejecting the strong inclination that the inference is valid and consequently SpV is strongly counterintuitive in this regard. So much for the resolution of the paradox. Analogous problems confront the SpV account of disjunction—problems which highlight the tension between vagueness and classical laws. After all, isn’t it the case, as many seem to suppose, that the presence of vague language within the scope of logic threatens the validity of some classical laws, e.g. the law of excluded middle? Not according to SpV , and nor should it according to supervaluationists. When the objector argues that, for vague A, A ∨ ∼A fails because neither A nor ∼A—e.g. ‘Tim is tall or Tim is not tall’ fails because Tim is neither tall nor not tall—the SpV theorist responds by admitting that neither A nor its negation are (super)true, and so
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were ‘∨’ truth-functional A ∨ ∼A would fail to be (super)true but, just as with ‘∃’, truth does not distribute over ‘∨’, i.e. the connective is not truth-functional and, as noted earlier, subjunction fails. Though the SpV theorist accepts failures of bivalence, this would only entail the failure of lem were disjunction to be truth-functional, which it is not. On this view the objection, like that which arose in response to the SpV solution to the sorites above, is based on the mistaken supposition that acceptance of lem commits one to semantic precision (if everything’s red or it isn’t then ‘red’ is precise), however, one should not confuse lem with the Principle of Bivalence. One should not confuse the claim that: (3) It is true that A ∨ ∼A with the claim that: (4) It is true that A or it is true that ∼A. SpV accepts the former whilst denying the latter. The latter is denied by virtue of the vagueness of A (just as (2) was denied by virtue of the vagueness of the predicate B) yet (3) is accepted (just as (1) was).⁹ As with (1), it is the acceptance of (3) that gives rise to misgivings and leads to doubts concerning the adequacy of SpV . Unlike (1) though, where the supervaluationist simply bit the bullet and sought to minimize the damage done by undermining the supposedly untoward consequences, (3) has been defended not as a cost worth paying but, rather, as a claim that all theories should embrace. (3) is straightforwardly true and follows from the logical truth of excluded-middle claims which themselves can be seen to be mandatory despite some peoples’ intuitions to the contrary. In the case of (3), objectors are quite simply wrong. Costs attend its rejection, not its retention. This is because (3) reflects supposed ‘penumbral connections’. Of course, to defer to talk of ‘penumbral connections’ as a means of defending a commitment to (3) is only successful to the extent that the existence of penumbral connections is independently defensible. Obviously then, to defend claims for the existence of penumbral connections it would simply beg the question to cite the need to retain classical laws. That is the very issue in question here, so argument independent of classical laws must be presented. And indeed it is. Keefe cites Edgington’s arguments for the non-truth-functionality of disjunction as evidence of such ‘penumbral connections’ (though Edgington is intent on developing an alternative, non-truth-functional logic to SpV ). Edgington presents a number of arguments for non-truth-functional disjunction. Initially, ‘ ‘‘Sibling’’ means the same as ‘‘brother or sister’’. There are sex changes; and they are not instantaneous. ⁹ Just as earlier concerns centering on the existential quantifier extend to the universal quantifier, so too here claims centering on disjunction extend to conjunction in the obvious way. Despite the non-falsehood of both A and ∼A, their conjunction is nonetheless always false, thus guaranteeing the validity of the law of non-contradiction. We will return to the issue of vagueness and lnc later.
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Therefore, at times, while someone is definitely a sibling, it is indeterminate whether they are a brother, or a sister. Therefore, someone can be definitely a brother or a sister, without being definitely a brother, or definitely a sister: a disjunction can be definitely true without either disjunct being definitely true’ (1997, 310). Problems attend the first assertion though. Arguably ‘sibling’ does not in fact mean the same as ‘brother or sister’. In the absence of borderline cases they are extensionally equivalent, however, ‘sibling’ simply means ‘having (at least one of ) ones’ parents in common’ and thus does not name a simple disjunctive category, but, rather, spans the categories of ‘brother’ or ‘sister’, or anything in-between. Edgington offers a further argument for non-truth-functional disjunction, one which would generalize to a defence of lem in the context of vagueness, and which seeks to explicitly force a disjunctive reading of the key term. ‘A library book can be such that it is not clear whether it should be classified as Philosophy of Language or Philosophy of Logic; but if we have a joint category for books of either kind, it clearly belongs there. It is not unusual for a term in one language to require a disjunctive translation in another. Suppose a language trivially different from English which has one word ‘‘bleen’’, for ‘‘blue or green’’. Something can be definitely bleen, but neither definitely blue nor definitely green. Therefore, something can be definitely blue or green, while neither definitely blue, nor definitely green’ (1997, 310). The argument is similarly unsuccessful. If one means by ‘a joint category for books of either kind’ a category that includes all those in the category Philosophy of Language and all those in the category Philosophy of Logic then the book in question is not clearly in this simple disjunctive category. ( Just imagine the librarian moving books onto the new shelves purchased to house books in the new category. Any book from either of the older categories is placed there. Is it clear that the contested book should be placed there? Surely not.) Of course, if the ‘joint category’ is one which spans books from Philosophy of Language, Philosophy of Logic and all in-between then the book will clearly belong in this category but this is no longer the simple disjunctive category required for the argument to establish the existence of a definitely true disjunction with admittedly indeterminate disjuncts. More generally, terms with ‘disjunctive translations’ like ‘bleen’ do definitely apply to objects which admittedly do not definitely satisfy either of the disjuncts if the ‘disjunctive translation’ names a span that covers each of the two disjunct categories and all in-between. But where the term’s ‘disjunctive translation’ is a mere disjunction and names a simple disjunctive category, as required for the argument to succeed, it is unclear whether the term definitely applies in cases where neither of the disjuncts definitely apply. Only by equivocating on exactly what one means by a disjunctive translation can the argument succeed. Both Edgington and Keefe have failed to establish (3) and have failed, more generally, to establish the acceptability of a non-truth-functional analysis of disjunction or the acceptability of penumbral connections. Opting for a weakly paracomplete response as opposed to a strongly paracomplete one that both abandons lem, restores subjunction and endorses a truth-functional analysis of disjunction, remains a costly option to pursue from a purely pragmatic point of view.
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As with concerns above arising from the retention of classical consequence, the retention of the classical laws themselves, especially lem, in conjunction with the resulting non-truth-functional account of disjunction might again be defended by claiming that such a logic follows from a representational account of vagueness. Fine seems to respond in this way, offering representationalism as a defence. Since we cannot precisely describe the precise world we cannot in general say precisely how it is, though we can say that A ∨ ∼A is the case if we can say that the world is precise. lem is indeed counter-intuitive in the context of vagueness, but the merely semantic nature of vagueness does not impugn lem. ‘Suppose I press my hand against my eyes and ‘‘see stars’’. Then lem should hold for the sentence S = ‘‘I see many stars’’, if it is taken as a vague description of a precise experience’ (1975, 285). If vagueness is merely semantic, as the representationalist takes it to be, then lem is prescribed and so defensible. SpV is again defended by appeal to a substantive theory of vagueness. Again, as with the defence of classical consequence and the behaviour of the existential quantifier, such an appeal needs careful elaboration and subsequent scrutiny. My own view is that, even if merely semantic vagueness is capable of justifying such anomalies, such an account is itself unacceptable. The supposed virtues of the conservatism encapsulated in (I) and (II) are overplayed.
22.1.3 Assertion, denial, and logical consequence In addition to overplaying the supposed virtues of the conservatism inherent in (I) and (II), the non-conservatism of the supervaluationist approach expressed through (III) is underplayed. Reflection on the nature of logical consequence ought convince us that (III) represents SpV as departing significantly from classical, regulative, logical principles. The non-coextensiveness of CL multiple-conclusion consequence and SpV multiple-conclusion consequence reflects a difference in respect of a relation that matters. Let me explain. Supervaluationism is a theory which explicitly recognizes truth value gaps, abandoning the Principle of Bivalence in the face of incompleteness as evidenced by applications of predicates to their borderline cases. There will be a sentence A which is not true and so too its negation. Given the plausible view that we should deny what is not true it follows that supervaluationists should deny both A and ∼A. Moreover, given that one ought not assert what one also denies, and thus ought assert only truths, supervaluationists should refrain from asserting either of A and ∼A. It follows then that the denial of A must be separated from the assertion of its negation, unless supervaluationists be required both to assert ∼A (by way of denying A) and not assert ∼A (since it is not true).¹⁰ Restall (2005) presents additional arguments for their separation independent of considerations of truth value gaps, but the weaker claim that ¹⁰ To be sure, as Keefe points out (2000, 155, fn. 1), denial of A may sometimes be expressed by a locution ‘which is hard to distinguish in practice from the assertion of the negation’—‘not A’ with an emphasis on ‘not’—nonetheless they are distinct. Keefe is ambivalent on the matter, suspecting they may amount to one and the same thing and viewing any resulting incoherence on what she sees as possible conflict arising from the absence of
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supervaluationists, at least, ought to separate the two notions is all that is relevant for current purposes. The distinctness of denial and assertion means that we cannot characterize constraints on denial by simply appealing to constraints on assertion. Constraints on denial must be independently characterized along with constraints on assertion. Such constraints are captured by an adequate theory of logical consequence. As Restall (2005, 191–3) argues, ‘[i]t is common ground that logical consequence—whatever it amounts to—has some kind of grip on assertion and denial, [the speech-acts associated with] acceptance and rejection. . . . Logical notions are nothing if they have no applicability to regulate cognitive states of agents like us, and the content of such states. . . . If an agent’s cognitive state, in part, is measured in terms of those things she accepts and those she rejects, then valid arguments constrain those combinations of acceptance and rejection. . . . [A valid] one-premise, one-conclusion argument from A to B constrains acceptance/rejection by ruling out accepting A and rejecting B’. More generally, a multi-premise entailment such as A constrains acceptance and rejection in the obvious way, ruling out accepting all of while rejecting A. As Restall goes on to point out, this understanding of the role of logical consequence ‘has the advantage of symmetry’. It does not privilege acceptance over rejection, assertion over denial. Recognition of the entailment equally mandates that those accepting all of cannot, on pain of cognitive incoherence, go on to reject A and that those rejecting A cannot, on pain of incoherence, go on to accept all of . This view of consequence and its role in regulating patterns of acceptance and rejection does, however, have repercussions for our understanding of logical consequence. Suppose an agent, a supervaluationist say, rejects both A and B. Can such an agent coherently accept their disjunction, A ∨ B? That, of course, depends on whether or not the multiple-conclusion consequence relation A ∨ B A, B holds—i.e. it depends on whether subjunction is accepted. To think that it is renders incoherent the acceptance of the disjunction while rejecting each of the disjuncts. Since supervaluationists, as we have seen, do accept some disjunctions (e.g. A ∨ ∼A) while rejecting each disjunct, they must, as we have also seen, fail subjunction. The generalized consequence relation describes the logical constraint that supervaluationists are forced to reject. Now, were denial the assertion of negation and rejection the acceptance of negation then rejection of each of A and B would be equivalent to acceptance of∼A and ∼B, and the coherence of accepting A ∨ B while rejecting both disjuncts would simply be equivalent to the coherence of accepting A ∨ B, ∼A a clear-cut rule for assertion. However she fails to heed the full force of the reductio argument just presented, and treats denial of non-truths and assertion of only truths as distinct and competing rules for assertion which may conflict rather than seeing the latter rule as directly following from the former. Parsons (2000, 20), on the other hand, goes on to treat denial as assertion of an alternative ‘exclusion negation’, ¬, but this will lead to an infinite regress. Given higher-order vagueness, there will be borderline cases between A and ¬A where denial of both is appropriate, and so we should refrain from asserting either. Denial of A then will need to be distinguished from the assertion of ¬A and attempts to cast denial as the assertion of a negation will require yet another species of negation. And so on.
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and ∼B. And that is representable without recourse to a multiple-conclusion consequence relation as A ∨ B, ∼A, ∼B ∅. Given the non-equivalence of denial and asserted negation though, there is no avoiding the generalized multiple-conclusion consequence relation as the means for representing the point at issue. Whether or not inference as ordinarily understood only ever takes one from premises to a single conclusion, as some seem to think (e.g. Keefe 2000, 198), it would be foolish to think that the logical notion associated with its evaluation—a single-conclusion consequence relation—was of paramount importance. Inference matters precisely because good inference generates constraints on acceptance and rejection, and this underlying value shows the importance of single- and multipleconclusion entailments equally. Just as multiple-premise, single-conclusion consequence establishes the incoherence of rejecting the conclusion while accepting all premises, single-premise, multiple-conclusion consequence establishes the incoherence of accepting the premise while rejecting all conclusions. The ‘symmetry’ associated with the cognitive-constraint account of logical consequence coupled with the relative independence of denial and negation thus points to another symmetry between premise-sets and conclusion-sets associated with logical consequence: each may be empty, a singleton, or many-membered. This having been said, those who, like Keefe, accept the failure of subjunction in SpV as an inevitable result of the acceptance of lem have, at least implicitly, manifested a preference for one logical principle (lem) over another (subjunction). Where it is explicitly acknowledged, the preference is justified by playing down the significance of the failure of subjunction on the grounds that it is not a principle employed in ‘ordinary life’. Keefe claims, for example that ‘ . . . [subjunction] fails according to supervaluationism . . . But we do not use multiple-conclusion arguments in ordinary life and it is reasoning in vague natural language that is in question’ (2000, 198, fn. 24). Even supposing that ordinary arguments are not multiple-conclusion arguments, we are now in a position to see that this focus on an asymmetric, single-conclusion consequence relation is not justified. There is a clear sense in which we do and must use multiple-conclusion consequence in ordinary life. The presumed relative irrelevance of subjunction as compared with principles capable of being represented by a single-conclusion consequence relation is illusory. Summing up then, a paracomplete response to vagueness must choose between lem and subjunction. Supervaluationists opt for a weakly paracomplete account, retaining lem. But we have seen their arguments for that theorem unconvincing or, in the case of the representationalist defence, as resting on an as yet untried assumption as to the merely semantic nature of vagueness. The supposed virtue of recognizing ‘penumbral connections’ is not established. Now we can also see that the cost of their recognition is significant indeed and the failure of subjunction would already represent a considerable departure from classical, regulative, logical principles. Unless a representationalist theory of vagueness can be defended (and even then, contra Fine and Varzi, we might doubt it sufficient to defend SpV ) a weakly paracomplete approach, in general—and supervaluationism, in particular—looks hard, if not impossible to defend.
The Prospects of a Paraconsistent Response to Vagueness 22.2
397
S U BVA LUAT I O N I S M
Closely allied to supervaluationism is another non-classical approach to vagueness—that revision of classical two-valued logic known as subvaluationism, SbV .¹¹ Let us briefly describe its key features. ( Those familiar with Hyde [1997] can simply skip the next section.)
22.2.1
Describing subvaluationism
Subvaluationism is the paraconsistent cousin of supervaluationism and SbV admits truth value gluts where SpV admits truth value gaps. Paraconsistency is, in fact, the dual of paracompleteness. A logic is said to be consistent if and only if, for any valuation or model of any contradictory pair of sentences A and ∼A, they cannot both be true in the valuation or model. i.e. A, ∼A . Classical logic is a paradigm of a consistent logic. A logic will then count as inconsistent just in case it is not consistent—i.e. for some sentence A, both it and its negation can be true together. i.e. A, ∼A . Assuming a sentence to be false if and only if its negation is true, approaches to the problem of vagueness like SbV that, as we shall see, postulate truth value gluts exemplify logics which are inconsistent. A, ∼A SbV . As with incompleteness, however, logics might only admit inconsistency in a trivial sense, so that whenever a sentence and its negation are both true in a theory every sentence and its negation is true. Not so here—in addition to the admission of inconsistency there is also a commitment to the non-triviality of the inconsistency. i.e. A, ∼A B. As is now standard, we shall say that a logic which admits non-trivial inconsistent theories is paraconsistent. The foundation of SbV as an approach to vagueness is that vagueness necessitates a paraconsistent response. Thus: A, ∼A SbV B. A paraconsistent response has been pointed to by a range of theorists in the past.¹² The main problem with many of these suggestions that vagueness warrants a paraconsistent analysis is that while they point in a paraconsistent direction they do not explain in any detail how vagueness is to be analysed from either a formal or philosophical point of view. Ja´skowski’s discussive logic reinterpreted as a dualization of supervaluationism presents us with both a formal analysis of vagueness and a philosophical interpretation as informative as its paracomplete rival.¹³ ¹¹ The formal system was first proposed as an account of vagueness in Ja´skowski (1948). Ja´skowski, a student of the Lvov–Warsaw School of philosophy, published his account a decade before Mehlberg, a former student of the same School, proposed the now popular supervaluationist account. ¹² See Hyde (2007, §4) for a discussion. ¹³ The formal duality between SpV and SbV was originally examined in detail in Varzi (1994). The suggestion that they can be seen as dual philosophical accounts of vagueness was presented in Hyde (1997).
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To begin, subvaluational semantics treat borderline cases for a vague predicate like ‘heap’ as cases to which both the predicate and its negation applies. That is, if a is a borderline case for ‘heap’ then ‘a is a heap’ is true and ‘a is not a heap’ is true (i.e. ‘a is a heap’ is false). Where supervaluational semantics defined truth simpliciter (or supertruth) as applying to a sentence just in case that sentence was true no matter how one admissibly precisified any vague constituents of the sentence (i.e. just in case the sentence was true for all admissible precisifications) subvaluational semantics defines truth simpliciter (or subtruth) as applying to a sentence just in case that sentence is true for some admissible precisification. Whilst determinate truths are still those sentences which remain true for all admissible precisifications, determinate falsehoods are still those which are false for all admissible precisifications, and indeterminate (vague) sentences still those which are true on some but not all admissible precisifications, this third class now consists of those sentences that are both true simpliciter and false simpliciter (as opposed to neither true nor false simpliciter). Indeterminate sentences take on both truth values. Unlike paracomplete responses to vagueness where indeterminacy is analysed as underdetermination, paraconsistent responses—and SbV in particular—analyse indeterminacy as overdetermination. Determinate truth is now considered a matter of truth only and determinate falsity a matter of falsity only. Since truth and falsity are taken to be exhaustive (i.e. completeness is assumed), indeterminate sentences are now considered neither true only nor false only but, rather, both true and false. It is easy to show that for such sentences, e.g. ‘a is a heap’, both it and its negation are true. Moreover, an evaluation which ascribes both truth values to such a sentence might nonetheless ascribe just the value ‘false’ to another sentence ‘b is a heap’. The logic is clearly paraconsistent. Like supervaluationism, subvaluationism seeks to minimize logical revision necessary to accommodate vagueness and so, against a classical background, will seek to preserve all classical tautologies. Subvaluationists thus require a paraconsistent logic of vagueness yet aim, in particular, to retain the law of non-contradiction (lnc). More particularly still, not only should contradictions therefore always be false (i.e. ∼(A & ∼A)) but they should also never be true (i.e. A & ∼A )—a non-trivial distinction in the current paraconsistent context. Consequently, the paraconsistent logic that is sought differs crucially from others that have been proposed to deal with the Liar Paradox, for example. There the most plausible candidate is the logic of Priest (1979), LP, where contradictions are sometimes true. The logic that is sought here, however, is what, following Arruda (1989), we may describe as a weakly paraconsistent logic; though it admits of non-trivial inconsistent theories which contain both A and ∼A (i.e. A, ∼A B), nonetheless no such non-trivial theory includes A & ∼A. (By contrast, strongly paraconsistent logics like LP do not distinguish between the truth of A & ∼A and the truth of both conjuncts, and consequently they admit of non-trivial valuations making A & ∼A true, i.e. A & ∼A B.) Thus: A & ∼A SbV B. Defining SbV -consequence in terms of preservation of truth simpliciter, i.e. subtruth, satisfies the foregoing constraints. Thus, SbV if and only if whenever all the
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members of are true then some member of is true (i.e. whenever all the members of are true in some admissible precisification then some member of is true in some admissible precisification). Or, equivalently for current purposes, it is impossible for all the members of to be true and all the members of to be not true. Such an account of logical consequence will obviously establish all classical (CL) theorems as theorems of SbV : (I )
SbV A if and only if CL A.
So, for example, though the principle governing the exclusivity of truth-values is rejected the law of non-contradiction is preserved in both of the following senses: lnc: SbV ∼(A & ∼A), and lnc : A & ∼A SbV . Where a restriction is placed on the consequence relation to the effect that the premise-set be a singleton set, the multiple-conclusion consequence relation is coextensive with classical consequence and admits as valid all and only those inferences that are classically valid. Thus: (II )
A SbV
if and only if A CL .¹⁴
But multiple-conclusion SbV -consequence, more generally considered, deviates from its classical counterpart. As a result of being paraconsistent but only weakly so, SbV fails adjunction: A, B SbV A & B. ¹⁵ This non-classical feature of SbV manifests the non-truth-functional account of conjunction required to underwrite a weakly paraconsistent theory of vagueness and is a major source of concern when evaluating the system’s adequacy. How can it make sense to accept two claims considered separately whilst rejecting them considered jointly? Classical multiple-conclusion consequence is preserved in SbV only in the following qualified sense: (III ) A1 & A2 & . . . & An SbV
if and only if A1 , A2 , . . . An CL .¹⁶
How then might such a paraconsistent approach to vagueness resolve the sorites paradox? To answer this question, consider the standard (i.e. many-conditionals) form of the paradox. A man with 1 hair on his head is bald. If a man with 1 hair on his head is bald then a man with 2 is. If a man with 2 hairs on his head is bald then a man with 3 is. ¹⁴ Proof follows from a simple generalization of the proof of (II) in Hyde (1997, 648). ¹⁵ Since, were adjunction to hold, substituting ∼A for B, lnc would mandate consistency (i.e. not both of A and ∼A could be true). ¹⁶ For a proof see Hyde (1997, 655).
400
Dominic Hyde .. . If a man with 9,999 hairs on his head is bald then a man with 10,000 is. ∴
A man with 10,000 hairs on his head is bald.
Since SbV does not preserve classical consequence unrestrictedly, the possibility arises of the paradox being discounted by virtue of its invalidity. One diagnosis available to the SbV theorist is exactly that—the premises, including conditional premises, are all true but modus ponens is not unrestrictedly valid. Consider the sentence ‘A pile of n grains of sand is a heap’ where a pile of n grains counts as a borderline case for ‘heap’. The sentence is true and false, so it is true. Since it is also false then the material conditional ‘If a pile of n grains of sand is a heap then a pile of n − 1 grains is a heap’ is true by virtue of the falsity of its antecedent. Nonetheless, a pile of n − 1 grains of sand might be determinately not a heap thus making the sentence ‘A pile of n − 1 grains of sand is a heap’ false.¹⁷ So: Heap(n), Heap(n) → Heap(n − 1) SbV Heap(n − 1), for some n. As is familiar from other paraconsistent logics then, modus ponens for material implication, i.e. disjunctive syllogism, is not valid in SbV . (Notice that what is claimed is that modus ponens is here denied for material implication—the implication relation typically assumed in modelling the sorites conditional. Beall and Colyvan (2001) are, of course, right to point out that this analysis of the sorites may hide the problem rather than solve it, since it assumes what many consider a very weak reading of conditionality. Stronger conditionals are definable and alternate responses are then available to the SbV theorist. Entailment is a much stronger conditional for which modus ponens clearly holds, but it does not provide an interpretation of the sorites premises which renders them true. A mid-strength connective, ‘ →’, just strong enough to satisfy modus ponens can be explicitly defined in SbV but it is easily shown that this does not provide an interpretation of the sorites-conditionals which renders all the premises true.¹⁸ There is, of course, an analogous problem lurking here for the supervaluationist who can equally be charged with assuming an overly strong reading of the conditional which is sufficiently strong to validate modus ponens but not weak enough to provide an interpretation of the sorites premises which renders them true.)¹⁹ ¹⁷ Higher-order vagueness may complicate matters here, but even the simplified approach will subsequently be found untenable. The simplification is thus harmless. Notice that the conditional premise will be counted both true and false in SbV under the conditions described, whereas SpV would count it neither true nor false. ¹⁸ Define ‘A → B’ as follows: A → B is true simpliciter iff either A is false in every admissible precisification or B. The aforementioned counterexample to modus ponens now renders the corresponding (ponendable) conditional ‘Heap(n) → Heap(n − 1)’ false. The newly defined connective is, in fact, just Ja´skowski’s discussive implication, ‘→D ’ suitably reinterpreted, which he explicitly introduced to recapture a conditional satisfying modus ponens. For further discussion of discussive implication see Priest and Routley (1989, 158f ). ¹⁹ See Hyde (2001).
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The foregoing simple paraconsistent reinterpretation of supervaluational semantics reproduces exactly the first formal system of paraconsistent logic—discussive logic—developed by Ja´skowski over fifty years ago, which already at the time was claimed to be applicable to vagueness. Ja´skowski’s long-standing proposal to treat vagueness from a paraconsistent perspective by means of a discussive logic is simply reinterpreted so as to be the dual of the dominant paracomplete supervaluationist approach. More exactly, where and are sets of sentences of the shared language of SpV and SbV and ∼ = df {∼A : for all A ∈ }: SpV
if and only if ∼ SbV ∼.²⁰
22.2.2 Defending subvaluationism Given the duality between supervaluationism and subvaluationism, it is unsurprising that subvaluationism faces objections that are the exact dual of those pressing against a supervaluationist account. Problems arise for SbV as regards both its resolution of the sorites paradox and its retention of classical laws. Before turning to these matters though, let us deal immediately with the very general objection that SbV must be inadequate merely by virtue of its very paraconsistency. Keefe claims that ‘many philosophers would soon discount the paraconsistent option (almost) regardless of how successfully it treats vagueness, on the grounds of the unappealing commitments and features of the logical framework as a whole, in particular the absurdity of p and ∼p both being true for many instances of p’ (2000, 197). As a sociological observation, this is quite possibly true. Many discount paraconsistency as an option on the grounds that it simply must be wrong. But the assumed absurdity does not obtain merely by virtue of its being presumed to obtain, and the arguments offered are not conclusive either.²¹ Paraconsistency per se has not been shown to be absurd any more than paracompleteness has. Both options are available for considered application. More particular concerns centre on subvaluationism itself. Like SpV , the nontruth-functionality of the subvaluationist response weighs heavily against it. More particularly, the failure of adjunction is a major concern. The feature is well known in discussive logic and its non-adjunctive nature has often been remarked upon and is frequently considered a major obstacle to the plausibility of this paraconsistent approach. (See, for example, Priest and Routley 1989, 158.) The implausibility transfers immediately to SbV . The failure of conjunction to satisfy this most basic of rules counts against its interpretation as a natural language conjunction. False conjunctions with no false conjunct are counterintuitive indeed. The most obvious SbV counterexample to adjunction is that which establishes it as a weakly paraconsistent system: A, ∼A SbV A & ∼A. ²⁰ For a proof see Hyde (1997, 656). ²¹ Those interested may wish to look at the debate in Sainsbury (1995), and Beall, Priest, and Armour-Garb (2004).
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Where A is vague, counterexamples to adjunction readily accrue. Keefe (2000, 198) takes this particular departure from classically acceptable reasoning to be a major obstacle to the acceptance of SbV , as indeed it is. Such a weakly paraconsistent account of logical consequence seems difficult, if not impossible, to defend. But we should be careful here, as elsewhere, to draw the appropriate lessons from such an anomaly. It is, in Keefe’s view, testimony to the inadequacy of a paraconsistent response, a weakly paracomplete one (supervaluationism) being acceptable. SbV must reject an instance of an acceptable multi-premise, single-conclusion consequence relation (adjunction), whereas SpV is only required to reject an instance of an acceptable single-premise, multi-conclusion consequence relation (subjunction). The former involves the rejection of an acceptable ‘ordinary’ inference properly counted as part of the provenance of ‘traditional’ logic, whereas the latter involves the rejection of a principle not part of ‘ordinary’ inference and beyond the scope of ‘traditional’ logic. Evaluating the two logical theories then as regards their ability to account for ‘ordinary’ inference and ‘traditional’ logic, SpV is supposedly superior. But we have already seen that claims for the relative irrelevance of subjunction are misplaced given a proper understanding of assertion and denial in the context of gaps and gluts. The proper lesson to be drawn from the objectionable failure of adjunction is that a weakly paraconsistent account of vagueness should be abandoned by virtue of its being weakly paraconsistent, just as the proper lesson to draw from the objectionable failure of subjunction is that a weakly paracomplete account of vagueness should be abandoned by virtue of its being weakly paracomplete. There is no relative difference here between the two approaches. Pressing the failure of adjunction more strongly, Keefe (2000, 200) also points to the related fact that SbV is forced to differentiate between seemingly equivalent forms of paradox. Although the standard sorites consisting of a categorical premise and many conditional premises is declared invalid, the closely related form where all premises are conjoined is valid but has a now false premise, and so too for the mathematical induction form of the paradox. This ‘unappealing lack of uniformity in locating blame results in denying most intuitions associated with the sorites argument: it is not valid, at least in some forms, one of the premises is not true, in other forms, and different ways of stating what is apparently the same argument are actually stating crucially different arguments’. Of course, the same is true of Keefe’s preferred SpV responses to the paradox in its many forms and no comparative disadvantage is manifested by the observation in relation to SbV . Supervaluationists similarly deny ‘most’ of our intuitions associated with the sorites argument: it is sound, at least in some forms (e.g. the line-drawing form);²² one of the premises is not true, in other forms (e.g. the standard sorites); and different ways of stating what is apparently the same argument are actually stating crucially different arguments (e.g. the standard sorites with no false premise and the ²² This form of the paradox derives the existence of a sharp cut-off point to the application of the relevant vague predicate F along an ordered sorites series, from the existence of an initial satisfier of F and final satisfier of ¬F .
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seemingly equivalent form with conjoined premises which does indeed have a false premise, the resulting conjunction). What the objection properly points to in relation to both a weakly paracomplete approach and a weakly paraconsistent approach is the fact that they are equally counterintuitive. Semantic anomalies already discussed in relation to supervaluationism find their analogue in subvaluational semantics. Naturally, defences can be mounted by analogy with supervaluationism. To wit, vagueness demands a modification of classical semantics, namely the admission of the truth of contradictory pairs of sentences yet, it might be contended, ‘penumbral connections’ must nonetheless be respected by the logic and thus contradictions themselves must always be false. However, the arguments for penumbral connection are as unconvincing in present circumstances as they were previously when considering the supervaluationist response to vagueness. Just as no compelling reason has been given for thinking lem should hold in the face of truth value gaps, so too arguments for lnc and, more particularly, lnc are unlikely to succeed in the face of truth value gluts. Opting for a weakly paraconsistent response as opposed to a strongly paraconsistent one that both abandons lnc , restores subjunction and endorses a truthfunctional analysis of disjunction, remains a costly option to pursue from a purely pragmatic point of view. As with supervaluationism though, lnc might again be defended by claiming that it follows from a representational account of vagueness. Adapting Fine’s response to the retention of lem in SpV , a representational view of vagueness might be appealed to by way of defence. Since we cannot precisely describe the precise world we cannot in general say precisely how it is, though we can say that A & ∼A is not the case if we can say that the world is precise. To be sure, lnc is counter-intuitive in the context of a paraconsistent approach to vagueness, yet the merely semantic nature of vagueness does not impugn lnc . Again, though I shall not argue for it here, this defence, even were it thought plausible assuming representationalism about vagueness, fails to the extent that arguments for representationalism fail. In addition to its being paraconsistent and non-adjunctive, a further concern might be expressed about the resolution of the sorites paradox described above. The failure of modus ponens might be objected to. But given that the failure is a direct result of the failure of disjunctive syllogism in paraconsistent circumstances, the objection amounts simply to scepticism concerning a paraconsistent approach and is no additional objection. Moreover, the subvaluationist can point to the fact that some intuition has to give way in resolving the problems attending the phenomenon of vagueness, and the failure of ‘→’ to satisfy modus ponens (i.e. the failure of disjunctive syllogism) is no more objectionable than the failure of ‘∃’ to satisfy standard semantic clauses (as required by SpV ), or the failure of conditional proof (as required by SpV when the language is extended to include a truth predicate or determinacy operator), or the rejection of the law of excluded middle (as required by strongly paracomplete approaches). Logical innovation appears inevitable. That modus ponens should be excluded from possible revision seems unprincipled.
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Dominic Hyde 22.3
C O N C LU S I O N
I have argued that super- and subvaluationism offer distinct, equally compelling, but equally inadequate responses to the phenomenon of vagueness. While non-classical semantics are, I think, required to deal with the phenomenon, the responses considered, in seeking to retain classical theoremhood, are thereby committed to weak paracomplete and paraconsistent responses respectively. The costs are high, as we have seen. Moreover, the costs seem equally weighted against each and do not speak against paracompleteness or paraconsistency per se. The lesson is that we should look to strongly paracomplete and strongly paraconsistent systems for an acceptable logic of vagueness. For all that has been said so far, the prospects of a paraconsistent response to vagueness are as good as those for a paracomplete response. With all this said, one might well wonder why it is then that paracomplete responses apparently enjoy such strong support, while paraconsistent responses do not. Keefe’s remarks (cited earlier) are suggestive of an underlying cause: many philosophers have discounted paraconsistency tout court on the grounds that it simply must be wrong. However, Sainsbury (1995, ix) is more frank: ‘To my regret . . . I do not accept it but cannot refute it.’ The same, I think, holds true in respect of paraconsistent vagueness more particularly. Re f e re n c e s Arruda, A. (1989), ‘Aspects of the historical development of paraconsistent logic’ in Priest, Routley, and Norman (1989), 99–130. Beall, Jc, and Colyvan, M. (2001), ‘Heaps of gluts and hyde-ing the sorites’, Mind 110, 401–8. Beall, Jc, Priest, G., and Armour-Garb, B. (2004), The Law of Non-Contradiction, Oxford University Press. Burgess, J. and Humberstone, L. (1987), ‘Natural deduction rules for a logic of vagueness’, Erkenntnis 27, 197–229. Edgington, D. (1997), ‘Vagueness by degrees’ in Keefe and Smith (1997), 294–316. Fine, K. (1975), ‘Vagueness, truth and logic’, Synthese 30, 265–300. Hyde, D. (1997), ‘From heaps and gaps to heaps of gluts’, Mind 106, 641–60. (1999), ‘Pleading classicism’, Mind 108, 733–5. (2001), ‘Reply to Beall and Colyvan’, Mind 110, 409–11. (2007), ‘Logics of vagueness’ in D. Gabbay and J. Woods, eds., Handbook of the History of Logic Vol. 8, North-Holland Press, 285–324. Ja´skowski, S. (1969) [1948], ‘Propositional calculus for contradictory deductive systems’, Studia Logica 24, 143–57. Originally published in 1948 in Polish in Studia Scientarium Torunensis, Sec. A II, 55–77. Keefe, R. (2000), Theories of Vagueness, Cambridge University Press. Keefe, R. and Smith, P. (1997), Vagueness: A Reader, MIT Press. Mehlberg, H. (1958), The Reach of Science, Toronto University Press. Parts reprinted in Keefe and Smith (1997), 85–8. Parsons, T. (2000), Indeterminate Identity, Oxford University Press. Priest, G. (1987), In Contradiction: A study of the Transconsistent, Martinus Nijhoff. Second edition, 2006, Oxford University Press.
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Priest, G. and Routley, R. (1989), ‘Systems of paraconsistent logic’ in Priest, Routley, and Norman (1989), 151–86. Priest, G., Routley, R., and Norman, J., eds. (1989), Paraconsistent Logic: Essays on the Inconsistent, Philosophia Verlag. Restall, G. (2005), ‘Multiple conclusions’ in Logic, Methodology and Philosophy of Science: Proceedings of the Twelfth International Congress, ed. P. Hajek, L. Valdes-Villanueva, and D. Westerstahl, King’s College Publications, 189–205. Sainsbury R. M. (1995), Paradoxes, Cambridge University Press (2nd edn.). Van Fraassen, B. C. (1966), ‘Singular terms, truth value gaps, and free logic’, Journal of Philosophy 63, 481–5. Varzi, A. (1994), Universal Semantics, PhD thesis, University of Toronto. Published as An Essay in Universal Semantics, Kluwer, 1999. (2001), ‘Vagueness, logic and ontology’, The Dialogue 1, 135–54. Williamson, T. (1994), Vagueness, Routledge.
23 Non-Transitive Identity Graham Priest
23.1
P RO B L E M AT I Z I N G I D E N T I T Y
The notion of identity has always been a problematic notion, especially when considerations of intentionality and change are around.¹ And though there is now a standard theory of identity—identity in ‘classical’ first-order logic—this can appear as unproblematic as it does only because it is normally presented in a way that is sanitized by the disregarding of such considerations. For example, suppose I change the exhaust pipes on my bike; is it or is it not the same bike as before? It is, as the traffic registration department and the insurance company will testify; but it is not, since it is manifestly different in appearance, sound, and acceleration. Dialecticians, such as Hegel, have delighted in such considerations, since they appear to show that the bike both is and is not the same.² A standard reply here is to distinguish between the bike itself and its properties. After the change of exhaust pipes the bike is numerically the same bike; it is just that some of its properties are different. Perhaps, for the case at hand, this is the right thing to say. But the categorical distinction between the thing itself and its properties is one which is difficult to sustain; to suppose that the bike is something over and above all of its properties is simply to make it a mysterious ding an sich. Thus, suppose that I change, not just the exhaust pipes, but, in succeeding weeks, the handle bars, wheels, engine, and in fact all the parts, until nothing of the original is left. It is now a numerically different bike, as even the traffic office and the insurance company will concur. At some stage, it has changed into a different bike, i.e. it has become a different machine: the bike itself is numerically different. ( This is a variation on the old problem of the ship of Theseus.) A version of this chapter was given at the third World Conference on Paraconsistency, Toulouse, 2003. Versions have also been given at the Universities of Melbourne and St Andrews. I am grateful to the audiences on those occasions for comments and helpful suggestions. ¹ In this chapter I shall concentrate on issues concerning change, and shall have nothing to say concerning intentionality. A discussion of identity in intentional contexts can be found in Priest (2002a) and ch. 2 of Priest (2005). ² See, e.g. Miller (1969), 413 ff.
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Non-Transitive Identity
407
True sentences of the form a = a and a = a are standard fare in paraconsistent theories of identity;³ but there is more to the matter than this. What is it for an object to be the same object over a period of time in which change occurs? The answer is, plausibly, different for different kinds of objects; for many kinds of objects, the answer is also likely to be contentious. But it is not uncommon to appeal to some kind of continuity condition. Thus, for example, Locke took personal identity to be given by continuity of memory.⁴ I am the same person that I was yesterday since I can recall most of what I could recall then, and some more as well. But continuity conditions of this kind are naturally non-transitive. Memories can be lost in trauma, or even in the simple process of ageing. There can therefore be objects, say people, a, b and c, such that there is sufficient continuity between a and b, and between b and c, but not between a and c. Thus, we have a = b and b = c, but not a = c. Identity fails to be transitive. Cases of fission and fusion can also give rise to similar problems. Suppose that between t0 and t1 , an amoeba, a, divides into two new amoebas, b and c; at t1 , b occupies location lb , and c occupies a distinct location lc . We may depict the situation as t1
t0
lb
lc
b
c
a
shown in the above diagram. At least arguably, a = b. (If c were to die on fission, this would be clear; and how can the identity of two things depend on what else exists?). Similarly, a = c. But it is not the case that b = c. Moreover, at t1 , b—that is, a—is at lb ; but c is not, even though a = c . We have a failure of the substitutivity of identicals, where the property in question has nothing to do with identity.⁵ There is, of course, much more to be said about all of these examples. But the discussion at least shows that various properties standardly taken to be possessed by identity (consistency, transitivity, substitutivity) are not to be taken for granted philosophically. One can, of course, simply specify by fiat that identity has these properties. But this is hardly satisfactory. The notion so produced will then certainly have those properties—and call it identity if you like; but it is all too obvious that the behaviour of the relationship involved in the above examples—and which we used to call identity before the word was usurped—still cries out to be understood. In what follows, I will provide a theory of a relationship that is naturally enough thought of as identity, but for which the properties that we have just seen to be problematic fail, though in a controlled and recoverable way. In the next few sections ³ See, e.g. Priest (1987), 5.3. ⁴ For references and discussion, see Parfitt (1984), 204 ff. ⁵ The example comes from Prior (1968), 83. See also Priest (1995).
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we will look at a formal specification of the relation. We will then return to the above examples.⁶
23.2
S E C O N D - O R D E R LP
The theory in question is based on a paraconsistent logic, LP.⁷ For reasons that will become obvious, we will work with the second-order version of this, though there are other ways to proceed, as we shall see in due course. Let us start, then, with a specification of the logic.⁸ The language has the connectives ∧, ∨ and ¬, and the first- and second-order quantifiers ∀ and ∃. The material conditional and biconditional are defined in the usual way: α ⊃ β is ¬α ∨ β; α ≡ β is (α ⊃ β) ∧ (β ⊃ α). There are predicates and function symbols, but we will suppose, for the sake of simplicity, that they are all monadic. First-order variables are lower case, and monadic second-order variables are upper case. I will avoid free variables. There are various forms that the semantics of second-order LP may take; importantly, there are various possible ranges for the second-order variables. I will choose one appropriate way here. An interpretation for the language, I , is a triple D1 , D2 , θ . D1 is the non-empty domain of first-order quantification. D2 is the nonempty domain of second-order quantifiers, and is a set of pairs of the form A+ , A− , where A+ ∪ A− = D1 . I will call A+ an extension, and A− a co-extension. We require that for every A ⊆ D1 , there is a B ⊆ D1 such that A, B ∈ D2 , but otherwise make no assumptions about how extensive D2 is.⁹ θ assigns every individual constant a member of D1 , every predicate constant a member of D2 , and every function symbol+ a (monadic) function from D1 to D1 . If P is a predicate, I will write θ (P) as θ (P), θ − (P) . θ can be extended to assign every closed term a denotation by the familiar recursive clause: θ (ft) = θ (f )(θ (t)). An evaluation, ν, is a function that maps each formula to {1} (true only), {0} (false only), and {1, 0} (both true and false), according to the following recursive clauses: ⁶ There are certainly other non-classical theories of identity to be found in the literature, even ones based on a paraconsistent logic. Thus, e.g. in Krause (1992) and Bueno (2000) there is to be found a theory in which substitutivity of ideniticals fails. The notion of identity of these papers is still an equivalence relation, however. In particular, identity is transitive. This makes the notion very different from that to be given here, and unsuitable for the major applications at issue. ⁷ See, e.g. Priest (1987), ch. 5. ⁸ For second-order LP, see section 7.2 of Priest (2002b). ⁹ In particular, we do not assume that every pair of the form A, B , where A ∪ B = D1 , is in D2 . This fact is, in itself, sufficient to give failure of substitutivity for molecular formulas. One might suggest that the only pairs that are in D2 are those which represent special properties of some kind, such as natural or intrinsic properties. Depending on how one interprets the notion, it may be natural to add extra closure conditions on D1 , such as closure under negation: A, B ∈ D2 ⇒ B, A ∈ D2 .
Non-Transitive Identity
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1 ∈ ν(Pt) iff θ (t) ∈ θ + (P) 0 ∈ ν(Pt) iff θ (t) ∈ θ − (P) 1 ∈ ν(¬α) iff 0 ∈ ν(α) 0 ∈ ν(¬α) iff 1 ∈ ν(α) 1 ∈ ν(α ∧ β) iff 1 ∈ ν(α) and 1 ∈ ν(β) 0 ∈ ν(α ∧ β) iff 0 ∈ ν(α) or 0 ∈ ν(β) 1 ∈ ν(α ∨ β) iff 1 ∈ ν(α) or 1 ∈ ν(β) 0 ∈ ν(α ∨ β) iff 0 ∈ ν(α) and 0 ∈ ν(β) To give the truth and falsity conditions for the quantifiers, we assume, for the sake of simplicity, that the language is expanded if necessary to give each member of D1 and D2 a name. If d ∈ D1 , I write its name as d; and if A ∈ D2 , I will write its name as A. The conditions may now be stated as follows. 1 ∈ ν(∃xα(x)) iff for some d ∈ D1 , 1 ∈ ν(α(d)) 0 ∈ ν(∃xα(x)) iff for all d ∈ D1 , 0 ∈ ν(α(d)) 1 ∈ ν(∀xα(x)) iff for all d ∈ D1 , 1 ∈ ν(α(d)) 0 ∈ ν(∀xα(x)) iff for some d ∈ D1 , 0 ∈ ν(α(d)) 1 ∈ ν(∃X α(X )) iff for some A ∈ D2 , 1 ∈ ν(α(A)) 0 ∈ ν(∃X α(X )) iff for all A ∈ D2 , 0 ∈ ν(α(A)) 1 ∈ ν(∀X α(X )) iff for all A ∈ D2 , 1 ∈ ν(α(A)) 0 ∈ ν(∀X α(X )) iff for some A ∈ D2 , 0 ∈ ν(α(A)) Finally, validity: I is a model of α iff 1 ∈ ν(α); if is a set of formulas, I is a model of iff it is a model of every member; and α iff every model of is a model of α. The first-order part of LP in the above semantics is entirely standard. The secondorder part is a natural extrapolation. I merely pause, therefore, to note a few of the properties of the material biconditional that will feature in what follows. In particular, it is easy to check the following. (I omit set braces in the premises.) α≡α α≡ββ≡α α, β α ≡ β ¬α, ¬β α ≡ β α, ¬β ¬(α ≡ β) β, ¬β α ≡ β α ≡ β ¬α ≡ ¬β α ≡ β, β ≡ γ α ≡ γ
(Make β both true and false.)
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DEFINING IDENTITY
With this background, we can now come to identity. Taking its cue from Leibniz’s Law, identity may be defined in second-order logic in the standard fashion. Thus, let us define t1 = t2 as: Def= : ∀X (Xt1 ≡ Xt2 ) Because the material biconditional is reflexive and symmetric, it follows that identity is too: t = t and t1 = t2 t2 = t1 . The material biconditional is not, however, transitive; identity inherits this property. Thus, consider the interpretation, I, where: • • • •
D1 = {a1 , a2 , a3 } θ (ti ) = ai (i = 1, 2, 3) {a1 , a2 }, {a2 , a3 } = A ∈ D2 For every other B ∈ D2 , B− = D1
Since At2 ∧ ¬At2 is true, so is At1 ≡ At2 ; and for every other B ∈ D2 , ¬Bt1 ∧ ¬Bt2 is true, so Bt1 ≡ Bt2 . Hence, ∀X (Xt1 ≡ Xt2 ), that is t1 = t2 is true. Similarly, t2 = t3 . But At1 ≡ At3 is not true; hence, neither is ∀X (Xt1 ≡ Xt3 ), that is, t1 = t3 is not true. Thus, t1 = t2 , t2 = t3 t1 = t3 . Since transitivity of identity is a special case of substitutivity of identicals, this, too, fails. For another counter-example, note that in I, both t2 = t3 and At2 are true, but At3 is not. Finally, note that identity statements may not be consistent. Thus, in I, since At2 ∧ ¬At2 is true, so is ¬(At2 ≡ At2 ). It follows that ∃X ¬(Xt2 ≡ Xt2 ), so ¬∀X (Xt2 ≡ Xt2 ), i.e. t2 = t2 .¹⁰ It might be objected that the account of identity just given is inadequate since what is required in Def= is not a material biconditional, but a genuine (and detachable) conditional, such as the conditional of an appropriate relevant logic. We would then have transitivity and substitutivity of identity (though maybe not consistency). However, this would be too fast. It is not at all clear that what is required for an expression of Leibniz’s Law is a genuine conditional. For example it is not clear that there is a relevant implication between, e.g. ‘Mary Ann Evans was a woman’ and ‘George Elliot was a woman’—at least, not without the suppressed information that Mary Ann Evans was George Elliot. What is required for Leibniz’s Law is that for every predicate, P, Pt1 and Pt2 have the same truth value; and this is what the material biconditional delivers. It might still be objected that this is not the case in LP, since α ≡ β is true (and false) if α is true only but β is both true and false. But again, this is too fast. Though the semantics are formulated formally as three-valued, there are, in fact, really only two truth values, true and false. It is just that sentences may have various combinations of these.¹¹ In particular, α ≡ β is true iff α and β are both true, or both false. It is ¹⁰ It is perhaps worth observing that if we drop the condition on interpretations that for all A ∈ D2 , A+ ∪ A− = D1 , and so base the theory of identity on FDE, then the Law of Identity, t = t, also fails. If we insist that A+ ∩ A− = φ, and so base the theory on K3 , the Law still fails, but transitivity and substitutivity hold. ¹¹ This comes out most clearly in the relational semantics for the logic. See Priest (2001), ch. 7.
Non-Transitive Identity
411
easy enough to check that α ≡ β is logically equivalent to (α ∧ β) ∨ (¬α ∧ ¬β). If α is true only and β is both true and false, both are true, hence one should expect the material biconditional to be true—and since one is true and the other is false, one should expect it to be false as well.
23.4
IDENTITY AND CONSISTENCY
Call an interpretation classical iff for every A ∈ D2 , A+ ∩ A− = φ. The classical interpretations are simply those where no atomic sentence—and hence no sentence at all—behaves inconsistently. The classical interpretations are, in fact, just the interpretations of classical second-order logic. And, restricted to those, the definition of identity just employed gives the classical account of identity. Thus, though some of the features of the classical account fail, they do hold when we restrict ourselves to classical models. Provided that we are reasoning about consistent situations, then, identity may be taken to behave in the orthodox fashion. I have argued elsewhere¹² that consistency should be taken as a default assumption. If this is right then the classical properties of identity may be invoked unless and until that default assumption is revoked. The idea may be turned into a formal non-monotonic logic, minimally inconsistent LP. The details for the first-order case are given in Priest (1991). How best to modify the idea so that it works in the second-order case, and so for identity, is not obvious. Here is one way. (I do not claim that it is the best.) If I is an interpretation, let I ! = {d ∈ D1 : ∃A ∈ D2 , d ∈ A+ ∩ A− }. I ! is the set of elements in D1 that behave inconsistently. If I1 and I2 are interpretations, define I1 ≺ I2 (I1 is more consistent than I2 ) to mean that I1 ! I2 !. I is a minimally inconsistent (mi) model of iff I is a model of and there is no J ≺ I such that J is a model of . Finally, minimally inconsistent consequence can be defined thus: m α iff every mi model of is a model of α If is classically consistent, its mi models are its classical models. Hence, its mi consequences are simply its classical consequences. In particular, since {t1 = t2 , t2 = t3 } is consistent, t1 = t2 , t2 = t3 m t1 = t3 . Similarly, t1 = t2 m α(t1 ) ≡ α(t2 ). More generally, m is a consequence relation where irrelevant inconsistencies do not prevent classical inferences from being employed. Thus: t1 = t2 , Pt1 , Qt2 ∧ ¬Qt2 m Pt2 . For if I is a mi model of the premises, θ (t2 ) must behave inconsistently, since θ (t2 ) ∈ θ + (Q) ∩ θ − (Q). But nothing forces θ (t1 ) to behave inconsistently, so θ (t1 ) ∈ θ + (P) and θ (t1 ) ∈ / θ − (P). But ∀X (Xt1 ≡ Xt2 ) is true, so Pt1 ≡ Pt2 . Since the left hand side of this is true only, the right hand side must be at least true. Hence, Pt2 is true. The relation is non-monotonic, however. In particular, if we add ¬Pt1 as an extra premise, the left hand side is now both true and false, and the right hand side may simply be false. ¹² See Priest (1987), 8.4.
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In closing this part of the discussion, it is perhaps worth pointing out the following. It is not uncommon for logicians and philosophers to distinguish a class of predicates for which the substitutivity of identity holds and ones for which it fails. Extensional predicates are usually taken to be among the former; intentional predicates among the latter. For the notion of identity at hand, substitutivity may fail for all sorts of predicates, even extensional ones. What determines whether substitutivity holds is not the kind of predicate in question, but simply the consistency of the situation.¹³ 23.5
S O M E A P P L I C AT I O N S
So much for the theory. Let us now turn to some philosophical applications, including the topics in section 23.1. Example 1 Let us start with an object that changes its properties. Consider some object, a; and suppose, for the sake of illustration, that its properties at some time are consistent. Let P be one of these properties. Suppose that at some later time it comes to acquire, in addition, the property ¬P, all other properties remaining constant. Call the object that results b. Then even after this time, Qa ≡ Qb for every Q. (Recall that Pa ≡ Pb ¬Pa ≡ ¬Pb.) Hence, ∀X (Xa ≡ Xb), that is, a = b. But since Pa and ¬Pb, ¬(Pa ≡ Pb); thus ¬∀X (Xa ≡ Xb). So a = b. Thus, a and b are both identical with each other and distinct from each other. Example 2 Now extend the example. Suppose that at a subsequent time again the object loses the property P, maintaining the property ¬P. Call the object that results c. Again, all other properties remain constant. Then, as before, a = b; similarly, b = c. But a has a property that c lacks. Hence, it is not the case that a = c. Transitivity has failed. Example 3 Next, consider the amoeba-fission case. Let B be the predicate ‘occupies lb at t1 ’; similarly for C. Take it that—consistently—Bb and ¬Cb; and that, similarly, ¬Bc and Cc. Take it also that Ba, ¬Ba, Ca, ¬Ca. Again, assume that these are the only relevant properties. Then a = b and a = c, but it is not the case that b = c; moreover, Ba and a = c, but we do not have Bc. Example 4 Finally, let us turn to the motor-bike of Theseus. Let us suppose that the bike goes through seven stages, at times t0 , ..., t6 . Let the motorbike at time ti be ai (0 ≤ i ≤ 6). Consider the predicate ‘is identical with a0 ’. Arguably, this is a vague predicate. a0 satisfies it; a6 does not; and somehow its applicability fades out in between. In a sorites progression of the kind produced by vague predicates, it is common enough to point out that there are borderline cases, and claim that these are cases of truth value gaps. But intuition is satisfied just as well by the thought that these are truth value gluts.¹⁴ Symmetry, after all, is what seems to be required. If we ¹³ For this reason, the construction will not deal with prima facie counter-example to substitutivity involving sentences such as ‘Clarke Kent entered the phone box and Superman came out’ (considered in Saul (2007)). Being in the phone box is (presumably) quite consistent. ¹⁴ See Hyde (1997).
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take the borderline cases to be gluts, we may expect the predicate to behave as follows. The predicate ‘is identical with a6 ’ behaves inversely, and is also shown. a0 = a0 a0 = a1 a0 = a2 a0 = a3 a0 ≠ a3
a0 ≠ a4
a0 ≠ a5
a0 ≠ a6
a4 = a6
a5 = a6
a6 = a6
a0 ≠ a6 a1 ≠ a6 a2 ≠ a6 a3 ≠ a6 a3 = a6
The bike undergoes various modifications, but it retains its identity as a0 until t4 , by which time it has already become (at t3 ) distinct from a0 , and identical with a6 . We also have a failure of transitivity. a0 = a3 , a3 = a6 ; but we do not have a0 = a6 . More generally, we would expect to have a0 = a1 , a1 = a2 , ..., a5 = a6 ; the failure of transitivity of identity stops us from chaining these together to obtain a0 = a6 .¹⁵ The Lockean example of personal identity, note, can be thought of as similar. Two persons are the same if they have a sufficient psychological continuity. But ‘sufficient psychological continuity’ is a vague predicate. So one should expect personal identity to be vague in just the required way. 23.6
VAG U E N E S S
Of course, there is a lot more to be said about sorites transitions. Vague predicates appear to be no more three-valued than two-valued. What is puzzling about sorites sequences is that there appear to be no semantically significant cut-off points at all. Thus, suppose that a0 , ..., a6 is a sequence of objects in transition from being red to not being red. Then if we treat borderline cases as semantic gluts, the associated truth a0
a1
a2
a3
Ra0
Ra1
Ra2
Ra3 ¬Ra3
a4
a5
a6
¬Ra4
¬Ra5
¬Ra6
values may go as shown in the box. And the cut-offs between simply true and both true and false (or both true and false and simply false) are just as counter-intuitive as any between simple truth and simple falsity. In Priest (2003) I argued that versions of the forced-march sorites demonstrate that, one way or another, we are forced to admit the existence of some sort of cut-off ¹⁵ The transition stages are to be expected to have other contradictory properties as well. Thus, if the bike is black at t0 and red at t6 then a3 has the property of having been black (qua a0 ), but also the property of not having been black (qua a6 ).
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points. All that is left for a solution to the sorites to do is to theorize the nature of the cut-off points and, crucially, explain why we find their existence so counter-intuitive. In that paper I suggested a solution in terms of metalinguistic non-transitive identity. We find the existence of a cut-off point counter-intuitive because whatever the semantic values of the relevant sentences on either side of the cut-off point, they are, in fact, the same. The failure of the transitivity of identity prevents the value bleeding from one end to the other. The theory of non-transitive identity given in Priest (2003) is based on a fuzzy logic. But the one outlined in this chapter would do just as well. Consider a language that can describe the semantic properties of the language of the red-sorites. The language has names Ra0 , ..., Ra6 , {1}, {1, 0}, {0}, and the one-place function symbol, ν (‘the truth value of ’). Take an interpretation for the language in which D1 = {Ra0 , ..., Ra6 , {1}, {1, 0}, {0}}, θ ({1}) = {1}, θ (Ra0 ) = Ra0 , etc., and θ (ν) is a function, f such that: f (Rai ) = = = f (t) =
{1} {1, 0} {0} {0}
if if if if
0≤i≤2 i=3 4≤i≤6 t is a truth value
The first three lines give an accurate description of the table for the Rai s. ( The last line is required since f must have values for its other arguments too; what these are does not matter for what follows.) By a suitable choice of D2 , we can ensure that for each i, the sentence in this language ν(Rai ) = ν(Rai+1 ) is true! This may be achieved in several ways. A simple one is to impose the following constraint on D2 : For every A ∈ D2 , {1, 0} ∈ A+ and {1, 0} ∈ A− ( Thus, the object {1, 0} is a highly paradoxical object.) If 0 ≤ i < 2, then the terms ν(Rai ) and ν(Rai+1 ) both refer to {1}. Hence, for any A ∈ D2 , Aν(Rai ) and Aν(Rai+1 ) have the same value, and so Aν(Rai ) ≡ Aν(Rai+1 ) is (at least) true. If i = 3, then the term ν(Ra4 ) refers to {1, 0}, so for any A ∈ D2 , Aν(Ra4 ) is both true and false, and so Aν(Ra3 ) ≡ Aν(Ra4 ). When 0 ≤ 4 ≤ 6, the arguments are similar.¹⁶ The problem with which Priest (2003) ends is how to obtain a metatheory for a vague object-language which has the same underlying logic as the object language. For fuzzy logic, this is still an open issue. But for the theory being deployed here, there are known solutions. In ch. 18 of Priest (2007), it is shown, using what the paper calls the ‘model-theoretic strategy’, how to formulate the metatheory for a language with underlying logic LP in a naive set theory which itself has underlying logic LP. The logic is not a second-order one, as is the case here, but the availability of sets gives the ¹⁶ Note that it is true that {1, 0} = ν(Ra3 ) = ν(Ra4 ) = {0}. But even if we extended the language to be able to express the fact that 1 ∈ {1, 0}, it would not follow that 1 ∈ {0}, due to the failure of substitutivity. This provides a solution to the extended semantic paradox given by Smiley, different from the ones given by Priest, in Smiley and Priest (1993). See 30 f. and 50 f.
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same effect. In particular, x = y may be defined as: ∀z(x ∈ z ≡ y ∈ z).¹⁷ Because of the use of a material conditional, this identity has exactly the same properties as the one we have been using here. Indeed, since the theory is a naive one, in which every condition defines a set, there is very little conceptual difference between this and the second-order approach. We could, in fact, have avoided using second-order logic by deploying set theory and this definition of identity, instead of the second-order one. I chose not to adopt that course here so as not to raise many important but, in this context, distracting questions.
23.7
C O N C LU S I O N
In this chapter I have outlined an account of identity and some of its applications. The notion of identity does not have all the properties of the orthodox notion. Especially, transitivity fails. However, the notion may be thought of as a generalization of the orthodox one, since, when restricted to consistent situations, the orthodox account is obtained. The idea was made precise with the notion of minimally inconsistent consequence. We have also looked at various applications of the notion, especially those that concern change. I have not discussed other approaches to the problems raised, which there certainly are; nor have I tried to mount a case that the approach deployed here is the best. But I do hope to have shown both the technical viability of this notion of identity and its potential philosophical fruitfulness. Re f e re n c e s Bueno, O. (2000), ‘Quasi-truth in quasi-set theory’, Synthese 125, 33–53 Hyde, D. (1997), ‘From heaps and gaps to heaps of gluts’, Mind 106, 641–60. Krause, D. (1992), ‘On quasi-set theory’, Notre Dame Journal of Formal Logic 33, 402–11. Miller, A. F. (trans.) (1969), Hegel’s Science of Logic, London, Allen and Unwin Ltd. Parfitt, D. (1984), Reasons and Persons, Oxford, Clarendon Press. Priest, G. (1987), In Contradiction, Dordrecht, Kluwer Academic Publishers, second edition, Priest (2007). (1991), ‘Minimally inconsistent LP’, Studia Logica 50, 321–31, reprinted as ch. 16 of Priest (2007). (1995), ‘Multiple denotation, ambiguity, and the strange case of the missing amoeba’, Logique et Analyse 38, 361–73. (2001), Introduction to Non-Classical Logic, Cambridge, Cambridge University Press, a slighly revised form is Part 1 of Priest (2008). (2002a), ‘The Hooded Man’, Journal of Philosophical Logic 31, 445–67. (2002b), ‘Paraconsistent Logic’, 287–393, vol. 6, of D. Gabbay and F. Guenthner, eds., Handbook of Philosophical Logic, second edition, Dordrecht, Kluwer Academic Publishers. (2003), ‘A site for sorites’, 9–23 of Jc Beall, ed., Liars and Heaps: New Essays on Paradox, Oxford, Oxford University Press. (2005), Towards Non-Being, Oxford, Oxford University Press. ¹⁷ As a matter of fact, identity is not defined in this way in that chapter: it is taken as primitive. But essentially the same construction goes through if identity is defined as indicated.
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Priest, G. (2007), In Contradiction, 2nd (extended) edition, Oxford, Oxford University Press. (2008), Introduction to Non-Classical Logic: From If to Is, Cambridge, Cambridge University Press. Prior, A. (1968), Papers on Time and Tense, Oxford, Clarendon Press. Saul, J. (2007), Simple Sentences, Substitution, and Intuitions, Oxford, Oxford University Press. Smiley, T. and Priest, G. (1993), ‘Can contradictions be true?’, Proceedings of the Aristotelian Society, Supplementary Volume 67, 17–54.
VII Many-Valued Logics
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24 Identity and the Facts of the Matter Graeme Forbes
24.1
OLD NUMBER ONE
In 1990, the specialty car company Middlebridge Scimitar Ltd contracted with the vintage car collector Edward Hubbard to buy the Bentley racing car known as Old Number One from him. Middlebridge agreed to pay Hubbard ten million pounds in cash and company assets. The price was so high because Old Number One was the most famous racing car in British history, dating from a period when motor racing was dominated by British cars and drivers. It was in Old Number One that Captain Wolf (‘Babe’) Barnato, diamond heir and leading light of the ‘Bentley boys’, had won his second and third Le Mans 24-hour races in 1929 and 1930. The 1929 race was a procession, with Bentley taking the first four places, but in 1930 there were more powerful German cars competing, and Barnato should not have been on the podium. But by a combination of skill and guile, he won again, ahead of Mercedes Benz. After(!) signing the agreement with Hubbard, Middlebridge did some more historical research, as a result of which they refused to perform the contract. Hubbard sued, and the case went to the High Court in London, Queen’s Bench Division, where it was heard before The Honourable Mr. Justice Otton, later Sir Philip Otton, Lord Justice of Appeal.¹ Middlebridge’s objection was that the car Hubbard was trying to sell to them wasn’t really Old Number One, the car they believed they had contracted
Some parts of this chapter are descended from material that was much improved by input from Kit Fine, Terence Parsons, Teresa Robertson, Nathan Salmon, and Nicholas J. Smith. This version is a revision of my paper for the 2007 Arch´e Conference on Vagueness at St. Andrews University. I thank my commentator, E. J. Lowe, and the audience, especially Peter Milne, Diana Raffman, Nathan Salmon, and Crispin Wright. I also benefited from the reactions of audiences in Paris, Kansas, Nottingham, and Frankfurt, where Johannes Ritter and Ede Zimmerman were especially helpful. Comments from Kathrin Koslicki and Teresa Rosen Peacocke led to last-minute improvements in the final draft. ¹ Much of the information Iam relaying here comes from the transcript of Otton’s verdict made by Cater, Wash & Co., and posted at http://www.gomog.com/articles/no1judgement.html. This document is the source of the quotes.
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to buy. After the 1929 race, the company argued, so many repairs, modifications and upgrades had been carried out that the car which came into Hubbard’s possession could not be said to be the car that won in 1929. Middlebridge was particularly concerned about events in 1932. After the 1930 race, Barnato had retired from competitive driving and Bentley had withdrawn from motor racing. Barnato bought Old Number One from the company, hired the Bentley mechanic, Wally Hassan, who had been responsible for the car, and raced it with mixed success. After entering it for the 1932 Brooklands 500, Barnato asked Hassan to upgrade the car substantially, which he did, and these changes were the ones Middlebridge found most objectionable.² Worse, the car crashed during the race, killing its driver, Clive Dunfee, and it seemed to be a write-off. However, Hassan testified that ‘The body was of course ripped off but all the mechanics, the mechanical parts, were all perfectly ok. We were just able to clean it up and we had a new body built for it, a coup´e body this time.’ In addition to the modifications of 1932, the car had undergone other changes after winning in 1929. The end result, according to Michael Hay (an expert on the history of the Bentley saga) was, as Otton reported in his verdict, that ‘None of the 1929 [car] survives [in Hubbard’s car] with the exception of fittings which it is impossible to date. Of the 1930 [car] Hay believes that only the following exist on the car as it is now, namely pedal shaft, gear box casing and steering column. Of the 1932 car, the 4 litre chassis and 8 litre engine form in which it was involved in the fatal accident, he believes that the following exist: the chassis frame, suspension (i.e. springs, hangers, shackles and mountings), front axle beam, back axle banjo, rear brakes, compensating shaft, front shock absorbers and mountings, the 8 litre engine, some instruments and detailed fittings.’ So Middlebridge had a point. On the other hand, there was plenty of testimony to the effect that Hubbard’s car was Old Number One, and that this had been Barnato’s own opinion. Some of this testimony came from people who had elsewhere said that Hubbard’s car was not Old Number One. But despite these conflicts, all based on the same information, Otton came down conclusively on Hubbard’s side. This was mainly because of the weight he gave to continuity considerations: ‘Here the entity which started life as a racing car never actually disappeared . . . Any new parts were assimilated into the whole at such a rate and over such a period of time that they never caused the car to lose its identity, which included the fact that it won the Le Mans race in two successive years. It had an unbroken period of four seasons in top-class racing.’ And perhaps with the possibility of reassembly of the 1929 car’s 1929 parts at the back of his mind, Otton concluded his verdict with the following Nozickian ² Hassan was still alive to testify in 1990 (at 85), and had remarkable powers of recall: ‘We started with a 4 litre chassis frame which was stronger than the old 6.5 litre because we feared that it would break or crack. We used all the existing parts of the older car—that is, the radiator, the clutch, the gear box, the axles, the scuttle, the electrical equipment and pedals, and we finished it up in the form it is now. It was ready for the 500 miles race in that September but Captain Barnato thought it would be a bit faster with a bigger engine, so we obtained an 8 litre engine and I built that into the car.’
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flourish: ‘There is no other Bentley, either extinct or extant, which could legitimately lay claim to the title of Old Number One or its reputation.’³ If a case like this did not already exist, we would have to invent one.⁴
24.2
A N A LT E R N AT I V E V E R D I C T
There are two other verdicts Otton might have reached. He might have decided in favor of Middlebridge, for we can certainly imagine courses of events concerning which Otton would judge that the rate and assimilation of new parts into the whole did cause the car to ‘lose to its identity’.⁵ And those who think he should have decided in favor of Middlebridge anyway can surely imagine courses of events with less radical amounts of change, spread out more gradually, which they would regard as making a pro-Hubbard verdict reasonable. But there is another option, which is perhaps the most reasonable of all, both in the actual circumstances and various mild variations of it. For in view of the disagreements over whether Hubbard’s car was Old Number One, disagreements which are not underpinned by any disagreement over facts that are independent of whether Hubbard’s car was Old Number One, Otton might well have concluded that there is simply no fact of the matter whether Hubbard’s car was Old Number One. We could put such a verdict into his mouth in these words: Sometimes there is no fact of the matter whether a statement is true or false. We are familiar with clear cases of people who are bald. Yul Brynner, for example. And with clear cases of people who are not bald. David Chalmers, for example. But there are people who have some but not much hair. They are not close enough to either paradigm for there to be sufficient similarity to settle that they are bald, or that they are not. Nor is there any linguistic rule required for mastery of ‘bald’ which we can apply to settle that they are bald, or that they are not. These are people for whom there is no fact of the matter whether or not they are bald. What holds for ‘bald’ holds for ‘being the same car as Old Number One’. We have certain clear cases of persistence through time. If Old Number One had been put in storage immediately after winning in 1929, had remained completely assembled since then, and had ³ Of all the larger than life characters figuring in this story—Hubbard, Barnato, his daughter, W. O. Bentley—perhaps none was so large as the car itself. Otton said ‘It was produced for my inspection in Lincoln’s Inn. It looked beautiful, and the magic and sheer power of its engine evoked excitement and nostalgic memories of the past.’ Anyone who was a British schoolboy of my generation or earlier will have no difficulty understanding how Otton’s pulse must have raced. ⁴ I may have taken some artistic license with my description of the case. It appears that at least to some extent the dispute was over whether Hubbard had in fact contracted to provide Middlebridge with the car that won in 1929, as opposed to, say, the car that he had acquired in such-and-such a way after a certain course of events (described neutrally vis a` vis identity with the 1929 winner). Still, I shall take Otton’s claim that the car never lost its identity because of the slow rate of assimilation of new parts to imply a philosophical view about persistence. ⁵ It is unclear from the transcript that Otton was right about the continuity facts. The upgrade for the 1932 Brooklands 500 race that Hassan described (see fn. 2) appears to have involved attaching some older parts to a new chassis—cannibalization of the 1930 car—as opposed to replacing an old part with a new part in a standing car.
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undergone no changes of parts, and Mr. Hubbard had purchased it but not causally interacted with it in any way, there would be no doubt that the car Middlebridge contracted to buy from him is Old Number One. If Old Number One had been completely destroyed in the 1932 crash and consumed in fire, so that only a few broken fittings and twisted pieces of metal were salvageable, and these pieces were incorporated in the construction of a car at Hassan’s workshop, which then passed on to Hubbard as the actual car did, there would be no doubt that the car Hubbard proposed to sell to Middlebridge, the one the company contracted to buy, is not Old Number One. But the actual history is not so obliging: it has aspects of both types of case. At some point, perhaps immediately after the preparations for the 1932 race, or immediately after the postcrash reconstruction, we find ourselves in a no-man’s-land between cases of persistence and cases of replacement by something new. Nor is there any linguistic rule required for understanding such phrases as ‘continues to exist’ or ‘ceases to exist’ to settle whether the post-crash reconstruction is the car that won in 1929, Old Number One. No sufficient condition for persistence holds, nor does any necessary condition fail. The case before us therefore concerns a dispute which has no correct resolution: the facts simply do not determine whether or not Mr. Hubbard’s car is Old Number One. The court therefore rules the contract ‘void for uncertainty’, and the case is dismissed.
This, I think, is what Otton should have said. And it has considerable initial plausibility, making it well worth our while to investigate whether there is a consistent account of identity through time which can accommodate ‘no fact of the matter’ in such cases.
24.3
OT H E R C A S E S A N D T H E U N I F O R M I T Y C O N S T R A I N T
Otton’s verdict for Hubbard rested on continuity considerations that were only available because a car, perhaps occasionally in a disassembled state, existed at each time. But if we replace repair and upgrade with gradual destruction, continuity considerations no longer suffice to stave off indeterminacy. Suppose, for example, that Hubbard had been trying to sell a Brancusi bronze to Middlebridge, but before delivering it, had melted it down a certain amount. Whether or not Middlebridge gets the statue it thought it was getting depends on how much melting down has happened; certainly, receiving a molten pool of bronze would entitle it not to perform the contract. But it is rather implausible that there is a precise moment in the melting-down process at which the original statue, or any statue at all, ceases to exist. Rather, there will be a range of points such that, if Hubbard stops at one of them, there is simply no fact of the matter whether the original statue still exists. There are other examples which do not involve temporal persistence but which seem to be puzzles of the same kind. There is a modal variant of the case of Old Number One, usually known as Chisholm’s Paradox (since it originates in Chisholm 1968), pithily summarized by Quine in the dismissive remark ‘you can change anything to anything by easy stages through some connecting series of possible worlds’ (Quine 1976, 861). For example, let g be the 8-litre engine that Hassan put into Barnato’s Bentley in 1932. We are unlikely to accept a conditional of the form ‘if g could have been
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originally built from these parts according to this design, then g could have been originally built from those entirely different parts according to that strikingly different design’, at least if we think that there must be restrictions on what de re stipulations make sense. But taking the conditional to be of the form φ1 (g) → φ100 (g), it is a logical consequence of a connecting chain of conditionals of the form φi (g) → φi+1 (g), 1 i 99. Here φ1 is a predicate specifying the actual parts of h and their actual configuration, or something very close to this, while φi+1 specifies parts and configuration very little different from φi . But as i increases, the degree of resemblance to the original configuration steadily decreases and the overlap with the φ1 -parts uniformly decreases. Each conditional in the chain is true, according to the tolerance principle that any artifact that could have originated from certain parts in a given configuration could also have originated from slightly different parts in a slightly different configuration. But the result of chaining the conditionals is false. Despite its being modal, the puzzle here is not much different from the one that confronted Otton. We would like to say about it that for some i, there is no fact of the matter whether φi (g). And there are other examples, superficially more different, of which the same seems to be true. For instance, (Salmon 1986, 113) has a case, the Storage Room puzzle, of the same type. Suppose that some furniture movers have to deliver n + 1 items of the same design to a storage facility. To place each item in the storage room, it is necessary to disassemble it, pass its pieces through the inconveniently narrow entryway, then reassemble it on the other side (evidently, a British storage facility). Things go well with the first piece, but as the day goes on, the movers get more and more careless, damaging more and more pieces of each item they try to store, and replacing them in the reassembly process from a cache of spare parts they brought with them. The last piece of the day is totally destroyed in disassembly, and is replaced in the storage room by a piece of furniture constructed there from the cache of spare parts. More formally, let ao , . . . , an be n + 1 distinct pieces of furniture, each with n + 1 parts (fixed n), and each capable of being disassembled and reassembled. For each ai , let bi be the object which results when ai is disassembled, then reassembled with replacement of i-many parts. We have, for each i, the seemingly true conditional ‘if ai = bi then ai+1 = bi+1 ’, yet we would hardly agree to ‘if ao = bo then an = bn ’, which looks straightforwardly false. But this last conditional is of course entailed by the others. Again, we would like to say that for at least one i, there is no fact of the matter whether ai = bi . There is a tolerance principle at work in this case too, namely, that if in case i the same piece of furniture is disassembled and reassembled, then in case i + 1 the same piece of furniture is disassembled and reassembled. And there was a tolerance principle at work in the case of Old Number One, namely that if Old Number One survived any repair or upgrade or modification, it survived the next one. ( Those who doubt this about the actual course of events in that example can produce a variant in which this principle is very plausible, though it leads to the conclusion that if Old Number One survived the first change it survived them all.) Since all the puzzles involve tolerance principles, a uniform approach to them will involve some way of preventing these principles from generating awkward
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consequences, or some persuasive reason to reject the principles or to accept their consequences. But uniformity requires rather more than this. For I have said that our three puzzles are essentially the same puzzle. If that is so, the apparatus we invoke to defuse them should be essentially the same apparatus in all three cases. I call this constraint the Uniformity Constraint. And what kind of apparatus are we envisaging? Since they are generated by tolerance principles, these puzzles appear to be of a familiar kind: they are Sorites paradoxes, of the same general sort as the Bald Man paradox, that if a man with n hairs on his head is bald, so is a man with n + 1 hairs, hence, if a man with no hairs on his head is bald, so is a man with thousands. The Uniformity Constraint would be met by taking some treatment of vagueness and applying it in the same way to the three puzzles. The Constraint immediately rules out certain approaches to Chisholm’s Paradox. Modal conditionals like the ones we considered may be translated into two extensional possible-worlds languages, one invoking relative possibility (‘accessibility’) and the other counterparthood, as illustrated in (1b) and (1c) below: (1) a. ψ(g) → θ (g) b. (∃w)(R @w ∧ ψ (g, w)) → (∃u)(R @u ∧ θ (g, u)) c. (∃w)(∃x)(Cxgw ∧ ψ (x, w)) → (∃u)(∃y)(Cygu ∧ θ (y, u)). According to (1b), (1a) has the truth condition that if for some world possible relative to the actual world (‘R @w’), g is ψ-at-that-world (‘ψ (g, w)’), then for some world possible relative to @, g is θ there. According to (1c), (1a) has the truth condition that if there is a possible world w and some x such that x is both a counterpart of g at w (‘Cxgw’) and ψ-at-w, then there is a possible world u and some y such that y is both a counterpart of g at u and θ -at-u. For (1b), see (Salmon 1981, 240–52), and for (1c), (Forbes 1983). The approaches to Chisholm’s Paradox which run into trouble with the Uniformity Constraint are ones which try to transfer certain non-classical semantics for languages with vague predicates to either of these extensional languages.⁶ There is no ⁶ Two approaches to vagueness which may apply uniformly to all our puzzles are epistemicism and contextualism. According to the epistemicist, the tolerance principles are simply false; at some point, a very small change tips the balance (there may be reasons in principle why we cannot know where that point is). According to some contextualists, there is also a tipping point, but we cannot say or think what it is without moving it. For epistemicism in general, see (Sorensen 1988), (Williamson 1994), and for a version restricted to puzzles about identity, (Salmon 2002). For contextualism, see (Raffman 1994), (Soames 1999, ch. 7), and also (Robertson 2000) for criticism of the latter. Much of the rationale for epistemicism depends on alleged shortcomings of (all) non-classical semantics. And contextualism does not seem to help with the purely conditional versions of the puzzles I use here. We assent to all the conditional premises of a Sorites on the very same non-truth-functional ground, that the states of affairs described by antecendent and consequent are too similar in relevant respects for the contentious condition to hold in the former state and fail in the latter. So there is no relativity to pairs, or to any ‘fluid’ psychological context, that would allow Raffiman’s apparatus for defusing forced-march Sorites to get a grip (my internal homunculus accepts all the conditionals, one after the other, since they are all equally plausible, even as their consequents grow increasingly implausible). Soames’s apparatus requires that we detach and assert the consequents (to change to context), but the apparent truth of the premise conditionals combined with the clear falsity of the
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reason why any of the vocabulary in ψ or θ should merit non-classical treatment, since we can make the specification of the parts and configuration of g arbitrarily precise. And a non-classical status for any of the conditionals (such as ‘neither true nor false’) would have to be inherited from their antecedents or consequents. So the modal operator must be at the root of the vagueness. In the extensional languages, this means looking to either the existential quantifier, or the special predicates R and C for relative possibility and counterparthood. And while an existential formula can have a non-classical status, this in turn is inherited from the non-classical status of its scope. So we are led to the proposal that such formulae as R @u or Cygu should have a non-classical semantics, one that makes room for there being no fact of the matter whether R @u or whether Cygu. In the simplest version, we allow R to be undefined for some pairs of worlds, or C to be undefined for some triples consisting in two worldbound individuals and a world. Generally, we have three truth value statuses, true, neither true nor false, and false, written and ordered as ⊥ < ∞ < !. Conditionals have a ‘sustaining’ semantics, on which (! → ∞) = (∞ → ⊥) = ∞, (∞ → ∞) = !, and ⊥ only results from (! → ⊥). So it might be that (1a) turns out to be neither true nor false, because the antecedents of (1b) or (1c) are true while the consequents are neither true nor false. In application to (1b), we may have the antecedent straightforwardly true, but worlds w where ψ (g, w) are on the verge of possibility relative to @, so that when we look at worlds u such that θ (g, u), we find that the best case is R @u undefined: because θ (g, u), there is no fact of the matter whether R @u. Treating ∃ as infinitary disjunction and disjunction as least upper bound, (∃u)(R @u ∧ θ (g, u)) would in such a case be neither true nor false. So (1b) is ! → ∞, that is, ∞. Thus we get the result that while none of the conditional premises in Chisholm’s Paradox is false, some are neither true nor false, so the Paradox is an unsound argument. The corresponding non-classical semantics for (1c) produces the same result, for while (∃w)(∃x)(Cxgw ∧ ψ (x, w)) may be true, it may also be that in any world u where some y is such that θ (y, u), we find Cygu either false or neither true nor false: at best, there is no fact of the matter whether such a y is a counterpart of g at u. Granted some u where Cygu is neither true nor false, the reasoning of the previous paragraph gets us to conditionals of the form (1c) which are neither true nor false, so this counterpart-theoretic interpretation of Chisholm’s Paradox also makes it unsound.⁷ conclusion conditional is by itself paradoxical. Of course, transitivity of the indicative conditional has been challenged, and some counterexamples may arguably be said to involve a shift in context; for example, with ‘‘if Jones doesn’t compete, Smith will win’’ and ‘‘if Smith wins, Jones will get the silver’’ it’s likely that the ‘‘Smith wins’’ worlds we consider in evaluating the second conditional are not among the ‘Jones doesn’t compete’ worlds that settle the first conditional. But a process of judging the sorites conditional premises, even one by one, does not involve anything like this. However, I agree with Edgington (1996, 309, n. 15), that there is a special case where a Raffimanstyle contextualism would be appropriate, namely, with ‘looks’ versions of Sorites conditionals: if x looks red/bald/tall and y looks the same as x in respect of color/head-hairiness/height, then y looks red/bald/tall. ⁷ Salmon (1981, 1986) argues that there are two puzzles, a Sorites-type one with material conditional premises ψ → θ, and a specifically modal one with strict implication premises
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But the relative possibility approach conflicts with Uniformity because it does not transfer to the case of Old Number One. This is because transitivity of ‘is in the future of ’ cannot fail. Hence all conditionals of the form Fφi (#1) → Fφi+1 (#1) are straightforwardly true, where the antecedent asserts Old Number One’s survival of the i’th change and the consequent, of the i + 1’th. Or, if this is not so, their semantics must be explained in very different terms. Either way, a generalization has been missed. A transfer of the counterpart-theoretic account might be objected to because it requires us to adopt a certain view of what identity through time consists in, the standing in a counterpart relation of uncountably many thing-stages. But this might be a way of meeting the Uniformity Condition (however unattractive), not a failure to meet it. The failure comes with the Furniture Storage puzzle. We would like to say that for some values of i, there is no fact of the matter whether the conditional ‘if ai = bi then ai + 1 = bi + 1 ’ is true or false, and it is now proposed to explain this in terms of there being no fact of the matter whether the counterpart relation holds between certain piece-of-furniture stages. The problem is that a judgement such as ‘a21 = b21 ’ is a plain-vanilla identity judgement, lacking any of the operators to whose semantics the intrusion of the counterpart relation can be attributed. We can imagine someone reading a document that uses only ‘a’-terms in listing the inventory of the factory where the furniture is first assembled, and a document that uses only ‘b’-terms in describing the contents of the storage room. To such a reader, the judgement ‘a21 = b21 ’ is entirely intelligible, though he has no reason to think it (or any other ai = bj ) true. We should be sceptical that there are hidden tense operators in the proposition that a21 = b21 which this person grasps.⁸ where paradox is obtained by repeated application of the rule (C), (ψ → θ), (θ → λ) (ψ → λ). The first paradox unsound, since some ψ → θ is untrue, and the second, though it has true premises, is invalid, since (C) requires that relative possibility be transitive, which it is not. He suggests (1989, 4–5) that a non-transitive R is demanded by intuitions about certain cases (also Peacocke 1999, 196): the idea is that even if, say, φ3 (g) is impossible as things stand, nevertheless, had φ2 (g) been the case, then φ3 (g) could have been the case: φ2 (g) φ3 (g). We also have φ2 (g), so we get φ3 (g) even though ¬φ3 (g). But the counterpart theorist can accommodate the intuition that the counterfactual is true. It means that some φ2 (g)-world where φ3 (g) holds is more similar to @ than any φ2 (g)-world where ¬φ3 (g) holds. In the framework of (Forbes 1983), this existential will have the highest degree of truth of its instances, a degree of truth that is indiscernibly close to absolute truth in cases where the counterfactual strikes us as true. So the intuitive plausibility of φ2 (g) φ3 (g) cannot differentially support an approach employing a non-transitive R . In fact, even those who are sure there are two paradoxes, a B-invalid modal argument and an unsound Sorites argument, may be better served by counterpart theory. For the counterpart-theoretic semantics can be recast to invoke counterparthood with each modal operator (as in Lewis 1968), so that φ3 (g) means that for some w and u, g has a counterpart at w that has a counterpart at u that satisfies φ3 . ψ → ψ now fails, but we have a better explanation why. All the counterexamples to the transitivity schema are de re: ψ contains either a name or a free variable. That it is a non-transitive counterpart relation that is doing the work explains why there are no de dicto counterexamples. ⁸ For someone happy to discern hidden operators in such identity statements, and willing to endorse the analysis of persistence in terms of stages and counterparts, the counterpart-theoretic approach remains quite appealing. Objections to it fall into two groups, (A) objections to the underlying extensional many-valued or partial logic (though the approach is consistent with using
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By contrast, the concepts Otton employs in my imaginary verdict about Old Number One transfer smoothly to the other cases. The crucial concept is that of there being no fact of the matter about a certain claim of identity. In the case of the Furniture Storage puzzle, the identity claims are quite explicit, and the idea is that indeterminacy can be attributed directly to the identity proposition, not to some element which only emerges on analysis. In Chisholm’s Paradox there are no explicit identities, but a trivial reformulation introduces them: replace ψ(g) with (∃x)(ψ(x) ∧ x = g). Certainly, this formulation is still amenable to an account cast in terms of counterparts or relative possibility of why there might be no fact of the matter about certain cases. But it also promotes the thought that, where (∃x)ψ(x) is true, (∃x)(ψ(x) ∧ x = g) may be neither true nor false because necessarily, anything that possibly satisfies ψ(x) is at best something that satisfies neither x = g nor x = g. In that case we can dispense with both counterparts and relative possibility, and use the simplest S5-semantics. On all three approaches, of course, we can say that what there is no fact of the matter about is whether possibly being ψ is a property of g, or whether being ψ is a possibility for g, or some such. The differences are in the underlying machinery that makes such ‘no fact’ claims true. In saying that there may sometimes be no fact of the matter about an identity judgement, are we making a claim about the concept of identity, the objects themselves, or about something else, such as the reference relation? The idea that the reference relation is the basic factor seems to get things exactly the wrong way round. It is implausible to hold that there is no fact whether Old Number One is Hubbard’s car because there is some indeterminacy in the reference of ‘Old Number One’ or ‘Hubbard’s car’. If there is no fact of the matter whether ‘Old Number One’ refers supervaluations instead), and (B) objections to the counterpart semantics for the intensional operators. (A) The main A-type objections are to allowing contradictions to be truth valueless, or to have an intermediate degree of truth (dt). In (Forbes 1983) I used the principle (∧) that dt(p ∧ q) = min{dt(p), dt(q)}. Because dt(¬p) = 1 − dt(p), we have dt(p ∧ ¬p) = .5 if dt(p) = .5 (but see Edgington 1996 for an alternative). However, (Williamson 1994, 136) insists that whatever the facts, they must falsify p ∧ ¬p. This appears to me to overreach from the correct ‘whatever the facts, they cannot verify p ∧ ¬p’ (Williamson says ‘ ‘‘He is awake and he is asleep’’ has no chance at all of being true’, but this is agreed to by everyone). We have no difficulty with the idea that if a sentence S is so anomalous that it fails to express a proposition, and so fails to possess a truth value, then S ∧ ¬S will also be truth valueless. If p and ¬p each fail to be verified by the facts, and fail to be falsified by them, the issue is what recursive implication this should have for p ∧ ¬p. If falsification of the whole has to flow through one or other conjunct, then p ∧ ¬p may be non-false. It appears that the critic of (∧) will have to employ some such notion as ‘false solely in virtue of meaning’. But see also n.11 below. (B) Fara and Williamson (2005, 18–20) object to counterpart semantics that it cannot accommodate an ‘actually’ operator. The particular counterexamples they offer depend on the semantics (i) permitting a single object to have multiple counterparts at a world, and (ii) introducing distinct counterpart quantifiers for distinct occurrences of a variable or name directly within the scope of a or . (ii), which makes (∀x)(x = x) invalid, I now think to be more trouble than it is worth, but so long as we have (i), there are likely to be difficulties in the bivalent case. In the present non-classical context, of course, the objectionable examples will simply be like p ∧ ¬p, sometimes non-false, and can be lived with. Alternatively, we could cut the Gordian knot by rejecting (i), since the option of providing an object with two same-world counterparts plays no role in the resolution of Chisholm’s Paradox.
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to Hubbard’s car, that indeterminacy would be because the facts of the case do not decide whether Old Number One, the car that won in 1929, is Hubbard’s car. So when identity judgements fail to be bivalent, a fundamental account will look to the concept, or the objects, not the terms. Here I am assuming that ‘Old Number One’ refers determinately to something, say, the car that crossed the finishing line in first position at Le Mans in 1929. So my point would be rejected by one who holds that there are uncountably many precise cars which did that, and the problem is that we have not settled on one as the referent of ‘Old Number One’. This is a view according to which the persistents of our ordinary ontology don’t really exist. But I am pursuing a reconciliation of indeterminate identity with our ordinary ontology. Deciding between concept and objects is harder, and it may be that these are equivalent descriptions of the same phenomenon. We can eliminate some indeterminacy by stipulating more precise conditions of persistence, conditions which, had they been in force in 1990, would have made the court case simple to decide. Since this is a conceptual fix, it suggests indeterminacy is in concepts. But the objects have to be a certain way as well; as the imaginary verdict says, it’s easy to imagine ways they could have been on which ‘Old Number One is Hubbard’s car’ would be true, or would be false, no argument. So I am unsure if there is anything of substance at issue here, resolving which would illuminate our way to a solution.⁹ ⁹ In the version of the chapter from which my St. Andrews talk was drawn, there followed a section on Leibniz’s Law (LL) and the use made of it by Evans and Salmon in arguing against indeterminate identity or vague objects. For reasons of space I have deleted this material, but I give a brief statement of my main points here. The basic argument underlying (Evans 1978) and (Salmon 1981, 2002) is that if there’s no fact of the matter whether a = b, and it’s a fact that b = b, then a = b, since b is such that it’s a fact that it is b, and a is not such that it’s a fact that it is b. So if there’s no fact of the matter whether a = b, then a = b. So there’s a fact of the matter, period. This appears to involve a contrapositive of LL, from (¬Pa ∧ Pb) to infer ¬(a = b). But, as emphasized in (Parsons 2000, §2.4), contraposition is not reliable when there is a third status for propositions. Define p q to mean that for every three-status valuation V, if V(p) = ! then V(q) = !. Then p q does not guarantee ¬q ¬p; for if V(¬q) = !, V(q) =⊥, and so, if p q, we can conclude V(p) = !, hence V(¬p) =⊥. But for ¬q ¬p we need the stronger V(¬p) = !, excluding ∞. The very case at issue illustrates this, and also the failure of the Leibniz Law conditional scheme LL→ , a = b → [φ(a) ↔ φ(b)]. Using " for being determinately the case, we have the instance a = b → ["(a = b) ↔ (b = b)]. But if V(a = b) = ∞, the biconditional is ⊥↔ !, so the whole conditional is ∞ ↔⊥, which is ∞ or ⊥ on any account. Since the classical principles used against indeterminate identity in the Evans-Salmon critique are put into question by the very cases under discussion, the critique seems to have no more force than a reductio of constructivism which boldly wields Excluded Middle. However (Salmon 2002, 245) writes that those who would reject the standard Leibniz schemes or the contrapositive of LL need to show that ‘a weaker alternative is independently intuitive, and . . . its historical omission was a logical oversight, akin to the Aristotelean logician’s inadvertently overlooking the fact that the inference from All S are P to Some S are P is invalid without the tacitly assumed premise Some things are S ’. I think we can meet this challenge. First, modern logic grew out of the attempt to formalize the canons of reasoning characteristic of classical mathematics, whose subject-matter is the domain par excellence where sharp cut-offs reign. When we move away from that domain and abandon bivalence, we bring to unfamiliar territory our near-automatic reflex to equate ‘not true’ and ‘false’ for meaningful statements. This carries over to our assessment of the significance of certain distinctions. In particular, we inadvertently overlook the possibility that while any difference at all between a and b with respect to properties establishes that a = b is untrue, there might be a special and unusual category of property, difference with respect to which only establishes untruth,
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H I G H E R - O R D E R I N D E T E R M I N AC Y
(Parsons 2000) defends a many-status logic in which there is one non-classical semantic status. However, there are reasons why we might prefer to use a semantics that generalizes this. One such semantics, fuzzy logic, has the real interval [0,1]R as possible semantic values, called degrees of truth, and as a result can offer a plausible explanation of why, for each conditional premise of an effective Sorites argument, that premise seems true to us (if not, there would be no Sorites paradox), and also, de dicto, why it seems to us that every conditional premise is true. A plausible explanation is one that attributes to each conditional an overwhelming appearance of unqualified truth. One · is cut-off subtraction: clause for → that does this is (2), in which — · (2) dt[φ → ψ] = 1 − (dt[φ] —dt[ψ]). In an effective Sorites argument, the worst case is that a conditional premise has an antecedent whose degree of truth (dt) is marginally greater than the dt of its consequent. For this case, clause (2) produces a dt for the whole conditional that is only marginally less than complete truth. So the conditionals falling under this case are semantically indiscriminable from all the others, which are themselves completely true.¹⁰ By contrast, a supervaluational account that identifies truth with supertruth makes it a gross error that we should think that every conditional premise is true, since ‘at least one is false’ is supertrue. And a three-status account is only a slight improvement over a two-valued account, since that there is a sudden transition from true antecedent to neither true nor false consequent does not seem much more likely than that there is a sudden transition from true antecedent to false consequent.¹¹ For our identity puzzles, the analogue of a many-conditionals Sorites paradox is the following style of argument, which Otton did not produce, either in fact or in my fiction: not falsity. Second, we have a near-automatic reflex to equate p and ‘p is true’ (an entire theory of truth is based on this reflex). So we fail to notice that there is a weaker version of LL→ which provides all we need in non-contested applications, namely, LL→ " , "(a = b) → [φ(a) ↔ φ(b)]. Uncontested applications of Leibniz’s Law are saved by LL→ " , which disagrees with the standard scheme only in the cases under discussion; so it is quite question-begging to use the standard scheme against indeterminate identity. Third, in all of philosophy there is no question more contested than that of the meaning of ‘if ’. On some approaches, e.g. the suppositional one of (Barnett 2006), evaluating a conditional requires supposing the antecedent to be true. No wonder the gap between LL→ and LL→ " goes unnoticed. So a ‘logical oversight, akin to the Aristotelean logician’s’ is not so far-fetched. ¹⁰ By ‘semantically indiscriminable’ I mean that a competent speaker in full possession of the facts (that don’t logically entail an assignment of statuses to antecedent and consequent) would be unable to provide good reasons for assigning antecedent and consequent different semantic statuses. ¹¹ Crispin Wright has emphasized that Sorites paradoxes can be formulated with premises of the form ¬(p & ¬q) (‘it’s not the case that this man’s bald and his neighbor isn’t’) which seem as plausible as their counterpart conditionals (‘if this man’s bald so’s his neighbor’), and thus should have as high dt’s; see, e.g. Wright 1987. But the standard treatment of & as min in fuzzy logic produces the wrong result, for ¬(p & ¬q) has a middling dt in a Sorites if p does. However, revisions to the fuzzy logical account of & along the lines of those proposed in (Edgington 1996, 306–8) seem to have good prospects of handling this difficulty.
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Old Number One is Old Number Four. Old Number Four is the car Dunfee crashed in. The car Dunfee crashed in is the car Hassan rebuilt. The car Hassan rebuilt is the car Hubbard bought. The car Hubbard bought is the car he sold to Middlebridge. Therefore, Old Number One is the car Hubbard sold to Middle bridge.
We can employ even more descriptions and descriptive names to increase the number of premises from which (3f ) is inferred, perhaps breaking down the controversial (3b) into multiple separate identities, with a new description or descriptive name for the car that exists after each new part that Hassan installs. The fact that neither Hubbard’s counsel nor Otton produced such an argument suggests, not that they missed an opportunity, but that they knew fallacious reasoning when they saw it. We would like to duplicate the success of fuzzy logic in explaining why all the premises of an effective Sorites seem true, even though some are untrue. However, the requirement that any proposal be applicable to (3) rules out degrees of truth, insofar as degrees of identity are unappealing. It would also be useful to work with something that can be explained more easily than degrees of truth seem to be (Keefe 2000, 91–3). My goal in the rest of this chapter will be to mimic degrees of truth with a different kind of semantic status that becomes available once we recognize the phenomenon of higher-order vagueness. Higher-order vagueness may be introduced by iterating the considerations that motivate some non-classical status for vague expressions in the first place. For standard examples of vague predicates, we deny that there is a specific point on the relevant spectrum (e.g. for the predicate ‘tall’, the spectrum of possible heights) at which they abruptly cease to apply and their fixed-point negations start to apply, because we cannot discern any feature of the world or any aspect of what is involved in mastery of the predicate in virtue of which some specific point would be singled out as the tipping point. So if F is such a predicate, it is conceivable that there is an object x and a proposition p saying that x is F , and there is no fact of the matter whether p, and no fact of the matter whether ¬p. Let us label the semantic status of such a p ‘indeterminate’. So conceivably, for some objects x, the proposition, that x is F , is indeterminate. But the main consideration of the previous paragraph applies over again: there is no specific point on the relevant spectrum at which F abruptly ceases to apply and ‘no fact of the matter whether F ’ starts to apply, and no specific point at which ‘no fact of the matter whether F ’ ceases to apply and ‘not F ’ starts to apply. In both cases, this is because, as before, there is no empirical or linguistic fact which could make any point such a tipping point. So it is conceivable that there is an object x and a proposition p saying that x is F , and there is no fact of the matter whether p and no fact of the matter whether it is indeterminate that p: there is no fact of the matter whether p or it is indeterminate that p.¹² This is also a semantic status, and it has a counterpart ¹² This embedding of a wh-complement induces ambiguity. I intend what Groenendijk and Stokhof call the alternatives reading (1982, 193).
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on the other ‘side’ of indeterminacy, the status of there being no fact of the matter whether it is indeterminate that p and no fact of the matter whether ¬p. So we now have five statuses linearly ordered, and the same considerations about the inconceivability of tipping points motivates the introduction of four more, one between each adjacent two of the first five. And so on; each time a new semantic status s is introduced between two statuses s1 and s2 , there is a refinement introducing two more statuses, one between s1 and s and the other between s and s2 , following the indicated pattern: the one between s1 and s is the status of being a proposition p such that it is indeterminate whether p has the status s1 or the status s, while the one between s and s2 is the status of being a proposition p such that it is indeterminate whether p has the status s or the status s2 .¹³ Suppose we use ! and ⊥ for the first two statuses. For convenience, we identify other statuses with pair sets, where {x, y} is the status introduced as that of being indeterminate between the status x and the status y. Thus, by the two previous paragraphs, there is also the status {!, ⊥}, which we regard as ‘above’ ⊥ but ‘below’ !. It is convenient to associate ! with 1 and ⊥ with 0. We may then construct a dense linear array of semantic statuses embedded in the rational interval [0,1]Q in an orderpreserving way, starting by associating {!, ⊥} with 0.5. Of course, almost all of these statuses are unintelligible, but we can grasp the first few and extrapolate: (4)
a. S 0 : Statuses s1 and s2 , s1 > s2 : s1 = !, s2 =⊥. b. S 1 = S 0 ∪ {s3 }; s3 = {!, ⊥}; s1 > s3 > s2 c. S 2 = S 1 ∪ {s4 , s5 }; s4 = {!, {!, ⊥}} = {!, s3 }, s5 = {{!, ⊥}, ⊥} = {s3 , ⊥}; s1 > s4 > s3 > s5 > s2 . d. S 3 = S 2 ∪ {s6 , s7 , s8 , s9 }; s6 = {!, {!, {!, ⊥}}} = {!, s4 }, s7 = {s4 , s3 }, s8 = {s3 , s5 }, s9 = {s5 , s2 }; s1 > s6 > s4 > s7 > s3 > s8 > s5 > s9 > s2 .
s3 is the status of the first order of vagueness, s4 and s5 are the statuses of the second order, and s6 , s7 , s8 and s9 are the statuses of the third order. The full set of statuses S produced by this construction is the union of a strictly increasing chain C of finite linearly ordered sets S 0 , S 1 , S 2 , . . .; the first four members of C are as in (4). Given S i , we form S i+1 by adding to S i a new status between each pair of adjacent statuses in S i : if s and s are adjacent in S i , we add s = {s , s }. With an indexing scheme starting ∗0 (!) = 1, ∗0 (⊥) = 0, we extend the indexing ∗i to new elements by ∗i+1 (s) = (∗i (s1 ) + ∗i (s2 ))/2. For any s ∈ S, ∗ (s) = ∗i (s) for the first (some, any) ∗i defined for s.¹⁴ ¹³ The existence of higher-order vagueness is nevertheless controversial; see (Wright 1992), (Heck 2003, 123–4), and also (Varzi 2003) and references therein. All I have done in these two paragraphs is gesture at how I would argue for the phenomenon. ¹⁴ The image of ∗ is the subsequence of the rational interval [0, 1]Q which has the same endpoints and includes in addition exactly the rationals m/2n , 1 m < 2n , m odd, n ∈ Z+ , the dyadic fractions in [0, 1]Q . Each S i is indexed by the set of rationals of the form m/2i , with m ranging from 0 through 2i . To accommodate quantifiers, limits should be added to S.
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Note that once we have the whole construction, none of the heuristic linguistic descriptions used in introducing the statuses are any longer applicable. For example, relative to S 2 , s4 is the status of being indeterminate between s1 and s3 . But this is not intrinsic to s4 : in S 3 , s4 lies between s6 and s7 , and so a proposition with status s4 determinately lacks the statuses s1 and s3 . Alternatively, we can think of each s as initially comprising a region of indeterminacy which shrinks to a point as the construction proceeds. Thanks to ∗ (the indexing by [0,1]Q ) we can use essentially clause (2) for the semantics of negation and the conditional. Where v is an assignment of statuses to sentence-letters, we define an extension [[ ]] of v to all formulae of L¬ →: (5)
a. [[π ]] = v(π ); b. [[¬φ]] = ∗−1 (1 − ∗ [[φ]]), where ∗−1 is the inverse of ∗ ; · ∗ [[ψ]])), where — · is cut-off subtracc. [[φ → ψ]] = ∗−1 (1 − (∗ [[φ]] — tion.
Because of (5c) we get the desired diagnosis of the irresistibility of an effective Sorites paradox: each conditional premise is either true, or has a status that is very close to true in the sense of ∗. But repeated chaining by transitivity of ‘→’ accumulates a large number of small departures from ! into a single large departure. If a valid finitepremise form never allows its conclusion to have a lower status than the ∗ –least-instatus of the premises then →-chaining is actually invalid (but see Williamson 1994, 124, against this definition of ‘valid’). If validity is simply guaranteed !-preservation, a standard Sorites is valid, but it still has an untrue premise. It is because the status difference between antecedent and consequent in an untrue premise is so slight that all the premises seem to us to be true. But so long as one premise is not quite true, the argument is unsound. It might be objected that we have only achieved the desired diagnosis by means of an arbitrary association of statuses with elements of [0,1]Q . To this I would reply that while the association has some stipulative aspects, the amount of arbitrariness is small, and smaller than in any genuinely different alternative. The crucial stipulations are two: first, that being the case is associated with 1 and being not the case is associated with 0; and second, that if s is introduced on S between members of an adjacent pair sa , sb from the previous S , then any extension of the indexing of the statuses on S to S must respect the constraint that s should be equidistant between sa and sb . Any violation of the second constraint would produce an unjustifiable asymmetry and be more arbitrary than the scheme we have chosen.¹⁵ So we have achieved our goal of capturing the advantages of the fuzzy logician’s diagnosis of standard Sorites paradoxes, and moreover, we have done so without saddling ourselves with having to explain degrees of truth, and surely worse, degrees of identity. However, before turning to the application of this apparatus to the various puzzles, we should address a point that many readers will have been wanting to interject for some while now: that if we have taken on board some of fuzzy logic’s advantages, we may have taken on board some of its disadvantages as well. ¹⁵ Thanks to Peter Milne for prompting this paragraph.
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The standard objection (e.g. Sainsbury 1991, 11; Tye 1994, 14) is that fuzzy logic simply replaces an implausibly exact classification of cases into two groups with an equally exact but vastly more incredible classification of cases into infinitely many groups. And it shares with all many-valued approaches the prediction that there is a specific premise in any Sorites paradox which is the first premise to be less than wholly true. I would argue, however, that the sharpness of the classification scheme is simply an artifact of the model, not a representational feature of it.¹⁶ What we have succeeded in modelling is how a sequence of true and almost but not quite true conditionals can carry us from a complete truth to a total falsehood. That we have traded notions such as ‘almost but not quite true’ for precise semantic statuses is simply to facilitate the proof of possibility: once we have seen how the paradox deceives us, using the precise framework, we can accept that the same process goes on when the statuses of propositions are themselves vague. We have also succeeded in making differences of semantic status reflect relevant quantifiable differences among objects, at least for countable sets of objects: for if a1 , a2 , a3 and a4 are all in some borderline area, and the difference between a1 and a2 with respect to F -ness is roughly the same as the difference between a3 and a4 , then the semantic status difference between the members of the two pairs will be about the same as well. So the status model has useful representational features, without committing us to there being a fact of the matter which propositions of the form Fai have exactly the status, say, s3 .¹⁷
24.5
T RO U B L E S W I T H T R A N S I T I V I T Y
To apply the apparatus of the previous section to the Storage Room paradox, we need interpretations whose domains of discourse include objects identity propositions over which sometimes have a status other than ! or ⊥. For example, we might have an interpretation with domain D including all the pieces of furniture a0 , . . . , an brought to the storage facility for storage, and all the pieces of furniture b0 , . . . , bn left in the storage room by the movers at the end of the day. [[=]] would be a function from D2 into the set of statuses S, and we would have the obvious clause (6) [[t1 = t2 ]] is identical to [[=]] ( [[t1 ]] , [[t2 ]] ). In a standard interpretation for many-status identity, [[=]] maps to ! exactly the pairs x, x , x ∈ D. A natural interpretation is a standard one which, as before, ¹⁶ See (Shapiro 2006, (50–4) on this contrast, and (Cook 2002) for extended discussion of how it might apply in the present context. ¹⁷ I have little to say about the problem of the first less than wholly true premise. For some premises, there will be no de re fact of the matter whether they are wholly true or slightly less, but it seems that it must be a de dicto fact that in a listing of Sorites premises in their natural order, some premise is the first to be less than wholly true. If there were no fact about this, there would be no fact whether a Sorites is unsound, but, since its conclusion is false, it had better be unsound. I suspect (de dicto) that when the workings of the status semantics are themselves the subject of discussion, there is some reason why it is appropriate to supervaluate over all natural assignments of statuses to the propositions in question.
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is faithful to the indeterminacies in the situation of the application. For the pieces of furniture, the statuses assigned by [[=]] to [[ai ]], [[bi ]] in a natural interpretation move from ! towards ⊥ tracking increases in i reasonably closely. So by (5c), some conditionals of the form ‘if ai = bi then ai+1 = bi+1 ’ are less than wholly true, because the status of ‘ai = bi ’ is higher than that of ‘ai+1 = bi+1 ’. In addition, there will be no difference in status between ai = bi and aj = bj that is very much larger or smaller than the difference between ak = bk and al = bl when the number of new parts in bj exceeds the number of new parts in bi by about the same as the number of new parts in bl exceeds the number of new parts in bk . So we can be confident that although natural assignments will make some conditionals of the form ‘if ai = bi then ai+1 = bi+1 ’ less than wholly true, they will only be slightly less than wholly true, and they will be closer to wholly true the larger the number of furniture-items that get stored. To meet the Uniformity Constraint, we have to extend this treatment to intensional puzzles such as Old Number One and Chisholm’s Paradox, so that these puzzles get defused in essentially the same way. We will use Chisholm’s Paradox for illustration. We let D be a set of possible objects, and as before, the identity or otherwise of some x ∈ D with some y ∈ D can have a non-classical status. We let W be a set of possible worlds, and we assign all of D to every w ∈ W as the domain of w. We want to arrange matters so that for some i, φi (g) → φi+1 (g) has a status slightly less than !. Since the φ-predicates simply record the parts from which g is made, we can assume them to be precise. So we need a world where φi (g) is closer to ! than is φi+1 (g) at any world. The basics can be exhibited just with monadic atomic predicates F and H . V assigns a rigid designation in D to each individual constant, and we let V (F )be a function which for each world as input, outputs a function from D into S. Each such function V (F )(w) is constrained by [[=]] in the following way: for each x ∈ D such that the status of Fx at w is non-classical, for each s ∈ S, the status of Fx at w is s iff for some y ∈ D, (i) the status of Fy at w is !; (ii) the status of x = y is s; (iii) ∃z ∈ D: the status of Fz at w is ! and the status of x = z is higher than s. In addition, the status of Fx at w is ⊥ iff ∃y ∈ D such that the status of Fy at w is ! and the status of x = y is higher than ⊥. And mutatis mutandis for H . So given V (F )(w)’s mappings to !, the rest of V (F )(w) is determined by [[=]] .¹⁸ Writing [[σ ]]w for the status of σ at w in , we then have the evaluation clause (7) [[Fg]]w = [V (F )(w)](V (g)) and mutatis mutandis for Hg. It should now be clear that we can arrange for Fg → Hg to have a status at w that is arbitrarily close to ! but still less than it. For example, we may have a w such that [[Fg]]w is s6 (see (4d)) because (i) ∃u ∈ W : [[Fg]]u = !; (ii) ∃x ∈ D, ∃u ∈ W : [[Fx]]u is ! and [[=]]( x, g ) is s6 ; and (iii) [[=]]( y, g ) s6 ¹⁸ The effect of these conditions is to make the extensions of F and G at w what Woodruff and Parsons (1999, 477–8) call tight sets: only by being indeterminately identical to a classical member of a set x is an object’s ∈-status with respect to x non-classical. Tight sets are the appropriate ones for the extensions of precise predicates over a domain with indeterminate identity.
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for any other y ∈ D, u ∈ W, with [[Fy]]u = !; but max({[[Hg]]u : u ∈ W }) is s4 (see (4c)) because the best [[=]] can do for any y ∈ D for which ∃u ∈ W with [[Hy]]u = ! is [[=]]( y, g ) = s4 . So [[Hg]]w is s4 . If we then generalize the account of conditionals in (5c) to intensional models, we will have [[Fg → Hg]]w = s6 . There are obvious affinities here with the counterpart-theoretic solution. But a counterpart relation is non-transitive: nothing prevents the degree of truth of ‘a is a counterpart of c’ being d , d < 1, even when the degrees of truth of both ‘a is a counterpart of b’ and ‘b is a counterpart of c’ are 1. Whereas, of course, a = b, b = c a = c. So if there is any non-zero amount of change in relevant respects consistent with transworld identity, we will not be able to get the result ! > [[Fg → Hg]]w >⊥ in any case where [[Fg]]w is !. We can illustrate the difficulty with the case of Old Number One. Suppose Hassan does the 1932 modifications over a 5-day period, making equal and accumulating modifications each day. We might like to say that the same car is in his workshop on adjacent days—the Monday car is the Tuesday car, the Tuesday car is the Wednesday car, and so on—but the same car is not present across a larger timespan—the Monday car is not the Wednesday car, say. Unfortunately, the Monday car being the Tuesday car and the Tuesday car being the Wednesday car entails that the Monday car is the Wednesday car. Nothing changes if we switch to a variable-domain semantics or relativize identity to times in the manner of non-logical predicates. The Tuesday car is Tuesday-identical to both the Monday car and the Wednesday car, so these cars exist on Tuesday, and by transitivity, the Monday car is Tuesdayidentical to the Wednesday car. So the one-day-of-modifications limit on persistence is violated. Hume would have said that this is to be expected. Identity, he held, is incompatible with any change: ‘in its strictest sense’ identity may be applied only to ‘constant and unchangeable objects’ (Treatise, Bk.1, Pt. 1, §5). There is a looser way of speaking, in which we attribute identity in a way that is tolerant to certain amounts and kinds of change. But Hume regards this looser way of speaking as erroneous, as resulting from overlooking, for this or that reason, the changes (Treatise, Bk. 1, Pt. 4, § 6). Within the framework developed here, a version of Hume’s view is significantly more palatable than ordinarily thought, for two reasons. First, Hume says that any change conflicts with identity, which presumably includes such changes as ones in location, size and shape. But the problematic changes are really only those a large number of which, each of undetectably small magnitude, can accumulate into a change so great that it threatens persistence through time or transworld identity (for statues, shape-change is admittedly one such). And these changes are equally ones which the accessibility theorist must regard as slightly reducing relative possibility and the counterpart theorist as slightly reducing degree of counterparthood. Secondly, Hume’s error thesis has not been well received because his accounts of the errors and why we are susceptible to them creak. However, fuzzy logicians have a better error thesis: those who are bewildered by Sorites reasoning accept certain conditionals—ones that involve standard vague predicates like ‘heap’ and ‘bald’—that are not strictly true, and their acceptance (however reluctant) is explained by the shortfall of
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the conditionals from truth being undetectable in normal circumstances. The credibility of this should carry over to the apparatus of statuses, for the same types of conditionals. But then, if we consider undetectably small amounts of change of the sort we have been concerned with, a parallel error thesis will have the same credibility: we are mistakenly taking an identity-judgement to have status ! when it only has status sn , where sn is as near true as to be true, for all we can tell. Once we combine a sequence of such judgements (e.g. the premises of (3)), with transitivity playing the role of modus ponens, the problem becomes evident, or, as we might put it, imperceptible errors have an evidently erroneous consequence. Or at least, if not evidently erroneous, then evidently debatable, something we might end up in court over. Re f e re n c e s Barnett, D. (2006), ‘Zif is if ’, Mind 115, 519–65. Chisholm, R. (1968), ‘Identity through possible worlds: some questions’, Noˆus 1, 1–8. Cook, R. (2002), ‘Vagueness and mathematical precision’, Mind 111, 225–47. Edgington, D. (1996), ‘Vagueness by degrees’ in (Keefe and Smith 1996). Evans, G. (1978), ‘Can there be vague objects?’, Analysis 38, 208. Also in (Keefe and Smith 1996). Fara, M and Williamson, T. (2005), ‘Counterparts and actuality’, Mind 114, 1–30. Forbes, G. (1983), ‘Thisness and vagueness’, Synthese 54, 235–59. Groenendijk, J. and Stokhof, M. (1982), ‘Semantic analysis of WH-complements’, Linguistics and Philosophy 5, 175–233. Heck, R. (2003), ‘Semantic accounts of vagueness’ in Jc Beall, ed., Liars and Heaps: New Essays on Paradox, Oxford University Press, Oxford. Keefe, R. (2000), Theories of Vagueness, Cambridge University Press, Cambridge. Keefe, R. and Smith, P., eds., (1996), Vagueness: A Reader, MIT Press, London. Lewis, D. (1968), ‘Counterpart theory and quantified modal logic’, The Journal of Philosophy 65, 113–26. Parsons, T. (2000), Indeterminate Identity, Oxford University Press, Oxford. Peacocke, C. (1999), Being Known, Oxford University Press, Oxford. Quine, W. V. O. (1976), ‘Worlds away’, The Journal of Philosophy 73, 859–63. Raffman, D. (1994), ‘Vagueness without paradox’, The Philosophical Review 103, 41–74. Robertson, T. (2000), ‘On Soames’s solution to the sorites paradox’, Analysis 60, 328–34. Sainsbury, M. (1991), ‘Concepts without boundaries’, Inaugural Lecture, Stebbing Chair of Philosophy, King’s College, London (published by the Department of Philosophy, King’s College, London). Reprinted in (Keefe and Smith 1996), page references to this printing. Salmon, N. (1981), Reference and Essence, Princeton University Press, Princeton. . (1986), ‘Modal paradox: parts and counterparts, points and counterpoints’, in P. A. French, T. E. Uehling and H. K. Wettstein, eds., Midwest Studies in Philosophy XI: Studies in Essentialism, University of Minnesota Press, Minneapolis. . (2002), ‘Identity facts’, Philosophical Topics 30, 237–67. Soames, S. (1999), Understanding Truth, Oxford University Press, Oxford. Sorensen, R. (1988), Blindspots, Oxford University Press, Oxford. Tye, M. (1994), ‘Sorites paradoxes and the semantics of vagueness’ in J. E. Tomberlin, ed., Philosophical Perspectives 8: Logic and Language, Ridgeview Publishing Company, Atascadero. Varzi, A. (2003), ‘Higher-order vagueness and the vagueness of ‘‘vague’’ ’, Mind 112, 295–9.
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Williamson, T. (1994), Vagueness, Routledge, London. (1999), ‘On the structure of higher-order vagueness’, Mind 108, 127–43. (2002), ‘Vagueness, identity and Leibniz’s law’ in A. Bottani, D. Giaretta and M. Carrara, eds., Individuals, Essence and Identity: Themes of Analytic Metaphysics, Reidel, Dordrecht. Woodruff, P. and Parsons, T. (1999), ‘Set theory with indeterminacy of identity’, Notre Dame Journal of Formal Logic 40, 473–95. Wright, C. (1987), ‘Further reflections on the sorites paradox’, Philosophical Topics 15, 227–90. (1992), ‘Is higher-order vagueness coherent?’, Analysis 52, 129–39.
25 Fuzzy Epistemicism John MacFarlane
It is taken for granted in much of the literature on vagueness that semantic and epistemic approaches to vagueness are fundamentally at odds. If we can analyze borderline cases and the sorites paradox in terms of degrees of truth, then we don’t need an epistemic explanation. Conversely, if an epistemic explanation suffices, then there is no reason to depart from the familiar simplicity of classical bivalent semantics. Thus, while an epistemic approach to vagueness is not logically incompatible with the view that truth comes in degrees, it is usually assumed that there could be no motivation for combining the two. My aim in this chapter is to question this assumption. After describing the way in which many-valued theories are usually motivated in opposition to epistemicism (Section 25.1), I give an argument for degrees of truth that even an epistemicist should be able to accept (Section 25.2). Unlike traditional motivations for degree theories, this argument is compatible with the epistemicist’s claim that we are irremediably ignorant of the semantic boundaries drawn by vague terms, and with nonsemantic (epistemicist and contextualist) approaches to the sorites paradox. Thus it opens up conceptual space for a hybrid between fuzzy and epistemic approaches, a ‘fuzzy epistemicism.’ According to fuzzy epistemicism, both uncertainty and partial truth are needed to understand our attitudes towards vague propositions. In Section 25.3, I consider how this hybrid theory can respond to some traditional objections to many-valued theories. I do not think that this all adds up to a compelling case for fuzzy epistemicism as the best approach to vagueness. As I will indicate, there are a couple of nonepistemicist approaches that seem at least equally promising. My aim here is to show that if one is inclined towards epistemicism, then (contrary to the conventional wisdom) one has good reason to accept degrees of truth as well. I presented versions of this chapter in June 2007 at the Arch´e Vagueness Conference in St. Andrews, Scotland, and the LOGICA Conference in Hejnice, Czech Republic. I am grateful to audiences at both conferences for their comments, and particularly to Dorothy Edgington, my commentator at St. Andrews. I would also like to thank Branden Fitelson, Michael Caie, Fabrizio Cariani, Elijah Millgram, Stephen Schiffer, Mike Titelbaum, and two anonymous referees for useful correspondence.
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If ‘tall man’ has a classical extension,¹ then there is a shortest tall man. Of course, we have no way of knowing how tall the shortest tall man is. And even if we could know, the placement of the line between the tall and the non-tall would appear arbitrary. Unlike ‘gold’ and ‘water’, ‘tall’ does not seem to pick out any kind of natural property. Nor does anything about our use of ‘tall’ make any particular cut-off point salient. So classical semantics is committed to unknowable and arbitrary-seeming semantic boundaries. Epistemicism is an attempt to bite this bullet, by explaining on general epistemological grounds why we should expect to be ignorant in just this way, and by rejecting as verificationist the idea that we should be in a position to know exactly where the semantic boundaries lie. According to the epistemic approach, what distinguishes vague language from non-vague language has nothing to do with truth-conditions. Formally, then, epistemicism is compatible with both classical and non-classical semantics. Typically, however, epistemicists defend classical semantics. One popular alternative to classical semantics is to suppose that truth comes in degrees. The most common form of this view represents these degrees by real numbers between 0 and 1, with 1 representing complete truth, 0 complete falsity, and the intermediate values various degrees of ‘partial truth.’ The extensions of predicates are then naturally understood as fuzzy sets, or mappings from objects to degrees of truth. Thus, ‘tall man’ may map a 7-foot man onto 1, a 6-foot man onto 0.75, a 5-foot-11 man onto 0.68, and so on. Small differences in height will yield small differences in the degree to which the predicate is satisfied. So as we look at shorter and shorter men, we will see a slow, steady decline in the degree to which ‘tall man’ applies, rather than a sudden, precipitous change from inclusion in the extension to non-inclusion. Such a theory affords an attractive analysis of the sorites paradox. Suppose we have a line of 100 men of gradually increasing height. Man 0 satisfies ‘tall man’ to degree 0, man 1 to degree 0.01, man 2 to degree 0.02, and so on up to Man 100, who satisfies ‘tall man’ to degree 1. Now consider the following sorites argument: (1) (C100 ) (C99 ) .. . (C1 ) (2)
Man 100 is a tall man. If Man 100 is a tall man, Man 99 is a tall man. If Man 99 is a tall man, Man 98 is a tall man.
If Man 1 is a tall man, Man 0 is a tall man. Therefore, Man 0 is a tall man.
On the Łukasiewicz semantics for the conditional, [[A → B]] = 1 if [[B]] > [[A]] and 1 − ([[A]] − [[B]]) otherwise (where [[φ]] denotes the degree of φ). So all of ¹ Relative to a context and an index of evaluation. I will not repeat this qualification in what follows.
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the conditionals C1 . . . C100 have degree 0.99. That is, they are all almost completely true, and that, the degree theorist proposes, is why we are inclined to accept them. But although modus ponens is valid in the sense of preserving degree 1, it is not valid in the sense of preserving degree of truth in general. Thus, when the premises of a modus ponens inference do not all have degree 1, the conclusion can have a lower degree than any of the premises. With each application of modus ponens, then, we lose a little truth, so that by the end of the argument we have none left at all. Notice how the degree theory is motivated as an alternative to epistemicism. By positing a smooth continuum of partial truth, we avoid the need to explain how our linguistic practices could fix a sharp boundary between the tall and the non-tall, and why we could not know where it lies. And by making it possible to say that the premises of the sorites are almost completely true, we avoid the need to explain why we should be inclined to accept a conditional that is just plain false (as one of C1 . . . C100 must be, if classical semantics is correct). The standard epistemicist response to such theories is to argue that they merely put off the pain, because the epistemicist’s resources will be needed anyway, at a later stage of analysis. So if the point of degree theories is to avoid having to tell epistemic stories, these theories are unmotivated. Let us look at some arguments to this effect.
25.1.1 Hidden boundaries One of the things that seemed objectionable about classical semantics was its commitment to unknowable, arbitrary-seeming semantic boundaries. But do degree theories do better? Just as on classical semantics, there will be a shortest man who falls into the extension of ‘tall man,’ so on a many-valued semantics, there will be a shortest man who satisfies ‘tall man’ to degree 1. A man 1 mm shorter than this man will not satisfy ‘tall man’ to degree 1. We have no way of knowing where this boundary lies, and even if we could know it, it would seem arbitrary. So the degree theory does not have any evident advantage over classical semantics in this respect. Roy Sorensen puts the point effectively: . . . advocates of alternative logics that use the sensitivity objection against the epistemic approach are guilty of special pleading. Given that the super-valuationists and many-valued theorists cannot use the sensitivity issue to claim an advantage over classical logic, what is left to recommend their positions? The central motive for appealing to these alternative logics was to avoid the commitment to unlimited sensitivity. Once it is conceded that this appeal cannot succeed, there is no longer any point in departing from classical logic. (Sorensen, 1988, 247)²
Degree theorists standardly respond that their precise assignments of degrees are meant as models of something imprecise. The sharp boundaries, they say, are just ² See also (Keefe, 2000, 115): ‘The best epistemic theorists offer detailed explanations of why we are ignorant in a borderline case . . .; a degree theorist taking option (i) similarly owes us an explanation of the ignorance it postulates, but one that does not at the same time justify the epistemic theorist’s position about first-order vagueness. It is far from clear that this can be done.’
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artifacts of the numerical models being used (Edgington 1997, 297, 308–9; Cook 2002). This is a plausible response, but more must be said. Degree theorists ought to say which features of their models are artifacts, and which are meant to represent real features of degrees of truth (Keefe, 2000). An obvious thought is that the ordering of the numerical degrees represents the real ordering of degrees of truth, even if it is an artifact which degree is represented by the number 0.5. But if the ordering is non-artifactual, so is the boundary between the maximal degree and all the others. So, also, is the question which of a series of successively taller women satisfies ‘tall’ to a greater degree than Sarah satisfies ‘short.’ Indeed, as Rosanna Keefe points out, the degree theorist cannot coherently hold that only ordinal relations between numerical degrees represent relations between real degrees of truth. For the Łukasiewicz semantics for the conditional makes the ordinal position of conditionals depend on the absolute difference of the numerical degrees of their antecedents and consequents (Keefe, 2000, ch. 5). So if we have conditional propositions, then the absolute distances between numerical degrees cannot be artifacts of the model unless some facts about ordering are also artifacts. A natural proposal, explored by Cook 2002, is that only large differences in numerical degree represent real differences in degrees of truth (cf. Edgington, 1997, 297–8). As Cook shows (244), this proposal is not strictly tenable: for example, on Edgington’s theory, if there are n mutually independent propositions, there will be at least some non-artifactual differences in degree less than or equal to 1/2n , and for plausibly large values of n, these differences will be very small. Importantly, though, these small differences will be knowable in principle, since they can be predicted from the semantics of the connectives. So perhaps it is a sufficient reply to the epistemicist’s tu quoque about unknown and arbitrary semantic boundaries to say that . . . truth (and falsity) do come in gradations, and both large differences in real number assignments and the logical relations between complex sentences and their constituents are indicative of real aspects of vague natural language. On the other hand, the assignment of particular real numbers to particular sentences, and the resulting sharp boundaries, are just conveniences, incorporated into the semantics for the sake of simplicity, but reflecting nothing actually present in the discourse being modeled. (Cook, 2002, 245)
As Cook notes, to say this is not to make the semantics itself imprecise, since the word ‘large’ is used not in the formal semantics, but in our informal description of how the semantics models linguistic reality. The fit between a formal model and the reality it models should not be expected to be precise. I won’t try to assess this response here. What’s important for our purposes is that both sides in the debate assume that, if the numerical degrees are viewed in a strongly representational way, and not as models with many artifactual features, then degree theory is unmotivated. Both sides agree that if we are going to accept hidden and arbitrary-seeming semantic boundaries, we might as well stick with a bivalent semantics. That is why the degree theorist must parry the classicist’s tu quoque by adopting the modeling perspective.
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25.1.2 The sorites It might be thought that the attractive many-valued analysis of the sorites paradox provides an independent reason for preferring many truth values to two. But on closer examination, this apparent advantage evaporates. As Weatherson 2005 observes, the sorites is no less compelling when run with negated conjunctions instead of conditionals: (1) (NC100 ) (NC99 ) .. . (NC1 ) (2)
Man 100 is a tall man. It’s not the case that Man 100 is a tall man and Man 99 is not. It’s not the case that Man 99 is a tall man and Man 98 is not.
It’s not the case that Man 1 is a tall man and Man 0 is not. Therefore, Man 0 is a tall man.
But with the usual many-valued semantics for the connectives,³ (NC50 ) gets degree 0.5—meaning that it is no more true than false. What this shows is that we can’t hope to explain the plausibility of the sorites argument solely by pointing to the very high degree of truth of its premises, since only in the conditional version of the argument do all the premises have a high degree of truth. This is not to say that a degree theorist can’t explain the plausibility of the sorites—just that the explanation cannot advert to the ‘near complete truth’ of the premises. Weatherson endorses Kit Fine’s suggestion that we are prone to confuse P with Determinately P, even when P occurs as part of a larger sentence. So we take (NC50 ) to be true because we conflate it with d ) It is not the case that Man 50 is determinately tall but Man 49 is determi(NC50 nately not tall.
But as Weatherson notes, ‘Fine’s hypothesis gives us an explanation of what’s going on in Sorites arguments that is available in principle to a wide variety of theorists’—supervaluationists, classical semanticists, and degree theorists alike. As a result, a degree theorist who makes use of this explanation cannot claim to have an advantage over any of these other theories in explaining the plausibility of sorites arguments.⁴ Other explanations of the pull of the sorites are also possible. Perhaps we mistake our inability to give a counterexample to (NC50 ) for evidence of its truth. Williamson ³ [[P & Q]] = max([[P]], [[Q]]) and [[¬P]] = 1 − [[P]]. This is the semantics that is usually discussed in the philosophical literature on degree theories (e.g. in Machina 1976, Williamson 1994, and Keefe 2000). Different choices are made in the fuzzy logic literature (see Hajek 2006). In ‘Łukasiewicz logics,’ strong conjunction is defined as follows: [[P & Q]] = max(0, [[P]] + [[Q]] − 1). If the conjunctions in our sorites are understood this way, the NCi ’s will all have degree 0.99. However, as we will see in the next section, there are strong reasons (independent of the sorites) for the degree theorist not to define conjunction this way. See note 5, below. ⁴ Weatherson, who is arguing for a kind of degree theory himself, concedes that he doesn’t ‘have a distinctive story about the Sorites in terms of truer.’
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1994 argues that, because of general ‘margin of error’ requirements on knowledge, we could never know that we had a counterexample (234). Contextualists argue that active consideration of a particular height changes the context so that the extension of ‘tall’ draws no boundaries there (Raffman 1996; Soames 1999; Fara 2000). Either of these strategies might explain why we are unable to refute (NC50 ), and hence why it seems plausible. There is no reason why a degree theorist couldn’t appeal to these explanations of the plausibility of the sorites. But then the degree theorist’s semantics would not be doing any work in explaining the apparent force of sorites arguments. So, one wonders, why not just stick with the simpler classical semantics? To sum up: the usual motivations for a degree-theoretic account of vague expressions assume that epistemic accounts of the sorites and of borderline cases are untenable. Both sides in the debate agree that if the degree theorist were to accept the epistemicist’s explanation of our ignorance of the locations of sharp semantic boundaries, the game would be lost. They agree that there would be no point being an epistemicist and accepting a many-valued semantics, since the epistemicism would deprive the many-valued semantics of any useful job to do. 25.2
A N EW A RG U M E N T F O R D E G R E E S
Having brought this assumption into the open, I now want to question it. I will present a new argument for a many-valued semantics for vague discourse. Unlike the standard motivations for degree theories, this one is compatible with epistemicism and does not require a ‘modeling’ perspective on numerical degree values. The core of the argument is an acute observation by Schiffer 2003. Though Schiffer himself rejects degree theories and argues instead for a complex ‘psychological’ theory, I will argue that the position that Schiffer’s observation really supports is a degree theory that accepts hidden semantic boundaries—a hybrid of traditional degree theories and traditional epistemic theories.
25.2.1
Combining uncertainties
Consider Borderline Jim. He’s just short of six feet tall, with a small tuft of hair on his head, and he’s pretty fast at solving sudoku puzzles, though not as fast as his brother Bill. He is, we might say, borderline tall, borderline bald, and borderline smart. Given Jim’s borderline status, it would be wrong for us to flat-out believe that he is tall, bald, or smart. But it would also be wrong to flat-out believe that he is not tall, not bald, or not smart. The appropriate attitude is something between full acceptance and full rejection, though what kind of attitude is less clear. Classical semantics would seem to commit us to a particularly simple answer to this question. Since according to classical semantics, there are facts of the matter as to whether Jim is tall, bald, or smart, our attitude toward each of these propositions should be one of uncertainty. If Jim is a paradigm borderline case—right in the middle between clear satisfiers of these predicates and clear non-satisfiers—we might take
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it to be 50% likely that Jim is bald, 50% likely that he is tall, and 50% likely that he is smart. Rather than full belief, we will have partial beliefs—credences of 0.5—in each of these propositions. But what should our attitude be to the conjunction of these propositions? Assuming (harmlessly, I think) that these propositions are stochastically independent, our credence in the conjunction ought to be the product of our credences in the conjuncts: 0.125. Classical semantics, then, recommends that we should endorse conjunctions of independent borderline propositions much less strongly than we endorse the conjuncts individually. But, as Schiffer observes, this just seems wrong (Schiffer, 2003, 204). It seems perfectly appropriate to endorse the conjunctive proposition that Jim is tall and bald and smart to about the same (middling) degree as we endorse the conjuncts separately. Certainly it seems wrong that we should be quite confident (0.875) that Jim doesn’t have all three properties. If you don’t have these intuitions, try increasing the number of independent properties. With seven independent properties, your credence that Jim has all of them should be less than 0.01, and your credence that Jim doesn’t have all of them greater than 0.99. That is, if Jim is also borderline fat, borderline old, borderline rich, and borderline nice, you should be very confident that he is not tall, bald, smart, fat, old, rich, and nice. Are you? The argument, then, runs as follows: 1. If classical semantics is correct for vague discourse, then borderline propositions are either true or false; no finer distinctions are made. 2. If borderline propositions are either true or false, then (since we don’t know which truth value they have) our attitudes toward them must be attitudes of uncertaintyrelated partial belief. 3. If our attitudes towards borderline propositions are attitudes of uncertaintyrelated partial belief, they ought to obey norms of probabilistic coherence. 4. We regard the propositions Jim is tall, Jim is bald, and Jim is smart as independent. That is, we don’t think Jim’s being bald (or smart, or bald and smart) would make it any more likely that he is tall, and so on. 5. Probabilistic coherence demands that our credence in the conjunction of several propositions we take to be independent be the product of our credences in the conjuncts. 6. But it is not the case that we ought to have much less credence that Jim is bald and tall and smart than we have that he is bald. 7. Therefore, classical semantics is not correct for vague discourse.⁵ ⁵ A similar argument can be used to rule out many-valued theories in which conjunction is understood as Łukasiewicz ‘strong conjunction’ (see note 3, above). On such theories, P & Q & R will have degree 0 when P, Q, and R each have degree 0.5. So this kind of fuzzy theorist will be even less well placed than the classical logician in accounting for our partial endorsement of the conjunction.
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Unlike the usual arguments against classical semantics for vague discourse, this argument is not aimed at the classicist’s commitment to unknowable and arbitraryseeming semantic boundaries, and it has nothing to do with sorites arguments. Instead, it is aimed at the idea that our attitude toward borderline propositions is one of uncertainty as to whether they are true or false.⁶ One might try to defend classical semantics by rejecting (2). This is essentially what Schiffer does. ( Though he does not present his view as a way of defending classical semantics, he emphasizes that it is a psychological solution to the sorites, and is thus at least consistent with classical semantics.) Schiffer argues that our attitude to borderline propositions is not standard uncertainty-related partial belief (SPB), but a special kind of vagueness-related partial belief (VPB): ‘It is a primitive and underived feature of the conceptual role of each concept of a vague property that under certain conditions we form VPBs involving that concept, and it is in this that vagueness consists’ (Schiffer, 2003, 212). VPBs are distinguished from SPBs in the following ways (198–207): •
SPBs represent uncertainty, while VPBs represent ambivalence. SPBs generate corresponding likelihood beliefs, while VPBs do not. If one has a SPB of 0.5 that one left one’s glasses at the office, one will take it to be 50% likely that one’s glasses are at the office. But if one has a VPB of 0.5 that Jim is bald, one will not take it to be 50% likely that he is bald. • Generally, if one has an intermediate SPB that p, one thinks that one is not in the best possible epistemic position to pronounce on p. But one can have an intermediate VPB that p and think that one could not be in a better epistemic position to pronounce on p. • SPBs are governed by norms of probabilistic coherence, whereas VPBs are governed by the Łukasiewicz many-valued truth tables. Thus, if one has a VPB of 0.5 that Jim is bald and a VPB of 0.5 that Jim is tall, one ought to have a VPB of 0.5 that Jim is bald and tall, even when the conjuncts are independent. •
⁶ Sorensen seems to reject the intuition that supports premise (6). He argues as follows against degree theories: ‘. . . suppose a speaker begins by describing Ted as short and then adds that he is also fat, bald, smart, athletic, and rich. We assign a degree of truth of 0.5 to ‘‘Ted is short’’ and 0.6 to each of the remaining attributions. But contrary to the conjunction rule [of many-valued semantics], we do not believe that ‘‘Ted is short, fat, bald, smart, athletic, and rich’’ equals the degree of truth of ‘‘Ted is short.’’ Our uncertainties compound making us assign a much lower degree of truth to the claim that Ted exemplifies the conjunctive predicate. . . . Also notice that ‘‘Ted is fat, or bald, or smart’’ is less of a borderline attribution than ‘‘Ted is fat,’’ (Sorensen, 1988, 235–6). Note that this argument just assumes that the degrees represent ‘uncertainties,’ which the degree theorist ought to deny. An alternative way of rejecting (6), suggested by an anonymous referee, would be to acknowledge the intuitions that are taken to support it, but claim that they are misleading and not to be taken as normative. Psychologists have shown that ordinary intuitions about probabilities frequently violate even the most basic norms of probabilistic coherence: in one famous case, a majority of subjects took a conjunction to be more likely than one of its conjuncts (Kahneman and Tversky 1983; for a different interpretation of the data, cf. Crupi et al. forthcoming). Could it be that the intuitions to which Schiffer has drawn our attention are the result of the ‘conjunction fallacy’ or something similar? That seems unlikely, since these intuitions can be found even in those who are not prone to probabilistic fallacies when vagueness is not in play. But there is room for further empirical investigation here.
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It’s this last feature that allows Schiffer’s theory to say that our degree of belief that Jim is tall and bald and smart shouldn’t be less than our degree of belief in any of the conjuncts singly, when Jim is a borderline satisfier of each predicate. Schiffer insists, reasonably, that (*) SPB(p) + SPB(¬p) + VPB(p) + VPB(¬p) = 1. Where p is a complete borderline case, SPB(p) and SPB(¬p) will both be 0, and VPB(p) and VPB(¬p) will sum to 1; where p is fully determinate, the VPBs will be 0 and the SPBs will sum to 1. But mixed cases are also possible, and on these Schiffer’s theory runs aground. Suppose, for example, that you think there’s a 50% chance that Sam is completely hairless and a 50% chance that he has about 50 hairs on his head. (You can’t remember which of two men he is.) If you knew he was completely hairless, you’d have an SPB of 1 that Sam is bald. If you knew that he had 50 hairs, you’d have a VPB of 0.8 that Sam is bald, and of 0.2 that he is not bald. But given your uncertainty, you’re in a mixed state, with some SPB and some VPB in both the proposition that Sam is bald and its negation. Schiffer gives some plausible principles for computing SPBs and VPBs in cases like this, but as I show in MacFarlane 2006, they are inconsistent with (∗).⁷ The basic problem should be evident: the norms governing SPBs and VPBs are fundamentally different, so they are not going to march in the kind of lockstep that would be needed to keep them summing to 1.⁸
25.2.2
Taking-to-be-partially-true
Let us return to the problem Schiffer’s theory was supposed to solve. Some kind of partial or qualified endorsement seems appropriate for borderline propositions. However, this partial endorsement does not seem to be standard uncertainty-related partial belief, since if it were, the degree of endorsement would drop dramatically as we added independent conjuncts. How, then, should we understand it? Here, at last, we have a task degrees of truth are well suited to perform. My proposal, to simplify slightly, is that we understand this partial endorsement not as partial belief in the truth of a proposition, but as belief in its partial truth. That is not quite the right thing to say, as it makes the attitude seem like a thought about a proposition, not about (say) Jim. In addition, it makes it seem as if the attitude requires deployment of a concept of degrees of truth—a concept many believers lack. But just as we might usefully understand first-order belief as taking-to-be-true, so we might ⁷ The fix Schiffer proposes in his reply (Schiffer, 2006) does not work. In fact, the first counterexample in Macfarlane 2006—SPB(p) = VPB(p) = SPB(q) = VPB(q) = 0.3, SPB(¬p) = VPB(¬p) = SPB(¬q) = VPB(¬q) = 0.2—is a counterexample to Schiffer’s revised proposal as well, and it is easy to generate others. ⁸ An alternative approach, due to Hartry Field (2003), is to avoid positing VPBs but allow SPB(p) + SPB(¬p) < 1. In cases we take to be completely indeterminate, SPB(p) + SPB(¬p) will be 0. Field’s approach agrees with Schiffer’s in predicting that one should have the same degree of belief in the proposition that Jim is tall and tall and smart that one has in the conjuncts separately, but disagrees about what this degree should be—for Field, it is 0. Schiffer objects (210 n. 38) that agents should not have the same degree of belief (0) in propositions they take to be borderline as they do in propositions they take to be determinately false.
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understand the first-order partial endorsement appropriate in borderline cases as taking-to-be-partially-true (for example, taking-to-be-true-to-degree-0.5). In describing the attitudes this way, we identify them by their constitutive aims. Mark Sainsbury puts the point well: Truth is what we seek in belief. It is that than which we cannot do better. So where partial confidence is the best that is even theoretically available, we need a corresponding concept of partial truth or degree of truth. Where vagueness is at issue, we must aim at a degree of belief that matches the degree of truth, just as, where there is no vagueness, we must aim to believe just what is true. (Sainsbury, 1995, 44)
An attitude towards p that a cognitive system normatively ‘aims’ to be in just in case p is true can justly be called ‘taking-to-be-true,’ even if the possessor of this attitude lacks an explicit concept of truth. Similarly, an attitude towards p that a cognitive system normatively aims to be in just in case p is true to degree N can be justly be called ‘taking-to-be-true-to-degree-N ,’ even if the possessor of the attitude lacks an explicit concept of partial truth. Attitudes of taking-to-be-partially-true, I suggest, can do all of the work Schiffer aimed to do with his VPBs: 1. They can be clearly distinguished from attitudes of uncertainty. They reflect, rather, ambivalence: in a case where I take p to be partially true and partially false, I am ambivalent about whether p. 2. They fail to generate likelihood beliefs. To take p to be true to degree 0.3 is not to take it to be 30% likely that p. 3. Taking p to be partially true is consistent with taking oneself to be in the ‘best possible epistemic position to pronounce on p.’ Partial truth is an objective status, not a feature of the thinker’s mental state or epistemic position. 4. Attitudes of taking-to-be-partially-true, unlike attitudes of partial belief, are not governed by norms of probabilistic coherence. If one takes the propositions that Jim is tall, that Jim is bald, and that Jim is smart to be true to degree 0.5, then one should take their conjunction to be true to degree 0.5 also. (On the Łukasiewicz semantics for continuum-valued logics, the degree of a conjunction is the minimum of the degrees of its conjuncts.) Schiffer’s VPBs look like a way of trying to get the benefits of a degree theory without accepting the idea that truth comes in degrees. But why not go for the original instead of this ersatz? Schiffer offers two arguments, neither of which is compelling.⁹ His first argument is that degree theories cannot capture what Crispin Wright calls ‘the absolutely basic datum that in general borderline cases come across as hard cases’ (Wright, 2001, 69–70). Schiffer argues that a degree theorist . . . is evidently constrained to hold that p is true just in case p is T to degree 1 (or—allowing for the vagueness of ordinary language ‘true’—to a contextually relevant high degree); false ⁹ Here I echo some of the discussion of MacFarlane 2006.
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just in case p is T to degree 0 (or to a contextually relevant low degree); and neither true nor false just in case p is T to some (contextually relevant) degree greater than 0 and less than 1. But suppose Harry is borderline bald. Then, since it would be definitely wrong to say that ‘Harry is bald’ is T to degree 1 (or to some other contextually relevant high degree), the theory entails that it would also be definitely wrong to say it is true that Harry is bald. But if Harry is borderline bald, it would not be definitely wrong to say that he’s bald, and thus not definitely wrong to say it’s true that he’s bald. (Schiffer, 2003, 192)
In assuming that a degree theorist is ‘constrained to hold’ that p is true simpliciter just in case its degree of truth exceeds some (perhaps contextually determined) threshold, Schiffer is thinking of a degree theory as a way of systematizing all-out truth and falsity assignments. That is one kind of degree theory. But on the more thoroughgoing degree theory recommended here, the degrees are given a significance directly, not indirectly through their role in systematizing ‘designatedness’ or all-out truth.¹⁰ According to this theory, when it is true to degree 0.5 that Harry is bald, it will be just as correct to believe that Harry is bald as it is to believe that Harry is not bald, and it will be just as correct to believe that it is true that Harry is bald as it is to believe that it is false that Harry is bald. This, I think, admirably captures the ‘ambivalence’ we feel in borderline cases. Schiffer mischaracterizes this ambivalence in representing it as indecision about whether to assert the borderline proposition. It simply isn’t correct to assert p when p is a borderline proposition, unless one is trying to effect some kind of ‘accommodation’ (Lewis, 1979) that would make it no longer count as borderline. Schiffer’s second argument against degree theories is that they allow that certain classically valid modes of inference (for example, reductio ad absurdum) can take one from premises that are true to degree 1 to a conclusion that is true to a degree very close to 0. His example is A person with $50 million is rich. A person with only 37¢ isn’t rich. Therefore, it’s not the case that, for any n, if a person with $n is rich, then so is a person with $n − 1¢. which, on the degree-theoretic analysis, has premises true to degree 1 and a conclusion true to a degree slightly greater than 0. This, he says, is ‘apt to seem flat-out unacceptable’ (193). But why? If we agree that the premises are true and want to reject (C) For some n, a person with $n is rich and a person with $n − 1¢ is not rich, then we have to give up some classically valid principle of reasoning. And a many-valued semantics gives an illuminating story about why reductio should fail in ¹⁰ Compare the discussion of M vs. MD in Weatherson, 2005, §1. The fact that normal talk of truth and falsity does not include degree qualifiers is no obstacle for this view, since on a natural semantics for ‘true,’ ‘It is true that Harry is bald’ will have exactly the same degree of truth as ‘Harry is bald.’ In fact, it must have the same degree of truth if the biconditional ‘Harry is bald iff it is true that Harry is bald’ is to get degree 1 on the Łukasiewicz semantics.
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vague contexts. If we derive a contradiction from premises S1 , S2 , S3 using valid (1preserving) inference rules, then we know that at least one of them has degree less than 1. If we also know that S1 and S2 have degree 1, then we can infer that S3 has degree less than 1. But all we can conclude about ¬S3 is that it has degree greater than 0. We certainly cannot conclude that it has degree 1. That’s why reductio fails in this context. Given that something needs to be done to block the reasoning that leads to (C), recognizing limits on the use of reductio seems well motivated and at least as moderate as Schiffer’s own solution, according to which it is indeterminate whether classical inference rules—including not just reductio but even modus ponens —are valid (Schiffer, 2003, 224). I suggest, then, that we explicate the kind of partial endorsement that is appropriate in borderline cases—what Schiffer calls ‘vagueness-related partial belief ’—as takingto-be-partially-true.
25.2.3 Combining partial truth with uncertainty As we have seen, Schiffer’s theory founders in its attempts to integrate two separate aspects of partiality of belief: the ‘ambivalence’ that stems from vagueness and the uncertainty that stems from incomplete information. Can the present approach do better in integrating taking-to-be-partially-true with partially-taking-to-be-true? This is a problem that any degree theorist must face in ‘mixed cases,’ where the degree of truth of a vague proposition (say, Sam is bald ) depends on some nonvague matter about which there is uncertainty (say, the number of hairs on Sam’s head). But the problem is especially acute for theorists who view all facts about the ordering of numerical degrees to be representationally significant (not artifacts of the model), since on their view every attitude towards a borderline proposition will combine ambivalence and uncertainty. We will never be in a position to know who is the shortest man who satisfies ‘tall man’ to degree 1, and we will have no good basis for taking the proposition that Jim is tall to be true to degree 0.653 rather than 0.649. We may be confident that he satisfies ‘tall’ to some intermediate degree, and perhaps we’d bet on 0.6 over 0.5, but there will remain some uncertainty. So, to model our attitudes towards borderline propositions, we will need to take into account both dimensions of partiality: ambivalence and uncertainty. The most straightforward way to do this, I think, is to represent our attitudes to vague propositions as probability distributions over degrees of truth (strictly speaking, over an algebra of precise propositions that ascribe degrees of truth to the vague propositions at issue). So, for example, your attitude towards the proposition Jim is tall might be depicted by Figure 25.1, where the horizontal axis represents degrees of truth and the height of the curve over any given degree represents the probability that Jim is tall has that degree of truth. This picture combines both dimensions of partial endorsement, taking-to-be-partially-true and partially-taking-to-be-true, in a unified representation.¹¹ ¹¹ This graph, and those that follow, was generated by a custom Haskell program using Tim Docker’s Charts library and Martin Erwig’s Probabilistic Functional Programming library
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Figure 25.1 Pr([[Jim is tall]] = x).
This approach can deal straightforwardly with the ‘mixed cases’ that proved troublesome for Schiffer’s theory. Suppose you aren’t sure exactly how many hairs Tom has on his head. Your credence function is represented by Figure 25.2, where the vertical axis represents probabilities and the horizontal axis the number of hairs. For each possible number of hairs x, there will be a probability distribution over degrees of truth that represents the attitude you would have towards the vague proposition Tom is bald if you knew that Tom had exactly x hairs. Three of these distributions are plotted in Figure 25.3. Taking into account your uncertainty about the number of hairs Tom has on his head, what should be your attitude towards the vague proposition Tom is bald ? Since Figures 25.2 and 25.3 both represent probability distributions, the solution is a simple application of probability theory. We construct a probability distribution over assignments of degrees of truth to Tom is bald as follows: Pr([[Tom is bald]] = x) = Pr(Tom has n hairs) × 0≤n<500
Pr([[Tom is bald]] = x | Tom has n hairs) (Erwig and Kollmansberger, 2006). To simplify the calculations in these charts, I use a finite set of degrees, {0, 0.01, 0.02, . . . , 0.99, 1}. This allows us to do probability calculations using simple algebra. Everything said here should generalize to real-valued degrees, but more complex methods would be needed to handle probability distributions over the degrees (see Zadeh 1968).
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Figure 25.2 Pr(Tom has x hairs).
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Figure 25.3 Pr([[Tom is bald ]] = x | Tom has n hairs), for n = 20, 250, 400.
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Figure 25.4 Pr([[Tom is bald]] = x).
The resulting curve, which represents your composite attitude of partial endorsement toward Tom is bald, is displayed in Figure 25.4. Given probability distributions over degree-assignments for atomic propositions, we can easily calculate distributions for truth-functional compounds. This is most straightforward in cases where the conjuncts are degree-independent: Definition: P1 , . . . , Pn are mutually degree-independent iff for all subsets {Pj , . . . , Pk } of {P1 , . . . , Pn } containing two or more elements, and for all 0 ≥ dj , . . . , dk ≤ 1, Pr([[Pj ]] = dj & . . . & [[Pk ]] = dk ) = Pr([[Pj ]] = dj ) × · · · × Pr([[Pk ]] = dk ).
If the propositions Jim is tall, Jim is bald, and Jim is smart are degree-independent (as seems plausible), then the probability that they will have degrees d1 , d2 , and d3 respectively is just Pr([[Jim is tall]] = d1 ) × Pr([[Jim is bald ]] = d2 ) × Pr([[Jim is smart]] = d3 ). And the degree of the conjunction Jim is tall & Jim is bald & Jim is smart on this assignment of degrees to the conjuncts is just the minimum of {d1 , d2 , d3 }. So we can compute the probability that the conjunction has degree x by summing the probabilities of combinations d1 , . . . , dn of degrees whose minimal member = x. The result is given in Figure 25.5.¹² ¹² Cases in which the conjuncts are not independent can also be handled, though with added complexity. We consider such a case below (Section 25.3.3).
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Figure 25.5 Pr([[Jim is tall and bald and smart]] = x).
Our complaint about classical semantics was that it predicts that one should have far less confidence in the conjunction Jim is tall and bald and smart than in any of the conjuncts. The view now being considered, by contrast, predicts that one should have a little less confidence in the conjunction than in the conjuncts. The more uncertainty there is about the degrees of the conjuncts, the larger the drop in confidence will be (see Figure 25.6). This seems to me about the right result: midway between Łukasiewicz and Williamson.¹³ ¹³ A similar view is defended by Nicholas Smith in his contribution to this volume. The main difference is that Smith defends the view that one’s ‘degrees of belief ’ in a proposition p should be identified with the expected value of the degree of truth of p—that is, with the average degree of truth weighted by probability—while I do not attempt to arrive at a single-number degree of belief. While it might be useful for some purposes to measure beliefs by the expected value of degree of truth, it seems rash to suppose that all the interesting quantitative differences between partial belief states can be boiled down to this one number. For example, compare (a) a belief state that assigns equal credence to each degree of truth (a flat line on our graphs), (b) a belief state that assigns near certainty to degree 0.5 (a sharp spike), and (c) a belief state in which credence clusters around two points, degree 0.2 and degree 0.8 (two humps with a dip in the middle). In all three cases the expected value of degree of truth will be 0.5, but these belief states can be expected to have different effects on behavior and inference. Hence I prefer to represent states of partial belief in two dimensions, rather than attempting to integrate the uncertainty and partial-truth aspects into one number. That said, Smith and I agree about much more than we disagree about.
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Figure 25.6 Conjunction with more and less uncertainty.
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TRADITIONAL OBJECTIONS RECONSIDERED
Traditionally, one of the most serious problems for degree theories has been the absence of any compelling motivation. Degrees of truth do not seem to be needed, as some thought they were, to understand the semantics of graded adjectives, hedges, or the ordinary predicate ‘true’ (see Lakoff 1973, Williamson 1994, Haack 1996, Kennedy 2007). Nor is it clear that degree theories provide a better diagnosis of the sorites paradox than is available to the classical semanticist. Finally, it is not clear that degree theories can avoid a commitment to arbitrary-seeming and unknowable semantic boundaries, so if that was what was objectionable about classical semantics, degree semantics fares no better. If we need epistemicism anyway, why should we abandon the elegant simplicity of classical semantics? We can now answer this question. Classical semantics should be rejected for vague discourse because it forces us to think of the partial endorsement appropriate in borderline cases as a kind of uncertainty. As Schiffer observed, this conception yields implausible recommendations about our attitudes towards conjunctions of independent borderline propositions. A many-valued semantics provides an elegant way to represent and work with the two kinds of partiality that can characterize our attitudes to borderline propositions: ambivalence (taking-to-be-partially-true) and uncertainty (partially-taking-to-be-true). It can be motivated on these grounds even if we accept hidden semantic boundaries and a diagnosis of the sorites paradox that is compatible with classical semantics. But not all of the worries people have had about degree theories concern motivation. It has been alleged that such theories run into problems with higher-order vagueness, and that their use of numerical degrees involves an implausible commitment to the comparability of degrees of truth. In addition, many criticisms have been raised against degree-functional semantics for the connectives, and specifically against the min rule for calculating the degree of a conjunction. In this section, I consider how a fuzzy epistemicist might respond to these objections.
25.3.1 Higher-order vagueness A classic objection to degree theories is that, even if they do give a nice story about borderline cases and the sorites paradox, all the problems come back at a higher level. For example, it is alleged that someone could be borderline between satisfying ‘tall’ to degree 1 and satisfying ‘tall’ to a degree less than 1. If so, we could construct a new sorites on the predicate ‘satisfies ‘‘tall’’ to degree 1,’ and the degree theory would have no distinctive diagnosis of this higher-order sorites. This is clearly not a problem for fuzzy epistemicism, which does not make use of degrees of truth to give a diagnosis of the sorites. As we saw in Section 25.1.2, the many-valued analysis does not work well for sorites arguments using negated conjunctions, so some other story about the sorites is needed anyway. The motivation for degrees of truth offered in the last section is consistent with a number of different possible accounts of the sorites paradox, including epistemic and contextualist accounts.
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Indeed, I’m not convinced that the higher-order sorites poses a serious worry even for standard degree theories. The predicate ‘satisfies ‘‘tall’’ to degree 1’ is sufficiently theoretical that it’s not clear why we should accept a sorites premise formulated with it. Perhaps that is why the objection is usually run using a sentential operator D (for ‘definitely’), stipulated to have the following semantics: [[Dφ]] =
1 if [[φ]] = 1 0 otherwise.
We do have a strong inclination to accept a sorites premise for ‘definitely tall.’ But it’s not clear that the ordinary meaning of ‘definitely’ matches that of D as defined above. More plausibly, ‘definitely φ’ means something like ‘φ is true enough, by a good margin, for present purposes,’ or ‘φ has degree 1 by a good margin,’ and on this understanding we should expect ‘definitely φ’ to take non-extremal degrees, since ‘enough’ and ‘good margin’ are vague. If that is right, then a degree theory can say exactly the same thing about a sorites with ‘definitely tall’ that it says about a sorites with ‘tall.’
25.3.2
Comparability
Any theory that represents degrees of truth as (real or rational) numbers, and takes the ordering of these numbers to be significant, not an artifact of the model, is committed to the degrees being totally ordered : for any sentences A and B, the degree to which A is true will be either less than, equal to, or greater than the degree to which B is true. Both critics and friends of degree theories have suggested that ‘multidimensional predicates’ pose a problem for the idea that degrees are totally ordered.¹⁴ Here is Williamson’s version of the complaint: Comparisons often have several dimensions. To take a schematic example, suppose that intelligence has both spatial and verbal factors. If x has more spatial intelligence than y but y has more verbal intelligence than x, then ‘x is intelligent’ may be held to be truer than ‘y is intelligent’ in one respect but less true in another. Moreover, this might be held to be a feature of the degrees to which x and y are intelligent: each is in some respect greater than the other. How can two real numbers be each in some respect greater than the other? On this view, degrees are needed that preserve the independence of different dimensions, rather than lumping them together by an arbitrary assignment of weights. (Williamson, 1994, 131)
Multidimensional predicates do pose a problem for degree theorists who understand degrees of truth in terms of comparatives. But the present theory explicates and motivates degrees of truth in terms of their role in explaining our attitude of declining partial endorsement of successive members of a sorites series, not by appeal to comparatives. A is F -er than B can be true to degree 1 even when A is F and B is F have the same degree of truth (Williamson, 1994, 126). Multidimensional predicates just give us one more reason not to tie our understanding of degrees of truth to comparatives. ¹⁴ In addition to Williamson, quoted below, see Goguen, 1969, 350–1, Forbes, 1985, 175, Keefe, 2000, 129, and Weatherson 2005.
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The same considerations provide a response to this argument, from Keefe 2000: . . . consider the case where p = ‘a is tall’ and q = ‘b is red’. Here we have no single comparative on which ‘true to a greater degree’ can piggy-back. The comparison may be read as ‘a is more clearly tall than b is red’ and if, for example, a is clearly tall and b is clearly not red, then this will be true. But in a wide variety of cases (e.g. with a 5-foot 10-inch man and a reddish-orange patch), neither disjunct of (CT ) [‘either p ≥T q or q ≥T p’] will be true. (129)
If our sole grip on the notion of degrees of truth came from comparatives, then in cases like these, where there are no comparatives to appeal to, there would be a case for saying that there is no fact of the matter about whether Joe is tall is truer than Patch #50 is red. But we have rejected the close tie between comparatives and degrees of truth. So why think that the difficulty in determining which proposition is truer is anything other than epistemic? Indeed, an epistemic difficulty is to be expected on the present theory, since our attitudes towards these propositions are represented as probability distributions over degrees. When their curves overlap, we will not be in a position to know which proposition has the higher degree. Of course, I am not opposed to the development of non-numerical degree theories that relax the requirement that degrees be totally ordered. I am just expressing skepticism about the usual motivations for such theories. There are, in addition, technical reasons for wanting degrees to be totally ordered. One is that it is unclear how to define negation with partially ordered degrees (see Williamson, 1994, 133). (Weatherson, 2005) ends up with Boolean negation, but this is patently unsuited to the purposes of a degree theory, as it allows the degree of A to be greater than the degree of ¬A only when A is determinately true and ¬A determinately false.¹⁵ Surely a degree theory ought to allow that, say, Man 60 is tall can be truer than it is false—truer than Man 60 is not tall —even if it is not completely true.
25.3.3 Degree functionality Another standard group of objections to degree theories is directed at the degreefunctional semantics for the connectives. These objections are often regarded as devastating, even by those sympathetic with degrees (most prominently, Edgington 1997, who argues for a degree theory with non-degree-functional connectives). Although the motivation for fuzzy epistemicism does not assume that the connectives are degreefunctional, it does assume that the degree of a conjunction of independent propositions whose degrees are 0.5 is 0.5, and none of the non-degree-functional degree theories I know of deliver that result.¹⁶ But I am not convinced that the arguments against degree-functionality are compelling, especially when one takes account of the ¹⁵ Proof: Suppose A ≥T ¬A. Then A & ¬A ≥T ¬A, because ¬A ≥T ¬A, and the degree of a conjunction is the greatest lower bound of the degrees of its conjuncts under the ≥T ordering. But then, since A & ¬A is determinately false, so is ¬A. See Weatherson, 2005, 67. ¹⁶ On Edgington’s theory, the degree of a conjunction of degree-independent propositions is the product of the degrees of the conjuncts.
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fact that our attitudes toward borderline propositions will generally be mixed with uncertainty. I will focus here on negation, conjunction, and disjunction, ignoring the conditional. This is fair, I think, for a couple of reasons. First, there are lots of worries about classical logic’s truth-functional conditional, so if there are problems with the Łukasiewicz conditional, it is not clear that they should reflect badly on the multivalued framework. Whatever improved technologies are devised to handle conditionals in a two-valued framework can presumably be ported to the multivalued framework as well. Second, the argument I’ve used to motivate degrees does not rely on any particular semantics for the conditional. (It may be contrasted, in this respect, with motivations for degree theory that focus on its treatment of the sorites paradox.) Perhaps the most obvious objection to degree functionality is that it forces us to accept that some contradictions are not completely false. Assuming degrees of truth make sense, it should be possible for there to be a sentence that is just as true as it is false. Let P be such a sentence: (1) [[P]] = [[¬P]]. Now, plausibly, (2) [[P & P]] = [[P]]. It follows immediately from these premises that (3) If & is degree-functional, then [[P & ¬P]] = [[P]] = [[¬P]] = 0. It is hard to see how one could have a meaningful degree theory while rejecting every instance of (1),¹⁷ and rejecting (2) would be very strange.¹⁸ Our choice, then, is clear. We can have a degree-functional semantics for conjunction, at the price of allowing some contradictions to be partly true, or we can ensure that all contradictions have degree 0, at the price of rejecting degree functionality. So can we grasp the nettle of half-true contradictions? Note, first, that a contradiction that is half-true is also half-false. In accepting such things, then, one is not committing oneself to the assertibility of any contradictions, since it is presumably not appropriate to assert half-falsehoods. One is merely accepting that some kind of ‘ambivalent’ attitude is appropriate towards certain contradictions. And this doesn’t seem wrong. Indeed, ‘It is, and it isn’t’ is just how we express our ambivalence in borderline cases.¹⁹ Moreover, if Jim is a borderline tall man, it seems wrong to assert, ‘Either he’s a tall man or he isn’t.’ This is explained nicely on the hypothesis that such disjunctions are only half-true. Granted, there are other possible explanations: for example, we may be ¹⁷ Though Weatherson 2005 seems to do just that. See note 15, above. ¹⁸ Though Goguen, 1969, 347 does reject it, defining the degree of a conjunction as the product of the degrees of the conjuncts. (2) is also rejected in Łukasiewicz logics with strong conjunction, where [[P & P]] = [[P]] only when [[P]] = 1 (see note 3, above). ¹⁹ When we say this, I take it, we are not asserting a contradiction; we’re finding words to express our attitude of partial endorsement. Note also that such uses cannot be construed as ‘It is in one sense, and it isn’t in another sense.’
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construing ‘Either he’s tall or he isn’t’ as ‘Either he’s definitely tall or he’s definitely not tall,’ or the assertion may carry some implicature about definiteness. But it should be registered that there is at least a prima facie case for assigning intermediate degrees of truth to some instances of excluded middle (the flip-side of assigning them to some contradictions). Finally, although many philosophers seem to think it’s obvious that a contradiction cannot have any positive degree of truth,²⁰ it’s not clear what the argument is supposed to be. Timothy Williamson writes: By what has just been argued, the conjunction ‘He is awake and he is asleep’ also has that intermediate degree of truth. But how can that be? Waking and sleeping by definition exclude each other. ‘He is awake and he is asleep’ has no chance at all of being true. (Williamson, 1994, 136)
But to say that a contradiction has degree 0.5 is not to say that it has a chance of being true. Indeed, if we are certain that it has degree 0.5, then we will take it to have no chance of being completely true. To say that waking and sleeping exclude one another is to say that if ‘x is awake’ is true to degree 1, ‘x is asleep’ is true to degree 0, and vice versa. Perhaps it is also to say that ‘x is awake’ is as true as ‘x is asleep’ is false. But on neither understanding does it rule out the possibility that both are true to some intermediate degree. So much for the objections to half-true conjunctions. It might be thought, however, that there are other, more powerful grounds for objecting to degree functionality. Here is an argument that can be found (in different versions) in Williamson 1994, Edgington 1997, and Keefe 2000. Let Jim be a borderline satisfier of ‘tall,’ so that Jim is tall is true to degree 0.5, and let Tim be just a bit shorter than Jim, so that Tim is tall is true to degree 0.45. Then, since Jim is tall has the same degree as Jim is not tall, by degree functionality we can conclude that Tim is tall and Jim is not tall has the same degree as Tim is tall and Jim is tall. This is counterintuitive, since Jim is taller than Tim. So we should reject degree functionality. But why does it seem counterintuitive that Tim is tall and Jim is not tall and Tim is tall and Jim is tall have the same degree of truth? It might be argued that when Jim is taller than Tim, Tim is tall and Jim is not tall must be completely false (degree 0). But then we must presumably say that Tim is not tall or Jim is tall is completely true (degree 1), and that seems odd in a case where, although nothing is hidden from us, we would not assert either that Tim is not tall or that Jim is tall. The issues here are, I think, very similar to the issues we considered above in connection with half-true contradictions. If we are willing to accept half-true contradictions, it does not seem too bad to accept that Tim is tall and Jim is not tall might be just slightly more false than true. Even if we do accept this, though, the feeling persists that we should have different attitudes toward Tim is tall and Jim is not tall and Tim is tall and Jim is tall, and that the former is bad in a way that the latter is not. That is why this objection is deeper ²⁰ For example, Williamson asks, ‘how can an explicit contradiction be true to any degree other than 0?’ (1994, 136).
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than the bare rejection of half-true contradictions. To resist it, we need to explain why different attitudes are appropriate towards these propositions, despite the fact that they have the same degree of truth. I think that fuzzy epistemicism can provide a kind of answer to this question. Our attitudes towards the propositions in question will not amount to certainty that they have such-and-such degree of truth. Rather, they will be representable as distributions of probability over a range of possible degrees. And, as we saw in Section 25.2, above, attitudes toward logically compound propositions will be determined by these distributions in a way that factors in both the uncertainty and the ambivalence that they reflect. So let’s see what happens when we add a bit of uncertainty about the degrees (using a normal probability distribution centered on 0.5 for Jim is tall and 0.45 for Tim is tall).²¹ As Figure 25.7 reveals, there is a clear difference in the recommended distribution of credences over degrees for our two propositions. Particularly salient is the fact that Tim is tall and Jim is not tall cannot be truer than 0.5, while Tim is tall and Jim is tall has a tail that goes up beyond 0.8. This, I suggest, may be enough to account for the persistent feeling that our attitudes toward the two propositions should differ, and that the latter is more strongly endorsable than the former.
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25.3.4 Edgington’s argument against the min rule The same resources can help defend against an argument, put to me by Dorothy Edgington, that the degree of the conjunction of two independent propositions each with degree 0.5 should be less than 0.5. The argument goes as follows.²² Suppose the degree of Jim is tall is 0.5. As we vary the degree of Jim is bald from 0 to 1, we should expect the degree of Jim is tall and bald to vary gradually from 0 to a maximum of 0.5. So the conjunction should have a degree less than 0.5 when Jim is bald has degree 0.5, contrary to the min rule. Edgington’s assumption that the conjunction is truer when Jim is bald has degree 0.7 than when it has degree 0.5 should, I think, be resisted. It draws intuitive support from the fact that, if asked to point to the ‘tall, bald man,’ I will point to the balder of two equally tall men, even if both are borderline tall and borderline bald. But this fact can be explained pragmatically; we need not conclude that the balder man satisfies the description ‘tall, bald man’ to a greater degree than the thinner one. If Yao Ming and Shaquille O’Neal are both on the court, everyone will understand who we mean if we talk about ‘the tall man,’ even though presumably both of them satisfy ‘tall man’ to degree 1. We can do more justice to the intuitions underlying Edgington’s argument if we bring in the dimension of uncertainty and represent the attitudes at issue by probability distributions over degrees. As Figure 25.8 shows, fuzzy epistemicism recommends a clear difference in attitude towards Jim is tall and bald as the degree of Jim is bald goes from 0.5 to 0.7. The effect, as before, is due to the uncertainties, and will be more prominent the greater the uncertainty about the precise degree of truth of Jim is bald. Thus we can vindicate the intuitions Edgington deploys against the min rule for conjunction without giving up the min rule itself—another nice illustration of the way in which the insights of epistemicism and degree theory can complement each other, once we give up trying to motivate degree theory as a way to avoid hidden semantic boundaries or solve the sorites paradox. Re f e re n c e s Cook, Roy (2002), ‘Vagueness and mathematical precision’, Mind, 111, 225–47. Crupi, Vincenzo, Branden Fitelson, and Katya Tentori (forthcoming), ‘Probability, confirmation, and the conjunction fallacy’, Thinking and Reasoning. Edgington, Dorothy (1997), ‘Vagueness by degrees’ in Rosanna Keefe and Peter Smith, eds., Vagueness: A Reader, 294–316, MIT Press, Cambridge. Erwig, Martin and Steve Kollmansberger (2006), ‘Probabilistic functional programming’ in Haskell, Journal of Functional Programming 16, 21–34. Fara, Delia Graff (2000), ‘Shifting sands: An interest-relative theory of vagueness’, Philosophical Topics 28, 45–81. Field, Hartry (2003), ‘The semantic paradoxes and the paradoxes of vagueness’ in Jc Beall and Michael Glanzberg, eds., Liars and Heaps, 262–311. Oxford University Press, Oxford. ²² Here I draw on Edgington’s comments on an earlier version of this chapter at the Arch´e Vagueness Conference. For related arguments, see Edgington, 1997, 304.
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Forbes, Graeme (1985), The Metaphysics of Modality, Oxford University Press, Oxford. Goguen, J. A. (1969), ‘The logic of inexact concepts’, Synthese 19, 325–73. Haack, Susan (1996), Deviant Logic, Fuzzy Logic: Beyond the Formalism, University of Chicago Press, Chicago, 2nd edn. Hajek, Petr (2006), ‘Fuzzy logic’ in Edward N. Zalta, ed., Stanford Encyclopedia of Philosophy. Fall 2006 edition http://plato.stanford.edu/archives/fall2006/entries/logic-fuzzy/ Kahneman, Daniel and Amos Tversky (1983), ‘Extensional versus intuitive reasoning: the conjunction fallacy in probability judgment’, Psychological Review 90, 293–315. Keefe, Rosanna (2000), Theories of Vagueness, Cambridge University Press, Cambridge. Kennedy, Christopher (2007), ‘Vagueness and grammar: the semantics of relative and absolute gradable adjectives’, Linguistics and Philosophy 30, 1–45. Lakoff, George (1973), ‘Hedges a study in meaning criteria and the logic of fuzzy concepts’, Journal of Philosophical Logic, 2, 458–508. Lewis, David (1979), ‘Scorekeeping in a language game’, Journal of Philosophical Logic 8, 339–59. MacFarlane, John (2006), ‘The things we (kinda sorta) believe’, Philosophy and Phenomenological Research 73, 218–24. Machina, K. F. (1976), ‘Truth, belief and vagueness’, Journal of Philosophical Logic 5, 47–78. Raffman, Diana (1996), ‘Vagueness and context relativity’, Philosophical Studies 81, 175–92. Sainsbury, R. M. (1995), Paradoxes, Cambridge University Press, Cambridge, 2nd edn. Schiffer, Stephen (2003), The Things We Mean, Oxford University Press, Oxford. (2006), ‘Replies’, Philosophy and Phenomenological Research 73, 233–43. Soames, Scott (1999), Understanding Truth, Oxford University Press, Oxford. Sorensen, Roy A. (1988), Blindspots, Oxford University Press, Oxford. Weatherson, Brian (2005), ‘True, truer, truest’, Philosophical Studies 123, 47–70. Williamson, Timothy (1994), Vagueness, Routledge, London and New York. Wright, Crispin (2001), ‘On being in a quandary: Relativism, vagueness, logical revisionism’, Mind, 110, 45–98. Zadeh, L. A. (1968), ‘Probability measures of fuzzy events’, Journal of Mathematical Analysis and Applications 23, 21–427.
26 Indeterminacy and Truth Value Gaps Mark Richard
My goal is to motivate, develop, and defend two ideas: that some claims are neither true nor false; that the claim that Jo is bald, when Jo is a borderline case of baldness, is without truth value. Section 26.1 tries to motivate the idea that truth value gaps are possible by giving an example of a meaningful predicate which clearly is neither true nor false of some objects. As I see it, the example is one whose semantics is very much like that of a vague predicate such as ‘bald’. So it should also motivate the idea that some natural language sentences containing vague predicates are neither true nor false. There are a host of objections to truth value gaps. For example, some say the existence of a truth value gap would lead to contradiction. Some say there is no motivation for gaps, as truth is the measure of success in assertion, and at the end of the day an assertion either succeeds (and is true) or it doesn’t (and is false). I think these arguments depend upon a narrow picture of what can count as ‘saying something’. Simply to recognize the possibility of denial as a sui generis speech act (one not to be defined as the assertion of a negation) defuses them. Frege and Geach give well-known objections to denial as a sui generis speech act. Section 26.2 sketches a response to Frege and Geach’s worries; section 26.3 then responds to objections to truth value gaps. Those who think that if Jo is borderline bald, then the claim that he is bald is without truth value often hold that vague predicates—indeed, all predicates—trisect their domains, in the sense that the sets—those of which the predicate is true; those of which it is false; the rest—are exclusive and exhaustive. I myself think this. There is a feeling that this can’t be right: it draws sharp boundaries (for example, in a sorites series) where there are none; it ignores the fact that it is no clearer where to draw the line between the bald and the borderline bald than it is clear where to draw the line between the bald and the not bald. Section 26.4 takes up such objections, arguing that the first misconstrues what it is for there to be a ‘sharp boundary’ between the As Thanks for comments to Jody Azzouni, Nancy Bauer, Alex Byrne, Elisabeth Camp, Matti Eklund, Patrick Greenough, John Hawthorne, Sarah Jane Leslie, Bernard Nickel, Agustin Rayo, Ted Sider, Tim Williamson, Crispin Wright, and readers for Oxford University Press. This chapter has benefited from spirited discussions at the 6th Arche conference on vagueness, at Princeton University, and in a Cambridge reading group. Thanks to all.
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and the Bs, that the second disappears once we reflect on what we are saying when we say (for example) that it’s indeterminate whether Jo is bald or borderline bald.
26.1 It’s sometimes said that truth value gaps can arise from a ‘deficiency of meaning’: if a predicate’s meaning is simply silent as to whether it is true or false of a particular object, then the predicate can be neither true nor false of the object. Scott Soames, for example, writes Imagine the predicate ‘smidget’ being introduced into a language by the following semantic stipulation: (i) Any adult human being under three feet in height is a smidget. (ii) Any adult human being over four feet in height is not a smidget . . . . The interesting thing about the predicate is, of course, that the defining conditions for something to be a smidget, and for something to fail to be a smidget, are not jointly exhaustive. Adults between three and four feet tall cannot be correctly characterized either as being smidgets or as not being smidgets.¹ I imagine that the skeptic about truth value gaps will respond so: Suppose that we simply list a set of objects {a1, a2, . . . ., ak}, stipulate that the predicate P is true of them, and say nothing more about P. Though this gives P semantic properties, why think it makes P non-bivalent? P is true of a1 through ak, not true of everything else. Adding to the original stipulation that P is false of b1, b2, . . . . bj (when the a’s and b’s don’t exhaust the universe) won’t changes matters. Why should we think that Soames’s example is any different? Soames will say that since he intends only to supply sufficient conditions for predicate application and sufficient conditions for predicate non-application, some cases are undecided. The skeptic agrees that Soames’s stipulation gives only sufficient conditions for application, sufficient conditions for non-application. But he will again raise the question of why we should think we have a truth value gap: Suppose that c is not among the a’s and b’s. Before Soames starting stipulating, P was not true of c. Nothing he said changed that. So P is still not true of c. Since predicating P of c is now saying something, and the predicate is not true of c, (shifting from mention to use) it’s not true that c is P; that is, it’s false that c is P. Where is the truth value gap? Whatever the merits of this as a response to Soames, there is something right, I think, about the idea that a procedure like Soames’s can result in truth value gaps. We seem to learn at least some vague predicates by learning that the predicate is true of things like these, false of things like those. In such cases, there is often a ragged line of possible cases from these to those, and things in the middle we find aren’t much like these or those in the relevant respects. Given no guidance beyond the examples ¹ Soames 1989, 584–5.
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of these and those, we in effect take the condition resembles these (in relevant ways) as sufficient and necessary for the predicate to be true of an object, the condition resembles those (in the relevant ways) as sufficient and necessary for the predicate to be false of the object. Since there are objects that resemble neither these nor those in the relevant ways, we seem to be saddled with truth value gaps. This suggests a modification of Soames’s example. We fix the meaning of predicate P so that (a) it’s necessary and sufficient, for the predicate to apply to x that x satisfy some condition C; (b) it’s necessary and sufficient for the predicate to fail to apply to x that x satisfy condition C ; (c) this stipulation is minimally coherent, in that it’s impossible for anything to satisfy both C and C ,² and; (d) there are objects that satisfy neither C or C . It’s easy to come up with an example that satisfies these conditions. Let B be a one place predicate. Stipulate that relative to a context (1) B is true of x if and only the context’s agent would judge, if presented with x and well situated to judge baldness, that x was bald. (2) B is false of x if and only if the context’s agent would judge, if presented with x and well situated to judge baldness, that x was not bald.³ There are many objects such that, if I were asked in favorable conditions whether they were bald, I would just demur, and say that I didn’t know, for they are sort of bald, but sort of not bald. For such objects, it’s not true, that I would judge them bald; not true that I would judge them not bald. Of these objects the predicate is neither true nor false.⁴ One might wonder about defining a predicate P with the stipulations (S) For any x: P is true of x if and only if C For any x: P is false of x if and only if C when it is known that there are (or might be) objects satisfying neither C not C . What would be the point? But there would be a point, if, for example: (1) there is (more or less of ) a continuum of states that objects might be in, running from a state of being definitely C, through states intermediate and indeterminate between C and ² If this weren’t satisfied, it’s not clear that we would succeed in giving a meaning to P. ³ Actually, to satisfy condition (c), the consequent of (1)’s right hand side should be extended to say that x would not judge that x was not bald; analogously for (2). Consider this added throughout the discussion. ⁴ Perhaps I should point out that the argument does not assume that any counterfactuals are without truth value. There is an argument for truth value gaps in Dummett 1959 (reprinted in Dummett 1978) that employs predicates whose application conditions are given with counterfactuals. The argument here doesn’t depend on the use of counterfactuals; all that’s needed is that application and non-application are defined in terms of conditions that aren’t jointly exhaustive. Do the truth value gaps from which B suffers arise from ‘deficiency of meaning’? In so far as such deficiency is a matter of the semantic facts leaving it open whether or not a predicate is true of an object, I don’t think B is an example of such deficiency. ( This on the assumption that there’s no ‘deficiency of meaning’ in the counterfactuals on the right hand sides of (1) and (2).) Given that I wouldn’t judge you bald, wouldn’t judge you not bald, and am in a position to evaluate the matter, (1) and (2) don’t leave your status with regard to B open in any way whatsoever.
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C , and on into states of being definitely C ; (2) we have interests and purposes that are well served by being able to signal when an object recognizably falls at one or the other end of this continuum, and (3) it seems natural to satisfy this interest by reading it on to the semantics of a single term, so the term is true of an object just when we are recognizably at one end of the continuum, false of an object just when we are recognizably at the other end. Doesn’t this in fact seem to be exactly how vague terms like ‘bald’ ought to be described? It is plausible to think that we apply ‘bald’ by reference to paradigm cases of baldness, paradigm cases of non-baldness; the bald are (roughly) those who resemble the paradigm bald enough, the not-bald are (roughly) those who resemble the paradigm not-bald enough. When someone doesn’t closely resemble the paradigms of the bald, we say he is not bald (and so it’s not true that he’s bald); if that person doesn’t closely resemble the paradigms of the non-bald, we also say that he’s not not bald either (and so it’s not false that he’s bald). A good deal of human classification seems well modeled by (S).⁵ One might argue that stipulations (1) and (2) are incoherent. Suppose there is someone j who is a borderline case of baldness. Suppose that were I presented with j, I would recognize j’s borderline status and judge j to be a borderline case of baldness. So if I were presented with j, and were in a position to judge baldness, I wouldn’t judge him bald and I wouldn’t judge him not bald. Then the right hand sides of (1) and (2) are (determinately) false. So we must accept their negations. But if we accept A if and only if B and not B, we must accept not A. So we must accept B is not true of j and B is not false of j. But if B is not true of j, then B is false of j. Contradiction. At least part of the problem here is the seeming incompatibility of the idea that indeterminacy involves lack of truth value—the idea that (I) What is indeterminate is not true and not false with the idea that (F) What is not true is false.⁶ I think this problem can be met. But it will require a bit of a detour to meet it. 26.2 We need to distinguish two uses of the idioms of negation. Words like ‘not’ (and particles like ‘in’ and ‘un’) may be used to contribute to the propositional content—the sense, if you like—of a sentence. When so used, they express truth functional negation, or some extension there of, such as that defined by the strong ⁵ Jody Azzouni objects that B and ‘bald’ differ so: epistemic conditions are built into the satisfaction conditions of B (because those conditions speak of whether one would judge that so and so), but not into those of ‘bald’. In response: If we try to articulate what it is that makes someone bald or not so, we will be driven to talk about a person’s resembling various paradigms of the bald and the not bald; but such resemblance is grounded in part in our propensities to judge. ⁶ The inconsistency arises modulo the assumption that some things are indeterminate, of course.
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Kleene table for ‘¬’. But ‘not’ et al. are also used as signals that a distinctive speech act—denial—is being performed. Denial is sui generis, not to be defined in terms of assertion; it is appropriate to deny a claim, to a first approximation, when that claim is not true—false or without truth value. (I am, of course, here using ‘not’ in denial.) Let ‘not’ represent the use of negation to signal denial, ‘¬’ to represent its use to introduce truth functional negation. Then (I) is schematically expressed so: (I’) If it is indeterminate that , then not and not ¬. Denying that is true—unlike asserting ¬—involves no commitment to the falsity of , for one situation in which denying is apt is the situation in which is truth valueless. Widerspruch aufgehoben. The well-known objection to all of this is that if we posit a sui generis speech act of denial, we mess up logic and generate seemingly unsolvable puzzles about meaning, as we cannot embed the ‘not’ of denial within a conditional or other sentence compounding device. How, Frege and Geach ask us, are we to make sense of an argument of the form (A) If not A, then B Not A So, B when the second premise is used in denial? After all, uttering the first premise needn’t involve denying anything. So it looks like positing a sui generis act of denial requires saying that the argument form (A) isn’t unequivocally valid; indeed, it looks like associating such an act with ‘not’ gives the word a meaning that can’t combine with the meaning of ‘if ’. The correct rejoinder to this is to observe that not just the idioms of negation but the other sentence compounding devices lead a double life, sometimes contributing to sense, sometimes to the determination of a force conventionally associated with a sentence.⁷ Think of speech acts as individuated in terms of commitments taken on by she who makes them. Assertion of p is committing to p’s truth; denial of p is committing to p’s not being true. Call these sorts of commitments—to the assertability or deniability of one or more claims—first order commitments (FOCs). Some speech acts are to be understood not as incurring FOCs, but commitments to the appropriateness of one or another of a family of FOCs. Call such commitments second order commitments (SOCs). Here’s an example of a speech act best understood in terms of second order commitment. Let J be the sentence ‘Jo is bald’, Jo possibly a borderline case of the bald. How should we understand what I say when I utter ( T ) J is true if and only if Jo is bald ? What’s said is right. But it’s not true, if sentence J is truth valueless. How can this be? Well, intuitively, I am committing myself to the claims, that J is true and that Jo ⁷ The balance of this section summarizes material developed at length in chapter 2 of Richard 2008.
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is bald, having ‘the same status’—either both are true or neither is. I am making a commitment to the appropriateness of one among the following FOCs: (i) it’s assertable both that J is true, and that Jo is bald (ii) it’s deniable both that J is true, and that Jo is bald. If one’s use of ‘if and only if ’ is to signal such a commitment, then what I say can be appropriate—it can be right —even if it’s not true. One needs at this point to tell a story which explains how the force associated with a sentence like if not A, then B is systematically determined by the meanings of its parts. But that’s not so hard. We want to tell a story on which every sentence has associated with it a second order commitment. On this story, the meanings of the connectives qua force indicators are functions mapping SOCs to SOCs—force functions, we might call them. SOCs are representable as sets of FOCs, so the meaning of the force indicators will map sets of FOCs to sets of FOCs. Think of a FOC as a commitment to the assertability of a set s of claims and the deniability of a set s of claims (where s or s might be null); represent such a commitment as the pair <s, s >. An SOC is then represented as a set {c1 , . . . , ck } of FOCs, a commitment that at least one of its constituent FOCs is apt. Assertion and denial may be thought of as particularly simple SOCs—the commitment incurred by uttering ‘snow is white’, for example, is the SOC {
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With these operations in hand, we can say that when ‘and’, ‘or’, ‘if ’, and ‘not’ are used as force indicators, they have associated with them force functions defined as follows: Where C(S) abbreviates the second order commitment associated with S, (R) C(A and B) is Conj (C(A), C(B)) C(A or B) is Disj (C(A), C(B)) C(not A) is Conv (C(A)) C(if A, then B) is Disj (Conv (C(A)), C(B)). Example. The SOC associated with if Jo is not bald, then Jim is not bald is the disjunction of: (a) the converse of the commitment associated with ‘Jo is not bald’; (b) the commitment associated with ‘Jim is not bald’. The commitment associated with ‘Jo is not bald’ is the commitment that the claim that Jo’s bald is deniable, so (a) is the commitment that we can assert that Jo is bald. So the conditional overall express a commitment that is appropriate just in case it’s apt to assert that Jo is bald, or to deny that Jim is. The example suggests—and it is not hard to show—that the definitions we have adopted indeed have as an upshot that the commitment associated with any sentence is a SOC—i.e. a set of sets of first order commitments.⁹ Imagine now that we have a propositional language L with truth functors interpreted in the strong Kleene way. Let L+ be the language whose formation rules are 1. Any sentence of L is a sentence of L+. 2. If A and B are sentences of L+, then so are not A; if A, then B; A and B; A or B. with (2)’s connectives intended as force operators. A semantics for this language: (i) associates claims with the sentences of L, thus inducing a partial assignment of truth values to the sentences of L; (ii) defines the (second order) commitment associated with the L+ sentence S (relative to an assignment of claims to L). When S is an L sentence expressing p, C(S) = {
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(iii) defines in the obvious way what it is for the commitment associated with a sentence to be appropriate in an interpretation: {<s, s >} is appropriate in an interpretation I provided all of s is true, none of s is; {c1 , c2 , . . . , ck } is appropriate in I provided one of the ci ’s is. This language has a straightforward logic, with validity defined not as truth preservation but as ‘commitment preservation’. Spelling this out: an interpretation is an assignment of claims (and thus a partial assignment of truth values) to the ‘atoms’—i.e. the L sentences—of the language. Once a (partial) assignment of truth values to L sentences is in place, the commitments associated with any sentence of L+ are determined as appropriate or otherwise by the semantics and the truth status of the L sentences therein. An argument is valid provided in every interpretation in which the commitments associated with its premises are appropriate, so is that associated with its conclusion. It is, I hope, tolerably clear how all of this constitutes the basis of an answer to Frege and Geach. The argument form (A) If not A, then B Not A So, B has a regimentation in L+ in which the ‘not’s and ‘if ’ are treated as force functors and a regimentation in which they are treated as truth functors. Both are non-problematically valid. The first, for example, is valid because the commitments associated with the premises are, respectively, {
26.3 At the end of Section 26.1, we observed that if we accept both (I) What is indeterminate is not true and not false and (F) What is not true is false ¹⁰ There are regimentations of (A)—ones in which ‘not’ is treated non-univocally (once as contributing to sense, once as affecting force)—which are invalid. But there are also arguments of English whose surface is mirrored by (A) that are invalid. For discussion see Richard 2008, section 2.7.
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it seems impossible that anything be indeterminate. Let’s concentrate on two instances of the general claims: (I*) If J is indeterminate, J is not true and J is not false (F*) If J is not true, then J is false. (I*) and (F*) rightly construed are correct. If J is indeterminate, we may deny that J is true and that J is false, though of course we may not assert the truth functional negations of ‘J is true’ or ‘J is false’. Using ‘not’ for denial, ‘¬’ for truth functional negation, (I*) is acceptable understood as (I**) If J is indeterminate, then not ( J is true) and not ( J is false). And if we can assert that J is not true—if it’s true that J isn’t true, then J must indeed be false. (F*) is right, understood as a claim about the connection between ¬ Sentence J is true and Sentence J is false: (F**) If ¬J is true, then J is false, where ‘if ’ is a force functor. There is no inconsistency among (I**), (F**), and the claim that J is indeterminate. J’s indeterminacy and (I**) imply not ( J is true). But one way this might be is for J to be truth valueless. If it is, so are J is true and ¬J is true. What of the objection that our stipulations about the predicate B are incoherent? We said that in any context: (1) B is true of x if and only if the context’s agent would judge, if presented with x and well situated to judge baldness, that x was bald. (2) B is false of x if and only if the context’s agent would judge, if presented with x and well situated to judge baldness, that x was not bald. A good part of the point of the last section was to put us in a position to understand these: as I see it, to introduce a predicate in the way (1) and (2) introduce B is to use the biconditional not as a truth functional connective, but in the way the force connective iff of section 26.2 is used.¹¹ This insures that B is neither true nor false of x, should x be such that, were I presented with him and in a position to judge baldness, I would neither judge x bald nor judge him not bald. One might object that this can’t be right: since the right hand sides of (1) and (2) are false, their left hand sides must be false, too. But surely if the claim S is true is false, then the claim S is false is true; likewise, if B is true of x is false, then B is false of x is true. But then the claim that B is false of x is true and false. ¹¹ Recall that we introduced the ‘force biconditional’ so that A iff B was appropriate just in case either both A and B are assertible or both are deniable (the latter meaning that each is either false or without truth value). A connective with these properties can be defined using the force connectives of the last section—either (if A, then B) and (if B then A) or (A and B) or (not A and not B) suffice to define it.
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The objection assumes that the argument A iff B ¬B So, ¬A is valid, when iff is our force biconditional, and ¬ is a truth functional negation operator.¹² But it’s not. A iff B is appropriate if either A and B are both assertable (and so both true), or both deniable (and so each is either false or truth valueless). So there is a way the premises can all be appropriate (A is without truth value, B is false) without the conclusion being so. So the objection fails. What’s valid is not the argument just displayed but the argument A iff B not B So, not A where not is our force operator. But the conclusion here doesn’t imply that A is false; rather it implies that A is either false or without truth value. What one can conclude from (1) and (2) and the falsity of their right hand sides are the denials of their left hand sides. And these denials are perfectly consistent. Indeed, they are just what we should maintain if we accept (1) and (2) and know their right hand sides are false.¹³ There is a worry about truth value gaps that arises from the idea that the instances of (t) ‘S’ is true if and only if S (f ) ‘S’ is false if and only if it’s not the case that S are in some important sense definitive of the notions of truth and falsity, and thus must be (tantamount to) logical truths. For if we say that ‘Jo is bald’ is not true and not false, the validity of (t1) ‘Jo is bald’ is true if and only if Jo is bald (f1) ‘Jo is bald’ is false if and only if it is not the case that Jo is bald leads immediately to contradiction. The right response to this objection is to observe that to say that (t) and (f ) are valid is not to say that all of their instances are invariably true, but that they are invariably ¹² The assumption is present because in the language L+ is false in an interpretation just in case ¬ is true therein. ¹³ It is possible to define a biconditional which validates the inference on which the original objection rests: A ⇔ B =df (A if and only if B) and (¬A if and only if ¬B). This biconditional is appropriate just in case A and B ‘have the same truth value status’: Both are true, or both are false, or both are indeterminate. If the biconditional used in stipulations (1) and (2) had been the one just defined, the objection in the text would have been sound. But this isn’t how I’m using ‘iff ’.
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appropriate in the sense sketched in section 26.2: the commitments associated with each side of an instance of each biconditional are either both appropriate or neither is. It is in fact fairly easy to show that if we introduce a truth predicate in the style of Kripke into a language containing the force operators, every instance of (t) will be valid, as will every instance of (f ), when: ‘false’ is taken as defined as ‘¬true’; (f )’s ‘not’ is treated as the denial operator; ‘if and only if ’ is defined in the obvious way in terms of the force functors ‘if ’ and ‘and’. When this is done, we can indeed infer from not: ‘Jo is bald’ is true and (t1) the denial of the claim that Jo is bald; we can indeed infer from not: ‘Jo is bald’ is false and (f1) the denial of the claim that Jo is not (viz., ¬) bald. But these denials are perfectly consistent, as they are both apt if ‘Jo is bald’ is neither true nor false. The last objection to truth value gaps that I’ll discuss here is an argument in favor of bivalence implicit in Michael Dummett’s paper ‘Truth’, one recently endorsed by Michael Glanzberg.¹⁴ Its lead idea is that truth values are first and foremost means for assessing assertions. An assertion’s purpose or point is to convey information. Truth is the measure of an assertion’s success—saying p is a success if, p being true, saying p is conveying information, not misinformation. From this perspective, the idea that there could be a claim that was neither true nor false is just plain puzzling: if an assertion is a success, it’s true; otherwise, it’s not a success. What could possibly be the point of positing a third possibility? At the least, anyone who thinks there can be assertions which are assertions —acts which make a claim, and thus are candidates for conveying information—but are neither true nor false, needs to be able to point to something about the purpose of assertion, that gives the point of a third classification. The argument moves from a statement of success conditions for assertions to the conclusion that there are only two possible ‘statuses’ for a claim—either is it true or it is false. This, it seems to me, involves a bit of legerdemain. Let us grant that the success of assertions and denials is measured in terms of truth. What does this mean? Well, an assertion is a success iff what is asserted is true; otherwise it is not a success. Likewise, a denial is a success iff what is denied is not true; otherwise it is not a success. It just does not follow from this that there are only two ‘truth value statuses’ for what is asserted or denied. At least it does not given that the ‘not’s in the characterizations of success for assertions and denials are the ‘nots’ of denial. To agree to the words ‘either an assertion is true or it is not’ just isn’t to embrace bivalence. What of the challenge to explain ‘the point’ of positing a third possibility for the contents of assertions, beyond being true and being false? The idea behind the challenge, I think, is that truth and falsity just are what we use to classify certain speech acts as successes or otherwise; so if there is need for a status beyond truth and falsity, its need should be evident when we consider (our tendencies to classify) those speech ¹⁴ In Beall 2003. There is a helpful sorting out of the arguments for and against truth value gaps in Dummett’s ‘Truth’ in the Postscript to its reprinting in Dummett 1978.
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acts whose success is measured in terms of truth. But as I see it, once we recognize that both assertion and denial are speech acts whose success conditions are to be given in terms of truth, and agree that denial is sui generis —it cannot be defined in terms of asserting negations—we will recognize that we must posit a status for claims other than true and false. Part of ‘the point’ of having denial in addition to assertion is so that we can characterize those situations in which we can’t assert p or p’s negation. Our wanting to do this requires positing a status beyond truth and falsity, since we need a status that p’s negation can have that (a) makes it apt to deny that negation, but (b) doesn’t make the assertion of p apt.¹⁵ 26.4 I do not identify the indeterminacy of a claim with its lacking truth value. Presumably truth value gaps may arise because of such things as presupposition failure or category mistakes, and these sorts of things needn’t involve indeterminancy. But I do think what is indeterminate is neither true nor false. So since the application of a vague predicate to a borderline case is not determinately true or false, such an application is not true or false, period. So, it seems to me, a vague predicate cleanly trisects its range into three classes, those of which it is true, those of which it is false, the rest. Call this the trisection thesis. It will be said that I am not ‘taking vagueness seriously’.¹⁶ Consider a sorites series for ‘bald’. What I have said commits me to there being numbers j and k such that the following is an accurate picture of such a series: # OF HAIRS: STATUS:
0, 1, 2, . . . . j − 1, BALD
| j, j + 1, . . . ., k − 1, | INDET.
| k, k + 1, . . . ., n | ¬BALD
¹⁵ Dummett and Glanzberg’s argument involves another questionable premise, that the point or ‘intrinsic purpose’ of assertion is to convey information. If we reflect on cases of Lewis style accommodation—in which a claim is ‘made true’ by raising or lowering the context’s standards of precision—we see that many of the speech acts we label ‘assertions’ are not attempts to convey information but rather suggestions that a topic would be best viewed in a certain way. To say, for example, that Farmer Brown’s field is flat in a context in which it is not obvious or even settled whether existing standards for flatness would grade the field as flat is obviously not an attempt to convey something that is true independently of the way the concept of flatness gets ‘fleshed out’ in a context. I would describe such an assertion as akin to a proposal that flatness (and Farmer Brown’s field) be thought of in a certain way, a proposal made by publicly thinking of them in that way. If this picture of assertion is correct, then the most general way to gloss assertive utterance of S is not ‘Here is how things are: S,’ but ‘This is the way we ought to think about things: S.’ Note that the latter subsumes the former: If Here is how things are: S, then indeed, This is the way we should think about things: S. But the converse does not hold, as is attested by the fact that we do not always accommodate what we ought. I think that when we think of assertion in this way, it becomes considerably more plausible that the idea of truth value gaps can be vindicated simply by appeal to the ‘intrinsic purpose’ or point of assertion. The claims I have just made about Lewis style accommodation are tendentious, and this is not the place to develop or defend them. Some attempt to do that can be found in Richard 2004 and chapter 4 of Richard 2008. ¹⁶ Terry Horgan has said this about the sort of position I’m defending. See, for example, Horgan1994.
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But surely this can’t be a correct picture of such a series. If a predicate is vague in the way that ‘bald’ is vague, there cannot be a boundary—well, there cannot be a sharp boundary—between those of whom the predicate is true (the bald) and those of whom it is not. The predicate can’t draw any sharp boundaries: there can’t be a sharp boundary between the bald and the borderline bald, nor between the bald and those on the borderline between bald and borderline bald, nor . . . . Mark Sainsbury puts the worry so: . . . the idea that there is a sharp borderline between the positive cases and the borderline ones . . . . can no more be sustained than can the idea that there is a sharp division between positive and negative cases . . . . [there must be] things which seem intermediate between being definite cases of children and being borderline cases of children. We decline to accept that there can be any sharp boundary here. If there were, it would remain true that there would be such a thing as the last heartbeat . . . . of my definite childhood, and that seems as crazy as the idea that the predicate ‘child’ divides the universe into a set and its complement within the universal set¹⁷
But why are we supposed to agree that in the sorites series pictured above, there is a ‘sharp boundary’ between the bald and the borderline bald? It is, after all, indeterminate whether or not j is bald. j is sorta bald, sorta not bald; there’s no saying whether or not he is bald. Given that it is indeterminate whether j is bald, it is not true that there is such a thing as the last bald man in the series. It is not true that no man after j is bald—i.e. it is not true that ¬∃x: x is a man with more than j − 1 hairs and x is bald. In what sense is that a sharp transition? It seems rather like a vague, fuzzy transition.¹⁸ One might grant the point but insist, looking at the picture above, that there is a sharp transition; one can see it, after all, at the point where the line is drawn. Looking at the picture we see that as one moves along the series it is clear that this man is bald, this man is bald, . . . . this one is bald, and then, suddenly, this one isn’t bald. If that’s not a sharp transition, what is? In response: Note first of all that we can know that our picture of the sorites series is correct without its being possible for us to determine which of the objects in the series is j, the first man who is indeterminately bald. That is, it’s perfectly possible that the picture is correct even though there is no point in the series at which one is presented with a clear case of baldness followed by a clear case of non-baldness, or with a clear case of borderline baldness. It is a familiar point that many vague predicates enjoy a sort of context sensitivity that typically makes whether the predicate is true of an object turn on whether it would seem to us that the object falls under it. When a predicate is like this, the borders in a sorites series will typically fall somewhere other than in the vicinity of objects at which we are looking—if we look at a particular object in the series, the similarity between it and its neighbors will tend to move boundaries away from the area we are ¹⁷ Sainsbury 1991, 168. ¹⁸ I am assuming that we have extended the formalism of section 26.2 so that existential quantifications are treated as infinite disjunctions; for a sketch of how this might be done see chapter 2 of Richard 2008.
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studying.¹⁹ So the idea that we can see the transition in the series pictured above— the idea that it is clear that j − 1 is bald, j is not bald—is in one important sense just wrong. Indeed, the epistemological border between the bald and the borderline bald is hopelessly blurred—that is, there will always be objects in a sorites series such that we don’t know whether to affirm that they are bald, affirm that they are not bald, or deny that they are either bald or not bald. But the absence of this sort of sharp boundary does not show that the sort of ‘boundary’ represented in our picture of the sorites series does not exist. What of higher order vagueness? Aren’t some objects on the borderline between being bald and being borderline bald, or some between being bald and being a borderline case of a borderline case, or some with an even more complex intermediate status? It will be said that the existence of such objects is inconsistent with the trisection thesis, for if it is (for example) indeterminate whether Jo is bald or borderline bald, then ‘bald’ cannot merely trisect its domain into the bald, the non-bald and the borderline bald. We need to separate two questions: (Q1) Is it possible for there to be an object x such that we can correctly deny: that x is bald; that x is ¬bald; that it is indeterminate whether x is bald? (Q2) Is higher order vagueness possible, in the sense that besides the bald, the not bald, and the indeterminately bald, there may be objects such that it is (for example) indeterminate whether they are bald or indeterminately bald? The answer to (Q1) is (a slightly qualified) obviously not. Suppose we can aptly deny that x is bald, and aptly deny that x is ¬bald. I suppose that this might be the case because of presupposition failure, or a category mistake in the claim that x is bald, or for some reason which has nothing to do with the vagueness of the concept of baldness. And in such a case perhaps it would wrong to say that it was indeterminate that x was bald. But barring presupposition failure and the like, if we can aptly deny that it’s true or false that x is bald, how can it apt to deny that it is indeterminate that x is bald? How can it fail to be indeterminate whether x is bald, if it’s not true that x is bald and not false either? The idea that there is a distinct category of objects which are indeterminately indeterminately bald—distinct in the sense that these are objects which are not bald, not ¬ bald, and not indeterminately bald—wobbles over the brink of incoherence. It doesn’t follow from this, however, that there isn’t such a thing as an object which is, say, on the borderline between being bald and being indeterminately bald. Let me explain. Talk of indeterminancy, it seems to me, is essentially contrastive: to say that it is indeterminate whether S is to say, for some T, that it is indeterminate whether S or T. Abbreviate it’s indeterminate whether S or T with ∇(S, T). Typically, though not invariably, when we say it’s indeterminate whether S, for relatively simple S (e.g. where S = ‘J is bald’) what’s meant is that it’s indeterminate whether S or ¹⁹ Raffman 1994, seconded in Soames 1999, and developed in Fara 2000.
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¬S.²⁰ With more complex values of S, there are often more possibilities. ‘it’s indeterminate whether it’s indeterminate whether S’ would naturally, I think, be understood as saying ∇(S, ∇(S, ¬S)). ‘it’s indeterminate whether it’s indeterminate whether it’s indeterminate whether S’—well, it’s hard to say out of any context what might be meant. When are such claims apt? Well, if it’s indeterminate whether S or T, then it’s not settled whether S or whether T. And if it’s not settled whether S or T, there must be some way of ‘tightening up’ S and T’s meanings, without affecting the non-semantic facts, on which it comes out apt to say S, unapt to say T. Likewise, there must be some way of tightening up these meanings so that it comes out not being apt to say that T, unapt to say S. For present purposes, we can think of such tightening up as a matter of extending a given assignment of extensions and anti-extensions to predicates, by putting some things for which a predicate is undefined in the predicate’s extension or anti-extension. Allow me the notion of an adequate extension of the semantics of a language. Such extensions will not subtract objects from the extensions or anti-extensions of predicates, but may add objects to these.²¹ Using the notion of an adequate extension, we may say that ∇(S, T ) is appropriate in a language L if and only if there are adequate extensions f and g of the semantics of L such that: in f, S is apt, T is not; in g, T is apt, but S is not. Here are some consequences of thinking of indeterminacy in this way. (1) ∇(S, ¬S) implies that neither S nor ¬S has a truth value. (For if S is true or false, it is in all adequate extensions.) (2) Assume Tightening: when x is, in interpretation I, a borderline case of the predicate F, there are adequate extensions I and I of I such that x is the extension of F in I and x is in the antiextension of F in I .²² ²⁰ But not always. One might say that it was indeterminate whether x was red, meaning that it was indeterminate whether x was red or orange. ²¹ Not any old way of extending extensions and anti-extensions will yield an adequate extension, if only because of what are often called ‘penumbral connections’ among predicate meanings. Such details are orthogonal to our present concern. ²² I think Tightening is correct. But it’s a pretty strong assumption. To see why, let L be a language containing the predicate B; let the domain of L be {1, 2, 3, 4}; suppose the extension of B in L is {1}, the anti-extension {4}. One might think that it could be indeterminate whether there is an adequate extension of the semantics of L in which 2 is in the extension of B. But this is not so if Tightening is (determinately) true, since it requires that there be an adequate extension of L in which 2 is in the extension of B (as well as an adequate extension in which 2 is in the anti-extension of B). Why do I think that Tightening is correct? Suppose that there is not an adequate extension of L’s semantics on which 2 is in the extension of B. ( The ‘not’ here is to be understood as the not of denial.) Then it is not (denial again) possible, given the facts, to sharpen the meaning of B so that it
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Suppose that S is a sentence, like ‘Jo is bald’, that is the application of a garden variety vague predicate to a borderline instance. Then ∇(S, ¬S) is apt. (3) Suppose ∇(S, ¬S) is apt. Any adequate extension that makes S apt or ¬S apt makes ∇(S, ¬S) inapt. And one ‘extension’ of the semantics is the null extension, in which ∇(S, ¬S) is apt, S is not. So if ∇(S, ¬S) is appropriate, so are (i) ∇(S, ∇(S, ¬S)) (ii) ∇(S, ∇(S, ∇(S, ¬S))) and so on up. In the present framework, (i) gives the form of the most likely regimentation of ‘it’s indeterminate whether Smith is bald or borderline bald’; (ii) gives the form of the most likely regimentation of ‘it’s indeterminate whether Smith is bald or on the borderline between being bald and being on the borderline between being bald and being borderline bald’. Thus, if there’s first order vagueness, there is higher order vagueness. (4) A unary indeterminacy operator can be defined as follows: ∇S is apt in a language L if and only if there are adequate extensions f and g of the semantics of L such that: in f, S is apt; in g S is not apt.²³ If S is a sentence that is truth apt—it is, say, the application of an ‘ordinary’ vague predicate such as ‘bald’ to an object—the sentence not (S and ∇S) is valid. (For if S is apt, it is true, and thus there will not be an adequate extension of the language’s semantics in which S is untrue and ∇S is apt.) (5) Given Tightening, when S is as in (4) the sentence (13) If ∇S, then ∇∇S is also valid. (For if ∇S is apt in I, then (i) S is apt and thus true in some adequate extension of I, and so ∇(S) is not apt in some adequate extension of I, and; (ii) ∇(S) is apt in the null extension of I.) Some might think that this result borders on paradox. Let S be a sentence such that ∇S. By (I3), ∇∇S. So ∇S and ∇∇S. But how can it be right to say anything of the form p and it’s indeterminate that p?²⁴ It is wrong to say anything of this form if p is something that might be true or false—if p, for example, is something like ‘Jo is bald’. And, as noted above, on the current account of indeterminacy, not (p and ∇p) is valid when p is a sentence like ‘Jo is bald’. But suppose p is a claim of indeterminacy—let’s is true of 2. But surely this means that it’s settled that B is not (truth functional not this time) true of 2. ²³ Perhaps you are wondering why we don’t define the operator so: ∇(S) is apt in a language L if and only if there are adequate extensions f and g of the semantics of L such that: ‘in f, S is apt; in g ¬S is apt.’ The reason to adopt the definition in the text is to allow the application of the indeterminacy operator to expressions (such as it is indeterminate whether S) that may be apt without being true or false. ²⁴ Crispin Wright expressed roughly this worry.
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say it’s the claim that it’s indeterminate that Jo is bald. In this case, p itself may be the right thing to say, but it won’t be determinate that this is so. After all, if it is indeterminate whether Jo is bald, it is possible to tighten up the meaning of ‘bald’ so that it is true, not indeterminate, that Jo is bald. And so it is possible to tighten up the meaning of ‘bald’ so that ‘it is indeterminate that Jo is bald’ is the wrong thing to say, as tightening up the meaning in the requisite way renders ‘Jo is bald’ determinately true. And so it is not determinate that it’s indeterminate that Jo is bald. If all this is on the mark, higher order vagueness poses no threat to the trisection thesis. But this is not because higher order vagueness is incoherent. There is no incoherence in the idea of higher order vagueness, only triviality: Of course, if B is borderline bald, it is not determinate whether B is bald or borderline bald. For if B is borderline bald, it’s not settled whether she is bald. And it’s also not settled whether she is borderline bald—for if it were—if there were no way of tightening the concept of baldness so that B counted as bald, it would be just false, not indeterminate, to say that B was bald. 26.5 Summing up. (1) There are perfectly possible meanings (ones of a sort one would think are possessed by many vague predicates) which would necessitate a predicate’s being gappy. (2) Many arguments against the coherence of truth value gaps depend on a very narrow picture of saying which ignores the possibility of such things as sui generis denial. (3) Frege / Geach objections to things like sui generis denial dissolve once we observe that ‘not’ and other sentence compounding devices lead a double life, sometimes contributing to sense, sometimes to force. There is a simple compositional story about how (for instance) embedding a denial operator within a ‘force conditional’ makes if not A, then B fit to perform a sort of speech act which, when combined with B’s denial, commits one to the aptness of asserting A. (4) The objection to the trisection thesis—that it is inconsistent with the idea that there are no sharp boundaries in a sorites series—is not compelling: there is no conception of a ‘sharp boundary’ on which it’s plausible both that there are no sharp boundaries in a sorites series and that trisection involves the creation of sharp boundaries. (5) Once we recognize that talk of indeterminacy is contrastive, we also recognize that higher order vagueness isn’t inconsistent with trisection. (6) We also, once we think of indeterminacy as contrastive, come to see that indeterminacy itself is indeterminate—if it’s indeterminate whether p, that indeterminacy itself is not something that is settled, but is itself indeterminate. Re f e re n c e s Beall, Jc, ed. (2003), Liars and Heaps, Oxford University Press. Dummett, M. (1959), ‘Truth’, Proceedings of the Aristotelian Society 59, 1959, 141–62. (1978), Truth and Other Enigmas, Harvard University Press. Fara, D. G. (2000), ‘Shifting sands: An interest-relative theory of vagueness’, Philosophical Topics 28, 45–81.
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Gabbay, D. and Guenthner, F., eds. (1989), Handbook of Philosophical Logic IV, Reidel. Glanzberg, M. (2003), ‘Against truth value gaps’, in Beall 2003. Horgan, T. (1994), ‘Robust vagueness and the forced-march sorites paradox’ in J. Tomberlin 1994. Raffman, D. (1994), ‘Vagueness without paradox’, Philosophical Review 103, 41–74. Richard, M. (2004), ‘Contextualism and relativism’, Philosophical Studies 119, 215–42. (2008), When Truth Gives Out, Oxford University Press. Sainsbury, M. 1991, ‘Is there higher-order vagueness?’, Philosophical Quarterly 41, 167–82. Soames, S. (1989), ‘Presupposition’ in D. Gabbay and F. Guenthner, eds., 1989. (1999), Understanding Truth, Oxford University Press. Tomberlin, J., ed. (1994), Philosophical Perspectives 8, Ridgeview Publishing.
27 Supernumeration: Vagueness and Numbers Peter Simons
There is a notable bifurcation between what philosophers think and say about vagueness and what people do who have to deal with it practically. There is a widespread consensus in the philosophical literature on vagueness that fuzzy logic, which essentially includes the assignment of numerical values to represent degrees of truth of vague sentences, is a flawed method, and that some other theory is to be preferred if we are to give a correct account of vagueness. When it comes to practical applications however, for people with actual problems to solve and computers and software to hand, fuzzy logic is the overwhelmingly predominant approach. Such applications include: Geographical Information Systems (GIS) Medical Diagnostics and Treatment (Expert Systems) Astrophysical Data Data mining and data fusion Control Systems
None of these is insignificant. By philosophical lights, this work is all either mistaken or concerned with something other than vagueness. By the lights of applied science, philosophers have their heads stuck well and truly either in the clouds or in the sand, or, paraconsistently, both. I advocate a way out of this impasse which addresses the concerns of both sides. F U Z Z Y LO G I C In fuzzy logic at its most basic, a vague statement is assigned a real number ν ∈ [0, 1] as its truth value: ν = 1 represents classical, complete or total truth, ν = 0 represents A previous version of this chapter presented at St Andrews benefited from critical discussion by Timothy Williamson, Dorothy Edgington, and other conference participants. I am also grateful for critical remarks by two anonymous referees for Oxford University Press. The approach has most affinities with that of Dorothy Edgington, who also believes we need numerical measures in considering vagueness: see Edgington 1997. Like Edgington, I exploit the analogy with probability theory, as in two previous papers, Simons 1997, 1999, but I stress that vagueness and probability are two different things, so the analogy must be exploited with care.
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classical falsity, and 0 < ν < 1 represents a non-classical or in-between case. This scheme has certain marked advantages: it gives a way of calculating with truth values with a simple extensional (value-functional) logic: if we symbolize the truth value of a statement p by |p|, the truth values for negations, conjunctions, and universal quantifications are given by |¬p| = 1 − |p| |p ∧ q| = min(|p|, |q|) |∀xA[x]| = minx (|A[x]|) Vague statements are those which do not have a classical truth value (0 or 1), and the numbers take account of the intuition that some statements are closer to truth (or falsity) than others. It also has a very simple and plausible account for Sorites Paradoxes. In a Sorites sequence a wholly true premise and a long sequence of almost true implications leads via many applications of modus ponens to a wholly false conclusion, the minute drops in truth value at each step cumulating to an overall drop from 1 to 0. Against these theoretical and practical advantages are two serious theoretical flaws. Firstly, contradictions need not be false, tautologies need not be true, and a contradiction may have the same truth value as a tautology: if |p| = 0.5 then |p ∨ ¬p| = |p ∧ ¬p| = 0.5. This makes statements appear vague which definitely are not, and therefore makes nonsense of hedging. If we are unsure whether someone is bald, for example, we can hedge, not only by saying something like ‘Well, he’s on the way to bald’, but in the extreme case, by retreating to ‘Well, at least he’s bald or he’s not.’ According to fuzzy logic, we may gain no security at all by so hedging, which is absurd.¹ Secondly, unclear cases are required to have a precise fuzzy truth value, one real number out of a continuum of others. This imparts vague statements, which seem to have no clear truth value, a spurious exactness. These seem to me as to many others to be finally damning reasons why fuzzy logic cannot capture the phenomenon of vagueness. Less crucially, fuzzy logic needs to resort to special tricks to cope with so-called higher-order vagueness.
W H Y P R AC T I T I O N E R S U S E F U Z Z Y LO G I C If fuzzy logic is theoretically such a no-hoper, why do practitioners use almost nothing else? Crucially, because it is numerical, it is easy to develop numerical and algorithmic methods for dealing with the data. There exist algorithms and software packages, help and discussion in superabundance. A simple Google™ search for ‘software’ + ‘fuzzy logic’ returned 1.65 million hits, whereas ‘software’ + ‘supervaluation’ returned 679.² Also, not all fuzzy reasoning concerns truth values. Many data are already quantitative and consist of approximate values for such parameters as mass, length, failure quota, and other statistical measures. Because it was the first approach to be used ¹ Thanks to a referee for this point. ² 18 August 2007. The catchiness of the term ‘fuzzy logic’ is only partly responsible.
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in programs, fuzzy logic cornered the market, and in engineering there is a strong founder effect.³ There is no effective alternative that practitioners can use, there exists a plethora of methods to suit different situations, and fuzzy logic gets results. That is not in itself an argument for the correctness of fuzzy logic’s analysis of vagueness, but it does show that philosophical alternatives have signally failed to produce tools for use outside the philosophy room, leaving practitioners with no alternative but to use what from a philosophical and theoretical point of view is regarded as a flawed theory. Use of numbers, even if they are not God-written, can be more useful than their non-use out of theoretical purity.
VAG U E O B J E C TS An object is vague when it is unclear where it starts and finishes, or what its parts are. The mountain Helvellyn has no clear boundaries. Of many a small object, from the atomic scale to larger chunks of rock, it is unclear whether it is part of Helvellyn or not. Object vagueness thus arises because of a special case of predicate vagueness: the source is the vagueness of the predicate ‘is part of ’. Object vagueness raises the further question whether there are vague objects in reality (ontic vagueness). Leaving that question unanswered, I note only that a viable account of vagueness ought to be able to cope with object vagueness, whatever its source, as well as with predicate vagueness.
S U PE RVA LUAT I O N S Philosophically the most favoured theory of vagueness employs supervaluations.⁴ Here the idea is that vague statements are treated by considering a range of admissible precisifications, each of which makes the statements involved classically precise, i.e. true or false. A statement’s truth status is the result of considering all admissible precisifications for it. If on all precisifications it is true, the statement is given the overall value of being true (sometimes called ‘supertrue’); if on all precisifications it is false, the overall value is false (‘superfalse’); if it is true on some precisifications and false on others, it receives no overall truth value. Logically complex statements are evaluated first within each precisification, using classical logic, and the overall outcome assessed in the same way as above. The advantages of this approach are that it retains classical logical tautologies and contradictions; vague statements do not have a sharp truth value; it is compatible with the world being sharp or exact, and it appears to locate vagueness in our concepts rather than in the world or in our beliefs. The disadvantages are that while it seems some statements are vaguer than others, and some vague statements closer to truth or falsity than others, the supervaluation approach provides no measure of how vague a statement is, or how true one is. The notion of an ³ Thanks to the same referee.
⁴ See Fine 1975, Keefe 2000.
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admissible valuation, which is standardly employed in supervaluational approaches, seems itself not to be exact, but to make the approach work there needs to be a sharp cut-off. This again runs into the issue of higher-order vagueness. Then the logic, while preserving classical tautologies and contradictions, is not truth-functional, and it is contended that inference patterns for some statements are no longer classical.⁵ W H AT I S TO B E D O N E ? One thing that cannot be done is to carry on as if the philosophy on the one hand and the science and practice on the other had nothing to do with one another. Since the scientific needs will not go away, I suggest it is adventitious to look again at the use of numbers in connection with vagueness, and see if we can come up with a way of providing materials for algorithmic treatments of inexactness which are less philosophically objectionable than fuzzy logic. E X PE C T E D T RU T H VA LU E S The approach I suggest combines aspects of supervaluations and fuzzy logic. From supervaluation theory it takes the idea of a range of different valuations, while from fuzzy logic it takes the idea of assigning numbers to truth and other values. It then puts the two together to give what I call an expected truth value for a statement. The term ‘expected truth value’ is adapted from probability theory, where the expected value (mathematical expectation or mean) of a random variable is the sum of the probabilities of each possible outcome multiplied by its outcome value or ‘payoff ’. ‘Expected’ in this context carries no epistemic connotations: an ‘expected’ outcome is not always even a possible outcome. For example the expected value for a single roll of a fair die is 3.5, which cannot be expected in any epistemic sense since there is no face with this value. If x is a sharp object, and a is a vague object, let the goodness of candidature of x to be a, |x for a| be a number in [0,1] with x |x for a| = 1. The summation is over all candidates x whose goodness is non-zero, or it could be over all objects. There is no need for Angst about a sharp cut-off between candidates and non-candidates, since on a numerical approach non-candidates have goodness zero and contribute nothing to the sum, but their nearest candidate fellows have goodness almost zero and contribute almost nothing to the sum. If R is a vague predicate and Z is an exact predicate, similarly define the goodness of candidature of Z to be R as a number |Z for R| such that Z |Z for R| = 1, where Z again ranges over all candidate relations. If Z is (say) two-placed then for any sharp objects x, y the statement xZy is true (1) or false (0): notate its truth value as [xZy]. The expected truth value (ETV) of the atomic statement aRb, written as ||aRb||, is defined as ⁵ This is controversial: against it, see Williams 2008.
486 ||aRb|| =
x
Z
y
Peter Simons |x for a|.|Z for R|.|y for b|.[xZy]
This is also a number in [0,1], so can be reckoned with. However the method for simple atomic statements does not generalize in the obvious way to complex statements, because we need to take account of what Fine calls penumbral connections, that is, logical relations among vague predicates.⁶ Consider two people: a, aged 41 and b, aged 39. It is absolutely and determinately true that a is older than b. Take now the two vague predicates ‘old’ and ‘young’. The ETVs ||a is young|| and ||b is old|| are both (we may suppose) non-zero. But if we attempt to calculate the ETV of ‘a is young and b is old’ in the obvious way as ||a is young and b is old|| = F G |F for young|.|G for old |.[Fa ∧ Gb] then since there are candidates for ‘young’ which make a young and candidates for ‘old’ which make b old, if these are allowed to vary independently we get that ||a is young and b is old|| > 0 which is absurd. It is wrong to allow these predicates to be precisified independently, since they are connected in meaning. Any precisification of the two predicates must respect the following three constraints No one is both old and young No one who is not young is younger than someone who is young No one who is not old is older than someone who is old⁷
which means they must be precisified together in a linked and constrained way. If we notate this linked precisification as ‘F ; G for old ; young’ then the ETV for ‘a is old and b is young’ is ||a is old and b is young|| = F ;G |F ; G for old ; young|.[Fa ∧ Gb] and we will obtain that ||a is young and b is old|| = 0, as required. There can be links also between candidates for vague objects, in particular if these are related by part–whole relations. For example if x and y are candidates for being a certain marsh m and x is a proper part of y and we are interested in the ETV of ‘m is a large marsh’ then under no precisification LM of ‘large marsh’ can we have LMx but not LMy. Similarly if a is a vague object and x and y are two candidates for a, even though x = y is absolutely true, and both x and y have non-zero candidature to be a, the ETV ||a = a|| must be 1: in other words we need to evaluate the ETV via the predicate λx[x = x] and not via the predicate λxy[x = y]. When considering the ETV of any complex statement, containing predicates and objects which are penumbrally linked, we must therefore evaluate each precisification for all the linked terms together, which is the method of supervaluations, and not separately, which is the method of fuzzy logic. In this respect the approach is much ⁶ Fine 1975. ⁷ Clearly the last two are instances of a general kind: Nothing which is not F is F -er than something which is F , where it is admissible to form the comparative.
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closer to supervaluationism than to fuzzy logic, the principal difference being in the assignment of ‘goodnesses’ to precisifications. AN EXAMPLE A commonly instanced example of a vague predicate is tall. In order to show the approach in action we shall look at a small range of interconnected vague predicates: tall, short, of medium height, very tall, very short. To avoid contextual complications we consider a single population of adult males at a single time. We assume that there is a height function h defined on the population so that at this time the height of an individual a is h(a), and we ignore diurnal height variation. If the population is large enough it is known that the height distribution approximates closely to the normal or Gaussian distribution given by the probability distribution function f (x) =
1 2ps 2
exp −
1 x−m s 2
2
where μ is the mean and σ is the standard deviation of the heights in the population. The actual values are furnished by the cumulated individual facts. We can consider a range of heights or, more practically, height-intervals, which are as precise as we need to make them, and look at the individuals falling in these height-intervals. The question is then, how we arrive at expected truth values for the vague predicates tall, short, of medium height, very tall, very short, so that the penumbral connections of the predicates is taller than, is shorter than, and is as tall as with one another and the mentioned vague predicates are all suitably respected. The first thing to note is that within the margin of error given by the width of the height-intervals, the binary predicates is taller than, is shorter than, is as tall as are precise (classical). Of these, the first two are asymmetric and transitive, the last is an equivalence. Their truth values in any case can be deduced by looking at the relative values of h(a) and h(b) for a given pair of men a, b. The constraints to be respected are that for any precisification of the predicates if a is very tall then a is tall if a is very short then a is short if a is tall then a is not short if a is very tall and b is tall but not very tall then a is taller than b if b is tall and c is of medium height then b is taller than c if c is of medium height and d is short then c is taller than d if d is short but not very short and e is very short then d is taller than e a is of medium height iff a is neither tall nor short if a is of average height (μ), a is neither tall nor short
The last constraint means that the cut-offs for tall and short must be above, respectively below, the mean. This strongly suggests we should treat the penultimate constraint as a definition of ‘is of medium height’.
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Suppose the mean height in the population is 176 cm and that we consider intervals of 2 cm from the odd-number cm heights, so the group of average height is in the range 175–177 cm. We look at the following five precisifications and tabulate their lower cut-offs: short 165 169 167 165 163
medium 169 171 171 173 173
tall 183 181 181 179 179
very tall 187 183 185 187 189
Goodness 0.1 0.3 0.3 0.2 0.1
The goodnesses assigned to precisifications in this mini-example have been done intuitively, but more methodical ways to do so would be to look at the height distribution curve and consider the percentiles assigned to the different categories, or, more empirically, to do a survey of people’s opinions on which men are tall, short and so on. We would also expect a much larger number of precisifications to be used. We are using a small number for illustration only, and while we have for simplicity arrayed tall respectively very tall symmetrically to short respectively very short about the mean, in real life this may not be what happens. Taking now a range of different heights for men, the expected truth values of their falling under the various predicates, given the precisifications above and their respective goodnesses, are as given in the table below, generated by simple spreadsheet calculation. Height 162 164 166 168 170 172 174 176 178 180 182 184 186 188 190
VS 1 0.9 0.6 0.3 0 0 0 0 0 0 0 0 0 0 0
S 1 1 1 1 0.9 0.3 0 0 0 0 0 0 0 0 0
M 0 0 0 0 0.1 0.7 1 1 1 0.7 0.1 0 0 0 0
T 0 0 0 0 0 0 0 0 0 0.3 0.9 1 1 1 1
VT 0 0 0 0 0 0 0 0 0 0 0 0.3 0.6 0.9 1
Again the numbers should not be taken too seriously, but they do illustrate how the gradations between for example very tall and tall or short and medium-sized give useful information about the gradual transitions in a way that a non-numerical supervaluational approach does not. At the same time the constraints respecting penumbral connections are reflected in the table.
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A DVA N TAG E S A N D P RO B L E M S This numerical-supervaluational approach, which for short I call supernumeration, as here outlined promises some clear advantages over the alternatives: it yields calculable numerical values (like fuzzy logic); it gives tautologies the value 1 and contradictions the value 0 (like supervaluations); unlike supervaluations, it quantifies goodness of case; it is iterable; and it minimizes the effect of higher-order vagueness because the numerical contribution of cases near to the extremes 1 and 0 is close to the values for those extreme cases. It further generalizes, using integration, to the infinite case, via the notions of candidature and truth value density functions, analogously to the way finitary probability generalizes to the infinite case; it can be applied equally well to quantities other than truth values, such as mass, size, etc.; and finally, it does not need to deny that the world in itself is sharp (whether we wish to affirm this is another matter). There are equally some obvious prima facie disadvantages with supernumeration: it is complicated, and harbours elements of arbitrariness. There are two obvious issues about arbitrariness in applying the method. One is where to get the numbers from. Too much should not be made of this issue: goodness of candidature is not something writ in the heavens, but is a constrained numerical estimate. Fuzzy logic is often accused of introducing spurious and indeed ridiculous hyperexactness into what is after all a vague and fuzzy matter. This is only a serious problem if the numbers are taken to be God-given real values existing independently of us. If the assignment of numbers is construed instrumentally, as a way we can work with otherwise intractable or unquantifiable properties, then they can be taken with metaphysical lightness. The more serious problems of fuzzy logic, concerning its value-functionality, remain even when the numbers are taken lightly in this way. In actually used fuzzy logic, computation typically allows numerical values to be varied and an algorithm run repeatedly to see how far the result deviates from other results with different values, and the same could be done here. Also the problem of infinite or large finite ranges of values is taken in hand by considering finitely many subranges of values, rather as taxation authorities divide incomes of taxpayers into different bands for the purposes of applying different rules, or statisticians divide continuous samples into bands for numerical treatment. Mapmakers provide contour lines cutting land surfaces at (e.g.) 10m intervals as ways to present complex relief: this is a necessary simplification, as is the more obvious device of colouring relief at different heights. False colour images from satellite and astronomical data are another presentational device that is frankly accepted as a necessary simplification. The other problem is that of discerning the constraints imposed on precisifications by penumbral connections. Here there appears to be no simple or uniform procedure or algorithm: it is not like logic. Again this mirrors what happens in applied probability. Each statement or type of statement needs to be looked at in its own terms, relying on the judgement, common sense, and accumulated semantic expertise of the investigator or investigators. Given the complexity of our vague language, this is only to be expected.
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Re f e re n c e s Edgington, D. (1997), ‘Vagueness by degrees’ in R. Keefe and P. Smith, eds., Vagueness: A Reader, Cambridge, Massachusetts, MIT Press, 294–316. Fine, K. (1975), ‘Vagueness, truth and logic’, Synthese 30, 265–300. Keefe, R. (2000), Vagueness, Cambridge, Cambridge University Press. Simons, P. M. (1997), ‘Vagueness, many-valued logic, and probability’ in W. Lenzen, ed., Das weite Spektrum der Analytischen Philosophie—Festschrift f¨ur Franz von Kutschera, Berlin/New York, de Gruyter, 307–22. (1999), ‘Does the sun exist? The problem of vague objects’ in T. Rockmore, ed., Proceedings of the XX World Congress of Philosophy, Vol. II, Metaphysics, Bowling Green, Philosophy Documentation Center, 89–97. Williams, J. R. G. (2008), ‘Supervaluationism and logical revisionism’, Journal of Philosophy 105, 192–212.
28 Degree of Belief is Expected Truth Value Nicholas J. J. Smith
This chapter presents a solution to a problem engendered by the following two claims: (A) Vagueness gives rise to degrees of belief. (B) These degrees of belief do not behave in the same ways as degrees of belief arising from uncertainty: they do not conform to the laws of probability. The problem is to give a clear account of the relationship between degrees of belief and subjective probabilities. The solution to be presented here also involves degrees of truth: in outline, the proposal is that one’s degree of belief in a proposition P is one’s expectation of P’s degree of truth. Those who already believe that vagueness should be handled using degrees of truth will believe (A) and (B). So the chapter can be read as solving a problem which arises for degree theorists. It can also be read as providing a positive argument in favour of degrees of truth, directed at those who do not start out believing that vagueness should be handled using degrees of truth, but do start out believing (A) and (B): the argument is that the best solution to the problem engendered by (A) and (B) employs degrees of truth.
28.1
VAG U E N E S S - B A S E D A N D U N C E RTA I N T Y- B A S E D DEGREES OF BELIEF
Suppose we have a Sorites series leading from tall men down to short men. Suppose also that we have accepted a degree-theoretic account of vagueness—so we think that An earlier version of this chapter—including a surrounding discussion of how degrees of truth can be incorporated into the framework of Stalnakerian pragmatics—was presented at the Arch´e Vagueness Conference in St Andrews on 8 June 2007 (see Smith (2008, §5.3) for that surrounding discussion). Other earlier versions were presented at the Annual Conference of the Australasian Association of Philosophy on 5 July 2007, in the Higher Seminar in Theoretical Philosophy in the Department of Philosophy at Lund University on 1 April 2008, and in the Current Projects Seminar in the Centre for Time at the University of Sydney on 25 August 2008. Thanks to the audiences on those occasions for useful feedback. For helpful discussions, I am grateful to Staffan Angere, Richard Dietz, Andy Egan, Michael McDermott, Peter Milne, Josh Parsons, Wlodek Rabinowicz, and Roy Sorensen. Thanks also to the two anonymous referees for their comments and to the Australian Research Council for research support.
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‘This man is tall’ goes gradually from 1 true, said of men at the beginning of the series, down to 0 true, said of men at the end. Then what attitude should we adopt to (the proposition expressed by) ‘This man is tall’ as we consider various men in the series? Surely we should go from being fully committed to the proposition at the beginning of the series, to fully rejecting it by the end of the series, via a gradually changing series of intermediate states of partial belief, which decrease in degree of confidence as we progress down the series.¹ So degree theorists should certainly accept (A). But it seems that non-degree theorists should accept (A) too. Consider Schiffer (2000, 223–4): Sally is a rational speaker of English, and we’re going to monitor her belief states throughout the following experiment. Tom Cruise, a paradigmatically non-bald person, has consented, for the sake of philosophy, to have his hairs plucked from his scalp one by one until none are left. Sally is to witness this, and will judge Tom’s baldness after each plucking. The conditions for making baldness judgments—lighting conditions, exposure to the hair situation on Tom’s scalp, Sally’s sobriety and perceptual faculties, etc.—are ideal and known by Sally to be such . . . Let the plucking begin. Sally starts out judging with absolute certainty that Tom is not bald; that is, she believes to degree 1 that Tom is not bald and to degree 0 that he is bald. This state of affairs persists through quite a few pluckings. At some point, however, Sally’s judgment that Tom isn’t bald will have an ever-so-slightly-diminished confidence, reflecting that she believes Tom not to be bald to some degree barely less than 1. The plucking continues and as it does the degree to which she believes Tom not to be bald diminishes while the degree to which she believes him to be bald increases . . . Sally’s degrees of belief that Tom is bald will gradually increase as the plucking continues, until she believes to degree 1 that he is bald. Although I’ll have a little more to say about this later, for now I’m going to assume that the qualified judgments about Tom’s baldness that Sally would make throughout the plucking express partial beliefs. After all, the hallmark of partial belief is qualified assertion, and, once she was removed from her ability to make unqualified assertions, Sally would make qualified assertions in response to queries about Tom’s baldness.
Other things that we might say about the case—things that would avoid claiming that Sally has degrees of belief—are (i) that Sally fully believes that Tom is not bald until a particular hair is removed, from which point on she fully believes he is bald; (ii) that Sally fully believes that Tom is not bald until a particular hair is removed, at which point she enters an indeterminate state in which she does not believe (to any degree, even 0) that Tom is not bald and does not believe (to any degree, even 0) that Tom is bald, and then when another particular hair is removed Sally comes to fully believe that Tom is bald; and (iii) that Sally does not have attitudes towards propositions such as ‘Tom is bald’, but only towards propositions such as ‘Tom is bald to degree x’ or ‘ ‘‘Tom is bald’’ is true to degree x’, each of which she either fully believes or fully rejects. The problem with these approaches is that they do not fit the phenomena. Contra (i) and (iii), Sally certainly seems to be unsure as to what to believe and say about Tom’s baldness, at various points in the process, and contra (ii), she does not have one catch-all ‘confused state’, which she enters, remains in, then leaves: rather, she seems clearly to become less and less sure that Tom is not bald, and then later more and more sure that he is. ¹ I treat the terms ‘degree of belief ’, ‘partial belief ’ and ‘credence’ as synonyms.
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The proponent of (iii) may reply that Sally’s qualified assertion that Tom is bald—behaviour which seems clearly to indicate that there is some P such that Sally is unsure whether P —should in fact be understood as a full-on assertion that Tom is bald to an intermediate degree. But this response is rather implausible on the face of it, and furthermore (iii) involves a strange separation between truth on the one hand, and belief and assertion on the other. The view involves a semantics which assigns degrees of truth to atomic propositions such as ‘Tom is bald’, but tells us that we cannot believe or assert such propositions. Rather, we must believe and assert meta-level propositions of the form ‘ ‘‘Tom is bald’’ is true to degree x’, or propositions about degrees, such as ‘Tom’s degree of baldness is x’. This kind of separation (between truth on the one hand, and belief and assertion on the other) should be regarded as a last resort, to be considered only if it were shown that we cannot, for some reason, adopt what should be the default position, namely that the very same things both have truth values and are the contents of beliefs and assertions.² So we need to countenance degrees of belief arising from vagueness. Doing so will not cause us any problem, however, if these degrees of belief are just the same as the kind with which we are already familiar: the kind that arise from uncertainty about the truth of propositions (in cases not involving vagueness), and are handled formally by means of probability theory. However it seems that degrees of belief arising from vagueness do not behave in the same ways as degrees of belief arising from uncertainty. To adapt and augment an example of Schiffer’s: Suppose that Sally is about to meet her long-lost brother Sali. She has been told that he is either very tall or very short, but she has no idea which (so she does know that he is not a borderline case), and she has been told that he is either hirsute or totally bald, but she has no idea which (so she does know that he is not a borderline case). As a result of her uncertainty, she believes both of the propositions ‘Sali is tall’ and ‘Sali is bald’ to degree 0.5. Suppose also that Sally regards these two propositions as independent: supposing one to be true would have no bearing on her beliefs about the other. Then, for familiar reasons, she should believe ‘Sali is tall and bald’ to degree 0.25. Now suppose that mid-way through Schiffer’s experiment, when Sally’s degree of belief that Tom is bald is 0.5, she also believes to degree 0.5 that Tom is tall—on the basis of looking at him and seeing that he is a classic borderline case of tallness.³ Then what should be her degree of belief that Tom is tall and bald? The answer 0.5 suggests itself very strongly: certainly the answer 0.25 seems wrong. If you don’t think so, then just add more conjuncts (e.g. funny, nice, intelligent, cool, old—where Sally knows of Sali only that he is not a borderline case of any of them, and of Tom that he is a classic borderline case of all of them): the more independent conjuncts you add, the lower the uncertainty-based degree of belief should go, but this does not seem to be the case for the vagueness-based degree of belief (Schiffer, 2000, 225), (MacFarlane, 2006, 6). So it seems that (B) is true, as well as (A). This means that we must abandon the familiar identification of degrees of belief with subjective probabilities, and offer a new ² John MacFarlane’s view (this volume) suffers from this problem. ³ Suppose, for the sake of the example, that Tom Cruise is borderline tall.
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account of their relationship. In Section 28.2 I critique one kind of account. In Section 28.3 I present my own view, and in Section 28.4 I reply to objections to this view. 28.2
T WO K I N D S O F D E G R E E O F B E L I E F ?
One thought in response to (A) and (B) is that there are two kinds of degree of belief: uncertainty-based degrees of belief and vagueness-based degrees of belief. Schiffer holds a view of this sort. He distinguishes SPB’s (‘standard partial beliefs’) and VPB’s (‘vagueness-related partial beliefs’). In his view, we have two distinct systems of degrees of belief: an assignment of SPB’s to propositions, which obey the laws of probability, and an assignment of VPB’s to propositions, which obey the laws of standard fuzzy propositional logic (i.e. VPB(¬p) = 1 − VPB(p), VPB(p ∧ q) = min{VPB(p), VPB(q)} and VPB(p ∨ q) = max{VPB(p), VPB(q)}). There is a grave problem for any proposal which posits two different systems of degrees of belief, where it is allowed that a subject may have a degree of belief of one kind of strength n in a proposition P, and a degree of belief of another kind of strength m = n in the same proposition P. The problem is that the very idea of degree of belief is made sense of via the thought that a degree of belief that P is a strength of tendency to act as if P. As Ramsey (1990, 65–6) puts it: the degree of a belief is a causal property of it, which we can express vaguely as the extent to which we are prepared to act on it. . . . it is not asserted that a belief is an idea which does actually lead to action, but one which would lead to action in suitable circumstances . . . The difference [between believing more firmly and believing less firmly] seems to me to lie in how far we should act on these beliefs.
But one simply cannot have two different strengths of tendency to act as if P, in a given set of circumstances. Consider, for example, the proposition that Fido is dangerous. When Fido enters the room, one will do some particular thing, for example sit still, or jump and run. When Fido looks at one, one will do some particular thing, for example tremble, or offer him some beef jerky. When Fido barks, one will do some particular thing, for example scream; and so on. One cannot both back away slowly and run screaming (at the same time), and it cannot both take Fido getting within two metres of one to make one run away, and require Fido getting within one metre to make one run. So one cannot both tend strongly to act as if Fido is dangerous, and tend weakly to act as if Fido is dangerous—at least not if there is to be any sort of transparent relationship between these tendencies and the way one actually acts. But given that a degree of belief just is a strength of tendency to act, this means that one cannot have two different degrees of belief in the same proposition. The proponent of two kinds of degrees of belief might offer a number of responses here. (1) She might deny that there is a transparent relationship between tendencies to act and the way one actually acts. So, in the case of Fido, one might have both a strong tendency to act as if Fido is dangerous, and a weak tendency, and these interact so as to make one behave in particular ways in particular situations (ways that we would like to describe as indicating that one has a mid-strength tendency to act as if Fido is dangerous—although on the current proposal, we cannot straightforwardly
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say this). But for this view to get off the ground, we would need to be told exactly how degrees of belief of the two sorts combine to produce certain behaviour, and furthermore, the view threatens to make it impossible for us ever to know (even roughly) someone’s degree(s) of belief in a given proposition. (2) She might say that although there are indeed two kinds of degrees of belief, they always have the same strength, for every proposition. But clearly this would run us headlong into the problem discussed above, that partial beliefs arising from vagueness do not and should not behave in the same ways as partial beliefs arising from uncertainty. (3) She might deny that degrees of belief are to be understood in terms of strength of tendency to act. But any view which disconnects degree of belief from tendency to act threatens to undermine the utility of the notion of degree of belief, and furthermore any candidate replacement proposal—for example, the view that the difference between believing more firmly and believing less firmly is a matter of strength of feeling⁴ —would seem to face the very same problem (one cannot have two different intensities of feeling about one proposition). (4) She might claim that one never has both kinds of degree of belief in the same proposition at the same time. For suppose, for reductio, that you have an uncertainty-based degree of belief of 0.3 that Dobbin wins the race, and a vaguenessbased degree of belief of 0.5 that Dobbin wins the race. How could you have acquired both these beliefs? In order to acquire the first, you would need to lack evidence concerning who wins. In order to acquire the second, you would need to have all the relevant evidence, and see that it—i.e. the world itself—leaves it unsettled who wins.⁵ So clearly you could not have both these degrees of belief at once. There are still problems for this view, however. First, we need to be told how to reason with several propositions—and compounds thereof—in some of which we have degrees of belief of one type, and in others of which we have degrees of belief of the other type. Second, what justifies saying that we have here two non-interacting systems of degrees of belief, rather than one system, which assigns degrees to all propositions, but where these degrees behave differently in different situations (e.g. sometimes they obey the laws of probability, sometimes they do not)? This is the remaining possibility regarding the relationship between vaguenessbased degrees of belief, and uncertainty-based degrees of belief: the suggestion that what we have is one univocal notion of degree of belief—one single system of assignments of degrees of belief to propositions—but where the degrees assigned sometimes behave in accordance with the laws of probability, and sometimes do not. This is the sort of view I shall advocate in the next section.⁶ ⁴ This is the view with which Ramsey contrasts his own view, in the discussion quoted earlier. ⁵ I am imagining a case where due to the vagueness of the boundaries of horses, two horses are equally good candidates for having crossed the line first. In practice this would no doubt be deemed a tie, but imagine that we are examining very high-resolution pictures of the finish, and that we are interested not in the practical question of distributing winnings, but purely in the question of which horse in fact crossed the line first. ⁶ Apart from my own view, another view which fits the description just given is that of Field (2000). Field supposes that an agent has a probability function P over propositions; he supposes also that the language includes a determinately operator D; and he then proposes that the agent’s degree of belief Q(α) in any proposition α is given by Q(α) = P(Dα). Thus my degree of belief that α is my subjective probability that determinately α. It may sound, then, as though we do have
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Nicholas J. J. Smith 28.3
D E G R E E O F B E L I E F A S E X PE C T E D T RU T H VA LU E
The picture I propose has three components: (1) an agent’s epistemic state; (2) the degrees of truth of propositions; and (3) an agent’s degrees of belief in propositions. (1) I take an agent’s epistemic state to be (represented by) a probability measure over the space of possible worlds. So, where W is the set of possible worlds, the agent’s epistemic state P is a function which assigns real numbers between 0 and 1 inclusive to subsets of W . Intuitively, the measure assigned to a set S of worlds indicates how likely the agent thinks it is that the actual world is one of the worlds in S. Given this understanding of P —together with the convention that assigning a set of worlds measure 1 means that you are absolutely certain that the actual world is in that set, and assigning a set of worlds measure 0 means that you are absolutely certain that the actual world is not in that set—the three probability axioms are well motivated: P1. For every set A ⊂ W , P(A) ≥ 0 P2. P(A ∪ B) = P(A) + P(B) provided A ∩ B = ∅ P3. P(W ) = 1. (2) At each possible world, each proposition has a particular degree of truth. Thus we may regard each proposition S as determining a function S : W → [0, 1], i.e. the function which assigns to each world w ∈ W the degree of truth of S at w.⁷ The relationships between the functions associated with various propositions will be constrained in familiar ways by the logical relationships between these propositions: thus, for example, (S ∨ T ) (w) = max{S (w), T (w)}, (S ∧ T ) (w) = min{S (w), T (w)} and (¬S) (w) = 1 − S (w). (3) We have a measure over worlds (the agent’s epistemic state P), and functions from worlds to real numbers (each proposition S). Thus S is a random variable, and I propose that we identify the agent’s degree of belief in S with her expectation (aka expected value) of S. To get an intuitive feel for the proposal, consider the case where there are finitely many possible worlds. One’s probability measure over sets of worlds is in this case determined, via the additivity axiom P2, by the values assigned to singleton sets: P({w1 , . . . , wn }) = P({w1 }) + . . . + P({wn }). So, treating probabilities assigned to singletons as probabilities assigned to their members, one can, in the finite case, think two different systems of degrees of belief: P-values and Q-values. But Field says that only Q-values are to be thought of as degrees of belief: ‘P should be thought of as simply a fictitious auxiliary used for obtaining Q’ (16); ‘P [should] not be taken seriously: except where it coincides with Q, it plays no role in describing the idealized agent’ (19). One worry I have about Field’s proposal concerns the appearance of a primitive determinately operator within the contents of beliefs. A second worry concerns the downgrading of P: I think Field takes this too far. In my proposal (Section 28.3), subjective probabilities do play an important role in describing an agent, but they are not to be identified with degrees of belief. Field on the other hand seems to be in the grip of the view that if subjective probabilities are allowed into the picture at all (as anything beyond fictitious auxiliaries) then they will automatically grab the mantle ‘degrees of belief ’. ⁷ For the sake of simplicity of presentation, I shall often conflate S and S , i.e. write of a proposition as being a function from worlds to degrees, rather than as determining such a function.
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of oneself as assigning each world a degree of likelihood: a number indicating how likely one thinks it is that that world is the actual world. Each world w itself assigns each proposition S a degree of truth S(w). Now, one’s degree of belief in S is one’s expectation of S, i.e. one’s expected value of S’s degree of truth. Let us denote this E(S). In this finite case, it can be calculated as follows, where w1 . . . wn are all the possible worlds: E(S) = P({w1 }) · S(w1 ) + . . . + P({wn }) · S(wn ) This is analogous to the calculation of expected utility in decision theory (with worlds playing the role of outcomes of acts, and degrees of truth playing the role of utilities of outcomes). The proposal has two particularly important features: it meshes perfectly with the guiding idea of one’s degree of belief that S as a measure of the strength of one’s tendency to act as if S; and it has the consequence that degrees of belief sometimes behave like probability assignments, and sometimes do not. I shall discuss these points in turn. First, consider the idea that one’s degree of belief that S is a measure of the strength of one’s tendency to act as if S. It is important to note that I am not claiming that two persons who have the same degree of belief that S will behave in the same ways, or even have the same tendencies to behave in certain ways. I am claiming that they will have the same tendency to act as if S. Whether a person’s behaving in a certain way constitutes her acting as if S depends on her preferences (desires, utilities) and on her other beliefs. For example, let S be the proposition that there is an especially fragrant rose in Bob’s garden. For a rose fancier, approaching Bob’s garden might constitute acting as if S, whereas for a person with an aversion to roses—or a rose fancier with false beliefs about the location of Bob’s garden—moving away from Bob’s garden might constitute acting as if S. So while two persons who have the same degree of belief that S will have the same tendency to act as if S —this is our guiding idea—in general they will only behave in the same ways (described at the level of bodily movements, for example—rather than in terms of whether they are acting as if S) if their other beliefs and desires are also the same. Consider now a simple example. There are three ‘open worlds’ w1 , w2 and w3 —i.e. three worlds such that one is not certain that one is not in them—i.e. P({w1 , w2 , w3 }) = 1. Suppose that S is the proposition ‘A tall person will win the race’. You don’t know who will win, but you do know that it is either the first man in our Sorites series leading from tall men to short men (this is the situation in w1 ), or the last man (this is the situation in w2 ), or the man in the middle (this is the situation in w3 ). You think that each of these three possibilities is equally likely, i.e. P({w1 }) = P({w2 }) = P({w3 }) = 13 . In w1 , S is 1 true; in w2 , S is 0 true; in w3 , S is 0.5 true. So your expectation that S is 13 · 1 + 13 · 0 + 13 · 0.5 = 0.5. This seems to be a true measure of the strength of your tendency to act as if S. Suppose you need a tall man for your basketball team, and you have a choice between signing up the race winner (whoever that should turn out to be), or Bill (whom you know to be of the same height as the first man in our Sorites series—hence ‘Bill is tall’ is 1 true, and you know this, and so your expectation of this proposition is 1), or Ben (whom
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you know to be of the same height as the last man in our Sorites series—hence ‘Ben is tall’ is 0 true, and you know this, and so your expectation of this proposition is 0), or Bob (whom you know to be of the same height as the man in the middle of our Sorites series—hence ‘Bob is tall’ is 0.5 true, and you know this, and so your expectation of this proposition is 0.5). You would sooner sign up the race winner than Ben, sooner sign up Bill than the race winner, and be indifferent between signing up the race winner and Bob. Thus, the strength of your tendency to act as if S mirrors your expectation of S. I have been making the assumption that your preferences regarding team members can be summed up thus: ‘The taller the better.’ If, on the other hand, you wanted only very tall players—so you are just as averse to signing up a borderline tall person as to signing up a short person—then of course you would have no tendency to sign up Bob. That is no problem for my view (even though, in this new case—in which you have different preferences—your expectation that Bob is tall is still 0.5). For if you wanted only very tall players, then signing up P would not constitute acting as if P is tall; rather, it would constitute acting as if P is very tall (recall the discussion on p. 497). In the situation described (in both cases—i.e. whatever your preferences), your expectation that Bob is very tall is 0. So in the second case—where (given your new preferences) signing up P now constitutes acting as if P is very tall, rather than acting as if P is tall—my theory correctly predicts that you will have no tendency to sign up Bob. In sum: as your preferences change from ‘the taller the better’ to ‘very tall’, your degrees of belief that Bob is tall and that Bob is very tall remain 0.5 and 0 respectively. However, the significance of signing-up behaviour changes. At first, such behaviour constitutes acting as if the signed-up player is tall; with the new preferences, it constitutes acting as if the signed-up player is very tall. That is why two people who have the same degree of belief in ‘Bob is tall’ might have different tendencies to sign up Bob. My claim is that they will have the same tendency to act as if Bob is tall. If their preferences differ, however, then what counts as acting as if Bob is tall for one person—say, signing up Bob—might not count as acting as if Bob is tall for the other person. The same kind of point applies in a host of other cases which, at first sight, might seem to pose a problem for my view. For example, suppose that persons A and B are faced with a choice of cups of coffee: cup 1, which they know is either freshly made or has been sitting there for several hours (they do not know which—but they do know each option is equally likely), or cup 2, which they know was made about fifteen minutes ago. In the circumstances, we may suppose that both A and B assign an expected truth value of 0.5 to both ‘cup 1 is hot’ and ‘cup 2 is hot’—i.e. on my view both A’s and B’s degrees of belief in both these propositions are 0.5. But A and B behave quite differently. A, who likes her coffee either very hot, or cooled to room temperature, reaches for cup 1 and has no tendency whatsoever to reach for cup 2. B, whose preference in coffee is ‘the hotter the better’, is equally inclined to reach for cup 1 as for cup 2. This is all grist for my mill. Both A and B believe to degree 0.5 that the coffee in cup 2 is hot, and believe to degree 0 that the coffee in cup 2 is very hot. Given B’s preferences, reaching for a cup is (other things being equal) a way of acting as if it contains hot coffee. Given A’s preferences, reaching for a cup is not (other things being
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equal) a way of acting as if it contains hot coffee; rather, it is (other things being equal) a way of acting as if it contains coffee which is very hot or at room temperature. A and B have the same degree of belief that cup 2 contains hot coffee. So my claim is that they will have the same tendency to act as if cup 2 contains hot coffee. But ‘acting as if a cup contains hot coffee’ amounts to doing something different in A’s case than in B’s. That is why A and B have different tendencies to reach for cup 2, even though they have the same degree of belief that it contains hot coffee, and the same tendency to act as if it contains hot coffee. The key point, then, is the one made on p. 497: whether a person’s behaving in a certain way constitutes her acting as if S depends on her preferences and on her other beliefs. So two people who have the same degree of belief that S, but differ in their other beliefs or in their preferences, might behave differently (described at the level of bodily movements), even though they have the same tendency to act as if S. Apart from meshing with the idea of one’s degree of belief that S as a measure of the strength of one’s tendency to act as if S, my proposal also has the desired feature that sometimes degrees of belief behave like probability assignments, and sometimes do not. Before showing this, I shall generalize the picture presented above. For so far we have considered only the special case where we have finitely many possible worlds, but of course we cannot, in general, suppose that there are only finitely many possible worlds—indeed we cannot suppose that there are only countably many. But if there are uncountably many possible worlds, then (i) we cannot assume that the agent’s probability measure is defined on all subsets of the space of possible worlds, and (ii) we cannot assume that every proposition determines a measurable function from worlds to truth values, i.e. a random variable. We shall handle this situation in the standard way. In regards to point (i), we suppose there to be a family F of subsets of the space W of all possible worlds which is a σ -field, i.e. it satisfies the conditions: 1. W ∈ F 2. For all A ∈ F, A ∈ F 3. For any countable number of sets A1 , . . . , An in F, n An ∈ F.⁸ Our probability measure will be defined on F, i.e. it will assign probabilities to sets in F, and not to other subsets of W ; the sets in F will be called the measurable sets of possible worlds.⁹ In regards to point (ii), for a function S from worlds to the reals to be measurable, i.e. a random variable, it must satisfy the condition that for any real x, {w ∈ W : S(w) ≤ x} ∈ F. If such a function is bounded, it will have a well-defined expectation E(S). All propositions are functions from worlds to [0, 1], and hence bounded. As for the condition that they be measurable, we henceforth restrict our attention to propositions which meet it. This means that we consider only propositions S such that it makes sense to ask ‘How likely do you take it to be that this proposition has a truth value within such-and-such limits?’ ⁸ By de Morgan’s laws, we could equivalently replace union with intersection in condition 3. ⁹ Once we have made this alteration to our set-up, it is standard also to change axiom P2 so that it applies not just to unions of two sets, but to unions of countably many sets—i.e. for any ∞ countable collection {Ai } of pairwise disjoint sets, P( ∞ n=1 An ) = n=1 P(An ).
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With the general picture now in place, we can make the following definitions: Definition 1 (vagueness-free situation). An agent is in a vagueness-free situation (VFS) with respect to a proposition S iff there is a measure-1 set T of worlds (i.e. a set T such that P(T ) = 1) such that S(w) = 1 or S(w) = 0 for every w ∈ T . ( That is, the agent may not know for sure whether S is true or false, but she does absolutely rule out the possibility that S has an intermediate degree of truth: for she is certain that the actual world is somewhere in the class T , and everywhere in T , S is either 1 true or 0 true.) An agent is in a VFS with respect to a set of propositions if she is an a VFS with respect to each of the propositions in . Definition 2 (uncertainty-free situation). An agent is in an uncertainty-free situation (UFS) with respect to a proposition S iff there is a measure-1 set T of worlds and a k ∈ [0, 1] such that S(w) = k for every w ∈ T . ( That is, it is totally ruled out that S has a degree of truth other than k: for the agent is certain that the actual world is somewhere in the class T , and everywhere in T , S is k true.) An agent is in a UFS with respect to a set of propositions if she is an a UFS with respect to each of the propositions in . We can now establish four results which show when degrees of belief behave like probability assignments, and when they do not. Proposition 1 (Degrees of belief equal probabilities in VFSs). If an agent is in a VFS with respect to S, then E(S) = P({w : S(w) = 1}).¹⁰ Proposition 2 (Degrees of belief equal degrees of truth in UFSs). If an agent is in a UFS with respect to S, then E(S) equals the degree of truth which the agent is certain S has.¹¹ Proposition 3 (Degrees of belief behave like probabilities in VFSs). Let be a class of wfs, closed under the operations of forming wfs using our standard propositional connectives ∨, ∧ and ¬, such that one is in a VFS with respect to .¹² Then one’s ¹⁰ Proof. We are given that there is a set T of worlds such that P( T ) = 1 and S(w) = 1 or S(w) = 0 for every w ∈ T . Divide T into two sets: T1 , containing the worlds in which S is 1 true, and T0 , containing the worlds in which S is 0 true. (We know these are both measurable as follows. Where S is a random variable and a is any real, {w : S(w) = a} is measurable. T0 = T ∩ {w : S(w) = 0}, and T1 = T ∩ {w : S(w) = 1}, and measurable sets are closed under intersection.) The expectation of a formula is not affected by its truth value anywhere outside a measure 1 set, so E(S) = P( T0 ) · 0 + P( T1 ) · 1 = P( T1 ). Let S1 be {w : S(w) = 1}. P(S1 ) = P( T1 ) + P(S1 \T1 ). But P(S1 \T1 ) = 0, because P( T ) = 1 and so if the measure of some set disjoint from T were positive, then by P2 the measure of W would be greater than 1, violating P3. So P(S1 ) = P( T1 ) and hence E(S) = P(S1 ). ¹¹ Proof. There is a measure 1 set T such that S(w) = k for every w ∈ T . The expectation of a formula is not affected by its truth value anywhere outside a measure 1 set, so E(S) = P( T ) · k = 1 · k = k. ¹² The closure requirement is no restriction, because if one is in a VFS with respect to a class of wfs, then one is in a VFS with respect to the closure of that class (because whenever the component wfs are 1 true or 0 true, so are the compounds).
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degrees of belief (i.e. expectations) of wfs in behave like probabilities, in the sense that they satisfy the following three conditions: 1. For all wfs γ ∈ , 0 ≤ E(γ ) ≤ 1.¹³ 2. For all tautologies γ ∈ , E(γ ) = 1.¹⁴ 3. If γ1 , γ2 ∈ are mutually exclusive, then E(γ1 ∨ γ2 ) = E(γ1 ) + E(γ2 ).¹⁵ Proposition 4 (Degrees of belief behave like degrees of truth in UFSs). Let be a class of wfs, closed under the operations of forming wfs using ∨, ∧ and ¬, such that one is in a UFS with respect to .¹⁶ Then one’s degrees of belief (i.e. expectations) of wfs in behave like degrees of truth, in the sense that they satisfy the following three conditions: 1. E(¬γ ) = 1 − E(γ ).¹⁷ 2. E(γ1 ∨ γ2 ) = max{E(γ1 ), E(γ2 )}. 3. E(γ1 ∧ γ2 ) = min{E(γ1 ), E(γ2 )}.¹⁸ ¹³ Proof. By proposition 1, E(γ ) = P({w : γ (w) = 1}. As this is a probability, it is between 0 and 1 (inclusive) by definition. ¹⁴ There are several possible definitions of ‘tautology’ in fuzzy logic. All we need for the proof is something they all agree on, viz. that a tautology never gets the value 0. Proof. By hypothesis we have a set T of worlds such that P( T ) = 1 and γ (w) = 1 or γ (w) = 0 for every w ∈ T . But as γ is a tautology, there are no worlds w such that γ (w) = 0, so we have a set T of worlds such that P( T ) = 1 and γ (w) = 1 for every w ∈ T . So E(γ ) = 1. ¹⁵ There are several possible definitions of ‘mutually exclusive’ in fuzzy logic. All we need for the proof is something they all agree on, viz. that two mutually exclusive propositions never both get the value 1. Proof. By hypothesis we have a set T1 of worlds such that P( T1 ) = 1 and γ1 (w) = 1 or γ1 (w) = 0 for every w ∈ T1 , and a set T2 of worlds such that P( T2 ) = 1 and γ2 (w) = 1 or γ2 (w) = 0 for every w ∈ T2 . So P( T1 ∩ T2 ) = 1. (For suppose it has measure 0 ≤ n < 1. Then T1 \T2 and T2 \T1 both have measure 1 − n. But then by P2, ( T1 \T2 ) ∪ ( T2 \T1 ) ∪ ( T1 ∩ T2 ) has measure (1 − n) + (1 − n) + n = 2 − n > 1, for these three sets are pairwise disjoint.) So we have a measure 1 set T1 ∩ T2 in which both γ1 and γ2 are 0 or 1 true at every world. But we also know γ1 and γ2 are mutually exclusive, i.e. there are no worlds where γ1 and γ2 are both 1 true. So we can divide our measure 1 set T1 ∩ T2 into three pairwise disjoint subsets, G, G1 and G2 , with G containing worlds at which both γ1 and γ2 are 0 true, G1 containing worlds at which γ1 is 1 true and γ2 is 0 true, and G2 containing worlds at which γ2 is 1 true and γ1 is 0 true. (We know these subsets are measurable by reasoning similar to that in n.10. Note also that if the set of worlds where each atomic formula is true is measurable, then by the conditions on a σ -field, the set of worlds where each propositional compound is true is also measurable.) The expectation of a formula is not affected by its truth value anywhere outside our measure 1 set T1 ∩ T2 (= G ∪ G1 ∪ G2 ). So E(γ1 ) = P(G1 ), E(γ2 ) = P(G2 ), and E(γ1 ∨ γ2 ) = P(G1 ) + P(G2 ) (because γ1 ∨ γ2 is true at worlds in G1 and G2 and false at worlds in G). Hence E(γ1 ∨ γ2 ) = E(γ1 ) + E(γ2 ). ¹⁶ Again, the closure requirement is no restriction, because if one is in a UFS with respect to a class of wfs, then one is in a UFS with respect to the closure of that class (because if one is certain that S is m true and that T is n true, then one is certain that S ∨ T is max{m, n} true, that S ∧ T is min{m, n} true, and that ¬S is 1 − m true). ¹⁷ Proof. There is a measure 1 set T such that at every world in T , γ is k true. So E(γ ) = k. At every world in T , ¬γ is 1 − k true. So E(¬γ ) = 1 − k = 1 − E(γ ). ¹⁸ Proofs. By hypothesis we have a set T1 of worlds such that P( T1 ) = 1 and γ1 (w) = m for every w ∈ T1 , and a set T2 of worlds such that P( T2 ) = 1 and γ2 (w) = n for every w ∈ T2 . So P( T1 ∩ T2 ) = 1, as in n.15. At every world in T1 ∩ T2 , γ1 is m true and γ2 is n true, hence γ1 ∨ γ2 is max{m, n} true and γ1 ∧ γ2 is min{m, n} true. The expectation of a formula is not affected by its
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Summing up my proposal: an agent’s degrees of belief are the resultant of two things: the agent’s uncertainty about which way the actual world is (represented by a probability measure over the space of possible worlds, with the measure assigned to a set of worlds specifying how likely the agent thinks it is that the actual world is in that set), and the facts about how true each proposition is in each world. Specifically, the agent’s degree of belief in a proposition is the agent’s expected value of its degree of truth: roughly, the average of its truth in all the worlds the agent has not ruled out, weighted according to how likely the agent thinks it is that each of those worlds is the actual one. In some situations, the agent will have ruled out vagueness: she may not know which world is actual, but she is certain that in the actual world, some propositions of interest are either fully true or fully false. In such situations, her degrees of belief will behave like probabilities (propositions 1 and 3). In other situations, the agent will be free of uncertainty with respect to some propositions of interest: she is certain of exactly how true they are in the actual world. In such situations, her degrees of belief will behave like degrees of truth (propositions 2 and 4). In situations which are neither vagueness-free nor uncertainty-free—that is, where the agent is unsure of the truth values of some propositions of interest, and cannot rule out vagueness, that is, cannot rule out that they might have intermediate degrees of truth—her degrees of belief in those propositions need not behave like probabilities or degrees of truth. (In situations which are both uncertainty-free and vagueness-free—that is, the agent knows of each of the propositions in question that it is 1 true, or that it is 0 true—degrees of belief behave both like probabilities and like degrees of truth. This is possible because the behaviours of probabilities and degrees of truth coincide in this special case.) In all cases, I maintain that an agent’s expectation of a proposition S’s degree of truth is an accurate measure of her tendency to act as if S, and this is why I identify degrees of belief with expectations. My proposal contrasts with the standard view, as expressed for example in the following passages: Let our degrees of belief be represented by a probability measure, P, on a standard Borel space (, F , P), where is a set, F is a sigma-field of measurable subsets of , and P is a probability measure on F . (Skyrms, 1984, 53) [By a reasonable initial credence function C ] I meant, in part, that C was to be a probability distribution over (at least) the space whose points are possible worlds and whose regions (sets of worlds) are propositions. C is a non-negative, normalized, finitely additive measure defined on all propositions. (Lewis, 1986, 87–8)
The crucial difference between the standard view and mine is that the former equates an agent’s degrees of belief directly with her subjective probabilities. My view, on the other hand, countenances the subjective probability measure—it models the agent’s epistemic state—but regards degrees of belief as resultants of this state and the truth truth value anywhere outside our measure 1 set T1 ∩ T2 , so E(γ1 ) = m, E(γ2 ) = n, E(γ1 ∨ γ2 ) = max{m, n} = max{E(γ1 ), E(γ2 )}, and E(γ1 ∧ γ2 ) = min{m, n} = min{E(γ1 ), E(γ2 )}.
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values of propositions at worlds. In the sort of cases Skyrms and Lewis were considering, in which bivalence was assumed, this difference makes no difference (propositions 1 and 3). However, if we want to add vagueness to the mix, then we will run into all sorts of problems, if we have already identified degrees of belief with subjective probabilities—for, as we saw at the outset, vagueness also gives rise to degrees of belief, but these degrees of belief do not behave like probabilities. On the other hand, if we identify degree of belief with expected truth value even in the bivalent case, then we can generalize smoothly to the case of vagueness, handled using degrees of truth.
28.4
OBJECTIONS AND REPLIES
(1) If your degrees of belief do not conform to the probability calculus, then you are subject to Dutch book, i.e. you are irrational. Reply: One should not bet at all on a proposition S unless one is in a vagueness-free situation with respect to S; if one does bet in a non-VFS, then it is for that reason alone that one is irrational. Suppose you are not in a VFS with respect to S. Suppose first that you know that S is k true, for some k ∈ (0, 1); say k = 0.5 for the sake of argument. Then you should not bet on S. For to bet is to agree to an arrangement whereby you get such-and-such if S turns out to be the case. But you already know what is the case—and you know that it is, in the nature of things, indeterminate whether S —hence indeterminate whether you get your payoff. Knowing all this, you should not bet in the first place. Second, suppose that you do not know whether S is true—and you cannot rule out that S has an intermediate degree of truth. In this case again you should not bet, because for all you know, the bet will not—for the sort of reason just seen—be able to be decided. Of course if there is in place some system for deciding bets on S when S has an intermediate degree of truth—say an umpire who rules one way or the other, or a rule that S will be deemed 1 true if it is more than 0.5 true—then one may enter into a betting arrangement on S. However in such a case the situation has, in effect, been turned into a VFS, by changing S’s intermediate degrees of truth in some non-ruledout worlds into 1’s or 0’s.¹⁹ ¹⁹ My comments about not betting in non-VFSs are concerned with standard bets—i.e. bets which do not specify what is to happen (who gets what) when the proposition in question is neither true nor false. Milne (2007) discusses a new type of betting arrangement, tailor-made for vagueness, on which one could legitimately bet in a non-VFS. The basic idea (although this is not the way Milne expresses it) is that if one bets on S, and S is n true, then one receives n times the stake. Of course this complements rather than conflicts with my comments above (Milne was not suggesting otherwise). I say that one should not accept an ordinary bet if one thinks that vagueness may be present—for when vagueness is involved, there is no way of deciding such a bet. This does not mean that one should not accept a new kind of bet—one designed precisely to avoid the problem faced by ordinary bets when vagueness is present, by explicitly building in a decision procedure which works even when the proposition on which one is betting has an intermediate degree of truth. (When I was writing the paper on which Milne (2007) was a comment, I considered the idea of introducing a type of bet along the lines discussed by Milne, designed specifically to handle vague outcomes. I did not pursue the idea, however, because in general we do not know the precise degrees of truth of vague propositions, so even if it is fixed that when S is n true, one receives n times the stake, still we will in general have no way of actually deciding and paying out the bet, because we will not know
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(2) Some writers have claimed that ‘The cunning bettor is simply a dramatic device—the Dutch book a striking corollary—to emphasize the underlying issue of coherence’ (Skyrms, 1984, 22). The idea is meant to be that one is internally incoherent if one’s degrees of belief do not conform to the probability calculus: the Dutch book idea simply serves to bring this incoherence into the open in a striking way; but even if one is not subject to Dutch book for some reason (e.g. because betting has been made illegal and this law is enforced absolutely) one is still internally incoherent. Reply: Why is one supposed to be incoherent in such a case? Well, here’s a way of bringing it out. Suppose I think A is 50% likely to occur (in 50% of futures compatible with the present, A occurs); I think B is 50% likely to occur (in 50% of futures compatible with the present, B occurs); I think A and B are incompatible (in no future do A and B both occur); and yet I think ‘A or B’ is not 100% likely to occur—i.e. I think that in (say) 50%, rather than 100%, of futures compatible with the present, ‘A or B’ will be true. When framed in this way in terms of sizes of sets of possible futures, this combination of beliefs is obviously incoherent. But my view endorses this assessment: in the situation envisaged, the agent is in a VFS (she does not know whether or not A or B will occur, but she assumes neither of them will sort-of occur), and so will not have these degrees of belief, on my view. On the other hand I do not think that, in itself, the following combination of degrees of belief is incoherent, even supposing the agent knows that A and B cannot both be fully true: A : 0.5,
B : 0.5,
A or B : 0.5
It all depends on how these degrees of belief arise. If you are in a VFS and have these degrees of belief, then you are indeed incoherent—as can be brought out either by Dutch book reasoning, or by reflections on sizes of sets of possibilities. But degrees of belief might arise in other ways—not just as a result of uncertainty; and when they do, this sort of combination can be perfectly reasonable. For example, suppose that A is the proposition that a certain leaf is red, and B is the proposition that it is orange; then A and B cannot both be fully true. Suppose also that the leaf in question is right in the middle of a Sorites series leading from red things to orange things. Then, I submit, the above combination of degrees of belief is perfectly reasonable: intuitively it is just fine, and neither the Dutch book nor the ‘sizes of sets of possibilities’ rationales can get a grip to show that there is something wrong with it. Dutch book reasoning does not get started because I will not bet (there is nothing to bet on—no outcome to wait and see about: I already have all the information about the leaf ’s colour before me). Similarly, the ‘sizes of sets of possibilities’ reasoning does not get started, because there is nothing I am uncertain about. (3) I claim that in non-VFSs, we have degrees of belief while not being prepared to bet (at all). The objection is that we cannot make sense of the idea of degrees of belief except in terms of fair betting quotients or odds. Reply: We make sense of the idea of degree of belief in S in terms of strength of tendency to act as if S, and ‘acting the actual value of n in question. However Peter pointed out to me that we are often in a similar position with regards to ordinary bets—i.e we cannot determine the outcome—but this does not reduce their theoretical interest in relation to degrees of belief.)
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as if S’ can be made sense of more generally than in terms of ‘betting on S’. After all, betting is essentially tied up with uncertainty—betting gets its life from the fact that we do not know what the outcome will be—but, I have argued, the idea of degree of belief gets a grip in circumstances in which there is no uncertainty at all. Consider an autumn leaf which is borderline red–orange. You have some tendency to act as if it is red, as discussed below (objection 5). But with the leaf in plain sight, you would not accept a bet that it is red, at any price: for we can all see quite plainly that the leaf is neither clearly red nor clearly non-red, and so we can see at the outset that the bet will misfire.²⁰ (4) Suppose my degree of belief in S is 0.5 because I am uncertain whether S is 1 true or 0 true. Then I might buy a bet on S, if the price and prize are right. But suppose my degree of belief in S is 0.5 because I am certain that S is 0.5 true. Then, for the reasons discussed above, I will not buy a bet on S, no matter what the price or prize. So the same state—a degree of belief of 0.5 in S —leads to different actions. How can this be, if my degree of belief measures my tendency to act as if S? Reply: These different actions are the results not of a single belief, but of complexes of beliefs, which are different in the two situations. A 0.5-degree belief that S combined with the belief that whatever further evidence comes in, I will not alter my degree of belief in S, leads to refusing to bet; a 0.5-degree belief that S combined with the belief that further evidence might come in leading me to believe to degree 1 that S, and that further evidence might come in leading me to believe to degree 0 that S, leads to accepting certain bets. (5) One’s expectation that S is not an accurate measure of one’s tendency to behave as if S. Suppose I know that a certain orangey-red autumn leaf is red to degree 0.5. Suppose also that I need a perfectly red leaf. Then I will have no tendency whatsoever to reach for this leaf, even though my expectation that it is red is 0.5. Reply: The key here is the presence of the word ‘perfectly’. Of course if I need a perfectly red leaf, then I will have no tendency whatsoever to reach for the orangey-red one. But this is quite compatible with the foregoing account, because my expectation that the leaf is perfectly red, i.e. red to degree 1, is 0. On the other hand, my expectation that it is red is 0.5; and if I need a red leaf, then I think I would have some tendency to reach for this one: less than for a perfectly red leaf, but more than for a green one.²¹ Re f e re n c e s Field, Hartry (2000), ‘Indeterminacy, degree of belief, and excluded middle’, Noˆus 34, 1–30. Lewis, David (1986), ‘A subjectivist’s guide to objective chance’ in Philosophical Papers, vol. II, 83–132, Oxford University Press, New York. MacFarlane, John (2006), ‘The things we (sorta kinda) believe’, Philosophy and Phenomenological Research 73, 218–24. ²⁰ Those who feel strongly that where there are degrees of belief there must be betting quotients can find comfort in the kind of betting arrangement discussed in Milne (2007) (see n. 19 above). Milne shows that the fair betting quotient a rational agent assigns to a bet on A of his kind perfectly matches the agent’s degree of belief that A in my sense, i.e. her expectation of A’s degree of truth. ²¹ Further objections to my view are considered in Smith (2008, §5.3.3).
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Milne, Peter (2007), ‘Bets and fuzzy propositions: Comments on Nicholas J. J. Smith’s ‘‘Degrees of truth, degrees of belief, and pragmatics’’,’ presented at the Arch´e Vagueness Conference, St Andrews, 8 June. Ramsey, F. P. (1990), ‘Truth and probability’ in D. H. Mellor, ed., Philosophical Papers, 52–94, Cambridge University Press, Cambridge. Schiffer, Stephen (2000), ‘Vagueness and partial belief,’ Philosophical Issues 10, 220–57. Skyrms, Brian (1984), Pragmatics and Empiricism, Yale University Press, New Haven. Smith, Nicholas J.J. (2008), Vagueness and Degrees of Truth, Oxford University Press, Oxford.
VIII Higher-Order Vagueness
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29 Demoting Higher-Order Vagueness Diana Raffman
Higher-order vagueness is widely thought to be a feature of vague predicates that any adequate theory of vagueness must accommodate. It takes a variety of forms. Perhaps the most familiar is the supposed existence, or at least possibility, of higherorder borderline cases—borderline borderline cases, borderline borderline borderline cases, and so forth. A second form of higher-order vagueness, what I will call ‘prescriptive’ higher-order vagueness, is thought to characterize complex predicates constructed from vague predicates by attaching operators having to do with the predicates’ proper application. For example, the predicates ‘mandates application of ‘‘old’’ ’ and ‘can competently be called ‘‘old’’ ’ are prescriptively higher-order vague. Higherorder vagueness appears in other guises as well,¹ but these two have been of particular interest to philosophers and will be my target here. I want to expose some misconceptions about them. If I am right, higher-order vagueness is less prevalent, and less important theoretically, than is usually supposed.² In what follows I am going to assume that vagueness is a semantic feature of natural language. For the most part I won’t discuss epistemic or pragmatic views, and I will say nothing about so-called metaphysical vagueness.
29.1
HIGHER-ORDER BORDERLINE CASES
That vague predicates have or could have higher-order borderline cases is largely taken for granted by theorists of vagueness. On the standard view, first-order borderline cases for a vague predicate ‘’ are neither-definitely--nor-definitely-not.³ Second-order borderlines (or anyway one set of second-order borderlines) are ¹ See e.g. Wright, this volume. A third form of higher-order vagueness is the generic (i.e. not necessarily prescriptive) vagueness of the metalanguage in which a theory of vagueness is formulated, where that metalanguage is a natural language like English. For example, perhaps vagueness is defined in terms of a certain kind of context-relativity. The word ‘context’ is probably vague. I take for granted that this kind of higher-order vagueness exists. ² For some of the important work that has been done on higher-order vagueness, see Deas 1989, Fara 2003, Heck 1993, Hyde 1994 and 2003, Sorensen 1995, Tye 1994, Varzi 2003, and Wright 1992, 1994, and this volume. ³ The hyphenation is only to avoid scope ambiguities.
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then neither-definitely-definitely--nor-definitely-not-definitely-; third-order borderlines are neither-definitely-definitely-definitely--nor-definitely-not-definitelydefinitely-; and so on. This hierarchy of ever higher orders of borderline cases is often said to continue ad infinitum, thereby constituting, or at least providing for, the blurred boundaries of the predicate ‘’. There are problems, however. For one, sharp cut-offs reappear in the end. Mark Sainsbury explains: [S]uppose we have a finished account of a [vague] predicate, associating it with some possibly infinite number of boundaries, and some possibly infinite number of sets. Given the aims of the description, we must be able to organize the sets in the following threefold way: one of them is the set supposedly corresponding to the things of which the predicate is absolutely definitely and unimpugnably true, the things to which the predicate’s application is untainted by the shadow of vagueness; one of them is the set supposedly corresponding to the things of which the predicate is absolutely definitely and unimpugnably false, the things to which the predicate’s non-application is untainted by the shadow of vagueness; the union of the remaining sets would supposedly correspond to one or another kind of borderline case. So the old problem re-emerges: no sharp cut-off to the shadow of vagueness is marked in our linguistic practice, so to attribute it to the predicate is to misdescribe it. (1988, 255)
Sainsbury’s reasoning seems to me decisive; and anyway there are simpler and more plausible ways to understand the blurred boundaries of a vague predicate (e.g. in terms of tolerance or soriticality). In addition, if an infinite hierarchy of borderline cases were required for blurred boundaries, then there would be sharp cut-offs in a sorites series.⁴ That can’t be right. My present aim is to articulate some further, mostly intuitive worries about higherorder borderline cases as standardly conceived. I will do this by setting out a series of informal questions and criticisms—I’ll call them ‘ruminations’—that help to reveal just how problematic the notion is. Rumination #1. Consider the set containing all possible borderline cases of any order for vague predicate ‘’, as in Sainsbury’s ‘finished account’. Why aren’t all of these items just first-order borderline cases? Don’t they all fall within a gap between the extensions of ‘’ and ‘not-’? Alternatively, why aren’t these items just more (first-order) borderlines, definitely items, and definitely not- items? In fact I think we have no grasp at all on the idea of an item that doesn’t fit into any of these three categories. Rumination #2. If a hierarchy of borderline cases doesn’t make for blurred boundaries, why else believe in them? If there can be borderline cases between and not-, the thinking goes, then surely there can be (second-order) borderline cases between and borderline ; and then surely there can be (third-order) borderline cases between and the second-order borderlines; and so on.⁵ This line of reasoning sounds plausible, but it overlooks a crucial possibility: viz., that there can be borderline cases ⁴ See my 2005, note 18; also Fara 2003. ⁵ I will underline when it is convenient to refer to the category (type, kind, property, class) named by a vague predicate, rather than to the predicate itself.
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between ‘’ and ‘not-’ only insofar as and not- are not themselves borderline categories. It may be that only non-borderline categories can have borderline cases. Notice that borderline cases are defined negatively, in terms of an absence or lack—specifically, a lack of category membership. Borderlines are possible only insofar as the (definite) extensions of vague ‘’ and ‘not-’ are not together exhaustive over the range of values in a relevant sorites series. This is why we say that borderline cases ‘fall within the gap’ between the extensions of ‘’ and ‘not-’. To put the point another way, there is nothing more to being borderline than failing to (definitely) belong either in the category or in the category not-. Consider how we classify items as borderline: presumably we measure them against, or judge their ‘distance’ from, the definite cases of and not- at the endpoints of a sorites series. But if there were definite borderline cases, surely we would classify items as borderline by judging their distance from those. (Indeed I think it is misleading to speak of being borderline as a ‘category’. Better to call it, say, a ‘status’.) My thought then is that ‘borderline ’ may not be the right sort of predicate to have borderline cases of application; it is not sufficiently centered or anchored, one might say. Thus when we talk of definite and borderline borderline cases, we are no longer treating the items in question as defined negatively, as falling within a gap. We are in effect transforming the (first-order) borderline cases into a new, non-borderline category with its own center of gravity—a full-fledged incompatible of and not-. As evidence of this transformation, consider where the putative second-order borderlines are supposed to be located in a sorites series—say, a series of heights progressing from a definitely tall height (e.g. 6 5 ) to a definitely average height (e.g. 5 9 ) compared to British men. And suppose that 5 10 1/2 is a borderline case (B1). (See Figure 29.1.) Then the second-order borderline cases would be located as in Figure 29.2. However, I predict that if a competent speaker were asked to proceed along the original tall/average series from the definitely tall height 6 5 to the definitely borderline height 5 10 1/2 , and to classify each height as definitely tall, definitely borderline, or borderline borderline (B2), she would locate any borderline borderline cases not as in Figure 29.2, but roughly as in Figure 29.3. Among other things, she would now classify as borderline borderline some heights that she previously classified as tall. The span of the first-order borderlines would spread out, as it were, pushing everything toward the tall end. Figure 29.3 still isn’t right, however; for in classifying the heights in this new, shorter sorites series, the speaker would not in fact be classifying them as tall, first-order borderline, and second-order borderline. Instead, she would be classifying them as tall, firstorder borderline, and, say, above average, as in Figure 29.4. She would be transforming what had been first-order borderline cases into a new height category, above average, with its own, new, first-order borderline cases that are neither-definitely-tall-nordefinitely-not-tall (above average). Unsurprisingly, some heights that were tall when ‘tall’ was opposed to ‘average’ are borderline or even not-tall when ‘tall’ is opposed to ‘above average’. (One might say that the standard for being tall as opposed to above average is higher than the standard for being tall as opposed to average.) I have not done a study to confirm this prediction, but some support comes from related observations. Sainsbury notes that ‘subjects asked to classify a range of test
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Figure 29.1
Figure 29.2
Figure 29.3
Figure 29.4
objects using just ‘‘young’’ and ‘‘old’’ make different assignments to these words from those they make to them when asked to classify using, in addition, ‘‘middle-aged’’ ’ (1997, 259). C. L. Hardin makes a similar claim about hue predicates: [T]he boundary of red in the broadest sense extends to the immediate neighborhood of unique yellow, and the breadth of that spread we acknowledge by our use of the modifier ‘reddish’. But, in a somewhat narrower sense, the boundary between red and yellow falls at the point at which the perceptual ‘pull’ of yellow is equal to that of red. This point is, of course, orange. But once we introduce orange as a distinct hue category, its boundary with red is at issue, and the extension of ‘red’ must be contracted to make room for the oranges. The natural red-orange boundary would seem to fall at the 75 per cent red, 25 per cent yellow region which was well within the scope we took ‘red’ to have when we were concerned to compare red with yellow. (1988, 184)
I expect that an analogous contraction of the extension of ‘’ (e.g. ‘tall’) would occur if a speaker attempted to locate second-order borderline cases in a sorites series. Rumination #3. Borderline cases are supposed to be of indefinite or indeterminate or uncertain status with respect to being . So borderline cases have a status other than being . Therefore definite borderline cases definitely have a status other than being . But intuitively, how could it be indefinite whether an item that is definitely other than is ? Allowing that ‘x is not ’ is true on a weak reading of the negation
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is not an adequate response, in my view. The intuitive question is: how could it be indefinite whether a definitely borderline item is , rather than just plain false? Rumination #4. The impossibility of higher-order borderline cases seems to follow from two intuitively plausible claims about vague predicates. For all vague predicates ‘’ and ‘ ’: (i) If an item is definitely , then failure to classify it as is mistaken or in some way improper or at least legitimately questionable.⁶ (ii) Failure to classify an item as borderline cannot be mistaken or in any way improper or even legitimately questionable. (Intuitively, one is never required to classify something as borderline; a judgment of ‘borderline’ is always optional.) If (i) and (ii) are true, then (iii) follows straightforwardly: (iii) Therefore no item can be definitely borderline . Given (iii), (iv) appears to follow (or so I will contend): (iv) Therefore no item can be borderline borderline . Call this the ‘Simple Argument’. It is so simple that it may seem to involve some sleight of hand; so I want to spell out the justification for each step. First, though, I want to acknowledge that one can of course define ‘definitely’, as a technical term, however one wants. But technical control risks estrangement from the ordinary meaning and application of vague words. The Simple Argument, and my ruminations in general, proceed on the assumption that the meaning of the definiteness operator in a theory of vagueness is grounded in the meaning of ‘definitely’ as used by ordinary speakers when they apply vague predicates. On this assumption, the behavior of the definiteness operator is in some measure constrained by ordinary linguistic intuition. Understanding ‘definitely’ in this way, let us consider how the premises and reasoning in the Simple Argument can be justified. Premise (i) makes an extremely weak claim about the character of definitely items. If, contrary to (i), definitely items can also permissibly be classified (e.g.) as not- or as borderline , then it is hard to see what definiteness comes to. Perhaps items that can competently be classified as borderline can also competently be classified as and as not-. But the analogous claim is not plausible for ‘’ and ‘not-’: it is not the case that any item that can competently be classified as (not-) can equally competently be classified as borderline or as not- (). Definitely (not-) items appear to carry some sort of requirement that they be so classified, thus making failure to do so mistaken or at least questionable. Premise (ii), it seems to me, can be found in ordinary linguistic intuition (but for experimental evidence that competent speakers proceeding along a sorites series do not always—indeed often do not—employ the category borderline even when it is ⁶ Here of course I refer to a hypothetical competent, sincere, cooperative speaker who fails to apply ‘’ upon being queried. Feel free to add whatever further specifications you think necessary for present purposes.
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explicitly made available, see Lindsey et al. 2009 [in progress].)⁷ Of course (ii) goes hand in hand with the thought that any item that can competently be classified as borderline can also competently be classified as and as not-. If (ii) is accepted, premise (iii) then follows. Premise (iv) is secured from (iii) not merely because ‘definitely ’ and ‘borderline ’ are interdefinable, but because, as I argued above, ‘borderline ’ is defined by ‘definitely ’ in a wholly negative fashion: there is nothing more to being borderline than failing to be either definitely or definitely not-. Hence if definite borderline cases are impossible, so are borderline borderline cases. Stewart Shapiro and Elia Zardini have pointed out (in conversation) that if definite borderline cases are impossible, then it seems to follow, absurdly, that all first-order borderlines are second-order borderlines. For if nothing can be definitely borderline, then, trivially, first-order borderlines are not definitely borderline. But first-order borderlines are also not-definitely-not-borderline. Therefore first-order borderline cases are not-definitely-borderline and not-definitely-not-borderline, which is just the definition of a second-order borderline case. The trouble with this clever objection is that if definite borderline cases are impossible, then second-order borderline cases are also not (first-order) borderline. Consider: first-order borderline cases come between the definitely items and the definitely not- items, being neither one nor the other (∼Defx & ∼Def∼ x). Secondorder borderlines are then supposed to come between the definitely-definitely- items and the definitely first-order borderlines, being neither one nor the other (∼Def Defx & ∼Def∼Defx). It would seem to follow, then, that if definite first-order borderlines are impossible—if there are only plain old regular first-order borderlines—any second-order borderlines must instead come between the definitely-definitely- items and the plain old regular first-order borderlines. ( There is nothing else for them to come between.) In other words, they must be neither-definitely-definitely--nor-borderline- (∼Def Defx & ∼(∼Defx & ⁷ An anonymous reviewer writes: ‘if I am [instructed] to classify various colours on the red-yellow continuum as either red, yellow or borderline then, if I failed to classify an orange patch as borderline, wouldn’t I be mistaken?’ I don’t see why. If the patch is orange, how could you be mistaken in failing to call it ‘borderline’? Unless ‘borderline’ just means ‘orange’ (in which case we are not speaking English), the correct response to the instruction is to say that it cannot be carried out because it fails to provide adequate response categories. By the same token, if you were instructed to classify colors on the ‘red–green continuum’ as either red, green, or borderline, you could hardly be convicted of error if you failed to classify a yellow patch as borderline. Such an instruction would be illegitimate (incompetent, if you like). Rejecting an instruction is not the same as making a mistake. Here’s another way to put the point. If you were required to call the orange patch ‘borderline’, then you would be using ‘borderline’ to name what would in fact be a non-borderline category—a category in its own right on a par with red and yellow, namely orange. In other words, you would be using ‘borderline’ to mean ‘orange’. A second anonymous reviewer writes that (ii) in the Simple Argument needs to be restricted to avoid counterexamples based on an abuse of privilege. ‘If I persistently characterize what you regard as borderline cases as clear cases, then I have at least abused a right. Think of prospective employees puffing their credentials.’ I don’t see why such cases would have to be classified as borderline. Couldn’t they fairly be classified as (definitely) not ?
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Neither Def Def TALL nor BORDERLINE nor Def BORDERLINE ~Def Def Tx & ~(~Def Tx & ~Def~Tx) & ~Def(~Def Tx & ~Def~Tx)
B2 Def Def TALL
B2 BORDERLINE (B1) ~Def Tx & ~Def~Tx
Def Def NOT-TALL
~DefBx
Figure 29.5
∼Def∼ x)). Then, finally, since every item in the series is not-definitely-borderline, the second-order borderlines must be neither-definitely-definitely--nor-borderline-nor-definitely-borderline- (∼Def Defx & ∼(∼Defx & ∼Def∼ x) & ∼Def(∼Defx & ∼Def∼ x)). Figure 29.5 provides an illustration, using ‘tall’. The trouble of course is that anything that is not-borderline is either definitely tall or definitely not-tall, which is incompatible with its being a borderline case of any order. Contrary to the objection, first-order borderlines are not second-order borderlines. Nothing can be second-order borderline. Perhaps we have thought that vague predicates could have higher-order borderline cases because we allowed technique to lead intuition: we were enchanted by the formal permissibility of generating certain expressions with the definiteness operator. What is possible, even coherent, in natural language may be a different matter. My own view (2005), which I will not elaborate here, is that borderline cases are not properly defined using a definiteness operator.⁸ In order to connect with the philosophical literature I have been going along with the standard definition; but in what follows I will no longer use that device. (What I want to say will not commit us to any particular analysis of borderline cases.)
29.2
P R E S C R I P T I V E H I G H E R - O R D E R VAG U E N E S S
Prescriptive higher-order vagueness appears to be a feature of certain metalinguistic predicates, such as ‘mandates application of ‘‘’’ ’ and ‘can competently be called ‘‘’’ ’, that have to do with the proper application of a vague word. I think the vagueness of these predicates has been misunderstood. (Actually, their being metalinguistic is probably inessential to the view I’ll sketch below; my argument may apply equally to ‘mandates being classified as ’ and ‘can competently be judged ’, for example. I ⁸ I propose an analysis of ‘borderline case’ that is bivalent, does not employ a definiteness operator, and eliminates the possibility of higher-order borderlines. On my view, borderlines are properly defined in terms of contrary or incompatible predicates, such as ‘old’ and ‘young’, or ‘old’ and ‘middle-aged’, rather than contradictory ones like ‘old’ and ‘not-old’.
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do not know how to distinguish the relevant family of terms in a principled way, but that should not cause trouble for us here.) To begin, consider the vague predicate ‘old’. Presumably a hundred-year-old person mandates application of ‘old’ (for a person): failing to call him ‘old’ would be linguistically unacceptable, incompetent, a mistake. In contrast, a 63-year-old person can be classified as old, but can also be classified as borderline old or as (e.g.) middle-aged. Different competent speakers, and each competent speaker on different occasions, may classify the 63-year-old differently. Similarly, a 50-year-old person mandates classification as old for a ballet dancer, whereas a 35-year-old could be called ‘old’ or ‘borderline old’ or ‘middle-aged’ for a dancer. I will say that cases like the 63-year-old and the 35year-old permit variable classification relative to the specified comparison class:⁹ we are free to apply the predicate ‘old’ and also free to withhold it. By the same token, variability of application in the neighborhood of its blurred boundaries is characteristic of—indeed, I would argue, essential to—competent use of a vague predicate. In a sorites series this variability is reflected in a multiplicity of permissible stopping places. In a sorites series of ages proceeding from 100 years to one year by increments of one year, on a given occasion you might stop applying ‘old’ (for a person) at seventy whereas I stop at sixty-five. And you might stop at 69 the next time. There is no question of error, because our particular stopping places are arbitrary: in every case we could as easily, as competently, have stopped elsewhere.¹⁰ To put the point another way, there is no reason, in the nature of the case, to shift at any particular place. If the predicate ‘mandates ‘‘old’’ ’ is vague, it too should permit arbitrarily variable application. But see what happens when ‘mandates ‘‘old’’ ’ is applied to the series of ages. Suppose that on a given occasion you stop applying ‘mandates ‘‘old’’ ’ at (after) 70. You stop there arbitrarily; indeed let us suppose that, as must typically be the case, you are well aware that your stopping place is arbitrary. To suppose that your stopping place is arbitrary is to suppose that ‘mandates ‘‘old’’ ’ could also permissibly be withheld from 70. But as a moment’s reflection reveals, if it is permissible to withhold ‘mandates ‘‘old’’ ’ from 70, it is permissible to withhold ‘old’ from 70. Hence you find yourself in the incoherent position of saying that 70 mandates application of ‘old’ while also granting that ‘old’ can permissibly be withheld from 70. It is one thing to judge that 70 is old while granting that it’s permissible to withhold ‘old’ from 70, and quite another thing to judge that 70 mandates ‘old’ while granting that it’s permissible to withhold ‘old’ from 70. The relevant difference is that the latter case involves a legislative judgment, viz., the judgment that 70 must be called ‘old’ on pain of incompetence. And the trouble is that the legislative force of ‘mandates’ and the permissible variability of the application of ‘old’ pull in opposite directions. The predicate ‘mandates application of ‘‘old’’ ’, which brings the two together, appears internally conflicted. ⁹ Make the comparison class or context as fine-grained as you like; the variability will persist. I elaborate, and provide some experimental evidence, in my 2009. ¹⁰ Consider that if our stopping places are not arbitrary, if reasons or argument can be given for stopping at one place rather than another, then the increments in the series are not small enough for a sorites series.
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An analogous difficulty seems to arise for the prescriptive metalinguistic predicate ‘can competently be called ‘‘old’’ ’. Suppose I stop applying ‘can competently be called ‘‘old’’ ’ (for a person) at (after) 50 years. I know that I stop there arbitrarily, which is to say that I grant the permissibility of withholding the predicate ‘can competently be called ‘‘old’’ ’ from 50. But now consider the speaker who chooses to withhold the latter predicate. She must grant the permissibility of applying it. But as a moment’s reflection reveals, if it is permissible to apply ‘can competently be called ‘‘old’’ ’ to 50, then 50 can competently be called ‘old’. Thus the speaker ends up in the incoherent position of supposing that 50 can competently be called ‘old’ while herself withholding the predicate ‘can competently be called ‘‘old’’ ’ from 50. It is one thing to withhold ‘old’ from 50 while granting that 50 can competently be called ‘old’, and quite another thing to withhold ‘can competently be called ‘‘old’’ ’ from 50 while granting that 50 can competently be called ‘old’. The legislative character of ‘can competently be called ‘‘old’’ ’ is what underwrites this distinction. The preceding discussion suggests that there can be no arbitrary permissible stopping places in a sorites series for ‘mandates ‘‘old’’ ’ or ‘can competently be called ‘‘old’’ ’. ( This seems to me independently plausible. How could what is mandatory or competent in the English language vary, arbitrarily, from speaker to speaker and time to time?) At the same time, there can be no permissible nonarbitrary stopping places: in other words, there can be no sharp boundary between the ages that mandate application of ‘old’ and the ages that don’t, or between the ages that can competently be called ‘old’ and the ages that cannot. It seems to follow, then, that there can be no permissible stopping places at all in sorites series for these legislative predicates. This is a baffling result, to say the least. How should we respond to it? The solution, I think, is to recognize that although the surface grammar of ‘mandates ‘‘old’’ ’ (simile ‘can competently be called ‘‘old’’ ’) has it applying to chronological ages, the predicate also, implicitly, makes reference to the verbal behavior of users of its embedded vague term ‘old’. In order to judge whether a given age mandates ‘old’, you need to know not only its number of years, but also how other speakers would classify it. A crude initial proposal might be this: a given age n mandates application of ‘old’ just in case, on average, almost all competent English speakers would apply ‘old’ to n.¹¹ (By ‘competent English speakers’ I mean only that they are generally competent at speaking English; the question whether they are competent specifically in the use of ‘old’ remains open for the moment.) What this means—here is the point that has been missed—is that a sorites series for ‘old’ is not a sorites series for ‘mandates ‘‘old’’ ’. Instead, a sorites series for ‘mandates ‘‘old’’ ’ may be a series of pairs, each containing a chronological age together with an average percentage of ¹¹ This crude proposal is doubtless incorrect, but it has the virtue of simplicity. Perhaps, for example, it would be better to say that a given age n mandates application of ‘old’ just in case, on average, almost all competent English speakers would respond to a failure to apply ‘old’ to n by, say, expressing bafflement, or correcting the speaker in question, or initiating an argument with the speaker in question.
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Average percentage of competent speakers who would apply ‘old’
age in years
99
97.
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50
40
30
10
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1
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Figure 29.6 Sorites series for ‘mandates application of ‘‘old’’ ’.
competent English speakers who would apply ‘old’ to that age.¹² Figure 29.6 illustrates such a series beginning with an age of 100 years together with an average of 99 percent of competent speakers applying ‘old’, progressing to an age of 1 year together with an average of 0.5 percent of speakers applying ‘old’. On a given occasion, if you proceeded along this series, you might stop applying ‘mandates ‘‘old’’ ’ at 97 percent, while I might stop at 95 percent. And you might stop at 90 percent the next time around. (Stopping at 97 percent would commit you to supposing that, on average, 3 percent of competent English speakers would use the word ‘old’ incompetently at any given time.) When ‘mandates ‘‘old’’ ’ is applied to the right kind of sorites series, we can see that, qua vague predicate, it behaves in the same manner as the lexical predicate ‘old’. Its competent application is arbitrarily variable. If you stop applying ‘mandates ‘‘old’’ ’ at 97 percent, you do so, and know that you do so, arbitrarily: you could as easily have stopped at 97.5 percent for example. No incoherence results, because you are no longer making a legislative judgment; you are making a merely descriptive judgment as to whether application of ‘old’ by 97 percent of competent English speakers is sufficient to make it the case that the corresponding age—whatever it may be—mandates application of ‘old’. As far as I can see, there is nothing incoherent about judging that 97 percent makes a given age mandate ‘old’, while also granting the permissibility of withholding that judgment. You and I may permissibly vary in our judgments as to whether a given percentage makes application of a predicate mandatory. Perhaps it seems that we can say ‘categorically’, without considering anyone’s verbal behavior, that 100 years mandates application of ‘old’. (Maybe the ages down through about 70 seem this way.) Intuitively: 100 years is old no matter what anyone else says. Similarly, it may see that we can say, without knowing anything about anyone’s verbal behavior, that a pure blue patch mandates application of ‘blue’. It may seem that the predicate ‘mandates ‘‘blue’’ ’ can competently be applied to such a patch upon inspection alone, just by looking. Judgments in very central cases do not seem to rest upon consideration of anyone’s verbal behavior. However, where an item is not a highly central case, we can see that a judgment as to whether it mandates application of a certain predicate, or whether a certain predicate can competently be applied to it, may be impossible apart from some knowledge of what other competent speakers would say. We can apply ‘blue’ just by looking, but not ‘mandates ‘‘blue’’ ’; we can apply ‘old’ just by considering the number of years, but not ‘mandates ‘‘old’’ ’. ¹² Presumably, as a matter of empirical fact, the percentage of competent English speakers who would apply ‘old’ to a given age (when queried, etc.) varies across time. Hence the reference to an average percentage.
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Our discussion of the prescriptive predicates suggests that in at least some cases, we cannot add conditions to the application of a vague predicate (e.g. that it be mandatory) without thereby generating a new vague predicate (e.g. ‘mandates ‘‘old’’ ’) requiring a new sorites series. To see what can happen if we overlook this rule, consider the following passage from Timothy Williamson’s influential book Vagueness: On the view that nothing is hidden [in particular a semantic view—DR], it should be harmless to imagine omniscient speakers, ignorant of nothing relevant to the borderline case . . . Accompanied by an omniscient speaker of English, you remove grain after grain from a heap. After each removal you ask ‘Is there still a heap?’. . . . For some number n, she says ‘Yes’ after each of the first n removals, but not after n + 1 . . . .You repeat the experiment with other omniscient speakers. . . . If they all stop at the same point, it evidently does mark some sort of previously hidden boundary. . . . [A non-epistemic view] must therefore hold that different omniscient speakers would stop at different points. They are conceived as having some sort of discretion . . . You can instruct the omniscient speakers . . . to use their discretion . . . conservatively, so that they answer ‘Yes’ to as few questions as is permissible . . . Now if two omniscient speakers stop answering ‘Yes’ at different points, both having been instructed to be conservative, the one who stops later has disobeyed your instructions, for the actions of the other show that the former could have used her discretion to answer ‘Yes’ to fewer questions than she actually did. But the omniscient speakers are cooperative. They will . . . obey your instructions . . . It is not as though, however many times they said ‘yes’, they could have said it fewer times, for the sorites series is finite . . . Thus if all [omniscient speakers] are instructed to be conservative, all will stop at the same point. You do not know in advance where it will come. It marks some sort of previously hidden boundary . . . (1994, 199–200).
Before I say anything about this argument in connection with prescriptive predicates, I want to make sure that its most obvious mistake is obvious. The instruction to ‘answer ‘‘Yes’’ to as few questions as is permissible’ is equivalent to an instruction to stop applying ‘heap’ at the earliest (most conservative) permissible place. But the semanticist about vagueness denies that there is such a place; the instruction cannot be carried out. (According to the semanticist, an omniscient speaker just is a competent speaker.) Only those who already believe in the existence of a sharp cut-off will imagine that it can. Now to the question of requiring a new sorites series. (We will need to work around the mistake just mentioned.) In instructing the omniscient speakers to stop applying ‘heap’ at the most conservative permissible place, Williamson is in fact asking them to apply a new predicate—something like ‘is a permissible stopping place for the predicate ‘‘heap’’ ’.¹³ Our reflections on ‘mandates ‘‘old’’ ’ suggest that whether a given collection of grains satisfies the latter predicate does not depend, or does not depend solely, on its number. Rather, the verdict depends also upon the way in which the community of English speakers applies the word ¹³ Actually, the predicate tacitly introduced by Williamson is ‘earliest permissible stopping place . . .’; but as I said, we must work around this mistake.
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‘heap’. We can imagine a sorites series analogous to the one for ‘mandates ‘‘old’’ ’ (see Figure 29.7). When ‘is a permissible stopping place for ‘‘heap’’ ’ is applied to the right sort of sorites series, the semanticist can say again that the omniscient speakers (who are, of course, just competent speakers!) have ‘discretion’, and so may diverge, in their applications of it. A speaker who starts applying ‘is a permissible stopping place for ‘‘heap’’ ’ at 90 percent knows that his judgment is arbitrary, so he acknowledges the permissibility of withholding that judgment. His judgment is not legislative, so no incoherence threatens.
29.3
A N OT H E R H I E R A RC H Y ?
What about the iterative predicates ‘mandates application of ‘‘mandates application of ‘old’ ’’ ’ and ‘mandates application of ‘mandates application of ‘‘mandates application of ‘‘old’’ ’ ’’ ’, etc.? Even supposing we’ve got the right kind of sorites series for each of these expressions, won’t we be stuck with an unending hierarchy of higherorder vague predicates? I don’t think so. To see why, consider again our sorites series for the predicate ‘mandates application of ‘‘old’’ ’ (see the bottom pair of series in Figure 29.8 below). If you were to proceed along series O, viz., the series of chronological ages, you would be judging whether a given age makes a person old. Proceeding along series MO, you would be judging whether application of ‘old’ by a certain average percentage of ordinary speakers makes it the case that application of ‘old’ to the corresponding age is mandatory. Consider now a further series, MMO, for the predicate ‘mandates application of ‘‘mandates application of ‘old’ ’’ ’ (‘mandates ‘‘mandates ‘old’ ’’ ’). MMO specifies average percentages of competent English speakers who would apply ‘mandates ‘‘old’’ ’, given percentages of speakers who would apply ‘old’ (as specified in MO). Speakers’ applications of ‘mandates ‘‘old’’ ’ reflect their judgments as to which of the various percentages specified in MO make it the case that application of ‘old’ to the corresponding age is mandatory. More strictly, MMO is a series of triples each containing a percentage of speakers who would apply ‘mandates ‘‘old’’ ’, together with the corresponding percentage from series MO and age from series O. Proceeding along MMO, you would be judging whether application of ‘mandates ‘‘old’’ ’ by a given percentage of speakers makes application of ‘mandates ‘‘old’’ ’ mandatory. As always, you would vary in your judgments from occasion to occasion. You might stop applying ‘mandates ‘‘mandates ‘old’ ’’ ’ at 95 percent one time, and at 90 percent the next. Analogously for further iterations (‘higher orders’).
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Now the scheme pictured in Figure 29.8 may appear hierarchical.¹⁴ But consider that, starting with the initial ‘metalinguistic’ series MO for ‘mandates ‘‘old’’ ’, you would always be judging whether application by a given percentage of English speakers makes it the case that application of the predicate in question is mandatory. As suggested in Figure 29.8, your judgments would vary from series to series (predicate to predicate), but in every series your classifications would depend upon answering the same question: does application of ‘’ by a certain percentage of speakers show that application of ‘’ is mandatory? One possibility, then, is that the variations in your classifications of the items (average percentages) in these different series (O, MO, MMO) are just the variations that would occur were you to make repeated runs along any one of them; in particular, your variations across these series may be just the variations that would occur over repeated runs along the MO series. (Perhaps this shows that all of the iterated ‘mandates’ operators are in effect semantically redundant upon the first.) For this reason I would suggest that, while we can regard the iterative predicates as higher-order vague insofar as they are metalinguistic and vague, the resulting structure may be better conceived as recurrent rather than hierarchical. In this connection it is interesting to note that whereas ‘mandates ‘‘old’’ ’ seems vague, ‘mandates ‘‘prime number’’ ’ and ‘mandates ‘‘richer than $110,000’’ ’ seem precise. ‘Mandates ‘‘blue’’ ’ seems vague while ‘mandates ‘‘6ft tall’’ ’ seems precise. In general, ‘mandates ‘‘’’ ’ seems vague just in case ‘’ is vague, and precise just in case ‘’ is precise. The same appears true for the predicate ‘can competently be called ‘‘’’ ’: it seems vague just in case ‘’ is vague. In view of this duality I suggest that, in these metalinguistic uses anyway, the terms ‘mandates’ and ‘competently’ are neither vague nor precise; if ‘mandates ‘‘’’ ’ and ‘can competently be called ‘‘’’ ’ are vague, their vagueness must derive entirely from ‘’. (I think we can say that the vagueness of ‘mandates ‘‘’’ ’ and ‘can competently be called ‘‘’’ ’ just is the vagueness of ‘’.) The distinction between vagueness and precision is often taken to be exhaustive for predicates, but I know of no good reason why. In fact, Russell says that ‘vague’ and ‘precise’ are contraries, not contradictories: ‘We are able to conceive precision; indeed, if we could not do so, we could not conceive vagueness, which is merely the contrary of precision’ (1999, 65). ¹⁴ It may also appear mind-numbingly complex; but that owes to the mind-numbing complexity of the predicates at issue. No one ever actually uses such crazy words.
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The point I want to make is that if ‘mandates’ is not itself vague, then iterating it, as in ‘mandates ‘‘mandates ‘’ ’’ ’ and ‘mandates ‘‘mandates ‘mandates ‘‘’’ ’ ’’ ’, etc., does not introduce any additional vagueness. Intuitively, the predicate ‘mandates ‘‘mandates ‘mandates ‘‘’’ ’ ’’ ’ is no vaguer than ‘mandates ‘‘’’ ’, which in turn is no vaguer than ‘’. This result lends further credence to the idea, floated above, that iterated ‘mandates’ operators are semantically redundant. And if that is right, then even if the structure pictured in Figure 29.8 is a hierarchy of some unobvious sort, it is not a hierarchy of vaguenesses. Of course, ‘mandates’ and ‘competent’ and their ilk have other uses, including object–linguistic ones, that may differ significantly from the metalinguistic uses we have been discussing. For instance, actions can be mandatory or permissible, and doctors and teachers can be competent. Of particular relevance to an understanding of vagueness is the fact that speakers may or may not be competent. Whether any conclusions drawn here about the metalinguistic uses of these words will transfer to their object–linguistic uses is a matter I leave for further investigation. Re f e re n c e s Deas, R. (1989), ‘Sorensen’s sorites’, Analysis 49, 26–31. Graff, D. (2003), ‘Gap principles, penumbral consequence, and infinitely higher-order vagueness’ in Liars and Heaps: New Essays on Paradox, ed. Jc Beall, Oxford, Oxford University Press, 195–222. Hardin, C. L. (1988), Color for Philosophers: Unweaving the Rainbow, Indianapolis, Hackett Publishing Company. Heck, R. (1993), ‘A note on the logic of (higher-order) vagueness’, Analysis 53, 201–8. Hyde, D. (1994), ‘Why higher-order vagueness is a pseudo-problem’, Mind 103, 35–41. (2003), ‘Higher orders of vagueness reinstated’, Mind 112, 46, 301–5. Keefe, R., and Smith, P. (1999), Vagueness: A Reader, MIT. Lindsey, D., Raffman, D., and Brown, A. (2009) (in progress), ‘Psychological hysteresis and the nontransitivity of insignificant differences’. Raffman, D. (2005), ‘Borderline cases and bivalence’, Philosophical Review 114, 1. (2009) (under review), Unruly Words: A Study of Vague Language. Russell, B. (1923), ‘Vagueness’, Australasian Journal of Philosophy and Psychology 1, 84–92. Reprinted in Keefe and Smith 1999, 61–8. Sainsbury, R. M. (1997) (1990), ‘Concepts without boundaries’, orig. Inaugural Lecture, King’s College London, 1990. Reprinted in Rosanna Keefe and Peter Smith, eds., A Vagueness Reader, MIT,1997, 251–64. Shapiro, S. (2006), Vagueness in Context, Oxford. Sorensen, R. (1985), ‘An argument for the vagueness of ‘‘vague’’ ’, Analysis 45, 134–7. Tye, M. (1994), ‘Why the vague need not be higher-order vague’, Mind 103, 43–5. Varzi, A. (2003), ‘Higher-order vagueness and the vagueness of ‘‘vague’’ ’, Mind 112, 295–9. Williamson, T. (1994), Vagueness, London: Routledge. Wright, C. (1992), ‘Is higher-order vagueness coherent?’, Analysis 52, 129–39. (1994), ‘The epistemic conception of vagueness’, Southern Journal of Philosophy 33, Spindel Supplement, 133–59.
30 The Illusion of Higher-Order Vagueness Crispin Wright
It is common among philosophers who take an interest in the phenomenon of vagueness in natural language not merely to acknowledge higher-order vagueness but to take its existence as a basic datum—so that views that lack the resources to account for it, or that put obstacles in the way, are regarded as deficient just on that score. My main purpose in what follows is to loosen the hold of this deeply misconceived idea. Higher-order vagueness is no basic datum but an illusion, fostered by misunderstandings of the nature of ordinary (if you will, ‘first-order’) vagueness itself. To see through the illusion is to take a step that is prerequisite for a correct understanding of vagueness, and for any satisfying dissolution of its attendant paradoxes.
30.1
THE INERADICABILITY INTUITION
One standard motive for acknowledging higher-order vagueness is given prototypical expression by Michael Dummett: Now the vagueness of a vague predicate is ineradicable. Thus ‘hill’ is a vague predicate, in that there is no definite line between hills and mountains. But we could not eliminate this vagueness by introducing a new predicate, say ‘eminence’, to apply to those things which are neither definitely hills nor definitely mountains, since there would still remain things which were neither definitely hills nor definitely eminences, and so ad infinitum [sic].¹
This thought—the ineradicability intuition—may be generalized like this. Take any pair of concepts, F and G, with a vague mutual border. If you attempt to eradicate the vagueness by introducing a new term, H, to cover the shared borderline cases of F and G, your nemesis will be that the F-H and G-H borders will be vague in their turn. It I am grateful to the members of Arch´e’s AHRC-funded project on Vagueness: its Nature and Logic (2003–6) for helpful discussion and critical comments during the seminars that saw the gestation of this chapter. My special thanks to Elia Zardini, who gave me detailed written comments on the draft I prepared for the 2007 Arch´e conference, and to Mark Sainsbury, my commentator on that occasion. A proper response to all their observations and suggestions would have demanded a much more extended and doubtless much improved treatment. ¹ From Dummett (1959), at 182 in Dummett (1978).
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follows, seemingly,² that the distinction between the Fs and the F-G borderline cases is itself already vague. Likewise for the Gs. So, iterating, we have a hierarchy of levels of borderline cases of F, and another hierarchy of levels of borderline cases of G, each continuing indefinitely. Notice how Dummett, like so many others, equates the lack of a sharp boundary between the Fs and the Gs with the (potential) existence of borderline cases, viewed as a kind of thing: things that are neither definitely F nor definitely G. I’ll henceforward term this characterization the Basic Formula. Moreover, Dummett does not, plausibly interpreted,³ intend to allow that things which are neither definitely F nor definitely G might yet be F or G all the same—only just not definitely so. He is thinking of the kind in question as cases that in some way come short of being either F or G: if x is an ‘eminence’, then it fails to qualify either as a hill or as a mountain. So for there to be no definite line between hills and mountains is for there to be (potential) things ‘in between’ that are, in some way, of a third sort. Thus the mutual vagueness of F and G, on this understanding, consists in the existence of a certain kind of buffer zone between their respective (potential) extensions. Yet this buffer zone had better be blurry on both edges in turn, or F and G will turn out to be not mutually vague but sharply separated by a mutual neighbour. And now it seems we have no option but haplessly to allow the blurred buffer-zone model to reiterate indefinitely. Dummett’s thought is closely related to, though distinct in detail, from that at work in these remarks of Russell: The fact is that all words are attributable without doubt over a certain area, but become questionable within a penumbra, outside of which they are again certainly not attributable. Someone might seek to obtain precision in the use of words by saying that no word is to be applied in the penumbra, but unfortunately the penumbra itself is not accurately definable, and all the vaguenesses which apply to the primary uses of words apply also when we try to fix a limit to their indubitable applicability.⁴
Here Russell envisages not the introduction of a new term but rather a moratorium on applying any term. If it is not certain that F is properly applied, then it is not to be applied—the penumbra is to be an exclusion zone. Still Russell’s idea, like Dummett’s, involves the notion of a kind of case separating those where the applications of F and not-F are respectively mandated, or ‘indubitable’. And it is clear that he confidently expects judgments about membership in this kind to involve no less ‘vaguenesses’ than we started out with. ² It does follow, provided we assume that the introduction of the new term effects no alteration in the respective extensions of the original concepts; I’ll come back to this point later. ³ In ‘Wang’s Paradox’, he writes: ‘For, in connection with vague statements, the only possible meaning we could give to the word ‘‘true’’ is that of ‘‘definitely true’’ ’—(Dummett 1978, 256.) No doubt here are no borderline cases of ‘Definitely P’ which are clear cases of P. The question is whether we should allow, as part of the intended meaning of the Definiteness operator, that it consists with something’s being a borderline case of ‘Definitely P’ that it yet be a case of P. Dummett is here saying no to that. We can call Dummett’s Principle the thesis that there are no truthful instances of the conjunctive form: P but not definitely P. As will emerge later, there is actually considerable pressure against the principle. ⁴ Russell (1923) at 63–4 of the Keefe and Smith reprint.
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The ineradicability intuition impresses as highly plausible. The linguistic stipulations respectively envisaged by Dummett and Russell would indeed—surely—not have the effect of introducing precision. But can that really be enough to enforce the vertiginous hierarchy of borderline kinds?
30.2
THE SEAMLESSNESS INTUITION
The ineradicability intuition provides one motive for postulating higher-order vagueness. A prima facie distinct motivation emerges from the idea that vagueness consists in the possession of borderline cases, together with one natural notion about how borderline cases, as characterized by the Basic Formula, come about and the apparent phenomenological fact of seamless transition. Consider a case where, as many would allow, something akin to vagueness is induced by deliberate definitional insufficiency. Suppose we characterize the notion of a pearl as follows.⁵ (i) It is to be a sufficient condition for being a pearl that a candidate have a certain specified chemical constitution and appearance and be naturally produced within an oyster. (ii) It is to be a necessary condition for being a pearl that a candidate have that same specified chemical constitution and appearance. What about artificial pearls? They satisfy the specified necessary condition but not the specified sufficient one. One thing we might say is this: since there is no sufficient basis for classifying them either as pearls (for they do not satisfy the only specified sufficient condition) or as non-pearls (for they do satisfy the only specified necessary condition), it is so far indeterminate whether artificial pearls are pearls.⁶ There is no fact of the matter. Now (this is the natural notion mentioned) suppose we think of borderline cases of naturally occurring vague predicates,—‘bald’, ‘heap’, ‘red’, and the other usual suspects—as relevantly like artificial pearls: cases which are left in classificatory limbo by a broadly analogous but naturally occurring kind of semantic incompleteness. Thus they are cases that do not meet any practice-established sufficient condition for satisfying the relevant predicate but do satisfy all practice-established necessary ones. This is, seemingly, a very intuitive way of thinking of the Basic Formula as being underwritten. The (definite) truths, and falsehoods, are what are determined as true, or ⁵ The example is John Foster’s from classes in Oxford in the early 1970s. Compare Kit Fine’s ‘nice1 ’ (Fine 1975, 266), Timothy Williamson’s ‘dommal’, (Williamson 1990, 107), and (1994, 213–14) and Mark Sainsbury’s ‘child∗’ (Sainsbury 1991, 173). ⁶ One who, like Timothy Williamson, believes that Bivalence, like the Articles of the United States Constitution, is a self-evident truth, has of course to move differently: to deny that ‘pearl’ has so far been endowed with a meaning, or—as proposed by Williamson himself—to regard artificial pearls as non-pearls purely by dint of their failure to satisfy any established sufficient condition for being pearls. See Williamson (1994, 213) and (1997, section 3). The availability of this proposal to Williamson is queried in Heck (2004, 112).
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false, by the facts and the semantic rules for the language in question. Borderline cases arise when the facts and semantic rules somehow fail to deliver.⁷ Next contrast the following two cases. Case 1: You have a collection of 2-inch square colour patches, each of a uniform shade, collectively ranging in hue from red to orange, and numerous and varied enough to allow that every patch is matched by something that matches something in the collection that it does not match.⁸ You have to arrange them in a ‘monotonic’ series; specifically, one such that the first patch is red and each subsequent patch is immediately preceded by something that is at least as red as it is. So your selection will consist in an initial batch of red patches followed by some which hover around the red-orange border followed by some orange ones, the whole giving the impression of a perfectly seamless movement, without regression, from red to orange. Case 2: You have a collection of pearls, artificial pearls and costume (plastic) pearls and, again, have to arrange them in a monotonic series; specifically, a series such that the first selection is a pearl and each subsequent selection is immediately preceded by something whose case to be a pearl is at least as strong. Then your selection will consist in a string of pearls, followed by a string of artificial pearls, followed by the fakes. The thought suggestive of higher-order vagueness is then simply this. Both series—we are currently supposing⁹—contain indeterminate cases, conceived as generated by semantic incompleteness. However, in the pearl series, the transitions from the pearls to the indeterminate cases, and from the latter to the non-pearls occur sharply, at specific places. And, associatedly, there is no second-order indeterminacy—no indeterminacy in turn in the pearl-indeterminate and indeterminate-fake pearl distinctions. So, the thought occurs, how to explain the manifest difference in the phenomenology of the changes occurring within the two series if not by postulating second and, indeed, indefinitely higher-orders of indeterminacy in the red-toorange series? How else to accommodate the fact that we are absolutely at a loss to identify specific first and last borderline cases of the red-orange distinction in that series, or indeed abrupt changes of any kind? The key thoughts again: the vagueness of pearl and red is held to consist in the existence of borderline cases of these concepts, conceived as items that are not definitely classifiable as ‘pearls’, or as ‘red’, and not definitely classifiable as something else, on account of the semantic incompleteness of the relevant expressions. The sharpness of the distinction between the pearls and the borderline pearls shows in the abruptness of the transition between them in the relevant monotonic series. By contrast, the ⁷ This type of view goes back to Frege and was for a long time regarded as datum, rather than theory. For modern exponents, see McGee and McLaughlin (1995, 209 ff); and Soames (2003, ch. 7, passim). For criticism, see Wright (2007, 419–23). Some of the criticisms there lodged are presented as depending on higher-order vagueness. I postpone to a future discussion the question whether they can survive in a qualified form if the conclusions of the present study are accepted. ⁸ At least one commentator (Fara 2001) has argued that this is impossible. I beg to differ—but the example could easily be reworked so as to finesse the issue. ⁹ In case it is not obvious, I do not think that this is the right way to conceive of the vagueness of the ‘usual suspects’.
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smoothness of the transitions between the reds and the borderline cases, and between the borderline cases and the oranges, enforces the idea that these distinctions are vague in turn. So it follows that they too admit of borderline cases. And so on ad infinitum. We can call the driving intuition here the seamlessness intuition.¹⁰ In general: unless we have an indefinite hierarchy of kinds of borderline case, it seems there will have to be sharp boundaries in any process of transition between instances of one vague concept and instances of another. Or so it anyway appears. But we’ll return to explore this thought in some detail. The resulting broad conception of full-blown higher-order vagueness: the conception of an infinite hierarchy of kinds, each potentially serving to provide an exclusion zone and thereby prevent a sharp transition, in a suitable series, between instances of distinctions exemplified at the immediately preceding stage of the hierarchy, may be termed the Buffering view. I shall argue for each of the following claims: (i) That the Buffering view is not well motivated by either the ineradicability or the seamlessness intuitions. (ii) That there is serious cause to question whether the Buffering view is fit for purpose. (iii) That for the kinds of vague concepts—the ‘usual suspects’—in which we are interested, the view that they exhibit higher-order vagueness on the model of the Buffering view is at odds with the broadly correct conception of their (‘firstorder’) vagueness. 30.3
P OT E N T I A L C O N F U S I O N S A B O U T H I G H E R - O R D E R VAG U E N E S S — T H R E E D I S T I N C T N OT I O N S
Within limits disrespected by Humpty Dumpty, philosophers are free to mean by the phrase, ‘higher-order vagueness’, whatever they choose. But the fact is that at least three distinct putative phenomena have been earmarked by it in the literature, without—perhaps—all of those who have so earmarked them being clear that their discussions concerned potentially different things. One is: (a) That the distinction between the things to which a vague expression applies and its first-order borderline cases—the cases where it is indeterminate whether it or its complement applies—does itself, in the cases that characteristically interest us, admit of borderline cases; that the distinction between the things to which a vague expression applies and this second-order of borderline cases also admits of borderline cases; that the distinction between the things to which a vague expression applies and this third-order of borderline cases also admits of borderline cases; and so on indefinitely. When, in the fashion noted, borderline cases are thought of as an intermediate kind, distinguished from the kinds of which they are borderline cases, this idea becomes the Buffering view. ¹⁰ I prefer ‘seamlessness’ to ‘continuity’. The relevant notion is pre-mathematical and intuitive. Compare Fara (2004).
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Standing apparently unrelated to that is (b) The vagueness of Vague: there are concepts which are borderline cases of the vague-precise distinction itself,—concepts which are neither definitely vague nor definitely precise,—and, further, there are borderline cases of membership of this range of concepts in turn, and borderline cases of those in turn. . . and so on.¹¹ Then finally there is the thought (c) That the usual kind of definiteness operator—that is: one introduced for the purpose of allowing us to characterize the borderline cases of F in accordance with the Basic Formula—ineluctably gives rise to a hierarchy of new, pairwise inequivalent vague expressions, ‘Definitely F’, ‘Definitely Definitely F’ and the like.¹² (Definitization modifies truth-conditions but does not eliminate vagueness.) It seems obvious enough that there is little connection between (b) and the other two. It seems quite consistent with holding to the Buffering view, or with thinking of ‘Definitely P’ as vagueness-inheriting though precision-increasing when applied to a vague claim P, that the notion of vagueness itself should divide all expressions into two sharply bounded kinds—that there is never any vagueness about the question whether an expression is vague or not. Conversely, one might think of the distinction between vague expressions and others as admitting of borderline cases but hold to a view of the nature of vagueness according to which there are no higher-order borderline cases; and one might simultaneously just repudiate any operator of definiteness, or take the view that any legitimate such operator generates only precise claims. At any rate, these are all prima facie compatibilities. If there are deeper tensions, that would be interesting—but they remain to be brought out. I will say nothing further here about thesis (b). Of potentially more importance for our purposes is the apparent distinctness of thesis (a) and thesis (c), the thesis that applications of the Definiteness operator, while they shift truth-conditions (since they take any originally indefinite claim to a false one), are nevertheless impotent to eliminate vagueness: if P is vague, so is Definitely P. Thesis (a) takes the distinction between F and (any order of) its borderline cases to be vague. F’s higher-order vagueness consists, at each nth order, n > 1, in the (potential) existence of borderline cases of the distinction between F and its borderline cases of the immediately preceding order. The thought embodied by thesis (c), by contrast, changes the terms of the relation of mutual vagueness. At second-order, for example, it is not F but ‘Definitely F’ that is assigned a vague borderline. More specifically, letting ‘Def ’ be the Definiteness operator, the ‘second order’ of borderline cases countenanced by thesis (c) may be schematized thus: ∼DefDef F&∼Def (∼Def F&∼Def ∼F) ¹¹ This discussion seems to originate in Sorensen (1985). See Hyde (1994) and (2003), and Varzi (2003). ¹² See, for example, Williamson (1999).
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And in general each successive nth order of vagueness, n > 1, is conceived as consisting in the vagueness of the boundary between the Def n−1 Fs—the things that are definitely . . . definitely (n-1 times) F—and the definite borderline cases of order n-1, that is, as consisting in the (potential) existence of cases satisfying the condition: ∼Defn F&∼Def (Borderlinen−1 F) Now, as a construal of the notion of higher-order vagueness as suggested by the ineradicability and seamlessness intuitions, thesis (c) initially just seems wayward. Those intuitions motivate a thesis about the existence of a hierarchy of orders of vagueness of a single originally targeted concept. Thesis (c) by contrast goes in for a hierarchy of kinds of first-order vagueness which successively concern different concepts: Definitely F, Definitely Definitely F, . . . and so on,—a hierarchy produced as an artifact of the introduction of the Definiteness operator. The preoccupation of much of the discussion with thesis (c) might therefore seem to offer one more example of philosophers taking their collective eye off the ball. It is hardly intuitively evident that natural language contains any operator that behaves like this. And even if it does, what can that have to do with the proper understanding of the nature of vagueness, which presumably comes fully formed, as it were—and therefore fully ‘higher-orderized’, if the phenomenon is indeed real,—even in languages lacking any Definiteness operator? Aspects of the behaviour of such an operator cannot constitute higher-order vagueness as originally motivated. What does thesis (c) have to do with anything? Here is one arguable connection. When the first-order borderline cases of the distinction between F and its negation are characterized by the Basic Formula, they will be, one and all, things that are not definitely F. So they will fall under the negation of ‘definitely F’ and will thus, none of them, be borderline-cases of ‘definitely F’.¹³ Now thesis (a) requires that there are borderline cases of the distinction between F and its first-order borderline cases. These will all, presumably, be clear cases of ‘not definitely not F’. So if they are borderline cases of the Basic Formula’s characteristic conjunction, they must be borderline cases of ‘not definitely F’. But if they were definite cases of ‘definitely F’, they would not be borderline cases of its negation. So they must be borderline cases of ‘definitely F’ too, which is therefore vague if thesis (a) is true of F and borderline cases are characterized by the Basic Formula. Very well. However thesis (c) involves two components: that definitization does not eliminate vagueness, just argued for, and that it generates statements which are not, in general, equivalent to those definitized. Since it is, intuitively understood, a factive operation, the second component is tantamount to the claim that a definitized statement is in general logically stronger than its prejacent. This too is, as will emerge, plausibly taken to be a consequence of thesis (a) and the characterization of borderline cases given by the Basic Formula. What about the converse direction? Is thesis (a) a consequence of thesis (c), assuming the Basic Formula? Again, arguably so. Let G be any predicate such that the F-G distinction is vague. Then F has borderline cases, characterized as cases which are not definitely F and not definitely G. But by thesis (c), ‘definitely F’ is vague if F is. ¹³ This step, nota bene, applies Dummett’s Principle. See note 3 above.
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And, since by hypothesis G is vague, so likewise is ‘definitely G’. Since vagueness is, presumably, preserved under negation, ‘not definitely F’ and ‘not definitely G’ are likewise vague. Since vagueness is presumably preserved under (consistent) conjunction, so is ‘not definitely F and not definitely G’—so the notion of a borderline case of F is itself vague, and hence has borderline cases. These cannot be definite cases of F or they would fail the first conjunct and hence not be borderline cases of the conjunction. So they must be borderline cases of F and of the notion: borderline case of F and G. The latter notion is then available for choice in place of ‘G’, and the reasoning can be iterated indefinitely. So, given that the vagueness of a predicate consists in its susceptibility to borderline cases and the thesis that these are one and all to be characterized as per the Basic Formula, there is a case—we can put it no stronger than that—that thesis (a) and thesis (c) are equivalent. If that is right, it offsets the charge of irrelevance against intended investigations of higher-order vagueness that have taken thesis (c) to be a constitutive matter. On the other hand, if thesis (a) depicts an illusion, the equivalence will mean that the illusion persists in thesis (c) as well. Work on the semantics and proof-theory of the definiteness operator directed towards the elucidation and stabilization of thesis (c) will then be so much misdirected effort.
30.4
T H E B A S I C F O R M U L A A N D L AC K O F S H A R P BOUNDARIES
So let’s assume for the sake of argument that borderline cases are felicitously described by the Basic Formula, and—thesis (a)—that certain concepts sustain an infinitely ascending hierarchy of orders of borderline case, each characterizable by a suitable application of the Basic Formula. What reason is there, in this setting, to think that the Definiteness operator should comply with the proof theoretic part of thesis (c): the claim that definitization increases logical strength? In fact there is quite powerful pressure towards that thought. It comes from reflection on that form of the Sorites paradox—what I once called the No-SharpBoundaries paradox—which seems to connect most directly with the very nature of vagueness.¹⁴ I’ll make the point in some detail over this and the succeeding section. The standard form of major premise for the Sorites is a universally quantified conditional, usually motivated by tolerance intuitions. But the major premise for the No-Sharp-Boundaries paradox takes the form of a negative existential, (i) ∼(∃x)(Fx & ∼Fx ), seemingly tantamount merely to the affirmation that F is indeed vague in the series in question. For vagueness is just the complement of precision, and precision (relative to the relevant kind of series) is, it seems, perfectly captured by (∃x)(Fx & ∼Fx ). ¹⁴ Wright (1987).
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But whereas it may be doubted that vague predicates really are tolerant, it hardly seems doubtful that they really are vague! In affirming (i), accordingly, we seem merely to have affirmed that F is vague.¹⁵ So vagueness appears paradoxical per se. Enter the Definiteness operator. What, it may be suggested, really constitutes precision is a sharp boundary between definite cases. Hence what is really tantamount to an expression of F’s vagueness in the relevant series is not the negative existential statement (i) above but rather: (ii) ∼ (∃x)(Def Fx & Def ∼Fx ) —the thesis that there is no last definite case of F in the series immediately followed by a first definite non-F. But (ii), unlike (i), gives rise to no immediate paradox. We can show of course by appeal to it that any n such that Def ∼Fn , must be such that ∼Def Fn. But then—absent further proof-theoretic resources for the Definiteness operator—we seem to have no means to commute the occurrences of ‘∼’ and ‘Def ’ to generate something soritical. What, though,—other than the reflection that we can apparently finesse the paradox thereby—is available to justify the claim that it is indeed (ii), rather than (i), that gives proper expression to F’s vagueness in the kind of series in question? There is a very good argument for that claim if we can legitimately have full recourse to classical logic. Take it that what F’s vagueness in the series consists in is the presence there of (first-order) borderline cases of F, and that these are suitably characterized by the Basic Formula. Specifically, suppose that there is such a borderline case of F: (iii) (∃x)(∼Def Fx & ∼Def ∼Fx) ¹⁵ We obtain a Sorites paradox from the negative existential major premise without reliance on any distinctively classical moves, by running right-to-left, as it were—by beginning with a minor premise of the form, ∼Fa, and reasoning through successive steps via the rules for conjunction, existential introduction and the (intuitionistically acceptable) negation-introduction half of reductio. It merits emphasis that the intuitive motivation for the major premises for Sorites paradoxes varies quite dramatically across forms that are classically equivalent. Consider for instance the three genres of premise: (i) (∀x)(∼Fx V Fx ) (ii) (∀x)(Fx → Fx ) (iii) ∼(∃x)(Fx & ∼Fx ) The last, as noted, is naturally motivated just by the thought that it is constitutive of the vagueness of a predicate that its extension in a suitably constructed series of objects not run right up against that of its negation. This thought involves no intuitive dependence on Bivalence. The second is driven, more specifically, by tolerance intuitions, of the kind discussed in Wright (1975), that in turn draw on folk-semantical ideas about observational and phenomenal predicates which have little explicit connection with vagueness. These ideas, again, involve no intuitive dependence on Bivalence but are stronger than the thought that motivates (iii) since someone who embraced a ‘Third Possibility’ view of borderline cases could accept (iii) while rejecting (ii): vagueness might be conceived as, in typical cases, intolerant of the distinction between some Fs and some borderline cases of F, even though sustaining no-sharp-boundaries principles in the form of (iii). (i), finally, is entailed by either of the other two if, but only if, Bivalence is assumed for predications of F. It is thus natural to conceive of (i) through (iii) as of decreasing strength. It is a significant weakness of the classical outlook that it stifles these intuitive differences.
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but also, for reductio, that there is a last definite case of F in the series immediately followed by a first definite non-F: (iv) (∃x)(Def Fx & Def ∼Fx ) Contradiction follows on the assumption of the monotonicity of the series (intuitively, that all the F-relevant changes manifested in it are one-directional), which we may capture by the pair of principles: (∀x)(Def Fx → Def Fx) —the immediate predecessor of anything definitely F is definitely F— and (∀x)(Def ∼Fx → Def ∼Fx ) —the immediate successor of anything that is definitely not F is likewise definitely not F. For suppose m is a witness of (iv); that is, Def Fm&Def ∼Fm , Then the monotonicity principles will ensure that every element preceding m in the series is Definitely F and every element succeeding m is Definitely not F; and hence that none satisfies the rubric for borderline cases given by the Basic Formula, contrary to (iii). We supposed that the vagueness of F in the series in question consists in the presence of borderline cases of F, as characterized by the Basic Formula. The reasoning we just ran through establishes that one who accepts that supposition thereby commits themselves to (ii). So in order to show that it is (ii), not the soritical (i), that is tantamount to an acceptance that F is vague in the series in question, we now require the converse direction: that someone who accepts that there is no last definite F element immediately succeeded by a first definite non-F element is thereby committed to the existence of borderline cases of F in the series concerned, as characterized by the Basic Formula. Straightforward—though classical—reasoning establishes the point. The series, we can take it, is such that (1) Def (F0) and (2) Def ∼(Fn) Suppose (ii) above and for reductio the negation of (iii): (3) ∼ (∃x)(∼Def Fx&∼Def ∼Fx), —there are no borderline cases of F in the series. Then (4) Def ∼Fx → ∼Def Fx, —from (ii). So (5) ∼Def (Fn-1), —from (2) and (4).
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Suppose (6) ∼Def ∼(Fn-1) Then (7) (∃x)(∼Def Fx & ∼Def ∼Fx), —contrary to 3. So (8) (∼∼)Def∼(Fn-1). This routine may be repeated eventually culminating in contradiction of 1. At that point (3) may be discharged by reductio, on (1), (2) and (ii) as remaining assumptions, a final step of double negation elimination then yielding (iii). Our result, then, is that—granted classical logic—F’s vagueness, identified with its possession of borderline cases as characterized by the Basic Formula, is equivalent not to the soritical (i) ∼(∃x)(Fx & ∼Fx ), but the apparently harmless (ii) ∼ (∃x)(Def Fx & Def ∼Fx ). It is the latter, then, which, we may accordingly be encouraged to think, is the canonical expression of F’s lack of sharp boundaries in the relevant kind of series. This result is the first point towards uncovering the advertised impetus towards the proof-theoretic component of thesis (c). I will pursue that further in the next section. It may also seem (as it once did to me) to be the first step towards a dissolution of the No-Sharp-Boundaries paradox. Obviously, however, it is at most a first step. For one thing, the reliance on classical logic is, of course, of some significance in this context. The question under review is whether, and if so, how a correct understanding of the nature of vagueness escapes a commitment to a soritical version, such as (i), of the NoSharp-Boundaries intuition. In exploring the matter, we therefore must resort only to principles of inference which are sound for vague languages. Those who share the doubts of the present author whether classical logic is in that case should therefore regard the reasoning just run through with at most qualified enthusiasm. Even were we satisfied that classical logic is fit for duty in this setting, however, there is a further issue. For unless we are prepared to allow that the boundary between the definite Fs and the borderline cases of F is sharp, there is the same intuitive motivation as previously to affirm (i∗ ) ∼ (∃x)(Def Fx & ∼Def Fx ), and this, if allowed, will in turn subserve a Sorites paradox (this time subverting the distinction between the borderline cases and the definite cases of F.) To be sure, the reply can be that the proper way to do justice to the vagueness of the second-order borderline is to affirm not (i)∗ but (ii)∗ ∼ (∃x)(DefDef Fx & Def ∼Def Fx )
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—there is no sharp cut-off separating the definite cases of ‘Definitely F’ from the definite borderline cases of F. And in general, for an arbitrary pair of mutually vague, contrary concepts, φ and ψ, exemplified in the series in question, it may be proposed, generalizing the reasoning above, that the proper way to give expression to a lack of sharp boundaries between them is to affirm the negative existential, (∗ ) ∼ (∃x)(Def φx & Def ψx ) So we need never, apparently, be committed at any level to a soritical claim. But where is this leading? If the seamlessness intuition is to be upheld, then it seems that it must be possible, in principle, so to describe a Sorites series that no abrupt transitions of any relevant kind take place between adjacent elements within it. So every pair of contrary concepts, φ and ψ, manifested in the series must sustain the truth in it of the relevant instance of (∗). More specifically: if the mutual vagueness of any pair of concepts, Def (. . .x. . .) and Def ∼ (. . .x. . .), is viewed as consisting in the existence of borderline cases as characterized by the Basic Formula, and if the seamlessness intuition is accepted, then we are committed to each of the following principles: ∼ (∃x)(Def Fx & Def ∼Fx ) ∼ (∃x)(DefDef Fx & Def ∼Def Fx ) ∼ (∃x)(DefDefDef Fx & Def ∼DefDef Fx ) . . . etc. Given the reliance on classical logic of the reasoning worked through above, it would be tendentious to proclaim these Gap principles¹⁶ to be respectively characteristic of the putative successively higher-orders of borderline case of the predicate F. But they are at least, it may seem, among our commitments if we accept that a series is possible in which a seamless, monotonic transition is effected from instances of F to instances of not-F, and in which any borderline cases of any distinction exemplified within it are characterized by the Basic Formula as applied to that distinction. Let us take stock. It is hard to reject the idea that the seamlessness intuition is sound in some form: the transition from Fs to non-Fs in a Sorites series can be effected without abrupt, noticeable change of status at any point. The thought that leads from seamlessness to the postulation of higher-order vagueness can be refined as follows. Define a monadic predicate (open sentence) as F-relevant if it is formulated using just F, the truth functional connectives and the definiteness operator. Conceive of seamless transition as the circumstance that the ranges of each pair of incompatible F-relevant predicates exemplified in a Sorites series running from instances of F to instances of its negation are buffered : between the instances of any such pair ¹⁶ Delia Graff Fara’s nice term in her (2004). Each such principle (Fara actually formulates them slightly differently) classically ensures that the instances in a suitable series of a pair of contrary concepts of the form, Def n φx and Def ∼Def n−1 φx, are separated by a gap—in our terminology above, a buffer zone.
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of predicates intervenes at least one element to which neither definitely applies: an element which is a borderline case of the distinction they express, according to the characterization of borderline cases given by the Basic Formula. As we saw, this conception, assuming monotonicity in the transition concerned, ensures that a Gap principle—an instance of (∗)—holds for any such pair of predicates. On classical assumptions, the holding of such a Gap principle is equivalent to the presence in the series of a borderline case, characterized as per the Basic Formula, of the original distinction. So the train of thought is this: •
Seamlessness requires buffering of all F-relevant distinctions exemplified in the series; • Such buffering requires the presence, in the series, of borderline cases (characterized as per the Basic Formula) of each such distinction; • The presence of such borderline cases requires (indeed, classically, is tantamount to) the holding of appropriate Gap principles. That said, though, note that a plausible connection between seamlessness and the Gap principles can of course be made out more directly. If any of the existential statements which the Gap principles respectively directly contradict is true in a Sorites series, then there is an abrupt, non-seamless change of status between the element that witnesses that statement’s truth and its immediate successor. So seamlessness, it appears, requires the Gap principles to hold anyway, whether or not we take that to be equivalent, as classically it is, to the presence of borderline cases of each appropriate higher order.¹⁷
30.5
T H E S I S ( C ) A N D T H E PA R A D OX O F H I G H E R - O R D E R VAG U E N E S S
Let us now connect the foregoing with the proof-theoretic component of thesis (c). I once argued that, so far from resolving the No-Sharp-Boundaries paradox, to corral our no-sharp-boundaries intuitions into an endorsement of principles of the (∗)-form merely generates new soritical problems.¹⁸ The argument utilized a proof-theory incorporating the rule: (DEF)
{A1 . . . An } ⇒ P {A1 . . . An } ⇒ Def P,
¹⁷ Note that anyone content with classical logic in this region who accepts the idea that seamless transition is possible and that it is correctly construed as requiring the Gap Principles to hold en masse, should worry about this: that no finite Sorites series can exemplify borderline cases of every higher order unless some borderline cases instantiate multiple, indeed infinitely many orders. This is noted in Fara (2004, 205). Given the ways, reviewed earlier, in which acceptance of higher-order vagueness is standardly motivated, this—egregious violation of Dummett’s principle—is an idea for which we are wholly unprepared, indeed an idea of questionable intelligibility. ¹⁸ Wright (1992). The argument was there presented as a reductio of the very idea of higher-order vagueness. In fact, what it puts under pressure is any set of assumptions entailing an nth-order Gap.
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where {A1 . . . An } contains only ‘fully definitized’ propositions (i.e. propositions prefixed by ‘Def ’.) Once Def ’s proof-theory incorporates this rule,¹⁹ each of the Gap principles corresponding to the successive higher orders of vagueness becomes soritical.²⁰ But the Gap principles, as we have seen, are seemingly imposed by the possibility of seamless transition across a Sorites series. Moreover, classically, each is tantamount to—and each is anyway a consequence of—an affirmation of the existence of a corresponding order of borderline cases, when characterized in accordance with the Basic Formula. So the postulation of any higher order of borderline cases is soritical unless the DEF-rule fails. And if seamless transition does indeed entail the Gap principles, then—even without classical logic—we must likewise accept that the DEF-rule fails provided we believe that seamless transition is possible.²¹ To reject the DEF-rule is to allow that Def P can be a consequence of a set of (fully definitized) premises, even though DefDef P is not. Since the entailment from DefDef P to Def P is unquestioned, to reject the DEF-rule is thus to regard the definitization of a sentence as potentially increasing its logical strength. That is the prooftheoretic component of thesis (c). Principle, n > 1. The picture of higher-order vagueness captured by the Buffering view incorporates one such set of assumptions, as we have seen. But we have also noted that the very idea of seamless transition appears to enforce the Gap principles as well. Focused on the case second-order Gap principle, presumed itself to be a Definite truth, the argument was this: 1 2 3 3 2,3 1 1,2 1,2 1
(1) (2) (3) (4) (5) (6) (7) (8) (9)
Def ∼(∃x)[Def (Def (Fx)) & Def (∼Def (Fx ))] Def (∼Def (Fk )) Def (Fk) Def (Def (Fk)) (∃x)[Def (Def (Fx)) & Def (∼Def (Fx ))] ∼ (∃x)[Def (Def (Fx)) & Def (∼Def (Fx ))] ∼Def (Fk) De(∼Def (Fk)) Def (∼Def (Fk )) → (Def (∼Def (Fk))
Assumption Assumption Assumption 3, DEF. 2, 4, ∃-intro. 1, Def -elim. 3, 5, 6, Reductio 7, DEF 2, 8 Conditional Proof
¹⁹ In effect, just an S4 rule for ‘Def ’. ²⁰ See the proof schema illustrated in note 17. Note that the general applicability of the schema assumes, in addition, that the Gap Principles are definite truths, and that there are definite borderline cases of the relevant order. These points would need defence in a fully rigorous presentation of the line of thought currently under development. ²¹ This pr´ecis ignores a number of subtleties. As Richard Heck (1993) pointed out, the reasoning of my original ‘paradox’ of higher order vagueness involved, besides the DEF-rule, free recourse to standard rules allowing for the discharge of assumptions, specifically reductio ad absurdum and conditional proof. The DEF-rule is under pressure from the paradox only if its combination with the standard introduction rules for the conditional and negation is acceptable. But one might independently doubt that. There are a variety of conceptions of the meaning of ‘Def ’ which will have the effect that the deduction theorem fails: for instance, any broadly many-valued set-up will underwrite a failure of the deduction theorem which (i) construes entailment as preservation of a designated value, (ii) regards Def P as designated if P is, but as taking a lower undesignated value than P when P is undesignated, and (iii) regards the conditional as undesignated just when its consequent takes a lower value than its antecedent. One of the interesting points about Fara’s (2004) reconstruction of the paradox is that it obviates the need for conditional-introduction steps.
The Illusion of Higher-Order Vagueness 30.6
537
A R EV E N G E P RO B L E M F O R T H E BU F F E R I N G V I EW
Let’s review the dialectic to this point. In the cases that interest us (the ‘usual suspects’), it is not, claimed Dummett and Russell, possible to eliminate vagueness by annexing a new expression to the borderline cases of a distinction, since the distinctions between items to which the new expression applies and those that fall under either of the original concepts will both remain vague. However it is typically possible so to arrange the elements of a soritical series for a concept φ that an apparently seamless transition is effected from instances of it to instances of some contrary concept, where seamlessness involves that no salient, relevant changes occur between any element of the series and its successor. Higher-order vagueness is meant to provide a natural and plausible explanation of both these putative items of data. Annexure of a new expression to the borderline cases of a distinction never results in precision because the concept to which the term is thereby annexed is itself a vague concept in it own right. Seamless transition is possible because it is possible so to engineer a soritical series that every pair of contrary concepts manifested within it are buffered by borderline cases of their contrast. This in turn requires the failure of the DEF-rule, if sorites paradoxes are not to recur. Where P is vague, DefDef P must in general be logically stronger than Def P, although still vague.²² There are a number of issues on which a fully satisfactory development of the Buffering view would have to elaborate. Three in particular are especially salient. First, it will not do, obviously, just to reject the DEF-rule on the grounds that paradox will otherwise be reinstated. Rather, an explanatory semantics is wanted for the Definiteness operator to underwrite the failure of the rule and explain more generally what form an appropriate proof-theory for the operator should assume. Second, any genuinely explanatory such semantics had better be grounded in further insight into the nature of borderline cases—an insight somehow serving to explain why the borderline cases of any vague distinction are themselves a vaguely demarcated kind. Third, it needs to explained how exactly a finite Sorites series can indeed provide for a seamless transition between incompatible descriptions. It is not enough to gesture at the idea of buffering by borderline cases: we need to be told in detail how a seamless transition may be fully adequately described, according to the Buffering view.²³ I do not believe that the Buffering view can deliver on these obligations. I shall not here, however, further consider what might be done to address the first.²⁴ For the ²² But see n. 21. ²³ This problem—what Mark Sainsbury christened the Transition Question (1992)—for any adequate account of vagueness has not drawn the attention in the literature meted out to other problems of vagueness. It is in effect the issue raised by the Forced March Sorites: the problem of explaining how a competent subject who is charged to give nothing but correct, maximally informative verdicts may respond, case by case, to the successive members of a soritical series without at any point committing himself to some kind of abrupt (and incredible) threshold. If the Buffering view can genuinely provide an account of seamless transition, it will provide the descriptive resources that the hapless subject of the Forced March needs. I shall pour cold water on the prospects—and, in a sense, on the problem—later. ²⁴ For development of some misgivings about the ability of supervaluational approaches, at least, to deliver on this aspect, see Fara (2004).
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second, the notion that the borderline cases of a vague distinction constitute a further vague kind taking a place, so to speak, in the same broad space of possibilities as the poles of that distinction,—this notion is exactly the illusion that I aim to expose. The third issue—the Transition Problem—will occupy us in the next section. The task for this section is to table an argument that, even before any further development is attempted, the Buffering view is susceptible to a new paradox. The paradox is a kind of ‘revenge’ problem, consequent on the possibility—as it appears—of defining a distinct operator of absoluteness in terms of that of definiteness as follows: Abs P is true if and only if each Def n P is true for arbitrary finite n. There seems no reason to contest that such an operator is well defined if Def is, nor that, intuitively, it should have some actual cases of application. Consider, for instance, Kojak, a man microscopic examination of whose scalp—under whatever degree of magnification—reveals no distinction, in point of the presence of hair fibres, from the surface of a billiard-ball. Does it make any sense to suppose that any of Def [Kojak is bald], Def 2 [Kojak is bald], Def 3 [Kojak is bald]. . . . Def n [Kojak is bald],. . . . . . fails of truth or is somehow less acceptable than a predecessor in the series? By its definition, AbsP entails Def P; so in particular any statement of the form Abs(At) entails Def (At), and therefore any statement of the form (∃x)(AbsAx) entails the corresponding (∃x)(Def Ax). Contraposing, any statement of the form, ∼ (∃x)(Def Ax) entails the corresponding ∼ (∃x)(AbsAx). Since any Gap principle for definiteness is—assuming that Def distributes across conjunction and collects conjuncts in the obvious way—equivalent to something of the former form, acceptance of any Gap principle for definiteness is a commitment to acceptance of the corresponding Gap principle for absoluteness. That is all as intuitively it should be. But now observe that, whatever the position with Def, the absoluteness operator, so defined, should be iterative across the conditional.²⁵ So the effect, just provided that the relevant Gap principle is itself absolute, ²⁵ This excellent observation is due to Elia Zardini. Here is a sketch of one plausible demonstration of it: 1 1 1 1
(i) (ii) (ii) (iv)
AbsA Def A & DefDef A & . . . . . .. DefDef A & DefDefDef A & . . . . . Def (Def A & DefDef A & . . . . . ..)
1
(v) (vi) (vii) (viii) (ix) (x)
DefAbsA AbsA → DefAbsA Def (AbsA → DefAbsA) DefAbsA → DefDefAbsA DefDefAbsA AbsA → DefDefAbsA
1
Assumption (i) Definition of Abs (ii) &E (iii) collection for Def over conjunction (iv) Definition of Abs (i), (v) Conditional Proof (vi) Def Intro—see below∗ (vii) Closure of Def over entailment (v), (viii), MPP (i), (ix) Conditional Proof
and so on. Thus each Def n AbsA can be established on AbsA as assumption. AbsAbsA is accordingly a semantic consequence of AbsA.
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and that the relevant polar verdicts are assumed absolute, is to reintroduce a version of the No-Sharp Boundaries paradox. The proof is just the obvious adaptation: 1
(1)
Abs∼(∃x)[AbsAbs(Fx) & Abs∼Abs(Fx )]
2
(2)
Abs∼Abs(Fk )
3 3 2, 3 1 1, 2 1, 2
(3) (4) (5) (6) (7) (8)
Abs(Fk) AbsAbs(Fk) (∃x)(AbsAbs(Fx)) & Abs∼Abs(Fx )) ∼ (∃x)(AbsAbs(Fx)) & Abs∼Abs(Fx )) ∼Abs(Fk) Abs∼Abs(Fk)
1
(9)
Abs∼Abs(Fk ) → Abs∼Abs(Fk)
Assumption—absoluteness of 2nd order Gap principle for Abs Assumption of polar absoluteness Assumption for reductio ((3), iterativity of Abs (2),(4), ∃-intro. (1), Abs-elim. 3,5,6, RAA. 7, iterativity and closure for Abs 2,8 CP.
In sum: The Gap principles may or may not be directly soritical when augmented by whatever may prove to be the appropriate proof-theory for Def. But even if they are not, there seems no objection to introducing the Abs operator as defined, if there is no objection to Def in the first place. If as argued, Abs is iterative, and if it is an absolute truth that a (first-order) borderline case of F is not an absolute case of F, and if the Gap principles for Def are absolute truths (whence those for Abs are also), then the Gap principles for Def do ultimately spawn a Sorites paradox in any case, even if they are innocent of paradox when worked on merely via the appropriate proof theory for Def. 30.7
T H E T R A N S I T I O N P RO B L E M
No doubt, there are lines of resistance for a defender of Gap principles to explore.²⁶ But we must delay no further in attending to a more basic difficulty which has been shadowing the discussion all along and is in the end, I suggest, decisive that the attempt to capture the seamlessness intuition by means of an apparatus of ascending Gap principles, a fortiori by means of limitless Buffering,²⁷ is fundamentally misconceived. Let’s step back. The seamlessness intuition, as interpreted by the Buffering view, has it that in any Sorites series for a concept F, no pair of adjacent elements are ∗ The
principle appealed to is that if | = A, then | =Def A. This should be uncontroversial—presumably all necessary truths are definite. ²⁶ One is to query the status of the minor premises. To treat the reasoning outlined as a Sorites paradox, properly so termed, requires that its conclusion—Abs∼Abs(F0)—confounds an acceptable such premise. Indeed it does if F(0) is absolutely true. But if F(0) were, say, merely definitely true (!), might that not be consistent with its also being an absolute truth that it is not absolutely true? For considerations in this direction, see Williamson (1997a) and Dorr (2009). ²⁷ Which, recall, is classically the same thing.
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characterized by incompatible F-relevant predicates.²⁸ Somehow a seamless transition is effected from (Definiten ) Fs at one end to (Definiten ) non-Fs at the other. The move to an apparatus of Gap principles is a response to this thought which interprets it as requiring that every incompatible pair of predicates, and , formulable using just F, Def and negation, which are exemplified in the series must be buffered—there have to be intermediate elements whose strongest F-relevant characterization is compatible with both and . These are the borderline cases of the − distinction. One direct corollary of this way of handling seamlessness which it is time—rather belatedly—to take proper note of is that if the Basic Formula is to offer a viable characterization of borderline cases, we have to think of ‘∼Def x & ∼Def x’ as compatible with both x and x. So ‘x & ∼Def x’ has to be a consistent description; and hence, it appears, we have after all to take seriously the possibility that there are items which satisfy it—things which while being a certain way, are not definitely that way. Dummett’s Principle has to be repudiated if the Buffering view is to have any chance of delivering seamlessness. And with it goes any Third Possibility interpretation of borderline status. The rejection of Dummett’s Principle can easily seem like nonsense. We might try to set aside that impression as owing to the intrusion of inappropriate resonances associated with the English word ‘definitely’. We are after all, it may be said, introducing a term of art for certain theoretical purposes. But that would be a pretty brass-necked response, given that it was exactly the resonances of the natural language word that made the Basic Formula seem apt in the first place. Be that as it may, the basic problem remains that, even after Dummett’s Principle is surrendered, the idea of limitless buffering in accordance with the Basic Formula, rather than providing for a lucid understanding of the possibility of seamless transition, seems, when pressed, merely to plunge into aporia. The difficulty is best elicited in the context of a version of the Forced March. Suppose you are the subject and that you have returned a correct verdict——concerning element m. If and ∼Def are compatible, then you now have the option of describing m as an instance of the latter without explicit concession of a change in -relevant status. Well and good. Nevertheless since Def is factive, some elements correctly describable as ∼Def will be so because they are . And m had better not be one of those, or the transition from m to m will mark a sharp boundary in the series after all. On the other hand, if m is also ,—as compatibly with its correct description as ∼Def it may, after the jettison of Dummett’s principle, now be—then the buffer zone is merely narrowed by one element and we can push on to m and raise the same possibilities again: is m an instance of ∼Def because it is ?—in which case there is a sharp boundary—or is it also an instance of ?—in which case the buffer zone narrows again. Obviously, the buffer zone must not narrow too far, or there will be a sharp cut-off between and in any case. So it appears that we have to think in terms of there being cases which are correctly describable as ∼Def but not because they are , and which also—if narrowing of the buffer zone is to be halted—do not exploit the ²⁸ Recall that a predicate is F-relevant if it is formulated using just F, negation, conjunction and the definiteness operator.
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compatibility of and ∼Def by being . These cases will constitute a distinctive kind of borderline case between and : cases that qualify for characterization in terms of the Basic Formula without exploiting the compatibility, after the surrender of Dummett’s Principle, of with ∼Def and of with ∼Def . It is essential that such cases occur if a seamless transition is to be effected. For if they do not, each case within the region characterized as ∼Def and ∼Def , will either be and or . So to solve the transition problem, you—the subject—need to be provided with the means in principle, whatever epistemological difficulties you might encounter in practice, to mark the occurrence of such cases. But how can that be done? This is already a fatal objection to the prospects for solving the Transition Problem using the resources at hand, since we now appear to be committed to recognizing a kind of indeterminacy for which the apparatus of -relevant and -relevant predicates and the Basic Formula provides no adequate means of expression—cases whose description in accordance with the Basic Formula masks their distinction from others which it also characterizes but which are, so to say, tacitly polar. There is therefore no prospect of your doing justice to seamless transition using just the notion of buffering by borderline cases, conceived in accordance with the Basic Formula, since we have given you no resources adequately to characterize the masked cases. But even if we had, a second lethal consequence looms large. In order to preserve seamlessness, we now need to avoid the postulation of a sharp boundary between a last and a first exemplar of this new genre of indeterminate cases, the non- tacitly polar instances of the Basic Formula applied to and (let’s call these the ’s.) So, on the Buffering view, we now need in turn to buffer the contrast between and , however exactly the instances of the latter are to be described. But strategically, the means at our disposal are just the same as—and hence no better than—those just deployed for the – distinction,—except that now, of course, there are fewer elements to subserve the buffering of the distinction, since the – series is shorter than the – one. Since exactly the same form of problem is going to recur at every stage and the series is finite overall, the strategy cannot succeed. The root of the trouble is that there is, simply, no satisfactory conception of what a borderline case is that is serviceable for the explanation of seamlessness. Obviously no ‘third possibility’ conception is to the purpose: if one is trying to explain seamless transition between contrasting situations, it doesn’t help to interpose a third category of situation contrasting with both. But if, recoiling from that, we essay to think of the interposed category as compatible with each of the originally contrasted statuses (so dropping Dummett’s principle), then in assigning an object to that category we fall silent concerning what if any shift from polar status it instantiates. To fall silent, is not to explain anything. Moreover, when pressed, as we saw, it seems we are forced to postulate a ‘Third Possibility’ type of case—-cases—after all. At which point, the game is effectively lost. We should conclude that there is no prospect of a stable elucidation of seamless transition by means of the conception of an endless hierarchy of orders of borderline cases. So far from being well motivated by the possibility of seamless transition between instances of incompatible vague predicates, the Buffering view winds up in compromise and confusion.
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Where does that leave the Transition problem? Well, it is striking that the kind of difficulty just outlined will afflict any attempt to do justice to the nature of the changes, stage by stage, involved in a process of seamless transition across a finite series of stages between contrary poles. It has nothing especially to do with vagueness or our having recourse to the notion of a borderline cases. For suppose we have somehow turned the trick: we have somehow succeeded in fully correctly describing, stage by stage, a process of seamless transition. We will have had to say incompatible things about some of the stages. Let m and n be a pair where we did that and which are as close together as any pair where we did that. They will not have been adjacent. Let F be the description given of m, and G that given of n. So m will have received a verdict, F , compatible with both F and G. Is F true of m ? If it is, then G isn’t. So, since compatible with G, F doesn’t do full justice, in relevant respects, to m , even if true of it. So if we did somehow do full justice to all the stages, F cannot be true of m . But then the series wasn’t seamless after all: there is a sharp boundary at m. Conclusion: the Transition problem is insoluble in any vocabulary if the ‘full justice’ requirement is enforced. So far from demanding recourse to a baroque apparatus of borderline cases of arbitrarily high orders, the requirement that seamless transition somehow allow of a fully adequate description, stage by stage, was unsustainable all along. When the task is to explain how seamless transition is possible in a way that involves doing full justice, in all relevant respects, to the elements in a finite series that manifests as effecting such a transition, it is about as helpful to believe in higher-order vagueness as to believe in fairies. Dissatisfaction may persist. Forget about doing full justice to seamless transition. Don’t we at least have invoke concepts of higher-order vagueness and buffering if we are to describe the relevant kind of series in a fashion consistent with seamless transition, even if the description does not do full justice to it? Well, no. Once the ‘full justice’ requirement is relaxed, and we need merely to avoid adjacent incompatibilities, we can perfectly well describe the stages of a seamless transition, without misrepresentation, using only precise vocabulary. Suppose Johnny grows seamlessly from 5 feet tall to 6 feet tall between his fourteenth and eighteenth birthday and consider a series of appropriately dated true descriptions: Johnny is now exactly 5 feet tall Johnny is now exactly 5 feet tall, give or take an inch Johnny is now exactly 5 feet 1 inch tall Johnny is now exactly 5 feet 1 inch tall, give or take an inch . . . and so on. If the ‘full justice’ requirement is in force, the spandrel-plagued apparatus of the Buffering view is to no avail; if the requirement is not in force, and we are allowed to give less than all relevant information, it is easy to turn the trick without involving anything of the kind. One last try. Notice that when the admissible substitutions for ‘F’ are restricted to predicates in the range used in the example in describing Johnny’s changing height,
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the result is not, of course, to provide a model of the original no-sharp boundaries principle, (i) ∼ (∃x)(Fx & ∼Fx ) —since for any choice of F in the range of predicates concerned, there will be a last case of which it is true. By contrast, isn’t it forced on us that each of the hierarchy of Fara’s Gap principles is true in a finite series exemplifying seamless transition between instances of contrary vague concepts? If so, then at least from a classical point of view, that enforces acceptance of the hierarchy of borderline kinds, even if we are thereby no better placed when it comes to doing justice to the phenomenon of seamless transition. But this has to be a bad thought. If, after we introduce the Definiteness operator, seamlessness enforces the Fara Gap principles, then before we introduced the Definiteness operator, it already enforced the major premise of the No-Sharp-Boundaries paradox. What we considered earlier was an argument, impressive in the context of classical logic, that (i) is not an adequate capture of F’s vagueness, which is rather canonically expressed by (ii) ∼ (∃x)(Def Fx & Def ∼Fx ). Let that conclusion stand. Then the vagueness of F, qua canonically expressed by (ii), does not impose (i). But nothing has been done to disarm the impression that the seamlessness of the relevant transition does. That is another matter. If seamlessness enforces the higher-order Gap principles, it enforces (i) too, and the No-Sharp boundaries paradox re-arises as a paradox of seamlessness. There are two directions on which to look for a response to the situation. One, proposed recently by Fine,²⁹ is to restrict the underlying logic of negation in such a way as to block the ‘right-to-left’ reasoning of the No-Sharp-Boundaries paradox. In that case, (i) and the members of the hierarchy of Gap principles will all be acceptable as mandated by seamlessness, however inchoately understood. But the needed weakening of the logic of negation is apt to impress as hugely counterintuitive, indeed as a betrayal of principles that are constitutive of the notion of negation. My own preference, accordingly, is to explore the thought that relevant instances of ‘unpalatable existential’ claims of the form, (∃x)(Fx& ∼ Fx ), are rendered ungrounded, rather than false, by the phenomenon of seamless transition, which is therefore in urgent need of a less inchoate understanding, and ²⁹ In his monograph [in progress]. Fine rejects the rule of ‘Conjunctive Syllogism’: A, ∼ (A&B) ∼B and therefore the intuitionistically acceptable half of classical reductio: , A ⇒ ⊥ ⇒∼ A
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that F’s vagueness in the relevant series likewise renders the unpalatable existential ungrounded. I have no space here to pursue these suggestions.³⁰ In any case, enough has been done, I trust, to discredit the Seamlessness intuition as a motive for the Buffering view.
30.8
THE INERADICABILITY INTUITION ONCE MORE
It remains to re-scrutinize the ineradicability intuition, expressed in rather different ways by Dummett and Russell. Both implicitly started from the idea of the vagueness of the borderline between and as consisting in a region of uncertainty—a ‘penumbra’ in Russell’s seminal image—and envisaged an additional stipulation to try to bring this region under linguistic control: a new predicate in Dummett’s case, a moratorium on description in Russell’s case.³¹ Both then simply asserted—plausibly but, notably, without any argument whatever—that the proper application of the new stipulation would itself be vague: that there would be cases where it would be uncertain how to apply the new term, or whether they fell within the scope of the moratorium. The assertion is plausible. But it should, on reflection, seem puzzling why it is plausible. The claim that there are borderline cases of a certain concept is, after all, partly an empirical sociological claim: to make it is to predict that possessors of the concept will not react with verdicts about its application that collectively converge on a sharp distinction between positive and negative cases. How do Russell and Dummett know this in advance, sitting in their armchairs? Who is to say that, after ‘eminence’, for instance, was introduced in the manner Dummett envisages, we would not in fact respond with a stable, consensual practice converging on an agreed range of applications for all three concepts—hill, eminence and mountain—and responding in no case with the characteristic manifestations of vagueness? So why is our reaction to the ineradicability claim not, ‘How do you know? What’s the evidence?’ Why don’t we feel it necessary to leave the armchair and try it out and see? The answer, presumably, is that we think we know already what the outcome of an experiment would be. But why do we think that?—It is not, after all, as if we have often made stipulations of the Dummett–Russell sort and experience has taught that they do not work. I suggest that the explanation of the armchair plausibility has to do with a sense of the limited guidance that the envisaged kind of stipulation would be able to give us. ³⁰ Wright (2001), (2004) and (2007) offer argument in some detail that acceptance of a predicate’s vagueness need not involve denial of a relevant unpalatable existential, i.e. endorsement of an instance of (i). Those arguments, if effective, equally militate against acceptance of higher-order Gap principles as a response to the vagueness of the predicates concerned. I have not elsewhere attempted to explain why seamlessness, properly understood, should not motivate acceptance of Gap principles. But the basic point that I believe that a proper treatment should develop is that seamlessness is an epiphenomenon of our discriminative limitations. It is merely a projective error to read it back into the characterization of the elements in a seamless series. ³¹ A third move in the same spirit would be to extend , if it is the complement of ,—or in any case, to extend the sphere of application of one of the concepts concerned.
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In going along with the prediction of uneliminated vagueness, we are reporting something about our own sense of limitation in response to the kind of stipulation hypothetically envisaged; the phenomenon is broadly—not exactly—of a piece with the ability to predict uncertainty in your application of rules which you know you have only partially understood; or the ability to knowledgeably say ‘No’ to the question, ‘Do you understand?’, when what is at issue is competence for some form of subsequent task. Our sense is that, in contrast to the corresponding Dummettian, or Russellian, stipulation for cases like ‘dommal’ or ‘pearl’, we are not clear enough about which the borderline cases are —which are the cases to trigger the stipulation—to be confident in general how to apply it. The key is to see that this uncertainty does not demand explanation in terms of the idea of higher-order vagueness. I’ll enlarge on that diagnosis in a moment. First, we need to consider an objection to the alleged connection between the ineradicability intuition and higher-order vagueness that that was prefigured at the beginning.³² The objection is that an additional presupposition is required before any connection with higher-order vagueness is even apparent. That presupposition is that the introduction of a linguistic stipulation of the kind envisaged by Russell and Dummett will have no impact on the identity of the concept——whose borderline cases it aims to provide means of denoting or otherwise differentially treating. This presupposition is actually quite implausible. Consider a small child tidying up his play-bricks, so far without any colour words save ‘red’ ‘blue’ ‘green’ and ‘yellow’, who is told to put the reds into one bin and the blues into another, although the bricks include many shades of red, blue, mauve, purple, pink, orange and so on. It seems quite expectable that he will place many reddish purples and bluish purples, for instance, in the red and blue bins respectively which, if we were to single out a few royal purple bricks and others of similar shades, and give him the word and a new bin with the instruction to tidy the purples into it, he would then prefer to house there. In general, it is to be expected that provision of the resources to mark an intermediate category will have the effect of disturbing—narrowing—the accepted extensions of the concepts which flank it to include fewer uncomfortable cases, and thereby of modifying the original concepts themselves. But if the effect of regulating the response to the borderline cases would be to modify the concepts concerned, then the ineradicability intuition provides no argument for thinking of them as being even second-order vague—rather we have a situation where the introduction of the new resources afforded by a Dummett/Russell stipulation merely generates three new concepts which then exhibit ordinary—first-order —vagueness in relation to each other. This is an important point. But I do not think that, on its own, it takes us to the heart of the issue. There is a second questionable assumption at work in Dummett’s and Russell’s line of thought—an assumption which indeed is still unchallenged even in the point just registered. It is the assumption that that the invitation to annex a new word to the borderline cases of a distinction, or to respond to them with a moratorium on classification, or some other kind of new, distinctive treatment, is in general one that can so much as be taken up. In order to respond to such an invitation, one ³² See n. 2 above.
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must first be able to corral the borderline cases—those, after all, are the only cases to which the new practice, whatever it involves, is to be applied. The question this goes past is whether the reactions that characteristically manifest the borderline status of a case involve the exercise of a concept somehow contrasting with the polar concepts; or whether what they betray is, rather, a subject’s difficulty in bringing it under one of the polar concepts—a ‘drying of the springs of opinion’, a slide into Quandary.³³ If it is the latter, then the reason why the invitation will not have the effect of generating precision—a new, sharply tripartite practice of some kind—is not because the separation between the cases to which the new convention is to apply and the rest is itself vague on both borders, but because we have no settled concept of those cases in the first place. We need to go carefully here. I am not, of course, denying that there is such a thing as the judgement that a case is borderline,—denying that we have any concept of what it is for a colour, for instance, to be a borderline case of red and orange. The question is: what is the content of such a judgement? Does regarding a case as borderline red-orange involve bringing it under a concept that competes, so to speak, within the same determinable space as the relevant polar concepts, red and orange? If so, it’s force, like theirs, will be normative and exclusive. The judgement will imply, e.g.: ‘Here you should not take either polar view—the case is too far removed from the clear cases of red and orange.’ Or is the judgement, rather, something that does not involve the application of a competitor concept in that way? It might, for example, be best interpreted as a projection of the characteristic phenomenology of attempted judgement in the particular case, so that its force is broadly sociological: say, ‘Here competent people in excellent epistemic position still have weak and unstable views, struggle to come to a view, etc.’ The difference is critical. The roots of the Buffering view of higher-order vagueness, when motivated by ineradicability, lie entirely in the former way of thinking. That may be fine for some cases—typified by the example of purple and the child’s toy bricks. But it cannot be the way to think about the general run of mutually vague concepts. Borderline cases of a vague distinction, − , are not in general things that form a kind unified under a concept that stands to the poles, and , as purple stands to blue and red. In all cases, the borderline region is indeed, as Russell stresses, one of uncertainty where we struggle to bring elements under either polar concept—but where basic vagueness is concerned, this is for reasons that have nothing to do with there being a third concept of the same broad kind, a competitor with the originals in the same determinable space, which seems preferable to both. When there is such a third concept, the invitation to annex a new word to it, or some other practice, will be intelligible enough. But the range of cases on the borders of this concept and the two originals will, again, be likely to defeat our powers of conceptualization—or if they do not, iteration of the process will anyway bring us eventually to mutual distinctions for which the model of purple, the model of an intervening kind, gives out. At that point, the reason why we will not be able to eradicate vagueness by proposing a differential form of classification, or treatment, of the ³³ I mean this notion only in an intuitive sense here, though the remark just made will bear interpretation in terms of the more specialized sense of ‘quandary’ developed in Wright (2001).
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borderline cases will not be because the concept—borderline case of and —that would control the new practice will itself be vague, but because we have no concept of such borderline cases that we can exercise in contradistinction to and , as we can exercise purple in contradistinction both red and blue. When borderline cases are exactly things that defeat our ability to apply any of the relevant concepts, borderline case of the − distinction is nothing we can regulate a new practice by. This is the point of connection, suggested above, with the phenomenon of avowably imperfect understanding. The reason why it may be confidently predicted that a Russell/Dummett stipulation will not have the effect of introducing precision is indeed broadly comparable to the reason why I can be confident that I will not be able to give the right answers when applying a rule I realize I have imperfectly understood. (Of course, in both cases there is the bare possibility that I will surprise myself.) Simply: I do not know how to apply such a stipulation because I lack any stable concept of the kind of cases which are meant to trigger it. My characteristic reaction to such cases is one of a failure to bring them confidently under either polar concept, but not because I am clear that I should bring them under neither. I do not, precisely, grasp them as a third kind. But that is exactly what I would need to do in order to be able to work the stipulation in a stable, discriminating way. Since I am not able to form a settled view about whether they are cases of the sort for which the new stipulation is not called for: that is, cases of or of , I cannot be confident about when to invoke the new stipulation. Again: if one’s characteristic reaction in the borderline area is a ‘drying of the springs of opinion’—an inability to bring a case under either polar concept that is not associated with a better alternative,—then of course the invitation to introduce a new predicate, covering cases whose status is to contrast with polar cases, will not result in clear guidance, let alone precision; that is, in confident and complete classifications across the range. The content of the quandary was precisely whether to apply a polar concept and if so which. So the invited new predicate, or new policy, the application of which will pre-empt either original polar judgement, will be bound to inherit that quandary. There is, as we noted, what we might term the sociological option: to annex a sociological conception of the borderline cases of a distinction to a stipulation of the Russell/Dummett kind. (In the case of a single judge, ‘borderline case’ will then become a concept grounded in his own characteristic psychological reactions.) But the obvious point to make in that case is that no such conception of the borderline cases of gives any literal sense to the idea of the boundary between the s and the borderline s being vague. As a first approximation: if the content of a judgement that a case is borderline is broadly sociological, or psychological, then whereas in judging that a case is , we are making a judgement about the case, in judging that a case is borderline , we are recording a judgement about us; so the idea that this distinction might itself be vague is incoherent—mutual vagueness requires a common domain of predication. I have been suggesting that it is a fundamental error to think of the borderline cases of a vague distinction as if they were shades of purple and the given distinction were like that between red and blue. Entrenched though the error is, it takes only a little
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reflection to see that this cannot be the nature of the general run of cases. In particular, it cannot be the nature of the distinction between the s and the non-s. Even setting that case to one side, there is an intuitive notion of adjacency for vague concepts that compete in a single space—in the way that red and orange, for example, or blue and purple are adjacent in colour space, or moderately uncomfortable and painful, perhaps, are adjacent in the space of sensations. Intuitively, when you move from red in the direction of yellow, the next thing you come to is orange. Where concepts are adjacent in this intuitive sense, we will have no third competitor concept to characterize a buffer zone between them, in the way in which purple buffers the blues and reds. We may indeed be able to master a narrower concept that applies in the borderline area (for example, blood orange), but this will not compete with the originals (red and orange) as they compete with each other. It will be open whether it is a determinate of either. And if we make it clear that it is not to be so viewed, and annex a word to it, the result will be the narrowing phenomenon we noted above. The root error in the Buffering view is to think of borderline cases as instances of what I have elsewhere called Third Possibility. I have given other arguments against that broad conception and will not rehearse them here.³⁴ The ineradicability intuition is indeed a commitment to the Buffering view when taken under the aegis of Third Possibility. And the lesson to learn is that the inference of buffering from ineradicability goes wrong by—draws the wrong conclusion as a result of—passing over a conception of mutually vague concepts not as demarcated from their neighbours by a borderline area conceived on Third Possibility lines but as, though adjacent—there is nothing of any other kind that separates them—characterized by the inability of those who have mastered the concepts concerned to run them right up against each other in stable judgement. The conflation of these two ideas—the failure to see that the second (the inability to run the extensions up against each other) does not require the first (a sensitivity to an intervening kind)—is the cardinal source of the illusion of second-order blurred boundaries. The second is the idea that Mark Sainsbury gestures at when he speaks of boundaryless concepts.³⁵ But I do not think the point of that perceptive piece of terminology has been generally understood. Re f e re n c e s Beall, Jc, ed. (2004), Liars and Heaps: New Essays on the Semantics of Paradox, Oxford, Oxford University Press. De Clercq, R. and Horsten, L. (2004), ‘Perceptual indiscriminability: In defence of Wright’s proof’, Philosophical Quarterly 54, 439–44. Dorr, C. (2009), ‘Iterated determinacy’, this volume, Chapter 31. Dummett, M. (1959), ‘Wittgenstein’s philosophy of mathematics’, Philosophical Review, 68, 324–48; reprinted in Dummett (1978) at 166–85. (1975), ‘Wang’s paradox’, Synthese 30, 301–24; reprinted in Dummett—(1978), at 248–68. (1978), Truth and Other Enigmas, London, Duckworth. ³⁴ For elaboration, see Wright (2001), (2004), and [forthcoming].
³⁵ Sainsbury (1990).
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Fara, D. G. (2001), ‘Phenomenal continua and the sorites’, Mind 110, 905–35. Published under the name ‘Delia Graff’. (2004), ‘Gap principles, penumbral consequence, and infinitely higher-order vagueness’ in Beall, Jc, ed. (2004), 195–221, Published under the name ‘Delia Graff’. Fine, K. (1975), ‘Vagueness, truth and logic’, Synthese 30, 265–300. (In progress), The Possibility of Vagueness. Heck, R. (1993), ‘A note on the logic of (higher-order) vagueness’, Analysis 53, 201–8. (2004), ‘Semantic conceptions of vagueness’ in Beall, Jc, ed. (2004), 106–27. Hyde, D. (1994), ‘Why higher-order vagueness is a pseudo-problem’, Mind 103, 35–41. (2003), ‘Higher-orders of vagueness reinstated’, Mind 112, 301–5. Keefe, R. and Smith, P., eds. (1993), Vagueness: A Reader, Cambridge, MA, Bradford MIT Press. McGee, V. and McLaughlin, B. (1995), ‘Distinctions without a difference’, The Southern Journal of Philosophy, supp. vol. 33, 203–51. Russell, B. (1923), ‘Vagueness’, The Australasian Journal of Psychology and Philosophy, 1, 84–92; reprinted in Keefe, R. and Smith, P., eds. (1996), 61–8. Sainsbury, M. (1990), ‘Concepts without boundaries’, London, King’s College, Inaugural lecture, reprinted in Keefe and Smith, eds. (1996), 251–64. (1991), ‘Is there higher-order vagueness?’ Philosophical Quarterly, 41, 167–82. (1992), ‘Sorites paradoxes and the transition question’, Philosophical Papers 2, 177–89. Soames, S. (2003), Understanding Truth, New York, Oxford University Press. Sorensen, R. (1985), ‘An argument for the vagueness of ‘‘vague’’ ’, Analysis 27, 134–7. Varzi, A. (2003), ‘Higher-order vagueness and the vagueness of ‘‘vague’’ ’, Mind 112, 295–9. Williamson, T. (1990), Identity and Discrimination, Oxford, Basic Blackwell. (1994), Vagueness, London: Routledge. (1997), ‘Imagination, stipulation and vagueness’, Philosophical Issues 8, Truth, 215–28. (1997a), ‘Reply to commentators: (Horgan, Gomez-Torrente, Tye)’, Philosophical Issues 8, Truth, 255–65. (1999), ‘On the structure of higher-order vagueness’, Mind 108, 127–43. Wright, C. (1975), ‘On the coherence of vague predicates’, Synthese 30, 325–65. (1987), ‘Further reflections on the sorites paradox’, Philosophical Topics 15, 227–90. (1992), ‘Is higher order vagueness coherent?’ Analysis 52, 129–39. (2001), ‘On being in a quandary: Relativism, vagueness, logical revisionism’, Mind 110, 45–98. (2003), ‘Vagueness: A fifth column approach’ in Jc Beall, ed. (2004), 84–105. (2007), ‘ ‘‘Wang’s paradox’’ ’ in The Philosophy of Michael Dummett, The Library of Living Philosophers vol. 31, ed. Auxier, R. and Hahn, L., Chicago, Open Court, 415–44. (forthcoming), ‘On the characterisation of borderline cases’ in Meanings and Other Things: Essays on Stephen Schiffer, ed. Ostertag, G., Cambridge, MA, MIT press.
31 Iterating Definiteness Cian Dorr
31.1
PRELIMINARIES
A central concept in the study of vagueness is the concept of a borderline case. This concept has its most basic application when we are faced with a question of the form ‘Is x F ?’, but are unwilling to answer ‘Yes’ or ‘No’ for a certain distinctive kind of reason. Wanting to be co-operative, we need to say something; by saying ‘It’s a borderline case’, we excuse our failure to give a straightforward answer while conveying some information likely to be of interest to the questioner. The nature of the considerations that make us unwilling to answer ‘Yes’ or ‘No’ in these cases is a topic of central importance in the philosophy of vagueness. Different views about this naturally lead to different answers to the question what it means to be a borderline case. Before we can even broach this question, we need to settle on a way of regimenting borderlineness-talk. There are two main approaches. On the first approach, the basic notion is metalinguistic in character, so that the task is that of making sense of locutions like these: Sentence S is borderline as used by community C. Sentence S is borderline as used by community C at possible world w. x1 , . . . , xn is a borderline case of predicate as used in contexts of type T by community C at time t at possible world w. On this approach, the philosophy of vagueness is clearly a branch of the philosophy of language. This is less clear on the second approach, which regiments ‘borderline’ as an operator (see Fine 1975: 148 ff). On this view, ‘It is borderline whether P’ is no more a claim about language than ‘It is contingent whether P’. The fact that it is borderline whether P, if it is a fact, is not a fact especially about any particular community or any particular linguistic expression; it can be expressed equally straightforwardly in many different languages.¹ ¹ Perhaps we should think of ‘it is borderline whether P’ as ascribing a property, borderlineness, to a proposition, the proposition that P, or some kindred abstract, non-linguistic entity. Or perhaps we should resist such attempts to impose a subject-predicate structure on sentences constructed using operators, as Prior did for modal and temporal operators (Prior 1968). This is an interesting
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The dispute is about priority: there is no reason for partisans of either approach to reject the vocabulary of the other approach as unintelligible. Suppose we already understand ‘it is borderline whether’ and the notion of a sentence being true as used by a community at a world. Then we can analyse ‘S is borderline as used by C at w’ as ‘It is borderline whether S is true as used by C at w’ (cf. Fine 1975: 296). Conversely, if we already understand ‘S is borderline as used by C at w’, we could define the borderlineness operator by stipulating that it is borderline whether should be synonymous with ‘’ is borderline as used by us at the actual world (when is a closed sentence) or v1 , . . . , vn is a borderline case of ‘’ as used by us at the actual world (when is an open sentence with free variables v1 . . . vn ).², ³ It is customary and convenient to treat ‘it is borderline whether’ as defined in terms of a ‘definitely’ operator, with ‘it is borderline whether P’ analysed as ‘Not definitely P and not definitely not-P’—in symbols, ¬P ∧ ¬¬P. We could either treat this as a first step in the analysis of ‘it is borderline whether’, or—if we prefer doing things the other way round—as an elementary logical consequence of the analysis of ‘definitely P’ as ‘P and it is not borderline whether P’. On the metalinguistic approach, it is similarly traditional to analyse borderlineness in terms of truth and falsehood: ‘S is borderline as used by C at w’ is analysed as ‘S is neither true nor false as used by C at w’. This is a controversial move: it is not so clear that our intuitive notions of truth and falsehood for sentences behave as they would need to behave for this analysis to be tenable (Williamson 1994: section 7.2). But those who are wary of this analysis should at least agree that there are two distinctively different ways in which a meaningful sentence can fail to be borderline, even if ‘true’ and ‘false’ aren’t the right labels for these ways. To preserve neutrality I’ll speak of subsidiary dispute, but as far as our overall conception of the place of vagueness in the scheme of things is concerned, the dispute between the metalinguistic and non-metalinguistic approaches is more central. ² I use Quine’s corner-quotes (Quine 1940). Since ‘’ is a variable ranging over linguistic expressions, ‘‘’ is borderline as used by us at the actual world’ is synonymous with ‘ ‘‘the result of writing‘‘ ‘ ’’ and then and then ‘‘ ’is borderline as used by us at the actual world’’ ’. ³ Opponents of the metalinguistic approach will complain that the operator defined in this way doesn’t interact in the right way with ambiguous expressions: it is borderline whether S should be ambiguous whenever S is, but if ‘borderline’ and quote-names are not ambiguous, ‘S’ is borderline as used by us at the actual world will never be ambiguous. One could attempt to fix this by treating ambiguity as homonymy, so that, e.g. the quote-name ‘ ‘‘Some banks are closed’’ ’ is ambiguous, referring on two different disambiguations to different linguistic entities. But it is not clear how this could work. If we are to understand borderlineness as a feature of a sentence’s use, we had better take sentences to be entities that can be used in different ways, with different meanings, by different communities at different possible worlds. This makes it hard to give an account of what the difference between the putatively distinct items named by ‘‘ ‘Some banks are closed’ ’’ could be, or of what would make it the case that one rather than the other of them was used by a given community. Moreover, adherents of the metalinguistic approach are liable to hold that ambiguity and vagueness are kindred phenomena, which should be treated in a unified fashion at the most fundamental level of theorizing. (For a radical version of this, see Braun and Sider 2007.) If so, even if we can make sense of a notion of sentences as uninterpreted but disambiguated, they will be entities of at best secondary theoretical importance. Nevertheless, it may be possible to find a way of talking about them that is adequate to the task of analysing ‘it is borderline whether’ so that it conforms to the rule that it is borderline whether S inherits any ambiguity in S.
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‘dtruth’ and ‘dfalsehood’. If you like, you can pronounce ‘dtrue’ as ‘definitely true’, and ‘dfalse’ as ‘definitely false’; but of course only followers of the operator approach will want to analyse these notions as the result of applying the ‘definitely’ operator to antecedently understood notions of truth and falsity. Followers of the metalinguistic approach should instead—at least as a first approximation—analyse Definitely S as ‘S’ is dtrue as used by us at the actual world. Both conceptions of the relation between ‘definitely’ and ‘dtrue’ vindicate the following principle, which we will need to refer back to later: T- For any sentence S, S is dtrue as used by us at the actual world iff definitely, S is dtrue as used by us at the actual world. On the operator approach, analysing ‘dtrue’ as ‘definitely true’, instances of T- can be derived from instances of the T -schema strengthened by a ‘definitely’ operator: Definitely: ‘S’ is true as used by us at the actual world iff S.⁴ On the metalinguistic approach, we can argue as follows, appealing to the analysis of (v) as ‘’ is dtrue of v as used by us at the actual world: (1) For any sentence S, S is dtrue as used by us at the actual world iff ‘S’ is dtrue as used by us at the actual world is dtrue as used by us at the actual world. (2) For any expression and unary predicate F , ‘’ is F is dtrue as used by us at the actual world iff F is dtrue of as used by us at the actual world. (3) So for any S, S is dtrue as used by us at the actual world iff ‘dtrue as used by us at the actual world’ is dtrue of S as used by us at the actual world. (4) So for any S, S is dtrue as used by us at the actual world iff definitely, S is dtrue as used by us at the actual world.⁵
31.2
INFINITE DEFINITENESS
Once we have introduced the ‘definitely’ operator in one way or another, it becomes natural to think about stronger operators defined by iterating it. We have the sequence , , . . . , i , . . . . And there are various ways we can introduce something like an infinite limit to that sequence. The most straightforward is to use infinitary conjunction, defining ω S as S ∧ S ∧ S ∧ . . . ∧ i S ∧ . . .. In my view, this is legitimate: while English is not itself an infinitary language, the ellipsis ‘. . .’ ⁴ If the semantic paradoxes force us to reject the claim that all instances of the T -schema are true, this won’t be enough for a general argument for T-. But since the semantic paradoxes don’t undermine T- itself, followers of the operator approach will presumably want to hold onto it in any case, just as with analogous Tarski-style principles for other operators. ⁵ This argument should still go through even if we reject the proposed analysis of Definitely S in favour of something subtler, in order to accommodate the claim that Definitely S inherits the ambiguity of S. For since ‘dtrue’ is not itself ambiguous, (1) will remain plausible, and the inference from (3) to (4) will remain valid, whether we understand the notion of dtruth applied to perhaps-ambiguous sentences as requiring dtruth on all disambiguations or merely on some.
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lets us express in English some definitions that would otherwise be formulable only in an infinitary language. Those who are suspicious of infinitary conjunctions can use quantification to achieve more or less the same effect. Say that S is ultratrue as used by C at w iff every finite definitization of S is dtrue as used by C at w, where a finite definitization of S is a sentence that consists of S preceded by zero or more ‘’s. As an alternative to an infinitary conjunction, we could define ω S as ‘S’ is ultratrue as used by us in the actual world.⁶ On either definition, the operator ‘ω ’ is a puzzling one. One source of puzzlement is the apparent validity of the following schema: Def
ω P → ω P
If we define ω using an infinite conjunction, each instance of Def will follow from Dist, which is the obvious extension to the infinitary case of the principle that definiteness distributes over conjunction: Dist
(P1 ∧ P2 ∧ . . .) ↔ (P1 ∧ P2 ∧ . . .)7
If on the other hand we define ω S as ‘S’ is ultratrue as used by us at the actual world, we can argue for Def by appealing to T-: (1) (2) (3) (4) (5) (6)
Every finite definitization of S is dtrue as used by us at the actual world (premise). T is a finite definitization of S (assumption). T is a finite definitization of S (definition of ‘finite definitization’). T is dtrue as used by us at the actual world (1, 3). Definitely, T is dtrue as used by us at the actual world (4, T-). Every finite definitization of S is definitely dtrue as used by us at the actual world (2–5). (7) If there are some things which are definitely all and only the F s, then every F is definitely a G iff definitely, every F is a G (premise schema).⁸ (8) There are some things that are definitely all and only the finite definitizations of S (premise).⁹ (9) Definitely, every finite definitization of S is dtrue as used by us at the actual world (6, 7, 8).
⁶ We could equally well have used ‘true’ instead of ‘dtrue’ in this definition; an argument similar to the one below shows that the definition with ‘true’ entails the one with ‘dtrue’. ⁷ Use conjunction elimination to get from ‘P ∧ P ∧ P ∧ . . .’ to ‘P ∧ P ∧ . . .’, then Dist to get ‘(P ∧ P ∧ P ∧ . . .)’. ⁸ (7) is a consequence of the principle that definiteness commutes with universal quantification: ∀x((x)) ↔ ∀x((x)). But unlike that principle, (7) leaves room for vagueness deriving from the quantifiers as well as vagueness derived from the predicates. ⁹ I assume that any vagueness in the reference of quote-names—which abstract entity, precisely, does ‘ ‘‘’’ ’ refer to?—can be harmlessly ignored.
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The key premise here is (7), which does the same work in this argument that Dist did in the previous argument, and seems similarly plausible.¹⁰ Def makes a certain kind of trouble for someone who thinks that ultratruth is a common phenomenon. Let’s say that x is ultrabald iff ω (x is bald). By Def, whenever one is ultrabald, one is definitely ultrabald. So whenever it is borderline whether one is ultrabald, one is not ultrabald. This is not yet to say that ‘ultrabald’ is precise, or that no one is borderline ultrabald. To show that, we would also need an argument that everyone who is not ultrabald is definitely not ultrabald.¹¹ But it does mean that we won’t be able to use the notion of a borderline case in the usual way to excuse our failure to give straightforward ‘Yes’ or ‘No’ answers to questions about ultrabaldness. If I am asked ‘Is so-andso ultrabald?’ and for some reason I don’t want to commit myself to the extent of saying ‘Yes’ or ‘No’, I should be just as unwilling to say ‘He’s a borderline case’; if I say this, I will have asserted something at least as strong as what I would have asserted by saying ‘No’. You will not be embarrassed by this if you think you know some precise necessary and sufficient condition for ultrabaldness. Otherwise, you may find it hard to respond co-operatively to questions about ultrabaldness without being able to appeal to borderlineness in the usual way. Inevitably, there will be cases where you will be unwilling to answer ‘Yes’ or ‘No’ to the question ‘Is this person ultrabald?’, no matter how much you might learn about the relevant precise facts. What should you say, given that saying ‘Borderline’ would commit you to saying ‘No’? Should you answer randomly? Should you just remain silent? These options are hardly consistent with the standard of co-operativeness to which you are trying to hold yourself. You will be tempted to say ‘I don’t know’. This is, after all, what we standardly say when we want to be co-operative but don’t want to give a straightforward answer to a question. But there are various reasons why we might be uncomfortable with such a response. In other work (Dorr 2003) I have argued that in many ordinary cases where it is borderline whether P, and one is reasonably well informed about the relevant underlying facts, it is borderline whether one knows that P. On this view, unless we can identify some special reason why knowledge would be harder to come by in ¹⁰ I don’t mean to suggest that Dist or (7) is beyond dispute. Hartry Field (2003b, 2008) has recently argued for rejecting such principles, on the grounds that doing so makes available a resolution of the semantic paradoxes that preserves the full intersubstitutivity of φ and True φ (where φ denotes the G¨odel number of φ) while validating the inferences (i) P | P and (ii) P → ¬P | ¬P. Where Qω is ¬ω True( Qω ), we have True( Qω ) → ¬ω True( Qω ) by intersubstitutivity, and hence ω True( Qω ) → ¬ω True( Qω ) by the factivity of ω . It follows by (ii) that ¬ω True( Qω ). Given Def, we could infer from this to ¬ω True( Qω ), i.e. to Qω itself. Then intersubstitutivity would give us True( Qω ), repeated applications of (i) would yield k True( Qω ) for each finite k, and finally, by an infinite conjunction introduction, we would have ω True( Qω ): a contradiction. It is not clear to me to what extent someone who accepts Field’s view should expect Def to fail even for ordinary non-semantic vague predicates. Incidentally, one moral of the argument I will be giving below is that (i) is not valid. ¹¹ The most obvious route to that claim would involve appealing to the controversial B (Brouweresche) axiom schema ¬P → ¬P, or perhaps to some weaker axiom schema of the form ¬P → ¬n P —see Fara 2002.
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borderline cases of ultrabaldness than in other borderline cases, ‘I don’t know’ is liable to be just as unacceptable an answer as ‘Yes’ or ‘No’. But even if you hold the more orthodox view that knowledge is inconsistent with borderlineness, you may still find there to be something unsatisfying about simply admitting that we don’t know who is and is not ultrabald and leaving it at that. Shouldn’t we philosophers who take a professional interest in questions of vagueness want to know more? If you have admitted that you don’t know whether someone is ultrabald, despite having been given as much time to reflect and as much access to other relevant facts as you have any use for, you will probably react with impatience to the suggestion that you undertake further inquiries. You will be tempted to protest that such inquiries would be pointless: you don’t just happen not to know; rather, neither you nor anyone else is even in a position to know, given any amount of further inquiry. But what could explain this inability? If borderlineness is a barrier to knowledge, your inability to know whether x is ultrabald might be explained by its being a borderline case. But since you don’t know that x is not ultrabald, you don’t know that it is borderline whether x is ultrabald, so you must leave open the possibility that the obstacle to your knowing whether x is ultrabald is of some other kind. But what other sort of obstacle to knowledge could be relevant in this context? Whatever it is, why can’t we tell it apart from the obstacle to knowledge characteristic of borderlineness? Wouldn’t it make more sense to adopt a more expansive use of the expression ‘borderline’, on which it applies to cases in which either sort of obstacle is present? If we did adopt this more expansive sense of ‘borderline’, along with corresponding senses of ‘definitely’, ‘dtrue’, ‘ultratrue’ and ‘ultrabald’, what then would be our epistemic situation with respect to the question ‘what does it take to be ultrabald?’ If there are people who are borderline ultrabald, and borderlineness is the only relevant obstacle to knowledge, we are doomed never to achieve a certain kind of theoretical satisfaction in our relations with them. So long as we form no opinion on whether they are ultrabald, we will never know whether, in failing to form an opinion, we are passing up knowledge which there is no obstacle to our possessing. If we do in fact give up on further inquiry, we will always be wondering if we could have resolved the question just by giving it a bit more thought. This would be an unsettling conclusion, I think. But these considerations don’t rise to the level of an argument that ‘ultrabald’ has precise and knowable conditions of application; at best, they show why it would be nice to have such an argument. The task of giving one will occupy the remainder of the paper. My conclusion will be radical: no one is ultrabald; in fact no sentence whatsoever is ultratrue, and no predicate whatsoever is ultratrue of anything.
31.3
N O S E N T E N C E I S U LT R AT RU E : F I R S T AT T E M P T
In this section I will take a first stab at arguing that no sentence is ultratrue. The result won’t be terribly hard to resist, but will serve as a basis for later refinements.
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It will help to make some simplifying assumptions about the laws of nature that enable us to define a well-behaved notion of distance between nomologically possible worlds. Suppose, then, that the actual world consists of finitely many point-particles in Newtonian absolute space. Where w and w are nomologically possible worlds with the same particles, define the distance between w and w at t as the sum, for each particle, of the distance between the point where the particle is located at t at w and the point where it is located at t at w .¹² Assume too that the laws of nature are deterministic and continuous, so that for any δ and t, there is a δ such that any two worlds which are less than δ apart now will remain less than δ apart until at least t units of time hence. With some such notion of inter-world distance in hand, we can state our argument. It has two premises: Series For every positive real number δ and sentence S, there is a sequence w0 , . . . , wn of possible worlds such that: S1 S2 S3 S4
S is not dtrue as used by us at w0 . wn is the actual world. For each 0 ≤ i < n, the distance between wi and wi+1 is less than δ. Our use of ‘’ at each wi is at least as stringent as it is at the actual world, in the following sense: for any sentence T , necessarily, if T is dtrue as used by us at wi , then definitely, T is dtrue as used by us at wi .
Margin There is a δ > 0 such that whenever a sentence S is definitely dtrue as used by us at w, and the distance between w and w is less than δ, S is dtrue as used by us at w . Now for the argument. Let δ meet the condition specified by Margin; let S be an arbitrary sentence; let w0 , . . . , wn be a sequence satisfying S1–S4. We show by induction that for each m ≤ n, m S is not dtrue as used by us at wm . The base step is just S1. For the induction step, assume that some sentence T is not dtrue as used by us at wm . By S3 and our choice of δ, T is not definitely dtrue as used by us at wm+1 ; and so (by the contrapositive of S4), T is not dtrue as used by us at wm+1 . So in particular, if m S is not dtrue as used by us at wm , m+1 S is not dtrue as used by us at wm+1 , which is what we need for the induction. Letting n = m, then, we have that n S is not dtrue as used by us at wn , i.e. at the actual world. A fortiori, S is not ultratrue, and ω S is not dtrue, as used by us at the actual world. Some comments, before we discuss how the premises Series and Margin might be justified: (a) At this point, it will be best to interpret predications of the form ‘S is dtrue as used by C at w’ as having to do with dtruth at the actual world rather than dtruth ¹² We could equally well use the notion of distance in configuration space standard in physics, which brings in the masses of the particles. Or, for an alternative that doesn’t require identifying points of space across possible worlds, see Barbour 2006: 116–17.
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(b)
(c)
(d)
(e) (f)
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at w.¹³ If we interpreted such predications as having to do with the dtruth at w of sentences as used at w, Margin would be implausible, since it would rule out our ever introducing perfectly precise sentences which express nomologically contingent truths. Framing such precise distinctions is difficult but not impossible. The expression ‘one second’, currently defined as ‘the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium-133 atom’ (BIPM 2006), is a reasonable candidate for being perfectly precise.¹⁴ If it is, and the duration of the universe is finite, there will be sentences of the form ‘The universe lasts for at least n seconds’ which are definitely dtrue at the actual world and definitely dfalse at worlds arbitrarily close to the actual world, not only as used at the actual world, but also as used at those worlds. Worries about the applicability of mathematical induction to vague predicates are not really to the point. It should be straightforward to argue for some large finite bound on the lengths of the sequences we need to consider, in which case we could reconstruct the argument using finitely many applications of modus ponens. We can run an exactly similar argument for the claim that no predicate is ultratrue of any sequence of arguments. But for simplicity I will continue to focus when possible on closed sentences. I have suppressed all mention of time. A-theorists about time shouldn’t mind this. B-theorists should either take every sentence as implicitly relativized to the present time, or else reinterpret all claims about ‘‘possible worlds’’ as claims about ordered pairs of worlds and times. I have suppressed the apparatus that would be necessary to deal with contextsensitive sentences. It should be easy to reintroduce. Since sentences need to be understood as items which can be used very differently at different worlds, they are presumably the sort of thing that can be ambiguous (as used by a given community at a given world). There are two ways of thinking about what it might mean to describe such entities as ‘dtrue’: it could mean ‘dtrue on every disambiguation’ or ‘dtrue on some disambiguation’. As far as I can see, it makes no difference which of these we adopt.
The case for Series is straightforward. Evidently we could have used any sentence S in such a way as to make it not be dtrue at the actual world, e.g. by using S in the same way that we actually use the sentence ‘0 = 1’, or by speaking a language in which S is not meaningful at all. We could have done this even at a world with the same kinds of laws we have been supposing to hold at the actual world, and with the same number of particles as the actual world. But any such world is a finite distance from the actual world, and thus can be reached from the actual world by way of finitely many steps of arbitrarily small size. The only remaining question is whether we can take these ¹³ I will reconsider this choice later, in section 31.5. ¹⁴ ‘One Planck time’ is an even better candidate.
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steps in such a way that all the worlds we visit along the way satisfy S4. By T-, the actual world satisfies S4. There is no trouble choosing w0 in such a way as to satisfy S4: for example, we could let w0 be a world where ‘’ is not meaningful at all, or where it is used so demandingly that no sentence of the form S is dtrue. And I see no reason to doubt that at least some such w0 can be connected to the actual world by a sequence of worlds all satisfying S4. If we think of ‘definitely’ as expressing a property of propositions, it would suffice for the property it expresses at each wi where it is meaningful at all to be the same as, or stronger than, the one it expresses at the actual world. If we think of S as meaning something like ‘S’ is dtrue as used by us at the actual world, it would suffice for this equivalence to be in place at each wi , and the use of ‘dtrue’ at each wi to be such that necessarily, whenever ‘dtrue as used by . . . at . . .’ is dtrue of some S, C, w as used by us at wi , it is dtrue of S, C, w as used by us at the actual world. In either case, it is hard to see how their could fail to be a topologically connected set of worlds satisfying S4 which contains both the actual world and some appropriate w0 . So much for Series. Why would anyone accept Margin? If you are anything like me, you will have noticed an affinity between Margin and certain claims characteristic of Timothy Williamson’s epistemic theory of vagueness (1994). In fact, I think I see a good argument from Williamson’s view to Margin. But the case is less straightforward than I initially supposed. For Williamson, the claim that it is definitely the case that P means, or at least entails, that there is no obstacle of a certain distinctive kind to our knowing that P. It is sufficient for the existence of such an obstacle for there to be a false proposition which we could very easily have expressed using the sentence we actually use to express the proposition that P. That is: W1 If we use S and no other sentence to express proposition p at w and to express proposition q at a world w that is close to w, then at w: if p is definitely true, q is true. If we could drop the ‘at w’ from W1, we would have something from which we could hope to derive Margin, at least restricted to communities in which each proposition is expressed by at most one sentence. It would just be a matter of putting a ‘definitely’ in front of W1, and arguing that for some δ, whenever the distance between two possible worlds is less than δ, they are definitely ‘‘close’’ in the relevant sense. But what could license eliminating the ‘at w’? We might try arguing as follows. First, insert actuality operators in the consequent of W1, so that it becomes W2 . . . at w: if definitely (p is actually true), then (q is actually true). Next move ‘at w’ inside the conditional: W3 . . . if at w, definitely (p is actually true), then at w, (q is actually true). Then interchange ‘at w’ and ‘definitely’: W4 . . . if definitely (at w, p is actually true), then at w, (q is actually true).
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Finally, appeal to the definite validity of the schema ‘P iff at w actually P’, to get W5 . . . if definitely (p is true), then q is true which is what we wanted. Unfortunately, two of these steps are dubious by Williamson’s lights. The step from W1 to W2 is problematic chiefly because it is hard to understand the question whether there is an obstacle of the relevant kind at w to our knowing that p is actually true. How are people at w supposed to pick out the actual world at all, in order to formulate the question whether p is true at it? On the most straightforward way of understanding what this would require (Williamson 1987, Soames 1998), the obstacles at w to our even entertaining p is actually true are so formidable that it is hard even to make sense of the question whether there are, in addition, any obstacles of the distinctive sort associated with vagueness to our knowing it. The second dubious step is from W3 to W4. While it is tempting in reasoning about vagueness to treat ‘definitely’ as commuting with ‘at w’, there is no obvious support for this in Williamson’s theory. In general, the claim that there is an obstacle of some given sort to our knowing that at w, P is independent of the claim that at w, there is an obstacle of that sort to our knowing that P. And there is no obvious reason why the particular sort of obstacle to knowledge that is distinctive of vagueness on Williamson’s view should be exceptional in this regard.¹⁵ Thus, the most obvious route from Williamson’s theory to Margin is fraught with difficulties. But we can do better, by focusing not on the use of the sentence S at w and w , but at the use of the predicate ‘dtrue as used by . . . at . . .’ at worlds close to the actual world. This predicate is manifestly vague. For any S, it is easy to construct Sorites sequences of possible worlds which take us in many small steps from a world of which it is clearly the case that S is dtrue as used by us there to a world of which this is clearly not the case. These sequences raise the same puzzles as the canonical Sorites sequences involving ‘bald’ and ‘heap’.¹⁶ While we can see that the accumulated tiny differences must somehow constitute the difference between a way of using S that makes it dtrue (at the actual world) and one that doesn’t, we have no more grip on the question how any one step along the sequence could constitute such a difference than we have on the question how a similarly tiny difference between two worlds could make it be the case that I am bald at one world and not at the other. We thus have as much reason to recognize that it is sometimes borderline whether S is dtrue as used by
¹⁵ The failure of ‘definitely’ to commute with ‘at w’ on Williamson’s view makes it surprisingly hard for propositions to be necessarily definitely true. For it to be false that necessarily definitely 0 = 1, it would suffice for there to be a pair of close worlds w, w , such that a sentence S that is used at w to express the proposition that 0 = 1 is used at w to express something false at w. But given that it must be possible for a sentence to express the proposition that 0 = 1 even though it could easily have expressed some other proposition, there is no obvious reason why it shouldn’t be possible for a sentence to express the proposition that 0 = 1 even when it could easily have expressed a false proposition. ¹⁶ Cf. Sorensen 1985.
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us at w as we do to recognize that it is sometimes borderline whether I am bald at w.¹⁷ Moreover, just as we have reason to believe, for sufficiently close worlds w and w , that it will never happen that I am definitely bald at w and definitely not bald at w , we have reason to believe that it will never happen that S is definitely dtrue as used by us at w and definitely not dtrue as used by us at w . That is: Borderline There is some δ > 0 such that whenever a sentence S is definitely dtrue as used by us at w, and the distance between w and w is less than δ, S is not definitely not dtrue as used by us at w . To help make this plausible, consider a very tiny value of δ, such that given the laws, whenever the distance between w and w is less than δ, it will take a trillion years before there is as much as a nanometre’s difference between the location of any particle at w and its location at w . Our conception of the way in which dtruth-conditions depend on use seems far too imprecise for such a tiny difference in use ever to make the difference between definite dtruth and definite lack of dtruth (at the actual world, or indeed at any given world).¹⁸ Since ‘not definitely not dtrue’ is weaker than ‘dtrue’, Borderline is, formally speaking, weaker than Margin. But Williamson’s theory of vagueness provides a way of closing the gap. For Williamson, the vagueness of any predicate consists in the fact that there are worlds close to the actual world where it expresses relations different in intension from the one it actually expresses. In principle, the actual world could be a ‘‘local maximum’’ with respect to the use of some predicate, so that the relation it actually expressed entailed all the relations it expressed at nearby worlds. But on any remotely plausible account of the connection between use and meaning, this will happen only in very special cases. Normally, if there are close worlds where a predicate expresses a relation weaker than the one it actually expresses, say because its use is slightly laxer in some respects, there will also be close worlds where it expresses a relation stronger than the one it actually expresses, because its use is slightly stricter in those same respects.¹⁹ I see no reason to think that ‘dtrue’ would be abnormal in this respect. In fact, on Williamson’s theory, there will plausibly be worlds close to the actual world where ¹⁷ John Hawthorne (2006) notices special problems with the idea that ‘true’ (or ‘true as used by . . . at . . . ’) expresses different things at nearby worlds. Unless instances of the disquotation schema are in danger of expressing falsehoods, each relation R that is expressed by ‘true as used by . . . at . . . ’ at some w near the actual world must be such that, necessarily, for any S, R(S, us, w) iff S is true as used by us at w. So there won’t be worlds where ‘true as used by . . . at . . . ’ expresses relations that are uniformly more demanding than the relation expressed by this predicate at the actual world. But the claim in the text concerns ‘dtrue’ rather than ‘true’. Even if the intension of ‘true’ were the same at all nearby worlds, we could still get the intension of ‘dtrue’ to vary by varying the use of ‘definitely’. ¹⁸ But see section 31.5 below for an important objection to this claim. ¹⁹ Plausibly, for this to fail, the intension actually expressed would have to be fairly ‘natural’, since a range of different patterns of use including the actual pattern together with all the ‘stricter’ patterns that obtain in close worlds result in the same intension being expressed. But the pull of naturalness cannot be too strong: the actual world must be perched near the edge of the set of worlds in which this intension is expressed, since different intensions are expressed at close worlds with ‘laxer’ patterns of use where different intensions are expressed.
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‘dtrue as used by . . . at . . .’ expresses a relation that is uniformly stronger than the one it actually expresses, in the sense that whenever the former relation holds between S, C and w, the latter holds between S, C, and any w within δ of w, for some δ. One would expect this to happen if people at the close world in question are, across the board, a bit more reluctant to apply the predicate ‘knows’ than they actually are, with the result that ‘knows’ expresses a uniformly stronger relation R, which a person bears to a proposition at w only if the person knows the proposition at all w sufficiently close to w. It is thus easier for there to be obstacles to the obtaining of R than for there to be obstacles to knowledge. If the connections Williamson posits between ‘knows’, ‘definitely’, ‘true’ and ‘dtrue’ remain in place at the world in question, this will make it harder for a proposition to have the property expressed there by ‘definitely true’, and thus harder for a sentence to stand in the relation expressed there by ‘dtrue’ to any given community and world. To get from Borderline to Margin, we only need a weaker claim: roughly, that if there are close worlds where the relation expressed by ‘dtrue’ is weaker along some dimension, there are close worlds where it is stronger along that dimension. To make this precise, let a ‘dimension’ be a triple S, C, λ , where λ is a straight path through the space of nomologically possible worlds, which starts with a world w+ such that S is definitely dtrue as used by C at w+ , ends at a w− such that S is definitely not dtrue as used by C at w− , and is such that for every other w ∈ λ, it is borderline whether S is dtrue as used by C at w. The actual world is a local maximum with respect to this dimension if the relation expressed by ‘dtrue’ at the actual world is one that fails to hold between S, C and w for any w ∈ λ other than w+ . Given that there are no especially natural relations in the vicinity of ‘dtrue’, it is plausible that this never happens. A stronger claim also seems plausible: that the actual world does not come arbitrarily close to being a local maximum—in other words, for some n, the actual cutoff for ‘dtrue’ occurs at least 1/n of the way along each S, C, λ , measured by our canonical notion of interworld distance. This gives us what we need: since by Borderline, there is a δ such that the length of λ is always at least δ, it follows that the distance along λ between w+ and any world w such that S is not dtrue as used by C at w is always at least δ/n. Thus δ/n witnesses the truth of Margin: whenever a sentence S is definitely dtrue as used by C at w, and the distance between w and w is less than δ/n, S is dtrue as used by C at w . The conclusion that no sentence is ultratrue need not be unwelcome or even especially surprising to Williamson. A central doctrine of Williamson 2000 is that only trivial conditions are luminous: such that necessarily, whenever they obtain, one is in a position to know that they obtain. This strongly suggests that the claim that P entails that P only when it is trivial that P. Since ‘ω P ω P’ is valid, and since it would have to be trivial that P for it to be trivial that ω P, it follows that that all ultratrue sentences express trivial conditions. The step from this to the claim that no sentence is ultratrue is relatively small. True, Williamson occasionally uses the methods of normal modal logic in modelling the logic of ‘definitely’. Since normal modal logics validate the rule of necessitation—φ is a theorem whenever φ is—these methods cannot be strictly correct if nothing is ultratrue. But the methods of normal modal logic can be useful tools without being strictly correct, as witness their
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widespread use in modelling knowledge even by those who reject logical omniscience. Almost none of the philosophical uses to which Williamson puts these methods require taking them any more seriously than this.²⁰
31.4
N O S E N T E N C E I S U LT R AT RU E : S E C O N D AT T E M P T
The foregoing argument from Borderline to Margin is highly specific to Williamson’s theory. I don’t know of any argument for Margin that nonepistemicists should find convincing. Here is a way of thinking about what it would be to accept Margin, once we expand it in the obvious way to cover ‘dtrue of ’ as well as ‘dtrue’. Where R is some quaternary relation between predicates, communities, worlds and sequences of arguments, let R ∗ be the quaternary relation that holds, necessarily, among , x1 , . . . , xn , C and w (in that order) iff R holds among the predicate ‘dtrue of . . . as used by . . . at . . .’ (‘dtrue’ for short), the sequence of arguments , x1 , . . . , xn , C, w , our community, and the actual world. Say that R is δ-modest iff whenever R ∗ (, x1 , . . . , xn , C, w), and the distance between w and w is less than δ, R(, x1 , . . . , xn , C, w ). Margin entails that the relation of dtruth is δ-modest for some positive δ. For Margin itself to be dtrue, then, it would have to be the case that every precisification of ‘dtrue’ is δ-modest for some positive δ. (Say that an n-ary relation R is a precisification of an n-ary predicate iff Necessarily, for all x1 . . . xn , R(x1 , . . . , xn ) iff (x1 , . . . , xn ) is not dfalse of that relation, as used by us at the actual world.) Once we set epistemicism aside, it is hard to see what aspect of our usage of ‘dtrue of ’ could constrain its precisifications in this way. By way of contrast, it is easy to see how our usage could impose the weaker constraint that R ∗ entails R whenever R is a precisification of ‘dtrue’. We treat ‘definitely’ as factive, in the sense that we treat sentences of the form ‘if definitely P, then P’ as obvious truths. It is unmysterious how these dispositions could render the sentence ‘Whenever ‘‘dtrue of ’’ is dtrue of , σ , C, w as used by us at the actual world, is dtrue of σ as used by C at w’ dtrue in our mouths. But given standard compositional rules, making that sentence come out dtrue requires preventing any R which fails to hold in some cases where R ∗ holds from being among the precisifications of ‘dtrue of ’. I don’t see any analogous facts about our usage of ‘dtrue of ’ that could, in a parallel way, constrain it to have only δ-modest relations as precisifications. ²⁰ One exception is Williamson 1999. On the definitions proposed in that paper, for to be ‘‘first-order precise’’, it is not enough for ∀x((x) ∧ ¬(x)) to be necessarily true: its truth must be ‘semantically guaranteed’, in a sense that requires it to remain true when prefixed by any number of ‘’s. Williamson motivates this by the desire to avoid what he deems ‘the counterintuitive situation of higher-order vagueness without first-order vagueness’. But once we realize that no sentence enjoys this kind of ‘‘semantic guarantee of truth’’, we will presumably have to admit, given any sensible definition of ‘nth-order vague’, that every sentence is nth order vague for some n. So if we want anything to count as precise, we had better get used to the situation that Williamson finds counterintuitive.
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On the other hand, it is, if anything, even more obvious that if Borderline is true, there is nothing in our usage of ‘dtrue of ’ that could render Margin dfalse —that could, that is, prevent any of the precisifications of ‘dtrue of ’ from being δ-modest for any positive δ. Fortunately, our argument that no sentence is ultratrue can be adapted so as to rely on this weaker claim instead of Margin. Roughly speaking: since Series and Margin jointly entail ‘no sentence is ultratrue’, the claim that Series is dtrue and the claim that Margin is not dfalse jointly entail that ‘no sentence is ultratrue’ is not dfalse, which in turn entails that no sentence is ultratrue. Let me restate that argument a bit more carefully, so as to forestall some distracting objections. Let ‘M (δ)’ stand for the claim that δ satisfies the condition specified in Margin, that is: Whenever a sentence S is definitely dtrue as used by us at w, and the distance between w and w is less than δ, S is dtrue as used by us at w . Let ‘W (S, n, δ)’ stand for the claim that there is a sequence w0 , . . . , wn which meets the conditions specified in Series, that is: S1 S2 S3 S4
S is not dtrue as used by us at w0 . wn is the actual world. The distance between wi and wi+1 is always less than δ. For any sentence T , necessarily, if T is dtrue as used by us at wi , then definitely, T is dtrue as used by us at wi .
The derivation at the beginning of section 31.3 shows that (∗ ) If W (S, n, δ) and M (δ), then n S is not dtrue as used by us at the actual world. is a logical (or at least a mathematical) truth.²¹ As such, (∗ ) is itself dtrue of each S, n, δ , as used by us at the actual world. But on almost any reasonable account of dtruth for conditionals and conjunctions, a dtrue conditional can have a dfalse consequent only if it has a dfalse antecedent, and a conjunction with one dtrue conjunct can be dfalse only if the other conjunct is dfalse. So if (∗ ) and W are both dtrue of S, n, δ (as used by us at the actual world) and M is not dfalse of δ, ‘n S is not dtrue as used by us at the actual world’ must not be dfalse of n, S . If so, ‘n S is dtrue as used by us at the actual world’ is not dtrue of n, S . So n+1 S is not dtrue. The conclusion that no sentence is ultratrue thus follows from a strengthened version of Series together with a weakened version of Margin: Series+ For each δ > 0 and sentence S, there is an n such that definitely W (S, n, δ). Margin− There is a δ > 0 such that not definitely not M(δ). The case for Series+ is not significantly weaker than the case for Series: the task of choosing w0 . . . wn in such a way that S1–S4 are definitely satisfied doesn’t seem ²¹ While it is controversial whether conditional proof is generally acceptable when vagueness is in question, none of the steps in the argument from section 31.3 has the features that are supposed to make for failures of conditional proof.
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especially harder than the task of choosing them in such a way that they are simply satisfied.²² And given Borderline, the case for Margin− is quite strong. For Margin− to hold, it is sufficient for even one of the precisifications of ‘dtrue’ to be δ-modest, for some positive δ. The non-existence of such precisifications would amount to a ‘‘penumbral connection’’ between claims about the dtruth-conditions of arbitrary sentences as used at arbitrary possible worlds and the dtruth-conditions of ‘dtrue’ at the actual world. As such, it would cry out for an explanation in terms of some distinctive feature of our use of ‘dtrue’: in general, when we don’t do anything distinctive to create penumbral connections, there aren’t any. One thing we could have done would have been to endow ‘dtrue’ with some precise cutoffs, so that there would be arbitrarily close worlds w and w such that some sentence S is definitely dtrue as used by us at w and definitely not dtrue as used by us at w . But according to Borderline, we didn’t do this. And we don’t seem to have done anything else relevant in the present case. 31.5
A P RO B L E M W I T H R E F E R E N C E - F I X I N G
This argument for Margin− depends essentially on Borderline: Borderline There is some distance δ such that whenever a sentence S is definitely dtrue as used by some community at w, and the distance between w and w is less than δ, S is not definitely not dtrue as used by that community at w . I motivated Borderline in section 31.3 by appealing to the idea that the dtruthconditions of a sentence depend on its use. So long as we focus on such aspects of ‘‘use’’ as people’s dispositions to affirm or deny a sentence under various conditions, and to behave in various ways in reaction to other people affirming and denying it, it will seem obvious that a sufficiently minuscule shift in use could never definitely make the difference between a sentence’s being dtrue and its not being dtrue (as evaluated at any given world). However, if we want the claim that dtruth-conditions depend on use to be uncontroversial, we had better make sure to understand ‘‘use’’ more broadly than this—broadly enough so that, for example, the sentence ‘there is water’ counts as being ‘‘used in different ways’’ on Earth and on Twin Earth. And once we pay attention to this sort of way for differences in the world to make for differences in ²² The only possible stumbling block is condition S4: it might be thought that the set of worlds of which it is definite that ‘definitely’ is used at least as stringently at them as it is at the actual world was too small or scattered to contain a path from actual world to an appropriate w0 . I doubt that there is anything to this worry: it seems easy to imagine ways of changing the use of ‘definitely’ that would definitely either strengthen it or leave it alone. But even if there were, it really wouldn’t matter much, since condition S4 is much stronger than it needs to be for the argument to work. All we really need is that the use of ‘’ at each wi should be similar enough to its actual use for something like Margin− to be true: that is, we only need there to be a δ such that for any of the wi , it is not definitely not the case that when S is dtrue as used by us at wi , and the distance between w and wi is less than δ, S is dtrue as used by us at w . It would be straightforward exercise to rewrite the argument using this weaker premise.
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the dtruth-conditions of sentences, potential counterexamples to Borderline come quickly to mind. Suppose we discover that the universe has a finite duration, from Big Bang to Big Crunch. We could then introduce the expression ‘aeon’ by issuing a stipulation: ‘Let ‘‘one aeon’’ name the duration of the universe.’ We thereby, let’s suppose, introduce a precise expression which is definitely, necessarily, dtrue of all and only those temporal intervals whose duration is the same as the actual duration of the universe. Thus the sentence ‘The universe lasts for at most one aeon’ is definitely dtrue as used by us at the actual world. But this same sentence is definitely not dtrue (at the actual world) as used by us at a close world where the Big Crunch happens a little earlier. Since we may suppose that the laws of nature allow such worlds to be arbitrarily close to the actual world, this is a counterexample to Borderline.²³ There are ways of fixing up our argument to make it proof against these counterexamples. The most obvious strategy is to put some ad hoc restriction into Borderline—something like ‘. . . so long as the use of S at w and w doesn’t involve a reference-fixing description which definitely denotes different things w and w ’—and to use this to argue for a correspondingly restricted version of Margin− . We could still argue for the conclusion that no sentence whatsoever is ultratrue, by appealing to the claim that there are some sentences—‘0 = 1’, say—which are ultratrue if anything is, and whose use doesn’t involve the kind of reference fixing that makes for exceptions to Borderline.²⁴ Still, it is interesting to see whether we can find any non-ad hoc, defensible principles in the vicinity of Borderline and Margin− . In the remainder of this section I will discuss two possible strategies for formulating such principles. The first strategy is to understand ‘S is dtrue as used by C at w’ in a way modelled on what the tradition of two-dimensional semantics calls the ‘primary intension’ as opposed to the ‘secondary intension’ of S. The idea is that while differences in the denotation of a reference-fixing description make a difference to the secondary intension of an expression, they make no difference to its primary intension, which ²³ Real-world uses of reference-fixing descriptions don’t pose any obvious problems for Borderline. ‘One kilogram’ is stipulated to be the rest mass of a particular platinum-iridium cylinder: but because of fluctuations over time in the mass of the cylinder, vagueness as regards the locations of its boundaries, and perhaps also further quantum-field-theoretic sources of vagueness in claims about the masses of particular material bodies, it is not plausible that this stipulation has made ‘one kilogram’ perfectly precise, and thus not plausible that there is any possible object of which ‘has a mass of at least a kilogram’ is definitely dtrue as used by us at the actual world and definitely not dtrue as used by us at worlds arbitrarily similar to the actual world. ‘One second’ may for all I know be completely precise, but unlike our envisaged definition of ‘aeon’, its definition (557 above) doesn’t seem to depend on anything nomologically contingent, so the ‘dtruth’-conditions of ‘one second’ wouldn’t vary between the actual world and other nomologically possible worlds where it is associated with the same description. Expressions constructed using ‘actually’ are the best real-world candidates to be counterexamples to Borderline. ²⁴ In fact, we need something a bit stronger than this: we need an assurance that such referencefixing is not a feature of the usage of any of the sentences i S, and we need this to be true not only at the actual world but at all of the worlds in some appropriate sequence w0 , . . . , wn . But this seems fine: none of the sentences i 0 = 1 seems to be anything like ‘The universe lasts for at least one aeon’ in the relevant respects.
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is always the same as the primary intension of the reference-fixing description. Thus, for any world w where the expressions ‘one aeon’ and ‘the duration of the universe’ are associated in the right way, it is a necessary truth that ‘the universe lasts for at most one aeon’ is dtrue in the primary sense as used by us at w if the duration of the universe is finite. The two-dimensionalist programme (Chalmers & Jackson 2001, Chalmers 2006) attempts to assign distinct primary and secondary intensions to many ordinary expressions, in such a way that the primary intension of each expression encodes certain facts about its epistemological properties. The application of two-dimensional machinery we are presently contemplating doesn’t require anything so ambitious. The main objection to assigning an expression like ‘water’ a primary intension distinct from its secondary intension is the sheer difficulty of finding any non-arbitrary principle for reading a primary intension off the use of this expression, let alone one that captures anything epistemologically significant. But ‘aeon’ poses a problem for Borderline precisely because the description ‘the duration of the universe’ plays such a clear and non-arbitrary role in regulating its use. If its use suffered from the sort of messiness that makes the project of assigning interesting primary intensions to most expressions so hard, it would no longer be plausible that an arbitrarily small change could make the difference between definite dtruth and definite lack of dtruth (at the actual world, in our old, ‘secondary’ sense). Thus, even a very modest dose of two-dimensionalism, on which primary and secondary dtruth-conditions diverge only in the rare cases where it is completely clear how to draw the distinction in a non-arbitrary way, should be enough to make Borderline proof against the counterexamples we have been considering in this section. This first strategy will, however, lead us into new difficulties if we combine it with the metalinguistic analysis of S as ‘S’ is dtrue as used by us at the actual world. Given the standard treatment of the primary intensions of sentences involving ‘actual’ and ‘actually’, this analysis makes the primary dtruth-conditions of S come apart from those of S in a surprising way. For example, consider a world w where we use ‘cat’ the way we actually use ‘lemur’, while using all other words just as we actually do. Since there are fewer than a million lemurs, ‘There are more than a million cats’ is dfalse (in both the primary and secondary senses) as used by us at w. But if ‘Definitely, there are more than a million cats’ is synonymous at w with ‘ ‘‘There are more than a million cats’’ is dtrue as used by us at the actual world’, it is dtrue in the primary sense, since the use of ‘ ‘‘There are more than a million cats’’ is dtrue as used by us at the actual world’ doesn’t vary between w and the actual world in any relevant way. Unsurprisingly, a notion of dtruth which made it this easy for ‘S’ to be dtrue without S being dtrue would make a mess of our argument.²⁵ ²⁵ The problem turns out to lie with S4, which becomes extremely demanding if ‘’ is analysed using ‘actually’, ‘actually’ is given the usual two-dimensional semantics, and ‘dtrue’ is understood in the primary sense. Suppose that ‘’ and ‘dtrue as used by us at the actual world’ are used in the same way at wi as they are at the actual world. Then whenever T is dtrue in the primary sense as used by us at the actual world, it is dtrue in the primary sense as used by us at wi . If wi also satisfies S4, it follows that whenever T is dtrue as used by us at the actual world, it is definite that T is dtrue in the primary sense as used by us at wi . For this to be true, our use of T at wi would have to
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At this point, the proponent of the ‘actually’-involving analysis of ‘definitely’ might consider simply eliminating all occurrences of ‘definitely’ from the argument in favour of the putatively more fundamental ‘dtrue as used by . . . at . . .’. There may be a workable argument to be found here, but things quickly get complicated, as we have to deal with claims about the dtruth of predicates of sequences of arguments inside which other sequences may be deeply nested. The task of formulating a compelling principle strong enough to play the role of Borderline in such an argument is quite challenging. The second strategy is to rethink our stipulation that claims of the form ‘S is dtrue as used by C at w’ are to be evaluated with respect to the actual world. As I pointed out (comment (a) in section 31.3) the argument would be hopeless if we had instead understood ‘dtrue as used by C at w’ as equivalent to ‘dtrue at w as used by C at w’: in that case, Borderline (and Margin, and Margin− ) would have been immediately refuted by the existence of precise, nomologically contingent sentences like ‘The duration of the universe is at least n seconds.’ But we can state a principle in the spirit of Borderline that allows for such expressions, by trading in our simple measure of similarity between worlds for a more complicated measure of similarity between ordered triples of sentences, communities and worlds: Borderline∗ There is a δ > 0 such that whenever a sentence S is definitely dtrue at w as used by C at w, and the distance between S, C, w and S, C, w is less than δ, S is not definitely not dtrue at w as used by C at w . How would the distance metric have to work for Borderline∗ to be defensible? Clearly, if S is ‘The universe lasts for at least n seconds’, and C is a community that uses ‘second’ much as we do at both w and w , the fact that the universe lasts for less than n seconds at w and more than n seconds at w must be sufficient for the distance between S, C, w and S, C, w to exceed some positive threshold, no matter how close w and w might be in our old sense. We can achieve this, I think, by thinking of the similarity relation between ordered triples as grounded, at least in part, in more or less natural relations, in the same way that the similarity relations among objects are generally thought of as grounded in their natural properties (see Lewis 1983). The more natural a relation, the more the fact that it holds between S, C and w but not between S, C and w will contribute to dissimilarity between S, C, w and S, C, w . As a special case, the more natural a function f from sentences, communities and worlds to propositions, the more the fact that f (S, C, w) holds at w while f (S, C, w ) fails to hold at w will make for dissimilarity. If C employs the actual definition of ‘second’ involving the number 9,192,631,770 at both w and w , then the fact that S is true at w and false at w on an interpretation on which ‘one second’ stands for 9,192,631,770 units of some fairly natural duration will make for substantial dissimilarity between S, C, w and S, C, w . If, instead, making S have different truth values at w and w required interpreting ‘one second’ be quite similar to our use of T at the actual world. Given this, there is no longer any clear reason to expect to be able to get from the actual world to an appropriate w0 via a chain of worlds all of which satisfy S4.
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as standing for 9,162,631,771 units of the same duration, this would make for much less dissimilarity between the triples. If you can make sense of the background ideology of degrees of naturalness, you should find Borderline∗ plausible. It is a instance of a plausible general schema, which captures the idea that we can only achieve precision along some dimension when the dimension contains sufficiently natural joints for our use to pick up on: (N) There is a δ > 0 such that whenever is it definitely the case that F (x1 , . . . , xn ), and the distance between x1 , . . . , xn and y1 , . . . , yn is less than δ, it is not definitely not the case that F (y1 , . . . , yn ). The picture is that by working hard—e.g. by formulating long and complicated definitions—we can make the use of a predicate sensitive to less and less natural distinctions, thereby reducing the maximum value of δ for the predicate in question. But we can only do a finite amount of this kind of work. If it turns out that objects of which some predicate is dtrue can be arbitrarily close to objects of which it is dfalse on some similarity measure, the measure in question must fail to represent all the natural joints in the relevant space. Can we use Borderline∗ to argue that no sentence is ultratrue? The argument from Borderline to Margin− in section 31.4 can be adapted to yield an analogous argument for Margin* There is a δ > 0 such that it is not definitely not the case that: whenever a sentence S is definitely dtrue at w as used by C at w, and the distance between S, C, w and S, C, w is less than δ, S is dtrue at w as used by C at w . But how are we to get from Margin* to the conclusion that no sentence is ultratrue? The space of S, C, w triples will turn out to be far from continuous under any metric that tracks natural properties and relations. On any reasonable way of thinking about degrees of naturalness, it is inevitable, leaving aside perfectly symmetric universes, that for any distinct triples S, C, w and S , C , w , there will be a relation R that has some positive degree of naturalness such that R(S, C, w) but not R(S , C , w ). Thus, if we want to trace a path from S, C, w to S , C , w in small steps, there will always be some lower bound to the size of steps we can allow ourselves. A space of possibilities with a metric based on naturalness is like a fractal landscape, crosscut so thoroughly with cracks that one can never get anywhere without stepping over a crack of some nonzero width. This makes it hard to state a premise that can take over the role of Series or Series+ in the new framework. But I don’t think that this problem is too serious. Even without anything like an articulated metasemantics, we can see that the power of naturalness to make for definite cutoffs in the extensions of our predicates falls off quite quickly. The property having a mass of more than 45955882 Planck masses is somewhat natural, but it is not natural enough for it to be at all plausible that ‘has a mass of at least kilogram’ as used in, say, 1800 was definitely dtrue of all and only the things whose mass was more than 4595582 Planck masses. This suggests that some δ0
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meets the condition specified in Margin* and is fairly big—big enough for the set of all S , C , w reachable from any given S, C, w by way of steps no bigger than δ0 to be, in general, quite extensive. So the following principle has some plausibility: Series∗ For each sentence S, there is a positive δ0 meeting the condition specified in Margin*, such that for some n, definitely, there is a sequence w0 , . . . , wn for which: S1 S is not dtrue at w0 as used by us at w0 . S2 wn is the actual world. S3 For each i, the distance between i S, us, wi and i S, us, wi+1 is less than δ0 . S4 For any sentence T , necessarily, if T is dtrue as used by us at wi , then definitely, T is dtrue as used by us at wi . Given Series∗ and Margin*, we can argue that no sentence is ultratrue in the same way as before. If you thought that the relation S being dtrue at w as used by C at w was itself fairly natural, you would have no reason to accept Series∗ (unless you already believed for some other reason that nothing was ultratrue). For in that case, if we started out assuming that S was ultratrue as used by us at the actual world, the stipulation that S is not dtrue as used by us at w0 would be enough to entail that for some i, the triples i S, us, wi and i S, us, wi+1 differ as regards the fairly natural relation being dtrue as used by . . . at . . ., and thus count as fairly far apart on the relevant similarity measure.²⁶ But let’s assume the more orthodox view that takes physics to be our best guide to the structure of natural properties. In that case, Series∗ seems quite secure for many values of S —‘0 = 1’, for example. It is not plausible, from the physicalistic perspective, that any of the remotely natural relations that hold between any sentence i 0 = 1, us, and the actual world is even sufficient for the sentence to be dtrue as used by us at the actual world. The physical facts about us that make some of these sentences dtrue in our mouths are just too complicated. So it should be possible to find a sequence w0 , . . . , wn satisfying conditions S1, S2 and S4, and such that whenever an even remotely natural relation holds between i 0 = 1, us, and wi , it also holds between i 0 = 1, us, and wi+1 . If so, we can make the distance between i S, us, wi and i S, us, wi+1 very small, on a naturalness-respecting metric. We will still have to step across tiny cracks, corresponding to relations which aren’t even ‘‘remotely’’ natural. But an account of dtruth that allowed even these very minor joints in nature to endow vague predicates with definite cutoffs would, I think, grossly overestimate the role of naturalness in metasemantics.²⁷ ²⁶ Hawthorne’s suggestion (2006) that ‘true’ (as opposed to ‘dtrue’) expresses a natural relation would also, I think, undermine the plausibility of Series∗ ; but the relevant considerations in that case are more intricate. ²⁷ Wait: isn’t being actualized (understood in such a way that it is contingent which world is actualized) a highly natural property? And isn’t being an S, C and w such that w is actualized
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Not all sentences are like ‘0 = 1’ in this respect. If the number of particles is finite, then there will be sentences which specify the position and momentum of each particle, using physically natural units, up to any desired degree of accuracy. If so, it will sometimes happen that a (ridiculously long!) sentence S has a highly natural interpretation on which it entails that it itself is dtrue as used by us. If so, it is also true of each sentence i S that it has a highly natural interpretation on which it entails that S is dtrue as used by us: we can always just interpret as a vacuous operator. In that case, if S is in fact dtrue as used by us, there will be no way to choose w0 , . . . , wn so as to satisfy S3: for some i, the triples i S, us, wi and i S, us, wi+1 will have to differ as regards whether their first member is true at their third member on the highly natural interpretation in question. So Series∗ cannot be defended in full generality. But we can still argue that no sentence is ultratrue indirectly, by first arguing, say, that ‘0 = 1’ is not ultratrue, and then arguing that if any sentence were ultratrue, ‘0 = 1’ would have to be. Or we could argue, first, that no sentence below some given length is ultratrue, and second, that since every sentence has short sentences among its logical consequences, if any sentence were ultratrue, some short sentence would have to be. This is a bit disappointing: it would be nicer to be able to argue by appealing to completely general premises. Still, for those who can stomach the hefty dose of vagueness involved in all this talk about degrees of naturalness and similarity relations that respect them, the present approach has dialectical advantages, in that it lets us avoid the task of formulating and defending an ad hoc restriction of Margin− that avoids counterexamples involving reference-fixing. therefore a highly natural relation? And won’t it follow, therefore, that whenever w is the actual world and w isn’t, the distance between S, C, w and S, C, w is fairly large, so that there can be no sequence of the kind required by Series∗ ? The point is well taken: the discussion in the main text implicitly assumes that the relevant distance relation is itself a necessary one, for which only necessary natural relations need to be taken into account. One way of fixing up the argument is to change Borderline∗ to make it explicit that only necessary relations are to be taken into account. So strengthened, Borderline∗ will no longer be an instance of a plausible general schema: for (N) to be plausible for ordinary contingent predicates like ‘is positively charged’, the distance relation must of course take contingent natural relations into account as well as necessary ones. But the stronger version of (N) does seem to be plausible for many predicates whose extensions are a necessary matter, for example predicates of the form ‘x is F at w’. The exceptions are predicates that achieve precision using devices like ‘actually’, like ‘x is the same length as it actually is at w’, and similar predicates introduced using reference fixing. But it is hard to see any relevant similarity between ‘dtrue’ and these. Another response is simply to drop the requirement that wn be the actual world, and replace it with the requirement that if n S is dtrue at the actual world, it is dtrue at wn . This seems safe—surely, if there are ultratrue sentences, there are sentences which would still have been ultratrue if things had been very slightly different. And once we no longer have to take a step from a non-actual world to the actual world, there is no obvious further reason why taking contingent natural relations into account in the distance metric should undermine the case for Series∗ .
Iterating Definiteness 31.6
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H OW FA R D O E S D E F I N I T E N E S S I T E R AT E ?
Once we have agreed that there is a finite i such that no sentence beginning with i is dtrue, it is natural to ask what the smallest such i is.²⁸ This much is clear: the vaguer we take ‘definitely’ to be, the fewer iterations we should expect it to sustain. But what sorts of numbers are we talking about? I don’t know how to make progress with this question in a way that is neutral between different theories of vagueness. So I will approach it from the standpoint of the following simple metalinguistic theory (Lewis 1969, Dorr 2003): for S to be dtrue as used by C at w is for there to be some true proposition P, such that there prevails among C at w a system of conventions that permits asserting S while believing P, and forbids asserting the negation of S while believing P. For short, let’s say in this case that S is ‘conventionally favoured’ by C at w. Those who have doubts about the very notion of a linguistic convention may worry that no sentence whatsoever—not even C0 0 = 1 —will get to be dtrue on this analysis. But if we bracket these doubts, I see no special reason to doubt that among the sentences that are conventionally favoured as used by us (readers of this paper) at the actual world, there will be some which characterize other sentences as ‘conventionally favoured’—for example, C1 C0 is conventionally favoured by us at the actual world. If our community has any linguistic conventions at all, then someone who insisted on asserting the negation of C1 while believing all relevant truths about how we treat C0 would surely manifest a failure to abide by them (including the conventions concerning the uses of the expressions ‘C0’ and ‘conventionally favoured’ that I have just instituted). Just as Yul Brynner is a paradigm case of the sort of person to whom it would be appropriate to apply ‘bald’, so C0, us, the actual world is a paradigm case of the sort of triplet to which it would be appropriate to apply ‘conventionally favoured by . . . at . . .’. And if dtruth is conventional favouredness, the claims of sentences like ‘C0 is dtrue as used by us at the actual world’ and ‘0 = 1’ to count as conventionally favoured, and thus dtrue, seem as strong as those of C1. The case that C2 C1 is conventionally favoured by us at the actual world is conventionally favoured by us at the actual world is not quite as strong, but is still compelling. Granted, no one would be tempted to use C1, us, the actual world as a paradigm case in introducing someone to the use of ‘conventionally favoured’: by comparison to our use of mathematical vocabulary, our use of words like ‘convention’ ²⁸ Opponents of classical logic may reject the presupposition that there is a smallest such i. Still, they will want to know how to answer various specific questions of the form ‘Are there dtrue sentences starting with i ?’
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is fluid, even anarchic. You can imagine someone being so impressed by this contrast that they insisted on applying the term ‘convention’ only to those regularities in linguistic activity that attained the level of rigidity encountered in domains such as mathematics, and thus was disposed to assert the negation of C2. But this would be an excessively finicky way to use ‘convention’. It would surely not be in accord with our actual conventions concerning the use of that word, at least in the kinds of contexts we are presently concerned with. This case becomes much harder to make when we turn to C3 C2 is conventionally favoured by us at the actual world. The level of finickiness in applying the word ‘convention’ that might lead someone to deny C3 is not nearly so high. Without actually being disposed to deny C2, one could still be exacting enough to insist that someone who did act on such a disposition would not thereby count as ‘violating a convention’ about the use of the word ‘convention’. This lower level of finickiness is considerably easier to feel sympathy with; it is less alien to our ordinary practice in applying the word ‘convention’. Would even this constitute a failure to abide by the conventions concerning the use of ‘convention’? At this point I don’t feel at all sure what to say. And I feel even less sure that if I were to answer ‘no’, I would thereby be violating any convention. Given the account of dtruth as conventional favouredness, doubts about whether C3 is conventionally favoured will carry over to sentences like ‘3 0 = 1’. A disposition to deny ‘3 0 = 1’ could arise from the combination of a degree of finickiness about the use of ‘convention’ sufficient to prompt the denial of C3 with an explicit endorsement of the theory that to be dtrue is to be conventionally favoured. It would thus be hard for a proponent of that theory to claim that ‘3 0 = 1’ is conventionally favoured while denying that C3 is. So if we adopt a metalinguistic theory of vagueness along these lines, we will find it hard to maintain that 3 S is dtrue for any S. This doesn’t mean that we will find it easy to argue that it isn’t ever dtrue: the point at which there starts to be overall theoretical pressure to claim that i S is not dtrue comes a bit later.²⁹ But in view of the sharp drop between the degrees of finickiness required to prompt the assertion of ¬i S and ¬i+1 S that emerged in the cases we examined, I can’t see how it could come much later. I think a fairly compelling argument could be made from the theory of dtruth as conventional favouredness to the claim that ‘5 0 = 1’ is not dtrue. I am not sure to what extent these considerations carry over to theories different in character from the convention-based theory. But one general point can be made. If we estimate the ‘degree of vagueness’ of expressions like ‘borderline’ and ‘definitely’ by comparing their use to the use of other expressions in the language, the natural conclusion is that they are extremely vague. The use of these expressions outside of philosophy is largely restricted to a few rather stylized contexts. Whereas our mastery of words like ‘causes’ or ‘believes’ involves impressive feats of co-ordination whose ²⁹ And of course the point at which there starts to be theoretical pressure to claim that i S is dfalse comes later still.
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inner workings are far from being transparent to us, our ability to use expressions like ‘borderline’ is not that big an achievement. There is conspicuously little discipline of the kind that is generally required for the range of cases where a vague expression is dtrue to outstrip the range of cases where its application is uncontroversial. Given these facts, merely admitting that a sentence like ‘5 0 = 1’ is controversial is already enough to put some pressure on the claim that it is dtrue. 31.7
W H Y D O E S I T M AT T E R ?
The question whether any sentences are ultratrue is not of merely technical interest. Its answer bears on several issues at the heart of the philosophy of vagueness. To begin with: if no sentence is ultratrue, there is just no sense in which the rule of ‘-introduction’, P P, is valid. The claim that -introduction is valid has had a wide following in the literature on vagueness (e.g. Fine 1975, Wright 1987, Heck 1993, Keefe 2000, Field 2000, 2003a). It provides one important motivation for thinking that vagueness requires revision of classical logic. For the notion of definiteness would be pointless if ¬P ¬P were valid; but given the classical metarule of proof by contradiction, the latter rule must be valid if -introduction is. It also constitutes a serious obstacle to the project of analysing borderlineness and definiteness. If -introduction is valid, any putative analysis of ‘definitely’ faces something like the open question argument: P follows from P; the putative analysans does not; so the putative analysis must be incorrect. The conclusion that no sentence is ultratrue undercuts any temptation to count -introduction as valid. Assuming that validity in the relevant sense is transitive, all inferences of the form ‘P n P’ must be valid if -introduction is. But even without proof by contradiction, the idea that something dfalse can be validly derived from every sentence cannot be taken seriously. The production of a valid argument from a sentence we have uttered assertively to something we regard as dfalse should, at the very least, compel us to retract or qualify our assertion. But it would be absurd to react to the realization that no sentence is ultratrue by abandoning the practice of making assertions. For the purposes of arguing against -introduction, the claim that very few sentences are ultratrue would do as well as more sweeping claim that none are, since the claim that almost all sentences have dfalse logical consequences is not significantly less absurd than the claim that all do. But the more sweeping claim still has important ramifications. While nobody would be tempted to think of the identification of the ultratrue sentences per se as a central goal of philosophy, the idea that the set of ultratrue sentences is small but nonempty is defensible only on the assumption that there is some more philosophically important feature which the ultratrue sentences share, and which explains why just they get to be ultratrue. This feature might be analyticity (though this would require an unusually restrictive conception of analyticity); it might be logical truth, conceived of as something less arbitrary than mere truth (or analytic truth, or necessary truth) in virtue of the meaning of some list of ‘logical
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constants’. Whatever the deep property that explains ultratruth is, philosophers will naturally be drawn to it as a standard of ultimate theoretical success. We will aspire not merely to express interesting truths (perhaps necessary truths) about the subject matters we investigate (causation, knowledge, right and wrong . . .): we will aspire to uncover the ‘logics’ of these domains, in some exalted sense. And to the extent that we adopt such a goal, we will see a difference in kind between our inquiry and inquiry in domains where ultratruth is not on the cards. But if no sentence is ultratrue, this way of distinguishing philosophical inquiry from inquiry of other kinds can be dismissed as chimerical. There may still be sharp categorical distinctions between different kinds of facts (or propositions); and philosophy may differ from other fields in the kinds of facts it aims to identify. But if there are no sharp discontinuities in the space of ways in which sentences can be used by communities, there can be no categorical distinction between the sentences philosophers aim to produce and sentences of other kinds. When it comes to putting our thoughts into words, we must all muddle along in the same way, doing our best to make ourselves understood with the limited verbal tools at our disposal. Re f e re n c e s Barbour, J. (2006), The End of Time, Oxford University Press, Oxford. Braun, D. and Sider, T. (2007), ‘Vague, so untrue’, Noˆus 41, 133–56. Bureau Internationale des Poids et Mesures [BIPM] (2006), SI Brochure, 8th edn., http://www.bipm.org/en/si/si brochure. Chalmers, D. (2006), ‘Two-dimensional semantics’ in E. Lepore and B. C. Smith, eds., The Oxford Handbook of Philosophy of Language, Oxford University Press, Oxford. Chalmers, D. and Jackson, F. (2001), ‘Conceptual analysis and reductive explanation’, Philosophical Review 110, 315–61. Dorr, C. (2003), ‘Vagueness without ignorance’ in J. Hawthorne and D. Zimmerman, eds., Philosophical Perspectives 17: Language and Philosophical Linguistics, 83–113, Blackwell, Oxford. Fara, D. G. (2002), ‘An anti-epistemicist consequence of margin for error semantics for knowledge’, Philosophy and Phenomenological Research 64, 127–42. (Originally published as ‘‘Delia Graff’’). Field, H. (2000), ‘Indeterminacy, degree of belief, and excluded middle’, Noˆus 34, 1–30. (2003a), ‘No fact of the matter’, Australasian Journal of Philosophy 81, 457–80. (2003b), ‘The semantic paradoxes and the paradoxes of vagueness’ in Jc Beall and M. Glanzberg, eds., Liars and Heaps, 262–311, Oxford University Press, Oxford. (2008), Saving Truth from Paradox, Oxford University Press, Oxford. Fine, K. (1975), ‘Vagueness, truth and logic’, Synthese 30, 265–300. Hawthorne, J. (2006), ‘Epistemicism and semantic plasticity’ in Metaphysical Essays, Oxford University Press, Oxford. Heck, R. G. Jr. (1993), ‘A note on the logic of (higher-order) vagueness’, Analysis, 53, 201–8. Keefe, R. (2000), Theories of Vagueness, Cambridge University Press, Cambridge. Lewis, D. (1969), Convention: A Philosophical Study, Blackwell, Oxford. (1983), ‘New work for a theory of universals’, Australasian Journal of Philosophy 61, 343–77.
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Prior, A. (1968), ‘Changes in events and changes in things’ in Papers on Time and Tense, 1–14, Oxford University Press, Oxford. Quine, W. V. (1940), Mathematical Logic, Harvard University Press, Boston. Soames, S. (1998), ‘The modal argument: Wide scope and rigidified descriptions’, Noˆus 32, 1–22. Sorensen, R. A. (1985), ‘An argument for the vagueness of ‘‘vague’’ ’, Analysis 45, 134–7. Williamson, T. (1987), ‘On the paradox of knowability’, Mind, 96, 256–61. (1994), Vagueness, Routledge, London. (1999), ‘On the stucture of higher-order vagueness’, Mind 108, 127–43. (2000), Knowledge and its Limits, Oxford University Press, Oxford. Wright, C. (1987), ‘Further reflections on the Sorites paradox’, Philosophical Topics, 15, 227–90.
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Index action 494–5, 497–9, 502, 504–5 actuality 558–9, 569 n. 27 additivity 496 countable 499 n. 9 adjectives, degree 229–31, 235, 237–44, 246–51; contrastive uses 242–3, 250 adjunction, see rules of inference, ‘adjunction’ admissible precisification, see precisification agnosticism 166, 172–4, 182, 184–5 ˚ Akerman, J. 10, 281 n. 26 Akiba, K. 155 Altham, J. 371 ambiguity 28, 86, 141, 290, 304, 307–8, 309–11, 551 n. 3, 552 n. 5, 557 analytic-synthetic distinction 161 Anderson, C. A. 131 n. anti-extension 8, 46, 47, 57, 59–60, 78, 83, 188, 191, 231, 233, 238–44, 241, 246, 248–9, 250, 252, 267, 269, 271, 276, 282, 373, 386, 478 anti-realism 9, 165–6, 180, 182, 185; see also realism appearance predicates 247–8 argument from cases, see rules of inference, ‘argument from cases’ assertion 5, 11, 16, 32 n. 16, 38, 39–45, 48–51, 53, 63 n. 1, 126, 191 n. 15, 193 n. 16, 289–91, 295, 304, 307, 310–11, 315–25, 330–3, 346, 348, 351, 352 n. 14, 354, 356, 357, 366, 393, 394–6, 402, 459, 464, 468, 469, 474–5, 492–3, 544, 572, 573 knowledge rule 332 norm of 5, 48–51 of undefined propositions 51–3 truth rule 331 assertoric content 41, 290–3, 302 Azzouni, J. 464 n., 467 n. 5 Bach, K. 368 n. 12 Barbour, J. 556 n. 12 Barnett, D. 92, 103–5, 429 n. 9 Beall, Jc 8, 191 n. 13, 194 n. 19, 400, 401 n. 21, 474 n. belief 15, 25–9, 32 n. 16, 36, 46, 53, 93–5, 101–2, 109, 111–13, 118, 122, 125–7, 150–2, 169, 201–4, 251, 269, 282, 283, 304, 311, 315ff, 353, 364–5, 368, 444–7, 449, 453 n., 484, 497–9, 505
degrees of 15, 491–505; see also uncertainty, ‘-based vs vagueness-based degree of belief’ reports 364–5 stable 201–3 see also partial belief Berry’s paradox 206–7 betting quotient 505 n. 20 bivalence 60, 63, 65, 99, 104–5, 113, 166, 169–73, 175–6, 181–2, 206, 208, 254, 266, 294–5, 297, 300–1, 331 nn. 6 and 8, 332 n. 9, 373–5, 378–80, 386–9, 392, 428 n. 9, 474, 503; see also truth, ‘classical’ Black, M. 23 borderline cases 1–2, 4, 6, 10, 11, 16, 17, 24, 30, 31, 40, 45, 74, 78, 82, 84, 89, 96, 98, 104, 110, 114, 115, 121, 123, 124, 129, 131, 132, 133, 134, 153, 166, 174, 181–4, 230, 248–9, 279, 280, 281, 287 n. 37, 289–91, 294–9, 302, 304, 313 n., 320–2, 327, 329–38, 374, 377–8, 382, 386, 393, 395, 398, 412–13, 438, 443, 447–9, 455, 458, 476, 509–15, 523–48, 555 higher-order 16, 509–15, 528 impossibility of 513 see also higher-order vagueness as optional 509–22 responses to: admissible responses 334–6 forced responses 336 hesitant responses 336–8 ‘macho’ responses 333–4 see also definiteness; operators boundaries 58, 65, 78, 81–2, 88–9, 91–2, 96, 116, 128, 146, 149, 196, 200–3, 206, 208, 259, 264, 268–9, 277–9, 282, 352, 377, 438–9, 443, 445, 455, 462, 476, 484, 494 n. 5, 510, 516, 548, 565 n. 23 hidden 440–3 sharp 9, 16, 153, 156, 166, 168, 169, 176, 182, 190, 200–5, 254–5, 268 n. 12, 276, 283, 296, 370, 480, 375, 377, 464, 474, 527, 530–5, 540, 541, 543; see also sharp cut-offs boundary shifting 276, 379, 382 Braun, D. 190 n. 9, 191 n. 15, 193 n. 17, 551 n. 3
578 Brouweresche (B) axiom 554 n. 11 Brown, A. 522 Brueckner, A. 131 n. Bueno, O. 408 n. 6 Burge, T. 24 n. Burgess, J. A. 150, 215, 220, 221 Burns, L. 67, 309 n. 9 Butler, B. 146 Campbell, R. 67 Cappelen, H. 254 n., 311 n. 11, 366 n. 9 Cargile, J. 232 central gap 254–5, 257–61, 265–71 certainty 496 Chalmers, D. 77 n., 79, 421, 566 change 406, 407, 412 Chisholm, R. 422 Chisholm’s Paradox 422, 424–5, 427, 434 classical solution, see No-Sharp-Boundaries paradox, ‘classical solution’ classicism 166, 182 n. 11, 184 closeness 71, 94–5, 102, 103, 106, 224, 249 closure 3, 168, 170, 222, 286, 287 n. 36, 408 n. 9, 500 n. 12, 501 n. 16, 539 Colyvan, M. 194 n., 400 commitment, first- and second-order 468–71 comparatives 456–7 comparison class 80, 228–30, 232, 237–44, 246, 250, 251, 279–80, 311, 317 n., 324 n., 516 conditional, Łukasiewicz semantics 439, 441, 447, 448 n. 10 conditional proof, see rules of inference, ‘conditional proof’ conjunction fallacy 445 n. 6 context dependence 200, 279, 312 n. 12, 364 context sensitivity 5, 28, 47, 63, 275, 279, 280, 281, 286, 311, 324, 476 and partial definition 57–62 contextualism 5, 9, 10, 44, 188, 194, 219, 233 n. 7, 248, 250, 264, 270, 275–88, 311–14, 317–19, 323–4, 324 n. 25, 328, 424–5 boundary-shifting 276–9, 284, 382 epistemicist 276, 282–7 extension-shifting 276, 277–9, 280 n. 21 indexical 328 non-indexical 328 radical 276, 285–8 contextualist hypothesis 214, 218 continuity through time 420–2 contradictions, half true 458, 459 contraposition 12, 49, 178, 375, 388, 428 n. 9 contrastive uses, see adjectives, degrees, ‘contrastive uses’
Index convention 5, 23, 24ff, 571–3 Cook, R. 23, 433, 441 Copeland, B. J. 155–6, 158, 159 counterpart semantics, see semantics, ‘counterpart’ credence, see belief, ‘degrees of’ crispness postulate 156–9 criteria of application 181–5 criterion of identity (for colour shades) 210, 219, 221 Crupi, V. 445 n. Davidson, D. 149–61, 363 n. 5 Davidsonian semantics, see semantics, ‘Davidsonian’ de Morgan’s laws 499 n. 8 Deas, R. 509 definite truth 11, 17 -introduction 573 dtruth 552, 556–7, 560, 562–7, 569, 571, 572 operator 456, 495 n. 6 definitely operator, see definite truth, ‘operator’ definiteness 286–7, 550–2 absolute 286–7 infinite 552–54 operator 513, 515, 524 n. 3, 528–30, 531–5, 537, 538–9, 540–1, 513, 515, 543 deflationism 68 n. 15, 202 degree functionality 457–62 degree-independent, see mutually degree-independent degree theories 4, 14–16, 233, 264, 379 n. 13, 438, 440, 442 n. 3, 443, 445 n. 6, 447, 448, 455–7 degrees of truth 31–2, 38–9, 87–8, 368, 429–30, 432, 435, 438–9, 441, 446, 449–50, 455–9, 482, 491–505 comparability of 456 as models 440–1 -introduction, see definite truth, ‘-introduction’ demonstratives 12, 360, 362, 365–71 denial 11, 16, 139, 159, 297, 331 n. 6, 379, 394–6, 402, 464, 467–7, 478 n. 22, 480, 544 n. 30, 572 density function 489 designate (designation) 132 n. 4, 143–6, 237 n. 15, 434 desire, see preference determinately operator, see definite truth, ‘operator’ Dietz, R. 23 n., 187 n., 228 n., 254 n., 275 n., 289 n., 327 n., 345 n., 360 n., 491 n.
Index
579
disagreement 87, 268, 327, 328, 331ff faultless 10–11, 327, 328, 329, 331, 332, 333 permissible 72 ff disambiguation 87, 100, 307, 308, 309, 310, 557 discriminability, see perceptual indiscriminability disjunction 12, 13, 14, 15, 53–6, 60, 99, 105, 127, 135, 140–2, 190, 191 n. 11, 201, 310, 375–8, 382 domain (of quantification, restriction of) 8, 237 n. 34, 250 n., 254–60, 266–72 Dorr, C. 17, 23 n., 67, 68 n. 14, 89, 301, 302 n., 345 n., 539 n. 26, 554, 571 double negation elimination 168–9, 178, 185, 533 dtruth, see definite truth, ‘dtruth’ Dummett, M. 5, 16, 48–51, 53, 167, 174, 234, 264–5, 466 n. 4, 474 n., 475 n. 15, 459, 466 n. 4, 474 n., 475 n. 15, 474, 523–5, 537, 544–5, 547 Dutch book 503–4 dyadic fraction 431
expectation 3, 485, 496–8, 499, 500–1, 505 of truth 491, 497, 502–3 expected truth value 15, 485–6, 487–8, 491–505 explanation: inference to the best 172–3 in terms of ignorance 184–5, 282–3 explosion principle, see rules of inference, ‘explosion principle’ extension 8, 17, 43, 46, 47, 54, 57–60, 65, 68–9, 78, 83, 143, 153, 188ff, 194, 196–8, 202, 208, 230ff, 267–9, 271, 275 n. 1, 276 n. 5, 278 n. 11, 279ff, 289–90, 292, 293, 296, 298, 300, 304ff, 319, 321ff, 355–6, 364, 365 n., 368, 373, 374, 377, 386, 408, 432, 439, 440, 443, 467, 478–9, 512, 531, 553 contextual adjustment of 58–9 fixing 230–1, 236–7, 239–44 gap 373 lack of 230–1, 244–9 externalism: about logical knowledge 168, 170–1
Ebbs, G. 150–1 Edgington, D. 91, 92, 103, 392–3, 425 n. 6, 427 n., 429 n. 11, 438 n., 441, 457, 459, 462, 482 n. Eklund, M. 5, 77, 84–8, 234, 464 n. Elder, Crawford (Tim) 187 n. elimination of alternatives 173 epistemic state 114, 496, 502 epistemic theories of vagueness, see epistemicism, vagueness, ‘epistemic theories of’ epistemicism 4, 64ff, 95, 100, 103, 131 n. 1, 153–4, 160–1, 166, 169, 184–5, 188, 194, 277, 283–5, 289, 296–7, 305–6, 312, 332, 333, 335, 337, 364–5, 370–1, 379 n. 13, 424 n., 443, 455, 457, 460, 462, 558–62; see also vagueness, ‘epistemic theories of’ fuzzy 438 equivalence-approximation 222–6 equivocation 278 eternalism 315–16 evaluation, circumstances of 281 n. 26, 304, 306–8, 309 n. 8, 314–18 Evans, G. 146 n. 27, 155–6, 225, 272, 316, 351 n. 12, 428 evidence 2, 9, 50, 53, 101–2, 111, 116, 141, 169, 171–3, 209, 211, 214–19, 225, 231, 233, 307 n. 6, 381, 389, 391–2, 394, 495, 505, 511, 513, 516 n. 9, 544 excluded middle 13, 53–7, 60, 61, 62, 99, 185, 204, 207, 377, 387
faultless disagrement, see disagreement, ‘faultless’ Fara, D. G. 2 n. 1, 3 n. 2, 12, 44, 77 n., 78, 85, 194, 209, 212–13, 218, 220, 228–9, 233, 264, 272 n. 1, 276 nn. 5 and 7, 278 n. 15, 300 n., 312 nn. 12 and 13, 313, 319 n. 22, 324 n. 25, 373–82, 427 n., 443, 477 n., 509 n.2, 510 n. 4, 526 n. 8, 527, 534 n. 16, 535 n. 17, 536 n. 21, 537 n. 24, 543, 554 n. 11; see also gap principles Field, H. 8, 9, 78, 99, 111, 126, 200–8, 289 n. 1, 304 n., 446 n. 8, 495 n. 6, 554 n. 10, 573 Fine, K. 12, 67 n. 9, 78, 92, 105, 134 n. 8, 194 n. 19, 195, 233, 264, 292 nn. 4 and 5, 309 n. 9, 330 n. 5, 360 n. 1, 373, 375, 378, 379 n. 14, 381, 387, 390, 394, 396, 403, 419 n., 442, 484 n. 4, 486, 525 n. 5, 543, 550, 551, 573 fission (and fusion) 407, 412 Fitelson, B. 438 n. Forbes, G. 16, 419–36, 456 n. 14 forced march, see sorites, ‘argument, forced march’ Frege, G. 71, 141, 156, 171, 190–2, 197–8, 210, 348, 350, 358, 361 n. 3, 363, 464, 468, 471, 480 Frege-Geach problem 464, 467–73, 480 gap principles 534–6, 538–9, 540, 543, 544 n. 30
580 Garc´ıa-Carpintero, M. 12–13, 228 n., 254 n., 298 n. 10, 327 n., 343–58, 363 n. 5 Gardner, M. 208 Gaussian distribution 487 Geach, P. 464, 468, 471, 480 genericism 232, 234–5, 243, 246–7 Gilbert, M. 24 n. Glanzberg, M. 5, 16, 48, 53, 474–5 Goguen, J. A. 283 n. 29, 456 n. 14, 458 n. 18 G´omez-Torrente, M. 8, 228–52 Goodman, N.: identity criterion for colour shades 221–3 matching relation 210 Grandy, R. 24 n. Greenough, P. 10, 23, 77, 81, 84–5, 88–90, 165 n., 171, 209 n., 254 n., 275-88, 464 n. Groenendijk, J. 430 n. Haack, S. 455 Hajek, P. 442 n. 3 Hardin, C. L. 214, 226, 512, 522 Harman, G. 118 n. 13 Hawthorne, J. 135–7, 138, 139 n. 16, 146–7, 560 n. 17, 569 n. 26 Heck, R. 66, 69, 233, 279, 281, 283, 285 n. 13, 431, 509 n. 2, 536 n. 21, 573 Hegel, G. 406 hidden boundaries, see boundaries, ‘hidden’ higher-order borderline cases, see borderline cases, ‘higher-order’ higher-order vagueness 11, 16–17, 37, 64, 66, 84–5, 88, 97, 110, 127–8, 204–6, 285–8, 304, 321–3, 330 n., 346 n., 349–51, 395, 400 n. 17, 29–31 455–6, 475–81, 483, 485, 489, 509–22, 523–49, 562 n. buffering view of 527–8, 535, 537–42, 544, 546, 548 and the ineradicability intuition 523–5, 527, 529, 544–8 metalinguistic 515, 517, 521, 522 paradox of 535–6 prescriptive 509, 515, 517, 519 and the seamlessness intuition 525–7, 529, 534, 539–44 Horgan, T. 85, 96, 152, 153, 264, 347, 475 Horsten, L. 9, 212 n., 222, 224 Horwich, P. 200–3, 205, 207–8 Hume, D. 435 Hyde, D. 13–14, 157–8, 194 n., 373, 397, 399 n. 2, 412 n. 14, 509 nn. 12 and 13, 522 Iacona, A. 11, 291 n. idealism 149–50, 152, 159
Index identity 6, 14, 16, 72, 146 n. 27, 152–3, 155, 176–7, 210, 219, 221, 222, 226, 227, 366, 545, 406ff, 419ff personal 407, 413 incoherence 201, 394 n., 395, 396, 477, 480, 504, 518, 520 incoherentism 3, 8; see also Beall, Jc; G´omez-Torrente, M.; Pagin, P. inconsistent predicate, see predicates, ‘inconsistent’ indeterminacy 5, 11, 55, 59, 60, 63ff, 78, 79, 82–5, 104, 110, 114–15, 125, 131, 133–5, 138–9, 143, 144, 146, 153, 155, 203–4, 305–6, 308–9, 312, 314–16, 323, 346, 352 n. 12, 362 n., 378, 386, 398, 422, 427–8, 429, 431–2, 467, 472, 475, 478–80, 526, 541 first-level 5, 64ff second-level 5, 64ff of reference 427 of translation 63 indexicality 308, 311–14, 323–4 indiscriminability, see perceptual indiscriminability ineradicability 17; see also higher-order vagueness, ‘and the ineradicability intuition’ inscrutability of reference 63 intensions 166–8, 175–85, 187ff, 230–1, 235–7, 240–1, 244, 269, 307, 560 n. 19, 566 classical 178, 181–3, 185 vague 187–9 intentionality 406 Jackson, F. 71, 566 Jamieson, D. 24 Ja´skowski, S 397 n. 11, 401 Johnston, M. 71, 83 Kahneman, D. 445 n. 6 Kamp, H. 44 n. 30, 312 nn. 12, 13, and 14, 318, 323, 324 n. 25, 373, 378, 379 n. 13 Kant, I. 149 Kant-Quine thesis 150–2, 154, 155, 157, 159–61 Kaplan, D. 281 n. 26, 306 n. 3, 309, 315 n. 17, 328, 353, 367–8 Keefe, R. 12, 64 n., 67 n., 92, 264, 279, 286–7, 301 nn. 13 and 15, 336, 354, 358, 360–71, 373, 377 n. 11, 378–9, 381, 385, 390–2, 396, 401–2, 404, 430, 436, 440 n. 2, 441, 456 n. 14, 457, 459, 573 Kennedy, C. 241 n. 22, 455
Index King, J. 315 n. 20 Kleene, S. 16, 135, 142, 191, 196 n., 198, 387, 468, 470 Klein, E. 241 n. 22 knowledge 2, 5, 7, 25–7, 37, 46, 50ff, 103, 116–17, 125, 137, 168, 183, 185, 201–4, 242, 265, 276–7, 282–5, 305, 309, 313, 320 n. 24, 327, 329, 332ff, 443, 518, 554–5, 559, 561–2, 574 safety-based account of 282–3 K¨olbel, M. 10, 307 n. 4, 309 n. 8 Koslicki, K. 419 n. Krause, D. 408 n. 6 Kripke, S. 69 n., 98, 235–6, 244, 474 LP 14, 398, 408ff, 414 Lakoff, G. 455 Larson, R. 348, 366 n. 10 least number principle 9, 200, 204, 206–7, 278 n. 12 Leibniz’s Law 136, 156, 410f, 428 Lepore, E. 311 n. 11, 366 n. 9 Lewis, D. 24, 26, 28, 29, 32, 36, 39–41, 67, 71, 83, 97, 99, 103, 133, 135, 135 n. 11, 136, 137, 138, 138 n. 14, 139, 139 n. 16, 144, 155, 267, 272, 306 n. 3, 309 n. 9, 323, 328, 335, 346–7, 351, 355, 368–70, 426, 436, 448, 475, 502–3, 567, 571 liar paradox, see semantic ‘paradoxes’ Lindsey, D. 514, 522 linguistic competence 23, 37, 42, 93, 265, 307; see also semantic competence linguistic vagueness, see vagueness, ‘linguistic’ logic 287 classical 3, 5, 9, 11, 13, 56, 63, 65, 67 n. 13, 81–2, 156, 166, 168–71, 173–8, 182, 197–8, 203, 206, 229, 240, 276 n. 7, 278 nn. 10 and 12, 279, 289, 295, 297, 301, 346, 374, 385ff, 397, 440, 484, 531, 533, 534, 535 n. 17, 536, 543, 571 n. 28, 573 fuzzy 14, 15, 16, 65, 72, 414, 429–30, 432–3, 435, 482–7, 489, 494, 496, 501–2 intuitionistic 166, 168, 178–9, 185, 277 many-valued 4, 14–16 modal 376 n., 377 n. 10, 561–2 paracomplete 387 weakly 388 paraconsistent 13–14, 397, 407ff weakly 398 Strong Kleene 16, 191 subvaluationist 13–14 supervaluationist 4, 11, 13 see also Łukasiewicz, J.; rules of inference
581 logical consequence 41, 175, 394–6, 399, 402, 423, 551, 570, 573 multiple-conclusion 386 n. 5, 395–6 logical expressions 230, 232–3, 240, 243–4, 246 L´opez De Sa, D. 11, 275 n., 327 n., 328 Lowe, E. J. 187 n., 419 Łukasiewicz, J. 14 continuum-valued logic 205, 208 strong conjunction 442 n. 3, 444 n. 5 MacFarlane, J. 46 n., 126, 127, 328 n. 2, 446, 447 n., 493 McGee, V. 67 n., 68 n., 92, 99–100, 103–5, 345–8, 353–6, 298 n. 11, 301–2, 325 n. 26 Machina, K. F. 233, 264, 442 n. 3 McLaughlin, B. 67 n., 68 n., 92, 100, 103–5, 298 n. 11, 301–2, 325 n. 26, 345, 354 Manning, L. 131 n. Manor, R. 268, 272 margin of error 443, 558–61 matching relation, see Goodman, N., ‘matching relation’ meaning 27ff, 255, 264, 268–9 relation to use 564–73 measurable: function 499 set 499 measure 260, 266–9; see also probability, ‘measure’ Merricks, T. 142 n. 23, 143 n. 24, 160–1 metaphysical realism, see realism, ‘metaphysical’ metaphysical vagueness, see vagueness, ‘metaphysical’ metasemantic acount (of vagueness), see vagueness, ‘metasemantic account of’ Milne, P. 419, 432 n., 491 n., 503 n., 505 n. 20 model-theoretic semantics, see semantics, ‘model-theoretic’ modus ponens, see rules of inference, ‘modus ponens’ Montague, R. 82 Moruzzi, S. 23 n., 46 n., 165 n., 187 n., 275 n., 327 n., 360 n. multidimensional predicates, see predicates, ‘multidimensional’ mutually degree-independent 452, 457 n. 16, 460 n. name-named relation: ‘Fidelle’-Fidelle model of 116–18 ‘Midtown’-Midtown model of 116–20, 124, 128
582 natural kind 47, 58 n. 14, 123, 181, 230, 244, 252, 368 terms 249, 368 natural properties 97 n. 9, 369, 567–8 naturalness 93, 357 n., 368–70, 560, 567–9 negation 457, 467–71 nihilism 167–8, 193 n. 17, 197, 235 strong 235 instability of 235, 252 no fact of the matter 7, 84, 131, 132 n. 4, 133, 137, 138, 140, 142, 144, 145, 245, 421ff no-no paradox 96–7 No-Sharp-Boundary paradox 167 ff, 530–5, 539, 543, classical solution 166, 168–71, 173, 182, 184–5; see also sorites Nolan, D. 77 n., 392 n., 398, 399, 79, 345 n. non-contradiction 61, 392 n., 398, 399 non-monotonicity 411 non-nihilist 188, 193–5, 196, 197, 199 nontransitivity 209, 210, 211, 213–16, 221 non-truth-status theories of vague properties, see vague, ‘properties’ nouns, scalar and non-scalar 229–30, 246–7 number uniformity 257, 268 objectivity 150–2, 159 occasions of use: irregular 230–1, 244–51 paradoxical 230–1, 234 n. 10, 246–52 regular 230–1, 249 closeness of 231, 249–50 odds 504 omniscience 201–2, 562 omniscient speakers 519–20 ontic vagueness, see vagueness, ‘ontic’ ontological vagueness, see vagueness, ‘ontological’ operators: borderline case 377–8 deontic 381 falsity 380 item satisfiability 376–7 truth 379, 381 see also definite truth, ‘operator’; definiteness, ‘operator’ optimism, see sorites, ‘optimism about’ Pagin, P. 8, 250 n., 275 n., 305 n. 1, 311 n. 11 paraconsistency 385, 397, 401, 404; see also logic, ‘paraconsistent’ paradigm cases 230–1, 236–7, 240, 242, 244–5, 248, 250–2, 327, 330, 333, 337–8, 467 Parfit, D. 407 n. 4
Index Parsons, T. 146 n. 27, 395 n., 419, 428–9, 434 partial belief 15 classical 112–13, 122, 126–7 standard (SPB) 445, 446 vagueness-related (VPB) 122, 126, 445, 494 see also belief, ‘degrees of’ partial definition 5, 48–51, 53–7, 59–62 Peacocke, C. 158, 426, 436 Peacocke, T. R. 419 Pelletier, J. 311 n. 11 penumbra 111 n. 7, 121, 128, 352, 524, 544 penumbral connections 16, 184, 195, 196, 268, 330, 354 n., 357 n. 18, 362–8, 370, 374, 486–8, 562–4 penumbral principle 257 n. perceptual indiscriminability 58 n.14, 209–10, 211, 214, 218 permissible disagreement, see disagreement, ‘permissible’ personal identity, see identity, ‘personal’ phenomenal continua 211, 212 Plato’s Heaven 115–19, 123 pleonastic properties 120 n.14 possibility 376 n. 8, 377–8 relative 424–7, 435 possible worlds 10, 28–9, 32–3, 39–43, 77, 98–9, 120–1, 237 n. 15, 241 n. 21, 269, 306, 308, 347, 352 n. 12, 363, 377 n. 10, 422, 424, 434, 496–7, 499, 502, 551 n. 3, 556–9, 561, 564, 565 n. 23 similarity between 556, 567–8 pragmatics 5, 24, 37ff, 191 n. 15, 304, 321, 491 n. conversational 491 n. precise boundaries, see boundaries, ‘sharp’ precisification 10, 11, 13, 67, 73, 133, 135, 360–71, 373–4, 376–8, 484; see also sharpening precision, see vagueness preconceptions 8, 230–1, 235–47 abstract 237–9 concrete 237–9 generic 230–1, 237–9 paradigm 230–1, 237–9, 244–5, 250–2 predicate-property relation 119 predicates: inconsistent 85–6 multidimensional 456–7 neither vague nor precise 521 tolerant 85, 89, 531 vague 525, 530 competent use 516 variable application 516 predication 132 n. 4
Index without correspondence 132 n. 4, 133, 134, 143, 144–5 preference 151, 231–4, 250–2, 316 n. 20, 396, 497–9 prescriptive higher-order vagueness, see higher-order vagueness, ‘prescriptive’ primitivism, see sorites, ‘primitivism’ probability 5–6, 15, 53, 95 n. 5, 102–3, 112, 235, 449, 450–3, 457, 460, 462–3, 482, 485, 487, 489, 491–505 axioms 112, 496 calculus, see probability, ‘axioms’ measure 496, 499, 502 problem of the many 106, 355–6 property complement 132 n. 3, 143 propositions 12, 15, 18, 28–34, 37–43, 48–57, 94, 97, 98, 101, 102, 104, 109–10, 113, 115, 121–2, 126–9, 132, 133, 137–42, 146, 160–1, 246 n. 26, 269, 283, 297, 306–7, 310–11, 315–20, 323–5, 348–57, 361–4, 377, 426–28, 430, 430–3, 438, 441, 443–58, 460, 462, 467, 470, 491–504, 536, 550 n., 558, 559 n. 15, 561, 567, 571, 574 Protagoras 149 Priest, G. 14–15, 23, 187, 275 n. 7, 398, 406 n. 1, 407 nn. 3 and 5, 408 nn. 7 and 8, 410 n. 11, 411 n. 12, 413, 414, 414 n. Prior, A. 315 n. 17, 407 n. 5, 550 n. 1 pseudo-true (pseudo-truth) 139–42 Putnam, H. 149, 152, 154–5, 158 quandary 73, 76, 106, 11–15, 121–2, 126–7, 129–30, 546 Q-Constraint 111, 113–15, 123, 129 quantifier 15, 88, 233, 254, 266, 315, 356 n. 17, 374, 380, 390–2, 394, 408, 428 Quine, W. V. O. 63, 66, 149, 150–2, 155–7, 159, 160–1, 422, 436, 551 n. 2 Raffman, D. 16, 44, 194, 213–17, 264, 272, 312 nn. 12, 13 and 14, 318, 324 n. 25, 378 n., 379 n. 13, 419, 424–5, 436, 443, 477 n., 522 Ramsey, F. 494–5 random variable 485, 496, 499, 500 Rayo, A. 4, 5, 23, 63 n., 68, 254 n., 464 n. realism 150–1, 154–5, 158, 162, 165–6 metaphysical 150–1, 154–5, 158 see also anti-realism Recanati, F. 306 n. 3, 315 nn. 18 and 20 reductio ad absurdum, see rules of inference, ‘reduction ad absurdum’ reference 9, 171, 174, 176–85 determination of 166, 174–6, 179–80, 182–3, 185
583 direct 135–6, 146 failure of 63, 71, 117, 121, 126 n. 18, 230, 232, 244–50 fixing 47–8, 231, 236, 238–9, 240–2, 564–5 theory of 166, 174–5 relativism 328–9 moderate 328 radical 328, 331 representational vagueness, see vagueness, ‘representational’ Rescorla, M. 131 n. Restall, G. 187 n., 394–5 Richard, M. 16, 315 n. 19 rigidity 236–7, 572 Robertson, T. 131 n., 419 n., 424 n. Rosen, G. 150, 155, 419 n. Rosenkranz, S. 8, 9, 166 nn. 1 and 3, 182 n.11, 183, 228 n., 250 n., 254 n., 275 n. rules of inference: adjunction 13, 399, 401–2 argument from cases 12 conditional proof 12, 388, 403, 536, 538, 563 n. 21 explosion principle 13 modus ponens 3, 13, 42, 318, 387, 388, 400, 403, 436, 440, 449, 483, 557 reductio ad absurdum 12, 136, 375, 448, 536 n. 21 subjunction 14, 388, 392–3, 395–6, 402–3 universal generalization 381, 382 universal instantiation 2, 3, 42 Russell, B. 141, 200, 201, 203–5, 207, 374 n. 5, 521, 524, 525, 537, 544–7 S4; 287, 536 n. 19, 556, 558, 563, 564 n., 566 n., 567 n., 569 SPB, see partial belief, ‘standard’ (SPB) Sainsbury, M. 64 n., 65 n., 78, 152–3, 155, 159, 233, 237, 239, 249, 253, 310 n. 10, 371, 433, 436, 447, 476, 510, 511, 522, 548 Salmon, N. 7, 131 n. 1, 134 n. 7, 138, 139 n. 17, 146 nn. 26 and 27, 147 n. 28, 236, 253, 419, 423–5, 428, 436 satisfaction 8, 121, 122, 177, 181, 185, 188, 189ff, 467 n. 5, 555 Fregean 190 Liberal 190 satisfiability 375; see also operators, ‘item satisfiability’ Saul, J. 360 n., 412 n. 13 Schiffer, S. 6–7, 12, 24, 65 n., 105 n., 109–11, 120, 123, 126, 233, 253, 360–7, 443–9, 492–4
584 scope confusion 378–81 seamless transition 17, 525–7, 535 n. 17, 536, 537; see also higher-order vagueness, ‘and the seamless intuition’; transition problem Segal, G. 366 n. 10 semantic: competence 85–6, 89; see also linguistic competence incompleteness 525–6 paradoxes 552 n. 4, 554 n. 10 uncertainty 67 vagueness, see vagueness, ‘semantic’ semantics 32ff counterpart 424–7 Davidsonian 83 model-theoretic 153, 156 set theory (paraconsistent) 414f Shapiro, S. 6, 187 n., 194, 264, 272, 276 nn. 4, 5, and 7, 278 nn. 13 and 14, 280 nn. 20 and 21, 312 n. 12, 334, 336, 433, 514, 522 sharp boundaries, see boundaries, ‘sharp’ sharp cut-offs 3, 5, 8, 64, 120, 131, 167, 171, 183, 219, 233 n. 7, 246–7, 250 n., 284 n. 31, 402 n., 428 n., 485, 510, 519, 534, 540; see also sharp boundaries sharp descendent (of a predicate) 192, 197 sharpening 99, 103, 133, 134–8, 140, 142 n. 22, 157, 169, 316–23, 325, 330, 369 coordinated 142 n. 22 Sider, T. 190 n. 9, 191 n.15, 193 n.17, 551 n. 3 σ -field 499, 501 n. 15, 502 Simons, P. 15–16, 283, 482 n. Skyrms, B. 502–4 Smiley, T. 414 n. 16 Smith, N. J. J. 15, 68, 77, 81, 84, 87–9, 155, 419, 453, 491 n., 505 n. 21 Soames, S. 4–5, 44, 48 n. 1, 52 n. 10, 55 n., 58 n. 15, 59 n., 60 n. 18, 61 n. 20, 233, 264, 275 n. 1, 276 n. 7, 311 n. 11, 312 nn. 12 and 13, 319 n. 22, 324 n. 25, 379 n. 13, 424 n. 6, 443, 465–7, 477, 526 n. 7, 559 Sorensen, R. 5, 68 n. 16, 74 n. 27, 85, 91–106, 131 n. 1, 166 n. 2, 171, 194, 277, 283, 284 n. 31, 305 n. 2, 424, 440, 445 n. 6, 509 n. 2, 528 n. 11, 559 n. 16 sorites 4, 9, 10, 14, 15, 17, 72, 91–3, 98, 106, 147 n. 28, 187, 254–8, 263–5, 269–72, 299–300, 304–5, 317–18 argument 13, 80, 85–6, 91, 101–2, 190, 254, 256–8, 264, 318, 374, 378, 381, 413 ff
Index forced march 264, 413f, 537 n. 23, 540 optimism about 232–3, 246–7 paradox 2–3, 7–9, 14, 42–4, 72, 80–1, 85, 100, 109, 167, 228–35, 247–50, 252, 255, 263, 269, 275–82, 284–5, 385, 389, 390, 399, 401, 403, 429–30, 432, 433, 438–9, 442, 455, 458, 462, 530–5, 539 contextualist solution 443 psychological solution 445 susceptibility to 280–2, 284–5 phenomenal 8–9 primitivism about 233 reasoning, see sorites, ‘argument’ series 2, 3, 7, 8, 232, 248, 464, 491, 497, 510–13, 515, 536, 537, 539 and sharp boundaries 476–80 see also No-Sharp-Boundaries paradox, tolerance speech acts: and compositionality 468–71 and logic 470–1 speech reports 313–14 indirect 12–13, 360–71 Stalnaker, R. 32 n.16, 39ff, 328, 491 n. Stanley, J. 270, 272, 310 n. 10, 324 n. 25 Stokhof, M. 430 n. Storage Room puzzle 423 strong conjunction, see Łukasiewicz, J. subjunction, see rules of inference, ‘subjunction’ subvaluationism 13–14, 188, 194, 385, 397–404 supertruth 136, 137, 139–40, 145, 300, 302, 429 supernumeration 15–16, 482–90; see also supervaluationism supertruth, see supervaluationism, ‘supertruth’ supervaluation, see supervaluationism, ‘supervaluation’ supervaluationism 11–13, 54–5, 67, 83, 89, 169–70, 264, 300–2, 319–21, 325, 373–82, 360–71, 386–9, 427, 429, 433 global validity 301 n. 15 local validity 301 n. 15 supertruth 136, 137, 139–40, 145, 300, 302, 429 supervaluation, 100, 134–5, 137, 140–1, 225, 300, 325, 375, 377–8, 483–6 see also vagueness, ‘supervaluational theories of’ supervenience 36, 175, 179, 181–2 on use 166
Index Tarski, A. 552 n. 4 temporalism 315–16 Tennberg, C. 131 n. Theseus, ship of 406, 412f; see also sorites, ‘argument’ Thomason, R. 133 n. 5 tolerance 2, 3, 4, 5, 7–9, 10, 12–15, 70, 71, 85, 86, 88, 90, 94–5, 102, 106, 158, 187–91, 194–9, 254–60, 264–7, 269–71, 276–7, 279 n. 15, 304, 311, 322, 423–4, 510, 530–1 epistemic 276 n. 6 full 187 level 256, 258, 260, 267, 270–1 relation 189 strong 277–8, 280 n. 22, 284 weak 276–7, 278 n. 15, 280, 283–4 tolerant predicate, see predicates, ‘tolerant’ transition problem 538, 539–44 Travis, C. 310 n. 10 truth 37–9, 41–2, 551–2, 560 n. 17 classical 183–4; see also bivalence correspondence vs disquotational 348 definition in a vague language 472–4 disquotational 289, 295, 297–8, 300–1 partial 439 ultratruth 553 see also definite truth; degrees of truth; pseudo-true (pseudo-truth); semantic, ‘paradoxes’; supertruth truth conditions 8, 29, 32, 48–50, 54, 141, 153, 165, 169, 189, 190, 194 n., 230–2, 243, 246, 249, 292, 320, 325, 345–8, 351–3, 357, 361, 363, 370, 375, 377 n. 10, 382, 439, 528, 560, 564–6 truth-status theories of vague properties, see vague, ‘properties’ truth-value gaps 11, 12, 16, 48, 49, 63, 96, 135, 230, 246–50, 274, 374, 387, 394, 397, 403, 414, 464–81 and assertion 464, 474–5 truth-value gluts 13, 14, 386, 397, 402, 403, 412 truth-value shift 375, 382, 390 truthmaker gaps 96–7, 99 Tversky, A. 445 n. 6 two-dimensionalism 565–6 Tye, M. 65 n. 8, 140 n. 19, 433, 509 n. 2 UFS (uncertainty-free situation) 500–2 ultratruth, see truth, ‘ultratruth’ uncertainty 491, 493–5, 502, 504–5 -based vs vagueness-based degree of belief 493–5 underspecification 292–6, 300–2 Unger, P. 167, 193 n. 17, 234 n.9, 253
585 uniformity constraint 422–8, 434 uniformity principle 257, 268 units of measurement 557, 565, 567 universal generalization, see rules of inference, ‘universal generalization’ universe of discourse 228–9, 237–9, 247 greatly unrestricted 243, 247, 252 utility 187, 190, 191ff, 191 n. 15, 487 expected 497 see also preference VFS (vagueness-free situation) 500–4 VPB, see partial belief, ‘vagueness-related’ (VPB) vague: ascriptions, de re 352–3, 356–7 moral predicates 79–80, 86–9 objects 349–51, 484; see also identity; problem of the many predicate modifiers 79, 87 properties non-truth-status theories of 115, 123, 129 truth-status theories of 113, 115 bivalent 113–14 non-bivalent 114–15 proposition 316–17 vague predicate, see predicates, ‘vague’ vagueness: arising from deficiency of meaning 465–7 as semantic indecision 347–8 characterization of 36–7, 249 and context sensitivity 476–7 contextualism about 368; see also contextualism definition of 77, 85–90 degree theory of 368 dual picture of 230, 232, 240–4, 246–7, 249–50, 252 epistemic theories of 110, 113–14, 558–62; see also vague, ‘properties’ inconsistency view on 70ff linguistic 149–61 linguistic theories of 550–1 metaphysical 149–61 metasemantic account of 23ff ontic 133; see also vagueness-in-the-world, vagueness, ‘ontological’ ontological 149–61, vagueness, ‘ontic’; vagueness-in-the-world primitivism about 103–6 representational 149–61 semantic 149–61, 304–26 supervaluational theories of 562–4; see also supervaluationism
586 vagueness-based degree of belief, see uncertainty, ‘-based vs vagueness-based degree of belief’ vagueness-in-language 7, 9, 131–40, 142, 143, 144, 145, 146, 149 vagueness-in-the-world 133, 134, 136, 143, 144, 145, 146, 147 n. 28, 160, 225; see also vagueness, ‘ontic’; vagueness, ‘ontological’ van Fraassen, B. 134 n. 8, 191 n. 13, 386 n. 3 Vander Laan, D. 131 n. Varzi, A. 187 n., 193 n. 16, 194 n., 301 nn. 13 and 15, 304 n., 390, 396, 397 n.13, 431 n.13, 509 n.2, 528 n.11 Weatherson, B. 5, 70 n.22, 187 n., 193 n. 18, 352 n.13, 354, 356, 358, 363 n. 6, 368–70, 442, 448 n. 10, 456 n. 14, 457, 458 n. 17 Wheeler, S. 187 n., 193 n. 17 Williams, J. Robert G. 187 n., 485 n. 5 Williamson, T. 3 n. 2, 16, 23 n. 1, 37, 37 n. 19, 50–1, 50 n. 9, 64 nn. 5 and 6, 65 nn. 7 and 8, 95, 96, 131 n. 1, 153–5, 194, 222, 232, 234, 239, 254 n. 264, 268 n. 12, 277, 282–4 284 n. 33, 287 n. 38, 289 289 n. 2, 295, 296 n. 6, 297,
Index 297 nn. 7, 8, and 9, 298 n. 300–1, 301 nn. 13, 15, and 16, 305 n. 2, 320 n. 24, 331 nn. 6 and 8, 332, 332 n. 9, 333, 347 n. 4, 364 n. 7, 367, 374 n. 6, 388, 388 n. 6, 424 n. 6, 427, 432, 442, 442 n. 3, 453, 455, 456, 456 n. 14, 457, 459, 459 n. 20, 482, 519, 519 n. 13, 525 nn. 5 and 6, 528 n. 12, 539 n. 26, 551, 558, 559, 559 n. 15, 560–2, 562 n. 20 Wilson, M. 150–1 Woodruff, P. 434 n. Wright, C. 9, 17, 23 n., 70 n., 72–4, 85, 88, 94 n., 105 n. 14, 111, 113–15, 122 n., 158, 165, 166, 169, 185, 187 n., 197, 209, 211, 212, 213, 214, 217, 220, 221, 225–6, 233 n. 8, 254–5, 265, 275 n., 320, 327 n., 328, 337–8, 345, 358, 419 n., 429 n. 11, 431 n. 13, 447, 464 n., 479, 509 n. 1, 526 n. 7, 530 n., 531 n., 535 n. 18, 544 n. 30, 546 n., 548 n. 34, 573 Zadeh, L. 450 n. 11 Zardini, E. 165 n., 275 n., 281 n. 26, 287 n. 39, 327 n., 360 n., 514, 523 n., 538 n. 25 Zimmerman, E. 419 n.
