Abstract
In this chapter, I present the Theory of Vagueness as Arbitrariness (VA). In Sect. 5.1 I consider some minimal constraints on the use of vague predicates. I argue that the principle of tolerance should not be considered a constraint, and that the clear-case constraint should be replaced by the ideal-case constraint. In Sect. 5.2, I argue that the notion of ideal case does not imply a violation of the criterion of precisification, at least if we accept the following intuition: all admissible precisifications of a vague predicate are equally arbitrary. My interpretation of this intuition is the first part of my theory: the Thesis of Arbitrariness (TA). TA is in line with some of the main theses about vagueness advanced by Raffman and Sainsbury. Nonetheless, I propose that we should augment it in order to achieve a final definition of vague predicates. The result is VA. According to VA, a vague predicate is an arbitrary predicate that must be precisified in order to contribute to sentences with truth-conditions. VA naturally leads us to Semantic Nihilism. Following Braun and Sider, I argue that Semantic Nihilism can be made viable by an account of how vagueness is typically and harmlessly ignored. Because Braun and Sider’s proposal depends on the existence of a clear-case constraint, an alternative proposal is outlined. I then argue that VA satisfies all three criteria of adequacy for an ideal theory of vagueness, and that it correctly systematizes the relevant intuitions. The chapter closes with replies to some possible objections (Sect. 5.3).
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Notes
- 1.
For the sake of simplicity, I consider only pairs of predicates. Nonetheless, it should be obvious that coordinate constraints are not limited to pairs. For example, there are similar constraints on the following set of predicates: “very small”, “small”, “medium”, “big”, and “very big”. In principle, coordinate constraints can hold for any number of predicates.
- 2.
I will not address the multi-dimensionality of “bald”. Not just the number of hairs, but also how they are distributed on someone’s head is relevant to the application of this predicate. Because of this, it is arguable that a person who has n hairs on her head is bald, while a person who has fewer than n hairs is not bald. We could handle this problem by saying that, all other things being equal, ∀n (Ban → Ban − 1) will hold. In so doing we simplify the discussion by isolating the effects of the number of hairs on the application of “bald”, while ignoring the effects of any other standards for the application of this predicate. For example, let <a0, a1, a2, …, a5,000, a5,001 …, a10,000> be a soritical sequence for “bald” such that a9,999 is the result of removing a single hair from a10,000’s head, a9,998 the result of removing a single hair from a9,999, and so on. In this case, the distribution of hairs on an’s head will be different from the distribution of hairs on an − 1’s head only to the extent that the former has one more strand of hair on her head. It seems clear that, in this case, the constraint ∀n (Ban → Ban − 1) will hold. The multi-dimensionality of vague predicates implies that constraints like that above hold only under certain conditions. Since it is usually difficult to state these conditions, the debate on vague predicates often goes under the simplifying assumption that they are one-dimensional. I will make this simplifying assumption here. Note, however, that this does not affect my further arguments, since all of them may be formulated so as to take into account the multi-dimensionality of vague predicates.
- 3.
I am not really sure about the plausibility of one discovering a linguistic rule that is systematically violated. Suppose a linguist claims that she has found a rule for the application of a predicate F. You then ask her what this rule is, and she replies it is R. You examine the way speakers use F, but you find that ordinary use of F systematically violates R. In this case, you should be suspicious about the claimed discovery. In fact, while there seems to be no problem in the idea of, for instance, a legal rule that is systematically violated, the idea of a linguistic rule that is systematically violated by ordinary speakers seems to be problematic. At least, an alternative explanation that did not predict such widespread mistakes would be preferable. As Raffman well notes, “it is a fundamental tenet of good theory construction that, all else being equal, a theory that does not have us chronically, irremediably (and hitherto unbeknownst to us) mistaken in the application of our own ordinary words is preferable to one that does” (Raffman 2014: 100). As we will see, my own account of the rules for the use of vague predicates, which is in line with some of Raffman’s main theses, is able to avoid this consequence.
- 4.
I am not really sure what Wright’s crucial thesis is, but I think it is something like this. His main goal is to argue against what he calls the “governing view”. The governing view is a conjunction of two theses: (i) our use of language can be correctly viewed as a practice in which the admissibility of a move is determined by rules; (ii) the general properties of these rules can be discovered by intuition and “working knowledge of the language from the inside, as users of it” (Burns 1991: 85). Wright’s reflections about the principle of tolerance led him to reject (ii). If we are supposed to make sense of ordinary use of vague predicates, talk of rules cannot tell the whole story, for we should also appeal to a behavioristic account of how people manage to apply vague predicates in certain ways so to preclude the consequence of unrestricted application. Burns (1991) discusses this solution at length.
- 5.
For present purposes, we may assume that to say that someone is not bald is the same as saying that it is false that someone is bald. We do not need to consider the possibility of a gap between truth and falsehood here. This is because the hypothesis under consideration is that the principle of tolerance is true, and the point of introducing gaps in this discussion is to reject the view that this principle is simply true. Besides, the arguments presented here could equally be formulated in terms of truth and a weaker interpretation of “not” in terms of untruth – according to which to be untrue is to be either false or indefinite.
- 6.
Curiously, the all or nothing view does not raise the foundational problem of precisification (Sect. 2.2.3). Whatever option we choose, there will be a clear explanation of how the sharp boundaries of vague predicates are determined. They are determined by some rules for the use of vague predicates, namely the principle of tolerance and the aforementioned relational constraint. However implausible this view may sound, it has this advantage.
- 7.
It should be noted that the existence of a gap between cases to which it is exclusively correct and cases to which it is exclusively incorrect to apply a vague predicate does not entail the existence of cases to which the application of the predicate yields propositions that are neither true nor false. For example, we have seen (Sect. 4.4.3) that Fara (2000) accepts the existence of a gap in the first sense without accepting the existence of indefinite cases. Moreover, to reject the existence of a clear-case constraint does not amount to rejecting the existence of clear cases. For example, an epistemicist would not accept the existence of a clear-case constraint – at least not in the sense considered here – but she can accept the existence of clear cases. In principle, one can accept that there are clear cases without accepting that they correspond to cases to which it is exclusively correct/incorrect to apply vague predicates. In Sect. 5.2.11, I will develop a notion of clear case that does not commit us to a clear-case constraint.
- 8.
It has been said that Tye accepts both vague sets and vague properties, and there might be reasons in favor of explaining the penumbra of vague predicates in terms of the penumbra of vague properties (Sect. 4.1). In this case, a vague predicate admits of borderline cases because the property it expresses admits of borderline cases. The distinction between objects to which it is exclusively correct and ones to which it is exclusively incorrect to apply a vague predicate will thus be understood in terms of the distinction between objects that clearly have the property and ones that clearly do not have the property expressed by the predicate.
- 9.
Some changes would be required in order to accommodate ideal cases to many-valued theories of vagueness. A degrees-of-truth theorist can say that F ideally applies to x iff the application of “bald to degree 0” to x violates some constraint or standard relevant to the use of this predicate. A three-valued theorist or a supervaluationist can say that F ideally applies to x iff the application of not-F (in the strongest sense of negation) to x violates some constraint or standard relevant to the use of this predicate. These modifications are required because, as we have seen, the argument against unrestricted application of vague predicates – on which the argument in this section is based – leads to different conclusions depending on what picture of vague predicates is under consideration. In any case, since my positive account of vagueness does not commit us to a many-valued picture, I will leave these details aside.
- 10.
As we will see in the next section, the notion of ideal case is much less inclusive than the notion of clear case. To be sure, only persons with no hair are ideal positive cases of “bald”. Perhaps even Williamson would accept that vague predicates admit of ideal cases. Just after presenting the above objection against Soames’s partially defined predicates, he went on to say that “it might be a convention of English that ‘bald’ applies to anyone who has no hair” (Williamson 2002: 427). Pagin says something similar: “I take it e.g. to be part of the meaning of ‘bald’ that a person with no hair 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” (2009: 269).
- 11.
In Sect. 5.3.1 I will argue that we should not require that clear cases correspond to cases to which it is exclusively correct/incorrect to apply vague predicates. In fact, we should require the opposite.
- 12.
This minimal modification does not affect in any significant way Imaguire’s notion of ideal case.
- 13.
The notion of admissible is explained in the next paragraph.
- 14.
Why should we call semantic arbitrariness “semantic”? In part, my choice of “semantic arbitrariness” is due to the lack of a better term. In part, it is due to the fact that I want to highlight that pragmatic arbitrariness is connected to speakers’ intentions in contexts of use in a way that semantic arbitrariness is not. Whether or not two precisifications are equally pragmatically arbitrary in a given context of use will depend on the intentions of the speakers in that context. Assuming, along with Bach (1997: 39) and Recanati (2001: 85), that information about speakers’ intentions have to do with pragmatics rather than with semantics, the determination of whether or not two or more precisifications are equally pragmatically arbitrary is not a matter of semantics. By contrast, the determination of whether or not two precisifications are equally semantically arbitrary does not depend on speakers’ intentions, and apparently has to do with information that is relevant to semantics. At any rate, the use of “semantic” here is not mandatory. If you find a better term, you can use it. The phrase “semantic” has been understood in many different ways, and it is unnecessary for present purposes to quarrel about this.
- 15.
Sainsbury does not highlight the permissibility of arbitrary stipulations. He says that speakers are allowed to precisify vague predicates for their own purposes, but he does not state that they can arbitrarily stipulate a boundary for vague predicates. This idea, however, can be easily accommodated by his approach. Raffman is more explicit in this respect. Like Sainsbury, she accepts both (i) and (ii) above. But she is more explicit in claiming that speakers can apply a vague predicate to an object “without independent or nontrivial reason or justification in the nature of the case” (Raffman 2014: 94). For example, we can set the voting age at 18 years, but in so doing we cannot justifiably appeal to a reason in the nature of the case, to the fact that the difference of a mere second would be relevant to the maturity required for voting. Similarly, there would be no rule for the use of “mature enough to vote” that would justify us to set the voting age at 18 years rather than 18 years less 1 s. The most we could say is that the former is more convenient or easier to remember than the latter, which is a practical reason. Now, Raffman is generous about what would count as a reason of this kind. Suppose one holds a gun to your head, ordering you to apply “rich” to a person whose annual income is 125,000 dollars. According to Raffman (2014: 94), this would be a case in which you have a reason to apply “rich” to that person. Along these lines, it would be natural to accept that speakers are allowed to apply a predicate even in the absence of any reason, for it is difficult to see what would make the forced application above more legitimate than an application made for no reason at all. Moreover, Raffman expresses the thesis that a vague predicate admits of many precisifications by saying that there are many admissible stopping places in a soritical sequence for the predicate, and she highlights the fact that the choice of any of them is arbitrary from the viewpoint of semantics (Raffman 2014: 92). So, it seems that she would accept that speakers can precisify vague predicates by means of arbitrary stipulations. In sum, it seems that Raffman would also accept (iii).
- 16.
For a brief but interesting analysis of Raffman’s positions, see Sainsbury (2015). I agree with some of Sainsbury’s criticisms of Raffman, especially with the objection that Raffman’s Multiple Range Theory of Vagueness does not avoid the violation of the criterion of precisification (Sainsbury 2015: 482). Raffman (2014: 105) solves the problem by appealing to the familiar assumption that the higher-order language in which a theory of vagueness is formulated must itself be vague. We have seen that this kind of strategy leads us to implausible results (Sects. 4.2.2 and 4.3.2), and I think the same will hold for Raffman’s theory of vagueness.
- 17.
If one thinks that the experiment should include a third option, I (= indefinite), then a perfect score would include a sharp boundary between Ys and Is, and a sharp boundary between Is and Ns.
- 18.
As I said above (Sect. 5.1.5), the plausibility of Three-Valued Theory probably depends on the more inclusive notion of clear case. Nonetheless, this does not affect my point in this section, for the existence of a clear-case constraint is consistent with TA, and my reasons against positing clear cases have nothing to do with TA itself. If one is not tempted to accept the stricter notion of ideal case rather than that of clear case, one can simply reformulate my statements above in terms of the latter. The same holds for what will be said about Supervaluationism in the course of this section. In Sect. 5.3.1, I return to the issue of whether or not a clear-case constraint is required to make an explanation of vagueness plausible.
- 19.
See Sect. 4.1 for a more detailed explanation of this conception.
- 20.
Braun and Sider do not explicitly state that the precisification of a vague predicate – and hence of a vague sentence – depends on contexts in the wide sense of the term. Yet, this supposition is consistent with their view, and I think it is the most natural way of interpreting them. They certainly accept that the linguistic meaning plus the relevant facts about the narrow context are not sufficient to determine a precise extension for an originally vague predicate (see, for example, footnotes 14 and 29). So, if there is a way in which these predicates could be precisified, it should involve speakers’ intentions. Even when we take wide contexts into account, however, they think that vagueness is rarely eliminated.
- 21.
In this case, they say that the sentence is approximately true. But they do not endorse the thesis that approximate truth is identified with genuine truth (Braun and Sider 2007: 144).
- 22.
See Sect. 5.2.9 for my solution to the sorites paradox.
- 23.
This version may be formulated with the assumption that each item in the soritical sequence is indistinguishable or indiscriminable from its immediate neighbors. I will not consider problems concerning non-transitive indiscriminability. See Sainsbury (2013: Sects. 3 and 4) for a solution that takes into account these problems. Sainsbury evaluates his own solution in comparison to that proposed by Scoreboard Contextualism.
- 24.
Paul Égré (2015) thinks examples of this kind should not lead us to the conclusion that some vague predicates are not tolerant. He contends that such examples show at most that there is a distinction between an expression that is universally tolerant and one that is existentially tolerant . The first is tolerant in all contexts, while the second is tolerant only in some contexts. An expression that is only existentially tolerant has potential vagueness, even when it is used in a non-tolerant way. An expression is potentially vague when there is a context in which it gives rise to the sorites paradox (Burnett 2017: 36). Now, the core idea is that we should not assume that vagueness is only universal tolerance. If there is a context in which “has few children” gives rise to a sorites paradox, then this expression is potentially vague, and hence it is vague. If there is no such context, then it is not vague. Since I reject the thesis that vagueness should be understood in terms of tolerance or sorites susceptibility, I would not accept this. Nonetheless, I can agree that this proposal can accommodate the alleged examples of vague predicates that are not tolerant. In any case, I am not claiming that predicates such as “has few children” provide us with a decisive argument against the thesis that all vague predicates are (apparently) tolerant. My point is only that this thesis is not obvious, and the above kind of example is adequate to support this.
- 25.
Importantly, my point here is not simply that arbitrariness can be easily mistaken for tolerance, but that this happens in the contexts in which the sorites is usually formulated.
- 26.
We have already seen that this is a mistake. Philosopher B in my earlier example (Sect.5.2.9) arbitrarily chose a sharp boundary, and there was nothing A could say to prevent him from doing this. B, however, was a philosopher with a theory of vagueness in mind (VA).
- 27.
As with many other explanations of why we are inclined to accept that vague predicates are tolerant, this is psychological speculation which depends on empirical confirmation. In any case, it sounds initially plausible to me.
- 28.
In response, one could argue that VA commits us to Semantic Nihilism, and this is an undesired consequence of VA. Yet, we have seen that Semantic Nihilism can be made a viable alternative by means of an account of how vagueness can be harmlessly ignored.
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Salles, S. (2021). Vagueness as Arbitrariness. In: Vagueness as Arbitrariness. Synthese Library, vol 436. Springer, Cham. https://doi.org/10.1007/978-3-030-66781-8_5
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