Abstract
This chapter provides a (biased) overview of analyses of vagueness within linguistics. First, the nature of vagueness is discussed, and contrasted with notions such as ambiguity and context dependence. After that, some reasons are given that could perhaps explain why vagueness is such a pervasive phenomenon in natural language. This is followed with a review of some more or less standard linguistic analyses of gradable adjectives. The chapter is focussed on approaches that take comparison classes into account. Because comparative constructions are ideally formed in terms of gradable adjectives, comparative ordering relations are discussed as well. It is argued that one specific ordering relation is crucial for any analysis of vagueness that wants to capture the notion of ‘tolerance’: semi-orders. A lot of attention is given to contextuallist’ approaches that want to account for the Sorites paradox, because these approaches are most popular within linguistics. In the final main section, the chapter discusses what some people have called ‘loose talk’. The main issue here is whether with loose use of language we say something that is strictly speaking false, but true enough in the particular conversational setting, or true, because the conversational setting loosens the requirements for a sentence to be true.
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Notes
- 1.
There are other problems as well, but they are not relevant for this chapter.
- 2.
At the level or individual words, the contrast between ambiguity and vagueness can also be denoted by the distinction between homonymy and polysemy.
- 3.
Roughly speaking, with the ‘core’ supervaluation theory account of vagueness, I mean the theory without something like Fine’s (1975) treatment of higher-order vagueness.
- 4.
Fuzzy logicians might respond by saying that with respect to a relative adjective like ‘tall’, no individual has this property to degree 1. But then, as noted by Williamson (p.c.), what to do with the sharp boundary between those objects that do and those objects that don’t have the relevant property P to degree \(> 0.5\)?
- 5.
Bosch (1983) refers here to Waisman’s (1968) notion of ‘open texture’, the feeling that for outlandish cases we wouldn’t know yet how to apply the predicate. I believe this notion is closely related with the notion of ‘unforeseen contingencies’ in economics.
- 6.
Franks Veltman stresses this in his inaugural lecture Veltman (2002).
- 7.
- 8.
But see also Kyburg and Morreau (2000).
- 9.
For some adjectives, like ‘tall’ this standard is either defined as the (arithmetical or geometrical, possibly weighted) mean of the height of all individuals in the class, for others, like ‘red’ this standard can be thought of as the prototype representative of the class.
- 10.
The idea that the ‘than’-clause of a comparative contains a negation goes back to Jespersen (1917).
- 11.
von Stechow (1984) proposed a somewhat different analysis to account for these sentences. It is fair to say that although von Stechow’s analysis has been improved on recently, it is the classic paper on comparatives, and still contains the most complete discussion of the analyses on the subject.
- 12.
In an early, but still very relevant study on grading, Sapir (1944), makes a distinction between adjectives for which inferences like that between (2) and (5) go through, and adjectives for which they do not. In the first class are things like ‘good’-‘bad’ and ‘far’-‘near’, while in the second class are antonyms like ‘brilliant’-‘stupid’: From ‘John is it less/more brilliant than Mary’ we conclude that both John and Mary are brilliant, and we can’t continue this sentence with ‘but both are stupid’. This in contrast with the appropriate discourse ‘From the point of view of America, France is on the near side of Europe, though actually far’.
- 13.
Proponents of the degree-based approach can propose that in comparatives ‘short’ means ‘not tall’, but not in the positive use of the predicates: ‘John is short’ doesn’t mean the same as ‘John is not tall’. They can account for this by making use of the POS-operator they use to account for positive use of adjectives.
- 14.
And for a related problem that you can’t say ‘John is 1.80 m short’.
- 15.
Well, this is Kamp’s (1975) proposal in case at least one of the two is considered to be a borderline tall individual.
- 16.
A relation R is irreflexive iff for all objects \(x \in I: \neg R(x,x)\). It is transitive iff for all objects \(x,y,z \in I: R(x,y) \wedge R(y,z) \rightarrow R(x,z)\).
- 17.
Notice that by adopting this constraint we can reformulate the above analysis of comparisons in terms of existential quantification: John is taller than Mary is true in a supervaluation frame just in case there is a complete valuation function, or world, in which John is tall, but Mary is not.
- 18.
Graff (2000) has argued that a comparison class should be an intensional instead of an extensional object. If we allow for individuals of different worlds to be part of the same comparison class, I don’t think this crucially undermines Klein’s proposal.
- 19.
Although it does not necessarily uniquely determine what counts as being tall. See the end of this section for some discussion.
- 20.
But see Section 6.6.
- 21.
Equatives can be analyzed in terms of comparison classes as well. Klein (1980) proposes that (i-a) should be interpreted as (i-b).
-
(i) a.
John is as tall as Mary.
-
b.
In every context where Mary is tall, John is tall as well.
Klein (1980) notes that on this analysis, the negation of (i-a), i.e. (ii-a), is correctly predicted to be equivalent with (ii-b):
-
(ii) a.
John is not as tall as Mary.
-
b.
Mary is taller than John.
Standard pragmatics can explain why in the context of question ‘How tall is John?’,(i-a) would come to mean that John and Mary are equally tall.
-
(i) a.
- 22.
Later we will see that this is guaranteed by van Benthem’s (1982) Downward Difference constraint (DD). However, some authors have argued that objects indistinguishable with respect to a small comparison class, might be distinguishable with respect to a larger comparison class. According to proponents of this view (see Section 6.5.2), constraint (DD) is not valid.
- 23.
But see von Stechow (1984) for an argument saying that also the degree-based approach is in line with Frege’s principle.
- 24.
Sapir (1944: 125) argues that we must regard grading from different points of view. ‘‘It is very important to realize that psychologically all comparatives are primary in relation to their corresponding absolutes (‘positives’) [...] Linguistic usage tends to start from the graded concept, e.g. good (= better than indifferent), bad (= worse than indifferent) [...] for the obvious reason that in experience it is the strikingly high-graded or low-graded concept that has significance, while the generalized concept which includes all the members of a graded series is arrived at by a gradual process of striking the balance between these graded terms. The purely logical, the psychological, and the linguistic orders of primacy, therefore, do not necessarily correspond.’ Perhaps this is so, but this doesn’t mean that making use of a theory that reflects such a correspondence wouldn’t be preferred.
- 25.
Of course, we do have ‘x is more of a heap than y’, but that seems to have a somewhat different meaning (or not?).
- 26.
There exists a striking syntactic difference between common nouns like ‘bank’ on the one hand and verbs and adjectives like ‘tall’ on the other: whereas common nouns combine with a determiner to form a noun phrase, verbs and adjectives must be nominalized first, before they can play that role. It has been argued that there corresponds a semantic difference with this syntactic difference: it is in general not determinate how to count things like ‘the tall ones’ or ‘the red ones’ (a red grapefruit, for instance, won’t have the same color as a red tomato), nor is it determinate how a thing which is tall or is red must be individuated and reidentified. In contrast to what falls under a common noun like ‘cat’ or ‘bank’, the general terms ‘tall’ and ‘red’ do not by themselves determine units which could underlie the possibility of counting: arbitrary many parts of a red object are red objects again. Of course, we can count red things, but only once we have determined beforehand what counts as an individual thing. Notice, though, that some philosophers have argued that even for nouns like ‘cats’ it is unclear exactly what should be counted (see Lewis, 1993 for discussion).
- 27.
According to Unger, stress forces a precise interpretation of the absolute adjective.
- 28.
Bolinger (1972) and others working on degree words observed another contrast between absolute and relative adjectives: while relative adjectives combine well with degree adverbs like ‘very’ and ‘rather’, absolute adjectives combine well with other adverbs like ‘completely’, ‘almost’, ‘hardly’, and ‘nearly’.
- 29.
It is standard in the literature to also denote a structure \(\langle I,\ge_P\rangle\) with ‘\(\ge_ {P}\)’ reflexive, transitive, and strongly connected by a weak order.
- 30.
The reader probably doubts whether weak orders are also appropriate for the analysis of vagueness. Indeed, I will argue in Section 6.5 that for vagueness, we need semi-orders, rather than weak orders.
- 31.
This is not always the case, it doesn’t hold for the pair full-empty.
- 32.
It should be noted, though, that Kennedy (2007) bases this general claim on the difference between the behavior of ‘long’ versus ‘full’. But the absolute adjective ‘full’ behaves crucially different from other claimed absolute adjectives, in that its antonym ‘empty’ is not contradictory with ‘full’. This might well be a crucial difference.
- 33.
In terms of the analysis of value judgments sketched in Section 6.3, one might think of each context structure M as a precise valuation function. Although all these structures give rise to the same ordering relation, they give the predicate that gives rise to this ordering relation different extensions with respect to the same comparison classes.
- 34.
As argued by many opponents of analyses like ours, making the meaning of the adjective context dependent doesn’t eliminate its vagueness. The phrase ‘old for a dog’ is just as vague as the adjective ‘old’ is. Thus, making the meaning of P, the positive form, dependent on both a comparison class and a structure M is not enough. We will come back to this in Section 6.5.
- 35.
The standard way to make the distinction between indistinguishability and incomparability is by starting with a structure like \(\langle I,>_P, \sim_P\rangle\) where I is a set of objects, ‘\(>_ P\)’ a primitive preference relation, and ‘\(\sim_ {P}\)’ a primitive indistinguishability relation. Given such a structure, and the natural definition of ‘\(<_ {P}\)’ in terms of ‘\(>_ {P}\)’, it is possible that \(>_P \cup \sim_P \cup <_P\not= I \times I\). Thus, it is possible that for two elements x and y of I, it is neither the case that \(x >_P y\), nor \(y >_P x\), nor \(x \sim_P y\). In that case, we call x and y incomparable. Now it is easy to rule out incomparability: just demand that \(>_P \cup \sim_P \cup <_P= I \times I\).
- 36.
For more-dimensional predicates, comparative formation is arguably more difficult as well. Kamp (1975), for instance, claims that ‘This is bluer than that’ is most of the time not a meaningful statement.
- 37.
See van Rooij (2011) for making precise these two approaches.
- 38.
Such mappings not only preserve differences, but also ratios between differences: \(\frac{f(x) - f(y)}{f(v) - f(w)} = \frac{g(x) - g(y)}{g(v) - g(w)}\).
- 39.
Schwarzchild (2005) noted that measure terms occur with some, but not all positive ‘measure’ adjectives. They occur in English with adjectives like ‘old’, ‘tall’, ‘high’, and ‘thick’, but not with ‘warm’, ‘heavy’, and ‘big’. We can say, for instance, ‘John is 5 years old’, and ‘The ice was 5 cm thick’, but we don’t say in English ‘The water was 75° warm’, ‘The suitcase is 20 kg heavy’, or ‘The apartment is 1,000 ft2 big’. We might try to account for this by saying that while adjectives like ‘old’ work on additive magnitudes, an adjective like ‘warm’ does not. Unfortunately, we can say things like ‘It is twice as warm in Amsterdam as it is in Berlin’. Moreover, there are cross-linguistic differences. In German some of the counterparts of the inappropriate examples above are acceptable: ‘Das Konzert war nur 40 Min lang’, ‘Der Koffer ist 20 Kilo schwer’, and ‘Die Wohnung ist 90 m2 gross’ are all appropriate. To account for this, Schwarzchild (2005) proposes a syntactic solution, but one might wonder whether a pragmatic solution is not more suitable. It seems that it is for some measure phrases much more natural to use the relative adjective than for others. It doesn’t make a lot of sense for mph, or degrees, or kilos, because saying that John went 50 mph doesn’t leave much room for the relevant (relative) adjective. The same for ‘It is 50°’, and ‘The suitscase is 20 kg’. It is different with centimeters, because there are many adjectives measured in terms of centimeters, so adding tall, or wide, makes perfect sense.
- 40.
Any relation that is irreflexive and satisfies the interval-order condition is called an interval order. All interval orders are also transitive, meaning that they are stronger than strict partial orders.
- 41.
Cf. Scott and Suppes (1958).
- 42.
As another example: consider the conditionals (c) ‘If Fred is tall, Bert is tall’ and (d) ‘If Fred is tall, Bert is not tall’. If the values of these sentences depend only on the values of their parts, they should have the same value. But this prediction is wrong: while (c) is plausibly true, (d) is certainly wrong. In defense of fuzzy logic, one might claim that conjunction and negation should not be analyzed as standardly assumed by Lakoff (1973) for instance. In fact, there exist many alternative ways to analyze the connectives in fuzzy logic. Arguably, however, these alternatives are less natural than the standard one (cf. Dubois and Prade, 1980).
- 43.
Williamson (1994) proposed that we should replace [P] by the weaker statement [P w ]: For any \(x,y \in I: (\Box P(x) \wedge x \sim_P y) \rightarrow P(y) \), where ‘\(\Box \phi\)’ means that φ is known. But, as Graff (2000) points out, this proposal does not explain why we are so inclined to believe [P]. Shapiro (2006) proposed to weaken [P] in yet another way. Making a difference between a classical ‘’ and a three-valued ‘\(\dot{\neg}\)’ negation, his principle of tolerance [\({\bf P}_{ws}\)] says that for any \(x,y \in I: (P(x) \wedge x \sim_P y) \rightarrow \neg \dot{\neg}P(y) \). I believe we want something stronger than this.
- 44.
This means that comparison classes are not scope-less (cf. Ludlow, 1989). Soames (1999) treats gradable adjectives as indexical, nevertheless. But also he might have good reasons for doing so, because even for indexicals constancy under VP ellipsis is disputable. Ellis (2004) gave the following example suggesting that this is not the case:
Thirty friends are standing in the middle of a very large field. One of them has the following idea: ‘Why don’t we each go and stand in any place we choose, and see where everyone goes. Jill, you go first.’. Jill walks a good distance away from the group and shouts, ‘I’m going to stand here!’ It’s Tom’s turn next, and being the tag-along Tom is, he goes straight for Jill and stands right next to her. Jill exclaims humorously, ‘And I guess Tom is too!’ Sally then goes and stands on the other side of Jill, who now says And apparently, so is Sally!’ Then Bill goes and stands behind Jill (‘and so is Bill’), and then Ann stands in front of Jill (‘and Ann’). Each of the other twenty-nine people walks towards Jill and stands as close to her as s/he can without touching anyone else. In each case, Jill amusingly shouts ‘And so is s-and-so!’
In this case, the interpretation of ‘here’, which appears in VP ellipsis, varies.
- 45.
This suggestion is close to one proposed by Kennedy (1999). He notes that something like this is needed to account for the intuition that for ‘Everybody in my family is tall’ to be true, we compare different members of my family to different sets: we compare men with other men, women with other women, and children with other children. Peter Bosch (p.c.) gave also the following example, which makes the same point: ‘Everything in America is big: The cars, the buildings, and even the Turkeys.’
- 46.
- 47.
In a very real sense, the solution Veltman and Muskens propose is based on the same intuition as the solution proposed by supervaluationalists: the Sorites paradox arises, because we equivocate ‘similarity’ with ‘sameness’. But Dummett’s point was to take the former notion more seriously.
- 48.
For a related recent critique of contextualist’s solutions to the Sorites paradox, see Keefe (2007).
- 49.
See in particular section 85–87: ‘A rule stands like a signpost ... The signpost in order in in normal circumstances it fulfils its purpose. See also Waismann’s (1968) notion of ‘open texture’.
- 50.
Also Graff (2000) has argued that not all sets of individuals can figure as appropriate comparison classes. Her reasoning, however, is quite different from ours. She argues, for instance, that comparison classes need to form a kind.
- 51.
Notice that also in discrete cases the relation ‘\(\sim_ {P}\)’ can be closed in \(c \times c\). In just depends on how ‘\(\sim_ {P}\)’ is defined.
- 52.
Although Graff (2000) seems to adopt a version of the standard contextualist’ approach to vagueness, her analysis can be thought of as being close to our pragmatic analysis as well. She makes two claims: (i) there exists cutoff-points, but (ii) if x is significantly P-er than z, \(x >_P^! z\), and y is similar to x, it must be the case that y is significantly P-er than z as well, \(y >^!_P z\). Although the similarity relation Graff (2000) seems to assume behaves like the similarity relation used in semi-orders, the relation ‘\(>_P^!\)’ can obviously not be the corresponding relation of a semi-order. Instead, the relation ‘\(>_P^!\)’ should have the properties of a weak order. In fact, if we start with a semi-order \(\langle I,>\rangle\), we can define a weak order \(\langle I^*, >^*\rangle\) that behaves just like the one Graff (2000) seems to assume. Given that ‘∼’ is defined in terms of ‘>’ as usual, we can define a new relation ‘≈’ in terms of it: \(x \approx y\) iff def \(\exists \vec{z} \in I^n: x \sim z_1 \sim \cdots \sim z_n \sim y\). Obviously, ‘≈’ is an equivalence relation. In terms of ‘≈’ we define equivalence classes like \([x]_{\approx}\) as usual, and take \(I^*\) to be the set \(\{[x]_{\approx}: x \in I\}\). Now we define the order relation \(>^*\) between the elements of \(I^*\) as follows: \(X >^* Y\) iff def \(X \not= \emptyset \wedge Y \not= \emptyset \wedge \forall x \in X: \forall y \in Y: x > y\). One can show that ‘\(>^*\)’ indeed is a weak order. Now, why is (our reformulation of) Graff’s analysis close to our pragmatic analysis? The reason is that in order to assume that the first element of a Sorites series has property P but the last one has not, she has to assume (if our reformulation of her ideas is faithful) that I is not closed under ‘\(\sim_ {P}\)’, and thus that the series allows for a cutoff-point.
- 53.
On this proposal, Stanley’s elliptical conjunction is claimed to be inappropriate. Or better, if we assume that [P] holds and that the property that remains constant under VP-ellipsis is \(\lambda x.T(x,f(x))\), where f was defined as \(f(x_1) = \{x_1,x_n\}\) and \(f(x_{i+1}) = f(x_i) \cup \{x_{i + 1}\}\), it is predicted that the sentence at the one but last step in the Sorites series is inappropriate.
- 54.
It turns out to be possible to characterize semi-orders in terms of the way relative adjectives behave with respect to appropriate comparison classes (see van Rooij, 2011).
- 55.
This is generally assumed in two-dimensional theories of presuppositions and conversational implicatures.
- 56.
- 57.
- 58.
I have argued above that in such a case it doesn’t make sense to use predicate P in context I. This doesn’t rule out, of course, that P can be used appropriately with respect to comparison classes smaller than I.
- 59.
Whether the assuming that predicate P give rise to (unlimited) higher order vagueness fully captures the intuition that the concept denoted by P is boundaryless is controversial, however.
- 60.
In Section 6.4 we have assumed that although statements involving predicates like ‘tall’ occurring positively are vague, comparatives are not. The vagueness of ‘tall’ was accounted for by interpreting sentences where the predicate is used positively with respect to a contextually given comparison class. Comparatives were not vague, because for their interpretation we existentially quantified over comparison classes. It is quite common among linguists to assume that comparatives are not vague. Philosophers like Kamp (1975), Williamson (1994) and Keefe (2000), however, have claimed otherwise. Kamp (1975) noted already that comparatives associated with more-dimensional predicates – for example ‘cleverer than’ – are typically vague. They have borderline cases: pairs of people about whom there is no fact of the matter about who is cleverer, or whether they are equally clever. This is particularly common when comparing people who are clever in different ways. Keefe (2000), argues that there can also be borderline cases of one-dimensional comparatives. She argues that although there is a single dimension of height, people cannot always be exactly placed on it and assigned an exact height. For what exactly should count as the top of one’s head? Consequently there may also be borderline cases of taller than. Even more interesting from our point of view is the fact that taller than can also be vague due to indeterminacy over exactly what should count as a point (or an equivalence class) in the tallness ordering: individuals whose height is 2 μm apart normally count as equally tall, although this would not be the case in contexts in which every micro-millimeter is important.
- 61.
In a recent paper, Sauerland and Penka (2007) studied the difference between modifiers like ‘exactly’ and ‘definitely’. Both could be used to modify measure phrases, but they do so in different ways: ‘exactly’ says something about how precise the measure phrase should be interpreted, while ‘definitely’ measures the speaker’s epistemic certainty. This is corroborated by the fact that the adjective ‘tall’ can be modified by ‘definitely’ but not by ‘exactly’.
- 62.
It is questionable whether compositionality should be assumed here, and it contrasts standard Gricean treatments of other pragmatic phenomena.
- 63.
But see Williamson (1994) for some discussion.
- 64.
van Lambalgen (2001) noted that this construction needs to be generalized, because it does not generalize to predicates of higher arity.
- 65.
According to Tim Williamson (p.c.), a sentence like ‘Enzo owns two of the same car’ is ok. This would be problematic for the Nunberg/Hobbs proposal.
- 66.
A function f from D to D' is surjective iff the range of f is D'.
- 67.
If one desires, one can think of this function as a counterpart function used in quantified modal logic.
- 68.
Technically, f is just a homomorphism from M to Mʹ. In general, homomorphisms don’t preserve negative sentences.
- 69.
In general, the truth conditions of sentences in course-grained model Mʹ are defined in terms of their truth conditions in fine-grained model M as follows:
$$\begin{array}{*{20}l}M',g\models P(x) \quad {\textrm{iff}} \quad \exists d \in f^{-1}({[[{x}]]}^{M',g}): M,g[^x/_d]\models P(x)\\ M',g \models \neg \phi \quad \ \ {\textrm{iff}} \quad M',g\not\models \phi\\ M',g \models \phi \wedge \psi \ \ {\textrm{iff}} \quad M',g \models \phi \, {\textrm{and}}\, M',g \models \psi\\ M',g\models \forall x \phi \quad \ {\textrm{iff}} \quad \, {\textrm{for\, all}} \, d \in I_{M'}: M', g[^x/_d]\models \phi.\\ {\textrm{Notice \, that}}\,\, M',g\models \neg P(x) \,\,{\textrm{iff}}\,\, \forall d \in f^{-1}({[[{x}]]}^{M',g}): M,g[^x/_d]\not\models P(x). \end{array}$$ - 70.
Instead, the system described here is much closer to the ‘inverse system’ used by van der Does and van Lambalgen (2000) to account for the logic of vision.
- 71.
In fact, this tradition goes back all the way to Leibniz. Technically, the actual world, the ding an sich, is seen as the inverse limit.
- 72.
In a sense this is just what Parikh (1994) argued for as well.
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Acknowledgement
I would like to thank Johan van Benthem, David Etlin, Ewan Klein, Frank Veltman, and Tim Williamson for commenting on (sometimes much) earlier versions of this chapter. Thanks also to Chris Kennedy, Manfred Krifka, and Roger Schwarzchild for discussion.
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van Rooij, R. (2011). Vagueness and Linguistics. In: Ronzitti, G. (eds) Vagueness: A Guide. Logic, Epistemology, and the Unity of Science, vol 19. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-0375-9_6
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