CHAPTER 20

VAGUENESS

SAM ALXATIB AND ULI SAUERLAND

20.1 INTRODUCTION

UNTIL recently, research on vagueness in logic and linguistics has more often relied on theoretical considerations and informal ‘armchair’ methods than on quantitative results. For example, a number of theoretical models of vagueness are based on a foundation that preserves the classical laws of Excluded Middle and Non-Contradiction, but the recent experimental turn has made it possible to reassess these (and many other) core assumptions. In this chapter, we discuss the contributions that experimental studies have made to the study of vagueness.

In contemporary literature, the problem of vagueness is most often presented using the sorites paradox, Eubulides’s paradox of the heap.¹ If we take a number n that makes it unquestionably true that n grains of sand make a heap (say one million), and add the unobjectionable inductive assumption that, for any number m, if m grains of sand make a heap, then m–1 grains of sand also make a heap, then we are led to the conclusion that even one grain of sand makes a heap.

The conclusion of the sorites argument is clearly false, so logically, something must be wrong in the way the argument is set up. Is the inductive premise in fact false? If so, why are we tricked into thinking that it is true? Questions like these are of central importance in theorizing about vagueness. And because vagueness affects the way we categorize things and the way we speak, much of its impact leaves a mark on our behaviour. There is therefore much to be learned about vagueness from investigating linguistic and inferential behaviour with experimental methods.

The experimental literature on vagueness is young when compared with the older, far more sizeable body of theoretical research. We do not intend to provide a comprehensive summary of theories here, but focus our review on theoretical distinctions that have been found amenable to experimental research so far.² We will pay particular attention to borderline cases with scalar adjectives such as tall since these are prototypical cases where the ascription of a vague predicate is neither clearly true nor clearly false.³ With these in mind, we review some prominent theoretical accounts of vagueness (section 20.2), and then discuss the experimental studies that were used to test their validity (section 20.3). In section 20.4 we discuss work on matters beyond borderline cases, focusing in particular on a typology of vagueness, and on the specific issues of number rounding, and the pragmatic utility of vagueness in that context.

20.2 THEORIES OF VAGUENESS

Our theoretical review will be organized roughly according to two main criteria: (i) valency and (ii) scope. Valency refers to the number of possibly different values an expression of propositional type may have. We distinguish three levels of valency: valency 2 (bivalence) allows only the values True and False; valency 3 (trivalence) makes available one additional truth-value for the proverbial ‘grey-area’; and multivalence allows more than three but typically infinitely many different truth-values, like the closed interval from 0 to 1.

Our ‘scope’ division is about the scope of operations that are used, in the various theories that concern us, to capture vagueness with metalanguage that closely resembles classical bivalent logic. For epistemic approaches, this operator is an epistemic quantification over different possible precisifications, and something similar is assumed for sub- and supervaluation theories. For trivalent theories such as Cobreros et al. (2012), the corresponding operation is contained within the assumption that speakers reject statements that are assigned the third truth-value. Finally, linguistic applications of fuzzy logic and other multivalent accounts generally assume that discourse pragmatics sets a truth threshold such that only statements whose truth-value exceeds this threshold are felicitously assertable. If we always view this operation as a silent operator O that is specific to the account, the question of scope arises. While older accounts predominantly assume that O applies only globally, Alxatib et al. (2013) argue empirically in favour of what we call a local account. Cobreros et al. (2015) accept this reasoning and develop a local account of their proposal, and locality can be combined with other accounts of vagueness too.

Readers familiar with the literature on vagueness will soon see that our strategy of classification is neither conventional nor comprehensive in its coverage. Nevertheless, the questions that it will help us bring up will be fitting for the theme of this chapter, and are ones that we consider important. The views we will discuss are: the epistemic view (section 20.2.1), super- and subvaluationism, and strict/tolerant models (section 20.2.2), and fuzzy logic (section 20.2.3). In each of these we will present a global and a local version, in preparation for our later discussion of the experimental literature. The landscape of theories, as guided by the valency and scope divisions, is summarized in Table 20.1.

Prominently missing from this table are paraconsistent accounts of vagueness (e.g. Weber, 2010) and contextualist accounts (e.g. Raffman, 1994; Fara, 2000). Our decision not to focus on these approaches comes from our belief that they are either not clearly relevant to the recent experimental literature, or are relevant, but in this context are not different crucially from the views that we discuss.⁴

Table 20.1. Theory classification by valency and scope

20.2.1 Valency two: Epistemicism

On the epistemic view, vagueness results not from any special semantic difference between vague and precise expressions, but from a difference in whether the extensions of these expressions are known or knowable (prominently Williamson, 1994). Epistemic theories of vagueness thus preserve the familiar bivalent semantics of classical logic—hence valency 2—and reduce vagueness to ignorance. This is stated succinctly in the Vagueness-as-ignorance hypothesis of Bonini et al. (1999), an experimental study that we will revisit in section 20.3.⁵

(1) The Vagueness-as-ignorance hypothesis:

Speaker S mentally represents vague predicates in the same way as other predicates with sharp true/false boundaries of whose location S is uncertain.

A general prediction of this hypothesis is that any instance of linguistic behaviour that is particular to (the use of) vague predicates should come ultimately from ignorance about their extension. And because vagueness is reduced to ignorance, the account makes it possible to preserve the laws of classical first-order logic: vague expressions are not formally represented in any different way from precise expressions, so any prediction that follows from the formal semantics of a precise term should in principle hold in the case of a vague term, provided that ignorance is shown to not make a difference. A case in point is the case of classically contradictory expressions like ‘c is neither P nor not P’ and ‘c is both P and not P’. If P has precise boundaries, and assuming that P is defined for c, then we expect both expressions to be false no matter where the boundaries of P are drawn.

Discussion of vagueness in epistemic theories, as indeed in most theories, does not hinge on any commitments about sentential structure or compositional semantics, which is why we categorize the standard epistemic account as global. The point is that the ignorance responsible for vagueness can be stated as ignorance of sentential truth conditions with no loss of crucial detail, even if the source of that ignorance is ignorance about the extension of some subpart of the given sentence. By contrast, imagine an alternative, ‘local’ account where the language of concern contains unpronounced epistemic operators ◊,□, whose semantics are like those of the adverbs possibly and certainly. Then there can be phonologically identical parses of the contradictory sentences seen earlier, but with embedded occurrences of these epistemic operators, for instance:

This variant of the epistemic view is, from a formal perspective, as conservative as the bareboned global one described earlier. It is different because it allows expressions of (un)-certainty in subsentential positions, and this in turn affords seemingly contradictory sentences with non-contradictory parses. For example, (2) is true provided that, in the given context, c is not certainly known (or thought) to be a case of P, but also not certainly known (or thought) to not be a case of P. The same context also makes (3) true.

We will call this local epistemic view Epist*. We do not know if such a view was ever proposed or discussed in the context of vagueness. But since unpronounced modal operators are established in other linguistic contexts (Meyer, 2013; and others), we feel it is important to keep this theoretical possibility in mind when we look at the experimental findings later.

20.2.2 Valency three: Super-/subvaluationism, strict/tolerant truth

Trivalent accounts of vagueness come in many flavours, and strictly speaking, the views that we classify here as trivalent are not built on semantics that make use of three truth-values. In supervaluations, for example, a vague predicate has an extension and an antiextension, as well as a gap between the two, while in subvaluations, the extension of a vague predicate P may overlap with its anti-extension, allowing a region where Pc is both true and false (a glut). If we define bivalence as the requirement that any sentence have exactly one of two possible truth-values, then neither super- nor subvaluations can be classified as bivalent. This is why we categorize them here as trivalent.

We also categorize Cobreros et al.’s (2012) strict/tolerant system as trivalent, because as we will see in section 20.2.2.2, the system allows three possible outcomes for any given sentence: the sentence may be strictly true, strictly false but tolerantly true, or tolerantly false. In what follows we expand on the building blocks of the two views, and on what their global/local versions might look like.

20.2.2.1 Super-/subvaluationism

To a supervaluationist, a vague predicate is associated with multiple ‘precisifications’ or ‘sharpenings'. In each of these precisifications, the predicate has a crisp separation between its extension and anti-extension, but the separating line may differ from precisification to precisification. This makes it possible for an individual c to belong to P in some of its sharpenings but not in others. On this account, a sentence Pc is true iff c belongs to P’s extension in every sharpening of P—this universal truth is what Fine (1975) calls ‘supertruth’. Conversely, Pc is false iff it is superfalse. If Pc is neither supertrue nor superfalse, then it is neither true nor false, hence the gap.⁶

Note that there may appear to be a formal similarity between epistemicism and supervaluations: the multiplicity of sharpenings in the case of vague predicates (according to a supervaluationist) may plausibly be taken to represent speaker ignorance about where the predicate draws its boundary. But despite this apparent parallel, the two views differ on significant conceptual grounds. To an epistemicist, the variation in where the line is drawn is represented in differing epistemically accessible possibilities. Vague predicates have sharp boundaries, but those boundaries are unknowable. To a super-valuationist, there is a distinction between ignorance of the location of a boundary—representable as a difference in epistemically accessible possibilities—and knowledge that the placement of a sharpening point is shiftable—representable as a difference across the precisifications of a predicate.

But there is a real point of similarity between epistemicism and supervaluationism. In both views, classical tautologies remain tautologous, and classical contradictions remain contradictory. As we noticed in our discussion of the global epistemic view, there is no need to know where the boundary of P lies in order to know that the sentences (Pc ∧ ¬ Pc), and ¬ (Pc ∨ ¬ Pc), are false. For similar reasons, both sentences are predicted to be contradictory in supervaluationism; because the sharpenings of P are all classical, there is no way for either sentence to hold in any precisification of P, which makes them both superfalse, regardless of what P’s extension is. (A similar point can be made for classical tautologies.)

The formal foundation of supervaluations is shared by the related subvaluationist framework, which differs in equating truth with subtruth, truth in at least one specification, and falsity with subfalsity.⁷ If P admits c in its extension in some sharpenings but not in others, then Pc will be true (because it is subtrue) and it will also be false (because it is subfalse). In this respect the account differs from supervaluationism, but it is like it in preserving classical contradictions—they are never subtrue and are always subfalse, therefore always false. (Again, a parallel prediction is made for tautologies.)

While contradictions are predicted to be trivially false in both super- and subvaluations, the accounts differ in what they assign to the predication Pc when c is a ‘borderline’ case of P—an example of P in some sharpenings but not in all of them. In supervaluations, Pc is neither true nor false in this case, while in subvaluations it is both true and false.

An imaginable local variant of the super-/subvaluational account is one where super-/subvaluation is encoded in unpronounced ‘definitely’/‘indefinitely’ operators (see Fine, 1975, for a discussion of the former). Like the modal necessity/possibility operators, ‘definitely’ and ‘indefinitely’ are duals, whose domain of quantification is the precisifications associated with vague predicates. Packaging these universal/existential truth conditions in syntactic operators effectively allows complex sentences to have ‘super’ truth-conditions for some of their subparts, and ‘sub’ truth-conditions for others. A proposal along these lines is made (informally) in Alxatib & Pelletier (2011a). The choice of local operation (whether ‘definitely’ or ‘indefinitely’) depends on whether the final meaning is the strongest one possible (among the consistent ones). In this, Alxatib & Pelletier adopt Dalrymple et al.’s (1998) Strongest Meaning Hypothesis (SMH), a principle that favours the most informative (strongest) of the readings that a certain sentence can have.

Setting formal details aside, we summarize the predictions of this account for borderline cases in the same format in Table 20.2. Note that, by the SMH, Pc and its negation take the stronger, ‘super’ interpretation, and on this interpretation they are each untrue of borderline cases. In the case of Pc ∧ ¬ Pc, only the subinterpretation of the conjuncts—individually through the ‘indefinitely’ operator—can produce a nontrivial reading (the reading that c is P in some sharpenings, and ¬P in others). This reading is true only of borderline cases (Table 20.3). Finally, assigning the superinterpretations (again individually, but via the ‘definitely’ operator) gives ¬ (Pc ∨ ¬ Pc) its strongest possible reading, and this reading is also true only of borderline cases.

Table 20.2. Predictions of (global) super-/subvaluationist accounts for borderline cases

Table 20.3. Predictions of local super-/subvaluationist accounts for borderline cases

20.2.2.2 Strict/tolerant evaluation

Cobreros et al.’s (2012) strict/tolerant system is built on a primitive notion of ‘tolerance’, in turn drawn from Crispin Wright’s writings on the sorites paradox,⁸ and from Van Rooij (2010).⁹ The idea is that the applicability of certain predicates (the vague ones) tolerates small changes in the relevant attributes of individuals in its domain. For example, the applicability of ‘tall’ does not intuitively change with very small changes in height, so when the predicate is ascribable to an individual of height n, it remains ascribable if n were slightly lower.

Using this intuition, Cobreros et al. (2012) introduce a primitive metalinguistic binary relation, ∼_P, which holds between any two individuals x,y iff x and y are sufficiently similar with respect to P. This, coupled with a classical conception of truth, allows two other derivative notions of truth: ‘strict’ and ‘tolerant’. A predicate P holds strictly of an entity x iff every entity y for which x ∼_P y, P holds classically of y. And dually, a predicate P holds tolerantly of an entity x iff there is at least one entity y such that x ∼_P y and P holds classically of y:

Let us also take note of how negation is defined in Cobreros et al., in preparation for the comparison with super-/subvaluationary models, and also for our later review of the work on borderline contradictions:

As (5) shows, a negated sentence is classically true iff its negatum is classically false; strictly true iff its negatum is tolerantly false, that is, iff its negatum does not even satisfy the (weaker) tolerant definition of truth; and tolerantly true iff its negatum is strictly false, that is, iff its negatum does not satisfy the (stronger) strict definition of truth.

To see an example, imagine a (classical) model with domain of entities D = {a,b,c,d}. Suppose predicate P has individuals a,b in its extension, and c,d in its anti-extension. Finally suppose that the binary relation ∼_P holds between a and b, b and c, and c and d: a ∼_P b ∼_P c ∼_P d. In this model, the boundary of P is drawn between individuals b and c, but only a is strictly P, because it is the only individual whose ∼_P-relatives all belong to P’s extension. Individuals b and c fall in the extension and the anti-extension of P, respectively, but they are both tolerantly P, because for each of them there is at least one ∼_P-relative that falls in P’s extension. Finally, d falls in the anti-extension of P and has no ∼_P-relative that does. This makes d strictly non-P. Figure 20.1 illustrates the strict/tolerant values of P and ¬ P in this example.

FIGURE 20.1. Example in tolerant/strict model. The shaded region is the extension of P

There is then some similarity between how borderline cases are represented in the strict view, and in the (global) supervaluationary view. In both cases, neither Pc nor ¬ Pc holds, because neither sentence is strictly (or super-) true. And there is a corresponding similarity between tolerant evaluation and (global) subvaluation when it comes to borderline cases: Pc is tolerantly true at the border (that is, when b is close to the boundary of P), as is ¬ Pc. Both sentences are also subtrue, as noted in the previous section.

These parallels break, however, when we look at the classical contradictions (Pc ∧ ¬ Pc) and ¬ (Pc ∨ ¬ Pc): while both sentences are false under strict evaluation, they are in fact tolerantly true when c is close to the ‘edge’ of P (i.e. a borderline case of P).

These predictions are summarized in Table 20.4, in the same format used for super-/subvaluations in the previous section.

Table 20.4. Predictions of strict/tolerant semantics for borderline cases

Cobreros and colleagues also consider the possibility of complementing their logic with a pragmatic principle like the SMH. Given that strict truth asymmetrically entails tolerant truth, it follows that strict evaluation, being stronger, is in general to be preferred whenever it yields a nontrivial reading. This is the case for the sentences Pc and ¬ Pc, but not for classically contradictory sentences, which can never be strictly true. In those cases, only tolerant evaluation can produce nontrivial readings, and that reading is true only for borderline cases (Table 20.5).

And here also, without elaborating on the technical details, we can see a variant, local account that encodes strict/tolerant evaluation in unpronounced operators, and that in combination with the SMH gives rise to a similar pattern for borderline cases (Table 20.6).

Table 20.5. Predictions of the (global) strict/tolerant+SMH model for borderline cases

Table 20.6. Predictions of the (local) strict/tolerant+SMH model for borderline cases

Note that in this local alternative, tolerant truth (through what we may call the ‘tolerantly’ operator) is used only in the conjunctive contradictory sentence; the disjunctive ¬ (Pc ∨ ¬ Pb) takes its strongest interpretation if its disjuncts are each prefixed with a ‘strictly operator. Both parses, in parallel to the local super-/subaccount, generate interpretations that hold only for borderline cases.

20.2.3 Valency greater than three: Fuzzy logic

We turn finally to fuzzy logic, as our exemplar of global multivalent systems (Zadeh, 1975a; Machina, 1976). We have no obvious reason to call classic fuzzy logic global other than by comparing it to variants where the truth-value of subsentences are ‘scaled’, so that their final truth-value is determined by dividing it over the range of possible truth-values (e.g. Osherson & Smith, 1982; and Alxatib et al., 2013). We discuss these systems in more detail later in this section.

A fundamental assumption in multivalued logics is that truth comes in degrees, infinitely many in the case of fuzzy logic. Traditionally the range is represented as the dense interval [0,1] (complete falsity to complete truth). The standard compositional semantics of the connectives is an extension of Łukasiewicz’s (1920) three-valued logic: a negation is as true as its negatum is false; a conjunction is as true as its falsest conjunct, and a disjunction is as true as its truest disjunct (we do not include the standard definition of the conditional here).

(7)a. [[¬ϕ]] = 1 – [[ϕ]]

b. [[ϕ ∧ ψ]] = min([[ϕ]], [[ψ]])

c. [[ϕ ∨ ψ]] = max([[ϕ]], [[ψ]])

In addition to this implementation of fuzzy logic, mathematical logicians have investigated a broader class of systems satisfying certain axioms (Hájek, 1998; and others). But for our current purposes, it is sufficient to focus on the system just introduced.

In borderline cases, where Pc is half-true ([[Pc]] = 0.5), the negation ¬ Pc is also predicted to be half-true, as are the conjunction, disjunction, and negated conjunction/disjunction, of the two sentences. There is thus no predicted distinction between these sentences in borderline cases (Table 20.7).

Table 20.7. Predictions of standard fuzzy logic for borderline cases

This prediction changes, however, in implementations of the fuzzy view that are not global. Non-global fuzzy systems of this sort were first discussed by Alxatib et al. (2013), who propose that the connectives use not just the truth-values of their sentential arguments, but also the range of values that the resulting coordination can have.¹⁰ As an example, take the case of Pc ∧ ¬ Pc. The lowest possible value for this sentence is 0 (e.g. when Pc is completely false), but its greatest possible value is 0.5; the higher the value of Pc, the lower the value of ¬ Pc, so the ‘truest’ that the conjunction can ever be is 0.5. So if Pc is half-true, the conjunction Pc ∧ ¬ Pc is in fact as true as it can possibly be. We might therefore define conjunction (or disjunction) by taking the minimum (or maximum) of the values of its arguments, subtracting the lowest value that such a minimum (or maximum) can reach, and dividing the result over the difference between the greatest possible minimum (or maximum) and the lowest.¹¹ This is Alxatib et al.’s (2013) scaling recipe for conjunction (their ) and disjunction (their ):

What does this change? Contradictory conjunctions are now predicted to be fully true in borderline cases, while the conjuncts that comprise them are each predicted to only be halftrue (simple sentences behave identically to the classical fuzzy system).

We summarize these results in Table 20.8. We will refer to this ‘local’ fuzzy logic as Fuzzy 2 in what follows. The account is local in the sentence that it scales conjunctions/disjunctions locally.

Table 20.8. Predictions of the ‘scaling’ fuzzy model for borderline cases

20.3 EXPERIMENTAL RESEARCH ON THE BORDERLINE

We now turn to our survey of the recent experimental studies that were relevant for theories of vagueness. We focus first on studies that pertain more closely to the theoretical predictions outlined earlier for borderline cases (section 20.3). In section 20.4 we provide a broader review of experimental work on other, vagueness-related phenomena. These will include a typology of vagueness, roundness, and pragmatic motivations for vagueness/imprecision.

In this section we will structure our descriptions of the different studies by focusing on (i) the theoretically relevant question(s) that the study brings up, (ii) the method(s) used in it to address those question(s), and (iii) the results that the study is claimed to have uncovered.

The first study we look, a milestone in the recent experimental work on vagueness, is Bonini et al. (1999)—BOVW hereafter. Broadly speaking, BOVW set out to determine whether the extensions/anti-extensions of vague predicates are intuited to be disjoint, rather than overlapping, by their experimental subjects. They did this by asking their participants to provide numeric answers to survey-style questions like the following:

(10) When is it true to say that a man is ‘tall’? Of course, the adjective ‘tall’ is true of very big men and false of very small men. We’re interested in your view of the matter. Please indicate the smallest height that in your opinion makes it true to say that a man is ‘tall’.

It is true to say that a man is ‘tall’ if his height is greater than or equal to ______ centimetres.

Recipients of surveys like (10) were labelled truth-judgers, in contrast with recipients of surveys like (11), termed falsity-judgers:

(11) When is it false to say that a man is ‘tall’? Of course, the adjective ‘tall’ is true of very big men and false of very small men. We’re interested in your view of the matter. Please indicate the greatest height that in your opinion makes it false to say that a man is ‘tall’.

It is false to say that a man is ‘tall’ if his height is less than or equal to ______ centimetres.

BOVW used six variant questionnaires of this style. In their Study 5, for example, the leadin text was simplified, and the metalinguistic character of the prompt was removed, as in:

(12) When is a person late for an appointment? Please indicate after how many minutes, in your opinion, a person is late for an appointment.

A person is late for an appointment if he shows up ______ or more minutes after the appointed hour.

The general findings from BOVW are claimed to confirm disjointness: by and large, the average of the answers given by truth-judgers was significantly higher than that of falsity-judgers. BOVW argue that these findings contradict the predictions of glut theories (e.g. subvaluations) and also the predictions of fuzzy logic. They also add that, though the results are compatible with gap theories like supervaluations, they are also compatible with the epistemic view of vagueness if it is augmented with the assumption that errors of omission are preferred to errors of commission.¹² To strengthen their argument in favour of the epistemic view, BOVW conduct a seventh study where they use a precise predicate with an unknown boundary instead of a vague predicate.¹³ An example is shown below: participants presented with (13a) were labelled ‘upper judgers’ (in parallel to truth-judgers from the other experiments); those presented with (13b) were labelled ‘lower judgers’.

(13)a. A man is at least of average height among 30-year-old Italians if his height is greater than or equal to ______ centimetres.

b. A man is not as tall as average among 30-year-old Italians if his height is less than or equal to ______ centimetres.

In these cases too, BOVW find that truth-judgers (upper judgers) provide significantly higher estimates than falsity-judgers (lower judgers). This leads them to conclude that vague predicates are mentally represented in the same way as precise predicates with unknown boundaries (recall their Vagueness-as-ignorance hypothesis in (1)).

Most of the subsequent literature has not accepted BOVW’s conclusion, primarily because of worries about the methodology. The most detailed critique of BOVW is offered in Serchuk et al. (2011; SHZ), who argue that the wording of BOVW’s survey introduces a bias towards gap-like responses. As they point out, the conditional used in queries like (10)-(12) may be understood to express a sufficiency condition: it is indeed true to say that a man is tall if his height is greater than (say) 150cm, or even 140cm. If participants interpreted the surveys in this way, they are more likely to have given safe (if extremely low/high) answers, and thus given gap-like responses.

SHZ set out to experimentally confirm this source of noise, by comparing a replication study of BOVW to a similar study in which the surveys are worded without the use of the conditional. A sample questionnaire from SHZ’s revised study is shown below.

(14) What is the smallest height a man can be so that he is still tall enough for it to be true to say that he is ‘tall’? ______ feet and ______ inches.

SHZ’s results confirm their hypothesis: their replication of BOVW’s experiment showed gap-like responses, but in their revised experiment where prompts like (14) were used, no gaps were found. SHZ conclude therefore that BOVW’s experimental findings do not favour any specific theoretical model of vagueness.

More generally, SHZ object that BOVW’s methodology is inherently flawed, since by eliciting precise (numeric) answers to their questionnaires, they presuppose that precise boundaries of vague predicates exist. By way of qualification, however, SHZ acknowledge that it isn’t clear how this ‘forced-filling in’ might affect the data, but they maintain that it is a ‘it is a source of noise that is inherent to the design of the experiment’ (p. 546). We agree with SHZ in principle, but we hesitate to say that design features like these are sources of noise. It is common practice in linguistics and philosophy to use artificial categories in investigating linguistic phenomena (e.g. ‘acceptable’, ‘grammatical’, etc). The use of such categories is not necessarily a commitment to a particular theoretical view. Rather, it provides a way of establishing an empirical typology of the phenomenon in question, with the aim of indirectly learning something about it. If the task that results from doing this was so unnatural as to be nonsensical, we would most likely expect chance responses. But if what we observe is not chance, then there must be some lesson that we can draw about the phenomenon from the way participants react, even if what they are reacting to is not a naturally occurring stimulus.

FIGURE 20.2. Visual context from Ripley (2011)

Reproduced from Ripley (2011), ‘Contradictions at the Borders’, in Nouwen et al. (eds.), Vagueness in Communication. © Springer 2011.

A second method to probe the status of borderline cases became prominent with the studies of Ripley (2011) and Alxatib & Pelletier (2011a,b). Ripley (2011), who introduced the term borderline contradictions for statements of the form ‘c is P and is not P,’ and ‘c neither is P nor is not P’, for c a borderline case of P, presented participants with a visual context (see Figure 20.2), and used a seven-point Likert scale to solicit agree/disagree responses to sentences of this type (see (15)).

(15)a. The circle is near the square and the circle isn’t near the square

b. The circle both is and isn’t near the square

c. The circle neither is near the square nor isn’t near the square

d. The circle neither is nor isn’t near the square

The main finding Ripley reports is a large majority of what he calls ‘hump’ responses: a pattern of answers where agreement peaks somewhere between the extremes, and rises (and declines) consistently into (and away from) the peak. As he points out, the precise location of the peak may not be relevant, given that different participants may differ on where they would place the center of the borderline region. What is more important is how participants evaluate the sentential prompts given where they view that borderline region to fall. Ripley takes the prevalence of hump response patterns to lend strong support to dialethic logics (like Priest’s (1979) LP). He adds that the similarity to fuzzy patterns is superficial, given that the majority of peaks in those hump response patterns reach 6 or 7, which is beyond the halfway-agree point that the fuzzy logician might predict. Of course, we take it that Ripley was considering logics of the Fuzzy 1 type we discussed earlier, not the ‘scaled’ Fuzzy 2 logic types where contradictions may be scaled to become ‘fully’ true.

A similar finding to Ripley’s was independently reported by Alxatib & Pelletier (A&P). Their goal was to compare the rate of acceptability of contradictory statements, of a similar type to Ripley’s, against the rate of acceptability of their non-contradictory subparts. Like Ripley, A&P presented their participants with a visual display (Figure 20.3) along with a number of sentences (examples in (16), where X was any of #1, #2, etc). Unlike Ripley, however, they used a three-way forced choice task instead of a Likert scale (the options were ‘True’, ‘False’, and ‘Can’t tell’).

FIGURE 20.3. Visual context from Alxatib & Pelletier (2011)

Reproduced from Alxatib & Pelletier (2011), ‘The Psychology of Vagueness: Borderline Cases and Contradictions’. Mind and Language 26. © John Wiley & Sons Ltd 2011.

(16)a. X is tall

b. X is not tall

c. X is tall and not tall

d. X is neither tall nor not tall

A&P also report a peak of ‘True’ responses to contradictions in the middle of the height range, specifically to the sentences ‘#2 is tall and not tall’ and ‘#2 is neither tall nor not tall’ (44.7% and 53.9%, respectively). They also find the hump response pattern among the participants that answered ‘False’ to ‘#2 is tall’ and to ‘#2 is not tall’: from among these subjects, over two-thirds (68.8%) simultaneously answered ‘True’ to ‘#2 is tall and not tall’.¹⁴

The acceptability of contradictory statements reported in Ripley and A&P is surprising on both the global epistemic account, and the global super-/subvaluationary accounts, since they both preserve classical contradictions. A&P attempt to explain the pattern by proposing the hybrid gap/glut+SMH system described in section 20.2.2.1. They also use their findings to argue against the epistemic view, where contradictions are predicted to be false, as well as the fuzzy view, where contradictions like ‘#2 is tall and not tall’ are predicted to be as acceptable as their conjuncts ‘#2 is tall’ and ‘#2 is not tall’.

In levelling these charges, however, A&P do not consider the local variants of epistemicism and of fuzzy logic. Consider first the local epistemic view described earlier, where the syntax is assumed to include (silent) epistemic possibility/necessity operators. On this account, contradictory sentences may have underlying representations where the conjuncts are prefixed with possibility operators, allowing a reading of ‘X is tall and not tall’ as ‘X is possibly tall and possibly not tall’, which is consistent and true of borderline cases. But note that on this view, parses like this should be available both for vague predicates and for precise predicates with unknown boundaries. So we expect the acceptability of contradictions reported in A&P to have a parallel in cases like (17):

(17)a. 170 cm is average height and not average height

Parsed as: 170cm is possibly average height and possibly not average height

b. The number of NYC subway trains running right now is more than 300 and not more than 300

Parsed as: The number of NYC subway trains running right now is possibly more than 300 and possibly not more than 300

We are sceptical that this parallel will be found, but with the many surprises that experiments have shown, we would be wise to suspend our intuitive judgements and await quantitative results.

A&P’s critique of fuzzy logic is stated more generally in Sauerland (2011). Sauerland points out that whenever two predicates P,Q are judged to ‘half-hold’ of an individual c, the sentence Pc ∧ ¬ Pc is predicted to have the same truth-value, and hence elicit the same responses, as the sentence Pc ∧ ¬ Qc. The generality of this inference pattern derives from the truth-functionality of all fuzzy logic systems in the sense of Hájek (1998). In Sauerland’s experiment, participants in different versions of an online questionnaire judged sentences such those below on a 0 to 100 scale.

Sauerland found that even when the constituent propositions Pc, ¬ Pc, Qc, and ¬ q all received a mean rating in the middle region (between 40 and 60), the mean rating for the ‘contradictions’ Pc ∧ ¬ Pc and Qc ∧ ¬ Qc was greater than that for the non-contradictory conjunctions Pc ∧ ¬ Qc and Qc ∧ ¬ Pc.

But in their criticism, both Sauerland and A&P assume that no extra-semantic principles can be added that would reconcile the fuzzy view with the findings they report. However, as we saw in section 20.2, the scaling version of fuzzy logic (Fuzzy 2 in Table 20.8) predicts exactly A&P’s and Sauerland’s response-patterns: Pc, Qc, and their conjunction, are half-acceptable at the border, as are ¬ Pc and ¬ Qc. But ‘contradictory’ conjunctions like Pc ∧ ¬ Pc are fully acceptable in borderline cases, via scaling.

With the increasing use of experimental methods in studying vagueness, it has become possible to formulate and address increasingly specific questions. In recent work, Égré & Zehr (2016) set out to replicate A&P’s results, and to compare the acceptability of conjunctive contradictory stimuli (of the form Pc ∧ ¬ Pc) to that of negated disjunctive analogues (of the form ¬ (Pc ∨ ¬ Pc)). To this end, Égré & Zehr presented their participants with a lead-in context and a sequence of (four) forced-choice prompts. A sample is shown below—the four prompts appeared one at a time, and each replaced the other as the participants clicked the arrow:¹⁵

A survey on heights has been conducted in your country. In the population there are people of a very high height, and people of a very low height. Then there are people who lie in the middle between these two areas.

Imagine that Betty is one of the people in the middle range. Comparing Betty to other people in the population, is it true to say the following?

[Click to see the first description]

[→ Click here to continue]

Égré & Zehr use the last two descriptions as true/false benchmarks, and report a near 100 per cent / 0 per cent mean acceptance rate for them (respectively). ‘Neither’ descriptions had a mean acceptance rate of roughly 82 per cent (significantly lower than that of the true description), and ‘and’ descriptions had a mean rating of about 47 per cent (significantly lower than ‘neither’ descriptions).

This preference for ‘neither’ descriptions calls for a finer semantic-pragmatic theory than any of those discussed in section 20.2, and Égré & Zehr propose a local strict/tolerant view that includes the SMH (as described in section 20.2.2.1) but where strict evaluation is assumed to be default but subject to replacement with tolerant evaluation when necessary. Tolerant evaluation (or subvaluation) is therefore dispreferred, and this causes preference for ‘neither’ descriptions over ‘and’ descriptions.¹⁶

Before we conclude our survey of borderline experiments, we briefly discuss the so-called ‘dynamic’ studies of vagueness. Here, unlike in the studies described earlier, judgements are elicited for a series of stimuli, and the order of presentation plays a central role as an independent variable. Building on work reported in Kalmus (1979) and Raffman (1994, 2014), Égré et al. (2013) presented their participants with a series of fifteen colour squares (displayed individually on a computer screen) that differ gradually from yellow to orange, and assigned them two kinds of tasks: linguistic and perceptual. In the linguistic task, participants were given two colour words, ‘yellow’ and ‘orange’, and asked to choose the one that better described the square they saw; the perceptual task was similar, but the choices were two coloured squares instead of colour words (the two squares corresponded to the yellowest/most orange squares in the series).¹⁷

Égré and colleagues’ main objective was to measure the point in the series at which participants switched their response (from one colour word to the other in the linguistic task, and from one square to the other in the perceptual task), and to test the effect of order on the location of that point. There were three conditions: two ordered (ascending and descending), and one random. Égré and colleagues’ results revealed a significant effect of order in the linguistic task, but no effect in the perceptual task: when participants were presented with an ordered sequence of colour stimuli, and asked to choose the colour word that better matched the stimulus, they changed their answers at an earlier point in the series than they did when the presentation was randomized. This result of enhanced contrast (or counterhysteresis¹⁸) was not found in the perceptual task, indicating that the effect was not due solely to the seamless progression of colour in the (ordered) stimuli. In another experiment (their Experiment 2), Égré and colleagues replicated the finding with a forced-choice ‘agree’/‘disagree’ linguistic task, in two kinds of conditions: one with positive sentences (e.g. ‘the square is yellow’) and one with negated sentences (‘the square is not yellow’). The results paralleled those of their first experiment: speakers switch the category of their response just as they enter the borderline region in the ordered series. For example, in a condition where the series proceeds gradually from orange to yellow, and where the participant is presented with the sentence ‘the square is orange’ (or ‘the sentence is not yellow’), the participant chooses ‘agree’ from the start but begins disagreeing in shades that are on the closer end of the orange-yellow (borderline) region.

To Égré and colleagues, this shows that an overlap theory is needed in the semantics, so that both colour descriptions are (roughly) equally applicable in the borderline zone. And while the account they propose is based on the strict/tolerant framework discussed in section 20.2, they note that other views are applicable so long as they allow for ‘gluts’ in the borderline range—recall, for instance, Epist* and the local super-/subvaluationary model presented earlier. To explain the counterhysteresis effect, Égré et al. propose a pragmatic principle that they call informativeness (akin to Grice’s maxim of quantity—Grice, 1967). The principle compels speakers to signal changes when they detect them, though only when that signal is accurate. In the gradual progression from yellow towards orange shades, the constraint leads participants to opt out of the ‘yellow’ categorization and in favour of ‘orange’ relatively early in the borderline range. This is (a) because in that area both descriptions are accurate (assuming a glut-like view), and (b) because the switch is encouraged by the principle of informativeness.¹⁹

Other findings from ‘forced-march’ experimental studies are more difficult to interpret, especially from the theoretical perspectives we outlined in section 20.2. In Raffman (2014; see also 1994), some participants were walked through a continuum of stimuli (like in Égré and colleagues’ ordered conditions), but as soon as they switched the category of their response (say from ‘yellow’ to ‘orange’), the direction of the march was reversed (towards the yellow endpoint again). Raffman reports that in these ‘reversal’ conditions, participants persist in applying the new category to stimuli that they had just applied the other category to. Response patterns like these are interesting because they exemplify hysteretic behaviour, and one might expect, on the basis of this behaviour, that complete marches of the sort used by Égré and colleagues induce analogous hysteresis rather than counterhysteresis (that is, a category switch at the end, rather than the beginning, of the borderline region). We must note here that counterhysteresis was reported in complete marches not only by Égré and colleagues, but by Raffman (2014), and Kalmus (1979) also. For discussion of the theoretical implications of forced-march experiments, we refer our readers to these studies and the works they cite.

This concludes our survey of borderline-case experiments.

20.4 BEYOND BORDERLINE CONTRADICTIONS

The studies we presented in the previous section all focused on borderline issues, and in most cases attention was paid to scalar adjectives like tall and rich.²⁰ Vagueness is a broader phenomenon, however, and in the limited remaining space we aim to do some justice to that breadth by discussing work on the general typology of vagueness. We later relate this discussion to the specific issue of rounding in numbers, and to relevant experimental work.

20.4.1 Imprecision and a typology of vagueness

One theoretical debate that we sidestepped so far concerns phenomena that are at least similar to vagueness with other expressions. Consider the natural concept terms heap, city, Beef Stroganoff, and sing. These, like the adjectives tall and rich, allow the construction of a sorites series. For example, it seems true that, if one vocal production qualifies as singing, another production that differs only slightly in acoustic properties also qualifies as singing.²¹ But then all vocal productions are predicted to qualify as singing. Even terms that have scientific uses such as bird or a mathematical use such as one thousand have been found to be gradient and susceptible to the construction of a sorites paradox on their natural use. For example, speakers can agree that, if it is acceptable to describe quantity X using the expression one thousand, it is also felicitous to describe quantity X’ which differs by one from X as one thousand. The two most relevant research questions about this broader class of phenomena that exhibit the hallmark property of vagueness are (i) whether they can all receive a uniform account, and (ii) if no uniform account is possible, whether accounts of this subclass of phenomena fall within the classes of accounts introduced in section 20.2.

The position that all phenomena are to be accounted for by one mechanism is taken most strongly by work in cognitive science. Extending Wittgenstein’s (1953) claim that our knowledge of a concept is constituted by knowledge of its central members rather than its boundary, Rosch (1975) and others developed prototype theory. In addition to vagueness, prototype theory also aims to account for the difference between more and less typical members of a category (e.g. a canary is a more typical bird than a penguin), and it proposes to do so by means of a graded membership function (Hampton, 2007; and others). Hence at the level of primitive concepts, prototype theory is analogous to fuzzy logic. But unlike fuzzy logic, most work in prototype theory does not commit to any compositional mechanism (see Kamp & Partee, 1995, Sassoon, 2013a, Del Pinal, 2016, for discussion). The Conceptual Spaces model of Gärdenfors (2000) is similar to prototype theory, but locates gradience explicitly in our conceptual representations. The proposal is that our categorization of objects is based on multidimensional spaces, for instance the perception of a colour by its hue, value and chroma in humans with typical vision on the threedimensional spindle of Munsell (1915). Furthermore Gärdenfors assumes a similarity metric; that is, a consistent measure of distance for any two points in a conceptual space. Together these two assumptions yield an account of the construction of conceptual categories where one or a few central members of a category are known by categorizing new points in the same category as the nearest known point. The view of concepts by Rosch and Gärdenfors is widely debated within cognitive science: for instance, Carey (2009) develops a quite different picture and Del Pinal et al. (2017) and others study gender stereotypes to show that concepts have more complex internal structure than predicted by prototype models. But the claim that a single theoretical model of vagueness is sufficient is not addressed in these critiques.

Most work in theoretical semantics and pragmatics, on the other hand, assumes that at least two different accounts are needed for all the different phenomena that give rise to a version of the sorites paradox. This view is implicit in earlier work such that of Kamp (1975) which focuses on adjectives, but only later work makes the typology explicit. Kennedy (2007) proposes a distinction between vagueness of scalar adjectives and imprecision of, for instance, round numbers, pointing to the intuition that only the latter allow a possibility of crisp intuitions for borderline cases. Sauerland & Stateva (2007, 2010) use the terms scalar and epistemic vagueness for an overlapping distinction. Rotstein & Winter (2004) and Kennedy & McNally (2005) propose a distinction within the scalar adjectives based on scale structure, in particular whether the scale contains an endpoint, and if so, whether it has one on both of its ends (see also Lassiter & Goodman, 2013). Finally Solt (2016) investigates one specific comparison in depth, namely the two quantifiers most and more than half, where she shows that the former is more vague than the latter using contrasts like (19) (Solt, 2016: 67) and other arguments.

(19)a. ??Most of the American population is female.

b. More than half of the American population is female.

The theoretical accounts of the subclasses in most cases make use of the theoretical proposal we introduced in section 20.2. For example, Kennedy (2007) accounts for imprecision by appeal to a proposal of Lasersohn (1999). This model is essentially a trivalent account, but Lasersohn assigns to an expression two meanings: its true denotation and a loose denotation. As illustrated later, Lasersohn furthermore proposes that approximators like exactly narrow the loose denotation, while loosely speaking equates the true denotation with the loose one.

Compared to the accounts discussed earlier, Lasersohn’s proposal is incomplete as, for example, it does not provide an account of negation. We now turn to the more specific discussion of round numbers.

20.4.2 Round numbers

Round numbers such as 90 provide less information about numerosity than non-round number such as 93. Roundness holds to different degrees for all numbers (Sigurd, 1988; Jansen & Pollmann, 2001)—100 is more round than 90 which in turn is more round than 93.5. Cummins et al. (2012) show that roundness also affects how modified numerals such as more than 90/93 are interpreted.²² Round numbers, as discussed earlier for 1,000, exhibit features of vagueness—specifically the sorites paradox—and in many accounts are viewed as one case of vagueness among others especially in the cognitive science literature. But of the expressions exhibiting vagueness-like phenomena, round numbers are on the opposite end of the spectrum from scalar adjectives such as tall: while tall intuitively has no precise interpretation, round numbers are always associated with a precise mathematical interpretation, and concepts like fruit occupy a middle ground where a precise interpretation is possible but not generally agreed upon. Sauerland & Stateva (2007, 2010) point out that the distribution of approximators differs for the two types of vagueness: for instance, precisely 1,000 is felicitous, but precisely tall is not (see also Morzycki, 2015). By this test, some other expressions like in the middle and at noon fall in the same category as round numbers, but the latter constitute the core case.

A novel model of round number denotation is proposed by Krifka (2009). He proposes that measurement scales at different levels of granularity are relevant for the interpretation of numbers. For example, the time scale may be divided into 15 minute segments or 30 minute segments. While in principle any number contained in it could be used to denote the interval (15,45), 30 is used for reasons of simplicity. Krifka argues that the relevant concept of simplicity is simplicity of representation, which associates 30 with the prominent conceptual unit of half an hour, edging out the equally short expressions 20 and 40. Krifka furthermore observes a general preference for vague interpretations: while 31 is interpreted as denoting an interval on the 1-minute scale, 30 must be interpreted on a coarser scale. He relates this preference to the greater utility of less precise interpretations. This hypothetical preference is supported by the finding that round numbers occur more frequently than non-round ones across languages (Dehaene & Mehler, 1992; Cummins, 2015), but more recent work has also found experimental evidence for the preference.

Initial experimental evidence for a preference for round numbers was provided by Van der Henst et al. (2002). They report on a series of studies where a researcher asks strangers in a public place for the time. In their Experiment 1, they found that even of the people using a digital watch, 57 per cent of the time responded with multiples of 5 in the minute position such as 4:25 or 4:40. Users of analogue watches gave round responses even more frequently (97% of the time), but what is noteworthy is that users of digital watches still round to the nearest multiple of 5, even though they could easily read the exact number off their watch displays—doing this at 57 per cent frequency is 37 per cent higher than what would be expected at chance since it is only 20 per cent of the time that the precise response is divisible by 5. Further experiments reported by Van der Henst et al. (2002) show that the frequency at which people round off is affected by the pragmatics of the situation—for instance, when the person asking them wants to set her watch, people tend to round less. Several other recent studies have found that speakers round even when precise information is available to them (see Solt et al., 2017, for an overview). However, other experimental work has found that sometimes people also give overly precise responses. Sauerland & Gotzner (2013) conducted an internet-based questionnaire study asking people sixty questions that they couldn’t know a precise answer to, such as How many houses are in your neighborhood? Though no precise responses are expected, they report that overall 20 per cent of the responses were greater than 20 and were not divisible by 5. However, they propose that such over-precise responses may serve to indicate a disdain for the questions, and may therefore flout the preference for vagueness.

Recently, Solt et al. (2017) addressed the motivation of the preference for vague expressions with two experiments designed to test whether round numbers are easier for cognitive processing. They also look at time expressions of either fine granularity (minutes not divisible by 5, e.g. 3:17), medium granularity (minutes divisible by 5, but not 15, e.g. 3:20), and coarse granularity (minutes divisible by 15; e.g. 3:15) using both a short-term memory task and a clock-time arithmetic task. In the memory-task, participants had to remember a sequence of between three and five clock times, and were then asked whether a probe time had occurred in the sequence. In the arithmetic task, participants had to compute additions or subtractions of a clock time and an increment in minutes; for instance, 3:21 minus 30. In this task, Solt et al. (2017) varied the granularity of the start time and the increment independently. In both experiments, Solt et al. (2017) found the expected effect that finer granularities are more difficult both in the percentage of incorrect responses and in the reaction times. In the memory task, however, there was no significant difference between the medium and fine granularities. Overall, the result shows that round numbers provide an advantage on some cognitive tasks. In sum, experimental evidence supports the existence of a preference for vagueness and the idea that a processing advantage may underlie this preference.

20.5 CONCLUDING REMARKS

Overall, we think that the recent experimental work has provided important insights for understanding vagueness in language. One likely reason for this is that, in contrast to other domains of semantic and pragmatic study, individual speakers’ intuitions about vague sentences are often not reliably reproducible, and speakers generally do not feel confident in their judgements. Controlled data collection from theoretically-naïve informants therefore offers the chance to increase reliability, and thus help guide theory development. The study of vagueness has been, and continues to be, a fertile ground for applying the methods of experimental pragmatics after the programme was launched by the publication of Noveck & Sperber (2004; see also the reviews of Noveck & Reboul, 2008; Sauerland & Schumacher, 2016). We hope to have demonstrated the rewards of this recent experimental turn in this chapter.

ACKNOWLEDGEMENT

The ideas and opinions presented in this chapter have benefited from discussions we’ve had, together and individually, with many colleagues. We wish to thank Alan Bale, Guillermo Del Pinal, Paul Égré, James Hampton, Nancy Hedberg, Morgan Mameni, Van McGee, Peter Pagin, Jeff Pelletier, Dave Ripley, Robert van Rooij, Hotze Rullmann, and Stephanie Solt. This work has benefited from the financial support of the BMBF (grant 01UG1411 to ZAS), DFG (grants SA 925/11-2 and SA 925/12-2 to Sauerland), and DAAD (grant 57316845 to Sauerland).

¹ Eubulides of Miletus (c.405–c.330 BC)—see Seuren (2005).

² For comprehensive theoretical reviews, see Williamson (1994); Keefe & Smith (1997a); Keefe (2000); Smith (2008); Van Deemter (2010); and Sorensen (2016).

³ The use of experimental methods in the study of adjective semantics is discussed extensively by Solt (Chapter 16 in this volume).

⁴ See for example Keefe (2000) for a discussion of the similarity between supervaluations and contextualist accounts of vagueness; and see Cobreros et al. (2012) for a detailed illustration of how tolerant evaluation is equivalent to Priest’s (1979) paraconsistent Logic of Paradox (LP).

⁵ Suppose for example that the term heap had a precise boundary n. Can we tell apart those things that have n or more grains in them from those that do not?

⁶ See also Kamp (1975). For precursors, see Mehlberg (1958) and Van Fraassen (1966). We use the terms ‘precisification’ and ‘sharpening’ interchangeably here. The corresponding term in Fine is ‘specification’. Much of the work on supervaluations and trivalent logics is motivated by puzzles concerning presuppositions and presupposition projection. Theoretical questions of this kind have also benefited from experimental studies in recent years, as discussed in Schwarz (Chapter 6 in this volume).

⁷ Jaśkowski (1969); Hyde (1997). See also Lewis (1982).

⁸ Wright (1975, 1976, 1987).

⁹ See also Ripley’s (2017) Vagueness-as-conflation hypothesis.

¹⁰ Alxatib and colleagues build on Zadeh’s (1978) and Osherson & Smith’s (1982) treatment of conjunction. Since these systems are not truth-functional, they do not constitute a fuzzy logic in the sense of Hájek (1998).

¹¹ We abstract away from the intensionality that is needed to define ‘lowest’ and ‘highest’ possible minima and maxima here. These correspond respectively to Alxatib et al.’s (2012) floor and ceiling in (8). It may be noticed also that these definitions effectively localize the product of Dalrymple and colleagues’ SMH, though this is a detail that need not concern us here (see Sauerland, 2012b, and Alxatib et al., 2013, for discussion).

¹² On this, BOVW cite Ritov & Baron (1990) and Spranca et al. (1991).

¹³ BOVW also argue against gap theories on theoretical grounds. We do not discuss their arguments here (but see Alxatib & Pelletier, 2011a,b).

¹⁴ This finding is reproduced in an Amazon Mechanical Turk (MTurk) survey reported in Alxatib (2010). Alxatib showed a nearly identical visual context to his participants, but asked them to select from the displayed men the ones that they thought would interest an imaginary woman who is looking for a date. In some cases, the woman was described as taking interest in a man ‘only if he is tall’; in others, ‘only if he is tall and not tall’, and so on. Alxatib treated instances of selecting, for instance, #3 for the ‘tall-preferring’ woman as a parallel to answering ‘True’ for ‘#3 is tall’ in A&P. The patterns Alxatib reported matched those of A&P.

¹⁵ Égré & Zehr (2016) used eight adjectives in total: tall, rich, heavy, old, loud, fast, large, and wide.

¹⁶ The reader will have noticed that many studies on borderline contradictions appeal to the SMH to explain their findings. If the SMH is related to pragmatic considerations, then it may be fruitful to test the proposals made in these studies from a developmental perspective. It is known in the literature on scalar implicature that children show systematic non-adult-like reactions to scalar terms (see Skordos & Barner and Breheny, Chapters 2 and 4, respectively, in this volume). If this is due to differences in mastery of communicative reasoning, we might ask if a similar difference can be found between adults and children in their use of vague terms, particularly in cases where the SMH is thought to play a role.

¹⁷ In another condition, Égré and colleagues also used a blue-green series.

¹⁸ Égré and colleagues use the terms enhanced contrast, counterhysteresis, and negative hysteresis interchangeably. The terms contrast with hysteresis, which in this context may be defined as the persistence of applying a category in presentations of gradually changing stimuli. If hysteresis were found in Égré and colleagues’ study, the category switch would occur at a relatively late point in the ordered conditions.

¹⁹ Égré and colleagues are careful to note that an overlap theory would still be consistent with hysteresis, rather than counterhysteresis. They favour the overlap view in light of the acceptability of conjunctive descriptions like ‘orange and yellow’ and ‘yellow and not yellow’. For reasons of space we cannot discuss their dynamic investigation of these stimuli.

²⁰ See also Solt (2015).

²¹ See Rabagliati & Srinivasan (Chapter 22 in this volume) for related discussions of ‘lexical flexibility’.

²² Semantic and pragmatic analyses of modified numerals have also benefited from recent experimental work. See Nouwen et al. (Chapter 11 in this volume) for a review.