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Approximators as a case study of attenuating polarity items∗
Stephanie Solt
Leibniz-Zentrum Allgemeine Sprachwissenschaft (ZAS)
1. Introduction
Approximators such as about and roughly exhibit a curious and little-recognized form of
polarity sensitivity. As illustrated below, in simple sentences they are positive polarity items
(PPIs), whereas when embedded in comparative quantifiers, they become negative polarity
items (NPIs):
(1) a. Lisa has about/roughly/approximately 50 sheep.
b. *Lisa doesn’t have about/roughly/approximately 50 sheep.
(2) a. *Lisa has more than about/roughly/approximately 50 sheep.
b. Lisa doesn’t have more than about/roughly/approximately 50 sheep.
As a caveat, (1b) and (2a) do have a use on which they are felicitous, namely when they
serve to contradict or deny an assertion involving the approximator in the prior discourse
context. If for example someone has previously claimed that Lisa has ‘about 50’ sheep, I
may counter that “no, she doesn’t have ‘about 50’” or “no, she has more than ‘about 50’ ”.
We will put this quotative or echoic usage aside, and focus on the occurrence of sentences
such as those in (1)-(2) in a neutral discourse context.
In their felicitous uses, approximator-modified numerical expressions make weaker as-
sertions than salient alternatives, notably their unmodified counterparts. Assuming that fifty
and about fifty describe the scalar ranges depicted in (3), we obtain the asymmetric entail-
ments indicated in (4).
∗For helpful comments and discussion, I would like to thank Clemens Mayr, Marie-Christine Meyer, An-
dreea Nicolae, Uli Sauerland, Benjamin Spector, Carla Umbach and Brandon Waldon, as well as the audience
of NELS48. This work was supported by the German Science Foundation (DFG) under grant SO1157/1-2.
Stephanie Solt
(3) Ranges corresponding to numerical expressions:
50
about 50
more than 50
more than about 50no more than 50
no more than about 50
(4) a. Lisa has 50 sheep. ⇒Lisa has about 50 sheep.
b. Lisa doesn’t have more than 50 sheep. ⇒Lisa doesn’t have more than about
50 sheep.
On this basis, we can assign approximators to the class of attenuating polarity items
(Israel 1996, 2006, 2011), whose characterizing feature is that they make relatively weak
or understated assertions relative to possible alternatives. Additional members of the class
include the NPI much (Homer *slept much / didn’t sleep much) and the PPI fairly (Bart is/
*isn’t fairly lazy).
Despite decades of research on the semantics and pragmatics of polarity items, attenua-
tors remain relatively understudied (notable exceptions are Israel’s work and more focused
contributions such as Matsui 2011 and Onea & Sailer 2013). With regards to approxima-
tors in particular, the PPI status of simple approximator examples is discussed by Rodr´
ıguez
(2008) and Spector (2014), while Solt (2014) observes that approximators in comparatives
pattern as NPIs. Yet to my knowledge the only discussion of the reversal in polarity sensi-
tivity illustrated in (1)-(2) is a brief mention in Israel (2006), and there is no fully worked
out existing account that is able to capture these data. The goal of this paper is to formulate
such an account, as a step towards a more general theory of attenuating polarity items.
The two central intuitions that form the basis of the proposal developed here are these:
• Approximators, like other polarity items, obligatorily introduce alternatives. Because
they are modifiers, their alternatives necessarily include the corresponding unmodi-
fied form.
• Being vague, approximators do not readily make stronger statements than salient
alternatives – notably the unmodified one. They are thus restricted to situations in
which the simpler unmodified form could not have been asserted.
The challenge is to justify these intuitive claims, and develop them into a rigorous theory
that accurately accounts for the polarity-based distributional restrictions that characterize
approximators – and other members of the attenuating class.
The structure of the paper is as follows: Section 2 develops the neo-Gricean framework
in which the present account will be couched. Section 3 applies this framework to the
polarity sensitivity of approximators. Section 4 considers in more depth the implicatures
that arise with approximators of different sorts in their felicitous uses. Finally, Section 5
Approximators as attenuating polarity items
contrasts the present approach to one based on a grammatical theory of implicature, and
Section 6 concludes.
2. Pragmatic framework
A wide variety of linguistic phenomena have been productively analyzed with reference
to alternatives to a linguistic expression that the speaker could have but did not utter. This
includes the semantics of focus-sensitive particles, scalar and other sorts of implicatures,
and – most relevant to the present topic – polarity sensitivity. I follow this broad tradition
here. More specifically, I adopt a neo-Gricean pragmatic approach to the role of alternatives
in linguistic meaning, based on proposals by Krifka (1995) and more directly Katzir (2007).
The central component of the formal system I assume is the following conversational
principle or rule of assertion from Katzir:
(5) Conversational principle: Do not use φif there is another sentence φ0∈ALT (φ)
such that both:
i. φ0is better than φ(φ0φ) ii. φ0is weakly assertable
Before delving into the definitions of the terms in (5), let us first note that this principle has
two important consequences:
1. In asserting φ, the speaker implicates that all better alternatives φ0cannot be asserted.
2. A sentence φis blocked when an implicature derived in this way contradicts the
original assertion, or equivalently, when φalways has a better alternative.
It is the second of these points that will form the basis of the account of the polarity-based
restrictions on approximators developed here.
Let us turn now to how the component parts of (5) should be understood. To start,
I adopt Katzir’s structural view of alternatives, according to which they are derived via
substitution and deletion. The formal details are the following:1
(6) Substitution source:
Let φbe a parse tree. The substitution source for φ, written as L(φ), is the lexicon
of the language.
(7) Structural alternatives:
Let φbe a parse tree. ALT (φ)– the set of alternatives to φ– is the set of parse
trees φ0that can be derived from φvia a finite series of deletions, contractions, and
replacements of constituents in φwith constituents of the same category taken from
L(φ).
1Note that Katzir discusses cases in which the substitution source cannot be equated with the lexicon, per
(6), but also includes elements from the prior discourse; as these issues do not come up in the present context,
I skip this refinement here.
Stephanie Solt
Also following Katzir, I take a sentence to be ‘weakly assertable’ if the speaker believes it
to be true, relevant and supported by the facts.
The definition of the ‘better than’ relation is where things become more interesting.
Building on a possibility discussed by Katzir (though not the option he ultimately adopts),
I propose that – at least in the case of attenuating polarity items – ‘better than’ should
be understood in terms of both informativity and simplicity. This position embodies the
often conflicting pressures towards informativity and simplicity embodied in Grice’s (1975)
maxims of Quantity and Manner, and in Horn’s (1984) Q- and R-principles. Formally, I
assume the following definition:
(8) φψiff φ%INF ψ∧φ%SIMP ψ∧(φIN F ψ∨φSIMP ψ)
In words, (8) states that φis better than ψif it is at least as informative and at least as
simple as ψ, and has an advantage on either informativity or simplicity (or both).
The simplicity relation φ%SIMP ψcan in turn be defined in structural terms:
(9) Simplicity:
a. φ%SIMP ψiff φcan be derived from ψvia substitution/deletion
b. φSIMP ψiff φ%SIMP ψand not ψ%SIMP φ
That is, φis simpler than ψif φcan be derived from ψvia a series of substitutions or
deletions, but not vice versa.
The informativity relation φ%INF ψturns out to be more complicated. In most work
in the alternatives-based tradition, informativity is understood in terms of asymmetrical
entailment: φis more informative than ψif the set of worlds in which φobtains is a proper
subset of the set of worlds in which ψobtains. This is fully adequate for the paradigm
cases to which such theories have been applied. For example, a relation of asymmetric
entailment holds between and and or, between (at least) 4 and (at least) 3, and between all
and some. But it becomes problematic when φor ψor both are vague (or, more accurately,
have constituents that are vague). The issue is that on one way of resolving the vagueness,
φmight asymmetrically entail ψ, but on another, the two might be equivalent, or it might
even be ψthat asymmetrically entails φ.
To deal with this issue, I propose that the ‘more informative’ relation φ%INF ψmust
be understood as ‘definitively more informative than’ (in way that will be made clearer
below). There is in fact independent motivation from the domain of scalar implicatures that
something like this is necessary. van Tiel et al. (2016) demonstrate that scalar implicatures
are more likely to arise between pairs of adjectives when the scalar distance between them
is greater (e.g. difficult tends to implicate not impossible, whereas content does not con-
sistently implicate not happy). As an even more relevant parallel, Leffel et al. (to appear)
observe an asymmetry in the interpretation of not very Adj: with minimum-standard abso-
lute gradable adjectives, this licenses the inference that the positive form Adj holds, while
with context-sensitive relative gradable adjectives, no such inference is drawn:
Approximators as attenuating polarity items
(10) a. John was not very late. John was late.
b. The antenna is not very bent. The antenna is bent.
(11) a. John is not very tall. 6 John is tall.
b. John isn’t very smart. 6 John is smart.
Put differently, the examples in (10) license the inference to the negation of the simpler
form not Adj, while the examples in (11) do not. Leffel and colleagues account for this
pattern via a constraint on the derivation of implicatures with vague predicates, according
to which such implicatures are not drawn if the conjunction of a sentence with its stronger
alternative would necessarily be a borderline contradiction (Alxatib & Pelletier 2011,
Ripley 2011). Intuitively, there are situations that are simultaneously clear cases of both
late and not very late; but any individual that is both tall and not very tall is necessarily a
borderline case of both predicates. Thus the vague standards of tall and very tall are not
sufficiently distinct for an implicature to be drawn from not very tall to NOT(not tall)=tall.
I retain the insight articulated by Leffel et al. (to appear), though giving it a somewhat
different implementation. Formally, I follow Krifka (2012) in taking expressions of lan-
guage to be interpreted relative to a pair of indices hw,ii, where wis a world index and
ian interpretation index. That is, the usual notion of a world parameter is decomposed
into world and interpretation components. If JαKw,i6=JαKw0,i, this means that there is some
factual difference in the state of affairs at indices hw,iiand hw0,ii. If on the other hand
JαKw,i6=JαKw,i0, the difference lies instead in how expressions of the language are inter-
preted at indices hw,iiand hw,i0i. The common ground can be modeled as a pair hW,Ii,
where Wis a set of worlds and Ia set of interpretations.
With this framework in place, it is possible to define relative strength or informativity
in several distinct ways, which differ in how stringent they are. Starting with the ‘at least
as strong as’ relation in (12), (13)-(15) represent some possibilities:
(12) φis at least as strong as ψ(φSψ) iff ∀i,{w:φi,w=1}⊆{w:ψi,w=1}
(13) φis weakly stronger than ψ(φWS ψ) iff:
a. φSψ
b. ∃w,i[ψw,i=1∧φw,i=0](i.e. ¬ψSφ)
(14) φis strictly stronger than ψ(φSS ψ) iff:
a. φSψ
b. ∀i∃w[ψw,i=1∧φw,i=0]
(15) φis definitively stronger than ψ(φDS ψ) iff:
a. φSψ
b. ∃w∀i,i0[ψw,i=1∧φw,i0=0]
Stephanie Solt
The definition in (13) states that φis (weakly) stronger than ψiff there is some inter-
pretation ion which it is stronger (understood in the usual way in terms of asymmetrical
entailment). That in (14) says that φis (strictly) stronger than ψiff it is stronger on all
interpretations. Finally, (15) is even stricter, specifying that φis (definitively) stronger than
ψiff there is some world for which across all possible interpretations of φand of ψ, the
latter is true while the former is false.2
I would like to argue that for the purpose of selecting between alternatives, the ‘more
informative’ relation INF in (8) must be understood as ‘definitively stronger’ DS, per
(15). The rationale is the following: In order for φto be preferable to ψon grounds of
informativity, there must necessarily be some situation or state of affairs that – regardless
of how any vagueness in the interpretations of each is resolved – makes ψtrue but φ
false. If this constraint did not hold, the proposition conveyed by an utterance of φon
some interpretation icould in fact be that conveyed by ψon some other equally acceptable
choice of interpretation i0; and furthermore, the negation of φrelative to some interpretation
icould actually contradict ψon some equally legitimate interpretation i0. Put differently,
on this quite stringent definition of greater informativity, there is at least one situation that
is definitively allowed by the weaker sentence but ruled out by the stronger one.
In the next section, we will see how the system as elaborated here accounts for the
polarity sensitivity of approximators.
3. Approximators and polarity
We begin with some preliminaries. Following Kennedy (2015), I take cardinal numerals
to have type-flexible semantics, with denotations as both degrees and quantifiers over de-
grees; the latter yields a 2-sided ‘exact’ rather than ‘at least’ interpretation for the numeral.
I further adopt the approach of Krifka (2007) in taking numerical expressions to be inter-
preted at a contextually determined level of granularity, which accounts for the imprecise
interpretations available to round numbers in particular. Formally, this may be modeled
via a granularity parameter gran, whose value is one of those determined by the interpre-
tation index i(see Sauerland & Stateva 2007 and Solt 2014 for alternate possible formal
implementations). Thus we have (16) as the interpretation of a simple numerical exam-
ple. Assuming a standard analysis of comparative quantifiers as degree quantifiers (see e.g.
Nouwen 2010) yields (17) as the interpretation of a comparative example:
(16) Lisa has 50 sheep. max{n: Lisa has nsheep}=50grani
(17) Lisa has more than 50 sheep. max{n: Lisa has nsheep}>50grani
Note that this analysis assumes a semantic rather than pragmatic approach to imprecision,
according to which imprecise interpretations are part of the truth conditions of sentences
containing numerical expressions, as opposed to being cases of pragmatic slack, where a
speaker says something that is literally false but ‘close enough’ to true (Lasersohn 1999).
We will revisit this point briefly in Section 6 below.
2Note that (14) and (15) separate specifically in the case where the set Iis infinite.
Approximators as attenuating polarity items
Turning to approximators such as about, these can be analyzed per Solt (2014) as refer-
ring to coarse-grained degrees, that is, intervals around the precise denotation of the modi-
fied numeral, with these intervals to be understood as unitary wholes. The width of the in-
terval is again determined at the interpretation index i. Thus the semantics of approximator-
containing examples are the following:
(18) Lisa has about 50 sheep. max{n: Lisa has nsheep} ∈ [50 −ki,50 +ki]
(19) Lisa has more than about 50 sheep. max{n: Lisa has nsheep}>[50 −ki,50 +ki]
Crucially, with these definitions, no relation of ‘more informative than’ (i.e. ‘defini-
tively stronger than’) obtains between (16) and (18), or between (17) and (19). The deno-
tation of the approximator-modified numeral corresponds to an interval around the precise
point-based denotation of the numeral, but the width of this interval can be wider or nar-
rower according to interpretation i; in the limiting case, ki=0, and the interval reduces
to a single point. Conversely, when the granularity parameter is set to be a coarse-grained
one, the bare numeral itself can be used to describe a value that deviates from its pre-
cise denotation. Essentially, bare and approximator-modified numerals are on this analysis
(potentially) equivalent.3Thus there is no numerical value that definitively (i.e. across in-
terpretations i) can be described as about fifty but not fifty (nor vice versa).
However, I propose that the usage of an overt approximator has the effect of establishing
the context to be one in which bare numerals are interpreted precisely, while approximators
introduce a non-trivial range around the precise point. This corresponds to the interpreta-
tions depicted in (3). Conceptually, the idea is that the choice to use an approximator is a
conversational move which establishes that imprecision (if intended) will be overtly sig-
naled. Formally, this may be modeled as an update to the common ground in which the set
Iof interpretations is restricted to a subset I0including just those interpretations iin which
bare and approximator-modified numerals are interpreted in this way. Restricted to such
interpretations, (16) asymmetrically entails (18), while (19) asymmetrically entails (17).
Let us return now to the polarity-based restrictions on the distribution of approxima-
tors that are the topic of this paper. Recall that the paradigm of interest is the following:
in simple numerical constructions, approximators such as about are PPIs, whereas when
embedded in comparative quantifiers they are NPIs:
(20) a. Lisa has / *doesn’t have about 50 sheep.
b. Lisa *has / doesn’t have more than about 50 sheep.
In each of these cases, the sentences with the approximator competes with the alternative
formed by deleting the approximator. The later alternative has an advantage in terms of
simplicity. Furthermore, as discussed above, there is no difference – in the relevant sense
– in informativity. Thus the unmodified alternative is the better one overall. This is repre-
sented formally below for the sentences in (20a):
3I thank Benjamin Spector for suggesting this characterization of the pattern in question.
Stephanie Solt
(21) φ=Lisa has about 50 sheep φ0=Lisa has 50 sheep
φ∼INF φ0φ0SIMP φ φ 0φ
φ0is not weakly assertable 4
(22) φ=*Lisa doesn’t have about 50 sheep φ0=Lisa doesn’t have 50 sheep
φ∼INF φ0φ0SIMP φ φ 0φ
φ0is not weakly assertable contradiction
By the conversational principle in (5), the result is that the utterance of one of the approximator-
modified sentences in (21) and (22) gives rise to the implicature that the speaker could not
have asserted its unmodified alternative. But as shown above, the effect of this implicature
is different in the positive and negative cases. At the contextually available interpretation
indices i, the positive φin (21) is asymmetrically entailed by its better alternative φ0, and
thus the assertion of the former results in the acceptable implicature that the speaker was
not in a position to assert that ‘(exactly) 50’ obtains. But the negative φin (22) asymmetri-
cally entails its alternative φ0, with the result that the corresponding implicature contradicts
the original assertion. This results in blocking, and thus the PPI status of approximators in
simple sentences.
The account works in parallel for cases with comparative quantifiers, but since the
entailment relation between the approximator-modified numeral and its unmodified alter-
native is reversed, so too is the observed pattern of polarity sensitivity. As in the simple
numerical case in (21)-(22), the ‘about’ sentence competes with the alternative formed
by deleting the approximator, with the latter better overall, because it is simpler and not
(definitively) less informative. We thus obtain the implicatures shown below. That in (24)
is informative (the speaker isn’t in the position to assert that ‘no more than exactly 50’
obtains), whereas that in (23) is contradictory and results in blocking.
(23) φ=Lisa has more than about 50 sheep
φ0=Lisa has more than 50 sheep is not weakly assertable contradiction
(24) φ=Lisa doesn’t have more than about 50 sheep
φ0=Lisa doesn’t have more than 50 sheep is not weakly assertable 4
Thus the present account explains the polarity sensitivity of approximators as arising
from contradictory implicatures relative to an unmodified alternative, which is simpler and
therefore (in this case) formally better than the original sentence. The account can be ex-
tended to other approximators such as approximately and roughly, if these are taken to
have interpretations parallel to about, and to other numerical constructions. For example,
we correctly predict the ungrammaticality of *Lisa has over about 50 sheep, since as in the
comparative quantifier case it gives rise to a contradictory implicature.
In the next section, we will look in more depth at the nature of the implicatures gen-
erated in the grammatical uses of approximators, where we will see one case that patterns
differently from about.
Approximators as attenuating polarity items
4. Approximators and implicature
As described in the preceding section, the use of an approximator-modified numerical ex-
pression gives rise to an implicature that the corresponding unmodified assertion is not
weakly assertable, that is, that it is not the case that the speaker believes it to be true,
relevant and supported by the facts. For example:
(25) Lisa has about 50 sheep
the speaker isn’t in the position to assert ‘Lisa has (exactly) 50 sheep’
This is an example of what Meyer (2013) calls a weak implicature, that is, an implicature
that the speaker does not know that φ. It is distinct from a true ignorance implicature,
i.e. an implicature that the speaker neither knows that φnor knows that ¬φ. This correctly
captures the facts: (25) could be asserted by a speaker who knows Lisa has (say) 47 sheep
but chooses to round off to the more salient value 50; however, it is less felicitous if the
speaker knows that exactly 50 obtains. The rounding use is possible for a range of ap-
proximators, as evidenced by the felicity of the following, which might be uttered by the
principal investigator of the study in question:
(26) About / roughly / around / approximately 70 patients took part in the clinical study.
Other approximating constructions, however, have a stronger ignorance effect, an ex-
ample being approximating disjunctions, as illustrated in (27a); we would tend to be con-
cerned if the researcher described her study as in (27b):
(27) a. Lisa has 40 or 50 sheep the speaker doesn’t know the exact number.
b. ?? 60 or 70 patients took part in the clinical study.
The present framework derives this ignorance implicature in a similar way to other accounts
of ignorance effects with disjunctions (e.g. Katzir 2007). Approximating disjunctions of the
form in (27a) have the two individual disjuncts as alternatives, as shown in (28). Both of
these alternatives are better than the original sentence, being simpler and arguably more
informative.4As described in Solt (2016), I take the use of such disjunctions to establish
that the contextual level of granularity at which numerical expressions are interpreted is
equal to the gap between the two values. In this case, this is gran =10, i.e. 40 is interpreted
as 40 ±5 and 50 is interpreted as 50 ±5. Formally, this again may be modeled via a restric-
tion on the set Iof available interpretations. We thus derive the implicatures in (29), which
taken together amount to an ignorance effect.
4In fact, in order for a ‘definitively stronger’ relation to obtain between these alternatives, it is necessary
to assume some upper bound on the degree to which bare numerals can be interpreted approximately, as
otherwise any value that could be described as forty or fifty could also be described as either forty or fifty. I
believe this is plausible, though I have to leave a formal derivation to future work. In any case, even without
informativity, the individual disjunct alternatives would be better than the original disjunction on account of
their greater simplicity.
Stephanie Solt
(28) φ=Lisa has 40 or 50 sheep. max{n: Lisa has nsheep}=40gran=10 ∪50gran=10
φ0=Lisa has 40 sheep. max{n: Lisa has nsheep}=40gran=10
φ00 =Lisa has 50 sheep. max{n: Lisa has nsheep}=50gran=10
φ0/φ00 INF φ φ0/φ00 SIMP φ φ 0/φ00 φ
(29) a. The speaker doesn’t hold the belief that 40gran=10 obtains
(≡the speaker doesn’t hold the belief that 50gran=10 doesn’t obtain)
b. The speaker doesn’t hold the belief that 50gran=10 obtains
(≡the speaker doesn’t hold the belief that 40gran=10 doesn’t obtain)
Finally, up to this point it has been sufficient to consider alternatives derived via deletion
of some part of the original sentence. On the structural approach adopted here, alternatives
may also be derived via substitution. This is relevant in particular in the case of approx-
imators in comparatives, where it accounts for certain additional interpretive effects. For
example, in some contexts the utterance of (30a) would convey that the speaker considers
it possible that Lisa has about 50 sheep. This inference can be accounted for by taking into
consideration a fuller set of alternatives such as those in (30b), derived via deletion the ap-
proximator and/or substitution of 50 by other values of a comparable level of granularity.
In particular, the alternative obtained by replacing 50 by 40 is a better one than the original
(30a), being equal in terms of structural simplicity and more informative; see (31). Thus
the use of (30a) generates the implicature that this alternative is not weakly assertable.
(30) a. φ=Lisa doesn’t have more than about 50 sheep.
b. ALT (φ) = {Lisa doesn’t have more than about nsheep : n=...40,50,60,...}
∪ {Lisa doesn’t have more than nsheep : n=.. . 40,50,60, . . .}
(31) φ=Lisa doesn’t have more than about 50 sheep.
φ0=Lisa doesn’t have more than about 40 sheep.
φ0INF φ φ0∼SIMP φ φ 0φ
The speaker doesn’t believe that Lisa doesn’t have more than about 40
sheep, i.e. considers it possible that she has about 50.
This is a variety of scalar implicature. Below we will see that the possibility of generating
this implicature sets the present account apart from a possible alternative.
5. Alternate analytical approaches
The present account of the polarity sensitivity of approximators has been framed in a prag-
matic theory of implicatures (Krifka 1995, Katzir 2007), the central component of which
is a conversational principle or rule of assertion that governs the choice between alterna-
tive expressions a speaker could use. The leading competitor to such an approach is the
grammatical theory championed by authors including Chierchia (2004, 2006, 2013), Fox
(2007), Spector (2014) and others, under which implicatures (and polarity-based distribu-
Approximators as attenuating polarity items
tional restrictions) derive from the operation of a covert exhaustification operator. In this
section, I briefly consider how the empirical data discussed here might be handled on such
an approach.
In one implementation of the grammatical theory, Chierchia makes use of the operator
O, a silent counterpart of overt only, whose effect is to negate all non-entailed alternatives
of a sentence containing an alternative-introducing element.
(32) JOK=λpλw.[pw∧ ∀q∈ALT (p)[qw→p⊆q]]
The operation of Oaccounts for scalar implicatures as well as the polarity sensitivity of
items such as any. However, a straightforward extension of its application to approxima-
tors yields incorrect results. Suppose, for example, that the alternatives to about 50 are
taken to be the individual values in the range denoted by about 50, as in (33)-(34). Then
application of Oin the positive case yields a contradiction, as these alternatives fully cover
the semantic territory of the approximator-modified numeral; the sentence is thus predicted
to be ungrammatical. The negative sentence, on the other hand, is predicted to be grammat-
ical, as all of its alternatives are entailed. This is of course the opposite of the pattern that
is actually observed.
(33) a. p=Lisa has about 50 sheep.
ALT (p) = {Lisa has nsheep : n∈50 ±ki}
b. O(Lisa has 50 sheep) contradiction!
(34) a. p=Lisa doesn’t have about 50 sheep.
ALT (p) = {Lisa doesn’t have nsheep : n∈50 ±ki}
b. O(Lisa doesn’t have 50 sheep) vacuous
If instead we assume that the only alternative is that based on the numeral 50 on its precise
interpretation (similarly to what was done in the present approach), the positive sentence is
predicted to be grammatical, but we incorrectly generate the implicature that ‘exactly 50’
does not obtain. As discussed in the preceding section, the actual implicature is a weaker
one, namely that ‘exactly 50’ is not assertable by the speaker. And once again, the negative
sentence is predicted to be fully acceptable, since this single alternative is entailed.
More promising results are obtained by incorporating some more recent proposals, in
particular the notion of obligatory exhaustification introduced by Spector (2014) as well
as the covert doxastic operator Kof Meyer (2013), where Kp can be interpreted as ‘the
speaker believes that p’. Assuming an exhaustification operator exh with broadly speaking
the semantics of the Ooperator above, and taking sentences with approximators to have a
single alternative formed by deleting the approximator, we correctly derive the following
results for simple sentences: the positive (35) is grammatical, and has the implicature that
the speaker does not know that ‘exactly 50’ obtains; the negative (36) is blocked, because
exh would be vacuous.
Stephanie Solt
(35) exh(K(Lisa has about 50 sheep))
ALT (p) = {K(Lisa has 50EX ACT sheep)}
K(Lisa has about 50 sheep)∧ ¬ K(Lisa has 50E XACT sheep)correct
(36) exh(K(Lisa doesn’t have about 50 sheep))
ALT (p) = {K(Lisa doesn’t have 50EXACT sheep)}
Single alternative is entailed; exh is vacuous. correct
However, other results are less satisfactory. In particular, consider again a comparative
example such as Lisa doesn’t have more than about 50 sheep, which as discussed above
tends to implicate that the speaker considers it possible that Lisa has about 50 sheep. On
the approach developed in the present paper, this inference is accounted for as a scalar
implicature, by taking into consideration not just the unmodified alternative but also those
such as Lisa doesn’t have more than about 40 sheep. This could be accommodated in a
parallel way in the grammatical approach. But when such alternatives are also included in
the case of the positive comparative sentence, exh is no longer vacuous, and the sentence is
predicted to be acceptable. For example, Lisa has more than about 50 sheep would have the
non-entailed alternative Lisa has more than about 60 sheep, to which the exhaustification
operator could apply non-vacuously. To salvage this approach, it seems we would need to
specify that exh is obligatory specifically with respect to the unmodified alternative. This
is equivalent to building a preference for simplicity into the system, just as done here.
This brief discussion has barely scratched the surface in terms of the analytical possi-
bilities available under a grammatical theory of implicature and polarity sensitivity. I have
no doubt that the insights that form the basis of the account developed in the present paper
could also be implemented within such an approach. However, I hypothesize that doing
so will require incorporating some of the same formal innovations that were adopted in
present account.
6. Conclusions and future direction
In this paper, I have proposed a pragmatic account of the polarity sensitivity of approxima-
tors such as about, according to which their ungrammatical occurrences are analyzed as be-
ing due to blocking by the unmodified form, or equivalently, an implicature that contradicts
the asserted content of the utterance. I have framed the analysis in a formal system whose
central components are a rule of assertion that calls for the comparison of alternatives in
terms of both informativity and simplicity, and a novel definition of relative informativity
as it pertains to sentences containing vague predicates.
There are a number of issues that due to space limitations I have not been able to pur-
sue here. In particular, I have not considered approximators in NPI-licensing contexts other
than sentential negation, nor have I been able to go into much depth regarding potential
difference between approximating constructions. One sort of example that merits further
investigation involves approximating disjunctions. The present analysis would seem to pre-
dict incorrectly that Lisa doesn’t have more than 40 or 50 sheep should be ungrammatical,
as its alternative Lisa doesn’t have more than 50 sheep is semantically equivalent but sim-
Approximators as attenuating polarity items
pler. Intuitively, what makes the disjunctive example acceptable is that its implicatures are
different than those of its simpler alternative, in that the former but not the latter conveys
that the speaker thinks it possible that Lisa has 40 or 50 sheep. This suggests that the sys-
tem developed here may need to be refined to take into consideration not only the semantic
interpretation of the alternatives to a given sentence, but also their pragmatic inferences.
Indeed, such a refinement may be necessary even for the data considered in the present pa-
per, if we take a pragmatic rather than semantic approach to the imprecise interpretations
of bare numerals. I leave the development of such a refinement to future work.
Also of interest is how the present approach might be extended to cases beyond approx-
imators. Many other members of the class of attenuating polarity items (e.g. much,fairly)
are also characterized by their vagueness, suggesting that they too might be amenable to an
analysis in terms of (lack of) informativity relative to the corresponding unmodified forms.
At the same time, not all vague modifiers are polarity sensitive, an example being very,
discussed briefly in Section 2. Thus an important challenge will be to characterize what
specifically sets apart the polarity sensitive class.
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Stephanie Solt
solt@leibniz-zas.de
