The logical syntax of number words: Theory, acquisition and processing
Julien Musolino
Department of Psychology and Center for Cognitive Science, Rutgers University, 152 Frelinghuysen Road, Piscataway, NJ 08854-8020, USA
a r t i c l e i n f o
Article history:
Received 22 August 2005
Revised 17 December 2008
Accepted 19 December 2008
Keywords:
Number words
Acquisition
Syntax
Quantification
Scope
Linguistics
Psychology
a b s t r a c t
Recent work on the acquisition of number words has emphasized the importance of integrating
linguistic and developmental perspectives [Musolino, J. (2004). The semantics and
acquisition of number words: Integrating linguistic and developmental perspectives. Cognition
93, 1–41; Papafragou, A., Musolino, J. (2003). Scalar implicatures: Scalar implicatures:
Experiments at the semantics-pragmatics interface. Cognition, 86, 253–282;
Hurewitz, F., Papafragou, A., Gleitman, L., Gelman, R. (2006). Asymmetries in the acquisition
of numbers and quantifiers. Language Learning and Development, 2, 76–97; Huang, Y.
T., Snedeker, J., Spelke, L. (submitted for publication). What exactly do numbers mean?]. Specifically,
these studies have shown that data from experimental investigations of child language
can be used to illuminate core theoretical issues in the semantic and pragmatic
analysis of number terms. In this article, I extend this approach to the logico-syntactic
properties of number words, focusing on the way numerals interact with each other (e.g.
Three boys are holding two balloons) as well as with other quantified expressions (e.g. Three
boys are holding each balloon). On the basis of their intuitions, linguists have claimed that
such sentences give rise to at least four different interpretations, reflecting the complexity
of the linguistic structure and syntactic operations involved. Using psycholinguistic experimentation
with preschoolers (n = 32) and adult speakers of English (n = 32), I show that (a)
for adults, the intuitions of linguists can be verified experimentally, (b) by the age of 5, children
have knowledge of the core aspects of the logical syntax of number words, (c) in spite
of this knowledge, children nevertheless differ from adults in systematic ways, (d) the differences
observed between children and adults can be accounted for on the basis of an
independently motivated, linguistically-based processing model [Geurts, B. (2003). Quantifying
kids. Language Acquisition, 11(4), 197–218]. In doing so, this work ties together
research on the acquisition of the number vocabulary with a growing body of work on
the development of quantification and sentence processing abilities in young children
[Geurts, 2003; Lidz, J., Musolino, J. (2002). Children’s command of quantification. Cognition,
84, 113–154; Musolino, J., Lidz, J. (2003). The scope of isomorphism: Turning adults into
children. Language Acquisition, 11(4), 277–291; Trueswell, J., Sekerina, I., Hilland, N., Logrip,
M. (1999). The kindergarten-path effect: Studying on-line sentence processing in young
children. Cognition, 73, 89–134; Noveck, I. (2001). When children are more logical than
adults: Experimental investigations of scalar implicature. Cognition, 78, 165–188; Noveck,
I., Guelminger, R., Georgieff, N., & Labruyere, N. (2007). What autism can tell us about every
. . . not sentences. Journal of Semantics, 24(1), 73–90. On a more general level, this work confirms
the importance of integrating formal and developmental perspectives [Musolino,
2004], this time by highlighting the explanatory power of linguistically-based models of
language acquisition and by showing that the complex structure postulated by linguists
has important implications for developmental accounts of the number vocabulary.
Ó 2009 Elsevier B.V. All rights reserved.
0010-0277/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved.
doi:10.1016/j.cognition.2008.12.008
E-mail address: julienm@ruccs.rutgers.edu
Cognition 111 (2009) 24–45
Contents lists available at ScienceDirect
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1. Introduction
This article investigates the way we use the combinatorial
power of natural language to express numerical relations.
To be more precise, I will be concerned with the
logical syntax of numerically quantified expressions
(NQE) (e.g. three boys, two balloons). Given the pivotal role
that numbers play in our lives, understanding how children
acquire numerical concepts and how they learn to express
these concepts through the medium of language represents
an important goal for developmental psychology. Reflecting
the importance of this goal, the past 30 years of research
in developmental and cognitive psychology have witnessed
the emergence of a growing body of work on the nature of
numerical concepts and the acquisition of the number
vocabulary (Carey, 2001; Dehaene, 1997; Gallistel & Gelman,
1992; Gallistel & Gelman, 2000; Gelman & Gallistel,
1978; Mix, Huttenlocher, & Levine, 2002; Sarnecka & Gelman,
2003; Spelke, 2000; Wynn, 1990; Wynn, 1992; Wynn
& Bloom, 1997; LeCorre and Carey, 2007, among many others).
Within this line of research, one of the key questions
has been to try to explain how children learn the meaning
of number words. In contrast, what has to be learned – that
is, what number words mean – has been taken for granted
from the start because it seems so obvious. To quote a recent
study by Huang, Snedeker, and Spelke (submitted for
publication) ‘‘The meaning of a word like ‘‘two” appears
self-evident: it functions as part of a phrase that picks out
sets with exactly two members . . .” (p. 1).
During the same period, number words have also figured
prominently on the research agenda of theoretical linguists
(Breheny, 2008; Carston, 1998; Gadzar, 1979; Horn,
1972; Horn, 1989; Horn, 1992; Koenig, 1991; Levinson,
2000; Sadock, 1984; Chierchia, 2004; Geurts, 1999, Geurts,
2006; Musolino, 2004). Surprisingly however, it is only
within the last few years that linguists and psychologists
have become aware of the existence – and the relevance –
of each other’s work (Musolino, 2004). What makes this
late convergence particularly interesting is that the very
question that psychologists have taken for granted regarding
the meaning of number words is one that linguists have
struggled over for more than two decades. Indeed, for linguists,
understanding what number words mean has been
one of the central questions since Horn’s (1972) seminal
analysis. Moreover, the standard orthodoxy in much of
the philosophical and linguistic literature is that number
words do not have an ‘exact’ semantics, as is assumed by
developmental psychologists. On this view, a phrase like
two cats does not literally mean exactly two cats, but rather
at least two cats (Barwise & Cooper, 1981; Horn, 1972; Horn,
1989; Levinson, 2000). Thus, John has two cats is strictly
speaking true, even if John has three, four, or fifty cats. On
the other hand, theoretical linguists have never really seriously
worried about how children manage to learn what
number words mean – a question that has driven research
in developmental psychology for over three decades.
Recently, the two lines of research described above have
made contact under the impetus of a small set of studies designed
to bridge the gap between linguistics and psychology
(Musolino, 2004; Papafragou & Musolino, 2003. See
also Huang et al., submitted for publication; Hurewitz,
Papafragou, Gleitman, & Gelman, 2006 for similar developments).
Specifically, these studies have shown that data
from experimental investigations of child language can be
used to illuminate core theoretical issues in the semantic
and pragmatic analysis of number terms. Thus, the thrust
of this new approach so far has been to highlight the implications
of developmental work for linguistic theory.
In this article, I focus on the other side of this equation
and show that the work of theoretical linguists has important
implications for developmental accounts of the number
vocabulary. In order to do so, I consider the logicosyntactic
properties of number words, focusing on the
way numerals interact with each other (e.g. Three boys
are holding two balloons) as well as with other quantified
expressions (e.g. Three boys are holding each balloon). On
the basis of their intuitions, linguists have claimed that
such sentences give rise to at least four different interpretations,
reflecting the complexity of the linguistic structure
and syntactic operations involved. Using psycholinguistic
experimentation with preschoolers and adult speakers of
English, I show that (a) for adults, the intuitions of linguists
can be verified experimentally, (b) by the age of 5, children
have knowledge of the core aspects of the logical syntax of
number words, (c) in spite of this knowledge, preschoolers
nevertheless differ from adults in systematic ways, (d) the
differences observed between children and adults can be
accounted for on the basis of an independently motivated,
linguistically-based processing model (Geurts, 2003). In
doing so, this work ties together research on the acquisition
of the number vocabulary with a growing body of
work on the development of quantification and sentence
processing abilities in young children (Geurts, 2003; Lidz
& Musolino, 2002; Musolino & Lidz, 2003; Noveck, 2001;
Trueswell, Sekerina, Hilland, & Logrip, 1999). At a more
general level, this work confirms the importance of integrating
formal and developmental perspectives (Musolino,
2004), this time by highlighting the explanatory power of
linguistically-based models of language processing and
by showing that the complex structure postulated by linguists
has important implications for developmental accounts
of the number vocabulary.
2. Theoretical background
The goal of this section is to describe the linguistic complexity
underlying the interpretation of NQE and to show
that NQE have a unique logico-syntactic profile that distinguishes
them from other quantificational expressions. Specifically,
I will show that NQE (a) allow distributive,
collective and cumulative readings, (b) give rise to scopedependent
readings, (c) also give rise to scope-independent
readings, (d) given two possible scope-dependent
readings, the one where the object takes distributive scope
over the subject is dispreferred, and (e) given two possible
scope-independent readings, the cumulative reading is dispreferred.
Having described the complexity underlying the
interpretation of NQE, we will then briefly turn to the
question of how such facts can be acquired by children,
and consider the hypothesis that the various readings of
J. Musolino / Cognition 111 (2009) 24–45 25
NQE described below are not learned per se, but rather that
they are implicitly deduced by young children. The purpose
of the psycholinguistic experimentation presented
in this article is to test this hypothesis by assessing knowledge
of the properties in (a–e) in preschool children and
adult speakers of English.
2.1. The interpretation of NQE
Let us begin with a simple observation regarding the
interpretation of NQE. Notice that the sentence in (1),
which contains the NQE three boys, is ambiguous between
a distributive and a collective reading, (1a) and (1b), respectively.
On the distributive reading, (1) is understood to
mean that there were three separate visits. In other words,
each of the three boys visited Mary on a specific occasion
(which was different for each boy). On the collective reading,
(1) is understood to mean that there was only one visit
and so that the three boys must have visited Mary
together.
(1) Three boys visited Mary.
a. There is a set M, M is a set of boys of cardinality 3,
and for each boy b in M, b visited Mary
(distributive reading)
b. There is a set M, M is a set of boys of cardinality 3,
and M visited Mary (collective reading)
Let us now consider what happens when a sentence contains
two NQE. Given that each NQE can in principle be
interpreted either distributively or collectively, it would
seem that the resulting sentence could receive, at least in
principle, four interpretations. However, this picture is
complicated by an additional factor, namely the fact that
NQE are scope-bearing expressions. The notion of scope,
in turn, can be illustrated using a simple mathematical
analogy. Consider the expressions in (2) and (3):
(2) 2 Â (3 + 5) = 16
(3) (2 Â 3) + 5 = 11
The scope of 2Â (the number 2 followed by the multiplication
sign) can be thought of as its domain of application. So
in (2), (3 + 5) falls within the scope of 2Â. By contrast, in
(3), 3 falls within the scope of 2Â whereas 5 falls outside
of its scope. Finally, notice that different scope relations
give rise to different results once the expressions are computed.
We can now turn to the concept of scope as it applies
to language by considering a sentence like (4)
which contains two NQE.
(4) Three boys are holding two balloons.
a. There are three x, x is a boy, and for each x, there
are two y, y is a balloon, such that x is holding y.
(subject wide scope)
b. There are two y, y is a balloon, and for each y, there
are three x, x is a boy, such that x is holding y.
(object wide scope)
In order to single out the effect of scope, let us hold the
interpretative value of the two NQE constant and assume
that both are interpreted distributively. Since NQE are
scope-bearing expressions, a possible interpretation of (4)
is one on which the object noun phrase, two balloons, is
interpreted within the scope of the subject noun phrase,
three boys, (4a). On this interpretation, the sentence can be
paraphrased as Three boys are such that they are each (distributive
interpretation) holding two balloons (picture 1).1
Notice
that this reading reflects a dependency between the two NQE
in that the choice of the set of balloons depends on, or varies
as a function of, which of the three boys is selected. Thus, the
quantity associated with the object NQE is multiplied by the
value of the subject NQE. Consequently, Three boys are holding
two balloons is true in a context in which six balloons are present
(picture 1) even though the number of balloons mentioned
in the sentence itself is only two.
Another possible interpretation of (4) is one on which
the object noun phrase takes scope over the subject noun
phrase, (4b). On this interpretation, (4) can be paraphrased
as meaning that there are two balloons such that each balloon
(distributive interpretation) is being held by a different
set of three boys (picture 2). In this case, the set of
three boys varies as a function of which balloon one
chooses. So as in the case of our mathematical analogy, different
scope relations give rise to different interpretations.
Picture 1
Picture 2
1
Notice here that due to the nature of the predicate, hold, and the way
the sentences are represented graphically (see picture 1), a collective or a
distributive reading of the object NP two balloons would yield the same
visual configuration. In other words, whether each of the three boys is
holding his two balloons together (collective reading) or one after the other
(distributive reading), would be represented as shown in picture 1. The
same holds of the wide scope reading of the object NQE, two balloons. Here,
once two balloons is interpreted distributively and takes scope over the
subject NQE, three boys, whether the subject is in turn interpreted
collectively or distributively yields the same visual configuration (see
picture 2).
26 J. Musolino / Cognition 111 (2009) 24–45
In addition to the scope-dependent readings described
above, linguists have also noticed that sentences containing
NQE allow scope-independent readings (Barwise,
1979; Beghelli, Ben-Shalom, & Szabolcsi, 1997; Gil, 1982;
Hintika, 1974; Sher, 1990). On such readings, the subject
and the object noun phrase are interpreted independently
of each other and a variety of connections are established
between the members of each set (in this case the set of
boys and the set of balloons). One such interpretation is
one on which each member of one set is connected to all
the members of the other (each-all reading). In other
words, each of the three boys is holding both balloons (picture
3). Another way to see this is to consider the fact that
when both NQE are interpreted collectively, scope becomes
irrelevant (hence the notion of scope-independence).
Thus, the reading that can be paraphrased as
There are three boys who, together, are holding two balloons
(wide scope reading of the subject) and the one that can be
paraphrased as There are two balloons which, together, are
being held by three boys (wide scope reading of the object),
are logically equivalent, as can be seen in picture 3.
Yet another possible scope-independent interpretation
is one on which the three boys, when considered
cumulatively, are holding a total of two balloons
(cumulative reading) (picture 4). At first sight, it might
seem that the cumulative reading is the same as the
collective reading. However, the example in (5) reveals
that the two interpretations are distinct. To see this,
consider what would have to be the case for (5) to
be true on a collective reading. Here, each of the three
men would have to co-own each of the houses. In
other words, house 1 would have to be owned by
all three men, and so would house 2, 3 and 4. By contrast,
the cumulative interpretation does not require
such co-ownership. So one way to make the sentence
true on the cumulative reading – but false on the collective
reading – would be to have two of the three
men each (uniquely) own a separate house and the
third man (uniquely) own the remaining two (see picture
5).
(5) Three men own four houses.
So far, we have seen that sentences like (4) give rise to (at
least) four possible readings which I will refer to as SU (for
the subject wide scope reading, picture 1), OBJ (for the object
wide scope reading, picture 2), EA (for the each-all
reading, picture 3) and CU (for the cumulative reading, picture
4). While all these readings are in principle available,
linguists have reported certain interpretive preferences.
First, given the availability of two scope-independent readings,
EA and CU, Gil (1982) reports a preference for EA. Second,
several investigators have noticed that object NQE do
not easily take scope over quantified subjects (Beghelli
et al., 1997; Liu, 1990; Liu, 1992. For experimental
evidence in children, see Lidz and Musolino (2002). For
evidence in adults, see Musolino and Lidz (2003)). Thus,
given SU and OBJ, SU appears to be the preferred
reading.
Another important aspect of the logical syntax of
NQE – and quantifiers more generally – is that these
expressions tend to be very choosy with respect to the
kinds of readings they allow as well as the scopal preferences
they induce. To be more precise, not all quantifiers
give rise to scope-independent readings, some
quantifiers do not allow collective or cumulative readings,
and finally, different quantifiers induce different
scopal preferences. Regarding the first of these three
facts, Liu (1990), Liu (1992 proposes a classification of
quantifier types and shows that depending on the kind
of quantifier involved, a sentence may or may not have
a scope-independent reading. Next, consider the fact
that only some quantifiers allow collective readings.
The contrast between (6) and (7) illustrates this property.
Recall that sentences like (6) are ambiguous.
On the collective reading, (6) can be paraphrased as
Picture 5
Picture 3
Picture 4
J. Musolino / Cognition 111 (2009) 24–45 27
meaning that three boys, when considered as a set, gave
a gift to Mary. In this case, Mary only receives one gift.
Alternatively, (6) may be interpreted on a distributive
reading in which case each of the three boys gave Mary
a different gift. So in this case, Mary receives three gifts.
By contrast, notice that if the relevant situation involves
three boys, (7) cannot receive a collective reading. It
must be interpreted as meaning that Mary received
three gifts – one from each boy. So each is obligatorily
distributive.
(6) Three boys gave a gift to Mary.
(7) Each boy gave a gift to Mary.
Let us continue to use each to illustrate the fact that different
quantifiers induce different scopal preferences. Consider
the contrast between (4), repeated here as (8), and
(9) in which two balloons has been replaced by each balloon.
Recall that for sentences like (8), the wide scope reading of
the object, corresponding to picture 2, does not obtain as
easily as the wide scope reading of the subject, picture 1.
In the case of sentences like (9), these preferences are reversed.
That is, (9) very naturally describes picture 2, but
not picture 1. The preferred scopal readings of these two
sentences are paraphrased in (8a) and (9a).
(8) Three boys are holding two balloons.
a. There are three x, x is a boy, and for each x, there
are two y, y is a balloon, such that x is holding y
(subject wide scope)
(9) Three boys are holding each balloon.
a. For each y, y is a balloon, there are three x, x is a
boy, such that x is holding y. (object wide scope)
Finally, let us briefly consider the kind of mechanism that
allows quantifiers to take their scope. In a sentence like
Three boys are holding each balloon, notice that each balloon
occurs within the scope of three boys in surface syntax. To
yield the interpretation on which each balloon takes scope
over three boys, the phrase each balloon must be displaced
from its surface syntactic position to a higher position from
which it will take scope over three boys. In the generative
literature, the standard mechanism which is responsible
for the displacement of quantifiers to their scope position
is called quantifier raising (QR) (Hornstein, 1995; May,
1977; May, 1985). The operation of QR is illustrated in
(10) in which the quantifiers have moved out of their argument
positions (subject and object) to a sentence initial position
where they can take their scope. (10a) corresponds
to the interpretation on which three boys takes scope over
two balloons, SU; in (10b) two balloons takes scope over
three boys and in (10c), each balloon takes scope over three
boys.
(10) a. [three boysi [two balloonsj [IP ti are holding tj ]]]
b. [two balloonsj [three boysi [IP ti are holding tj ]]]
c. [each balloonj [three boysi [IP ti are holding tj ]]]
The kind of displacement illustrated in (10) is ‘covert’ since
there are no phonological consequences associated with it.
The level of linguistic representation at which covert displacement
takes place is called logical form (LF). Finally,
it should also be pointed out that several alternatives to
QR have been proposed in the linguistic literature (e.g.
Hornstein, 1995; Reinhart, 1997). For our purposes however,
whether quantifiers take their scope via QR or some
other mechanism is of no direct consequence.
2.2. The learning question
As the previous section illustrates, the linguistic properties
associated with NQE are complex and yield an intricate
array of interpretive options. Indeed, to say that a sentence
like Three boys are holding two balloons can receive at least
four different interpretations does not sound immediately
obvious. To be sure, explaining why this is the case requires
several pages of theoretical background and explanation.
This fact raises an obvious question: to what
extent are the interpretations described in the previous
section ‘psychologically real’? In other words, would a
naïve speaker – i.e., a person who does not have an advanced
background in linguistic theory or logic – nonetheless
be aware (if only implicitly) of all the interpretations
described above? To the extent that the answer to this first
question is affirmative, a second, important question
arises: how are these facts acquired?
One possibility is that such knowledge, because it is
intricate and linked to knowledge of logic, may be acquired
late, perhaps as a byproduct of instruction in arithmetic
and basic logic. If so, we would certainly not expect preschoolers
– because of their obvious lack of experience in
the two domains just mentioned – to display knowledge
of such facts. However, it should be noted that although
possible in principle, to the best of our knowledge, this
type of account has not been explicitly endorsed in the linguistic
or psychological literature. Another possibility is
that such knowledge, however intricate it may be, is a
manifestation of core linguistic knowledge. In other words,
knowledge of the grammar of English (along, of course,
with knowledge of the meaning of English words), would
entail (tacit) knowledge of the complex array of facts described
above. In this regard, there exist a number of proposals
in the semantic literature bearing on how learning
of such complex properties might take place. One such proposal,
by Gennari and MacDonald (2005/2006), is that such
knowledge is mostly derived from experience with the relevant
expressions and the contexts and situations in which
they are used. To quote these authors ‘‘ We argue that children
. . . are sensitive to the distributional patterns of language
use . . . and their pairing with specific situations,
and that children’s experience of such patterns shapes
their comprehension of scope ambiguous sentences” (p.
128/129).
Here, we explore an alternative hypothesis regarding
how such learning might take place. In fact, on this scenario,
the logico-syntactic properties of NQE need not be
learned per say but rather they are tacitly deduced by young
children. This prediction follows straightforwardly from a
core property of natural language, namely the fact that
28 J. Musolino / Cognition 111 (2009) 24–45
semantics is compositional. In other words, the meaning of
a sentence can be systematically deduced from the meaning
of its part and the way they are arranged. To be sure,
language, to reiterate a classic observation, allows its
speakers to make infinite use of finite means, and so there
is little doubt that brains, which are finite, must contain
general principles for generating sentences – what linguists
call grammars - rather than long lists of possible
and impossible sentences. A direct consequence of this
observation is that semantics must be compositional. This
follows from the fact that language allows its speakers to
interpret a potentially unbounded number of sentences.
What we predict then is that to the extent that children
have acquired (a) the lexical meaning of NQE – i.e. the
meaning of expressions like two N, three N, etc. – and to
the extent that (b) they have a command of the core grammatical
principles of their language and know that semantics
is compositional, they should be able to (tacitly)
deduce the range of meanings arising from the interaction
of multiple NQE. If so, the only learning that would have to
take place would be indirect (i.e. learning the meaning of
the words, and the core properties of English grammar).
There is massive evidence in the literature on language
acquisition that (a) and (b) are in place by the age of 5.
We then predict that by the age of 5, children should have
knowledge of the range of interpretations arising from
interacting NQE. As we will see, this prediction is indeed
borne out.
3. Developmental background
As mentioned earlier, since the publication of Gelman
and Gallistel’s (1978) seminal study (henceforth G&G), a
voluminous body of work on children’s growing conception
of number has emerged. A review of this literature,
in turn, reveals that developmental psychologists have
been concerned with two fundamental and interrelated
questions, namely (a) what is the nature of the conceptual
apparatus underlying our ability to grasp numerical relations,
and (b) how do children acquire the meaning of
number words. The first question, while no doubt extremely
interesting, is not directly relevant to our current
purposes (see Dehaene (1997) for an overview). In regard
to the second question, much of this literature has grown
out of a reaction to G&G’s proposal that the representation
of integers is part of our built-in cognitive endowment and
that children’s acquisition of the number vocabulary is
guided by innate counting principles. While there is universal
agreement that G&G defined the principles governing
the domain, there is great controversy over the role
of these principles in explaining development. One view,
sometimes referred to as the ‘principles before skill’ (e.g.
Gelman & Meck, 1983; Freeman, Antonucci, & Lewis,
2000) or the ‘continuity hypothesis’ (Le Corre, Van de Walle,
Brannon, & Carey, 2006) endorses G&G’s nativist position
vis-à-vis the representation of integers and
knowledge of the counting principles (Cordes & Gelman,
2005; Gallistel & Gelman, 1992; Gelman, 1993; Gelman &
Greeno, 1989; Greeno, Riley, & Gelman, 1984, among others).
An alternative position, sometimes called the ‘skill before
principles’ view or the ‘discontinuity hypothesis’ holds
that knowledge of integers and the counting principles is
in part derived from experience, or, as Le Corre et al.
(2006) put it that ‘‘the acquisition of the verbal count list
may involve the construction of a system that is not innately
available” (p. 133). Although the details of how this
new representational system emerges are still of matter of
debate, many seem to have adopted this general position
(Bialystok & Codd, 1997; Briars & Siegler, 1984; Carey,
2004; Cooper, 1984; Fuson, 1988; Karmiloff-Smith, 1992;
Mix et al., 2002; Spelke & Tsivkin, 2001; Starkey and Cooper,
1995; Wynn, 1990; Wynn, 1992).
From the perspective of the present study, there is one
important thread that runs through the developmental literature
in the domain of number word acquisition: to the
extent that this body of work is concerned with the linguistic
properties of number words, the focus has almost exclusively
been on how such expressions are used in the
counting system. However, as discussed in the previous
section, the linguistic behavior of number words extends
far beyond counting and into the realm of syntax, semantics,
and pragmatics. Thus, any account of how children
learn to express numerical relations through the medium
of language would be incomplete without a consideration
of the linguistic properties described earlier. Moreover, it
is only very recently that the two perspectives described
earlier, represented by work in theoretical linguistics and
developmental psychology, have been brought together
into a single framework. The first study to do so is by Papafragou
and Musolino (2003) (henceforth P&M).
P&M’s work was motivated by the recent observation
that preschoolers, who are otherwise semantically competent,
often show a remarkable lack of sensitivity vis-à-vis
scalar implicatures in tasks designed to assess language
comprehension (Chierchia, Crain, Guasti, Gualmini, &
Meroni, 2001; Musolino & Lidz, 2006; Noveck, 2001). The
generalization emerging from this recent line of research
is that although adults tend to favor the pragmatic interpretation
of weak scalar terms (e.g. some as being incompatible
with all), young children often interpret the same
terms semantically (e.g. some as being compatible with
all). P&M’s goal was to determine whether this observation
would extend to all scalar terms, and in particular to NQE.
P&M tested preschoolers (and adults) on their interpretation
of scalar terms such as some, two and start in contexts
which satisfied the semantic content of stronger terms on
each scale, i.e. all, three and finish. For example a situation
in which all/three horses jumped over a fence would be described
as some/two horses jumped over the fence. What
P&M found is that while preschoolers overwhelming accepted
such descriptions in the case of sentences containing
some and start, they massively rejected these
descriptions in the case of sentences containing two. Thus,
P&M’s results show that preschoolers treat NQE differently
from other scalar terms (also see Hurewitz et al. (2006) for
a similar conclusion).
Building on P&M, Musolino (2004) further explored the
way preschoolers represent NQE semantically. Musolino’s
study grows out of the observation that in addition to their
‘exactly n’ interpretation, NQE also give rise to ‘non-exact’
readings, namely ‘at least/most n’ (Carston, 1998; Horn,
J. Musolino / Cognition 111 (2009) 24–45 29
1972; Koenig, 1991; Sadock, 1984). Imagine for example a
story in which one of the characters has to throw hoops at
a pole and is told that he needs to get two hoops on the
pole in order to win a prize. Suppose now that the character
in question gets three hoops on the pole. Should he win
the prize? If two hoops in you need to get two hoops on the
pole is interpreted as exactly two hoops, then the character
should not win. On the other hand, if two hoops is interpreted
as at least two hoops (two hoops or more), then
the character should win the prize. Imagine now that in
the same situation our character is told that he can miss
two shots and still win the prize. Suppose now that the
character only misses one shot. Should he win the prize?
If two shots is interpreted as exactly two shots, then the answer
is no (the character missed one shot, not two). On the
other hand, if two shots is interpreted as at most two shots,
then the answer is Yes. Using scenarios like these, Musolino
(2004) showed that by the age of 5, preschoolers have
implicit knowledge of the fact that expressions like two
hoops can, in addition to their exact reading, be interpreted
as at least two hoops and at most two hoops.
Results from these two studies have been used to tease
apart competing analyses of the semantics of NQE. Specifically,
these results have been taken as evidence against
the standard, neo-Gricean analysis of NQE (Horn, 1972;
Horn, 1989; Levinson, 2000) according to which NQE have
a lower bounded (i.e. at least n) semantics (Musolino,
2004; also see Hurewitz et al. (2006), Huang et al. (submitted
for publication), Geurts (2006) and Breheny (2008) for
similar conclusions). Thus, these recent studies have
shown that the integration of linguistic and developmental
perspectives is not only feasible but that it is also a fruitful
endeavor. So far, this work has focused on the semantic
and pragmatic properties of NQE. However, as we saw in
the previous section, NQE are also associated with complex
logico-syntactic properties. The next step in this integrative
research program is therefore to determine when
and how such properties are acquired by young children.
The research presented in this article is designed to address
these questions.
Before turning to the experimental section of the paper,
one more study deserves mention.2
This is a study by Lidz
and Musolino (2002) which examined the way preschoolers
(and adults) interpret sentences containing negation and an
object NQE, e.g. The Smurf did not catch two birds. Notice that
this sentence is scopally ambiguous. On the wide scope
reading of negation, it can be paraphrased as meaning that
it not the case that the Smurf caught two birds (so perhaps
the Smurf caught only one bird). When the object NQE takes
wide scope over negation however, the sentence can be
paraphrased as meaning that there are two specific birds
that the Smurf did not catch. The main finding from Lidz
and Musolino’s (2002) study is that children, unlike adults,
display a strong preference for the narrow scope reading
of the object NQE, i.e. It is not the case that the Smurf caught
two birds. This result might suggest that preschoolers do not
yet have a complete mastery of the logical syntax of NQE, or,
alternatively, that they have a strong preference for the
interpretation of quantificational expressions which corresponds
to their surface syntactic position. For evidence supporting
this latter conclusion, see Lidz (2004), Musolino and
Lidz (2003), and Musolino and Gualmini (2003).
4. Experimental investigation
The goal of this experiment is two-fold. First, for adults,
the aim is to determine whether the intuitions of linguists
regarding the logical syntax of NQE can be verified experimentally.
As mentioned earlier, the relevant facts are that
NQE (a) give rise to scope-dependent readings, (b) also give
rise to scope-independent readings, (c) given two possible
scope-dependent readings, the object wide scope reading
(OBJ) is dispreferred, and (d) given two possible scopeindependent
readings, the cumulative reading (CU) is dispreferred.
The second, more important, goal is to experimentally
investigate the learning question, and in
particular test the hypothesis that by the age of 5, children
have knowledge of the logico-syntactic properties associated
with the use of NQE.
In order to achieve these goals, I will use an experimental
technique which has proved to be very successful
in assessing children’s (and adults’) interpretation of a
broad range of complex linguistic constructions, often
involving ambiguous sentences and intricate interactions
between quantificational expressions (Lidz & Musolino,
2002; Musolino, Crain, & Thornton, 2000; Musolino &
Lidz, 2003; Musolino & Lidz, 2006). This technique is
called the Truth Value Judgment Task (TVJT) (Crain &
Thornton, 1998). In a nutshell, I will show that knowledge
of the facts in (a-d) can be assessed by asking child and
adult participants to judge sentences like Three boys are
holding two/each balloon in a range of configurations corresponding
to the different interpretations that such sentences
are claimed to give rise to. The details are
provided below.
4.1. Method
4.1.1. Participants
We tested 32 English-speaking preschoolers (16 boys
and 16 girls) between the ages of 4;2 and 6;2 (year, month)
(M = 5;0, SD = 7 months). The children were recruited at
daycare centers in the Bloomington, Indiana area. We also
tested 32 adult native speakers of English, all undergraduate
students at Indiana University.
4.1.2. Procedure
We tested participants’ interpretation of sentences containing
NQE using the TVJT. Participants watched short
2
There is also a study by Lee (1996) which would be relevant here.
Although Lee’s study is on the acquisition of quantification in Chinese, and
not specifically on the acquisition of NQE, it does include sentences
containing two NQE, e.g. you sange shushu tiaozhe liang tong shui (Three men
are carrying (on their shoulder) two buckets of water). Lee reports two main
findings. The first is that Chinese-speaking adults accept the subject wide
scope reading only about 20% of the time whereas Chinese-speaking
preschoolers do so much more often. The second is that the acceptance rate
for the cumulative reading was lower in children than in adults. Although
these results are certainly interesting and relevant, they are difficult to
evaluate because certain critical experimental details are omitted from
Lee’s study and no statistical analyses of the data are provided.
30 J. Musolino / Cognition 111 (2009) 24–45
animated vignettes involving various characters and objects
imported from Microsoft ClipArt. The vignettes were
created and animated using Microsoft PowerPoint software
and they were displayed on a computer monitor. A
prerecorded female voice described each vignette as events
unfolded, and at the end a statement describing the outcome
of each vignette was made. All the statements were
of the form ‘‘I know . . . statement . . . am I right?”, e.g. ‘‘I
know, three boys are holding two balloons, am I right?”.
The participants’ task was to determine whether the computer
was ‘right’ or ‘wrong’. The task was identical for both
children and adult participants. Adult participants were
told that the task was designed for use with young children
and that we needed to obtain a baseline measure in order
to make sense of the responses we would get from
children.
It is worth pointing out that there is now overwhelming
evidence that children in the 4–5 age range experience no
difficulty whatsoever with the TVJT and that they are perfectly
capable of giving either Yes or No answers when
appropriate, including appropriate justifications for their
answers. Moreover, the TVJT has now been used successfully
to test young children’s knowledge of complex linguistic
constructions in different languages, including
English (Musolino, 2004), Greek (Papafragou & Musolino,
2003), Kannada (Dravidian) (Lidz & Musolino, 2002) and
Korean (Han, Lidz, & Musolino, 2007).
The children were first introduced to the task as a group
during ‘circle time’ and then tested individually at their
preschools in a quiet room away from the class or in the
Psycholinguistics Laboratory at Indiana University. Each
child and adult participant was first shown two pretest
vignettes and if they could answer those appropriately
(including appropriate justifications for their responses),
they were then given the complete experimental package
which consisted of a total of 36 vignettes.
4.1.3. Materials
Participants were tested on their interpretation of sentences
like (11) and (12) which differ only with respect
to the type of the quantified object (two N vs. each N).
Quantifier type was manipulated because, as discussed in
Section 2, object NQE differ from other quantified objects
in their ability to take scope over the subject. Thus, this
condition was included to test the claim that different
quantifiers induce different scopal preferences and to
determine whether children distinguish NQE from other
quantifiers in this regard. Specifically, recall that while an
object NQE does not easily take scope over a quantified
subject, quantifiers like each N seem to require a wide
scope interpretation.
(11) Three boys are holding two balloons
(12) Three boys are holding each balloon
For both sentence types, four experimental conditions and
four control conditions were created. The four experimental
conditions were designed to assess participants’ ability to assign
sentences containing NQE scope-dependent (subject
wide scope, SU and object wide scope OBJ) and scope-independent
readings (each-all, EA and cumulative, CU). The four
control conditions were designed to test children’s ability to
correctly interpret phrases like two N, three N and each N in
contexts in which they do not combine with each other and
to ensure that children would correctly reject sentences containing
two NQE in situations which do not satisfy the truth
conditions associated with such sentences (control false condition).
Pictures depicting configurations corresponding to
the four experimental conditions, SU, OBJ, EA and CU, as well
as the two control false conditions are shown below.3
3
Recall that participants were not shown static pictures but rather
animated vignettes in which who was holding what was made very clear so
as to remove any possible doubt or confusion regarding the final configurations.
In doing so, great care was of course taken to not use phrases like
two N or each N or anything else that might give away the answer. Instead,
as objects and characters appeared, the voice would say things like ‘look,
here’s a red balloon and this boy is holding it. This boy is holding it too’ etc.
J. Musolino / Cognition 111 (2009) 24–45 31
In all eight conditions (four experimental and four control),
four repetitions of each outcome/configuration were presented
involving different combinations of characters and
objects (e.g. boys and balloons, girls and kites, clowns and
flowers etc.) (see Appendix A for a complete list). An additional
four repetitions were presented in the control false
condition, thereby yielding a total number of 36 trials
(eight conditions  four outcomes + four additional repetitions
for control false). The reason for adding four additional
trials to the control false condition was motivated
by the need to ensure that the number of Yes and No responses
was balanced. For both quantifier types (two and
each), the 36 trials were arranged in two orders of two
blocks of 18 trials (eight experimental and 10 control trials).
In each case, the two most accessible and least accessible
readings – as per the intuitions of linguists – were
blocked. Thus, in the case of two, one block contained SU
and EA (the two most accessible readings) and the other
block contained OBJ and CU (the two least accessible readings).
Within each of these two blocks, order of presentation
of the material was randomized. In the case of each,
one block contained OBJ and EA (the two most accessible
readings) and the other block contained CU and SU (the
two least accessible readings). Within each block, order of
presentation was randomized. Finally, order of presentation
of the two blocks was counterbalanced across subjects.
These manipulations gave rise to a 2 (age: children/
adults) Â 2 (quantifier type: two/each) Â 4 (interpretation:
SU/OBJ/EA/CU) design where age and quantifier type were
treated as between subjects variables and interpretation
was treated as a within subjects variable. Finally, the 32
adults and 32 preschoolers were randomly assigned to
the two levels of quantifier type (i.e. each and two). The
group of 16 preschoolers (8 boys and 8 girls) assigned to
the two N condition ranged in age between 4;2 and 6;1
(M = 5;0, SD = 7 months) and the 16 children (eight boys
and eight girls) assigned to the each N condition ranged
in age between 4;2 and 6;2 (M = 5;0, SD = 7 months).
4.2. Predictions for adults and children
The facts reviewed in Section 2 allow us to make a number
of predictions regarding the way adult participants will
interpret sentences like Three boys are holding two/each balloon(s).
For sentences like Three boys are holding two balloons,
three different readings have been claimed to be
available, namely SU, EA and CU and a fourth reading, OBJ,
is believed to be much more difficult to access. One can
therefore expect relatively high acceptance rates for SU,
EA and CU (at least significantly above chance) and a comparatively
lower acceptance rate for OBJ. Quantifiers like
each N, unlike NQE, are known to have an inherent tendency
to take wide scope with respect to other logical expressions.
Therefore in sentences like Three boys are holding each balloon,
the quantified object will have a strong tendency to
take scope over the subject NQE. The resulting interpretation
is one on which each of the two balloons – in a scenario
that involves two balloons – is being held by three boys. We
therefore expect high acceptance rates for OBJ and EA (at
least significantly above chance) because in both cases,
the two balloons are indeed each being held by three boys
(see pictures 2 and 3). On the other hand, we expect comparatively
lower acceptance rates for SU and CU where each
balloon is not being held by three boys (see pictures 1 and
4). These predictions are summarized in Table 1 where high
and low stand for predicted high or low acceptance rates.
Turning to children, a number of specific predictions
can also be made. Recall that there is solid evidence from
the developmental literature that by the age of 5, children
know the meaning of NQE like two N and three N (Gelman
& Gallistel, 1978; Wynn, 1990; Wynn, 1992; Le Corre and
Carey, 2007 among many others). Moreover, there is also
good evidence that children in that age range know the
grammatical mechanisms involved in the logical syntax
of quantified expressions, e.g., QR (Crain & Thornton,
1998; Lidz et al. 2004, Syrett and Guasti, 2000; Lidz et al.,
2004). On the compositional hypothesis described in Section
2.2, these observations lead to the prediction that preschoolers
should know that NQE are scope-bearing
expressions, and thus that sentences like Three boys are
holding two balloons can be interpreted along the lines of
SU. Moreover, based on the results of the study by Lidz
and Musolino (2002) showing that preschoolers have difficulty
assigning an object NQE wide scope with respect to
other logical expressions, one would expect a low accepPicture
6.
Picture 7.
32 J. Musolino / Cognition 111 (2009) 24–45
tance rate for OBJ. In the case of EA and CU, the compositional
hypothesis would again predict relatively high
acceptance rates. However, an additional consideration
suggests that the acceptance rate for CU may in fact be
lower than expected. Indeed, based on the results from
Lee’s (1996) study on Chinese – and to the extent that Chinese
and English are comparable in the relevant respects –
one would expect to find acceptance rates for CU lower
than those found in adults. Finally, given the well-known
difficulties that preschoolers experience with universal
quantification (every, all and each are all universal quantifiers)
(Crain et al. 1996; Donaldson & Lloyd, 1974; Drozd &
van Loosbroek, 1999; Geurts, 2003; Inhelder & Piaget,
1964; Philip, 1995), one might expect a less adult-like profile
for sentences like Three boys are holding each balloon as
compared to sentences like Three boys are holding two balloons.
Moreover, work by Brooks and Braine (1996) suggests
that preschoolers do indeed experience difficulty
with quantifiers like each.
4.3. Results
Let us first consider participants’ performance on the
control conditions designed to test knowledge of the
meaning of individual quantifiers (each N, two N and
three N). Here the dependent measure was the proportion
of correct responses. On average, across both quantifier
types, adults responded correctly 98.9% of the time to
statements containing two N, 100% of the time for three
N and 100% of the time for each N. On average, across
both quantifier types, children responded correctly
98.9% of the time to statements containing two N,
99.35% of the time for three N and 97.65% of the time
for each N. Since participants’ performance was virtually
flawless in these control conditions, no further statistical
analyses were conducted. Finally, recall that a fourth
control condition was added in order to ensure that children
were able to reject sentences containing two NQE in
situations which do not satisfy the truth conditions associated
such sentences. In the two N condition, children
gave correct responses 98.4% of the time, compared to
100% of the time for adults. However, in the each N condition,
children displayed significantly higher acceptance
rates compared to adults (85.1% vs. 23.4%, respectively,
t(30) = 6.5, p < 0.001, two-tailed). We return to this interesting
difference in Section 5.3.3.
Let us now turn to the experimental conditions. Here
the dependent measure was the proportion of Yes responses.
Table 2 provides the percentages of Yes responses
for children and adults in both quantifier type conditions
(two N and each N) and for each interpretation (SU, OBJ,
EA, CU).
The proportions of Yes responses were entered into
a 2 (age: children/adults) by 2 (quantifier type: two/
each) by 4 (interpretation: SU/OBJ/CU/EA) mixed design
ANOVA in which age and quantifier type were treated
as between subjects variables and interpretation as a
within subject variable. The analysis revealed a significant
main effect of interpretation (F(3,180) = 32.17,
p < 0.001), no main effect of age or quantifier type
(F(1,60) = 0.9, p = .34 and F(1,60) = 1.6, p = .19, respectively).
There were also significant interactions between
age and quantifier type (F(1,60) = 10.26, p < 0.01), interpretation
and age (F(3,180) = 4.8, p < 0.01), interpretation
and quantifier type (F(3,180) = 39.34, p < 0.001)
and interpretation, quantifier type and age (F(3,180) =
16.66, p < 0.001).
In order to further examine these interactions and test
the predictions made for adults, separate analyses breaking
down the results by age and quantifier type are presented
below. Recall that for adults, high acceptance rates for SU,
EA and CU were predicted, as well as a comparatively lower
acceptance rate for OBJ in the two N condition. By contrast,
high acceptance rates for OBJ and EA and low
acceptance rates for SU and CU were predicted in the each
Table 2
Percentages of Yes responses for children and adults in both quantifier type
conditions (two N and each N) and on each interpretation (SU, OBJ, EA, CU).
Adults Children
Two N (%) Each N (%) Two N (%) Each N (%)
SU 82.8 31.2 78.1 90.6
OBJ 7.8 100 28.1 84.3
EA 100 85.9 98.4 79.6
CU 78.1 17.1 23.4 54.6
Table 1
Predicted response patterns for adult participants.
Quantifier type SU OBJ EA CU
Two N High Lower than SU, EA and CU High High
Each N Lower than OBJ and EA High High Lower than OBJ and EA
Pictures
J. Musolino / Cognition 111 (2009) 24–45 33
N condition. As can be seen from Figs. 1 and 2 below, these
predictions are born out.
The percentages of Yes responses for adults in the two N
condition were entered into a separate, repeated measures
ANOVA which revealed a significant effect of interpretation
(F(3,45) = 46.08, p < 0.001). Post-hoc, pairwise comparisons
(with Bonferroni adjustment) between the different
means indicated that the mean for OBJ is significantly lower
than each of the other three means (SU, EA and CU),
p < 0.001. The mean for CU was also found to be significantly
lower than the mean for EA, p < .05. No other significant
differences were found. Further analyses revealed
that the acceptance rates for SU, EA and CU (82.8%, 100%
and 78.1%, respectively) were significantly above what
would be expected by chance performance (p < 0.001)
and that the acceptance rate for OBJ (7.8%) was significantly
below what would be expected by chance
(p < 0.001).
The percentages of Yes responses for adults in the
each N condition were also entered into a separate repeated
measures ANOVA which revealed a significant effect
of interpretation (F(3,45) = 39.86, p < 0.001). Posthoc,
pairwise comparisons between the different means
(with Bonferroni adjustment) indicated that the mean
for SU and CU, while not significantly different from
each other (p = 0.5), were each significantly different
from the means for OBJ and EA (p < 0.001). The means
for OBJ and EA, in turn, were not significantly different
from each other (p = 0.5). Further analyses revealed that
the acceptance rates for OBJ and EA (100% and 85.9%)
were significantly above chance (p < 0.001) and that
the acceptance rates for SU and CU (31.2% and 17.1%)
were significantly below what would be expected by
chance performance (p < 0.01). Figs. 3 and 4 illustrate
the pattern of responses for children in the two N and
each N condition for each of the four interpretations,
SU, OBJ, EA and CU.4
0
0.2
0.4
0.6
0.8
1
SU OBJ EA CU
Proportions of Yes responses
Fig. 1. Adults, two N.
0
0.2
0.4
0.6
0.8
1
SU OBJ EA CU
Proportions of Yes responses
Fig. 2. Adults, each N.
0
0.2
0.4
0.6
0.8
1
SU OBJ EA CU
Proportions of Yes responses
Fig. 3. Children, two N.
0
0.2
0.4
0.6
0.8
1
SU OBJ EA CU
Proportions of Yes responses
Fig. 4. Children, each N.
4
As one of the reviewers pointed out, blocking the various readings in
the design of the experiment might be a concern here, given children’s
notorious tendency to perseverate. In other words, blocking might lead to
undesirable order effects. However, further statistical analyses revealed
that no such effects can be found in the data. For each quantifier condition,
Two N and Each N, a separate mixed design ANOVA was conducted with
ordering as a between subjects variable and reading (SW, OBJ, EA and CU)
as a within subjects variable. In the case of Two N, the analysis revealed no
significant effect of order (F(1,14) = 0.697, p = 0.41) and no interaction
between ordering and reading (F(3,14) = 2.1, p = 0.11). In the case of Each N,
the analysis revealed no significant effect of order (F(1,14) = 0.756, p = 0.39)
and no interaction between ordering and reading (F(3,14) = 0.65, p = 0.58).
34 J. Musolino / Cognition 111 (2009) 24–45
The percentages of Yes responses for children in the two
N condition were entered into a separate, repeated measures
ANOVA which revealed a significant effect of interpretation
(F(3,45) = 18.75, p < 0.001). Post-hoc, pairwise
comparisons (Bonferroni adjustment) indicated that the
mean for OBJ and CU, while not significantly different from
one another, were each significantly different from the
mean for SU and EA (p < 0.05). SU and EA, in turn, were
not significantly different from one another (p = 0.43). Further
analyses revealed that the acceptance rates for SU and
EA (78.1% and 98.4%) were significantly above what one
would expect by chance (p < 0.001) and that the acceptance
rates for OBJ and CU (28.1% and 23.4%) were significantly
below what one would expect by chance (p < 0.01).
The percentages of Yes responses for children in the
each N condition were also entered into a separate, repeated
measures ANOVA which revealed a significant effect
of interpretation (F(3,45) = 4.05, p < 0.05). Post-hoc,
pairwise comparisons (Bonferroni adjustment) indicated
that the only significant difference was between the means
for SU and CU (p < 0.05). Further analyses revealed that the
means for SU, OBJ and EA (90.6%, 84.3% and 79.6%, respectively)
were significantly above what one would expect by
chance performance (p < 0.001) but that the mean for CU
(54.6%) was not different from chance performance
(p = 0.5).5
Figs. 5 and 6 directly compare patterns of responses obtained
for children and adults in each of the two quantifier
type conditions (two N and each N).
We can see that the age by quantifier type interaction is
due to the fact that children have a more adult-like profile
for two N than for each N. The interaction between interpretation
and age is driven by the way children and adults
differ on their acceptance rates for CU in the two N condition
and for CU and SU in the each N condition. The interaction
between interpretation and quantifier type arises
from the fact that two N and each N give rise to different response
patterns for children and adults. Finally, children
display a more adult-like response pattern on two N than
on each N only on certain interpretations, which accounts
for the interaction between age, quantifier type and
interpretation.
5. Discussion
5.1. Adults
Theoretical linguists, on the basis of their intuitions,
have claimed that sentences containing two NQE, such as
Three boys are holding two balloons, can receive four different
interpretations. This means that such sentences are
true in at least four different types of configuration involving
boys holding balloons. Two of the readings involve scopal
interactions between the two NQEs, namely SU and OBJ
and the other two, EA and CU are scope-independent readings.
Explaining what these readings are and precisely how
they arise from the interaction of the expressions involved
requires several pages of theoretical background (see Section
2) and a fair amount of exposure to formal linguistic
theory and logical analysis. Yet, as the results of the present
experiment demonstrate, individuals who do not hold
advanced degrees in logic or linguistics routinely access
such interpretations in the course of language comprehension.
To be sure, as predicted, adult participants accepted
SU, EA and CU 82.8%, 100% and 78.1% of the time, respectively
and in all cases, significantly above what would be
expected by chance performance (p < 0.001). Another prediction
was that given SU and OBJ, two scope-dependent
readings, OBJ would be much more difficult to access and
that given two scope-independent readings, EA and CU,
EA would be preferred. Here again, these predictions are
confirmed by the data, albeit with noticeable differences
in the size of the preferences: EA was preferred over CU
100% vs. 78.1% (p < 0.05) whereas SU was preferred over
OBJ 82.8% vs. 7.8% (p < 0.001). In interpreting these results,
one might be tempted to conclude that OBJ is simply not a
possible reading for sentences like Three boys are holding
0
0.2
0.4
0.6
0.8
1
SU OBJ EA CU
Adults
Children
Proportions of Yes responses
Fig. 5. Adults vs. children, two N.
0
0.2
0.4
0.6
0.8
1
SU OBJ EA CU
Adults
Children
Proportions of Yes responses
Fig. 6. Adults vs. children, each N.
5
Children’s acceptance rate for CU, namely 54.6%, might at first glance
suggest that they were performing at chance in this condition. A closer look
at the pattern of responses however, reveals a bimodal distribution with 13
of the 16 children either accepting the relevant sentences 100% of the time
(7 children) or 0% of the time (6 children). The other three children
accepted the sentences 25%, 75% and 75% of the time.
J. Musolino / Cognition 111 (2009) 24–45 35
two balloons. However, this conclusion would be premature
as the very low acceptance rate in this case might simply
reflect the fact that OBJ is massively dispreferred. In
this regard, a recent study by Musolino and Lidz (2003)
shows that adult speakers of English very seldom accept
sentences like Two frogs did not jump over the rock on a narrow
scope interpretation of the NQE with respect to negation
(It is not the case that two frogs jump over the rock).
However, when certain contextual factors are manipulated,
the acceptance rate for the same sentences is almost
at ceiling. The moral here is that low acceptance rates on a
given reading should not automatically lead to the
conclusion that the reading in question is in principle
unavailable.
Another observation made by linguists is that different
quantifiers induce different scopal preferences. Thus,
while an object NQE cannot easily take scope over a
quantified subject, a quantifier like each seems to require
a wide scope interpretation. This prediction is confirmed
by adults’ different acceptance rates for OBJ in the two N
and each N condition, namely 7.8% vs. 100%, respectively.
More generally, the fact that each seems to require a
wide scope reading predicts that sentences like Three
boys are holding each balloon (in a situation in which
there are two balloons) will be interpreted as meaning
that for each of the two balloons, there are three boys
holding the balloons. This, in turn, predicts that in addition
to OBJ, adults should accept any configuration in
which each of the two balloons is being held by three
boys and tend to reject configurations in which this is
not the case. This predicts a high acceptance rate for
EA and comparatively lower rates for SU and CU which
is confirmed by the data (85.9% acceptance rate for EA
vs. 31.2% and 17.1% for SU and CU, respectively with
the mean for EA significantly higher than those for SU
and CU, p < 0.001)
5.2. Children
Children’s near perfect performance on the control conditions
designed to test knowledge of the individual meaning
of expressions like two N, three N and each N (98.9%,
99.35% and 97.65% correct, respectively) indicates that preschoolers
experienced no difficulty with the task, and that
they were able to provide both Yes and No answers when
appropriate (since approximately half of the control items
required a Yes answer and the other half a No answer).
Turning to the experimental conditions, recall that this
experiment was designed to test knowledge of the core
logico-syntactic properties of NQE shown in Table 1.
Children’s performance in the two N condition suggests
that by the age of 5, knowledge of these core properties is
in place, as predicted by the compositional hypothesis discussed
in Section 2.2. Specifically, children’s adult-like
acceptance rate in the SU condition (78.1% acceptance rate
vs. 82.8% for adults, t(30) = À0.34, p = 0.73, two-tailed)
demonstrates knowledge of the scopal properties of NQE.
Recall that in this condition, each of the three boys is holding
two balloons and so that the total number of balloons is
six. Nevertheless, children readily accept sentences like
Three boys are holding two balloons, even if the total number
of balloons, six, is different from the one explicitly mentioned
in the sentence, two. This shows that preschoolers
truly understand the scopal nature of NQE and the fact that
in this case, the interpretation of the object NQE is dependent
on that of the subject NQE.6
Children’s adult-like acceptance rate in the EA condition
(98.4% for children vs. 100% for adults) indicates that preschoolers
can assign NQE scope-independent readings.7
The asymmetry observed in children’s acceptance rate of
SU and OBJ (78.1% vs. 28.1%, respectively, p < 0.05) demonstrates
that given two scope-dependent readings, children,
like adults, display a strong preference for SU over OBJ. Finally,
given two scope-independent readings, EA and CU, children,
like adults, displayed a preference for EA (98.4%
acceptance rate vs. 23.4%, respectively, p < 0.05). In this regard,
it should be noted that the size of this preference is different
in the two populations. For adults, EA is preferred over
CU (100% vs. 78.1%, respectively), but the acceptance rate for
CU nevertheless remains high and significantly above chance
(as high as SU, 82.8% and significantly higher than OBJ, 7.8%).
For children, EA is also preferred over CU (98.4% vs. 23.4%,
respectively) but the acceptance rate for CU is significantly
lower than that for SU (23.4% vs. 78.1%, p < 0.05) and not different
from that for OBJ (23.4% vs. 28.1%).
Given the availability of both distributive and cumulative
interpretations for NQE, the previous observation suggests
that preschoolers tend to assign NQE a distributive
reading. Whether children’s low acceptance rate in the case
of CU reflects an inability to assign NQE a cumulative interpretation
or a strong preference for a distributive reading is
a question that cannot be answered given the data available.
On the assumption that this pattern reflects preschoolers’
preferences – and not a lack of ability – we may be witnessing
in children an exaggerated version of the adult preference
for EA/SU over CU. A similar phenomenon, also
involving NQE, has been recently reported in the developmental
literature on quantification (Musolino & Lidz,
6
This result is noteworthy, especially in light of recent findings on the
scalar behavior of NQE showing how reticent preschoolers are to deviate
from the ‘exactly n’ interpretation (Huang et al., submitted; Hurewitz et al.,
2006; Musolino, 2004; Papafragou & Musolino, 2003). For example,
Musolino (2004) tested 4 and 5-year-olds on their interpretation of phrases
like two N in contexts which license an ‘at least n’ interpretation. This was
done in the context of a game in which the main character had to get two
hoops on a pole to win a prize. At the end, the character in question
managed to get four hoops on the pole. The test question was whether he
should win the prize. Adult participants almost always answered affirmatively
thereby assigning the phrase two N an ‘at least n’ interpretation (You
need to get two hoops on the pole was thus interpreted as You need to get at
least two hoops on the pole). Preschoolers on the other hand answered
negatively two thirds of the time, arguing that the character should not win
because the number of hoops on the pole was four, and not two. This result
shows that preschoolers, unlike adults, have a strong tendency to assign
NQE an ‘exactly n’ reading. In spite of this bias, the children in the present
experiment, who were the same age as those used in Musolino (2004), had
no difficulty accepting sentences containing the phrase two balloons in a
context in which the total number of balloons was six.
7
Here again, one may argue that EA is just a special case of OBJ and so
that children (and adults) know EA by virtue of knowing OBJ. However,
what would be puzzling on this view is the asymmetry between EA and OBJ
seen in both children and adults. If EA is just a special case of OBJ, then
what would explain that both children and adults find it so difficult to
access OBJ as compared to EA?
36 J. Musolino / Cognition 111 (2009) 24–45
2003). To begin, Lidz and Musolino (2002) showed that preschoolers,
when given a truth value judgment task, display a
strong preference for the narrow scope reading of the NQE
with respect to negation in sentences like The Smurf did
not catch two birds (i.e., It is not the case that the Smurf caught
two birds). To be more precise, in a context in which the narrow
scope reading is true and the wide scope reading is
false, children readily accept sentences like The Smurf did
not catch two birds, on the grounds that the narrow scope
reading is true. However, in a context in which the wide
scope reading is true and the narrow scope reading is false,
children overwhelmingly reject the same sentences on the
grounds that the narrow scope reading is false. In contrast,
adults in the same task can access the two scope readings
equally often (the wide scope reading of the NQE can be
paraphrased as There are two birds that the Smurf did not
catch). What Musolino and Lidz (2003) showed is that in a
context in which both scope readings are true, adults justify
their (necessarily) affirmative answers by invoking the fact
that the narrow scope reading is true. This result shows that
the preference seen in children for the narrow scope reading
of an object NQE is an exaggerated version of a preference
also observable in adults.
One way to interpret these results is in the context of recent
findings on the development of sentence processing
abilities in young children (Trueswell et al., 1999). What
emerges from the work of Trueswell and collaborators is
the observation that preschoolers do not yet resolve ambiguity
as efficiently as adults. The main difference between
the two populations lies in their respective ability to revise
an initial parse which at a later point in time turns out to be
incorrect. While this process is quick and efficient in adults,
the same is not true of preschoolers who often end up getting
stuck with the initial (incorrect) parse. If so, the kind of
exaggerated preference described above is precisely what
one would expect to find. In the case of sentences like The
Smurf did not catch two birds, children and adults initially
access the narrow scope reading but only adults are later
able to revise their initial interpretation and access the
wide scope reading. The same kind of reasoning would apply
to sentences like Three boys are holding two balloons. In
both populations, the initial parse would correspond to a
distributive interpretation of the NQE which would explain
high acceptance rates for SU and EA.8
On this view, adults
would be able to revise their initially distributive parse and
assign the NQE a cumulative interpretation, but this would
be much more difficult for children and would explain the
lower acceptance rate for CU compared to EA and SU. The initial
preference for a distributive parse of the NQE, in turn,
might come from differences in the frequencies with which
NQE are used on the distributive/cumulative readings.
Next, let us consider children’s performance in the each
N condition. This condition was included mainly to test the
claim that different quantifiers induce different scopal
preferences. Specifically, recall that while an object NQE
does not easily take (distributive) scope over a quantified
subject, quantifiers like each N seem to require a wide
scope interpretation. This effect was thus expected to manifest
itself in the OBJ condition and was verified for adults
(7.8% acceptance rate for OBJ in the two N condition vs.
100% in the each N condition). The data obtained from children
indicate that the distinction between NQE and quantifiers
like each N is also in place by the age of 5 (28.1%
acceptance rate for OBJ in the two N condition vs. 84.3%
in the each N condition, p < 0.05). This result is worth
emphasizing because it suggests that young children have
knowledge of the mechanism whereby quantifiers are
interpreted in a position that is different from their surface
syntactic position, namely quantifier raising (QR) – or
whatever mechanism may be responsible for its effects. Indeed,
recall from the discussion in Section 2 that in order to
obtain an interpretation of the sentence Three boys are
holding each balloon on which each balloon takes scope over
three boys, the phrase each balloon must be (covertly) displaced
from its surface syntactic position to a higher position
from which it will take scope over three boys. We thus
have evidence that by the age of 5, young children have
knowledge of one of the core linguistic mechanisms underlying
the logical syntax of natural language quantification
(for related evidence, see Brooks & Braine, 1996; Guasti,
2000 and Lidz et al., 2004).
The other noteworthy observation regarding children’s
performance in the each N condition concerns their acceptance
rates for SU and CU which are significantly higher
than those observed in adults (90.6% for children vs.
31.2% for adults in the case of SU, t(30) = 4.9, p < 0.001,
two-tailed, and 54.6% for children vs. 17.1% for adults in
the case of CU, t(30) = 2.7, p < 0.01, two-tailed). The importance
of this difference as well as its implications are the
topic of the next section.
To summarize, the results of our experimentation with
preschoolers reveal that, by the age of 5, children know
that (a) NQE give rise to scope-dependent readings, (b)
NQE also give rise to scope-independent readings, (c) given
two possible scope-dependent readings, the wide scope
reading of the object (OBJ) is dispreferred, and (d) given
two possible scope-independent readings, the cumulative
reading (CU) is dispreferred. Moreover, children’s behavior
in the each N condition indicates that they have knowledge
of QR – or its theoretical equivalent – one of the core mechanisms
underlying the logical syntax of natural language
quantification. Notice that these results are compatible
with the compositional and the learning hypothesis described
in Section 2.2. We come back to this question in
Section 5.4. Finally, in spite of their impressive command
of the logical syntax of NQE, we have also seen that preschoolers
differ from adults in their ability to assign NQE
a cumulative reading and in their elevated acceptance
rates for SU and CU in the each N condition.
5.3. Explaining differences between children and adults
An important feature of the integrative approach advocated
in this article is the claim that linguistically-based
models can be used to explain developmental phenomena.
In order to substantiate this claim, I will now show that the
differences between children and adults uncovered in the
8
Notice that both SU and EA, even though one reading is scopedependent
and the other is not, can be interpreted as distributive in the
sense that in both cases, each of the three boys is indeed holding two
balloons.
J. Musolino / Cognition 111 (2009) 24–45 37
preceding section can be accounted for by an independently
motivated, linguistically-based, processing model
originally proposed to account for children’s non-adult
behavior in the domain of universal quantification (sentences
containing every) (Geurts 2003). In order to do so,
I will proceed in the following manner. First, I provide a
summary of the main features of Geurts’ (2003) model.
Second, I show that this model naturally extends to the
kind of sentences that we have been concerned with and
that it provides a principled explanation of some of the differences
we observed in the behavior of children and
adults. Third, I consider further predictions of Geurts’
(2003) model and show that they can also be verified using
the data reported here. Finally, I briefly discuss alternative
accounts of the differences under consideration as well as
related issues arising from the use of Geurts’ (2003) model.
5.3.1. Geurts’ (2003) processing model
It is well-known in the developmental literature on
quantification that preschoolers – and even older children
– often give strikingly non-adult responses to statements
containing the universal quantifier every. For example,
when shown a picture in which three boys are each ridding
a different elephant and a fourth elephant is left without a
boy, and asked whether every boy is riding an elephant,
children, unlike adults, often answer ‘No’ and justify their
answer by pointing out that one of the elephants is sans
boy (Drozd et al., 1999; Philip, 1995, among many others).
The most recent account of this phenomenon is a proposal
by Bart Geurts (2003). Although the account itself
is fairly technical, and thus requires a good working knowledge
of linguistic theory to fully appreciate, the core intuitions
are relatively easy to grasp. I will therefore focus
here on these core intuitions and try to make them intelligible
to readers who do not possess the relevant linguistic
background. Readers interested in a more thorough – and
more technically accurate – exposition of Geurts’ ideas
can turn to his original 2003 article. As a first step, let us
try to think about what a sentence like (13) means. Given
a contextually determined set of individuals (say the ones
shown in a picture), (13) expresses the idea that all the
individuals that happen to be boys have at least one property
in common, namely that of being an elephant rider.
This, in turn, tells us that there are three important parts
to the meaning of a sentence like (13): the quantifier itself,
every, the noun that the quantifier combines with, boys and
finally, the property that all the boys have in common,
namely that of being elephant riders. Accordingly, linguists
find it useful to represent the logical structure of (13)
roughly as given in (14). In plain English, the representation
in (14) is simply saying that for every x, if x is a boy,
then there is a y, y is an elephant, such that x is ridding y.
(13) Every boy is riding an elephant.
(14) a. Every (x) [boy (x) ? (y: elephant (y) and x
rides y)]
b. Every (A) (B)
More generally, we can think of a sentence like (13) as having
the structure in (14b) where every is the quantifier, (A)
is the noun that combines with the quantifier (called the
restriction) and (B) is the property that Every (A) must satisfy
for the sentence to be true.
The next step is to observe that quantifiers do not all
give rise to the tripartite structure described above. In fact,
linguists have long recognized (for reasons that need not
concern us here) that quantifiers fall into two categories:
those that give rise to a tripartite structure like (14b),
called strong quantifiers, and those that only give rise to
a two-part structure, as shown in (15), called weak
quantifiers.
(15) Quantifier (B)
We are now in a position to express the key insight underlying
Geurts’ proposal. Based on the assumption that a
three-part structure is more costly to build than a two-part
structure, Geurts proposes that when children parse a sentence
which contains a strong quantifier like every, they
initially treat the quantifier as though it were weak, and
thus only ‘build’ a two-part structure of the type shown
in (15). Crucially however, children know that every is
strong, and thus requires a tripartite structure. This means
that the final representation for a sentence like (13) will
contain three parts indeed but that one of them, the (A)
part, will be left underspecified. This is shown in (16a).
(16) a. Every (. . ..) (boy, elephant, boy rides elephant)
b. Every (boy) (elephant, boy rides elephant)
c. Every (elephant) (boy, boy rides elephant)
Given the representation in (16a), there is now a choice
as to which of the two nouns, boy or elephant, will end
up serving as the restriction on the quantifier every. On
Geurts’ account, pragmatic/contextual factors ultimately
determine which of the two nouns ends up serving as
the restriction on the quantifier. In case the noun boy
ends up in the restrictor, as shown in (16b), we get the
adult representation, and the sentence means that for
every boy there is an elephant that that boy is riding.
However, if the noun elephant ends up serving as the
restriction on the quantifier, as shown in (16c), we get
a representation which would make the sentence true
just in case, for every thing that happens to be an elephant,
there is a boy riding that elephant. This latter representation
is what accounts for children’s classic
interpretative ‘error’. Indeed, recall that preschoolers often
reject a sentence like Every boy is riding an elephant
in a context in which one of four elephants is sans boy.
On the account developed here, this makes sense if children
incorrectly take the sentence Every boy is riding an
elephant to mean that for every elephant, there must be
a boy riding that elephant, (16c).
5.3.2. Using Geurts’ (2003) model to account for children’s
non-adult behavior
Let us now see how Geurts’ account extends to sentences
like Three boys are holding each balloon. What needs to be explained
here is why children’s acceptance rates for pictures 1
38 J. Musolino / Cognition 111 (2009) 24–45
and 4, corresponding to SU and CU, are significantly higher
than those of adults (31.2% and 17.1% for adults, vs. 90.6%
and 54.6% for children, p < 0.01). First it is important to point
out that each, like every, is a strong quantifier.9
Also recall
from Section 2 that each has a strong tendency to take wide
scope with respect to other logical expressions. Thus, in the
case of a sentence like Three boys are holding each balloon, each
balloon will tend to take scope over three boys. With these facts
in mind, consider now the adult representation of Three boys
are holding each balloon given in (17a). Since each combines
with the noun balloon, balloon serves as the restriction on the
quantifier and thus, on a wide scope reading of each balloon,
(17a) can be paraphrased as meaning that for each balloon,
there are three boys holding that balloon. Clearly, this is not
true of pictures 1 and 4, corresponding to SU and CU, which explains
adults’ low acceptance rates in these cases.
(17) a. Each (balloon) (three boys, boys hold balloon)
b. Each (. . ..) (balloon, three boys, boys hold balloon)
c. Each(threeboys)(balloon,threeboysholdballoon)
(17b) shows the representation corresponding to a weak
construal of each – with the restriction left underspecified
– and (17c) illustrates the representation in which boys –
and not balloons – serves as the restriction on each. (17c),
also on a wide scope reading of each, can be paraphrased
as meaning that for each of the three boys, there is a balloon
that that boy is holding, which is true of pictures 1
and 4 and thus accounts for children’s higher acceptance
rates in these cases.10
5.3.3. Testing further predictions of Geurts’ (2003) model
As we have just seen, Geurts’ (2003) model, initially formulated
to account for children’s non-adult interpretation
of every, naturally extends to other strong quantifiers and
nicely captures the non-adult pattern reported here.
What’s more, Geurts’ account makes another specific prediction
in the present context. Recall that pictures like 6
and 7 were used as controls in the two N and each N condition.
These configurations were introduced to make sure
that participants could reject sentences like Three boys
are holding two balloons when the truth conditions associated
with such sentences were not satisfied.
If the account developed so far is correct, we should expect
to find that children and adults respond differently to pictures
6 and 7 in the each N condition, that is when sentences
like Three boys are holding each balloon are used to
describe these pictures. Let us see why. If Three boys are
holding each balloon tends to be interpreted by adults as
meaning that for each of the balloons there are three boys
holding it, (17a), we would expect adults’ acceptance rate
to be rather low in the case of the configurations corresponding
to pictures 6 and 7. By contrast, if children tend
to interpret the same sentence as meaning that for each
of the three boys there is a balloon that that boy is holding,
(17c), then we would expect children’s acceptance rate to
be comparatively higher for pictures 6 and 7. As already
mentioned in the results section, an analysis of the data
obtained from preschoolers and adults in response to configurations
corresponding to pictures 6 and 7 indicate that
this prediction is indeed borne out. Recall that adults accepted
sentences like Three boys are holding each balloon
23.4% of the time in the context of configurations corresponding
to pictures 6 and 7 whereas children accepted
the same sentences significantly more often, i.e. 85.1% of
the time (t(30) = 6.5, p < 0.001, two-tailed). Moreover, notice
that the difference in size between these two
acceptance rates, roughly equal to a factor of 3.5, nicely
corresponds to the size of the difference in acceptance
rates between children and adults for pictures 1 and 2
9
As witnessed by the unacceptability of a sentence like There is/are each
student(s) in the room. See Appendix 2 for further discussion.
10
Notice that three boys in (17c) would need to be interpreted distributively,
a fact that can be captured in the system proposed by Geurts (Bart
Geurts, personal communication).
J. Musolino / Cognition 111 (2009) 24–45 39
(31.2% vs. 90.6% and 17.1% vs. 54.6%, respectively). Finally,
notice that this difference emerges only in the each N condition
(in the two N condition, recall that children’s (correct)
rejection rates were not different from that of
adults (98.4% vs. 100%, respectively).
5.3.4. Qualifications and further predictions
Anybody familiar with the literature on the acquisition
of quantification will be aware of the fact that Geurt’s is
not the only account of children’s non-adult interpretation
of the universal quantifier currently available. Since Piaget
and collaborators uncovered the basic phenomenon some
40 years ago (Inhelder & Piaget, 1964), a number of accounts
have been proposed, sometimes leading to vigorous
debate (Crain et al., 1996; Drozd et al., 1999; Geurts, 2003;
Philip, 1995 among others). My aim here is not to add to
this discussion by trying to decide between these competing
accounts – a task which clearly falls beyond the scope
of the present article – nor even to suggest that Geurts’ is
the only account capable of accounting for the data presented
here. Rather, in using Geurts’ (2003) account, my
goal has been to illustrate the explanatory power of linguistically-based
models when it comes to explaining
developmental patterns. To the extent that the results presented
here are compatible with a different linguisticallybased
account of children’s non-adult interpretation of
sentences containing the universal quantifier, my point remains
unaffected.11
What is worth emphasizing here in the context of the
relationship between linguistic theory and linguistic development
– and more generally the relationship between linguistics
and psychology – is that the explanatory force of
the type of account proposed by Geurts comes from two
separates sources. First, the formal constructs and technical
vocabulary involved in the proposal, such as quantifier,
domain, nuclear scope, weak, strong, etc., are independently
needed to account for the competence of mature
speakers. The fact that an explanation of the behavior of
children can be formulated in terms of the same constructs
and vocabulary illustrates the explanatory power of formal
accounts of language through what is known as the continuity
assumption (Pinker, 1984). Second, the model proposed
by Geurts is independently motivated in another
way as well: it is needed to explain the kind of non-adult
responses that children often give to sentences like Every
boy is riding an elephant.
Having said that, I would now like to briefly consider an
alternative account of the developmental pattern under
consideration and discuss two further issues arising from
the analysis discussed above. Recent work on the acquisition
of quantification shows that children often interpret
sentences containing multiple quantificational elements
on the basis of the surface syntactic position of these elements.
This phenomenon has come to be known as ‘isomorphism’
(Lidz & Musolino, 2002; Musolino, 1998;
Musolino & Gualmini, 2003; Musolino & Lidz, 2003; Musolino
et al., 2000; Noveck, Guelminger, Georgieff, & Labruyere,
2007). The question I would like to consider is
whether the difference observed between children and
adults in the each N condition may be a manifestation of
the isomorphism effect. Recall that for adults, sentences
like Three boys are holding each balloon require assigning
each balloon wide scope and thus interpreting it in a position
that is different from its surface syntactic position.
Suppose now that an isomorphic child were to interpret
each balloon where it occurs in surface syntax, that is in
the scope of three boys. If so, children would tend to interpret
Three boys are holding each balloon as meaning that
three boys are such that they are holding all the balloons
(recall that each is universal quantifier). At this point,
two further options would present themselves. Given a
narrow scope reading of each balloon, children could still
in principle interpret three boys distributively or cumulatively.
If the former option is chosen, one would expect
children to accept Three boys are holding each balloon only
in configurations in which each of the three boys is holding
all the balloons. This in turn would fail to account for children’s
elevated acceptance rates in the SU and CU conditions
since in these configurations it is not the case that
each boy is holding all the balloons. Suppose now that children
assign three boys a cumulative interpretation. In this
case, one would expect children to accept Three boys are
holding each balloon only in configurations in which the
three boys, when considered cumulatively, are holding all
the balloons. This would indeed account for children’s elevated
acceptance rates in SU and CU since in these two
cases, the boys, when considered collectively, are holding
all the balloons. The problem with this explanation though
is that it would require that children assign the NQE a
cumulative interpretation. However, results from the two
N condition indicate that children have a massive preference
for the distributive interpretation of NQEs. Another
problem with this account is that isomorphism arises in
cases involving the interaction of quantified NPs and negation
and not in cases involving the interaction of two quantified
NPs.
Another issue that deserves clarification is the fact that
accounts like Geurts’ (or any other for that matter) do not
predict that children will always resort to the strategy described
in these accounts. In other words, as a matter of
empirical fact, children do not always interpret statements
containing the universal quantifier in a non-adult manner
(see Philip, 1995 for discussion). Nor is it the case that
adults never do so (Freeman, Sinha, & Stedmon, 1982. Also
see Musolino and Lidz (2003) for related evidence) – the
term ‘non-adult’ here can thus be somewhat misleading.
The main thrust of the kind of accounts under consideration
is not to predict the frequency with which children
will resort to non-adult interpretations but rather to explain
the nature of the representations and processes involved
when they do, as well as specify the contexts in
which such non-adult interpretations are expected to occur.
Thus, given the availability of a strategy S, which deviates
from the ‘standard’ strategy used in interpreting
11
It is now generally accepted that the kind of errors involving the
universal quantifier originally reported by Piaget and his collaborators are
not due to conceptual deficits in the young child, as originally proposed. It
is now clear that the problem is linguistic, not conceptual (see Philip (1995)
and Lidz and Musolino (2002) for relevant discussion).
40 J. Musolino / Cognition 111 (2009) 24–45
universally quantified statements, the observation is that
children resort to S more often than adults.
This last point leads to an important question regarding
the results presented here, namely whether children resort
to the kind of interpretation predicted by Geurts’ model
across the board, that is in all four conditions, SU, OBJ,
CU and EA, or just in the two conditions in which they differ
from adults, namely SU and CU (as well as the control
conditions discussed above). This issue arises because children’s
apparent adult-like behavior in the OBJ and EA condition
would also be what one would expect on the
analysis based on Geurts’ model. Recall that on this view,
the key difference between children and adults is that preschoolers
tend to interpret sentences like Three boys are
holding each balloon as meaning that for each of the three
boys, there is a balloon that that boy is holding. Since this
is obviously true of the configurations corresponding to
OBJ and EA, one could argue that children are giving the
right answer for the wrong reason. This, in turn, one might
think, would undermine the conclusions reached earlier on
the basis of children’s apparent adult-like behavior in the
OBJ condition. Recall that we took this observation as evidence
for knowledge of the mechanism of quantifier raising
– or its equivalent – in preschoolers.
Fortunately, a closer look at the phenomenon reveals
that this potential objection should not be a concern. On
the account discussed so far, children are claimed to often
interpret Three boys are holding each balloon to mean that
each of the three boys is holding a balloon. Notice now that
in order to get to this interpretation, two things need to
happen. First, as described by Geurts’ model, children need
to ‘incorrectly’ let the noun boys serve as the restriction on
each. But this is not all. The issue of the relative scope of
(incorrectly derived) each boy and a balloon still needs to
be settled. On a wide scope reading of each boy over a balloon,
we get the desired reading, namely that for each of
the boys there is a balloon that that boy is holding. This
would account for the high acceptance rate in the OBJ condition
since in this case, each boy is indeed holding a balloon.
Consider now the narrow scope interpretation of
each boy with respect to a balloon. Here, we would get a
reading that would be true just in case there is a particular
balloon that all the boys are holding (recall that each is a
universal quantifier). This, however, would not be true of
the configuration corresponding to the situation depicted
in the OBJ condition where three boys are holding one balloon
and another balloon is being held by a different group
of three boys. We can therefore conclude that for children
to behave in a seemingly adult-like fashion in the OBJ condition,
they must, even if they resort to the kind of interpretation
predicted by Geurts’ model, interpret each as
taking wide scope. In other words, children must interpret
each in a position which is different from its surface syntactic
position. This, in turn, amounts to knowing the
mechanism (QR or other) which covertly displaces quantifiers.
Thus, even if children behave in a adult-like fashion in
the OBJ condition for the ‘wrong reasons’ the conclusion
that preschoolers know QR (or its equivalent) remains a
valid one.
Finally, the reader will also have noticed that in the each
N condition, adults accept sentences like Three boys are
holding each balloon in the configurations corresponding
to SU and CU approximately a third and a fifth of the time
(31.2% vs. 17.1%, of the time, respectively, to be more precise).
One way to think about this observation is that the
difference between children and adults is only quantitative,
and not qualitative, which brings us back to considerations
of sentence processing abilities in the two
populations and again ties the present research to other
work on the acquisition of quantification and the development
of sentence processing abilities. One could argue that
both children and adults are prone to the kind of misparse
described by Geurts’ (2003) model, but that children resort
to this strategy much more often than adults do. This interpretation,
advocating for continuity of representation and
process, would be compatible with the finding that in certain
contexts, children can be made to behave more like
adults when it comes to their interpretation of universally
quantified statements (Crain et al., 1996) and also that the
‘errors’ observed in children can be induced in adults (Freeman
et al., 1982. Also see Musolino and Lidz (2003) for related
evidence). The interesting question in the case at
hand is whether children could be ‘turned into adults’ –
and vice-versa – if the context and the experimental conditions
in which the relevant sentences are presented were
appropriately manipulated. For example, Geurts (2003)
predicts that the contextual salience of the various characters
and objects determines which of the nouns in the sentence
ends up serving as the restriction on the universal
quantifier. In the case of sentences like Three boys are holding
each balloon, this would predict that making the balloons
more salient than the boys should have the desired
effect.
5.4. Implication for development and developmental accounts
In addition to experimentally verifying the intuitions of
linguists regarding the logical syntax of NQE, highlighting
the explanatory power of linguistically-based models,
and tying the present results to recent developments on
the acquisition of quantification and the development of
sentence processing abilities in young children, these findings
also raise serious developmental questions. What I
hope to have shown here is that a grasp of the complexity
associated with the logico-syntactic properties of NQE is
not the exclusive privilege of individuals holding advanced
degrees in logic or linguistics. College undergraduates, and
more importantly preschool children, who have had little
to no formal education, possess the intricate knowledge
described by linguists. Consequently, developmental accounts
are now faced with the new challenge of having
to explain how this knowledge is acquired.
To put this challenge in perspective, recall that the main
puzzle that developmental psychologists have been trying
to solve over the past three decades is how children learn
that expressions like two dogs are used to describe sets of
two dogs. As the work of linguists makes it clear however,
mastery of the number vocabulary goes far beyond the recognition
that two is used in the presence of two-membered
sets. NQE also give rise to a range of interpretations
depending on the context in which they are used (Musolino,
2004), a rich set of pragmatic inferences and scalar efJ.
Musolino / Cognition 111 (2009) 24–45 41
fects (Papafragou & Musolino, 2003), and when combined,
they interact with each other and other logical expressions
in intricate ways. Thus, mastery of the number vocabulary,
just like any other aspect of language, involves knowledge
of complex semantic, pragmatic and syntactic properties.
Although a detailed account of how these complex linguistic
properties are acquired by young children is a task
that lies far beyond the scope of the present article, I would
nevertheless like to consider, if only briefly and in a speculative
and abstract manner, how this might be accomplished,
focusing on the properties of NQE discussed in
this article. What I would like to consider are two options
at opposite ends of the nativist/empiricist continuum,
without necessarily ascribing these views to any particular
account or investigator. Specifically, I would like to consider
the idea that the complex logico-syntactic properties
discussed in this article are directly learned from an analysis
of the input, and the idea that these properties are
not learned – and in fact that they do not even need to
be learned – but rather that they are (implicitly) deduced
from knowledge of the lexical meaning of NQE combined
with knowledge of the general rules of syntax for one’s language,
as discussed in Section 2.2.
According to the first approach, children would come to
know that NQE can give rise to scope-dependent/independent
readings, inter alia, because they have heard NQE
being used by the people around them in the relevant situations.
First, notice that this scenario would require that
mothers and caretakers routinely use sentences containing
multiple quantificational expressions (NQE combining
with NQE and other quantifiers) in highly constrained situations
which correspond to the configurations verifying
the truth conditions of the readings that we have been discussing
throughout this paper. However remote this may
sound, this is of course not impossible, and only a detailed
analysis of the input will settle the issue one way or the
other. A second, perhaps more serious, problem is that
the inferences and conclusions that children would draw
about the behavior of NQE on the basis of examples provided
by caretakers would have to be highly constrained.
Imagine for example that a child were to hear a sentence
like Three boys are holding two balloons in configurations
corresponding to SU, OBJ and CU. One logically possible
conclusion that the child may reach on the basis of such
evidence is that sentences like Three boys are holding two
balloons are true when (a) the number of boys is either 3
(SU) or 6 (OBJ), and (b) the number of balloons is either 6
(SU) or 2 (CU). The problem here is that such a conclusion,
perfectly reasonable as it may be given the nature of the
evidence, would nevertheless lead to massive overgeneralization.
For example, the child would now erroneously assume
that a sentence like Three boys are holding two
balloons is true when six different boys are each holding
one balloon (number of boys = 6 and number of balloons
= 6) or when three boys are such that two are holding
one balloon each and the third is holding four balloons
(number of boys = 3 and number of balloons = 6). Examples
of this sort are just a small fraction of the set of logically
possible conclusions that a child might reach on the basis
of the positive evidence that would seem to be required
for the appropriate learning to take place. And so unless
the child receives explicit instruction regarding possible
and impossible configurations, it is hard to see how learning
could be successful. This, of course, is nothing more
than the familiar problem of unconstrained induction applied
to language (Chomsky, 1975).
As mentioned earlier, another possibility is that the
properties under consideration are not learned but rather
that they are (implicitly) deduced from knowledge of the
meaning of the individual words in a given sentence along
with knowledge of the basic rules of syntax for one’s language.
This would not be unreasonable since it is unthinkable
that children learn the meaning of every single
sentence they hear on the basis of a pairing of each
sentence with a situation that would verify its truth conditions.
Fortunately for children, semantics is compositional.
On this view, the problem would boil down to having a rich
enough theory of the lexical semantics of NQE, which,
when engaged by independently motivated, general syntactic
principles would yield, without any further stipulation,
the desired set of logico-syntactic properties. The
main developmental issue on this view would be to explain
how children solve the mapping problem for NQE –
namely, how they learn to pair the relevant pieces of phonology
with the right pieces of lexical semantics. Needless
to say, this problem needs to be solved regardless of one’s
theoretical inclination.
In this regard, some linguists working on natural language
quantification seem to have opted for an approach
of the kind just described in order to account for the principled
– and yet highly idiosyncratic - behavior of different
quantifiers. For example, Beghelli and Stowel (1997), based
on their own work and that of others, have partitioned the
class of natural language quantifiers into a number of separate
categories such as, to give a few examples, interrogative,
negative, distributive-universal, counting and group
denoting quantifiers (with further subclassifications in certain
cases). Moreover, on this view, the lexical entries for
quantifiers include specific features such as [+interrogative],
[+negative], [+distributive], [+universal], etc. All this
structure is in part motivated by the need to account for
the intricate logico-syntactic behavior displayed by different
types of quantifiers, including NQE, without having to
massively complicate the rules of logical syntax. On this
view, the child’s main problem would not be to have to
learn the complex logico-syntactic properties of different
quantifiers, but rather to recognize what kind of quantifier
she is dealing with. Once phonological sequences have
been appropriately mapped onto the correct lexical concepts,
the rest would come for free and follow from the
compositional nature of semantics and the relationship between
syntax and semantics. I take the results presented
here to be compatible with this hypothesis, but it should
be recognized that they are also compatible, at least in
principle, with the ‘learning’ hypothesis discussed above.
The facts discussed throughout this article, namely that
NQE, while displaying some of the properties of other
quantified expressions, are nevertheless associated with a
very specific logico-syntactic profile has further consequences
for developmental accounts of the number vocabulary.
In order to illustrate this point, I will focus on a
recent proposal by Susan Carey (2004). In doing so, my
42 J. Musolino / Cognition 111 (2009) 24–45
aim is not to argue against this particular proposal, or even
to review it in full detail, but simply to show that the facts
and results presented in this article impose severe constraints
on accounts involving general claims of the kind
made by Carey (2004) (for related discussion and similar
conclusions, see Hurewitz et al., 2006).
An intriguing aspect of the acquisition of the number
vocabulary is its unusually protracted nature. A number
of studies on the acquisition of number words have shown
that English-speaking children first seem to learn the
meaning of the word one and take all the other number
words to contrast with one and mean something like ‘more
than one’. About half a year to nine months later, children
learn the meaning of two but they still do not seem to
know the exact meaning of numbers words greater than
two. Some time later, children come to be known as
‘three-knowers’ which means that they know the meaning
of one, two and three but take all the other number words
to mean something like ‘more than three’. Eventually, after
a few more months, children finally understand the correct
relationship between the count list and the integers
(Wynn, 1990; Wynn, 1992).
On the basis of these observations, Carey (2004) proposes
that preverbal number representations are not representations
of integers and, more importantly for our
present purposes, that children initially represent number
words as quantifiers (to use Carey’s terminology). Since
NQE share some of their properties with other quantifiers,
this much does not seem particularly controversial. Let’s
take a closer look at what exactly is being proposed
though. Carey’s second conclusion seems to hold of the
number words whose meanings children know ‘‘As described
above, the child learns the meanings of the first
number words as natural language quantifiers. Children
learn the meaning of ‘one’ just as they learn the meaning
of the singular determiner ‘a’ . . .” (p. 66). This conclusion
also applies to those number words whose exact meaning
children do not yet appear to know ‘‘In the early stages of
being a one-, two- or three-knower, the child represents
other number words as quantifiers, meaning ‘many’, where
‘many’ is more than any known number words.” (p. 66). So
the claim seems to be that children initially ‘mismap’ NQE
– if one takes Carey’s proposal literally – for example initially
assigning two N or three N the semantics of many, because
the child does not yet have the appropriate
conceptual wherewithal to do the correct mapping
(namely the concept of integer and cardinality).
The problem here is that even though a and one, some
and two, three and many may seem conveniently interchangeable
when their intuitive meaning is concerned, as
soon as their more complex linguistic properties are taken
into consideration it is clear that NQE are quite different
from other quantifiers. Consequently, a child who has initially
identified two as meaning some or three are meaning
many, will have a lot of unlearning and fine tuning to do in
the course of development if s/he ever hopes to arrive at
the correct conclusions regarding the set of semantic, pragmatic
and logico-syntactic properties associated with NQE.
To be sure, as we have seen here, NQE are associated with
specific logico-syntactic properties that sets them apart
from other quantifiers. Moreover, this conclusion also applies
to the semantic and pragmatic properties of NQE
which are different from those of other quantifiers, like
some, for example (see Papafragou and Musolino (2003),
Musolino (2004), Hurewitz et al. (2006), and Huang et al.
(submitted for publication) for relevant theoretical and
experimental evidence).
Moving beyond particular developmental accounts of
the number vocabulary, the general point that I would like
to emphasize here is that by providing a richer specification
of the end point of development, the integrative approach
advocated in this article imposes constraints on
developmental explanations. Moreover, these constraints
may ultimately serve as a basis to tease apart competing
developmental accounts just as previously obtained developmental
data can be – and have indeed been – used to
tease apart competing theoretical analysis of the semantics
and pragmatics of number words (Huang et al., submitted
for publication; Hurewitz et al., 2006; Musolino, 2004;
Papafragou & Musolino, 2003).
6. Concluding remarks
The main goal of this paper has been try to bridge the
gap between theoretical linguistics and developmental
psychology and in doing so to extend the integrative approach
developed in Musolino (2004). In principle, these
two disciplines share, as a common goal, the task of
explaining how children acquire language. In reality, each
side approaches this task rather differently and too often
without much influence from the other. A review of the
work on number words that linguists and psychologists
have carried out over the past few decades provides a good
illustration of this observation. The complex structure postulated
by linguists raises important developmental questions
which, unfortunately, are rarely addressed within
linguistics. On the other hand, the experiments and theorizing
of developmental psychologists are based on
assumptions about linguistic structure that often do not
match the level of complexity advocated by linguists.
In this paper, I have tried to show that it is not only possible,
but also desirable, to approach questions pertaining
to the acquisition of the number vocabulary from a perspective
which combines the concerns of both linguists
and psychologists. Specifically, I have shown that the tools
of developmental psychology can be used to experimentally
verify the complex structure postulated by linguists
and that the models proposed by linguists, in turn, can
be used to account for developmental patterns emerging
from the results of such experimentation. Finally, I have
shown that the results presented here have important
implications for developmental accounts of the number
vocabulary.
Acknowledgments
I would like to thank Kristen O’Connor, Lindsay Wood,
and Aubrey Messier, at the time all undergraduate students
at Indiana University, as well as Michaela Rose, a
graduate student in the Linguistics department at Indiana
University, for their precious help with data collection.
J. Musolino / Cognition 111 (2009) 24–45 43
Thanks are also due to the children and the staff of the following
daycare centers/preschools in the Bloomington,
Indiana area: Bloomington Developmental Learning Center,
Monroe County Jack and Jill Daycare Inc., Edgewood
Early Childhood Center, and Saint Charles Daycare.
Appendix A
Target sentences for both quantifier type conditions
(two N and each N)
Condition Target sentences Configurations
Two N Three boys are holding two
balloons (Â4)
SU, OBJ, EA,
CU
Three girls are holding two
kites (Â4)
SU, OBJ, EA,
CU
Three men are walking two
dogs (Â4)
SU, OBJ, EA,
CU
Three clowns are holding two
flowers (Â4)
SU, OBJ, EA,
CU
Each N Three boys are holding each
balloon (Â4)
SU, OBJ, EA,
CU
Three girls are holding each
kite (Â4)
SU, OBJ, EA,
CU
Three men are walking each
dog (Â4)
SU, OBJ, EA,
CU
Three clowns are holding
each flower (Â4)
SU, OBJ, EA,
CU
Control sentences used to test knowledge of the meaning
of individual quantifiers (two N, three N, each N) in both
quantifier type conditions (two N and each N)
Control sentences True/false
Three bunnies are eating carrots False
The clown is holding three flowers False
The boy is holding three balloons True
There are three monkeys in the tree True
Each boy is holding a balloon False
The woman gave each boy candy False
Each boy has an ice-cream cone True
The man gave each dog a bone True
Two cows are eating grass False
The man is walking two dogs False
The girl is holding two kites True
Two cats are playing with the mouse True
Control sentences used to create configurations that falsify
target sentences in the two N condition. The same sentences
in the same configurations were used in the each
N condition except that two N was replaced by each N.
Condition Other control sentences Configurations
Two N Three boys are holding two
balloons (Â2)
A, B
Three girls are holding two
kites (Â2)
A, B
Three men are walking two
dogs (Â2)
A, B
Three clowns are holding two
flowers (Â2)
A, B
Each N Three boys are holding each
balloon (Â2)
A, B
Three girls are holding each
kite (Â2)
A, B
Three men are walking each
dog (Â2)
A, B
Three clowns are holding
each flower (Â2)
A, B
‘A’ configurations correspond to situations in which all
three characters are each Ving one N. ‘B’ configurations
correspond to situations in which two of the three characters
are each Ving two Ns and the third character is only
Ving one N.
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