BUCLD 35 Proceedings
To be published in 2011 by Cascadilla Press
Rights forms signed by all authors
Quantifier Spreading is not Distributive
Thomas Roeper, Barbara Zurer Pearson, and Margaret Grace**
“Quantifier spreading” has captured the imagination of researchers on
different sides of the grammar interface: syntax, semantics, and pragmatics
(Philip, 1995; also Brooks, Braine, Jia, & da Graca Dias, 2001; Crain et al.
1996; Drozd, 2001; Guerts, 2003, among others). Piaget (Inhelder & Piaget,
1958) was among the first to have reported this phenomenon over fifty years
ago, yet much remains unknown about the micro-path of quantifier acquisition.
The classic experimental example for spreading is shown in Figure 1.
Figure 1. Example spreading item from the Dialect Sensitive Language Test
(Seymour Roeper & de Villiers, 2000).
Shown three girls riding bikes plus an empty bike, and asked, “Is every girl
riding a bike?” many children point to the empty bike, saying “no, not this
bike.” The child exhaustively distributes all girls to all bikes and all bikes to all
girls in a one-to-one fashion. Thus, both distributivity (one-to-one) and
exhaustivity (including all present entities) appear to be involved. It also
appears that the scope of the quantifier is not restricted to the NP in which it
occurs, but can apply either to the object NP or both NPs, as in “Is a girl riding

























































*

 University of Massachusetts Amherst. Contact Author: Thomas Roeper, Dept. of
Linguistics, 226 South College, Amherst MA 01003; roeper@linguist.umass.edu. Some
of the materials for the current experiment were assembled and piloted while Pearson
was a visiting researcher at the ESRC Centre for Research on Bilingualism in Bangor,
Wales.
every bike?” or “Is every girl riding every bike?”
In this paper, we present evidence that exhaustivity, not distributivity is at
the root of children’s quantifier spreading. The experimental protocol also
produces examples that demonstrate the syntactic operation of quantifier
“floating” (Bobalik, 1998) that Roeper, Strauss, and Pearson (2006) argued
gives rise to the spreading interpretation.
1. The Acquisition Path
Quantifier spreading is very persistent. The following data from the
Dialect Sensitive Language Test (DSLT, Seymour, Roeper, & de Villiers, 2000)
are consistent with the general finding. Figure 2 charts the results for the 333
typically developing European-American general American English speakers in
the nationwide DSLT field testing sample. They show that spreading was more
frequent than the target response through age 8 years. As shown in the figure,
the percentage of children who gave spreading responses (in the gray bars)
increased through age 7-8. Although such responses declined after age 8, they
did not disappear. Forty percent of the participants at ages 9 to 10 years and
28% at ages 11 to 12 years gave at least one answer analogous to “not that
bike.”
Figure 2. Spreading Answers by Age, Typically Developing General
American English Speakers (N= 333)
It is important to note that although there has been disproportionate attention to
every, the phenomenon of spreading has been found with other quantifiers, e.g.
some, most, and both (Roeper & de Villiers, 1991; Roeper & Mattei, 1974;
Stickney, 2006).
1.1. The acquisition challenge
To show its semantic, lexical, and syntactic sides, we enlarge on the
challenge facing the child in learning not to spread. We hint at pragmatic
factors involved, but they are beyond the scope of this paper.
Consider first the semantic contrast between collectivity and distributivity
which is illustrated in the difference in meaning between every and each. While
every allows distributivity, it does not require it like each does. For example,
when a waiter lifts a tray of glasses, the sentence in (1a) is true, while the
sentence in (1b) is not (Tunstall, 1998).
(1) a. The waiter is lifting every glass.
b. *The waiter is lifting each glass.
The sentence with each requires that each glass be lifted individually while
every allows the glasses to be lifted collectively.
We know from corpora and diary studies of early speech that children begin
early with collective readings, such as allgone, allgone milk, or allgone cookies
(Roeper et al., 2006). These are also exhaustive, a concept which has been
shown to be part of children’s very early repertoire (Strauss, 2006). With
respect to distributivity, 3- and 4-year-olds have given evidence of the cognitive
ability to distribute, for example, with plurals (Avrutin & Thornton, 1994), but
it is not clear how and when this distinction is lexically linked to quantifiers.
Thus, beyond the semantic concepts, there are lexical, and as we shall see,
syntactic elements to be acquired.
Children’s first uses of every occur in collective compounds like everybody
and everything (Roeper et al., 2006). Within the compound, a collective reading
may be required as in (2a), whereas in (2b) outside the compound, the collective
reading is impossible.
(2) a. Everybody surrounded the house.
b. *Every person surrounded the house.
In addition to the semantic distinction between collective and distributive,
several other distinctions are illustrated by the contrast of every and each, as in
examples (3) to (5):
(3) a, Every => Generic
Does every cow have one tail => every cow in the world
b. Each => Specific
Does each cow have one tail => presupposes a defined set
and only each allows a partitive:
(4) a. each of the boys
b. *every of the boys
Furthermore, each can float, but every cannot. By “floating” we mean that the
quantifier is not tied to one location in the sentence, but can occur elsewhere
(Bobaljik, 1998), as in (5a and 5b), or even be copied as in (5c), with no change
in meaning.
(5) a. All the children are here
b. The children are all here
c. All the children are all here
Moreover, floating applies generally to all quantifiers that can appear outside
the DP, but it is lexically specific. It does not apply to every in English (6b), but
it does apply to jeder [‘everyone’] in German (6c), indicating that the properties
are not universal, but must be learned within the context of specific languages:
(6) a. Every boy is here
b. *the boys are every here
c. Die Kinder singt jeder allein [‘the children sing every alone’]
Floating does not entirely constrain scope assignment, which remains
somewhat ambiguous. For adults, the floated element typically has scope over
the object, imposing 1-to-1 distributivity as can be seen in the contrast in (7).
(7) a. Each of the children has one angle on the view
b. The children have one angle each on the view
In (7a), several children could share one angle, but in (7b), there is 1-to-1
distribution of angles to children. The scope of the quantifier in the typical
every, or each, sentence is ambiguous for a child. For example, the child’s
quantifier often applies entirely to the second NP, or can be applied to both, as
we will see in the discursive examples below in 2.6.4.
These facts involving scope, plus the semantic diversity of the other
quantifiers that undergo spreading, like most and some, suggest that a semantic
approach alone will not capture all of the facts.
1.2. Semantic approaches
A prominent semantic approach, Weak Quantification, for example, has
been followed to explain the transfer of scope from the subject NP to the object
NP. It proceeds from the well-known semantic phenomenon in which weak
quantifiers (those that do not presuppose a fixed domain, such as one sense of
many) in subject position are allowed to apply to objects as in (8):
(8) Many Scandinavians won the Nobel Prize
=> many Nobel Prize winners are Scandinavian (Drozd, 2001)
This is contrasted with strong quantifiers, which obey conservativity for
adults, such that the Q applies only to the NP and requires the truth of the VP.
Hence Weak Quantifiers (many) involve a Context variable c (an unfixed
context) with the formula (in Smits, 2011):
(9) many3
(A)(B)= many3
A(A ∩ B) where |(A ∩ B)| > c · |B|
where A = set of Scandinavians, B = set of Nobel prizes and c = set of
contextually relevant Scandinavians. In other words, it is pragmatically
conditioned on how the speaker fixes the relevant context.
Smits (2011) directly investigated how children determine the relevant sets
and found that children ages 4 to 7 years allowed both strong and weak readings
but with no preference according to context for examples like (10):
(10) 5/20 parrots are wearing hats
a. Are many parrots are wearing hats?
b. Are many hat-wearers parrots?
The pragmatic context was changed with gesture and by varying the other
characters in the scene. Under the Strong reading, the answer to (10a) is “no”—
most parrots are not wearing hats. But if it is read “weakly” as (10b), then it is
true, and the relevant set is hat-wearers and not parrots. It is conceivable that the
child begins without the assumption of a fixed specific set and therefore the
weak reading is available, but it is not weak quantification per se that is the
explanation for spreading.
1.3. Syntactic approaches
In general, semantic accounts of spreading, such as Event Quantification
(Philip, 1995) and Weak Quantification (Drozd, 2001; Guerts, 2003), have
utilized a non-syntactic LF representation to capture the child’s meaning.
Looking more to syntax, Roeper and de Villiers (1991) proposed that the
quantifier behaves like an adverb which can move through a clause and appear
in many positions, although with subtle scope effects, as Cinque (1999) has
shown. This would be similar to a quantifier like only which can apply either to
NPs, as in (11a) or clauses (11b):
(11) John needs to go. Only he needs a snack.
a. [IP [only HE] needs a snack.
b. Only [IP he needs a snack]
If children first adjoin high, as proposed by Lebeaux (2000) among others, then
it is not unnatural that the child could project every as a sentence-level adverb as
well. The child must then re-analyze the quantifier as applying to the NP only
and as limited to NP scope.
Roeper, Strauss, and Pearson (2006) make a more refined syntactic
proposal based on syntactic studies of focus in Hungarian by Kang (1999),
where quantifiers can be lifted out of NPs to a higher focus node where the
quantifier c-commands an entire clause, as in the tree in Figure 3.
Figure 3. Quantifier as a higher Operator as argued for Hungarian (see
Kang,1999; Brody, 1990).
In English, every, like only, could attach inside the DP or higher up. We argue
that the child first attaches it high and then later learns to put it lower. In the
higher position if can function as an Operator which c-commands the second
NP. The first stage is identical to quantification in Hungarian. A syntactic
misanalysis that involves high attachment could allow floating, which in turn
has diverse semantic consequences, such as imposing distributivity on NP2 as in
example (7b).
Our general hypothesis is that every syntactic, semantic, and pragmatic
feature will have an impact on the acquisition path. Some may be linked and
acquired in concert---following the deeper notion of parametric links—and
careful experimental work should reveal what they are.
2. Experimental Explorations
2.1. Objectives
Separating distributivity and exhaustivity. Since experiments on
spreading like the one in Figure 1 entail both exhaustivity and distributivity, the
two concepts are not distinguishable. The experiments do not reveal whether
children seek to satisfy distributivity, or exhaustivity, or are targeting both
properties. In this experiment, we separated distributivity and exhaustivity in
spreading examples as follows. Three picture choices of vases and flowers as in
Figure 4 were presented (adapted from Brooks et al., 2001):
A
 B
 C




 

Figure 4. Stimuli for every and each
A and B both invite spreading with empty vases, but differ on distributivity.
C has no opportunity for spreading, and its distributivity is not individual-byindividual,
(although it can be construed as “partial distributivity” [Lima,
2010]). Thus, there were three conceptual options:
• Collective (NOT distributive)/ NOT exhaustive
• Distributive/ NOT exhaustive
• Exhaustive/ NOT 1-1 distributive
Crucially, no scene satisfied both exhaustivity and distributivity, (although that
condition was tested separately). Sentence prompts tested each as well to push
the child more toward distributivity (Tunstall, 1998) with a quantifier less fixed
within its NP that would permit floating.
Eliciting spreading in the children’s own words. We proposed an
experimental protocol that would allow children to describe the prompts at
greater length than is typical and so they could produce sentences spontaneously
that might show the ambiguous scope of the quantifiers in their grammars, in
particular, the movement of every out of the first NP in the surface structure.
2.2. Hypotheses/ predictions
In the current experiment, we predicted 1) that children would give evidence
of incomplete lexical learning, to the extent that each and every would not be
distinguished and would have the same semantic properties for them; 2) that
exhaustivity, the more general property would be more prominent for the
children than distributivity, whose subtle properties make it harder to learn; 3)
that children would adjoin the quantifier high where, as an Operator in the CP, it
could c-command the whole sentence, and therefore allow a “floated”
interpretation on the object noun; and 4) there would be no strong age effect, but
exhaustivity would diminish among the older children, and distributivity would
be more salient to older children than younger children.
2.3. Participants
Participants were 38 children, ages 5;4 to 9;4 in kindergarten through third
grade in a middle to lower middle-class school in western Massachusetts.
Forty native English-speaking adults (3/4 from the U.S.; 1/4 from the U.K. and
Canada) provided a baseline for response patterns to the prompts used.
Table 1. Participants by age
Age in years 5 6 7 8 9 20+
N = 2 10 12 10 4 40
(For the statistical analyses, the 5- and 6-year-olds were combined, as well as
the 8- and 9-year-olds, giving three age groups.)
2.4. Materials
The prompt for this analysis was embedded in a larger sequence of
questions testing aspects of quantifier interpretation in several everyday
contexts and in mathematics word problems: a 22-slide powerpoint for the
children, and a 39-item web survey for the adults1
. The actual display was from
Brooks et al. (2001), which depicts the alternatives shown above in Figure 4.
2.5. Procedures
Children were questioned individually in a quiet room in their school. They
were shown the display as in Figure 4 and asked which picture or pictures
matched the sentences, “Every flower is in a vase” and “Each flower is in a
vase,” in that order. They were told explicitly that they could point to zero, one,
two, or three pictures. If they pointed to just one picture, that was considered
their best response and they were then asked if any of the others could also “be
okay.” If they picked out more than one picture, they were asked which one
was best. Following the question, they were asked “Why?” or “Why not?”
Responses were written down by a second examiner and video-recorded for
later confirmation.
The experiment explicitly pursues preferences and explanations as
important indicators of interpretations.
2.6. Results
2.6.1. Adult baseline
Table 2. Percentage of people who gave responses of each type.
Every (flower) Each (flower)
All ok 94% 17%

























































1 The web survey is available at http://www.kwiksurveys.com/onlinesurvey.php?surveyID=OIHKG_7f21b1b7
(accessed 1/25/2011)

Prefers A 21% --
Prefers B 36% 90%
Prefers C 8% --
Only B -- 20%
Rejects B -- 0
2.6.2. Children’s responses
Figures 5 and 6 show the percent of children with each response, compared
to the adult baseline.
Figure 5. Children’s Every (with adult as reference)
Figure 6. Children’s Each (with adult as reference)
Not shown in the chart, children chose the “best” responses for each relatively
equally: A, B, and C, 24%, 26%, and 32%, respectively.
Summary of differences. Thus, the strongest adult-child differences for
every stemmed from the level of acceptance of all pictures: almost all adults
accepted all and almost no children did. C was the clear favorite for the
children, but it was not a preference and was even slightly dispreferred by the
adults. For each, adults clearly preferred B, whereas over 60% of children
specifically rejected B for it. Although many adults found A and C acceptable
for each, no adult chose either of them as best; for children both A and C were
strong choices, but not disproportionately so.
Adults showed clearly different preferences for every and each. By
contrast, 15 (40%) of the children treated each and every the same. When asked
“why?” after the each questions, several said “I just told you,” referring to their
answer for every.
2.6.3. Age patterns in children’s responses
There was no age effect for target interpretations of either quantifier (χ2
[2,
N = 38] = 2.1, p = .35 for both) but exhaustive answers were significantly less
frequent among the older children, (χ2
[2, N = 37] = 6.1, p = .05). Six-year-olds
were not different from 7’s (p=0.14) but 7- and 8-year-olds were significantly
different (p= 0.02). Ten children (26%) showed at least a rudimentary
sensitivity to distributivity, 17% of 6- and 7-year-olds and 43% of 8-year-olds,
but the age difference did not reach significance, χ2
(2, N = 38) = 3.1, p = .21.
2.6.4. Discursive responses
2.6.4.1. Responses showing non-adult interpretations: quantifier floating
and exhaustivity
The discursive responses to the “why” questions gave further, more direct
evidence of the children’s interpretations of the quantifiers. The most striking
result was that the questions elicited spontaneous productions of the floated
quantifier (applying to a second NP) and exhaustivity as in the following
examples.
Copying the quantifier in the second NP.
* “looks like each flower is in each vase” (age 8;0 in years and months)
Floating every and/or each to NP2 (vases).
* “[C], it’s the only one with flowers in every vase.” (9;4)
* “Not B, there’s just one in each [vase]” (6;1)
* (for every flower in a vase), “could be 1 flower in each vase” (9)
* “one [flower] in each [vase]” (8;1)
Children’s answers emphasized exhaustivity (or the lack thereof).
* “all vases are full” (8)
* “flowers in all [vases]” (7;9)
* “could be C, if there was just one flower in each, in all the vases” (7;1)
* “not A or B, no flowers in those two vases” (6;2) (7;8)
* “No, they don’t have flowers in all vases.” (9)
* “[C, It’s] the only one where vases are filled with flowers” (8;0)
* “these two vases don’t have flowers” (6;2) (6;5) (7;8)
* “no, the others have empty vases” (6;11) (7;4) (8;4)
* “no, because some of the vases are empty” (7;9)
* “not A, [it has] only one filled vase” (8;2)
2.6.4.2. Responses showing adult interpretations
Focus on flowers. Four children (two 5-year-olds and two 8-year-olds)
gave adult interpretations for every to the extent that the justifications for their
answers focused on the first NP, flowers, not the vases.
* “All the flowers have vases that they’re in” (5;4) or
* “There are empty vases, but where there are flowers, they are in a vase” (8;1).
Articulating sensitivity to the configuration of flowers (distributivity) for
each. Three of the four “every-knowers” also gave somewhat adult
interpretations for each in their rejection of A for each. Their explanations
showed an awareness of its distributive requirement, such as saying that each
couldn’t have “all in one vase” (6;5). Another child (8;1) explained that “[B is
better] because it’s spread out.” The fourth every-knower also responded that
each is better “when it has [only] one flower,” but then described the picture
inaccurately. Six other children (ages 6;5; to 9;0) preferred B and also
articulated an awareness of whether the flowers were distributed or not.
However, distributivity was not always associated with each. Two 9-yearolds
showed they were paying attention to the distributive configuration, but
they did not associate it with each. That is, they preferred A for each because in
B and C “[the flowers] weren’t all in the same vase.” (which to them would
have been preferable). One 7-year-old appealed to configuration to explain his
choice of C for every: ”only C, [because] all in same is wrong and 1 in 1 is
wrong.”
3. Discussion
A primary finding in this analysis is that children—unlike the adults---
preferred exhaustivity as a defining feature for both every and each, as we
predicted. Thus, they appear to identify this property first as a lexical property
of these quantifiers. The distributive property, though recognized elsewhere, is
not yet projected as a lexical feature. This is not surprising. Many semantic
notions appear in syntactic environments before they are represented lexically
or on all relevant lexical items. For example, notions of time may be found in
tense phrases, nouns (such as noon), prepositions and conjunctions (before,
after) and adjectives (late) without a necessary order of triggering (Roeper,
2007).
3.1. Acquisition theory
3.1.1. Proposed developmental sequence
These results are consistent with the syntactic account provided by Roeper
et al. (2006) where a quantifier is initially projected as an Operator and is
overgeneralized, an explanation that converges with many other traditional
observations in acquisition. Syntactic overgeneralizations may produce
negative concord, “I don’t want none” and tense concord, “did lifted” or “had
came.” Lexical plurals (e.g. feet) overgeneralize to forms like “feetses.”
Finally, recent results on wh-forms (Schulz, 2010) show that children who do
pairing for double wh- (who ate what) immediately extend it to triple wh- as in
who gave what to whom (Schulz, 2010). This, too, suggests that the pairing
property is linked to an Operator that will control every wh-word in its ccommand
domain.
3.1.2. How do children eliminate quantifier spreading?
If our representation is correct, it immediately raises the question of how
children advance to the realization that every must be a Determiner within the
Determiner Phrase with scope only over its NP. The option is, of course,
provided by UG, but we need triggering evidence that will block the higher
Operator projection. One possibility is that the child grasps the impact of two
quantifiers which is immediately incompatible with spreading, as in (12):
(12) Every dog has some bones
The prediction is that if there is an extra bone, then the child will not say “not
this bone.” If this perspective is on the right track, the child will simultaneously
pragmatically grasp a situation, semantically grasp quantifier interaction, and
syntactically block an Operator. Thus, every step is an “interface” action. But
this experiment remains to be done.
3.2. General conclusion
What emerges from our discussion is how intricate the interface for
quantification is, engaging lexical, semantic, and syntactic phenomena, even
before considering the extensive presence of implicatures in the comprehension
of quantification. The acquisition path must reflect all of these threads. We
need, ultimately, to imagine every ingredient as a part of each stage, and then
see how evidence will push the child forward.
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