Quantifying kids prefer intersecting sets – Draft version, July 2014 (subm. to Language Acquisition)
Quantifying kids prefer intersecting sets
Hubert Haider & Christina Schörghofer-Essl & Karin Seethaler
Dept. of Linguistics & Centre for Cognitive Neuroscience, Univ. Salzburg1
Abstract
Children between 4 and 6 years of age are known to encounter problems with universal quantifiers. In this paper we present the experimental answer to a key question that has not been
asked in any previous work. We confronted the standardly used universally quantified sentences with truth-conditionally equivalent sentences with a negated existential quantifier.
Universal quantifiers are strong quantifier; existential quantifiers are weak ones. Given that
the strong vs. weak difference is likely to be the essential factor, we predicted a significantly
higher success rate for the stimuli with a weak quantifier in comparison with their truthconditional equivalents with a strong quantifier. The results confirm this prediction and provide immediate support for the hypothesis that the distinction between weak and strong quantifiers is a major causal factor for the phenomena under discussion.
1. Introduction
The phenomenon is at issue since Inhelder & Piaget (1958). A generally accepted explanation
is still wanting. The puzzle is this. In picture selection tasks with stimuli that involve universally quantified statements, the reactions of children of about 4-6 years of age differ from older children, whose reactions are adult-like. For utterances with a universally quantified subject, these children seem to prefer exhaustive pairing answers. They construe a sentence like
„Every boy rides a bike‟ as if it also entailed that every bike is ridden by a boy. In particular,
they frequently tend to reject a statement if the picture shows one bike more than boys. In
general, the test items have the logical form (1a). The kids have to decide whether a given
statement of the form (1a) is correct relative to one of the depicted situations (1b) or (1c):
(1) a. For every A, A is in a particular relation to B
b. Situation 1: several A, all in relation to B; one unrelated B.
c. Situation 2: each B in relation to A; one A unrelated to B.
One of the items used in our experiments is figure (1) and it is representative of (1b). When
asked “Every car is parked in a garage. Correct?” a relatively high percentage of children
would wrongly say “No” and tell the experimenter that there is an empty garage. A test case
for (1c) is figure (2). When confronted with this picture and the same question, a certain percentage of children will wrongly say “Yes”, as if the question had been “In every garage,
there is a car. Correct?” The percentage of mistakes triggered by (1c) is typically lower than
in the case of (1b).
1
This paper presents findings of two related studies. The second author studied the effect of substituting a universal quantifier by a negated existential quantifier. The third author investigated the effects of the linguistic rendering of universal quantification by „alle’ (allpl.) vs. „jeder’ (eachsg.). The division of labor was as follows: The
experiments were carried out and analyzed by the 2nd and 3rd author, respectively. The first author was the thesis
supervisor, proposed the research strategy, and put this paper in writing.
Thanks go to Lilli Schörghofer-Essl for providing the graphic stimuli (shared by the experiments), to Ms. Christina Brandauer (Kindergarten Kuchl) and Ms. Irene Mellmer (Pfarrkindergarten Herrnau) for their kind authorization of the testings.
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Quantifying kids prefer intersecting sets – Draft version, July 2014 (subm. to Language Acquisition)
The very same children would not have problems with existentially quantified utterances like
„Some cars are in a garage” or „In some garages, there is parked a car.‟2
Figure 1:
Figure 2:
The explanations for the failures proposed in the literature take one of three alternative circumstances to be the major causal factor, namely (i) the pragmatics of testing (pragmatic felicity conditions, distraction by the test situation, with kids being bothered by the experimental set-up), (ii) syntactic deficiencies in handling NP structures with quantifiers, or (iii)
semantic deficiencies in composing the meaning of universally quantified sentences.
As for pragmatics, there are claims that insist on the children‟s full semantic competence and
locate the source of errors in the test situation. “We conclude that young children have full
competence with universal quantification” (Crain et als. 1996:83). According to their view,
children do not lack knowledge of any aspect of quantification. The kids‟ incorrect responses
are declared as errors due to flaws in the experimental design. The main objection is that the
test situation presents utterances in a context that does not offer a chance of „plausible dissent’
or „plausible denial’ (Freeman & Stedmon 1986). “In the contexts for yes/no questions, felicitous usage is said to dictate that both the assertion and the negation of a target sentence
should be under consideration.” (Crain et al. 1996: 102) If this possibility is not given, children are believed to be puzzled as to why the experimenter wants to know if a statement is
true.
In several experiments, Brooks and colleagues have specifically provided conditions for plausible dissent and shown spreading effects with them (Brooks & Braine 1996; Brooks et. al.,
2001). Sugisaki & Isobe (2001) explicitly tested against „plausible dissent‟ as the crucial factor and demonstrate that children expected to fail show adult-like performance even in situations in which the condition of plausible dissent is not satisfied.
Geurts (2003) objects, too, and emphasizes that the main point is missed: “Their diagnosis
still leaves open some of the most intriguing issues raised by the empirical facts. Why is it, for
example, that older children and adults aren’t bothered by an experimental set-up that, by
Crain et al.’s lights, is hopelessly flawed? Crain et al. have very little to say about this.”
In the experiments to be reported below, the pragmatic felicity was procured as follows. The
experimenter would tell the children that her set of pictures had previously gotten messed up.
Each child is explicitly encouraged to assist the experimenter in deciding which picture
should go to the „is wrong’ pile, and which one to the „is correct’ pile.
2
Smith (1979, 1980) confirms a clear contrast between „some‟ and „all‟. Stimuli with „some’ nearly always
prompt adult-like responses.
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As for a syntactic causality, Roeper & de Villiers (1993) conjectured a syntactic deficiency
behind the high failure rate. They argue the children fail to identify the quantifier as part of
the noun phrase. Rather it is regarded syntactically like a sentential adverb with scope over
the whole sentence. This is seen as the source of the preference of exhaustive-pairing readings.
This cannot be the right explanation for a simple reason. In German (and in all other Germanic languages, except English), the first position in a declarative clause is reserved for a single
syntactic unit. Hence, „jedes Auto’ (every car) is easily identified as a single constituent in a
clause like „[Jedes Auto] stehtV-fin. in einer Garage‟ (every car sits in a garage). Germanspeaking children master the verb-second constraint at latest by the age of three but their failure rates with universally quantified subjects are parallel to English-speaking children at the
age of four and later. Hence, it can be definitively excluded that they misidentify „alle Autos‟
(all cars) by something like „[[always] [[cars]…]]‟
A semantic deficiency was the initial guess, with the very same suspected source. According
to Philip (1995:53), children tend to construe universal quantifiers like adverbials, quantifying
over events rather than individuals. For whatever reason, children tend to assign to a sentence
with a universally quantified subject the logical form which older children and adults assign
to a sentence containing an adverb of quantification. This characterization introduces a central
premise without independent evidence. Geurts (2003) emphasizes that “in accounts as Philip’s, a child has to negotiate an entirely different style of quantification in order to master the
intricacies of nominal quantifiers like ‘every’ and ‘all’, and quite apart from the question why
this detour has to be taken in the first place, it is not so easy to see how the child achieves it.”
Geurts (2003) elaborates on Drozd (2001), who suspects that children are unable to handle the
appropriate distinctions between weak and strong quantifiers. Geurts locates the complications in the mapping of syntactic form to semantic representation. The children are well aware
of the meaning of „all’, „each’, and „every’ in isolation. They fail when it comes to produce
the appropriate compositional semantics of a universally quantified subject (that is, a universal quantifier and its restrictor) and a predicate.
“In my view, the child’s problem (if there is one) lies in the mapping from grammatical form
to semantic representation, which is more complicated for universal quantifiers than it is for
others. Children have a certain tendency to interpret all quantifiers as if they were weak, because it is easier to do so.” (Geurts 2003:209).
Weak and strong quantifiers differ in an essential aspect. A weak quantifier amounts to an
intersective relation, whereas a strong quantifier amounts to an inclusion relation. In logical
terms, this is the difference between the logical form of a conditional (2a) and that of a conjunction (2b).
(2) a. [All A are B]
 For every x [if (x is an A) then (x is a B)]
b. [Some A are B]  There are some x such that [(x is an A) and (x is a B)]
In set-theoretic terms, the restrictor set of a universally quantified noun phrase (3a) is a proper
subset of the predicate set (viz. set inclusion). On the other hand, in weakly quantified noun
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phrase denotations (3b), the restrictor set intersects with (the extension of) the predicate (viz.
set intersection) in their extensive interpretation.
(3) a. „All A are B‟ equals:
(B  A) = A, that is, A  B,
b. „Some A are B‟ equals: (A  B)  A (if A is not included in B)
For children, the canonical relation for combining the denotation set of the subject with that
of the predicate is set intersection. For arriving at the meaning of „Some cars in this box are
yellow‟ a child must intersect the set of cars in the box with the set of yellow objects and decide that the intersection is not empty. Children are good at these tasks already at an early
age.3 Problems arise, however, when there is no proper intersection (because of inclusion):
In order to decide on „Are all cars in this box yellow?’ the set of cars must be a subset of the
set of yellow objects. The special case is the situation in which all the cars contained in the
box are yellow and there are no other yellow items in the box. In this case the intersection of
the set of cars with the set of yellow items is the same set. This is the limiting case of intersecting two sets such that no set is a proper subset of the other.
It has been our guiding idea that kids try to stick to genuine intersection relations as long as
possible. Note that this triggers the preference of exhaustively paired answers. If kids try the
same intersective strategy that works for „Put every yellow car into this box’ on „Every cat sits
in a tree‟ they will end up at an exhaustively paired reading.4 They expect a canonical intersection5, that is, an intersection set that is a subset of each of the intersecting sets. The limiting
case for proper intersection is two matching sets. This yields the paired reading.6 No set is a
proper subset of the other but each intersecting set is an improper subset (viz. identical) of the
intersection set. Since „all cats’ must be in the set, the intersection must not leave a cat alone,
and since the cats must not be a subset, there must not be an empty tree.
However, universal quantification canonically requires an inclusion relation with a (proper)
subset. For kids, a subset relation or inclusion is a non-canonical intersection and hence they
stick to their canonical case of intersection for some time. This is what kids seem to have in
mind if they say “No!” when they have to decide on a picture with four trees, and three cats,
each on one of the trees, as answer to the question “Is every cat sitting on a tree?”
Let us recapitulate. Mistakes triggered by the situations (1b) when confronted with a figure
like figure (1) result from the attempt to stick to an intersective interpretation instead of recognizing the inclusion relation. This invites the supposition of a bijective relation.
3
This shows in play situations when kids correctly handle the appropriate items when asked “Put (all) the yellow
cars into this box”. They would not object – as in the case of universally quantified noun phrases in picture
judgment tasks – that there are other yellow items that are not cars.
4
“Perfect one-to-one correspondence appears to be the canonical representation for each and its counterparts on
other languages, including the English quantifier every.” (Brooks et als. 2001:337)
5
A „canonical‟ intersection is a proper intersection, that is, an intersection for which neither set is included in the
other. In other words, (A  B)  A, and (A  B)  B.
6
In terms of extensional semantics, Every car is in a garage is true in a given situation if the extension of be
in a garage, which is the set of all discourse participants with the property of being in a garage in this situation,
contains the entire set of cars of this situation as a subset.
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The mistakes of the type (1c) with figure (2) are transition effects on the way from proper
intersection to inclusion.7 Intersection is commutative, inclusion is not. In the corresponding
predicate logic representation, conjunction (viz. “”) as triggered by the logical from of weak
quantifiers is commutative; implication (viz. “”), as triggered by the logical form of universal quantifiers, is not.
If a child applies intersection, it correctly equates “Some pupils are girls” with “Some girls
are pupils”, but when it is about to recognize inclusion, it simultaneously must inhibit commutativity. Hence, the mistakes of the type (1c) may turn out as mistakes on the way from
intersection to inclusion. In this situation, „Every car is in a garage‟ is wrongly equated with
“In every garage there is a car”. Depending on the stimulus material, other factors may additionally facilitate this. Freeman et als. (1982) found that relative salience tends to bias children‟s responses one way or the other, a finding that was confirmed in experiments by Drozd
and van Loosbroek (1999). The clause final position is the position for salient elements because it is the position for the nuclear accent. So, „in a garage‟, gets interpreted as salient by
virtue of being accented.
And here comes the leading idea of the paper in support of weak versus strong quantification
as the core effect: There is a crucial issue that has been neglected in all previous studies. It is
an obvious issue in the context of quantificational semantics and it helps clarifying the quest
for the true causality of the observed behavior of children at the given age. We tested this issue.
The issue is this. It is a logical truth that every universally quantified expression (= strong
quantifier) can be turned into a truth-conditionally equivalent existentially quantified expression (= weak quantifier). Universal and existential quantifiers can be inter-defined by means
of negation (4).
(4) [x (A(x)  B(x))]  [ x (A(x)  B(x))]
A pair of sentences like those in (5a) is a linguistic instantiation of the equivalence law in (4).
Corresponding stimulus sentences used in the experiment in combination with appropriate
pictures that had to be judged are (5b,c).
(5) a. [Every girl has a cat] = [No girl is without a cat]
b. Jedes Mädchen hat eine Katze. Ist das richtig?
„every girl has a cat. Is this correct?‟
c. Kein Mädchen ist ohne eine Katze. Ist das richtig?
‚no girl is without a cat. Is this correct?‟
The proponents of pragmatic deficits that allegedly show when testing quantification, as discussed above, are bound to predict that (5b) and (5c) will pose the very same pragmatic difficulties. The stimuli types (5b) and (5c) are semantically equivalent, that is, they are verbalizations of truth-conditionally equivalent logical forms. Whatever context „defect‟ in terms of
„(im)plausible dissent‟ or „(im)plausible denial‟ holds for (5b) will equally hold for (5c).
7
This correlates with the lower failure rate of this type. In experiment 1 (see result section, below), this mistake
virtually did not show up (3,6% mistakes). In Experiment 2, the failure frequency of this type (viz. 1c) was 55%
lower than the frequency of the type triggered by situation (1b).
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On the other hand, if Drozd (2001) and Geurts (2003) are on the right track and strong versus
weak quantification is indeed at the root of the problem, the prediction is that the pattern (5c)
will be mastered more easily than the pattern (5b). (5b) involves a strong quantifier (inclusion
relation) while the equivalent version (5c) involves a weak quantifier (intersection relation),
that is, the negated existential quantifier (viz. the intersection set is the empty set).
In sum, the leading question of the paper is this: How do kids perform on structures (5c) in
comparison with (5b)? If it turns out that patterns like (5c) are significantly easier despite the
occurrence of two instances of negation, then this is strong support for the semantic account
along the lines proposed by Drozd (2001) and Geurts (2003).
2. Hypotheses put to test
The two hypotheses that have been tested are these. First, replacing universal quantification
by truth-conditionally equivalent negated existential quantification is expected to have a significant positive effect on the mastering of the task, since this amounts to replacing an inclusion relation by an intersection relation, and a strong quantifier by a weak one.
Second, the linguistic form (singular vs. plural) of the quantifier expression might matter. According to Drozd and Philip (1993) and Philip (1995), English-speaking children are indifferent to the distinction between „all‟ and „every‟. We wanted to see whether this can be replicated in German, given that number agreement and case is more explicitly coded in German than
in English. The item „jeder„ (every) as in „jeder Hund‟ (everyNom-sg. dog) could not be mistaken as a kind of adverbial quantifier – as presumed for English by Roeper & de Villiers (1993).
„Jeder’ is clearly inflected (case, number) as determiner of a noun phrase. It cannot be misunderstood as an adverbial element.
Taken all together, it is not implausible that the following variants (6a,b) pose different degrees of difficulties to children acquiring the construal of quantifier expressions. In one case,
the quantified expression is singular (6a), in the other case it is plural (6b). Plural may facilitate the construal of set relations.
(6) a. Jeder Bub reitet auf einem Elefanten
everyNom-sg. boyNom-sg. ridessg. on an elephant
b. Alle Buben reiten auf einem Elefanten
allpl. boyspl. ridepl. on an elephant
Since singular vs. plural receives a distinct semantic interpretation, this nourished the following expectations for grading these structures in terms of their processing difficulty. The correct processing of a universal quantifier may be facilitated if it is part of a plural NP since this
directly invites the construal of a plural restrictor set. The noun „boys‟ in (6b) denotes a set of
boys, and hence „all boys‟ denotes the entire set of boys. „Each‟ (in 6b) on the other hand is
part of an NP marked for singular as can be read off from verbal agreement. So, the construal
of the restrictor set might be hampered by the fact that the NP that contains the quantifier is
singular. So, we were prepared to see an increase in difficulty:
(7) Hypothesis of a singular vs. plural effect: The semantic processing of a universally
quantified plural NP differs from the semantic processing of a universally quantified singular NP.
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As for the other hypothesis, the assumption we start from has been introduced already in the
preceding section: The core of the children‟s difficulties with universal quantification is supposed to be connected with the computation of inclusion relations (as an improper intersection
of sets) since this amounts to a semantic representation with an „if-then’ relation (8a). On the
other hand, children easily master intersection relations. Logically, this amounts to a conjunction relation (8b) rather than an implication relation (8a). Eventually, there is the well-known
equivalence relation between universal and existential quantification (8c,d).
(8) a. All A are B  For every x [if (x is an A) then (x is a B)]
b. Some A are B  There are some x such that [(x is an A) and (x is a B)]
c. [x (A(x)  B(x))]  [ x (A(x)  B(x))]
d. No A is not B  There is no x such that [x is A and x is not B]
Our hypothesis invites the expectation that children react differently to the two truthconditionally equivalent linguistic renderings of the very same situation (8c). The formulation
in terms of a negated existential quantification is expected to be mastered more easily than the
logically equivalent version in terms of universal quantification. The version (8d) can be
computed by a conjunction relation on set intersection whereas a universally quantified utterance requires computing an inclusion relation that involves a conditional form (8a). The minimal pairs that we tested consisted of stimuli like (9a) and (9b):
(9) a. Every snowman has a broom
= [x (S(x)  have-B (x))]
b. No snowman is without a broom = [x (S(x)  have-B(x))]
Judging the overall complexity of (9a,b) we note that (9a) and (9b) involve two independent
complicating factors. Mastering (9a) presupposes the mastering of a conditional form as the
semantic representation of an inclusion relation. The mastering of (9b) should be easier since
the conjunction in the logical form triggers an intersection relation which is the canonical relation that kids apply for combining a subject with a restrictor and the predicate. On the other
hand, (9b) also necessitates the simultaneous mastering of two occurrences of negation. One
negation, the negation of the existential, is explicitly coded by „no‟ while the second negation
of (3d) is provided by the preposition „without‟. “Without x” means “not with x”.
If the ease of processing an intersection, which amounts to a conjunction relation (9b), overrides the additional complexity incurred by the two instances of negations, children will perform significantly better on stimuli of the type (9b) than on the universally quantified type
(9a). Taken together, this would strongly support Geurts‟ (2001) and Drozd‟s position. What
really matters would be strong vs. weak quantification.
A basic developmental step in the children‟s competence at the crucial age would be the ability to handle universally quantified subjects by constructing the proper semantic forms and
their denotations. This requires enlarging the tool kit by set inclusion (with a conditional in
the logical form) in addition to the favorite combinatorial strategy of set intersection (based
on conjunction in the logical form). Therefore, the hypothesis to be tested was this:
(8) Hypothesis of a weak vs. strong quantifier effect: Stimuli whose linguistic form represents a strong quantifier provoke more deviances than truth-conditionally equivalent
stimuli whose linguistic form represents a weak quantifier.
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3. Method and materials
The experiments were carried out in a kindergarten in Salzburg and one in the neighboring
town Kuchl. Each child was tested individually by a female experimenter in a quiet room.
Testing a child took 10-15 minutes. During the testing, the child received no contingent feedback from the experimenter. All children were natively speaking German, with no indications
of language impairments or neuro-cognitive deficits.
Number of participants and age distribution:
Experiment 1: N = 21 (of 25);8 mean age 5;06 (youngest participant: 4;10; oldest one 7;04)
Experiment 2: N = 15 (of 16);9 mean age 4;09 (youngest participant: 4;04; oldest one 5;05)
Procedure:
In order to create a pragmatically natural situation, each child was first told by the female experimenter that her set of pictures had gotten in disorder and that she asks the child for assistance when putting the pictures in order again, as correct or incorrect. Next, the experimenter
would read the respective stimulus sentence, show a picture and ask the child “Stimmt das?”
(Is this correct?). The stimuli were presented in a pseudo-randomized order.
Stimuli: The two experiments shared professionally designed graphic stimuli that served as
the reference pictures for judging whether the respective verbal stimulus is true or false relative to the depicted situation shown to the child.
Experiment 1:
12 pairs of sentence-plus-picture; 4 distractor pairs served as control for testing the understanding of the number (three or four) of objects or protagonists on the pictures. 8 pairs tested
the situation (1b) and (1c) respectively. 4 pairs contained „jede/r/s (eachNom-/f/m/n), 4 pairs „alle‟ (all) as the quantifier element.
Experiment 2:
16 pictures (8 for situation (1b), 8 for situation (1c)). Each picture was employed in random
order with one of three stimulus sentence types, that is, i. with a universal quantifier (9a,c); ii.
with a negated existential plus „without‟ (9b); iii. with a negated existential plus a lexically
converse predicate (9d). Here are examples from the stimulus set:
(9) a. univ. Q:
b. neg. exist. Q + „without‟:
c. univ. Q:
d. neg. exist. Q + converse:
Jeder Hund hat einen Knochen
Kein Hund ist ohne Knochen
Jeder Vogel sitzt auf einem Baum
Kein Vogel ist in der Luft
(every dog has a bone)
(no dog is without a bone)
(every bird sits in a tree)
(no bird is in the air)
The set (9d) served both as a control for the understanding of negated existential quantification as well as for pragmatical adequacy since (9d) is the polar opposite of „Every bird is in
the air‟. If the latter suffered from a defect w.r.t. „plausible denial‟, (9d) would suffer, too.
4. Results
Both experiments produced clear-cut results. Experiment 1 shows that the manipulation of the
quantifying element (all vs. each in German) had no detectable effect on the distribution of
8
9
The data of four children had to be excluded from the study because they did not finish testing.
The data of one child had to be excluded from the study because the child did not finish testing.
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correct or incorrect answers. The results of experiment 2 shows a strongly positive effect of
the semantic difference between a strong and a weak quantifier, that is, the difference between
proper set inclusion and proper set intersection.
Experiment 1:
The aggregated answers to stimuli testing situations (1b) numbered 252 (=21x12). The percentage of deviant judgments is 47,23%. This is within the bandwidth of previous studies. The
percentage of correct answers is 52,77%. The mean number of correct answers was 6.34, with
a standard deviation of 4.23. 58,82% of all deviant responses come from the subgroup of children who are younger than 5;06, which is the mean age of the group.
The stimuli for type-(1c) situations triggered a very low number of deviant answers: 3,57%.
This is a percentage that comes close to nearly perfect mastering and differs from the typical
figures found in previous studies for situations of type (1c).
As for the 77 deviant responses to situation-(1b) stimuli, they are precisely halved by the two
factors that were manipulated. 38 were triggered by „alle X‟ (all x) and 39 by „jeder X‟ (every
x); see table (1):
Table 1: Deviant answers per type for situation-(1b) stimuli
Sentence type
‚Alle X sind Y„
‚Jedes X ist Y„
Type (1b)
38 (of 84)
39 (of 84)
%
45,24
46,43
mean
1,80
1,86
stand. dev.
1,54
1,19
‚all X are Y„
‚each X is Y„
The low number of deviant responses to situation-(1c) stimuli, namely 9 out of 252, makes a
finer-grained analysis dispensable.
Experiment 2:
The experiment tested three conditions, namely (i) the responses to stimuli with universally
quantified subjects, (ii) with negated existentially quantified subjects plus the preposition
„without‟, and (iii) with negated existentially quantified subjects plus lexical converse predicates of predicates in (i). The raw data are presented in table 2 (appendix).
Condition (i) produced the expected deviances along the lines of previous findings in the literature.
Situation (1b): 23,3% correct; Possible max. score: 8. Mean 1,87. Median 1,0. Standard
dev.: 1,41.
Situation (1b): 57,5% correct; Possible max. score: 8. Mean 4,60. Median 5,0. Standard
dev.: 2,10.
Condition (ii), the crucial condition for our hypothesis, yielded the following results:
Situation (1b): 47,7% correct; Possible max. score: 4. Mean 1,87. Median 2,0. Standard
dev.: 1,46.
Situation (1b): 70,0% correct; Possible max. score: 4. Mean 2,80. Median 3,0. Standard
dev.: 0,94.
Condition (iii), yielded the following results:
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Situation (1b): 96,7% correct; Possible max. score: 4. Mean 3,87. Median 4,0. Standard
dev.: 0,35.
Situation (1b): 93,3% correct; Possible max. score: 4. Mean 3,73. Median 4,0. Standard
dev.: 0,46
5. Discussion
Experiment 1 amounts to a replication of what Drozd and Philip (1993) and Philip (1995)
found for English. English-speaking children are indifferent to the distinction between „all’
and „every’ and so are the German-speaking children we tested, for the distinction between
„alle’ and „jede/r/s’. The grammatical number (singular vs. plural) of the quantified phrase or
the grammatical number of the quantifying element has no influence on the success rate.
Experiment 2 was the crucial experiment. The success rate for condition (ii), viz. the stimuli
with a negated existential plus „without‟, is twice as high as the rate for the stimuli with universal quantification, viz. condition (i). This amounts to a strong confirmation of the hypothesis that the crucial causal factor reflected in the children‟s reaction patterns is the handling of
weak vs. strong quantifiers. The quasi-control condition (iii) turned out as nearly perfectly
mastered.
We feel entitled to conclude that the factor strong vs. weak quantifier (with the concomitant
relations in an appropriate semantic model, namely set inclusion vs. set intersection) is a decisive factor for the failures with stimuli in situations of the type (1b).
Children at the given age span have to find out that universal quantification necessitates the
contemplation of set inclusion rather than set intersection. The data suggest that it takes some
time for children to correctly differentiate between the preferred mode of set intersection in
the building of semantic interpretations and the required mode of set inclusion for the restrictors of universal quantification.
As for the extremely low frequency of situation-(1c) failures in experiment 1, this seems to be
at least partially promoted by the fact that the sample of test subjects had a wider age distribution from 4;10 to 7;04, with the mean age of 5;06. On the other hand, the total number of failures of this type produced by participants between age 4.10 and the mean age of 5.06 was
only 6 (out of the total 9). In experiment 2, the mean age was 4;09 (4;04 - 5;05). Their success
rate for situations of the type (1c) was 42,5% and hence within the frequency band known
from other studies.
6. Summary
The factor weak vs. strong quantifier has a strong effect on the success rate of children of
about 4-5 years of age when confronted with decision tasks. The (widely observed low) success rate for universal quantifiers improves by nearly 100% when the stimulus is replaced by a
semantically equivalent stimulus with a weak quantifier.
The factor singular vs plural quantifier or quantified phrase has not effect on the success rate.
Arguably, the causal factor for the failures is the overextension of the strategy of proper set
intersection into contexts that require proper set inclusion.
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Quantifying kids prefer intersecting sets – Draft version, July 2014 (subm. to Language Acquisition)
References
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quantifiers „all‟ and „each‟? Cognition 60(3). 235-268.
Brooks, Patricia. J., Braine, Martin D.S., Jia, Gisela & Dias, Maria da Graca. 2001. Early representations for all, each, and their counterparts in Mandarin Chinese and Portuguese. In
Melissa Bowerman & S. C. Levinson (eds.), Language acquisition and conceptual development, 316-339. Cambridge: Cambridge University Press.
Crain, Stephen, Rosalind Thornton, Carole Boster, Laura Conway, Diane Lillo-Martin &
Elaine Woodams. (1996). Quantification without qualification. Language Acquisition 5.
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den Universalquantoren „alle“ und „jeder“. Salzburg: Univ. of Salzburg MSc.Thesis.
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Sugisaki, Koji and Miwa Isobe. 2001. Quantification without qualification without plausible
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Quantifying kids prefer intersecting sets – Draft version, July 2014 (subm. to Language Acquisition)
Appendix 1: Table 2 (raw data of experiment 2).
Correct x
Child 1
Child 2
Child 3
Child 4
Child 5
Child 6
Child 7
Child 8
Child 9
Child 10
Child 11
Child 12
Child 13
Child 14
Child 15
mean
median
Sd
max score 8
Abs.
% corr
Type (1b)
4
0
1
4
3
0
2
3
1
1
1
1
4
1
2
1,87
1,00
1,41
Type (1c)
5
5
6
0
2
3
6
5
5
8
5
7
2
4
6
4,60
5,00
2,10
Correct x
Child 1
Child 2
Child 3
Child 4
Child 5
Child 6
Child 7
Child 8
Child 9
Child 10
Child 11
Child 12
Child 13
Child 14
Child 15
Type (1b)
4
0
3
0
3
2
4
2
1
2
0
1
4
1
1
1,87
2,00
1,46
Type (1c)
4
3
2
4
2
2
1
2
3
4
3
4
3
3
2
2,80
3,00
0,94
28
46,67
42
70,00
max score 4
28
23,33
69
57,50
% corr
Corr. lex. conv.
Child 1
Child 2
Child 3
Child 4
Child 5
Child 6
Child 7
Child 8
Child 9
Child 10
Child 11
Child 12
Child 13
Child 14
Child 15
Type (1b)
Type (1c)
4
4
4
3
4
3
3
4
4
4
4
4
4
3
4
4
4
4
4
3
4
4
4
4
3
4
4
4
4
4
3,87
3,73
4,00
4,00
0,35
0,46
max score 4
% corr
58
96,67
56
93,33
Note that the maximum score a child can reach for the first set (stimuli with universal quantification) is 8. For the two sets with negated existentials (plus „without‟ or plus a converse
verb), the maximum score is 4 per set and 8 in toto.
Appendix 2: Table 3 (test protocol form: stimulus set of experiment 2)
“Bild A” = picture depicting a situation (1b). “Bild B” = picture depicting a situation (1c)
For the full set of stimulus picture please consult: http://www.uni-salzburg.at/fileadmin/oracle_file_imports/1859246.PDF
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