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Edited by Barbara H. Partee, University of Massachusetts at Amherst, Amherst, MA, and approved June 20, 2016 (received for review February 18, 2016)
Significance
Although much research has been devoted to the acquisition of number words, relatively little is known about the acquisition of other expressions of quantity. We propose that the order of acquisition of quantifiers is related to features inherent to the meaning of each term. Four specific dimensions of the meaning and use of quantifiers are found to capture robust similarities in the order of acquisition of quantifiers in similar ways across 31 languages, representing 11 language types.
Abstract
Learners of most languages are faced with the task of acquiring words to talk about number and quantity. Much is known about the order of acquisition of number words as well as the cognitive and perceptual systems and cultural practices that shape it. Substantially less is known about the acquisition of quantifiers. Here, we consider the extent to which systems and practices that support number word acquisition can be applied to quantifier acquisition and conclude that the two domains are largely distinct in this respect. Consequently, we hypothesize that the acquisition of quantifiers is constrained by a set of factors related to each quantifier’s specific meaning. We investigate competence with the expressions for “all,” “none,” “some,” “some…not,” and “most” in 31 languages, representing 11 language types, by testing 768 5-y-old children and 536 adults. We found a cross-linguistically similar order of acquisition of quantifiers, explicable in terms of four factors relating to their meaning and use. In addition, exploratory analyses reveal that language- and learner-specific factors, such as negative concord and gender, are significant predictors of variation.
Number words and quantifiers are abstract words that denote properties of sets rather than individuals. Twoness and allness in “two/all of the black cats in the street” are not true of any individual cat, whereas blackness and catness are. Children display knowledge of number words and quantifiers around their second birthday, comparatively long after they have acquired concrete nouns (1, 2). As far as number words are concerned, a range of cognitive and perceptual systems supports their acquisition. These systems include an object-tracking system, which enables the precise representation of small quantities, and an analog magnitude system, which enables imprecise and approximate comparisons (1), as well as general principles of word learning (3). The role of language in the acquisition of number is manifold: it can be viewed as a system of labels for expressing numerical concepts (4), a system that allows the combination of information from diverse sources (5), or a provider of cues for acquisition (6⇓–8). For example, children learning languages that distinguish between singular and plural or between singular, dual, and plural morphology learn the meaning of “one” and “two” earlier than children learning languages that do not (9, 10). There are also cultural practices, such as the verbal count list, the recital of number words in a fixed order (“one, two, three, …”), and finger or other body part counting routines, that are widely practiced across many languages (11, 12). These systems and practices converge toward a universal order of acquisition, starting with “one” and proceeding in line with increasing cardinality. The order itself is stable and not affected by differences between languages as regards the specific timing of the acquisition of each number word (9, 10, 13).
Quantifiers (e.g., “none,” “some,” and “all”) too are properties of (or relations between) sets. The onset of the acquisition of quantifiers coincides with the acquisition of number words, and some systems are likely to be implicated in the acquisition of both (principles of word learning and the role of language as a system of labels among others) (3). However, what about the order of acquisition of quantifiers? Is it fixed, like that of number words, or does it vary? Which systems constrain it? The perceptual object-tracking system that supports the acquisition of numbers is largely neutral to the order of acquisition of quantifiers. A set of five and a set of ten individual objects could both be referred to as “some,” “most,” or “all” in different contexts. Moreover, there is no known routinized practice for quantifiers, such as the verbal count line or body part counting for numbers. Even if there were to be a “verbal quantifier line,” which quantifiers would it include and in which order? The choice is not trivial (e.g., consider “none,” “many,” “not all,” and “fewer than half”), and there are multiple intuitively plausible orderings. If we were to suppose that, just as numbers are acquired in order of increasing cardinality, quantifiers are learned as a function of their increased proportion of overlap between two sets, we would predict that “a few” and “some” would be acquired from a very early age and that “most” and “all” would be acquired last. However, the evidence from corpora (14) and experiments (15, 16) reveals that, although many 2-y-old children have acquired “all,” even some 7-y-old children are not fully competent with “most.”
Overall, a simple parallelism between the order of acquisition of numbers and that of quantifiers is not fruitful and furthermore does not make sense of the available evidence. Although the acquisition of number words and quantifiers is supported by some shared systems, there are constraints in the order of acquisition of numbers that are not as relevant for quantifiers (such as a verbal routine). Moreover, there may well be constraints in the order of acquisition of quantifiers that do not extend to numerals.
In this paper, we hypothesize that a major constraint in the order of acquisition of quantifiers comes from the meaning of each term. Unlike number words, which have meanings that vary as a function of cardinality alone, the meanings of quantifiers are varied and rich. Specific features among these word meanings are likely to play a role in their acquisition. To give substance to this distinction, consider statements such as “all/none of the students are playing football.” “All” is a positive and monotone increasing quantifier that licenses inferences to supersets (e.g., “all of the students are playing a sport”), whereas “none” is a negative and monotone decreasing quantifier that licenses inferences to subsets (e.g., “none of the students are playing football in the rain”). We will shortly describe this distinction formally to argue later that it is one of the features of meaning to play a role in acquiring quantifiers in a fixed order across languages. Of course, some languages could offer specific cues to support acquisition. For instance, they may offer additional cues that a quantifier is negative by marking negation twice (once on the quantifier itself and once with a negative particle on the verb phrase), a phenomenon known as negative concord (as in French “aucun des élèves ne jouent au football”). In what follows, we turn to aspects of quantifier meaning and use that we argue are relevant to their order of acquisition.
Cross-Linguistic Similarities and Differences
Quantifiers predicate properties of members of sets. For example, the meanings of the English quantifiers “all” and “some” are traditionally taken to correspond to set theoretical logical concepts (17). Under this view, the truth conditions of many quantified sentences are given as relations between sets as shown below, where “iff” is if and only if, ∩ is the intersection of two sets, − is their difference, and ∅ is the empty set.
i) “All of the As are Bs” is true iff A ∩ B = A.
ii) “Some of the As are Bs” is true iff A ∩ B ≠ ∅.
iii) “None of the As are Bs” is true iff A ∩ B = ∅.
iv) “Most of the As are Bs” is true iff |A ∩ B| > |A – B|.
v) “Some of the As are not Bs” is true iff A − B ≠ ∅.
Quantified sentences have systematic entailment properties. If the sentences in i, ii, and iv are true, then it is guaranteed that, for any set B′, which is a superset of B, the corresponding sentence is also true (e.g., if it is true that “all/some/most of the students are playing football,” then it is guaranteed that “all/some/most of the students are playing a sport”). Quantifiers that guarantee inferences from sets to supersets in this way are known as monotone increasing. Conversely, if the sentences in iii and iv are true, then it is guaranteed that, for any set B′, which is a subset of B, the corresponding sentence is also true. Quantifiers with this property are monotone decreasing.
Typological research in semantics suggests that many human languages contain these quantifiers and others and that the entailment properties of these quantifiers exhibit similarities (18). These similarities extend to considerations of quantifier use, such as the need to be informative. For instance, speakers should not describe a situation in which all students are playing football by saying “some students are playing football.” Under the definition in ii, this description would be (strictly speaking) true, but the speaker would be underinformative and potentially inviting the listener to draw additional conversational inferences. These word choices rely on norms of human rational behavior (19) and cost–benefit optimization in information exchange (20, 21). The existence of such norms is widely reported in the world’s languages (22, 23).
Language-specific factors are also evident among quantifiers (contributions are in the work in ref. 24). In the following section, we specify four developmental patterns that follow from cross-linguistic similarities. We then outline some of the language-specific factors that may affect acquisition. We focus on the set of four quantifiers that are the English equivalents of “all,” “some,” “some…not,” and “none.” These quantifiers are the basis of Aristotle’s theory of syllogisms, and they have held a special status in Western thought for more than two millennia (25). For reasons mentioned below, we also include “most.”
Developmental Generalizations
Focusing on single languages, previous studies in the processing of quantifiers (14⇓–16, 26, 27) have made several generalizations that could be expected to have cross-linguistic relevance for the order of acquisition of quantifiers. Here, we hypothesize that these generalizations have the status of cross-linguistically applicable constraints (Discussion). Constraint 1 concerns monotonicity, which we defined above. According to this constraint, children will be more successful at comprehending monotone increasing compared with monotone decreasing quantifiers (26, 28, 29). For this study, we would expect children to show greater competence with “all” compared with “none” and “some” compared with “some are not.”
Constraint 2 (totality) is that children are more successful at acquiring quantifiers that attribute a property to all or none of the members of a set than they are at acquiring those that attribute a property to only a part of the set (30, 31). In our dataset, this constraint will facilitate the acquisition of totality quantifiers “all” and “none” compared with partiality quantifiers “some” and “some…not.”
Monotonicity and totality are independent properties. They will sometimes align to render a quantifier particularly easy or difficult for children and sometimes diverge and compete. We predict that “all,” which is a monotone increasing and totality quantifier, will be the easiest of four Aristotelian quantifiers, whereas “some…not,” a monotone decreasing and partiality quantifier, will be the hardest. The acquisition of “none” and “some” is a matter of the relative strength of the two constraints. If the advantage bestowed by totality outweighs the disadvantage of monotone decreasing, “none” will be easier than “some” and vice versa.
Constraint 3 (complexity) is that children are more successful at comprehending “some” than “most.” To understand “most of the As are Bs,” children need to be able to restrict the domain of quantification to some relevant set of As in the universe of discourse and then, compare the cardinalities of the set of As that are Bs with those of the set of As that are not Bs (32). However, “some As are Bs” is simpler, because in this case, children do not need to restrict the quantifier to a specific set of entities or compare cardinalities. They can simply treat “some students like football” as logically equivalent to “there is at least one entity that is both a student and likes football” (33).
Finally, constraint 4 (“informativeness”) is that children will be stricter toward violations of truth than toward violations of pragmatic felicity. That is, children do not reject utterances that are underinformative (e.g., saying “some” when “all” is true) to the same extent as utterances that violate truth (e.g., saying “some” when “none” is true) or the same extent as adults (27, 32, 33). We, therefore, expect that children will accept underinformative utterances more often than false ones, regardless of the language that they speak. In our dataset, this constraint predicts that children are more likely to reject a false statement with “some,” “some…not,” and “most” than an underinformative one (and at rates that are distinguishable from those in adults).
These predictions are summarized below (>> implies higher performance, and / indicates no prediction).
i) Constraints 1 and 2: “all” >> “none”/“some” >> “some…not.”
ii) Constraint 3: “some” >> “most.”
iii) Constraint 4: false >> underinformative for “some,” “some…not,” and “most.”
In addition to these four factors that may affect the acquisition of quantification in similar ways across languages, language-specific properties may have an important role as well. The explicit presence of a partitive marker (such as “of the” in English) may positively affect children’s performance with underinformative utterances (27) by drawing attention to the divisibility of the reference set. Syntactically, negative concord may be a significant predictor, with the presence of two negative markers highlighting the fact that the utterance contains a negative quantifier. Finally, a range of nonlinguistic factors may also be important predictors of children’s performance. These potential predictors include biological factors, such as gender and age, and social factors, such as socioeconomic and educational status (e.g., whether children are enrolled in formal schooling at the time of testing).
Experiment
As part of a larger project known as the European Cooperation in Science and Technology Action A33 (Acknowledgments), the empirical investigation focused on the comprehension of quantified sentences by 768 children (mean age = 5.5 y old; age range = 5.00–5.11 y old; 398 of them were female) and 536 adult participants (all adults were over 18 y of age; 293 adults were female; because of experimenter error, the gender of 46 adults was not recorded). The participants spoke 1 of 31 languages: Basque, Cantonese (Yue) Chinese, Catalan, Croatian, Cypriot Greek, Danish, Dutch, English, Estonian, Finnish, French, Georgian, German, Greek, Hebrew, Italian, Japanese, Korean, Lithuanian, Malay (Kuala Lumpur variety), Maltese, Mandarin Chinese, Norwegian, Polish, Russian, Serbian, Slovak, Spanish, Tamil, Turkish, and Urdu. This sample contains representatives of 15 language genera (Baltic, Chinese, Finnic, Germanic, Greek, Indic, Japonic, Karto-Zan, Korean, Malayo-Sumbawan, Romance, Semitic, Slavic, Southern Dravidian, and Turkic). These languages belong to 11 language types [8 of the main language families in the world (Afro-Asiatic, Altaic, Austronesian, Dravidian, Indo-European, Kartvelian, Sino-Tibetan, and Uralic/Finno-Ugric] as well as 3 language isolates (Basque, Japonic, and Korean)] (classified according to the work in ref. 34). Details of the languages’ properties are given in Table S1. In the main part of the task, participants were presented with five boxes and five objects. Between zero and five of the objects were inside the boxes for any test item. Participants then heard a description containing one of five quantifiers and had to judge if the description was right or wrong for the visual display. Details of the test procedure are presented in Methods.
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Table S1.
Coded properties of languages tested
Results
The results for child and adult participants per language are presented in Tables S2 and S3. Across all languages and expressions, adult responses were, on average, 99% correct in the true or false conditions. These ceiling adult data validate the task as a test of competence with quantification and are not discussed further; 84% of adult responses to underinformative items were rejections, and this less than perfect consistency accords with previous literature (32) and is discussed in the context of constraint 4.
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Table S2.
Summary child data by language
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Table S3.
Summary adult data by language
Across all languages and expressions, child responses were, on average, 82% correct in the true or false conditions, and 51% of responses in underinformative conditions were rejections. Starting with constraint 1 (monotonicity), we first report child performance with each of the monotone increasing quantifiers in the dataset (“all” and “some”) compared with the performance with each of the monotone decreasing quantifiers (“none” and “some…not”). Performance with “all” was numerically higher than that with “none”—the monotone decreasing quantifier that is matched with “all” for totality—in 29 of 31 languages. The exception was Korean (we consider exceptions those languages in which the numerical difference was the opposite of the one expected), while there was no numerical difference in English. Turning to “all” and “some…not”—the monotone decreasing expression that is not matched to “all” for totality—children performed better with “all” in 30 of 31 languages, with no differences in Georgian.
In 28 of 31 languages, children performed better with monotone increasing “some” compared with “some…not,” the monotone decreasing quantifier that is matched for totality (Catalan was an exception, with no difference in English and Georgian). Children performed better with “some” than with “none” in 15 of 31 languages (the exceptions being Cantonese, Catalan, Dutch, English, Estonian, Finnish, French, German, Greek, Japanese, Polish, Serbian, Slovak, and Turkish; no differences in Cypriot Greek and Georgian).
Overall, when keeping the setting of totality constant (that is, comparing the two totality quantifiers, “all” and “none,” with each other and comparing the two partiality quantifiers, “some” to “some…not,” with each other), the monotone increasing quantifiers give rise to better performance than the corresponding monotone decreasing ones in 27 of 31 languages (Catalan, English, Georgian, and Korean being exceptions).
Turning to totality, performance with “all” was higher than that with “some” (which is the quantifier with the same setting of monotonicity) in 26 of 31 languages (with Korean, Malay, Maltese, and Russian as exceptions and no differences in Georgian). Children performed higher with “all” than with “some…not” (which is the quantifier with a different value for monotonicity) in 30 of 31 languages, with no differences in Georgian.
Performance with “none” was higher than with “some…not,” which is matched for monotonicity, in 29 of 31 languages (with Tamil as the exception and no differences in Georgian) and higher with “none” than with “some,” which has a different setting for monotonicity, in 14 of 31 languages.
Overall, when keeping monotonicity stable, totality quantifiers “all” and “none” give rise to better performance than the corresponding partiality ones (“some” and “some…not,” respectively) in 25 of 31 languages (Georgian, Korean, Malay, Maltese, Russian, and Tamil being exceptions). Visual inspection of Table 1 shows that the order predicted by constraints 1 and 2 is, indeed, upheld, with “all” being the easiest quantifier for 5-y-old children across the languages in our sample and “some…not” being the hardest. The two constraints have relatively equal weight, with no consistent order of acquisition between “some” and “none.”
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Table 1.
For all quantifiers, N languages and types where children’s performances with true and false statements were numerically higher (>>)
Multivariate analyses were also performed. The analyses revealed main effects of language, monotonicity, and totality along with higher performance when the correct answer was rejection. A small effect of gender (boys outperforming girls) was also obtained, but we found no significant effect of age (Table S4).
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Table S4.
Analysis of hypotheses 1 and 2
We also conducted parallel analyses using language genus (n = 15) and language type (n = 11; family or isolate) in place of individual languages along with analyses without any language variable at all. The analyses returned a significant effect of language genus and type, but in all cases, model comparison using the Akaike Information Criterion (35) revealed that the inclusion of any one of the language variables resulted in the model being overfitted compared with a model with no language variables (hence, that the inclusion of language, genus, or type in the model was not statistically justified). Likewise, models positing an interaction of monotonicity or totality with the language variables were overfitted. Therefore, the data are most appropriately modeled by positing effects of monotonicity and totality but no effect of language, regardless of whether at the level of each individual language, genus, or type. Put another way, children were more successful with the acquisition of quantifiers in some languages compared with others, but the main effects on the order of acquisition that we hypothesized (monotonicity and totality) were upheld in the dataset, regardless of the specific language (or language genus or type) that the children were learning.
Turning to constraint 3, the hypothesis that “some” would be mastered earlier than “most” on account of its semantic simplicity was borne out numerically in all 31 languages in our sample. The effect of complexity was corroborated through multivariate analyses as with constraints 1 and 2. Model comparison indicated that models that included language, genus, or type (or an interaction of complexity by language, genus, or type) were overfitted by comparison with models that did not. A small effect of gender (boys outperforming girls) was obtained, but there was no significant effect of age. Details are in Table S5.
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Table S5.
Analysis for hypothesis 3
Finally, we consider constraint 4 (underinformative uses of “some,” “most,” and “some…not”). Compared with the false statements with the same expression, children rejected underinformative uses less often in all 31 languages. Looking at each expression on its own, underinformative “some” was rejected less often than false “some” in every language. This preference held for “some…not” in 25 of 31 languages (the exceptions being Croatian, Hebrew, Malay, Maltese, Mandarin, and Tamil) and “most” in 24 of 31 languages (the exceptions being Danish, English, Finnish, French, Norwegian, Polish, and Slovak) (Table 2).
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Table 2.
N languages and types where children rejected false statements more often than underinformative ones
For constraint 4, we also discuss the adult data, because the adults rejected underinformative statements more frequently than children did (84% compared with 51%, respectively); however, they did not reach the ceiling. Looking at all three quantifiers, adults rejected underinformative uses less often than false ones in 28 of 31 languages. Cantonese was an exception because of two erroneous responses among false statements and ceiling performance in the underinformative conditions. Russian and Urdu showed no differences, with both false and rejected underinformative conditions being at the ceiling in both languages. Furthermore, constraint 4 held in 25 of 31 languages for the case of “some” (with Basque, Croatian, Cantonese, Georgian, Russian, and Urdu showing no differences), 27 of 31 languages for “some…not” (with Cantonese as an exception and Georgian, Russian, and Urdu showing no differences), and 25 of 31 for “most” (with Cantonese as an exception and English, Mandarin, Russian, Turkish, and Urdu showing no differences). Therefore, not only do the child data support constraint 4, the adult data do as well.
We performed multivariate analyses for each of the quantifiers “some,” “some…not,” and “most” for the child data. In each case, highly significant main effects of language and informativeness were shown, with underinformative statements being rejected less often than false ones. No effects of gender or age were obtained (Table S6). Model comparison again suggested that models including language, genus, or type or their interactions with informativeness were overfitted.
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Table S6.
Analysis for hypothesis 4
The analyses for constraints 1–4 for the child data can be supplemented by comparisons with what would be expected if performances were guided by chance. Everything else being equal, 27 of 31 languages accorded with monotonicity (Catalan, English, Georgian, and Korean being exceptions), 25 of 31 languages accorded with totality (Georgian, Korean, Malay, Maltese, Russian, and Tamil being exceptions), and all 31 accorded with complexity and informativeness for all quantifiers. Each of these patterns is more consistent than if the distribution was random (P < 0.01 by the sign test) (Figs. S1 and S2).
Fig. S1.
N = 768. Percentages based on correct responses to true and false conditions for the English-equivalent expressions for “all,” “none,” “some,” “some…not,” and “most,” excluding underinformative conditions for “some,” “some…not,” and “most.” Languages are classified following the work in ref. 34. Table S7 shows the English-equivalent expressions in our sample.
Fig. S2.
Children (n = 768) and adults (n = 536). Percentages for the average of “some,” “some…not,” and “most” are presented in A, for “some” in B, for “some…not” in C, and for “most” in D. Languages are classified following the work in ref. 34. Table S7 shows the English-equivalent expressions in our sample.
Having shown our effects of interest and further documented that there is variability between languages, we then explored whether this latter variability is explicable by other linguistic factors or features of the learners in our sample. Exploratory analyses suggest that attending formal school at the time of testing was a significant facilitating factor (P < 0.001) along with learning languages that use negative concord (P < 0.001) and learning expressions with a partitive marker in the case of “some” (P < 0.05). Because our language sample is not balanced with respect to these properties, we do not draw firm conclusions here.
Discussion
The descriptive reports and the statistical modeling analyses suggest that our hypothesized constraints 1–4 are valid generalizations about the order of acquisition of quantifiers across the languages in our sample. These constraints were posited on the basis of generalizations made in previous research in single languages (14⇓–16, 26, 27), and these findings confirm their relevance to acquisition more widely. However, additional research is required to elucidate their nature and produce theoretical models from which they would follow. For example, constraint 1 (monotonicity) is closely related to negation (29) in that all negative quantifiers are monotone decreasing but not vice versa. Because both monotone decreasing expressions in our sample (“none” and “some…not”) contain negation, additional work could reveal whether the effects that we obtained here are because of monotonicity, negation, or both.
With regards to the exceptions in our sample, an important question is whether there was systematicity among the languages that did not conform to the hypothesized constraints. Two observations suggest that this is not the case. First, no language or language type violated more than one constraint, except Georgian, which violated two. Second, in Georgian (as well as in other languages), the violations were evidenced in cases of ceiling performance.
This observation leads to the issue of generalizability of the patterns in other languages and for other quantifiers. Our sample consists of representatives of 11 language types. Although there is an overrepresentation of Indo-European languages in our sample, the diversity of distinct language types in our sample is squarely within the range used for state of the art comparative linguistic (24) and psycholinguistic research (22). Of course, extrapolating from patterns observed in this sample to universal patterns should always be done with caution and as a working hypothesis only.
Similar considerations apply when extrapolating to quantifiers not tested here. For example, many languages have more than one universal quantifier, including the English equivalent of an each quantifier that is used for distributive quantification (ref. 36 reports eight different universal quantifiers in Malagasy that differ on the dimension of “distributivity”). The prediction is that the effects that we obtained here should hold as long as the appropriate considerations are taken into account. Turning to the case of each, monotonicity and totality should facilitate its acquisition across different languages, but distributivity itself may be an additional important—facilitating or hindering—factor.
In terms of explaining the cross-linguistic variation, where the acquisition of quantifiers was more successful in some languages compared with others, exploratory analyses found that language-specific features, such as using negative concord and partitive markers, had a facilitating effect. We hypothesize that negative concord may serve to better highlight that a quantifier is negative and additionally, highlight the contrast between negative and positive quantifiers. Partitives highlight that these expressions are related to parts of sets. Cross-linguistic variation may also be caused by linguistic factors that we did not model in our analyses (e.g., agreement, the number of competing expressions, and the overlap of their meaning). Clearly, additional research on this topic is needed.
Exploratory analyses also revealed an effect of attending school at time of testing. We do not believe that the effect is related to explicit instruction about quantifiers, because all of the teachers and caregivers of the children who we recruited reported that quantifiers were not part of the curriculum or any extracurricular activity. Instead, we hypothesize that attending school raises the children’s readiness for activities of the kind that we administered. We also found that age was not a significant predictor of success. We believe that this was because of the restricted age range that was part of the selection criteria (5.00–5.11 y old).
Our analyses also found a gender effect, whereby boys in this study outperformed girls in the acquisition of the true or false meaning of the quantifiers (Tables S4 and S5), but there were no differences when it came to informativeness (Table S6). Linguistic skills are generally more advanced among girls than among boys (37, 38). An investigation of over 13,000 children in 10 European linguistic communities suggests that these advantages are robust across different languages (38), although the level of overall linguistic attainment differed. Research on gender and mathematical competence suggests that there are widespread similarities between boys and girls (39). Nevertheless, a specific and small advantage is reported for boys for mathematical reasoning, perhaps reflecting higher aptitude with logical and set theoretical concepts (39). Conversely, an advantage specific to arithmetic is reported for girls, which seems to be attributable to the girls’ higher verbal skills that are implicated in arithmetical processing (40).
To the extent that these gender differences are robust, the language of quantification brings them into competition. Girls in our sample may have benefitted from an overall advantage in language skills as well as in arithmetic and counting, whereas boys may have benefitted from an advantage with set theoretical concepts, with the latter being more critical for the specific task than the former. We should note that our analyses for gender effects were exploratory and that future studies should take into account several potentially confounding factors (40).
Before we conclude, we need to address an alternative interpretation of the findings. That is, perhaps the patterns obtained here reflect competence with counting and checking the objects that need to be verified as belonging to a set (rather than competence with the meaning of a quantifier). We can reject this interpretation for two reasons. First, counting and verifying sets with up to five members, the maximum required in this task, were parts of the selection criteria (Methods). Moreover, increased demands on counting and verification complexity do not make correct predictions in this dataset. To take but one example, consider “none” and “some…not.” When “some…not” is true (that is, when two of five objects are in the boxes) in a random selection checking procedure given five objects, “some…not” requires checking the position of 1.5 objects, on average, against the boxes. When false (that is, five of five objects are in the boxes), “some…not” requires checking the position of five objects. For “none,” this requirement is five objects when “none” is true (and five of five objects are outside the boxes) and two objects when “none” is false (when two of five objects are in the boxes). In sum, to give the correct response to “some…not” in true and false conditions, participants need to check 6.5 objects, on average, against the boxes, and for “none,” they need to check seven objects. If it were the case that counting and verification complexity were primarily responsible for performance, “some…not” should be easier than “none.” At the very least, there should be no major difference. However, “none” is easier than “some…not” in 29 of 31 languages and 9 of 11 types as predicted by constraint 2 (totality). Of course, verification and counting are important components of success with tasks like our task, and additional research could identify their role for younger children to determine which specific verification strategy is implemented for each quantifier (26, 41).
Conclusion
In this paper, we investigated the order of acquisition of five common quantifiers and hypothesized four cross-linguistic constraints on their acquisition based on considerations of their meaning and use. A cross-linguistically similar order of acquisition emerged in a sample of 31 languages. This order accorded with the constraints that we posited, supporting the claim that they are potential universals in the acquisition of quantification. This claim is in line with recent proposals favoring the existence of extensive cross-linguistic similarities in language meaning and use (22, 42). However, we also found that language-specific features, such as whether a language uses negative concord, have a significant effect on the learners’ performance along with social and biological factors.
Methods
Tables S2 and S3 show details of child and adult participants per language. The actual quantifiers used in each language were selected by researchers who were native speakers of that language. Where more than one lexical item was available, the choice was guided by considering which item would be most familiar to children. Where possible, this decision was informed by investigating corpora of child-directed speech; in other cases, researchers consulted colleagues and/or school teachers. Table S7 shows materials and glosses.
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Table S7.
Materials and glosses (English-equivalent lexical item, person, number, and/or gender where available) for all languages tested
Informed consent for participation to the experiment was given by the adult participants and by the caregivers of the child participants. Children also assented to participating in the experiment. This research was approved by each researcher's institutional review board. Children were tested at nurseries or primary schools. Participants were administered the “cavegirl task,” which was designed to test the comprehension of quantified sentences (16). In this task, the cavegirl is asked to say, “How many toys are in the boxes” in visually presented situations. In each trial, the cavegirl produces a single utterance of the type, “[Quantifier] (of the) [objects] are (not) in the boxes.” Children are then asked to evaluate whether what the cavegirl said was right or wrong and if they say wrong, justify why. Two types of visual situations are used for each quantifier tested: one that renders an utterance with this quantifier true and informative and one that renders an utterance false. For “some,” “most,” and “some…not,” there is also a third type of display that renders an utterance true but pragmatically underinformative (where all of the objects are in the boxes for “some” and “most” and where none of the objects are in the boxes for “some…not”).
The task is preceded by a warmup session, where children are familiarized with the cavegirl, the task demands, and the pictures of the objects mentioned in the sentences. The first five items of the task test (the comprehension of number words one to five) ensure that children can make correct judgments about quantity when simple counting is involved. Children who did not perform correctly with all five number words did not continue with the main task. This criterion resulted in less than 5% of children not continuing. All justifications of rejections in the main task, whether correct or incorrect, mentioned a quantity-related word or deictic expression often combined with a spatial expression (e.g., “because these are out”), which suggests that children responded based on the appropriateness of the quantifier rather than some other aspect of the sentence. Ref. 16 has additional details of the task administration and a full list of items in their respective visual situations as well as sample visual displays.
Acknowledgments
This research was funded by European Cooperation in Science and Technology Action A33 “Cross-Linguistically Robust Stages of Children’s Linguistic Performance.” In addition, N.K., C.C., and I.N. were supported by the European Science Foundation Euro-XPrag Network; N.K., C.C., and N.S. were supported by the United Kingdom Economic and Social Research Council XPrag-UK Network; N.K. was supported by United Kingdom British Academy Grant SG090676; A.G. was supported by Spanish Ministerio de Economía y Competitividad Project FFI2014-56968-C4-1; A.G. and K.K.G. were supported by University of Cyprus Project 8037-61017; K.J.d.L. and L.S. were supported by Danish Council for Independent Research (Humanities) Grant 09-063957; M. Vija and S.Z. were supported by Estonian Science Foundation Grant ETF7492 and Estonian Research Council Grant SF0180056s08; K.Y. and U.S. were supported by European Commission for Education and Culture Grant 135295-LLP-2007-UK-KA1SCR and German Federal Ministry of Education and Research Grant 01UG0711; A.A. and J.v.K.T. were supported by a grant from the L. Meltzers Høyskolefond; E.H. and A.M. were supported by Grant 809/N-COST/2010/0 from the Polish Ministry of Science and Higher Education and National Science Centre; and D.A., M.S., and S.J. were supported by Grant ON179033 (2011-2014) from the Serbian Ministry of Education, Science, and Technological Development.
Footnotes
- ↵1To whom correspondence should be addressed. Email: nk248{at}cam.ac.uk.
↵2Deceased February 17, 2014.
Author contributions: N.K. designed research; N.K., M.-J.E., A.G., J.K.K., G.H., K.K.G., A.S., K.J.d.L., L.S., A.v.H., B.H., J.O., M.F., M.v.K., N.S., M. Vija, S.Z., S. Kunnari, T.M., M.R., K.Y., A.F., S.V., K.K., S.F., M.T.G., M. Vernice, R.O., M.I., P.C., Y.H., I.B., Y.M.A.N., H.G., D.G., W.N.C., A.A., J.v.K.T., E.H., A.M., N.G., J.P., D.A., M.S., S.J., D.S., S. Kapalková, T.B., D.Ö., S.H., C.Y.H.C., H.v.d.L., U.S., and I.N. performed research; N.K., C.C., and T.O. analyzed data; and N.K., C.C., U.S., and I.N. wrote the paper.
The authors declare no conflict of interest.
This article is a PNAS Direct Submission.
Data deposition: The dataset is freely available at DSpace@Cambridge, https://www.repository.cam.ac.uk/handle/1810/256762.
This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1601341113/-/DCSupplemental.
References
Acquisition of quantifiers
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