Elsevier

Cognition

Volume 205, December 2020, 104307
Cognition

Testing theories of plural meanings

https://doi.org/10.1016/j.cognition.2020.104307Get rights and content

Abstract

Plural morphology in English is associated with a multiplicity inference. For example, “Emily fed giraffes” is typically interpreted to mean that Emily fed multiple giraffes. It has long been observed that this inference disappears in downward-entailing linguistic environments, such as in the scope of negation. For example, “Emily didn't feed giraffes” does not merely suggest that she didn't feed multiple giraffes, but rather that she didn't feed any. There are three main approaches to explaining this puzzle: the first proposes that the plural is ambiguous, and invokes a preference for stronger meanings; the second derives multiplicity inferences as implicatures; and the third provides a homogeneity-based account. These different approaches can all account for the interpretation of the plural across upward- and downward-entailing environments. They differ, however, in what they predict for three further aspects of the plural: the status of positive and negative plural sentences in singular contexts, children's acquisition of plural meanings, and the relationship between plural meanings and scalar implicatures. In this paper, we report on three experiments investigating adults' and preschool-aged children's interpretation of plural morphology in English. The experiments reveal that participants distinguish positive and negative plural sentences presented in singular contexts, and that adults assign a different status to these positive and negative sentences. It is also observed that children, unlike adults, tend to accept underinformative positive plural sentences in singular contexts — in parallel with their behavior on standard scalar implicatures — while they are relatively more adult-like when it comes to negative plural sentences in the same contexts, showing a tendency to reject the negative sentences. We discuss how the findings of the three experiments are expected on a scalar implicature approach to multiplicity inferences, and the open challenges they pose for the ambiguity and homogeneity approaches.

Introduction

The plural-singular distinction is the source of a long-standing puzzle (see Sauerland, 2003; Sauerland, Andersen, & Yatsushiro, 2005; Spector, 2007; Zweig, 2009; Farkas & de Swart, 2010 and Magri, 2014 for discussion). Consider the English sentence with a plural noun phrase (1a). This sentence seems equivalent in meaning to one in which the plural noun phrase is replaced by “more than one giraffe”, as in (1b), and it differs in meaning from a sentence containing a singular noun phrase, such as (1c). These simple observations suggest that English plural morphology is associated with the meaning ‘more than one’ (Chierchia, 1998; Lasersohn, 1995, among others).

  • (1)

    a. Emily fed giraffes.

  • b. Emily fed more than one giraffe.

  • c. Emily fed a giraffe.

The paradox arises when the sentence with the plural noun (1a) appears under negation, as in (2a). This is because we would expect the negative sentence with the plural noun to yield a ‘not more than one’ reading, as in (2b). Instead, the sentence is better paraphrased as the negation of the corresponding sentence with the singular noun, (1c). That is, the better paraphrase of (2a) is (2c).

  • (2)

    a. Emily didn't feed giraffes.

  • b. Emily didn't feed more than one giraffe.

  • c. Emily didn't feed a (single) giraffe.

This is not an isolated fact. The absence of the expected ‘more than one’ meaning of a plural noun is also observed in other downward-entailing environments, that is, linguistic environments that license inferences from sets to subsets.1 Again, in these environments, the plural morphology is better paraphrased using singular morphology than it is by substituting ‘more than one’. That is, examples (3a) and (4a), containing plural morphology, are semantically equivalent to (3b) and (4b), containing singular morphology.2

  • (3)

    a. If there are books on Stephen's desk, Robin should lock the door.

  • b. If there is a (single) book on Stephen's desk, Robin should lock the door.

  • (4)

    a. Are there books on Stephen's desk?

  • b. Is there a (single) book on Stephen's desk?

To restate the paradox: the interpretation of positive sentences with plural morphology gives rise to a ‘more than one’ inference, but this multiplicity inference typically disappears under negation, in the antecedent of a conditional, and in questions. This pattern is clearly problematic for a semantic account of multiplicity inferences, which simply encodes this inference in the literal meaning of plural morphology (e.g., Chierchia, 1998; Lasersohn, 1995).

Three main approaches to this puzzle have been proposed in the formal semantics literature. The first, defended in Farkas and de Swart (2010), is a semantic account that proposes a polysemous meaning for the plural and invokes a principle for selecting which meaning is preferred in a given context. The second is an implicature treatment of multiplicity inferences, implemented in different forms in Sauerland (2003), Sauerland et al. (2005), Spector (2007), Zweig (2009), Ivlieva (2013), and Mayr (2015). The third approach accounts for the different interpretations of the plural by appealing to homogeneity (Križ, 2017). As we discuss below, these different approaches can all account for the interpretation of the plural across upward- and downward-entailing linguistic environments. They differ, however, in what they predict for the following three aspects of the plural: the status of positive and negative plural sentences in singular contexts, children's acquisition of the different interpretations of the plural, and the relationship between plural meanings and scalar implicatures.

In this paper, we report on three experiments investigating adults' and preschool-aged children's interpretation of plural morphology in English. The experiments reveal that adults assign a different status to positive and negative plural sentences presented in singular contexts. The experiments moreover show that children are relatively more adult-like in their interpretation of negative plural sentences, compared to their interpretation of positive plural sentences — a pattern similar to that observed for classical scalar implicatures. We discuss how the findings of the three experiments are more easily captured under a scalar implicature approach to multiplicity inferences than under ambiguity- or homogeneity-based approaches.

The rest of the paper is organized as follows. In the next section, we sketch the three accounts and discuss their predictions for adults' and children's behavior. In Section 2, we summarize two previous studies on the multiplicity inference and discuss how these previous results motivate the present experiments. We present our experiments in 3 Experiment 1, 4 Experiment 2, and 5, and in Section 6, we discuss the results in the context of the theoretical predictions outlined in Section 1.3. Section 7 concludes the paper.

In response to the monotonicity puzzle described in the previous section, Farkas and de Swart (2010) enrich the semantic account in order to explain the sensitivity to monotonicity exhibited by multiplicity inferences. Simplifying, their approach is essentially based on two main ingredients: a polysemous meaning for plural morphology and a principle for choosing between two readings of plural sentences in different contexts. In brief, the plural morpheme can have a weak inclusive meaning (compatible with both singular and plural entities) or a strong exclusive meaning (which excludes singularities). A plural sentence like (1a) will then correspondingly be associated with two possible readings, a weak reading along the lines of (5a) (compatible with Emily having fed a singularity or a plurality of giraffes) and a strong reading along the lines of (5b) (compatible only with Emily having fed more than one giraffe). The reading in (5b) is stronger than that in (5a), as it entails it — if (5b) is true, (5a) must necessarily also be true.

  • (5)

    a. Emily fed one or more giraffes. (weak)

  • b. Emily fed more than one giraffe. (strong)

Similarly, the negative (2a) is then associated with the two meanings in (6a) and (6b).

  • (6)

    a. Emily didn't feed one or more giraffes. (strong)

  • b. Emily didn't feed more than one giraffe. (weak)

The proposed ambiguity of the plural morphology alone will not account for the fact that (1a) tends to be interpreted as (5b) while (2a) tends to be interpreted as (6a). To explain this pattern, Farkas and de Swart (2010) assume a version of the Strongest Meaning Hypothesis, formulated as in (7), which essentially regulates the choice between the two possible readings of plural sentences, favoring the strongest between the two (see Dalrymple, Kanazawa, Kim, Mchombo, & Peters, 1998; Winter, 2001, among others).

  • (7)

    The Strongest Meaning Hypothesis for Plurals: for a sentence involving a plural nominal, prefer that interpretation of plural which leads to the stronger overall interpretation for the sentence as a whole, unless this interpretation conflicts with the context of utterance.

As can clearly be seen, (7) will favor the interpretation (5b) for the positive (1a) and (6a) for the negative (2a), as these correspond to the strongest possible interpretations of these sentences. (7) therefore allows the ambiguity approach to capture the monotonicity effect.

In addition, the principle in (7) allows for the weaker interpretations to re-emerge if the stronger ones are in conflict with the context of utterance. This predicts that we should be able to force a weak interpretation by adding a continuation that is in conflict with the strong counterpart. This is indeed the case, as illustrated in (8); here the negated plural receives the weak interpretation along the lines of (6b), rather than the strong interpretation in (6a).

  • (8)

    Emily didn't feed giraffes, because she fed only one!

In sum, the ambiguity approach can explain the different readings of the plural and their distribution across upward- and downward-entailing contexts. In the next subsection, we turn to a different take on the puzzle, which involves treating multiplicity inferences as implicatures.

A different response to the pattern associated with multiplicity inferences is to treat them as a type of implicature (Ivlieva, 2013; Mayr, 2015; Sauerland, 2003; Sauerland et al., 2005; Spector, 2007; Zweig, 2009). This is because implicatures, and in particular scalar implicatures, exhibit a similar monotonicity pattern. To illustrate, a familiar example of a scalar implicature is (9). In interpreting a positive sentence containing a disjunction phrase “A or B”, we typically infer the exclusive ‘A or B but not both.’ For example, (9a) implies (9b).

  • (9)

    a. Leo ate the apple or the orange.

  • b. ⇝ Leo did not eat both the apple and the orange.

When disjunction is embedded under negation, however, it is typically interpreted inclusively, rather than exclusively: (10a) is typically interpreted as (10b) and not as (10c); the latter corresponds to the negation of the exclusive meaning of disjunction and is compatible with Leo eating both the apple and the orange. But (10a) doesn't seem compatible with such a situation.3

  • (10)

    a. Leo didn't eat the apple or the orange.

  • b. ⇝ Leo didn't eat the apple and he didn't eat the orange.

  • c. ⇝ Leo either ate both the apple and the orange or he ate neither of them.

Both multiplicity inferences and classical scalar implicatures, then, share the property of arising in upward-entailing environments and disappearing in downward-entailing environments. This parallelism in their sensitivity to monotonicity has led some researchers to capture the ‘more than one’ meaning of the plural in the same manner as classical scalar implicatures.

A standard approach to deriving scalar implicatures like (9) is to treat them as arising from the hearer's reasoning about what the speaker actually said compared to what she could have otherwise said, assuming she was being a cooperative conversational partner (see Grice, 1975 and much subsequent work). Simplifying, the implicature above would arise from the comparison of the original assertion with alternative assertions that could have been uttered but were not. In particular, a sentence with the weak scalar term “or” is compared to the stronger alternative sentence containing “and”:

  • (11)

    a. Leo ate the apple or the orange.

  • b. Leo ate the apple and the orange.

Assuming the speaker is being as informative as she can be, the fact that she uttered the assertion containing “or” rather than the stronger, more informative alternative containing “and”, invites the listener to conclude that the speaker was not in a position to assert the stronger alternative. If the listener further believes that the speaker is well-informed with respect to the alternative, she will infer that the stronger alternative must therefore be false; the inference in (9b) is thus entered into the listener's mental model of the conversational context.

This reasoning also naturally explains why (10a) doesn't give rise to an implicature and, as we observed, is instead interpreted as the negation of an inclusive disjunction. The corresponding alternative that the speaker might have instead uttered would be (12), but (12) is not more informative than (10a); it is in fact weaker than (10a). The listener therefore does not draw an implicature from the speaker's use of (10a).

  • (12)

    Leo didn't eat the apple and the orange.

The implicature approach to multiplicity inferences extends this proposal to plural sentences like (1a) and (2a). Roughly, the idea is the following. A sentence like (1a), repeated below in (13), is assumed to unambiguously mean that Emily fed one or more giraffes. Upon hearing (13), the listener reasons about why the speaker didn't instead say something to the effect of (14).4

  • (13)

    Emily fed giraffes.

  • (14)

    Emily fed exactly one giraffe.

Given that (14) would have been more informative than (13), the listener concludes that the speaker must believe that (14) is false. But if (13) is true and (14) is false, the result is then exactly the multiplicity inference we are after, as illustrated in (15):

  • (15)

    Emily fed one or more giraffes and it's not true that she fed exactly one giraffe = Emily fed more than one giraffe.

This approach can also explain why, as with the disjunction case above, the negative (2a), repeated below in (16), is not associated with any inference. This is because the listener will compare (16) to its alternative in (17). (17), however, is weaker than (16) and therefore the listener will not draw any implicature from (16).

  • (16)

    Emily didn't feed giraffes.

  • (17)

    Emily didn't feed exactly one giraffe.

Finally, the scalar implicature approach can also capture the additional reading of the negated plural in (8), repeated below in (18). (18) typically involves stress or emphasis on the syllable containing the “-s” morpheme, and results in a reading that excludes the singularity. Notice that this kind of reading also arises with standard scalar items like disjunction; for instance, (19), when pronounced with stress on “or”, also has a marked reading compatible with Leo eating both the apple and the orange. The implicature approach unifies the two cases, extending the same mechanism from the standard scalar (19) to the plural (18). Generally, the way of generating these marked readings is to postulate a local scalar implicature that is computed under the scope of the negation. Given that implicatures tend not to arise under negation, the readings in (18) and (19) are correctly predicted to be marked.

  • (18)

    Emily didn't feed giraffes, because she fed only one!

  • (19)

    Leo didn't eat the apple or the orange, he ate both!

Plural definite descriptions such as “the giraffes” typically give rise to so-called homogeneity effects: the positive The giraffes are tall and the negative The giraffes aren't tall are neither true nor false when the group of giraffes is not homogeneous with respect to the property of being tall, i.e. when only some of the giraffes are tall (Križ, 2015; Löbner, 1987; Schwarzschild, 1994). Križ (2015) provides an account of this homogeneity effect, and Križ (2017) extends it to account for the multiplicity inference of bare plurals. The general idea under the homogeneity approach is that predicates applied to pluralities can not only be true or false, but also undefined under certain conditions.

On the homogeneity theory, homogeneity arises as a lexical property of predicates, in combination with a general homogeneity principle. These two ingredients together yield trivalent truth conditions for sentences involving predicates applied to pluralities. For example, the nominal predicate giraffes has the meaning in (20):

  • (20)

    giraffes=λx1iffxisaplurality of giraffes0iffxdoesn'tcontainanygiraffe#otherwise

As Križ (2017) shows, when a predicate like (20) appears in an episodic sentence such as (21), it gives rise to the following trivalent truth-conditions: it is true when both (22a) and (22b) are true, false when both are false, and undefined otherwise. This gives us the intuitively correct reading of the sentence in (21), namely that it's true if and only if Emily fed more than one giraffe.

  • (21)

    Emily fed giraffes.

  • (22)

    a. Emily fed one or more giraffes.

  • b. Emily fed multiple giraffes.

When (21) is negated, as in (23), the undefinedness is unaffected by negation, so that the conditions for (23) are as follows: (23) is true when both (24a) and (24b) are true, false when both are false, and undefined otherwise. These appear to be the correct conditions for (23) and in particular they capture the intuition that the sentence is true if and only if Emily didn't feed any giraffes.5

  • (23)

    Emily didn't feed giraffes.

  • (24)

    a. Emily didn't feed one or more giraffes.

  • b. Emily didn't feed multiple giraffes.

In sum, the homogeneity approach can account for the alternation between positive and negative cases.

For the cases of cancellation that we have seen above, Križ (2017) appeals to a pragmatic principle for dealing with undefinedness, which allows the use of a sentence even if undefined in the context, as long as the actual situation that makes the sentence undefined is equivalent to a situation that would make the sentence true. This principle is summarized in (25).

  • (25)

    An undefined sentence can be used when the situation described in the context is, for current purposes, equivalent to the situation in which the sentence is true. [Križ, 2017, p. 22]

Returning to the example in (26), the prediction is that it can be used felicitously to the extent that we can accommodate in the context that the distinction between Emily feeding just one giraffe and her feeding no giraffes is irrelevant (i.e. it only matters whether she fed more than one giraffe or not).

  • (26)

    Emily didn't feed giraffes, because she fed only one!

In sum, all three approaches can account for the different readings associated with plural morphology in English and in particular for the observed sensitivity to monotonicity. In the next subsection, we turn to the different predictions they make for children and adults, which will motivate the experiments in 3 Experiment 1, 4 Experiment 2, and 5.

All three approaches are designed to capture the observation that multiplicity inferences arise in upward-entailing environments but generally not in downward-entailing environments. Under all three approaches, then, we should expect to observe an effect of monotonicity on the rate of multiplicity inference computation. The three approaches diverge, however, along three dimensions: the status of positive versus negative plural sentences in singular contexts, the predictions they make in relation to the acquisition of the readings of the plural, and the relationship between multiplicity inferences and implicatures more generally. We turn to each of these next.

As Križ (2017) points out, the status of positive vs. negative plural sentences is an important point of divergence in predictions among the alternative approaches to the plural. Consider the sentences in (27a) and (27b) in a context in which Emily fed only one giraffe.

  • (27)

    a. Emily fed giraffes.

  • b. Emily didn't feed giraffes.

The homogeneity and ambiguity approaches make symmetric predictions here. The former predicts both to be undefined. The latter predicts both to be false (or both to be true, if the Strongest Meaning Principle is overridden). The implicature approach, on the other hand, is compatible with — and in fact even predicts — a difference between the two polarities. This is because the positive case is literally true in the given context, but gives rise to a false implicature, while the negative case is plainly false.6 The status of (27a) versus (27b) is therefore an important data point to test the predictions of the three different approaches.

The three approaches also differ in the predictions they make for the acquisition of the plural and how children might, at a certain stage of development, differ from adults in this area.

On the implicature approach, the expectation is that children should behave roughly as they do with implicatures more generally. If, as has been reported in much previous developmental literature, 4–6-year-old children differ from adults in their computation of scalar implicatures, then everything being equal, we should expect a similar difference between the two groups when it comes to multiplicity inferences as well. We will discuss the predictions about the relation between scalar implicatures and multiplicity inferences in the next subsection.

On the ambiguity approach, the predictions for children's acquisition of the plural depend on what assumptions are made about the acquisition of the proposed meanings for the plural and the Strongest Meaning Principle. On this approach, children need to have acquired the two proposed meanings of the plural and be able to rely on a Strongest Meaning Principle in order to behave in an adult-like manner. Given these prerequisites, there are three main scenarios in which children might not be adult-like. They might go through a developmental stage where they have only acquired one of the two meanings of the plural (either the weak or the strong). Or, they might go through a stage where they have acquired both meanings for the plural, but are not yet able to deploy the Strongest Meaning Principle in an adult-like way. These three possible scenarios will lead to different predictions for how children will respond to plural sentences, compared to adults. For instance, if children have only acquired the strong meaning of the plural, they should appear adult-like on the plural in upward-entailing linguistic environments but not in downward-entailing environments; if they have only acquired the weak meaning of the plural, they should only appear adult-like in downward-entailing environments. On the other hand, if they have acquired both meanings of the plural but cannot yet make use of the Strongest Meaning Principle in selecting a reading, they might not be guided by the relative strength of the two meanings of the plural in the same way that adults are. They might instead, for example, always charitably go with the interpretation that is made true in the context.

Finally, in the case of the homogeneity approach, adult-like behavior is dependent on children having acquired the homogeneity principle and the pragmatic principle for dealing with undefinedness. If they have acquired both, they should perform like adults; if they are missing either ingredient, they will not perform like adults. Importantly, there isn't an obvious way to distinguish between upward- and downward-entailing contexts in this respect; either children will be adult-like in both upward- and downward-entailing contexts, or they will be non-adult-like in both.

The three approaches also make different predictions regarding how multiplicity inferences compare to standard scalar implicatures. Consider first the implicature approach. If multiplicity inferences are merely scalar implicatures, one might expect to observe the same kind of behavioral profile for multiplicity inferences as we observe for standard cases of scalar implicature. The nature of this uniformity prediction is somewhat complicated, however, by the observation that adults (as a group) vary quite widely in how much they compute different kinds of scalar implicatures (van Tiel, van Miltenburg, Zevakhina, & Geurts, 2014 observe that implicature rates vary quite widely across lexical scales, e.g., “some”/“all”, “try”/“succeed”, “cool”/“cold”, etc.). One way to tackle this complication is to compare inference rates across populations that are reported to differ in their performance on implicatures. In particular, children have been reported to compute fewer standard implicatures than adults (barring special experimental manipulations and/or the use of certain scales). If multiplicity inferences are merely scalar implicatures, we should expect to see similar between group differences when we compare the two scales. For example, we should expect to see fewer of both kinds of inferences from children compared to adults, even if adults may compute the two inferences at different rates. Let us formulate this uniformity prediction of the implicature approach as in (28).

  • (28)

    Uniformity prediction of the implicature approach: If multiplicity inferences and scalar implicatures are of the same nature, we expect to observe the same pattern of between-group differences (or between-group similarities) across the two kinds of implicatures.

In this respect, a comparison between 4- and 5-year-old children and adults is particularly useful, as it has been widely reported in the developmental literature that these two groups differ considerably in how often they compute implicatures. While there has been considerable variation in the reported rates of children's success with scalar implicatures, a consistent finding is that children compute fewer standard scalar implicatures than adults do, unless special experimental manipulations are implemented (Noveck, 2001; Gualmini, Crain, Meroni, Chierchia, & Guasti, 2001; Chierchia, Crain, Guasti, & Thornton, 2001; Papafragou & Musolino, 2003; Barner, Brooks, & Bale, 2011; Katsos & Bishop, 2011; Tieu, Romoli, Zhou, & Crain, 2016, among many others). For instance, children typically accept sentences such as (29a) in situations where the stronger alternative (29b) is also true. This observation has usually been taken to indicate that children fail to compute the implicature in (29c).

  • (29)

    a. Emily fed some of the giraffes.

  • b. Emily fed all of the giraffes.

  • c. Emily didn't feed all of the giraffes.

If multiplicity inferences are scalar implicatures, then we expect children to likewise compute fewer multiplicity inferences from plural morphology, compared to adults.7

The ambiguity and homogeneity approaches, on the other hand, make no particular predictions with respect to the relationship between multiplicity inferences and implicatures, since they do not relate the two phenomena. A convincing case could therefore be made against the implicature approach, if non-uniformity of the two phenomena were observed across populations. Any other result would be compatible with all three approaches.

In sum, while all three approaches to the plural make the same prediction regarding an effect of monotonicity (namely, more multiplicity inferences in upward- than downward-entailing linguistic environments), there are other areas where their predictions diverge: the status of positive versus negative plural sentences in singular contexts, the relationship between multiplicity inferences and standard implicatures, and the relative performance of children and adults on the two kinds of inferences. Before moving on to our experiments, we will briefly describe one previous study that set out to test the implicature theory in children and adults, which will allow us to raise some potential issues that we then address in our own experiments.

Section snippets

Previous studies

There has been greater experimental investigation of multiplicity inferences in recent years. In particular, there have been studies investigating the suspension of multiplicity inferences compared to that of standard implicatures (Pearson, Khan, & Snedeker, 2011), the sensitivity of the multiplicity inference to contextual manipulations (Grimm, 2013), the different readings associated with plural sentences (Dieuleveut, Chemla, & Spector, 2019; Patson, 2016; Patson, George, & Warren, 2014;

Experiment 1

We designed a Truth Value Judgment Task (Crain & Thornton, 1998) to assess the interpretations that adults and children assign to declarative sentences with singular and plural descriptions, in both upward and downward-entailing environments.9 Testing the interpretations of declarative sentences allowed us to avoid the potential problem that was associated with the disappearance of multiplicity inferences in polar questions in Sauerland et al.

Experiment 2

While the finding that children computed fewer multiplicity inferences than adults in Experiment 1 is consistent with a general implicature approach, the reported results in the existing literature reveal variation across both scales and studies. We reasoned that a within-subject design would enable us to better compare children's performance on multiplicity inferences and classical scalar implicatures, and conducted a second experiment to further target the relationship between the two.

Experiment 3

Experiment 3 used a ternary judgment task to test the different predictions of the three approaches regarding the status of positive versus negative plural sentences in singular contexts. As we have discussed, the ambiguity and homogeneity approaches do not predict a difference in the status of such positive and negative sentences; they are either both undefined or both false. The implicature approach, on the other hand, is compatible with a difference between the two; it in fact predicts a

General discussion

The findings from our experiments shed light on the predictions discussed in Section 1.3. In the following, we discuss how our results relate to the different theoretical approaches.

All three experiments revealed a significant effect of monotonicity, with (both adult and child) participants computing more multiplicity inferences in response to positive plural sentences like (55a) than in response to negative plural sentences like (55b). This effect of monotonicity is expected on all three

Conclusion

There are three main theoretical approaches to the ‘more than one’ meaning of plural morphology. The goal of the present study was to experimentally test the predictions of the ambiguity, homogeneity, and scalar implicature approaches. In particular, the three approaches diverge on three points: (i) the status of positive plural sentences versus negative plural sentences in singular contexts, (ii) the acquisition of the different readings of plural sentences, and (iii) the relationship between

Credit author statement

Lyn Tieu: Conceptualization, Methodology, Formal analysis, Investigation, Resources, Data curation, Writing - original draft, Writing - review & editing, Visualization, Project administration, Funding acquisition. Cory Bill: Conceptualization, Methodology, Investigation, Writing - review & editing, Project administration. Jacopo Romoli: Conceptualization, Methodology, Writing - original draft, Writing - review & editing, Funding acquisition. Stephen Crain: Conceptualization, Methodology,

Supplementary material

The raw data and R scripts for analysis for all three experiments can be accessed at https://osf.io/9sek5.

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      This was addressed by Experiment 2, which found that many younger children had not yet acquired a robust meaning for the dual and also failed to treat singular forms as exact, despite associating the singular with singleton sets. Note that in this regard, the younger Slovenian children tested in Experiment 2 performed much like English-speaking children in previous studies (Barner et al., 2009; Tieu et al., in press), readily associating the singular with sets of one (in some cases by as early as 24 months; Kouider et al., 2006; Wood et al., 2009), but also accepting its use for larger sets. Only older Slovenian-speaking children who had mastered the dual also exhaustified their interpretation of the singular.

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    For helpful feedback and discussion, we would like to thank Scott AnderBois, Emmanuel Chemla, Alexandre Cremers, Giorgio Magri, Uli Sauerland, Florian Schwarz, Jesse Snedeker, and Benjamin Spector. We would also like to thank Dorothy Ahn for allowing us to use her illustrations in our stimuli. Finally, we are grateful to the families who participated in the study at the CCD at Macquarie University. The research leading to these results was supported by the Australian Research Council Centre of Excellence in Cognition and its Disorders (CE110001021), the European Research Council under the European Union's Seventh Framework Programme (FP/2007–2013)/ERC Grant Agreement n. 313610, ANR-10-IDEX-0001-02 PSL*, ANR-10-LABX-0087 IEC, the Leverhulme Trust, grant RPG-2016-100, and Western Sydney University through the University's Research Theme Champion support funding.

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