1 Missing Implicatures in Adult Language

1.1 Disjunctions: Free Choice and Distributive Inferences

Disjunctions in the scope of a possibility modal trigger so-called free choice inferences: (1a) gives rise to the inference (1b) (e.g., Kamp 1973).

(1)

  • John can read Article 1, Article 2, or Article 3.

  • John can read Article 1, and he can read Article 2, and he can read Article 3.

In the literature, such inferences are typically (but not always) derived as implicatures, crucially relying on the assumption that a sentence with a disjunction activates domain alternatives. To understand what domain alternatives are, note that a disjunction can be described by the set of elements (objects or propositions) that it covers. For the disjunction in (1a), that set of elements, which we will refer to as the domain D of disjunction, would be D = {Article 1, Article 2, Article 3}. Domain alternatives of a disjunction are other disjunctions that differ from the original in that they are constructed on smaller domains D′⊆D. In other words, a sentence with a disjunction, schematically P(A1 or A2 or A3), with P(_) standing for the environment in which the disjunction is embedded, activates domain alternatives of the form P(A1 or A2), P(A1 or A3), P(A2 or A3), P(A1), and so on. These alternatives have been argued to serve as input for a mechanism that derives free choice inferences, as in (1b). All details and motivation can be found in Fox 2007.

Leaving aside the technical details, assume that one can thus consider free choice inferences as evidence that sentences with disjunction activate domain alternatives. Given most theories of implicatures—say, that of Chierchia, Fox, and Spector (2008) for concreteness—this not only explains the free choice inference in (1), but also makes predictions about the interpretation of a disjunction in the scope of a universal quantifier: a sentence such as (2a) intuitively gives rise to the inference (2b).

(2)

  • Every girl took Apple 1, Apple 2, Apple 3, or Apple 4.

  • Each of the four apples was taken by some girl.

This is indeed a prediction of this approach; let us see why. Schematically, in (2a) the disjunction appears in the environment P(_) = Every girl took _, and therefore one of the domain alternatives that (2a) triggers is P(A2, A3, or A4). This alternative is logically stronger than the original sentence and, according to standard assumptions about implicature derivation, it therefore ends up being negated; that is, the original sentence gives rise to the inference not-P(A2, A3, or A4). One thus obtains the inferences that ‘it is not the case for every girl that she took Apple 2 or Apple 3 or Apple 4’, which, together with the original utterance, entails that ‘some girl took Apple 1’. Similarly for every apple x in the domain, one obtains the inference that ‘some girl took Apple x’. All of these together amount to the inference stated in (2b).

In sum, given the alternatives evidenced by free choice inferences (see (1)), one may derive implicatures and judge (2a) as not true in the situation depicted in figure 1. This prediction is borne out. These implicatures are called distributive inferences (see Spector 2006 as well as quantitative data in Crnič, Chemla, and Fox 2015 and the experiment reported in online appendix B, https://www.mitpressjournals.org/doi/suppl/10.1162/ling_a_00340).

Figure 1

Every girl took an apple, but not every apple was taken

Figure 1

Every girl took an apple, but not every apple was taken

1.2 Indefinites: Free Choice but No Distributive Inferences

Indefinite noun phrases also trigger free choice effects in the scope of a possibility modal: in a context with three salient articles, a possible reading of (3a) is (3b).1

(3)

  • John can read an article.

  • John can read Article 1, and he can read Article 2, and he can read Article 3.

Indefinites introduce existential quantification over a contextually supplied domain; assume for (3a) that this domain is D = {Article 1, Article 2, Article 3}. One can then postulate that the indefinite activates domain alternatives, obtained by exchanging the contextually supplied D with its subsets. For the same reasons as above, this would generate free choice inferences with indefinites. As was done above for disjunctions, one may thus reason inductively and assume that the free choice inferences observed in (3) can be taken as evidence that indefinites (and not only disjunctions) activate domain alternatives. This automatically makes a prediction for cases such as (4a), in which the indefinite appears in the scope of the universal quantifier: namely, that because of the implicatures derived due to domain alternatives of the indefinite, (4a) could be interpreted as (4b) and therefore could be judged not true in the situation depicted in figure 1. Interestingly, there is no experimental or introspective evidence for such a reading in adults, as shown by the experiment reported in online appendix A.

(4)

  • Every girl took an apple.

  • Every (relevant) apple was taken by some girl.

1.3 Summing Up: A Dissociation in Adult Data

If free choice effects with disjunctions and indefinites are implicatures and can thus be taken as evidence that disjunctions and indefinites activate domain alternatives, then distributive inferences are predicted for both disjunctions and indefinites in the scope of a universal quantifier. However, distributive inferences are only attested for disjunctions. There are several ways to go from here. First, one may abandon unifying (implicature) theories of indefinites/disjunctions for free choice/distributive inferences. For instance, there are theories of free choice inferences that treat them as entailments rather than implicatures (see Aloni 2007, Barker 2010) and do not make predictions about how indefinites and disjunctions will be interpreted in the scope of a universal quantifier. Alternatively, as sketched in section 3, one may try to amend implicature theories of free choice so that one can explain the observed dissociation (the absence of distributive inferences with indefinites). We will now discuss a child language phenomenon that may be taken as an argument for the latter approach.

2 Children: Q-Spreading as Distributive Inferences

Strikingly, children seem to have the exact interpretation of the indefinite in the scope of a universal quantifier that is predicted, yet unattested, in adult language. That is, they report that (4a) is false in a situation like that in figure 1. This phenomenon in child language has been variously called quantifier spreading (hereafter, q-spreading), Type A error, and exhaustive pairing error. It has been extensively studied at least since Inhelder and Piaget 1964.

2.1 Q-Spreading as Distributive Inferences

Given the previous discussion, an account of q-spreading is readily available: whatever is responsible for the different behaviors of indefinites and disjunctions in the scope of a universal quantifier in adult language is not operative in child language. Q-spreading in child language may thus be the result of implicatures, derived by negating domain alternatives that indefinite noun phrases ought to activate, since they give rise to free choice inferences.

Such an account is coherent with and similar in spirit to the account of Singh et al. (2016) for the conjunctive interpretation of disjunctions by children in English (see Tieu et al. 2017 for French, German, and Japanese). The two approaches offer accounts of phenomena that are usually considered to be language performance errors in child language in terms of results of a sophisticated and legitimate inference mechanism. They thus call for a research program investigating to what extent different children’s language performance errors are in fact misclassified linguistic inferences.2

2.2 More Empirical Data Points

We cannot review all existing accounts of q-spreading. In short, though, the issue has been attributed to numerous domains such as the syntax or the syntax-semantics interface (e.g., Roeper and de Villiers 1991, Philip 1995, Drozd and van Loosbroek 1998, Geurts 2003), performance (Crain et al. 1996), or problems with cognitive resources and shallow processing (Freeman, Sinha, and Stedmon 1982, Brooks and Sekerina 2006). Most of the cited authors agree, however, in that they locate the problem in the universal quantifier (its syntax, semantics, or complexity). The account we describe (hereafter, the implicature account) does not and thus represents a distinct option that ought to be evaluated in depth. The implicature account of q-spreading also has another simple merit: a full account of children’s grammar is obtained—it is simply the typical account already motivated for adults—and the explanatory burden is shifted from the question of why children have q-spreading to why adults do not. Let us therefore review how this account connects with empirical knowledge on the topic more broadly.

2.2.1 Developmental Path

Aravind et al. (2017) conducted a longitudinal study in which children were tested four times between ages 4 and 7. They found that children at first showed little q-spreading, but that the amount of q-spreading increased with each subsequent testing (the q-spreading errors thus seem to disappear after the age of 7). The same effect was found in two cross-sectional studies reported in Kang 2001 (although not in the experiment reported in Philip and Takahashi 1991). This U-shaped trajectory is difficult to accommodate under accounts that attribute q-spreading to limited cognitive resources and processing (e.g., Brooks and Sekerina 2006). On the other hand, it has been shown that young children start out with low rates of implicatures (e.g., Noveck 2001), and if one could explain how they move from these low rates of implicatures to high rates, one might be able to explain the initial increase in q-spreading errors, which are nothing else than (unwarranted) implicatures under the current view.

Explaining the increase of q-spreading interpretations over development as an increase in implicature derivation is thus tempting, but at this stage it requires some caution. One complexity is that the typical finding that children do not show high rates of implicatures concerns implicatures requiring scalar alternatives. Free choice inferences and distributive inferences, if they are implicatures, are instead based on domain alternatives, and such inferences are more frequently derived by children (see Tieu et al. 2015, Singh et al. 2016, Tieu et al. 2017, Pagliarini et al. 2018). So, the rate of derivation of these inferences is initially higher than that of regular scalar implicatures for young children, but it is not known whether these inferences are at ceiling early on. If they are not at ceiling, they could also increase over time, which is what would be needed to explain the developmental path for q-spreading in the current framework. Overall, a direct comparison between the rate of derivation of free choice inferences and the rate of q-spreading interpretations in different populations would be the most direct test of the implicature approach.

2.2.2 Contextual Effects

It has been noticed that certain manipulations of experimental context can significantly reduce the amount of q-spreading (Crain et al. 1996, Philip 2011). For instance, Philip (2011) and Hollebrandse (2004) report that what is considered to be a topic in the experimental context can influence q-spreading. When the context surrounding the experimental question is about the subject noun that restricts the universal quantifier, there is less q-spreading than when the context is about the indefinite noun. An implicature account of q-spreading has the potential to capture this context dependency, since context naturally influences the derivation of implicatures. More specifically, what might be behind the reduction of quantifier spreading when the experimental context is about the subject noun is the following: the more salient the alternative trigger (in this case, the indefinite), the more likely a child (or adult) might be to generate its alternatives and proceed to the derivation of implicatures. The manipulation in Philip’s and Hollebrandse’s experiments might be drawing children’s attention to the subject noun and away from the indefinite noun: in other words, their manipulations might be reducing the salience of the alternative trigger, thereby reducing the impact of the alternatives.

Online appendix C reveals one way in which context dependency can be observed in the adult counterpart of q-spreading—that is, in distributive inferences with disjunction. This gives further credence to the idea that q-spreading and distributive inferences have much in common.

2.2.3 Superficially Similar Errors

One may also wonder whether q-spreading (also called Type A error) is similar to other types of “errors” children make with universally quantified sentences. Children make other errors with sentences such as (4a), and the implicature account is only able to explain q-spreading. But the other relevant errors are different from q-spreading in ways that may justify a separate account. First, children sometimes accept (4a) when not every girl took an apple, but all of the apples in the image shown to them were taken. But these errors (Type B) have been shown to have different developmental trajectories than q-spreading. In Aravind et al’s. (2017) aforementioned study, with each subsequent testing children made fewer Type B errors, but showed more q-spreading. Second, children may reject (4a) when every girl took an apple, but the image also displays, for instance, a boy who took a banana. This type of error (called Type C error or the perfectionist response) seems to be the least common of the three types of errors and is restricted to very young children (Geurts 2003), which suggests that q-spreading is different in nature from Type C errors as well.

2.2.4 The Role of the Indefinite

The current account locates the issue in the presence of the indefinite and therefore does not make predictions in the absence of an indefinite (or an alternative trigger). However, to date the literature has focused almost exclusively on how children interpret sentences with indefinite noun phrases in the scope of a universal quantifier. If q-spreading were consistently found without indefinites, the implicature account would have to justify that children could believe that some elements, like indefinites, also trigger disjunction-like alternatives.3

Conversely, the implicature account makes systematic predictions when an indefinite is present: children should exhibit q-spreading with indefinites whenever adults exhibit q-spreading-like effects with disjunctions. This is unlike existing accounts of q-spreading, which may locate the problem in the universal quantifier and not in the indefinite. One could thus systematically compare children’s interpretation of an indefinite and adults’ interpretation of disjunctions in new environments, such as the scope of modals and quantifiers other than a universal one. In particular, there are data suggesting that q-spreading does not occur with definite subjects (Drozd 2001), and one can find at least indirect reports of tests of sentences headed by other determiners such as both or cardinal determiners (Takahashi 1991, Roeper, Pearson, and Grace 2011). At this point, it is fair to say that more systematic tests would be informative, with indefinites in various environments and with the new view that q-spreading effects may take the specific form discussed here, that of distributive inferences (and not, for example, confusion about which noun is meant to restrict the universal quantifier).

3 Challenges and Directions concerning Adults and Children

Under the current view, child language facts become simple, as children’s q-spreading behavior is captured by what semanticists have proposed for adults. Adult language facts are more puzzling: the existence of free choice effects and the absence of q-spreading with indefinites creates a tension. Exploring possible ways to resolve this tension is important from the child language perspective, because understanding why distributive inferences with indefinites are absent in adults is crucial for understanding what it is that children must learn in order to achieve adult-like competence in terms of inferences generated with indefinites.

One possibility is that the current account of q-spreading is correct for children, while the analysis of free choice effects as implicatures is incorrect for adults, at least for indefinites. The theory of indefinites, independently of that of disjunctions, would then have to provide an analysis of the free choice effects for indefinites, one that does not lead to the expectation of distributive inferences. One could then argue that children are initially mistaken and believe that indefinites activate domain alternatives, possibly due to exposure to cases such as (3)—in other words, for the same reason that some semanticists are mistaken about their theory of alternatives activated by indefinites. One remaining question would then be how and what (presumably rare) input helps children recover from this assumption and reach adult-like knowledge.

Another possibility is that the analysis of free choice effects as implicatures is entirely correct, but that distributive inferences with indefinites are blocked or masked for adults by some independent mechanism, which children are not yet aware of. Let us discuss two such possible mechanisms.

3.1 Q-Spreading Masked by Domain Restriction of Indefinites

A salient difference between indefinites and disjunctions is that, for indefinites, the relevant domain of individuals—also known as the domain restriction for quantified phrases more generally (see, e.g., von Fintel 1994)—is left implicit and is thus potentially subject to more variability. This could indeed affect the inferences we are interested in. For instance, there are some differences in free choice effects with indefinites as compared to disjunctions (see (5)),4 which become less mysterious once the variability of domain restriction is factored in.

(5)

  • John can present an article. But not Article 3.

  • John can present Article 1, Article 2, or Article 3. #But not Article 3.

In (5), it looks as if the free choice inference is absent or weaker with the indefinite, but this is readily explained by noting that, quite generally, the domain can be flexibly adjusted with the indefinite, and not with the fully explicit disjunction. Concretely, in (5a) Article 3 may be dropped from the implicit domain of the indefinite if the second clause makes this necessary. Domain restriction may thus explain the difference in strength of free choice effects between indefinites and disjunctions.

But the challenge is to explain a difference between free choice inferences and distributive inferences with indefinites (i.e., presence/strength of the former and absence/weakness of the latter). A theory of domain restriction would presumably be based on pressures in favor of minimally/maximally large domains, or pressures to end up with a maximally weak/strong meaning. But such pressures would probably not help to distinguish between the cases in (3) and (4): to the extent that domain restriction may weaken free choice inferences, it would also weaken distributive inferences. If such a theory of domain restriction could be formulated, though, one could then go on to seek independent evidence showing that indeed domain restriction works differently in child language than in adult language.

3.2 Q-Spreading Blocked by Intervention Effects

There may be a piece of adult grammar that blocks the distributive inferences with indefinites for adults that is missing both in children’s grammar and in semanticists’ theory of grammar. This piece of grammar would be missed by children for lack of evidence up to a certain age or for lack of the competence needed to deploy it. Let us see what this piece of grammar could be like.

Chierchia (2013) develops an account for the distribution of negative polarity items (NPIs), such as the word any in English. He proposes that NPIs activate domain alternatives, which are then exhaustified by a syntactically present exhaustivity operator exh. Simplifying considerably, the semantic import of exh is to negate the alternatives activated by its prejacent that are not entailed by this prejacent. According to the proposal, NPIs are licensed when the exhaustification does not result in a contradiction. For instance, if one considers a domain D with three apples, D = {Apple 1, Apple 2, Apple 3}, any is licensed in (6a) because all of the domain alternatives of (6a), of the form shown in (6b), are entailed by (6a). However, this account faces a challenge. Consider the ungrammatical (7a). Its domain alternatives are as in (7b). Clearly, the exhaustification of (7a) does not result in a contradiction—why, then, is the NPI not licensed there? To address this challenge, Chierchia (2013) proposes that every creates intervention effects: in a configuration such as (7a), in which the relevant operators are in the syntactic configuration schematized in (8), the alternatives of the NPI cannot be exhaustified for syntactic reasons (simplifying somewhat, Chierchia argues that the common syntactic features of the universal quantifier and the indefinite create a minimality violation for the exhaustivity operator). This, by assumption, makes the NPI ungrammatical in the scope of every.

(6)

  • Mary didn’t take any apples.

  • Mary didn’t take an apple in D′ (with D′ ⊆ D).

(7)

  • *Each of the three girls took any apples.

  • Each of the three girls took an apple in D′ (with D′ ⊆ D).

(8) *[exh [every [any]]]

The absence of q-spreading with indefinites in adult language may be the same type of situation: the universal quantifier creates intervention for the exhaustification of the alternatives of indefinites in the same position as any above, and therefore implicatures for sentences such as (4a) would be blocked. However, as already mentioned, adults do exhibit q-spreading with disjunctions (i.e., distributive inferences; see online appendix B). To the extent that this phenomenon results from exhaustification, this exhaustification happens within the same syntactic configuration as in (8), with the disjunction occupying the position of any. Hence, the universal quantifier does not intervene for all elements that activate alternatives and are in its scope: it does not intervene for the exhaustification of disjunction, and this selectivity of intervention would have to be explained.5

If the implicature account of q-spreading in child language is correct and indeed adults do not exhibit q-spreading with indefinites due to intervention effects, then the difference between children and adults would be that children are not sensitive to intervention effects. It seems to us that this account involves many nontrivial pieces. But it has the virtue of being testable: children might accept polarity items in the scope of the universal quantifier, at least to the same extent that they show q-spreading effects.

4 Conclusion

Given current assumptions regarding the similarity of alternatives activated by indefinites and disjunctions, the fact that disjunctions, but not indefinites, trigger domain implicatures in the scope of a universal quantifier is puzzling. We have proposed that even if current semantic theories turn out to be incomplete for adults, they may be entirely correct for children, with q-spreading effects revealing the expected presence of domain implicatures. We discussed two possible reasons why children and adults behave differently: (a) children and semanticists are both wrong to assume that indefinites activate the same alternatives as disjunctions; (b) indefinites do activate the same alternatives as disjunctions in child language, but children are missing a piece of adult grammar (perhaps flexible domain restriction of indefinites or sensitivity to intervention effects) that blocks some of their effects.

Notes

1 The indefinite in (3) can also get a specific interpretation; (3) thus has a reading in which there is a particular article that John can read, but this is not the reading that concerns us here.

2Mascarenhas (2014) discusses a related research program of sorting adults’ reasoning “errors” into actual performance errors and legitimate linguistic inferences.

3Inhelder and Piaget (1964) report that children may reject sentences like All the circles are blue in a situation in which all the circles are blue but there are also blue squares. Studies by Philip and Takahashi (1991) and Takahashi (1991) have shown some degree of q-spreading (a) with indefinites; (b) with transitive sentences with null objects like Every boy is driving, with children saying “false” when some car has no driver; and (c) with intransitive sentences like Every dog is sleeping, with children saying “false” when there are some extra beds in which no dog is sleeping. While one may argue that there is a null indefinite hidden in these cases, what would be more relevant is to understand, by means of systematic investigations, whether a unified account of these errors with q-spreading with indefinites is justified to begin with (see section 2.2.3 for other types of errors children make with universally quantified sentences, though arguably errors with different properties).

4 Thanks to an anonymous LI reviewer for encouraging us to discuss these data.

5 Perhaps the universal quantifier intervenes only when the element that activates alternatives is of some particular type; surely any and indefinites are more similar to one another than they are to disjunctions. But note that items like some do trigger implicatures when in the position of any in configurations such as (8), so one would need to explain a broader pattern: the universal quantifier intervenes with any and indefinites, but not with disjunctions and some.

Acknowledgments

We would like to thank Amir Anvari, Moshe E. Bar-Lev, Gennaro Chierchia, Danny Fox, Martin Hackl, Naomi Havron, Daniel Hoek, Manuel Križ, Jeremy Kuhn, Mora Maldonado, Philippe Schlenker, Florian Schwarz, Jesse Snedeker, Andres Soria Ruiz, Benjamin Spector, Yasutada Sudo, Lyn Tieu, and the audience at SALT 28 for helpful discussion, critiques, and questions. The research leading to this work was supported by the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement n.313610, and by ANR-17-EURE-0017.

Authors’ affiliations: Milica Denić: Laboratoire de Sciences Cognitives et Psycholinguistique (ENS, EHESS, CNRS); Institut Jean Nicod (ENS, EHESS, CNRS); Département d’Etudes Cognitives, Ecole Normale Supérieure, PSL University

Emmanuel Chemla: Laboratoire de Sciences Cognitives et Psycholinguistique (ENS, EHESS, CNRS); Département d’Etudes Cognitives, Ecole Normale Supérieure, PSL University

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