Quantifier Spreading in Child Language as Distributive Inferences

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)

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

b. 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)

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

b. 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).

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

Figure 1.

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

(3)

a. John can read an article.

b. 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...