Domain widening and at least Jon Ander Mendia – UMass Amherst

T HE P ROBLEM . Sentences with the superlative numeral modifier at least give rise to an ignorance inference that the speaker doesn’t know exactly how many students came (Geurts & Nouwen 2007). In some contexts, notably under necessity modals, these inferences may disappear, giving rise instead to an “authoritative” reading (Büring 2008). (1)

[The teacher]: You must read at least 3 books to pass the class, # but I don’t know how many.

P ROPOSAL . This paper argues for a treatment of at least in line with epistemic indefinites in a Hamblinstyle semantics. In my account at least serves the purpose of widening the domain by introducing a set of alternatives that keeps expanding until a suitable operator closes it off (as in Kratzer & Shimoyama 2002), and ambiguities can arise as a product of the kind of operator that captures the set. BACKGROUND . Ignorance inferences with at least have been analyzed by drawing a parallel between at least n and disjunction, assuming the disjunctive meaning in (2a). The ignorance is then derived just like with disjunctions (cf. Sauerland 2004): a weak utterance like x ≤ 20 implicates that the speaker does not believe that any of the stronger alternatives are true (2b). This explains why “bare” at least only has the ignorance implicature and is incompatible with full knowledge of the quantity in question (Nouwen 2010). (2)

a. At least 20 A’s B ⇔ exactly 20 A’s B or more than 20 A’s B b. ¬B(x = 20) ∧ ¬B(x > 20) ∧ B(x ≥ 20) c. There are exactly 15 speakers, # so that’s at least 10.

epistemic

The ambiguity of (1) with  + at least n has been proposed to be either reducible to a scope ambiguity (Büring 2008, Coppock & Brochhagen 2013) or a strong implicature (Nouwen, to appear). In the scopal approach, the authoritative reading emerges when at least n scopes over the modal, whereas the epistemic meaning is attributed to a logical form where at least takes scope over the necessity modal. Implicature based approaches account for the epistemic reading as a weak implicature (4a), whereas the competence assumption (Geurts 2010) derives a strong implicature which conveys the authoritative reading (4b). (3)

a. Weak Implicature: ¬B(x > 3) ∧ ¬B(x = 3) ∧ B(x ≥ 3) b. Strong Implicature: B¬(x > 3) ∧ B¬(x = 3) ∧ B(x ≥ 3)

(4)

a.  [ 3 = max(λd. you read d-many books) ∧ 3 > max(λd. you read d-many books)] b. 3 = max(λd.[you read d-many books]) ∧ 3 > max(λd.[you read d-many books])

epistemic authoritative auth. epi.

Neither analysis is problem free. The scopal analysis predicts that (i) at least is able to QR and (ii) in environments where at least cannot raise there is no ignorance effect. However, (5a) shows that at least (i) cannot raise in the prototypical QR cases, and that (ii) those sentences are still compatible with ignorance. (5)

A student wrote at least three papers, ok but I don’t remember how many

*[x ≤ 3] > ∃

The implicature based account lacks a better motivation for why only comparatives are compatible with full knowledge, even though in principle they can also be part of informativity scales just as well as superlatives. (6)

The are exactly 15 speakers, ok so that’s more than 10.

A NALYSIS . I take at least to be a domain widener: just as Free Choice indefinites pick a member from a given set (Alonso-Ovalle & Menéndez-Benito 2010), so does at least pick a cardinality from a set of numbers. The widening of the domain happens for different reasons: the speaker is unsure and wants to avoid committing to a stronger statement (epistemic reading), or because there are several values that satisfy a requirement (authoritative reading). In both cases, the speaker widens the domain to avoid committing himself to a stronger statement (Kadmon & Landman 1993). This captures the intuition that at least signals an anti-specificity requirement (Nouwen 2010): the meaning of at least is associated to a set Jat-least nK = {n, n+1, n+2, ...}, and the set keeps expanding by pointwise functional application (7a), until a suitable operator is applied, returning a singleton set. In declarative sentences –and in the absence of any other operator– alternatives are bound by existential closure [∃] (7b). (7)

a. Jat least 2 men cameK = {p ∈ Dhsti : ∃x[x is a set of men in w ∧|x| ≥ n ∧ p = [λw.x came in w]]} = A = {2 men came in w, 3 men came in w, 4 men came in w,. . . } w,g b. [∃](JAK ) = λw’.∃p[p ∈ JAKw,g ∧ p(w’)]} = {the proposition that is true in all worlds in which some proposition in A is true}

In the case of +at-least, A may be closed either by [∃] or [∀], defined as follows (from Kratzer 2005): (8)

a. J∃Kw,g (JAKw,g )={λw’.∃p[p ∈ A ∧ p(w’)]} = q b. J∀Kw,g (JAKw,g ) = {λw’.∀p[p ∈ A → p(w’)]} = q

If it is closed by [∃], the ignorance effect arises: [∃] takes the set of alternatives A generated by at least and returns the singleton q containing the proposition that is true in all worlds w if one of the propositions in the set A is true in w (9b). Then the necessity modal takes q as its argument and returns the singleton set containing the proposition that for every accessible world w the proposition q is true in w (9c). That is, the requirement is such that for every accessible world w, there is some proposition from A that is true: you read 4 books in w1 , you read 5 books in w2 , you read 6 books in w3 , etc. Thus, the speaker knows that a value x may meet the requirement just in some worlds, but not in others, giving rise to the ignorance effect. (9)

a.  > [∃] (you read at least 3 books) b. J[∃]Kw,g (JAKw,g ) = {λw0 .∃p[p ∈ A ∧ p(w0 )]} = q c. J[]Kw,g (JqKw,g ) = {λw.∀w0 [w0 ∈ Accw → q(w0 )]}

[=(1)]

a.  > [∀] (you read at least 3 books) b. J[∀]Kw,g (JAKw,g ) = {λw0 .∀p[p ∈ A → p(w0 )]} = q c. J[]Kw,g (JqKw,g ) = {λw.∀w0 [w0 ∈ Accw → q(w0 )]}

[=(1)]

The authoritative reading arises when [∀] applies to the set of propositions A and returns a singleton q that contains the proposition that is true in all worlds w if every proposition in A is true in w (10b). The modal then applies to q and returns the singleton containing a proposition that is true if for every accessible world w, the proposition q that is true in every world if every proposition p in A is true (10c). That is, it requires every proposition from A to be a valid value in every world: that is, the weakest alternative will be true in every world w, and so there is certainty about the lower bound of the value. (10)

Treating at least in a Hamblin-style semantics offers a less stipulative solution to ignorance inferences. There are cases where the [∀] is overtly present in the structure (11a), or is introduced by other kind of covert speech act operators (11b-c); in those cases the presence of an authoritative reading is correctly predicted,. (11)

a. Every student wrote at least two papers, John wrote 2, Mike 3 and Ethan 10. b. Write at least 3 papers! (Although more is OK too.) c. Cars have at least one door

overt ∀ imperatives generics