6.5 On too in alternative questions
6.5.1 The felicitous cases
Let us start by discussing the felicitous cases. Since (86) and (87) are symmetric, we will focus on (86).
We first give an entry foronly based on our definition ofexh. The only difference is that we takeonly to presuppose its prejacent, whileexhasserts it.
(89) ⟦only⟧o=λXT.λs.∃s′∶s⊑s′∧X(s′⊺) ∧ ¬∃t∶ [X(t) ∧t<Cs′] ⟦only⟧f= {⟦only⟧o}
S Int ○6 Closed ○5 2 ○ 1 ○ A Mary danceF ∼C1 too(C1) or ○4 3 ○ B Mary singF ∼C2 only(C2)
The relevant values for the meaning of each disjunct are:
1. ⟦A⟧o=Dˆ(m) 2. ⟦B⟧o=Sˆ(m) 3. ⟦A⟧f = ⟦B⟧f= {λs.αs⊆πs∧ P(m) ∶ P ∈D⟨e,T⟩} 4. ⟦○1⟧o=λs.∀i∈πs[1Cgi⊆ ⟦A⟧f∧ ⟦A⟧o∈C1gi∧ ∣C1gi∣ ≥2] ∧Dˆ(m)(s) 5. ⟦○3⟧o=λs.∀i∈πs[C2g i⊆ ⟦B⟧f∧ ⟦B⟧o∈C2gi∧ ∣C2gi∣ ≥2] ∧Sˆ(m)(s) 6. ⟦○2⟧o=λs.∀i∈πs∃P[P∈C1g i∧ ⟦A⟧o∈C1gi∧C1gi⊆ ⟦A⟧f∧ (⟦○1⟧o⊭P) ∧true(P, i)] ∧Dˆ(m)(s) 7. ⟦○4⟧o=λs.∃s′[s⊑s′∧ ∀i∈π s′[C2gi⊆ ⟦B⟧f∧ ⟦B⟧o∈C2gi∧ ∣C2gi∣ ≥2] ∧Sˆ(m)(s⊺)] ∧ ¬∃t[∀i∈πt[C2gi⊆ ⟦B⟧f∧ ⟦B⟧o∈C2g i∧ ∣C2gi∣ ≥2] ∧Sˆ(m)(t) ∧t<C2 s′]
For simplicity, let us first assume thatC1andC2are uniformly assigned the same value, a set consisting of only two elements: ˆD(m)and ˆS(m). This means that the only possible antecedent forP in the first disjunct is ˆS(m), and the only effect ofonlyin the second disjunct is to negate ˆD(m). On top of that,onlypresupposes the truth of ˆS(m). The net result is that both disjuncts presuppose that Mary sings, the first disjunct asserts that she (also) dances, and the second disjunct asserts that she doesn’t dance. This disjunction is therefore equivalent to the polar question “Does Mary dance?”, but with a stronger presupposition (namely, that she sings).
Now, let’s see what other valuesC1andC2could take. Let us first note that any value other than ˆS(m)
in C2 is negated byonly. Assuming that the presupposition of the first disjunct projects, this means that
resolvingP to anything other than ˆS(m)would result in a contradiction when evaluating the second disjunct. Conversely, since the second disjunct presupposes ˆS(m), which is not supported by the assertive content of the first disjunct, the disjunction as a whole will anyway presuppose that Mary sings, so the presupposition of the first disjunct is necessarily satisfied as long as ˆS(m) is in C1. In short, the disjunction as a whole will only ever presuppose that Mary sings, and adding more alternatives toC1 andC2 has no effect but to
strengthen the meaning of the second disjunct.
The disjunction in place, we can now turn to the top of the tree. Because this is an alternative question and not a polar question, we first apply Closed. There’s a problem however: both disjuncts contain a focus-operator, and given the definition of the∼-operator, no alternative projects (or to be precise, the focus value of the whole disjunction is just the singleton of its ordinary value). This means that attaching a new
C3to the tree will result in a presupposition failure, and so we cannot give a new restrictor to the exhaustifier introduced byClosed.
We see two solutions to this problem. The first would be to assume that focus-sensitive operators (more precisely, the∼-operator) do not reset the focus alternatives. That is to say, each disjunct would project its own alternatives, which would be combined by disjunction by point-wise function application as usual. The problem is, these alternatives would then combine withtoo andonly, resulting in a focus value consisting of arbitrary disjunctions of the form “Atoo or only B” with possibly complex presuppositions.
Another solution is to recycle the focus value of the disjuncts, under the assumption that it is identical across disjuncts. While this may seem far-fetched at first, Abenina-Adar and Sharvit (2018) have already proposed a very similar constraint to account for the contrast in (90).
(90) a. Did John even eat the cakeF?
b. #Did John even eat the cakeF, or the candyF?
They argue for a constraint ofDomain Uniformity, which requires all disjuncts in an alternative question to share the same focus value. Without going into the details of their argument, we can adopt this constraint, which in the present case means that C1 and C2 must be assigned the same value. It is only natural then,
to assume that this shared restrictor can be picked up byClosed.
We can then applyClosed to the disjunction, but this is almost vacuous. Theonly-disjunct is already exhaustive of course. The too-disjunct presupposes the truth of the prejacent of only, so this cannot be negated by Closed. The result is simply to exclude any third alternative from the first disjunct (to the effect that Mary may be dancing on top of singing, but nothing else).
Finally, we can applyInt. It only adds the presupposition triggered byQto the meaning of the argument as the argument is already inquisitive.
A final note is in order. We haven’t discussed the fact that droppingonlyfrom the disjunct not carrying the additive particle seems to be slightly degraded (though not completely ungrammatical). There may be several reasons for this. One could be very superficial, e.g., a parallelism requirement between the two disjuncts. We may have a more principled explanation however. The application of Closedadds to the disjunct with the additive particle the negation of all alternatives except the one that is presupposed by too. From the other alternatives, it removes all alternatives, including the antecedent oftoo. . . unless the antecedent oftoo is the second disjunct itself. Because of presupposition projection, the only way for the second disjunct not to be contradictory (and therefore vacuous) is to be picked as the antecedent oftoo. The problem now may have
to do with the way presuppositions project from disjunction (a problem we have largely ignored so far): the presupposition of each disjunct may be evaluated against the negation of the other disjunct (alternatively, this may be done incrementally, with the presupposition of the first disjunct projecting as is). If this is the case, the presupposition oftoo would be contradicted by the negation of the second disjunct. Insertingonly
would transform the assertive part of the second disjunct into a presupposition, which happily projects out of the whole disjunction.