Weak quantifiers

114 A semantic procedure for scope construal 5 (5.48) [The god of whose ancestors]did Shamhat pray to?

With all this freedom to establish scope, let us now look at restrictions.

First, we return to the clause boundedness of quantifiers, that we briefly men-tioned when introducing the complementizer denotations. The quite robust observation from Section 2.3 is that quantifiers can take scope only over the clause they occur in, especially they cannot outscope quantifiers occurring in a higher clause. This is illustrated in (5.49), which only has the linear scope reading (5.49a) but not the inverse scope reading (5.49b).

(5.49) [CP1Everyone knows[CP2that Ishtar is angry with most humans]].

a. Everyxknows that for most humansy, Ishtar is angry withy.

b. For most humansy, everyxknows that Ishtar is angry withy.

In the account of scope construal sketched so far, this follows directly from the fact that the complementizers involved in building both CPs introduce a delimiter. For example in (5.49),most humanscan capture the context only up to the nearest enclosing reset, which is introduced by the embeddedthat. The way we specified the evaluation contexts used in the reduction rule for shift, there is no way to skip this reset. Hence most humans can take scope only over CP2 but not over CP1. This captures that (5.49) does not have a reading where most humansoutscopeseveryone.

Also, scope islands like relative clauses and complex NPs, as in (5.50a) and (5.50b) respectively, are straightforward, because in both cases the quantifica-tional noun phrase occurs inside a CP which delimits its scope.

(5.50) a. A beast[which slaughtered every sheep]was hunted down.

b. Anu heard the rumor[that every beast died].

Nevertheless, the theory of scope we have right now is both too permissive and too restrictive. First, it is too permissive, because scope ambiguities do not arise with all quantifiers. Recall from Chapter 2 that weak quantifiers such as no humancannot outscope other quantifiers. And second, our treatment of scope is too restrictive, because indefinites are extremely free in taking scope at an arbitrarily high point in the structure. We will first focus on modelling the behavior of weak quantifiers and turn to indefinites only in Section 5.4.3.

5.4 Quantificational noun phrases 115 rise to two readings. So fixing an evaluation order would leave only one of the possibilities and rule out the other one. This is done by specifying a slightly different versionD⁰ of subcontextsD, whereFranges over pure expressions:

D⁰::= [ ]|(E D⁰)|(D⁰ F)

I will call such contextsweak delimited evaluation contexts, or weak contexts for short.

The way D⁰ is defined now, an applicand can be reduced only when the argument is a pure expression, i.e. does not contain any shifts. That is, the reduction of impure expressions in an application has to proceed from outside to inside. Suppose we have the application of a two-place predicatepredto two ξ-expressions:

h((pred ξk1.E1) ξk2.E2)i

Changing the context a shift can capture to D⁰ leaves only one possibility to reduce this expression: Firstξk2.E2has to be evaluated and only thenξk1.E1

can also be evaluated. This is because if ξk1.E1 was evaluated first, it would capture the context ((pred [ ]) ξk2.E2), which is not a licit weak contextD⁰. If you consider pred as a verb denotation, ξk₁.E₁ as the object noun phrase of the verb and ξk₂.E₂ as the subject noun phrase, then the fixed evaluation order amounts toE₂taking scope overE₁, i.e. derives the linear scope reading ifE₁ andE₂ contain scope taking operators. We will see an explicit example in more detail a bit later.

Now we have two subcontexts that we could use in the reduction rule for shifts. Accordingly, we want to have two shifts at our disposal: one that captures contexts D, and one that captures weak contexts D⁰. To this end, the definition ofξ-expressions is changed such that it encodes which context is captured. We do this by means of a superscriptMode.

E::=. . .|(ξ^Modek::τ →α.E::β)::τ_α^β Mode::= weak|strong

Now we need two reduction rules, one forξ^strongusingD, and one forξ^weak usingD⁰:

C[hD[ξ^strongk.E]i]BC[hE{k7→λx.hD[x]i}i]

C[hD⁰[ξ^weakk.E]i]BC[hE{k7→λx.hD⁰[x]i}i]

Note that the reduction rule itself is exactly the same; the only difference is which kind of context is captured. In the following, I will often writeξas short

116 A semantic procedure for scope construal 5 forξ^strong. This way, all instances ofξfrom the last section rightly correspond to theξ^strongof this section. Furthermore, I will usually writeξ^weakasξ⁰. This is a form that is slightly better readable and corresponds to the use of the apostrophe in the definition of contextsD andD⁰.

The main consequence for the semantics of our grammar fragment is the following: For quantifier denotations usingξ^weak, the evaluation order is fixed, for quantifier denotations usingξ^strong, the evaluation order is free. That is, the former derive only linear readings, while the latter allow for scope ambiguities.

To see weak quantifiers in action, consider example (5.51a), which has a linear reading (for most gods there is no human they admire) and does not have an inverse scope reading (no human is such that most gods admire him).

When deriving the corresponding semantic interpretation, we arrive at (5.51b), usingξ^weak. The denotation ofnois given by¬∃and the denotation ofmostP areQis represented asMostx:(P x).(Q x). Thusmostcan be seen as relating two sets; the exact modeltheoretic interpretation, however, is not of concern here.

(5.51) a. Most gods admire no human.

b. h((admire ξ⁰k.¬∃x.(humanx)∧(k x)) ξ⁰k.Mosty:(gody).(k y))i There is only one possibility to reduce the expression in (5.51b): First, the denotation of most gods has to be evaluated. This is because reducing the denotation ofno humanfirst would capture the following illicit weak context:

((admire[ ]) ξk.Mosty:(god y).∧(k y))

The reduction proceeds as follows, according to the reduction rule for ξ^weak: h((admire ξ⁰k.¬∃x.(humanx)∧(k x)) ξ⁰k.Mosty:(god y).(k y))i

BhMosty:(god y).(λz.h((admire ξ⁰k.¬∃x.(humanx)∧(k x))z)iy)i BhMosty:(god y).h((admire ξ⁰k.¬∃x.(humanx)∧(k x))y)ii

Note the role of static scoping here: The reduction rule introduces a new reset around the captured context. This limits the context the yet unreduced ξ-expression will capture to the context below the operator Most. Hence, no human is not able to outscopemost gods. The reduction proceeds as follows:

hMosty:(gody).h((admire ξ⁰k.¬∃x.(humanx)∧(k x))y)ii BhMosty:(gody).h ¬∃x.(humanx)∧(λz.h((admire z)y)ix)ii BhMosty:(gody).h ¬∃x.(humanx)∧ h((admirex)y)iii Finally the resets can be deleted and the result is:

Mosty:(gody).¬∃x.(humanx)∧((admirex)y)

5.4 Quantificational noun phrases 117 This is the linear scope reading and it is the only reading that can be derived with the evaluation order fixed like above.

The results are accordingly when considering sentences with a strong quan-tifier and a weak quanquan-tifier: the strong one can outscope the weak one but not vice versa.

Let me finally note that the approach to modeling different scope behavior demonstrated here is similar to one that Shan ([99],[100]) proposed. He employs a hierarchy of generalized shifts and resets superscripted with strength levels.

Quantifiers can take control at different levels and depending on the level, they can or cannot outscope each other. Only if they take control at the same level, scope ambiguities occur. Shan’s approach thus differs from ours in maintaining a fixed evaluation order and relying on the hierarchy of quantifiers to determine scope behavior. My approach in this chapter, on the other hand, invokes two different shifts (for weak and strong quantifiers) together with one delimiter for both of them, and enforced an evaluation order for weak quantifiers which prevents them from outscoping other quantifiers.

A yet different possibility was proposed by Stabler [105], who adopted the idea that different quantifiers check features in different functional domains. In our terms, this can be modeled by introducing a different shifts and resets for all those domains. We would then derive that quantifiers can take scope only in their specific domain and cannot outscope quantifiers in higher domains.

Now, before moving on, let us turn to the exceptional wide scope behavior of indefinites. I will sketch how the mechanism employed for strong and weak quantifiers can be extended to also capture those cases. The strategy should be familiar by now: leave the reduction rule for shift as it is but change the context that the shift captures.