Free scope

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.

118 A semantic procedure for scope construal 5 As already anticipated, this can be achieved by changing the context that the shift can capture, more specifically by specifying the reduction rule not by of usingDorD⁰, which both do not contain any resets, but usingC, which was defined as being arbitrary contexts containing any number of resets. In order to keep the shift operator with this exceptional wide scope behavior apart from the shift operators ξ^weak and ξ^strong from above, we add a new mode that we call free.

Mode::=. . .|free

We therefore have a new operator ξ^free with the following reduction rule:

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

This way, an expression of form ξ^freek.E is not restricted to capturing the context up to the nearest enclosing reset, but can capture the context up to an arbitrary reset.

Let us illustrate this with an example. Consider (5.53a). The existential some zombie can either take narrow scope, yielding a reading where everyone takes scope over the existential, or wide scope, yielding the reading that there is some specific zombie that everyone believes to be able to run. That is, the intermediate clause boundary does not restrict the scope of the existential, although it would, in contrast, restrict the scope of the universal (see (5.53b), which only has a linear scope reading).

(5.53) a. Everyone believes[that some zombie is able to run].

b. Someone believes[that every zombie is able to run].

The denotation of the noun phrase will be an impure generalized quantifier denotation as familiar from the previous sections, using the operatorξ^free:

ξ^freek.∃x.(zombiex)∧(k x)

Let us assume that the denotation ofis able tocan be represented as a predicate isAbleToof type (e→t)→(e→t), i.e. that applies to a property (presumably expressing some action like running or hunting bears) as well as to an individual, and predicates of this individual that it is able to do the specified action. Then the denotation of the embedded clausethat some zombie is able to runamounts to the following expression:

h((isAbleTo run) ξ^freek.∃x.(zombiex)∧(k x))i

5.4 Quantificational noun phrases 119 I abbreviate the denotation ofsome zombieasξ^freek.E_zombie. And analogously, I abbreviate the denotation ofeveryone,ξk.∀y.(persony)⇒(k y), asξk.E_person.The semantic expression corresponding to the whole sentence then is:

h((believeh((isAbleTo run) ξ^freek.Ezombie)i) ξk.Eperson)i

The expressionξk.Eperson captures the context up to the nearest enclosing de-limiter, which is the outer reset. The existential ξ^freek.Ezombie, on the other hand, captures the context up to an arbitrary enclosing delimiter, which hence can either be the inner reset or the outer reset. Capturing the inner reset leads to the narrow scope reading (5.54a), and capturing the outer reset leads to the wide scope reading in (5.54b).

(5.54) a. ∀y.(persony)⇒((believe ∃x.(zombiex)∧((isAbleTo run)x))y) b. ∃x.(zombie x)∧ ∀y.(persony)⇒((believe((isAbleTo run)x))y) For the unambiguous sentence (5.53b), however, we would derive only one reading. The sentence denotation would be like above with the difference that the strong quantifier resides in the embedded clause and the existential resides in the matrix clause. Abbreviating the denotation of the universalevery zombie as ξk.Ezombie and the existentialsomeoneas ξ^freek.Eperson, it amounts to:

h((believeh((isAbleTo run) ξk.E_zombie)i) ξk^free.E_person)i

Again, the universal (here ξk.Ezombie) captures the context up to the nearest enclosing delimiter, which is the inner reset. The existential, on the other hand, captures a context up to an arbitrary delimiter. Since there is only one in this case, namely the outer reset, only one reading, given in (5.55), is derived.

(5.55) ∃y.(persony)∧((believe ∀x.(zombiex)⇒((isAbleTo run)x))y) A desirable consequence of the treatment of indefinites sketched here is that it automatically derives the right truth conditions for sentences like (5.56), which prove problematic for unselective binding approaches.

(5.56) If some human is sacrificed, Cthulhu will awake.

First note that (5.56) has two readings, depending on whether the existentiala humantakes narrow scope over the if-clause, which is schematically presented in (5.57a), or wide scope over the whole sentence, which is represented in (5.57b).

(5.57) a. (∃x. xis a human andxis sacrificed) ⇒Cthulhu will awake b. ∃x. xis a human and (xis sacrificed ⇒Cthulhu will awake) c. ∃x.((xis a human andxis sacrificed)⇒Cthulhu will awake) The interpretation that unselective binding approaches usually derive is the one in (5.57c). However, this is not correct because it would already be true in case there is anxwhich is either not a human or is not sacrificed.

120 A semantic procedure for scope construal 5 Let us look at what happens in our approach. The denotation forsome hu-manis the expected impure generalized quantifier using the free shift operator:

ξ^freek.∃x.(humanx)∧(k x)

An important fact here is thatk occurs only in the second conjunct. That is, whatever context is captured, it will be plugged into that second conjunct and not affect the restriction (human x). Let us see what this means for the con-struction of the sentence denotation. Assume thatif. . . thensubcategorizes two CPs, i.e. there are three resets introduced: one enclosing the whole sentence, one enclosing the if-clause, and one enclosing the then-clause. Then we derive the following denotation for the whole sentence:

h h(sacrificed ξ^freek.∃x.(humanx)∧(k x))i ⇒ h(awake Cthulhu)i The ξ^free-expression now captures a context up to some enclosing delimiter.

This is either the reset enclosing the if-clause or the reset enclosing the whole sentence. The former gives (5.58a) corresponding to the narrow reading (5.57a) and the latter gives (5.58b) corresponding to the wide scope reading (5.57b).

(5.58) a. (∃x.(humanx)∧(sacrificedx))⇒(awake Cthulhu) b. ∃x.(humanx)∧((sacrificedx)⇒(awake Cthulhu))

Thus, the restriction (humanx) ends up in the right place in both cases. This is due to the above mentioned fact that the captured context is plugged into the second conjunct in the noun phrase denotation, separate from the restriction.

Something like in (5.57c) could therefore not happen.

Let me end this subsection with a remark concerning the intention of this subsection. I do not want to make a claim with respect to the question whether wide scope existentials are in fact quantifiers or rather referential expressions.

This subsection rather served to demonstrate how exceptional wide scope be-havior can in principle be modeled in the advocated approach to quantifiers.

Whether one wants to adopt this mechanism or rather rely on different tools is a matter of taste and conviction. In either case, the free scope account of this subsection will become useful again in the next section, when we treat in situ wh-phrases.

So let us now concentrate on yet another kind of scope-taking operators:

wh-expressions.