Scope (formal semantics)

There are two possible readings which are called i. the surface reading(SR) and ii. the inverse reading(IR).[1] In the following both readings are described in different contexts in English.

(1) A rabbit has verried every tree.

SR: There exists only one rabbit, who had verried every tree.

The surface reading has its roots in the overt syntax and shows that Quantifier phrase 1(QP1) has scope over QP2 visualized by the linear form of the sentence.

To make the relation between the quantified phrases more understandable it is formalized like: ∃ < ∀

IR: For every tree there exists a rabbit (potentially different), who is veering around it.

The inverse reading creates a quantifier-shift, which makes it possible that QP2 has scope over QP1.

This is formalized in the following way: ∀ < ∃

For IR, investigated under the base of logical form(LF), it is important, that the more embedded version of the QP’s is chained to quantifier Raising. Using movement, QP2 is reaching a hierarchically higher position. This position occupied by the QP2 makes it possible to have scope over QP1.

(2) Every driveway is blocked by a tree.

In (2) the SR is possible if there is for every driveway a (potential different) tree who has blocking status. The IR is semantically restricted because it is impossible that a real tree is blocking every driveway in the world, maybe at least three driveways (if the tree comes from the canadian forest where the existence of driveways is mostly rare).

(3) A water bottle in every backpack is leaking.

In (3), the SR is semantically restricted, because (at least) one water bottle that is located in every backpack is leaking. The water bottle is not splittable and therefore the appearance of it in every backpack at the same time is impossible. In contrast the IR is possible after the quantifier raising.

The ambiguity behaves different if the QP's show independent status:

(4) Some man loves every woman. [2]

SR: There is a single man x who loves every woman.

IR: For every woman z there is a (potentially different) man x such that x loves z.

In (4) the semantical independence of the quantifiers makes it possible that no restrictions on both reading appear.

Using a quantifier with a pronoun, the following three examples [3] show the boundaries in which different scopal readings are possible:

(1) a. Every boy read his book.

b. I showed every boy his book.

c.* That every boy left upset his teacher.

In a-b there is for every boy one potential book. In c it is restricted, because the pronoun is not referring to a contextually specified boy, every boy could be in the role of upsetting the teacher.

Split scope is the phenomenon where different components of an expressions item's meaning take scope in different places. Negative quantifiers are one category of expression which have been argued to take split scope.

1. The company need fire no employees.

On the de re (non-split) reading, this sentence means that there is no employee such that the company needs to fire that employee. This is a non-split scope reading since "no" simply takes scope above the modal "need". On the split scope reading of this sentence, it means that it is not the case that the company needs to fire any employees. On this reading, "no" decomposes into a negation scoping above "need" and an existential quantifier scoping below it.[4]

Indefinites have been argued to have split scope, having separate existential scope and distributive scope. This fact can be seen in the following example:[5][6][7]

1. If three relatives of mine die, I will inherit a house.

Among this sentence's reading is one which means "There exists a set of three relatives such that, if those three relatives die, I will inherit a house." On this reading, the indefinite "three relatives of mine" takes existential scope outside the conditional–– it asserts unconditionally that those three relatives do in fact exist. However, it the indefinite takes distributive scope inside the conditional–– the speaker will inherit a house if three relatives die, not if x dies where x can be any of those three relatives.

Definite descriptions have also been argued to have split scope. Definites are classically considered to presuppose that their referents are unique. For instance, the definite description "the cat" is infelicitous in a context where there are multiple cats which the speaker could have in mind. However, this generalization seems to be contradicted by Haddock descriptions such as the following:

1. Context: In front of the speaker are numerous hats, one of which contains a rabbit.
   Haddock description: The rabbit in the hat

This noun phrase is felicitous to use in this context, even though there is no unique hat. What seems to license this surprising use of the definite description is the fact that the context contains a unique rabbit-containing hat. To cash out this idea, it has been proposed that the uniqueness presupposition of "the hat" takes scope separately from the rest of the definite's meaning. In other words, a witness set is establishes low in the structure, but is checked for singletonness higher up.[8][9]

While operators can often take scope above their surface position, there are not entirely free to take scope wherever they want. For instance, as illustrated by Sentence 1 below, quantifiers that originate inside an if-clause usually cannot take scope outside of that "if"-clause. This sentence cannot mean that Beth will inherit one house for each dead relative.[10][11]

1. If every relative of mine dies, I will inherit a house.

This fact parallels the fact that a wh-phrase cannot be extracted from an "if"-clause, as shown in Sentence 2.

1. Which relative_i will you inherit a fortune if t_i dies?

Examples of this sort have been used to argue that scope relations are determined by syntactic movement operations.

Aside from their theoretical significance, scope islands are also practically useful since they can be used to construct unambiguous paraphrases of sentences with scope ambiguities.[11]

While most operators are unable to scope out of an island, others can.[12] For instance, the indefinite "a" in the sentence below can take scope outside of its surface position inside an "if"-clause. This sentence can mean that there is a particular relative who must die for the speaker to get a house.

1. If a relative of mine dies, I will inherit a house.

Examples of this sort have been used to argue that indefinites do not have standard generalized quantifier denotations. On the choice function approach proposed by Tanya Reinhart, indefinites contribute a variable over choice functions which can be existentially closed at any point higher in the structure.[13] Angelika Kratzer proposed another choice function-based theory, which is similar to Reinhart's except that the choice function variable is left free.[14] Recent work such as Charlow (2020) treats indefinites as denoting sets of individuals which can be type shifted so that they take scope in a manner similar to Karttunen's (1977) alternative-based mechanism for wh-questions.[15][16]