Quantifier Raising: Entailment and Inverse Scope (Syntax II, Spring 2025)

Consider the following two sentences, each involving an existential quantifier phrase and a universal quantifier phrase.

1.

a. Some boy loves every girl.

b. Every boy loves some girl.

In both cases, we can ask what the possible logical form representations of the sentence are, and what the truth conditions of those representations are. If we adopt the syntactic operation QR, then (1a) with inverse scope yields truth conditions that clearly distinguish it from (1a) with surface scope. The question of whether inverse scope is needed for sentences like (1b) is more intricate. This note goes into these issues in detail.

Applying QR to (1a), there are two logical form representations:

2.

LF1: (surface scope)

[TP <[some boy]1> [TP <[every girl]2> [TP DP1 loves DP2]]

LF2: (inverse scope)

[TP <[every girl]2> [TP <[some boy]1> [TP DP1 loves DP2]]

In these structures, the notation <…> indicates that the occurrence is not pronounced. In LF1, for example, [some boy] has two occurrences, but only the one in subject position is pronounced. Furthermore, in these representations the possibility of VP adjunct is ignored (see Fox 2002 for discussion).

These LF representations can be paraphrased as follows:

3. LF1 (paraphrase): There is some boy who loves every girl.

LF2 (paraphrase): For every girl, there is some boy who loves her.

Now consider the following two situations:

4.

S1: John love Sue, Mary and Kathy.

(and there are no other people or relationships involved)

S2: John loves Sue, Bill loves Mary, Chris loves Kathy.

(and there are no other people or relationships involved)

Calculating the truth value of each LF representation in each situation (actual semantic calculation omitted), we have the following results (SS = surface scope, IS = inverse scope):

5. S1 S2

LF1 (SS) true false

LF2 (IS) true true

This table shows that LF2 (inverse scope) is true in S2, whereas LF1 (surface scope) is not. These truth values provide an important argument for the existence of inverse scope. If inverse scope were not allowed, there is no way we could capture the intuition that (1a) (with structure LF2) is true in S2.

In the theory I presented in class (from May 1977, Fox 2002), inverse scope is accounted for in terms of QR. In inverse scope, the object undergoes QR to a scope position higher than the subject scope position (see LF2 above). Therefore, the data in (5) provides evidence supporting the covert movement operation QR.

This table shows that only S2 distinguishes the two LF representations, S1 does not. Therefore, it is clear that LF1 is stronger, in the sense that whenever LF1 is true, so is LF2, but not vice versa. In fact, using formal semantic tools, it is possible to prove the following:

6. a. LF1 entails LF2 (LF1 ⊨ LF2)

b. LF2 does not entail LF1 (LF2 ⊭ LF1)

When there is an entailment relation like this between two LFs, it is sometimes hard to distinguish them empirically. For example, in 5, both LF representations were true in S1. And so S1 could not be used to distinguish them.

With this background, consider now (1b) above. Applying QR to (1b), there are two logical form representations:

7.

LF1: (surface scope)

[TP <[every boy]1> [TP <[some girl]2> [TP DP1 loves DP2]]

LF2: (inverse scope)

[TP <[some girl]2> [TP <[every boy]1> [TP DP1 loves DP2]]

These LF representations can be paraphrased as follows:

8. LF1 (paraphrase): For every boy, there is some girl who he loves.

LF2 (paraphrase): There is some girl who every boy loves.

Now consider the following two situations:

9.

S1: John loves Sue, Bill loves Mary, Chris loves Kathy.

(and there are no other people or relationships involved)

S2: John love Sue, Bill loves Sue, Chris loves Sue.

(and there are no other people or relationships involved)

Calculating the truth value of each LF representation in each situation (calculation omitted), we have the following results:

10. S1 S2

LF1 (SS) true true

LF2 (IS) false true

In this case, LF2 is stronger than LF1, in the sense that LF1 is true whenever LF2 is true. In other words, we have the following entailment relations:

11. a. LF2 entails LF1 (LF2 ⊨ LF1)

b. LF1 does not entail LF2. (LF1 ⊭ LF2)

Now let’s ask what the evidence is for the inverse scope representation in LF2 of the sentence in (1b). Unfortunately, because of the entailment relations, there is no situation (such as S1 or S2) where LF2 (inverse scope) is true and LF1 (surface scope) is false. This makes it difficult to justify the inverse scope representation purely on the basis of situations where the LF representations are true. Therefore, the question is whether in this case one needs an inverse scope representation of (1b) at all.

To put matters more intuitively, concerning (1b), if it is true that there is some girl who every boy loves (inverse scope) then it is also true that for every boy, there is some girl who he loves (surface scope). So perhaps in this case we could say that only the surface scope representation in LF1 is possible, but it is vague as to whether everybody loves the same girl or not.

We can summarize as follows:

12.

a. Some boy loves every girl.

i. Inverse scope does not entail surface scope.

ii. There is a situation where inverse scope LF is true,

but surface scope LF is not true.

b. Every boy loves some girl.

i. Inverse scope entails surface scope.

ii. If inverse scope LF is true in some situation,

then so is surface scope LF.

Notice that LF1 and LF2 in (7) are not equivalent, and so they are false in different cases (see (10)). In particular LF1 is true in S1 but LF2 is false in S1. This leads to the suggestion that we could use falsity to distinguish the interpretations of LF1 and LF2.

So, given S1, consider the following dialogue:

13. (in S1)

A: Isn’t it strange that every boy loves some girl? (with structure LF2)

Intended: Isn’t it strange that there is some girl that every boy loves?

B: That is false, they all love different girls.

If this were an acceptable discourse it would provide evidence that both LF1 and LF2 are needed for (1b). Since it would show that LF2 (inverse scope) is possible, and is false in a particular situation. Unfortunately, it is not very easy to access the inverse scope interpretation in 13A, and the discourse seems forced.

Another similar way to distinguish LF1 and LF2 is through constructions like “It is false that” and “It is not the case that”. These constructions negate the embedded clause:

14. a. It is false that every boy loves some girl.

Intended: It is false that there is some girl that every boy loves.

b. It is not the case that every boy loves some girl.

Intended: It is not the case that there is some girl that every boy loves.

If LF2 (inverse scope) were a possible representation here, then the sentences would be true in S1. Once again, the facts are not that clear.

There are ways to bring out inverse scope. For example, adding the word particular makes it much easier:

15. a. It is false that every boy loves some particular girl.

Intended: It is false that there is some particular girl that every boy loves.

b. It is not the case that every boy loves some particular girl.

Intended: It is not the case that there is some particular girl that every boy loves.

This kind of sentence where one is negating the inverse scope LF representations suggests that inverse scope is also needed in examples such as (1b).

In summary, based on truth conditions, it is easier to justify an inverse scope LF representation for (1a) than it is for (1b). However, given the clear need for an inverse scope LF representation for (1a), the null hypothesis is that such an inverse scope representation should also be available for (1b). That conclusion leaves open the difficult status of the judgments in (13) and (14).