It is well known that any sentence with two or more quantifiers will result in in multiple possible readings depending on the ordering of the quantifiers. To take a known example (1), there will be two readings: In (1.a) the universal takes wide scope while in (1.b) the existential takes wide scope and the universal takes narrow scope.
(1) Everybody loves someone. (1.a) ∀ > ∃ [For every person, there is a potentially different person they love.] (1.b) ∃ > ∀ [There is a specific person and everyone loves that specific person.]
(1.a) is what is known as a surface scope, i.e., the logical form matches up with the phonological form while (1.b) is known as an inverted scope reading i.e., the logical form is the invert of the phonological form. Most of the times, the surface scope entails the inverse scope but not vice versa:
(2) A boy climbs every tree. (2.a) Surface scope: There is a boy who climbed every tree. (2.b) Inverse scope: For every tree, there is a boy who climbed it.
This is fairly straightforward, but are long distance inverted scopes also possible? If there are three or more quantifiers, does an inverted scope exists for them too? This is what I am struggling with.
(3) Some man bought two cars in every car dealership. (3.a) Surface scope: ∃ > two > ∀ [There is a specific man and that man brought two potentially different cars in every car dealership.] (3.b) Intermediate scope: two > ∃ > ∀ [There are two specific cars such that there is a potentially different man who bought the two specific cars in every dealership.] (3.c) Wide scope: ∀ > ∃ > two [For every car dealership, there is a potentially different man who bought two potentially different cars.] (3.d) Inverted scope (?): ∀ > two > ∃ [For every car dealership, there are two potentially different cars such that a potentially different man bought them.]
Is a reading such as (3.d) possible? The more I think about it, the more plausible such a reading is, but is it an actual possible reading of (3)? Are long distance inverted scopes even possible or are there cases where they don't exist. Theoretically, they should be possible given that quantifiers behave similarly to wh-movements in that they move successive cyclically, thus not violating the phase boundaries.