English is not logic. Logic is in fact a skeletal model of some parts of language, but it is only a skeleton, and the bones can be articulated in a number of ways. The Universal and Existential quantifiers are the only two used in philosophy, logic, and mathematics, but they are not the only ones in natural language, which does not share the minimalist esthetics of mathematics. Being evolved and alive, language bristles with variations on everything, some of which survive to the next generation.
Quantifiers are one such type of variation. As you point out, the linear order of quantifiers in a logical proposition is criterial for determining their relative scopes. I.e,
- (∃y: BOOK (y)) (∀x: BOY (x)) READ (x, y)
and("there exists some y which is a book, such that for every x which is a boy, x reads y") - (∀x: BOY (x)) (∃y: BOOK (y)) READ (x, y)
are each unambiguous("for every x which is a boy, and display both of the readings of Every boy read some book.there exists some y which is a book, such that x reads y")
are each unambiguous, and display both of the readings of Every boy read some book.
However, the syntax of natural language outranks the artificial syntax of logic; words that have quantification built in are also usually modifiers of one sort or another, and word order is quite fixed in English syntax, whatever the requirements of logic might prefer.
One of the results is that there are syntactic rules in English that move pieces of sentences around without preserving their original order or precedence.
One of the rules that does this is Quantifier-Float, which produces
The boys all read some book
from
All the boys read some book.
Another is Passive, which replaces the subject NP with the object NP:
Some book was read by all the boys.
As can be seen, word order is not the same as quantifier order.
Another result is that virtually any English sentence with two Operators (anything with a focus and scope -- typically Modals, Negatives, Quantifiers) in it will be ambiguous, because their scopes can become entangled. I.e,
- more than one type of quantifier (like Every boy read some book)
or - any quantifier plus a negative (like All the boys didn't leave)
or - any negative plus a modal (like She may not go tonight)
One of the ways to get around this is with qualified quantifiers, like (∃y: BOOK (y)).
Such that is the phrase one uses for this, but again, this is an idiomatic relative clause construction and has its own syntax; you can't just put such that clauses any old where.
