Generalized Quantifiers as Boolean Objects

Fact: Coordination applies freely to proper names and other NPs alike.

(1)

Mary and/or John, neither Mary nor John, every woman or every man, most women and most men, many students but few teachers, one student and five teachers, the teacher and every student etc.

The denotation of these NPs is easily derived using GQs and the boolean treatment of coordination.

(2)

a.

Mary and John smiled.

b.

Mary smiled and John smiled.

(3)

a.

Mary or John smiled.

b.

Mary smiled or John smiled.

(4)

a.

Neither Mary nor John smiled.

b.

Mary didn't smile and John didn't smile.

Analyzing what these propositions amount to easily shows that (2a) is treated as equivalent to (2b), and similarly for (3) and (4). This is of course correct.
The same hold for all NP coordinations:
Prediction: NP and/or/nor NP VP NP VP and/or/nor NP VP
Reason: iff and
iff or
iff neither holds nor does
This is in agreement with the old (and ill-defined) transformational rule of conjunction reduction (CR). However, consider the following:

(5)

a.

NP sang and danced NP sang and NP danced
NP = some man, no man, not every man, Mary or John, at least/most five women, exactly five women, most women

b.

NP sang and danced NP sang and NP danced
NP = every man, Mary, Mary and John

(6)

a.

NP sang or danced NP sang or NP danced
NP = every man, no man, not every man, at least/most five women, exactly five women, most women

b.

NP sang or danced NP sang or NP danced
NP = some man, Mary, Mary and John

Boolean semantics and GQs account for these (non-)equivalences. For example:

(7)

Some man danced and sang.

This can be false when both and are non-empty.

(8)

Every man danced and sang.

This holds iff and .

Conclusion: The boolean semantics of GQs is much more fine-grained than any syntactic account of the semantics of coordination.