Formal Approaches to Semantic Microvariation: Adverbial ...

2.2.1 The Unreducibility of BCP SF
As discussed in chapter 1, a DP like Every student, in an English sentence like Every
student danced, denotes a type < 1 > generalized quantifier expressing a property of
a unary relation, in this case, DANCE. Within the extension of Generalized Quantifier
theory to relations of arity greater than one, sentences like No student likes every book
can be analyzed as involving a type < 2 > generalized quantifier, one that expresses a
property of a binary relation, in this case LIKE. Of course there is no need to propose
that No student likes every book involves a binary quantifier, since such a quantifier is
straightforwardly reducible to the iteration of two unary quantifiers: first every book
is applied to LIKE and then no student is applied to the result. Binary quantifiers that
are equivalent to iterated application of unary quantifiers are known as Fregean (or
reducible) in the literature. However, it has been know since Keenan (1987) that some
binary quantifiers found in natural language are unreducible, i.e. not equivalent to the
composition of two unary quantifiers.
In this section, I show that the binary extension of BCP 1 proposed above is unreducible.
The proof employs Keenan (1992)’s reducibility equivalence theorem (84).
(84) Reducibility Equivalence (RE) (Keenan (1992:211))
For F,G reducible functions of type < 2 >, F = G iff for all subsets P,Q of E,
F(P × Q) = G(P × Q).
(84) says that, for two reducible binary quantifiers, if they behave the same way on
the product relations, then they are the same function. This theorem is frequently
used to show unreducibility in the following way: the binary quantifier in question is
sentences, they use non-standard notation and functions that are never defined.
It is for this reason that, although Dekydspotter, Sprouse & Thyre and I share Obenauer’s idea that
QAD is quantification over both the denotation of the verb and the denotation of the object, the analysis
presented in this thesis bears no other similarity to the analysis proposed in their (2001) paper.
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