  The semantics of determiner phrases, be they definite de- scriptions,
indefinite descriptions or quantified noun phrases, is often as- sumed to be a
fully solved question: common nouns are properties, and determiners are
generalised quantifiers that apply to two predicates: the property
corresponding to the common noun and the one corresponding to the verb phrase.
We first present a criticism of this standard view. Firstly, the semantics of
determiners does not follow the syntactical structure of the sentence. Secondly
the standard interpretation of the indefinite article cannot ac- count for
nominal sentences. Thirdly, the standard view misses the linguis- tic asymmetry
between the two properties of a generalised quantifier. In the sequel, we
propose a treatment of determiners and quantifiers as Hilbert terms in a richly
typed system that we initially developed for lexical semantics, using a many
sorted logic for semantical representations. We present this semantical
framework called the Montagovian generative lexicon and show how these terms
better match the syntactical structure and avoid the aforementioned problems of
the standard approach. Hilbert terms rather differ from choice functions in
that there is one polymorphic operator and not one operator per formula. They
also open an intriguing connection between the logic for meaning assembly, the
typed lambda calculus handling compositionality and the many-sorted logic for
semantical representations. Furthermore epsilon terms naturally introduce
type-judgements and confirm the claim that type judgment are a form of
presupposition.
