  The integration of lexical semantics and pragmatics in the analysis of the
meaning of natural lan- guage has prompted changes to the global framework
derived from Montague. In those works, the original lexicon, in which words
were assigned an atomic type of a single-sorted logic, has been re- placed by a
set of many-facetted lexical items that can compose their meaning with salient
contextual properties using a rich typing system as a guide. Having related our
proposal for such an expanded framework \LambdaTYn, we present some recent
advances in the logical formalisms associated, including constraints on lexical
transformations and polymorphic quantifiers, and ongoing discussions and
research on the granularity of the type system and the limits of transitivity.
