  The idea of Universal Grammar (UG) as the hypothetical linguistic structure
shared by all human languages harkens back at least to the 13th century. The
best known modern elaborations of the idea are due to Chomsky. Following a
devastating critique from theoretical, typological and field linguistics, these
elaborations, the idea of UG itself and the more general idea of language
universals stand untenable and are largely abandoned. The proposal tackles the
hypothetical contents of UG using dependent and polymorphic type theory in a
framework very different from the Chomskyan ones. We introduce a type logic for
a precise, universal and parsimonious representation of natural language
morphosyntax and compositional semantics. The logic handles grammatical
ambiguity (with polymorphic types), selectional restrictions and diverse kinds
of anaphora (with dependent types), and features a partly universal set of
morphosyntactic types (by the Curry-Howard isomorphism).
