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We present a framework, named the Montagovian generative lexicon, for computing the semantics of natural language sentences, expressed in many sorted higher order logic. Word meaning is depicted by several lambda terms of second order lambda calculus (Girard's system F): the principal lambda term encodes the argument structure, while the other lambda terms implement meaning transfers. The base types include a type for propositions and many types for sorts of a many sorted logic for expressing restriction of selection. This framework is able to integrate a proper treatment of lexical phenomena into a Montagovian compositional semantics, like the (im)possible arguments of a predicate, and the adaptation of a word meaning to some contexts. Among these adaptations of a word's sense to the context, ontological inclusions are handled by coercive subtyping, an extension of system F introduced in the present paper. The benefits of this framework for lexical semantics and pragmatics are illustrated on meaning transfers and coercions, on possible and impossible copredication over different senses, on deverbal ambiguities, and on "fictive motion". Next we show that the compositional treatment of determiners, quantifiers, plurals,... are finer grained in our framework. We then conclude with the linguistic, logical and computational perspectives opened by the Montagovian generative lexicon.