While presenting and specifying the characteristics, potential and limitations of Montague grammar, it analyses a number of significant differences between a formal and a cognitive semantics. It therefore deals which many of the issues that have been given alternative treatment in cognitive semantics (structure, syntax and semantics, the compositionality of meaning, competence, and so on). The main thesis, in fact, is that Montague semantics is not a theory of meaning for natural language (see also Langacker’s contribution), but a particular, or even complete, theory of an idealization of inferential competence. The essay clarifies in particular the different relationships between semantics and syntax in formal and cognitive semantics, explaining how a Montague grammar is a theory of the semantic effects of composition and how, consequently, it affects meaning.