Montague Semantics is really interesting — and totally inadequate! Still, it's a very good approach, and all formal systems for natural language semantics we know of today are still totally inadequate. It's proving to be very difficult to model natural languages as formal languages.
The basic idea of MG (and CCG) is to treat lexical items ("words") as function in the lambda calculus, and use higher order functions to compose these lexical functions, yielding truth conditions in a logical language (typically anything from predicate logic, over intensional or two-sorted type logic, to even higher order logic, dynamic logic, probabilistic logic, etc.)
Some of the more interesting extensions of traditional MG are introduced by Combinatorial Categorial Grammar, which, you guessed it, uses combinators to enable compositional analyses of lexical items even in complex syntactic constructions. One more thing that is very interesting is that continuation passing style transformations on these combinators and lexical functions seems to be rather effective! Read Barker 2004 for an overview.
It's a very interesting field of study, but almost entirely academic. There is very little commercial interest nowadays, as everybody is all about statistical NLP.
Absolutely, it is inadequate for modeling language, but then everything is! (My assertion is that so much contextual information is required to converse with humans that one needs to build a proper AI in order to have unrestricted conversation. That or you have to reliably constrain conversation to a domain)
The importance of systems like Montague Semantics are to kind of scope out what sort of thing we need to be able to model language in the abstract.
Statistical NLP is definitely way better for building engineering systems that are deployed to accomplish particular tasks.
Yes, just because something is incredibly hard, doesn't mean you have to stop pursuing it. We thought in the fifties and sixties that with rapidly increasing computing power, modelling natural would be within reach soon.
Oh, how wrong we were. Natural language (and human thought for that matter, because ultimately, AI and NLP might be two facets of the same problem) is so much more complicated than we imagined.
Researching human language and thought gives us insight not only sufficient to engineer interactive systems, but also to understand the human condition as a whole. Just take the entire discussion about rigid designators, and naming across possible worlds in intensional logics (read "Naming and Necessity" by Saul Kripke.) It is but one of the ways in which the need for a good formal approach to language resulted in an amazing philosophical discussion that isn't just about models, but our understanding of the world. Ultimately, natural language semantics can quickly transcend into deep philosophy. It sometimes takes me completely by surprise, actually :-)
The basic idea of MG (and CCG) is to treat lexical items ("words") as function in the lambda calculus, and use higher order functions to compose these lexical functions, yielding truth conditions in a logical language (typically anything from predicate logic, over intensional or two-sorted type logic, to even higher order logic, dynamic logic, probabilistic logic, etc.)
Some of the more interesting extensions of traditional MG are introduced by Combinatorial Categorial Grammar, which, you guessed it, uses combinators to enable compositional analyses of lexical items even in complex syntactic constructions. One more thing that is very interesting is that continuation passing style transformations on these combinators and lexical functions seems to be rather effective! Read Barker 2004 for an overview.
It's a very interesting field of study, but almost entirely academic. There is very little commercial interest nowadays, as everybody is all about statistical NLP.