Chapter 6 of Kornai's "Mathematical Linguistics" considers ways of treating natural semantics formally. While the syntax chapter was in many ways a run-down of other treatments, this chapter appears to make some new proposals. The first section discusses some of the classic problems of the philosophy of language such as the Liar paradox, and largely dispenses with a great deal of the angst that philosophers and logicians have put into their treatments of these matters. K. basically shows how a lot of the technical problems with these paradoxes disappear when one deals with meanings in a way that is more appropriate for natural language modeling. He finishes by proposing that natural semantics has to have a paraconsistent logic as a formal model, which allows for inconsistency and contradiction without a total breakdown of reasoning---seems like a great idea to me.
The second section gives a very nice overview of Montague grammar (of course a must for any book on mathematical semantics). But this overview introduces some new ideas and draws in other methodologies. Prominent is the idea of a "construction," which is a semantically interpretable constituent structure analysis. K. discusses the importance of allowing "defeasible" information into a meaning, which then calls for a paraconsistent logic. The seven-valued logic of Ginsburg (1986) is then alluded to for application, though it is never really explained fully. Since Kornai seems to see this book as a textbook, he might consider really discussing the system of logic he has settled on for applications. K. then lays out in just a few pages a variety of apparently novel suggestions for treating natural language constructions of various kinds using some blend of Montague grammar with a paraconsistent logic. While intriguing, this little precis of how to do semantics in a different way really deserves to be fleshed out into a big paper of some kind that is put forth to the community as a stand-alone publication. This new approach seems like a pretty big deal to me, but here it is kind of hiding in the middle of a purported textbook, without the benefit of a thorough presentation.
The final substantive section connects K's ideas about formal semantics with the sorts of syntactic theories he earlier called "grammatical," which rely on notions of case, valence, and dependency more than traditional syntactic structures. The proposed methodology is still presented as a variant of Montague grammar, but now he is putting forth another novel set of proposals. Do these complement the proposals of the previous section, or do they replace them? I was a bit confused about what K. is really advocating we should do at this point. The brevity of this section, together with the unfamiliarity of the methods, makes it seem almost telegraphically concise.
My feeling about this chapter is that it makes a great many new proposals, but it is too short as a result. We really deserve a whole book that expands Chapter 6, I think. I, for one, would gladly read it. One thing that struck me as a little surprising, too, was the glib way in which Kornai sidesteps the hyperintensionality problem that has been long known to afflict Montague-style intensional logic treatments of semantics. This is widely regarded as a very big deal, and several researchers have spent large portions of their time working on solutions. Witness the recent work of Carl Pollard to fix Montague's IL, as he presented in a course over several days at the European Summer School in Logic, Language and Information (Dublin 2007). There was a contemporaneous publication by Reinhard Muskens in the Journal of Symbolic Logic detailing a different approach to fixing intensional logic. Does Kornai feel these people are wasting time on a non-issue? Or perhaps he would welcome joining forces between his proposals and those just cited.