The title of this post alludes to a very deep and interesting paper from Marcus Kracht, which was published in Linguistics and Philosophy (2007). It is sort of embarrassing that it took me this long to notice the paper, but I am not in the loop sometimes, and this is better late than not at all I suppose.
In this paper, Kracht applies his considerable intellect to fundamental problems of compositional semantic representations and the syntax-semantics interface. I dearly love fundamental problems papers, and they are so rare and hard to get published, so this one is really a treat. Kracht begins by defining what a compositional semantics should involve, and more importantly, what it should not involve. It should not involve indices, in the sense that variables in logic can have indices to tell them apart. I think it is more than reasonable from a cognitive perspective, to say that if human semantic representations use "variables" in any way, it is highly unlikely that any sort of named or indexed variables are used. Without this commonplace crutch, Kracht has to design a new kind of semantic representational system. He proposes further that, if this can be done properly, then semantic representations provide motivation for evident syntactic constituents in sentences. Basically, the idea is that the semantics is too weak to get the right meaning without some assistance from syntactic constituency, I think.
In the rest of the paper, Kracht dismantles modern syntactic theory and Montague grammar, and puts the pieces back together in a new and interesting way. As a kind of large aside, he proves an interesting result about Dutch context-freeness (or lack thereof). I can't bring myself to try to summarize the technical details, so just take a look at Kracht's paper if you want to see a very useful alternative perspective on many things that are frequently taken for granted without much worry.