The principle of compositionality is often held up as fundamental to natural language semantics. This principle has been discussed and fine-tuned in plenty of papers in the literature (including one of my own, see Fulop and Keenan 2002 in the book Semantics edited by Hamm and Zimmermann). In a rough form, it is usually stated something like "the meaning of an expression is a function of the meanings of its parts and the way they are syntactically combined." This gets technically messy when one has to specify what a "meaning" is supposed to be, but there is pretty much a consensus view that some form of this principle is behind the functioning and learnability of natural language. In a language completely devoid of compositionality, any given sentence could take on any arbitrary meaning, so that "I painted a picture" might mean "the moon is a green cheese."
I guess that most linguists working on formal semantics use some kind of type-theoretic system, but the importance of type theory is much more questionable. Some authors dislike type theory as a semantic model. An interesting paper by Erdelyi-Szabo, Kalman, and Kurucz "Towards a natural language semantics without functors and operands" was published in the Journal of Logic, Language and Information (2008). This paper proposes a new scheme of formal semantics that is compositional but does not employ the typical type-theoretic system wherein a verb is viewed as a "functor" requiring an argument, for example. In their view, "compositionality presupposes that we are able to compare arbitrary meanings in terms of information content." They sketch a formal system in which the basic entities are "pieces of evidence" which might engender a belief in something. The formal language involves just a single binary non-associative construction operator used for creating structured complex terms representing pretty much anything. Maybe such a scheme should be seriously considered, given the many difficulties with the "idealization" of semantics as type-theoretic in nature.
But we have to keep it compositional.