I'm going to open this post with an explicit caveat on not being as up on the relevant literature as I should be (i.e., I'm even further out of the loop than usual). In particular, Kent Bach has an important piece on the myth of conventional implicatures ; and idea which was developed by Chris Potts in his dissertation (forthcoming from OUP). There is also Stephen Levinson's book on generalized conversational implicatures. I haven't gotten around to reading any of these things--though they are all on the to do list.
That said, I've been kicking around the possibility of explicating (at least some) conventional implicatures as construction-level idioms. From a construction-based perspective, typical linguistic constructions are highly productive. For instance, the Subject-Predicate Construction is the linguistic basis for almost all atomic sentences of the language. [Aside: in my view, there are actually a number of distinct Subject-Predicate Constructions which differ wrt the predication relations invoked. The distinctions to be drawn here are closely related to the sorts of issues that arise in a Montague-style type theoretic setting and which generate the need for flexibility principles.] So the compositional meaning of a sentence involving the S-P Construction is read-off from the meanings of the component parts plus the linguistic form (or informational structure) of the sentence.
At the same time, however, there are certain sentences in English which appear to have the following characteristics: (a) their apparent meanings outstrip the meanings of the lexical items involved plus any previously posited linguistic constructions, such as the S-P Construction; and (b) the general linguistic form isn't highly productive. Here is a familiar example:
(1) George is smart, but unpretentious.Of course, in standard logic, the English word but is interpreted as logical conjunction, &. If this were correct, then (1) would be true iff George were both smart and unpretentious. However, most people understand (1) to involve a further condition; namely, that being smart and being unpretentious are not usually coinstantiated. And this suggests that the meaning of but outstrips the meaning of logical conjunction.
As Grice noted, however, the prima facie richer semantics of but doesn't seem to hold up under scrutiny. For instance, the implications that intelligence and unpretentiousness are not usually coninstantiated is cancellable; for instance, by adding as are most intellectuals. This makes the "unusualness" connotation look like an implicature. But, unlike standard conversational implicature, it is not one that requires a great deal of conversational stage-setting to go through. Thus, Grice concluded that there is a standing or conventional implicature associated with English but. Unfortunately, spelling out the mechanics of such conventional implicatures is not easy.
Here is my suggestion. Typically, linguistic constructions involve fairly general bits of linguistic information that are common to an entire class of lexical items. But it is at least possible that some constructions could involve specific lexical items, say, for instance, the word but! The idea would be that English contains both a general conjunction construction:
CP + & + CP (ignoring cross categorial issues)as well as an associated idiom:
CP + but + CP.And only the latter idiom-like construction gets the richer semantic interpretation. And the cancellability phenomena aren't really implicature concellations; they are simple examples of disambiguation since the sequence George is smart but unpretentious can be formed from either of the two constructions.