More on Recanati in fact. This is from his Embedded Implicatures. It’s an argument against the King and Stanley view that implicatures are computed globally.

For example we may say:

John ate some of his cake but Jim ate all of his

Here the scalar enrichment of ‘some’ takes place within the first conjunct, in the scope of ‘but’, yet it cannot be explained away in terms of some process of saturation that independently takes place in interpreting this construction.

At first this looks like it goes by a little quickly. After all, there’s a fairly simply scale here:

John ate some of his cake but Jim ate all of his
John ate all of his cake but Jim ate all of his

The weaker claim, the higher on the list, was asserted, implicating the negation of the other. So King and Stanley’s view is not threatened.

At second glance maybe this reply doesn’t work, because the second sentence on our scale is not well-formed. There is, we might notice, no contrast between the conjuncts.

At third glance, this response fails twice over. We can say John ate all of his cake but Jim ate all of his if, for example, it matters a lot whether John ate more than Jim. Say Jim is to be rewarded if he eats less than John, and I ask you whether I’m going to have to give Jim a reward. Then you can say John ate all of his cake (indicating I probably will have to reward Jim) but Jim ate all of his (indicating that in fact no reward is necessary).

More importantly, it’s consistent with the King/Stanley view to say that implicatures are computed sequentially, as long as the sequence starts after semantic interpretation. So it’s consistent with their view (I think) that we first do the semantic interpretation of the sentence, getting that John is a part of cake eater and Jim is a whole of cake eater, then do one Gricean inference to infer that John is a proper part of cake eater, then another to infer that John and Jim contrast in a particular way, licencing ‘but’.

This is not to say I believe the King/Stanley line, because I’m not convinced their theory can adequately handle conditionals whose antecedents are true but contain false implicatures. But that’s a different matter to the argument involving John and Jim, which I don’t think works.