Chapter 5 of Soames' Reference and Description (previous entries: introduction and chapter 4) concerns Stalnaker's two-dimensional framework. I'm by no means an expert on Stalnaker's system, but Bernard Nickel of MIT was there to help out. There are a few odd features of the chapter: Soames doesn't cite anything by Stalnaker after his 1978 paper 'Assertion', and expresses puzzlement at the relation between Stalnaker's and Kaplan's framework, even though this is something that's been clarified greatly in more recent literature. Nevertheless, Soames raises an interesting problem for Stalnaker's framework that's worth addressing.
Stalnaker's use of the two-dimensional apparatus is intended in part to explain how utterances such as 'Hesperus is Phosphorus' can be informative. On Stalnaker's general framework (simplifying slightly), an utterance is informative iff it reduces the "context set" -- the set of contexts compatible with the presuppositions of the speakers. Here contexts are represented as possible worlds. But the proposition expressed by 'Hesperus is Phosphorus' is true in all possible worlds. If an utterance of 'Hesperus is Phosphorus' simply imposes the constraint that this proposition be true in all contexts in the context set, it won't narrow down the context set at all. So it won't be informative.
In 'Assertion', Stalnaker handles this by saying that the relevant utterances of 'Hesperus is Phosphorus' communicate not the ordinary proposition expressed by this sentence but the diagonal proposition: here, roughly the proposition that 'Hesperus is Phosphorus' expresses a truth. (Stalnaker's proposition expressed is more or less what I call a secondary intension. His diagonal proposition is a formal analog of what I call a primary intension, but it is defined metalinguistically rather than epistemically.) There are complex mechanisms governing why the diagonal proposition is expressed in certain cases, but these mechanisms don't matter here. What matters is that when the mechanisms are in play, it is the diagonal proposition, rather than the proposition expressed, that narrows the context set. According to Stalnaker, the speakers don't presuppose antecedently that 'Hesperus' and 'Phosphorus' corefer, so there are worlds in the context set in which 'Hesperus' and 'Phosphorus' refer to different objects. The utterance will narrow down the context set by eliminating these worlds. This is why the utterance is informative.
Soames' objection is that 'Hesperus is Phosphorus' can be informative even when the speakers have already made certain presuppositions that eliminate the relevant worlds from the context set. Here Soames appeals to certain de re presuppositions that Stalnaker doesn't discuss. It seems that speakers may presuppose the de re knowledge they'd express by saying "'Hesperus' refers to that object" in the evening, and the de re knowledge they'd express by saying "'Phosphorus' refers to that object" in the morning. But the former presupposition is true in all and only worlds where 'Hesperus' refers to Venus, and the latter presupposition is true in all and only worlds where 'Phosphorus' refers to Venus. So these two presuppositions jointly exclude all worlds where the two terms fail to corefer. If so, then the diagonal proposition of an utterance of 'Hesperus is Phosphorus' won't narrow down the context set any further. So Stalnaker's principles entail that such an utterance won't be informative. But nevertheless, it seems that such an utterance can be informative.
To respond, it looks like Stalnaker has to find a way to exclude these alleged presuppositions, or to argue that they don't exclude all worlds where 'Hesperus' and 'Phosphorus' fail to corefer. It's not easy to see how this will go, though. Perhaps Stalnaker might stipulate that de re knowledge is never relevant for constraining the context set, but as Soames points out, this seems incompatible with things that he says at various points. Furthermore, it looks like the same object-involving constraints can be generated from de dicto knowledge of propositions expressed using rigid designators such as ordinary names, so that sort of knowledge would need to be excluded too. But that would seem to thin out the relevant presuppositions too much. One might also try just excluding metalinguistic de re knowledge, but then it looks like the relevant constraints can still be generated by conjoining metalinguistic de dicto knowledge with first-order de re knowledge. Bernard suggested (drawing on remarks in Stalnaker's paper on the objective self) that one could suggest that in the case of 'Hesperus is Phosphorus', the relevant bits of de re knowledge are never operative at the same time. But as he pointed out, this won't help with analogous cases where everything happens at once.
It also seems that the problem can arise quite independently of issues about de re knowledge. For example, although Soames doesn't discuss this explicitly, one can get the problem going with disquotational knowledge. For example, it seems that speakers can presuppose that 'Hesperus' refers to Hesperus and that 'Phosporus' refers to Phosphorus. But any world in which these presuppositions are jointly true will be one in which the two terms corefer. So again, presuppositions rule out the worlds that Stalnaker wants to rule in. The same goes in non-name-involving cases. For example, Stalnaker wants to use the 2D apparatus to explain how 'Ophthalmologists are eye doctors' can be informative to non-expert speakers, but it seems that such an utterance can be informative even when such speakers presuppose disquotationally that 'ophthalmologist' refers to ophthalmologists and that 'eye doctor' refers to eye doctors, thereby ruling out worlds where the utterance expresses a false proposition (given that ophthalmologists are eye doctors in all worlds). The problem here seems just as acute.
Note that problems of this sort won't arise for the sort of two-dimensionalism that Jackson and I endorse. On this view, a context set will always be seen as a set of worlds considered as actual (or a set of epistemic possibilities), and beliefs and knowledge will narrow down this set according to their primary intensions. So when one has a presupposition that one would express by saying "'Hesperus' refers to that", this imposes the constraint that the context be one in which 'Hesperus' refers to the object to which one is pointing on a given occasion in that context (or something like that, depending on just how one understands the primary intension of a use of 'that'). But now, there will be plenty of contexts that satisfy both the 'Hesperus' and 'Phosphorus' constraints -- they'll be worlds where one is pointing at different objects on the different occasions. Likewise, the presupposition that 'Hesperus' refers to Hesperus imposes the constraint that 'Hesperus' refers to the evening star (or something like that, depending on just how one understands the primary intension of Hesperus). These constraints yield more intuitively reasonable results concerning the contexts that are epistemic possibilities for the speaker.
But this sort of move doesn't seem to be available to Stalnaker. The reason is that the two-dimensional content (if any) of presuppositions and other mental states plays no role in his framework. The two-dimensionalism enters later in the day, to associate propositions with utterances after presuppositions have already narrowed down the context set ("one-dimensionally", so to speak). In this respect, Stalnaker's two-dimensionalism seems somewhat half-hearted, from my perspective: it applies here to language but not to thought, and it isn't fully integrated into the way that presuppositions constrain epistemic possibilities. (Conversely, from Stalnaker's perspective, my sort of two-dimensionalism looks like a wild overextension.) I'd suggest that this lack of integration is what's causing the problems.
I'm sure there are other moves available to Stalnaker in response, though. I'd be interested to hear about possible responses from those more familiar with Stalnaker's program.