Stalnaker seems to have one hell of a time defending his view. I think this is partly due to his view being false (and vague), but let's see if we can help him out anyhow.
Mark Richard has a pretty clever argument against him. He considers an argument, and evaluates Stalnaker's methods of avoiding the deduction problem with respect to it. The argument is as follows (p.14)
(C) Barbers shave only those who do not shave themselves,
(D) The barber Jones shaved all those who attacked Lionel,
(E) Anderson shaves himself
(C&D&E) -> (A)&(C&D&E)
(A) Anderson did not attack Lionel
But also (C&D&E) -> (J)&(C&D&E)
(J) Jones did not attack Lionel.
Richard's argument runs roughly as follows:
(1) Stalnaker's view
(2) (1) -> (3)
(3) deductive inference is acheived when one considers two or more belief states, and integrates them by having as his/her new belief state the intersection of the states considered
(5) There is only one deductive consequence of considering (C&D&E)
(6) (A) is distinct from (J)
(7) (A) and (J) are deductive consequences of considering (C&D&E)
(8) ~(6) (5, 7)
I'd like to apologize to Chelsey for my rampant use of reductio.
(2) is supported in Stalnaker, I'll throw in a couple of quotes
"A person may be disposed, in one kind of context, or with respect to one kind of action, to behave in ways that are correctly explained by one belief state, and at the same time be disposed in another kind of context or with respect to another kind of action to behave in ways that would be explained by a different belief state."(p.83)
And Richard quoting Stalnaker:
"There may be propositions whose truth might be discovered by a purely deductive inquiry... The thesis [is] that acquiring deductive knowledge is putting one's seperate belief states together"
(4) is derived by presuming that the only (or best) way of integrating one's beliefs is to take the intersection of them (the possible worlds). This is the natural way of looking at belief integration under this model, and moreover it's not clear how else one could integrate beliefs.
(6) is supposed to be obvious. Intuitively there are two distinct propositions under question. (7) is assumed by hypothesis). The rest follows.
I believe Stalnaker already has a response to this up his sleeve. Recall Stalnaker chapter (4), in which he discusses the sentence 'Jim is a doctor' as said in the mouth of a child and an adult. I left that paper at school, so I won't quote. However, Stalnaker seems to have the view that the child does not understand the propositions that Jim is a doctor as well as an adult because there are many situations under which the adult could determine the truth value of the proposition, but the child could not. For instance, if the child is unaware that philosophers are Doctors (or has some dim notion of it) then the child wouldn't know the truth value of the proposition if Jim had been a philosopher.
This is all kind of rough and ready, and I think it conflicts with other things Stalnaker says, but let's run with it. On this picture the child and the adult are actually grasping (understanding, whatever) a different set of worlds when considering the proposition that Jim is a doctor. This seems a lot like some sort of descriptivism about that-clauses. There's a set of worlds that a speaker associates with a that-clause. If this is true, there can be multiple associations. This could tell against (4). It may be true that one performs deductive inference by taking the intersection of belief states. However, these belief-states don't match up directly to propostions in the ways that (4) requires. When one considers the consequences of (C&D&E), one takes the intersection of the sets of worlds one currently associates with (C&D&E). This set of worlds is not, however, the set of worlds determined by (C&D&E). So for (4) to be false one merely has to make the deduction twice, each time associating (C&D&E) with different sets of worlds.
This sort of approach can also account for deductive error, and various hooded-man type situations. However, the drawback is that it is descriptivism, and falls prey to the 100,000,000 lethal objections to descriptivism. It also makes it nearly impossible for propositions to be shareable.
On a side note, I can't decide whether or not this is actually Stalnaker's view.