1) A proposition asserted is always true in some but not all of the possible worlds in the context set.
In other words, a proposition asserted is always true in a proper subset of the context. What this means for a partitioned common ground is that an assertion of [[A or B]] is a bid to reduce the common ground--to throw out some worlds that are neither-A-nor-B worlds. I'm not sure this is right. "Or" can just as well be used to introduce possibilities, or [in Stalnaker's own words] to ''make explicit'' that neither A-possibilities nor B-possibilities are being overlooked in the common ground.
Perhaps a better thing to say here would be: an assertion of a disjunction is always (like an assertion of anything) a bid to change the common ground in some way.
2) Any assertive utterance should express a proposition, relative to each possible world in the context set, and that proposition should have a truth-value in each possible world in the context set. (Otherwise, it wouldn't be determinate how the context set was to be modified.)
I'm somewhat worried about this one--but this worry seems to be of a piece with worries about what possibilities are supposed to do. If I say (demonstrating some object in the room): "that might not be here," I guess the claim is just...well, false. It's hard to think of a case where what I say might not express any proposition at all.
For the "expansive" function of disjunction, it can be used to cancel presuppositions: "the King of France is bald, or there is no King of France." C.f. Heim on presupposition projection for [[A or B]]:
3) The same proposition is expressed relative to each possible world in the context set.
Now, this can't carry over, since conditions on the information state don't (directly) put conditions on individual worlds. But the need for this maxim is reduced if the effect of a sentence is specified directly in terms of its effects on the context set; then, the detour via expressing a definite content at each world is unnecessary.
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