Content as Possible Worlds

One argument for a possible-worlds analysis of content is that it helps us make sense of partial understanding, as found for example in children and animals (Stalnaker, Inquiry, p.65):

The child who says that his daddy is a doctor understands what he says, and knows it is true, to some extent, because he can divide a certain, perhaps limited, range of possibilities in the right way and locate the actual world on the right side of the line he draws. As his understanding of what it is for Daddy to be a doctor grows, his capacity will extend to a larger set of alternative possibilities or, rather, the extension of this capacity will be the growth of his understanding.

A second issue is that one may have a belief 'by default', so to speak, taking something for granted even if one has never explicitly thought about it -- and sometimes only because one has never thought about it, as Stalnaker insightfully notes (p.69):

With riddles and puzzles as well as with many more serious intellectual problems, often all one needs to see that a certain solution is correct is to think of it--to see it as one of the possibilities.

Stalnaker accounts for this as follows (pp.68-9):

Attitudes are primarily attitudes to possible states of the world and not to the propositions that distinguish between those states. A belief state can be represented as a set of possible worlds. Individual beliefs are properties of such a belief state: to believe that P is for the proposition that P to be true in all the possible worlds in the belief state. If one conceives of beliefs in this way, they look like something negative: to believe that P is simply to be in a belief state which lacks any possible world in which P is false.

Can these two challenges be met just as well without appeal to possible worlds?