In his book "Modal Logic as Metaphysics," Timothy Williamson argues that anything existing in a possible world has an identical counterpart in any other possible world. As one reviewer summarizes it, "every possible thing is a necessary thing."
My question is, does this principle also apply to propositions?
The reason I ask is because I feel like, if it did, it seems like it would make logic impossible. If every possible sentence is true in every possible world, then every possible sentence and its negation is true in every possible world. But if every possible sentence and its negation is true, then nothing could ever be false, and this would inflate the notion of truth to the point of absurdity.
Is this something necessitists like Williamson embrace somehow? Or is distinction drawn between the kinds of things that exist in possible worlds and the kinds of things that are true in possible worlds, with necessitism only being about the former?
As Andreas points out in the comments, the reason this is not a problem is that not all sentences need be possibly true. You appear to be conflating "possible sentence" in the sense of "the language contains this sentence as a well-formed formula" and "possible" in the sense of "possible according to some appropriate Kripke model."
Consider the Kripke frame with one possible world and the reflexive accessibility relation. Anything that holds in any possible world with this frame holds necessarily, but it does not follow that every sentence holds in this world. In fact we can slap any maximal consistent set of sentences onto this world as the set of sentences that holds there, and anything not included among these will be impossible, in the sense of our semantics, even if it's well-formed.
What necessitism is arguing is, roughly, just that the metaphysically appropriate Kripke frame is the one above.