How are possible worlds in modal logic qualitatively different from valuations? Are they just kinds of valuations?

Question

How are possible worlds in modal logic qualitatively different from valuations? Are they just kinds of valuations? It seems to me that what Kripke semantics does is just quantifying over valuations. Is that so, or in what respect is it not? Or is it just nomenclature, do we just call them possible worlds so that it is clearer what we are quantifying over?

Answer 1

An individual world in propositional modal logic is no different from a valuation. However, a Kripke frame is more than just a collection of worlds - it also describes a way that the worlds relate to each other.

To see this at play, consider two frames with the same set of worlds but different accessibility relations. The simplest example would be the frame with one world and no accessibility (the world doesn't even see itself!), and the frame with one world which does see itself.

The first validates the sentence $\Box p$ - that is, no matter how we attach valuations to the worlds involved (granted, there's only one) the sentence $\Box p$ is true at ever world. This is a vacuous truth: "$\Box$" means "is true in every world accessible from the current one," and there aren't any of those.

The second, meanwhile, validates $p\rightarrow \Diamond p$. Meanwhile, the second frame does not validate $\Box p$ and the first frame does not validate $p\rightarrow\Diamond p$.