Similarity between PWS and predicate logic

It seems to me that the formulas:

1. ◊P

2. (∃x)(Fx)

and:

3. □P

4. (∀x)(Fx)

are quite similar, if we translate the modal propositional ones into PWS. Here is the result:

1*. (∃w)(Pw)

3*. (∀w)(Pw)

Translated into a formal-ish english language, they should be read as:

1*. There exists a possible world, w, such that P is the case in w.

2. There exists an x such that Fx.

3*. For all possible worlds, w, P is the case in w.

4. For all x, Fx.

The similarity becomes stronger when an interpretation is added to the predicate logic formula though I leave that up to the reader to do.