I have the following question.
If it is not possible that p is not possible in K, does it follow that p is possible in K?
Thanks in advance!
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No, that does not follow within K. You are giving as a premise, it is not possible that it is not possible that P, i.e.
From this it follows by the equivalence of ¬◇¬ to □
But in K, this does not entail ◇P. It fails to hold because without axiom T, you cannot derive φ from □φ. Within the frame condition of K, a countermodel exists when there there is a world with no worlds accessible to it, not even itself.
If K is closed (and does not include, under itself, sentences referring to K as a whole), then the "is/not possible" of the outer layer of the sentence will not be the same as the "is/not possible" of the inner layer. So consider:
I'm not sure about (4) (I feel like there's a bit of an "affirming the consequent" danger in its expression, like if some different system, L, is possible2 and p is possible1 in L, then p might be possible2 because of L).
For better relevant analysis, see Dominic Gregory, "Iterated Modalities, Meaning, and A Priori Knowledge" as well as Timothy Williamson, "Counterpossibles in Metaphysics".