If it is not possible that p is not possible in K, does it follow that p is possible in K?

Question

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!

Answer 1

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.

¬◇¬◇P

From this it follows by the equivalence of ¬◇¬ to □

□◇P

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.

Answer 2

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:

1. (In K) it is not possible₁ that it is not possible₁ that p.
2. Therefore, in K, p is possible₁.
3. If K is possible₂, then p is possible₂.
4. If K is not possible₂, then p is not possible₂.

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".