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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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    Can you specify what p and K are? What does "possible in K" mean? Do you have possible worlds in mind?
    – Frank
    Commented Feb 5, 2023 at 16:33
  • 1
    Thanks for your comment. p is a variable; K is a logical system Commented Feb 5, 2023 at 16:37
  • If (false && (false)) return true
    – estinamir
    Commented Feb 10, 2023 at 2:51

2 Answers 2

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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.

¬◇¬◇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.

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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".

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