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Modal Logic question about provability/ countermodels.

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This statement is provable. You can turn it into an equivalent implication: ¬□(□A → B) → □(□B → A)

Then make the hypothesis that the first term is true and show that the second follows.

The first term is equivalent to

◇¬(□A → B)

◇(□A ∧ ¬B)

From which follows that

□A ∧ ◇¬B

□A

Then in any possible world, we have

A

From which we can trivially show that in this possible world,

□B → A

This is true in any possible world, so we've shown that the second term of the implication follows from the first.

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