Modal logic question

[Matthias123]

1. □ B ⊃ □A
2. □A ⊃ ~□ (□ B)
3. □ (B ⊃ A) ⊃□(□A)
4. □ (A ⊃ □A) • □ (B ⊃□A)

If B is necessary then A is necessary – if A is necessary then it is not necessary that B is necessary – if it is necessary that if B then A then necessarily A – If it is necessary that If A then necessarily A and if B then necessarily A, then the conclusion is true.

In reference to premise 2, is it true that the axiom □P → □□P but this axiom cannot be used to prove that ~□□P → ~□P?

 

[OliveiraBR]

1. □ B ⊃ □A
2. □A ⊃ ~□ (□ B)
3. □ (B ⊃ A) ⊃□(□A)
4. □ (A ⊃ □A) • □ (B ⊃□A)

If B is necessary then A is necessary – if A is necessary then it is not necessary that B is necessary – if it is necessary that if B then A then necessarily A – If it is necessary that If A then necessarily A and if B then necessarily A, then the conclusion is true.

In reference to premise 2, is it true that the axiom □P → □□P but this axiom cannot be used to prove that ~□□P → ~□P?

Concerning to the end of your post, it doesn’t matter the premises you take as instance to prove, the veracity of the reflexive property of the “if…then” operator. You can verify it building the truth table, the result is only T through the → column, indicating the relation in this case is a tautology