Modal logic: deriving K* from K' and RM

Question

I'm reading an article on modal logic which says that K* can be derived from K' and RM. (Note that ⊃ is the material conditional symbol and ≡ stands for material equivalence.)

[K'] □(p ⊃ q)⊃(□p ⊃ □q)

[RM] ⊢p ⊃ q => ⊢□p ⊃ □q

[K*](□p ∧ □q) ≡ □(p ∧ q)

Can someone please help me understand why this is the case?

Many thanks in advance.

Answer 1

Here’s a short proof using the rules you listed:

1. ⊢P⊃(Q⊃(P∧Q)) (Prop. Taut.)
2. ⊢□P⊃□(Q⊃(P∧Q)) (1,RM)
3. ⊢□P⊃(□Q⊃□(P∧Q)) (2,K’ Prop Logic)
4. ⊢(□P∧□Q)⊃□(P∧Q) (3, Prop Logic)
5. ⊢(P∧Q)⊃P (Prop. Taut.)
6. ⊢(P∧Q)⊃Q (Prop. Taut.)
7. ⊢□(P∧Q)⊃□P (5, RM)
8. ⊢□(P∧Q)⊃□Q (6, RM)
9. ⊢□(P∧Q)⊃(□P∧□Q) (7,8 Prop Logic)

10.⊢□(P∧Q)≡(□P∧□Q) (4,9 Prop Logic)