Exercise 2.2b from A New Introduction to Modal Logic

I am kind of stuck on page 48 on exercise 2.2b in Hughes and Cresswell; In case you don't have the book at hand here's the question:

Let K** be K but with N and K replaced by

LT: L( p→ p),

R*: ⊢ a → b ⇒ ⊢ La → Lb

and

K2*: (Lp&Lq) → L(p&q).

Show that K and K** have the same theorems.

What I thought is that in 2.2a we have showed that LT+R* ⊢ N, so it's enough to show that LT+R*+N+K2* ⊢ K.

Now I thought of old trick of math to look from the end back to the start of the proof, i.e:

We need to prove:

L(p → q) → (Lp → Lq)

But I don't see how to use the other rules and axioms, any hints?

• It's not enough to show that the axioms of K** imply the axioms of K (that would only prove that the theorems of K are a subset of the theorems of K**), you also have to show the reverse. (Also: formatting your question better might help.) – Eliran Apr 30 '16 at 8:12
• Yes, I know but in part a of this question I have shown that LT+R |- N and that N|- LT, and since R is DR1 in K system I've showed that K and K* have the same theorems in in 2.2a. Now for 2.2b I am left with only showing that K** implies K since it's evident that K implies K**. – MathematicalPhysicist Apr 30 '16 at 8:36
• @EliranH do we have latex capabilities in the stackexchange? – MathematicalPhysicist Apr 30 '16 at 8:37
• No, unfortunately not (because it's in beta I guess). – Eliran Apr 30 '16 at 8:44
• @EliranH then how to edit my post for better formatting? – MathematicalPhysicist Apr 30 '16 at 8:49

I contacted one of the authors of the textbook, Max Cresswell and he kindly suggested me a solution to the problem.

All I need to prove is that K follows from K**.

So here's a proof:

1. ((p->q)&p)->q PC axiom.

2. L((p->q)&p)->Lq 1,R*.

3. (L(p->q)&Lp)->L((p->q)&p) K2*.

4. L(p->q)->(Lp->L((p->q)&p)) 3,PC.

5. L(p->q)->(Lp->Lq) 2,4, PC.

Where PC is axiom of propositional logic, and in 5 I used a sort of hypothetical syllogism.