I shall first post a couple of screenshots to make it clear what I'm talking about.
I am reading A New Introduction to Modal Logic by Hughes and Cresswell. They answer the question in title very explicitly here, but I want to know why this is the case:
But how come the rule of necessiation isn't derivable? The necessiation rule states that if a is a theorem, then so is □a. I will offer a proof and would greatly appreciate it if anyone could point out my mistake.
1. ⊢ a Given
2. ⊢ (p => p) <=> a PC
3. ⊢ □((p & q) => p) AS1.2
4. ⊢ □(-(p & q) v p) PC
5. ⊢ □(-p v -q v p) PC
6. ⊢ (-p v -q v p) <=> ((p => p) v -q) PC
7. ⊢ ((p => p) v -q) <=> (p => p) PC
8. ⊢ □(p=>p) eq 5,6,7
9. ⊢ □a eq 8,2
So given ⊢ a, we reach ⊢ □a, which is the rule I shouldn't be able to get in S1. What am I doing wrong?