## Proof by using multiple theorems

Hi all,

I have the solution to this question, but i don't understand how it was derived. And i hope someone could further explain it to me. A lot of theorems have been used such as modus tollen, proof by contradiction, etc.

Qns:
Knights always tell the truth, knaves always lie.
Can we tell what A and B are, from what they say below?

A: B is a knight
B: A and I are of opposite type.

Let a = "A is a knight"
Let b = "B is a knight"

Solution:
1. a
2. therefore b -- A tells the truth
3. therefore (a^~b) v (~a^b) -- B tells the truth (**)
4. therefore a^~b -- elimination
5. therefore ~b -- specialization