I have these in my lecture notes, its about the rules where knights always tell the truth and knaves always lie:
If A says “The statement ‘there is gold on the island’ and the statement ‘I am a knight’ are either both true or both false” he is asserting A ≡ G where A is the assertion A is a knight and G the assertion there is gold on the Island. Any assertion by a native has the same truth value as A so: A ≡ (A ≡ G)
(A ≡ A) ≡ G
Since A ≡ (A ≡ G), it can only be the first case or third case. Hence, G is true and A can be true or false
The notes end here, I just don't understand why is it not possible for A to lie about being a knight and about gold being on the island, I mean it just shows that A is a knave, it should still be legal, even though the truth table says its illegal for the that to happen(last row), I just can't internalize why.