Solving Knight and knaves using rules of inference

Question

There are two tribes living on the island of Knights and Knaves: knights and knaves. Knights always tell truth and knaves always lie. You encounter two people A and B. What are A and B if A says, “B is a knight”, and B says, “The two of us are opposite types”?

So I got the first possibility of;

A:A is knight

B:B is Knight

Assuming A

B

A and -B

A^B->-b Simplification

Contradiction therefore statement false

However upon the 2nd I'm having trouble representing what this means for A,B overall and what might be the conclusion for this. I know that A,B are both knaves for this statement to be true but don't know how to prove it.

1.Assuming -A

-B

-A and -B

...?

Answer 1

So, in these island problems, the proposition that the speaker is a knight must be logically equivalent to the proposition that the speaker said. So if $A$ is the proposition that A is a knight and $B$ is the proposition that B is a knight, then the first speaker's statement means that $$A\equiv B$$ is true. B says that the two of them are of different types, i.e. $A\not\equiv B$, so $$B\equiv(A\not\equiv B)$$ is also true.

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So, since we know $A\equiv B$ is true, then $A\not\equiv B$ is false. Therefore, since $B\equiv(A\not\equiv B)$, $B$ must be false, so B is a knave. Finally, since $A\equiv B$ is true, $A$ must be false, so A is also a knave.

Let's double-check.

• A said that B is a knight. That is false, since B is a knave. Since A is a knave, telling a lie is what we expected.
• B said that the two of them were of opposite types. That is false, since they are both knaves. Since B is a knave, telling a lie is what we expected.

Therefore, our conclusion that they are both knaves is valid.

Answer 2

Not a completely formal proof, but an attempt at using inference rules to solve this problem