Knights and Knaves, both accuse each other of being Knaves.

Question

I'm having a bit of an issue with figuring this one out. I ended up saying one has to be a knave, but I feel like it doesn't have a concrete solution, from the contradiction.

Knights always tell the truth, and knaves always lie.    
E says: F is a Knave.
F says: E is a knave.
How many knaves are there?

My thinking is if you assume E is a knight, So what E said must be true. So, F is a knave. Then assume F is a knight, so what F says is true. This gives a contradicition, so E is not a knight and must be a knave. Thus what F says must be true, and F is a knight. But, it feels like this question could fall into the liar's paradox making what ever solution is found both false and true.

So, what's the logical answer for this one?

What does it mean to be a "knight" or "knave"? The former always tells the truth and the latter always lies? — user7530, Feb 16 '15 at 06:30
Suggestion, if you are talking about truthteller vs liar questions: search this site for other questions about the knight-knave domain, and use the input from all those answers. (E.g., my answers.) That should give you ways to approach questions like this. — MarnixKlooster ReinstateMonica, Feb 16 '15 at 06:49

score 2 · Answer 1 · answered Feb 16 '15 at 06:54

If both were knaves, both would be telling the truth => contradiction.

If both were knights, both would be lying => contradiction.

Therefore, one of them is a knight and the other a knave, which it turns out does not lead to a contradiction: The knight tells the truth about the other being a knave, and the knave lies about the other being a knight, thus both claim the other to be a knave.

score 0 · Answer 2 · answered Feb 16 '15 at 06:49

You were on the right track at the beginning but wandered off it. If $E$ is a knight, then he’s telling the truth, so $F$ is a knave. Is that consistent with $F$’s statement? Yes: when $F$ says that $E$ is a knave, he’s lying (since we assumed that $E$ is a knight), which is what he must do as a knave. Thus, if $E$ is a knight, exactly one of them is a knave, namely, $F$.

Now suppose that $E$ is a knave. Then he’s lying, so $F$ is a knight. And that’s also fine, because $F$’s statement that $E$ is a knave is in fact true, as it should be if he’s a knight. Again we find that exactly one of them is a knave, but this time it’s $E$.

These are the only two possibilities, since $E$ must be either a knight or a knave. Thus, we can be sure that exactly one of the two is a knight and the other a knave, even though we have no way to tell which is which.

Knights and Knaves, both accuse each other of being Knaves.

2 Answers2