# $A$ says "I am a knight" and $B$ says "$A$ is a Knave?" therefore what is $A$ and $B$?

$$A$$ says "I am a knight" and $$B$$ says "$$A$$ is a Knave?" therefore what is $$A$$ and $$B$$ ?

The logic is Knights always tell the truth and Knaves always lie.

What I'm thinking is that $$A$$ is knight and $$B$$ is knave because if what $$A$$ says is true then $$B$$ says the opposite and that makes his statement false and $$B$$ becomes a knave and $$A$$ is a knight.

Although what if $$A$$ is lying then can't we similarly say that $$B$$ is telling the truth? Hence making $$A$$ to be a knave and $$B$$ to be a knight.

Can I have some help please?

• It's not determined, for the reason you mention.
– lulu
May 18 at 3:00
• Can I write that as an answer? Can you elaborate why this is so? I want to write a formal answer. Can you help me with that? May 18 at 3:01
• You already wrote it out. We know one is a Knave and the other a Knight but it could be either way.
– lulu
May 18 at 3:02
• @lulu so in this case it is undetermined. I'm asking that is "undetermined" a legit option for these kind of questions? I'm new at this stuff. May 18 at 3:03
• The problem is undetermined, there's really nothing more to say
– lulu
May 18 at 3:05

You can only conclude that one is a Knight and the other a Knave, but not which one among A and B is the Knight

As mentioned in comments, both a Knight and a Knave can say "I am a Knight" so A's statement gives no information.

If B says "A is a Knave", then you can conclude that A and B are of different "type" (Knight or Knave). Indeed, if B is telling the truth, then B is a Knight and A is a Knave. But if he is lying, he has to be a Knave, and A is a Knight.

Note that $$A$$ saying that "$$A$$ is a knight" gives you no information at all: both a knight and a knave would claim themselves to be a knight.

So, all you effectively have to work with is $$B$$ saying that $$A$$ is a knave. Which has exactly the two solutions you mention: either $$B$$ is a knight and thus $$A$$ is indeed a knave, or $$B$$ is a knave and thus $$A$$ is a knight.

If you want to to this a bit more formal, note that we can nicely use biconditionals with these kinds of Knights and Knaves puzzles: one is a knight if and only if what they are saying is true.

Thus, if we use $$A$$ to represent the claim that "$$A$$ is knight", then $$A$$ saying that "$$A$$ is a knight" can be symbolized as $$A \leftrightarrow A$$ ... which is a tautology ... and thus, as pointed out above, effectively says nothing at all.

$$B$$ saying that $$A$$ is a knave becomes $$B \leftrightarrow \neg A$$ ... and if you have any experience with the logical operators at all, you know that that simply means that $$A$$ and $$B$$ have different truth-values, i.e. that the persons $$A$$ and $$B$$ are of opposite types.

Enumeration by cases is the clearest way to see what is going on.

There are 4 cases to consider:

Case 1: A is a Knight, and B is a Knight

Case 2: A is a Knight, and B is a Knave

Case 3: A is a Knave, and B is a Knight

Case 4: A is a Knave and B is a Knave

For each case, check the statement against the characters of A and B.

It is quickly seen that cases 2 and 3 are consistent with what is said, while cases 1 and 4 lead to inconsistency.