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

Question

$A$ 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?

Answer 1

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.

Answer 2

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.

Answer 3

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.