0
$\begingroup$

$A$ 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?

$\endgroup$
8
  • 2
    $\begingroup$ It's not determined, for the reason you mention. $\endgroup$
    – lulu
    May 18 at 3:00
  • $\begingroup$ 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? $\endgroup$
    – Itachi
    May 18 at 3:01
  • 1
    $\begingroup$ You already wrote it out. We know one is a Knave and the other a Knight but it could be either way. $\endgroup$
    – lulu
    May 18 at 3:02
  • $\begingroup$ @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. $\endgroup$
    – Itachi
    May 18 at 3:03
  • $\begingroup$ The problem is undetermined, there's really nothing more to say $\endgroup$
    – lulu
    May 18 at 3:05

3 Answers 3

1
$\begingroup$

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.

$\endgroup$
1
$\begingroup$

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.

$\endgroup$
0
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.