Three people, a, b and c are in a garden. Each person is either a knight or a knave. A stranger asks a, "are you a knight or a knave?" a answered indistinctly, so the stranger could not make out what he said. The stranger then asked b, "what did a say?" b replied, "a said that he is a knave." at this point, the third man, c, said, "don't believe b; he is lying!". What are b and c?
Answers
Answer:
I supposed that truth tables can be used, and composed the following:
| | | F1 | F2 | G
===|===|===|==============|========|=========
A | B | C | B ↔ (A ↔ ¬A) | C ↔ ¬B | F1 ^ F2
===|===|===|==============|========|=========
1 | 1 | 1 | 0 | 0 | 0
1 | 1 | 0 | 0 | 1 | 0
1 | 0 | 1 | 1 | 1 | 1
1 | 0 | 0 | 1 | 0 | 0
0 | 1 | 1 | 0 | 0 | 0
0 | 1 | 0 | 0 | 1 | 0
0 | 0 | 1 | 1 | 1 | 1
0 | 0 | 0 | 1 | 0 | 0
Provided that:
We use A, when A is a knight, and ¬A, when A is a knave.
1.F1 is what B said (A↔¬A), i. e. B said that A said he's knave. Therefore, B is telling the truth if and only if he's a knight (B).
2.F2 means that C is a knight if and only if he's telling the truth, i. e. B is a knave (¬B).
3.G allows us to select only those claims amongst F1 and F2 which are true.
can I safely say that we have only two cases, when G is true and the following conclusions can be made:
1.B is a knave, because there are 0s (false) in the appropriate rows.
2.C is a knight, because B is telling lies, and there are 1s (true) in the appropriate rows.
3.We cannot say what is A exactly, because we couldn't make out what he said, and there are two cases in the table with 0 and 1 in the appropriate rows, where G is true.
Explanation: