I have the following issue #26 from What Is the Name of This Book? of R. Smullyan:
There is a wide variety of puzzles about an island in which certain inhabitants called "knights" always tell the truth, and others called "knaves" always lie. It is assumed that every inhabitant of the island is either a knight or a knave. I shall start with a well-known puzzle of this type and then follow it with a variety of puzzles of my own.
According to this old problem, three of the inhabitants — A, B, and C — were standing together in a garden. A stranger passed by and asked A, "Are you a knight or a knave?" A answered, but rather indistinctly, so the stranger could not make out what he said. The stranger than 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!" The question is, what are B and C?
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
- We use $A$, when A is a knight, and $\neg A$, when A is a knave.
- $F1$ is what B said ($A \leftrightarrow \neg A$), i. e. B said that A said he's knave. Therefore, B is telling the truth if and only if he's a knight ($B$).
- $F2$ means that C is a knight if and only if he's telling the truth, i. e. B is a knave ($\neg B$).
- $G$ 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:
- B is a knave, because there are $0$s (false) in the appropriate rows.
- C is a knight, because B is telling lies, and there are $1$s (true) in the appropriate rows.
- 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.
Please tell me if my calculations and the truth table are right, not only the conclusion. The best answer is one, which either explains what I'm missing in my truth table, or contains a correct one instead of mine, being supposedly wrong. I'm trying to figure out how they can be used, and I guess this issue is quite simple to play with, after all you have the same reasoning in your mind.
Thanks in advance.