Can anyone please help me with this knights & knaves logic problem? It is from Raymond Smullyan's Forever Undecided.
P= Proposition, and Q = Different Proposition.
Properties:
1) R(P) -> B(P)
2) B(P) -> B(B(P))
3) B (P->Q) & B(P) -> B(Q)
Premises:
1) B(B(k) -> c)
2) B(B(c) -> c)
3) k -> (B(k) -> c)
Conclusion = B(c) & B(k)
Best Answer
Assume that the native (I'll call him A) is a knave. This means that he lies, i.e. the sentence "If you ever believe that I'm a knight, then the cure will work" is false, or that its negation -which is "You will at some point believe I am a knight, but the cure will not work"- is true. This is because $\lnot(P\to Q)\equiv(P\land\lnot Q)$. Therefore it is true is that the reasonable believer (B) will believe that A is a knight and he will not be cured.
But if B believed that A is a knight then he will believe that what A said is true (since knights are by definition truth tellers), which means that B will believe that he will be cured (since he believes that A is a knight). Believing that he will be cured (according to B's doctor whom we trust) means that B will be cured. We have reached a contradiction (namely that he will both be cured and not be cured). Thus A is a knight.
The reasonable believer trusts the results of his reasoning, and thus using the above reasoning he will be convinced that A is a knight. This is enough to show that B will also believe that the cure will work (to see this follow the reasoning of the above paragraph) and if the doctor is to be trusted B will indeed be cured.
EDIT: To answer to your edit, my second paragraph is talking about the world as the reasonable believer sees it. I did this because I didn't want to abuse the word "believe". This is the reason that I used freely the fact that the doctor is correct, though no such information is given. So it will be inconsistent for B to believe that A is a knave so B will have to believe that A is a knight. Then B will believe what A said, which would give that B will believe that the cure will work.