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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)

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1 Answer 1

up vote 4 down vote accepted

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.

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you've confused knight with knave in second sent. –  Milosz Wielondek May 10 '11 at 8:00
    
@Milosz: Where exactly have I confused knight with knave? –  Apostolos May 10 '11 at 8:22
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@Milosz: I assume now that you mean in the second sentence. I haven't mixed them up. The negation of "If you believe I am a knight then the cure will work" is "You will believe that I am a knight and the cure will not work" since $P\to Q$ is logically equivalent to $\lnot P\lor Q$ and its negation is $P\land\lnot Q$. I will edit my answer to spell it out. –  Apostolos May 10 '11 at 11:10
    
@Apostolos Your answer is correct. I'd only add that the premisses are $k \leftrightarrow (B(k) \rightarrow c)$, and $B(c) \rightarrow c$. –  Marc Alcobé García May 10 '11 at 12:10
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@Eric: I'm sorry, I meant something completely different. What I mean is that if B believes that A is a knave then B believes that he will at some point believe that A is a knight but B will not be cured. But if he believes that he will believe that A is a knight B will believe that he will believe what A said, which means that B will believe that he will believe that he will be cured since he will be believing that the native is a knight, but that would mean that he will believe that he will be cured because he trusts his doctor, which is contradictory from the believers standpoint. –  Apostolos May 11 '11 at 20:33
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