At 19:47 23/07/04 +0200, I wrote:
Big Problem 5:
Could a native tell you "You will never know that I am knight" ?
Very Big Problem 6:
Could a native tell you "You will never believe that I am knight" ?
It was perhaps not pedagogical to say "big" and "very big". Here John Mikes would be accurate to say those are not problems, but koans. So you can *meditate* on it ... My intent in the use of the word "big" was more to point on the fact that ,going from the preceding problems to those new one is, as you can guess, the passage from ordinary logic to modal logic, giving that "knowing" and "believing" are modalities. Still we do have intuition on what knowledge and believe can be, and we can try to get some conclusion.
Let us try the problem 5. Just to be sure let me know if you agree that the proposition:
You will not know that Lance Amstrong has win the "Tour de France", is false. I mean the negation of "You will not know p" is "you will know p".
OK then. Suppose the native is a knave. Then it means he is lying when
he say that "you will never know I am knight". But that means I will know
he is a knight. But I cannot *know* he is a knight if he is a knave, so he cannot be
a knave, and thus he is a knight. But having reach that conclusion I know
he is a knight, but then he was lying (giving he told me I will never
know that. So he is a knave, yes but we have already show he cannot
be a knave. It looks like we are again in an oscillating state of mind.
Is it a tourist again? No. I told you it is a native and all native are,
by the definition of the KK island, either knight or knave. So what?
I will give many critics later on some "less good" chapter of FU, (ex: the chapter "the heart of the matter" is too short, or when he identify a reasoner with a world in his possible world chapter, etc.) But Smullyan's discussion on the problem 5 is really quite instructive. Its attack of the problem 5 is quite alike my attack on the mind body problem, Smullyan will interview "reasoner" (actually machine) on those questions. And he goes on very meticulously by defining a hierarchy of type of reasoner, making those problem, mainly the problem 6 more and more interesting. Indeed, with the problem 5, we are quickly done, giving that the problem 5 leads toward a genuine contradiction even with the reasoner of lowest type: the one Smullyan calls the "reasoner of type 1".
A reasoner, or a machine, or a system (whatever) will be said to be of type 1 if by definition the following conditions hold:
1) he believes all classical tautologies (our lowest type of reasoner is already a platonist!) 2) if the reasoner ever believes both X and X ->Y, he will believe Y.
I will also say, following Smullyan and ... Theaetetus, that a reasoner know p, in the case the reasoner believes p and p is true, that is when the reasoner correctly believes p. There is nothing metaphysical in our use of the word "believe". You can substitute it by "prove" or even just "print". That would mean you are in front of a sort of theorem prover which will, for any classical tautology, print it one day or an other (condition 1), and, in case it prints X and X->Y, it will, soon or later, print Y. That is he prints a proposition if and only if he believes it.
To transform the "koan 5" into a genuine problem, I must explain what it means for a reasoner to believe in the rules of the island. It means that if he met a native asserting a proposition p, then he believes the proposition "the native is a knight if and only if p is true". We will write Bp for the reasoner believes p, and Kp for the reasoner knows p. For exemple we have Kp <-> p & Bp
problem 5' (ameliorated version): A visitor of type 1 meets, on the KK island, a native telling him "you will never know I am a knight". Convince yourself that this really cannot happen. Derive a contradiction.
Problem 6' (new version): A visitor of type 1 meets, on the KK island, a native telling him "you will never believe I am a knight". Convince yourself that this can happen. Derive as many conclusions as you can (note that all FU will follow, and even my thesis!!! (if you are patient enough). The idea is to give more and more self-awareness to the visitor ... Curiously enough that path converges.
PS Those who feels overwhelmed can wait I come back on problem 4, when I will recall a little bit more matter on propositional logic. Apology for my post to Jesse which could look a little bit "advanced" for many (it presupposes FU and a little bit more). Actually writing this stuff helps me for my paper. I thank you in advance.