Bruno, (and Class<G>) We have an overwhelming ignorance about Ks and Ks. We don't know their logical built, their knowledege-base, their behavior. Is the K vs K rule a physical, or rather human statement, when - in the latter case there may be violations (punishable by jail - ha ha). Do K & K abide by 100.00% by the ONE rule we know about them, or ~99.999%, when there still may be an aberration? Are they robots or humans? Looks like machines. Are machines omniscient?
To your present 2 problems: none of them CAN be a knight, because in all fairness, nobody knows what may a person 'know' or 'believe' in the future. If I go to the Office of Records, I may learn what that 'K' is. John Mikes ----- Original Message ----- From: "Bruno Marchal" <[EMAIL PROTECTED]> To: <[EMAIL PROTECTED]> Sent: Monday, July 26, 2004 12:49 PM Subject: Re: ... cosmology? KNIGHT & KNAVE > Hi, > > > 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. > > Bruno > > 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. > > http://iridia.ulb.ac.be/~marchal/ > >
