At 17:19 26/07/04 -0400, John M wrote:
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).
Neither physical, nor human ... (see below).
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?
Interesting question (not addressed by Smullyan!). But easy though. From Godel's incompleteness (which we have not yet proved, except in the diagonalisation post some time ago, but on which we will come back: it is the "heart of the matter" in FU's term), it will be easy to prove that:
- Machine cannot be omniscient.
- Both knight and knaves are omniscient, and so they cannot be machine.
I expect, but will not argue now, that knights cannot exist at all, even in platonia (and this with or without comp).
Does this throws doubts on what we can infer from FU's puzzles? No, because the KK island is just a pedagogical tool for building a fictive but easily imaginable situation where reasoners must believe some self-referential propositions. But with the "diagonalization lemma" (alias the heart of the matter) we will eliminate the need of the KK island. It is the logical fate of the correct machine to meet inescapably true and believable (provable) self-referential propositions, from which we can derive true but unbelievable propositions, ... and much more.
PS: "Thanks" to those who have send me hard puzzles! I will try to solve them after 16 August. I will be busy until then. I will just answer Hal Finney KK Posts, and then finish my paper. I hope I will get the authorization to make it public soon for it will be a good base to proceed on. It is a step toward the "English paper" I promised to Wei Dai, a long time ago.