> -----Original Message----- > From: [EMAIL PROTECTED] [mailto:[EMAIL PROTECTED] On > Behalf Of jdiebremse > Sent: Thursday, October 12, 2006 10:33 AM > To: Killer Bs Discussion > Subject: Re: Paradox, or, Breaking the mind of logic > > > > --- In [EMAIL PROTECTED], "Alberto Monteiro" <[EMAIL PROTECTED]> wrote: > > > But how does this work for N(blue) = 4? > > > > > The key point is that the natives are omniintelligent and > > know that all other natives are also omniintelligent. > > > > > The initial state is that each native has two cases: > > > > > > 1) There are three blue-dot natives, and each blue dot native sees > > > two blue dot natives. > > > > > > 2) There are four blue-dot natives, including himself, and each blue > > > dot native sees three blue dot natives. > > > > > > In this case, I don't see how the naturalist provides any additional > > > information. In the initial state, every native knows that every > other > > > native knows that there is at least one blue dot. > > > > > He does. Because of the omniintelligence hypothesis, each native > > can reason like this: > > > > (a) If there is only one blue dotted native, then, seeing that > > everybody else is red dotted, this native will commit ritual > > suicide in the first night. > > > Maybe I'm exhibiting my ignorance here, but if N(blue) = 4 then all the > natives *know* that there is *not* "only one blue-dotted native" before > the anthropologist even arrives.
That's certainly true. But, what they don't know is what the other native can deduce. After the anthropologist makes his statement, even though it is a statement that every native knows to be true, the natives can surmise what the actions of other natives would be. Considering the case where there are 4 natives with a blue dot. The 4 with a blue dot know that there are either 3 natives with a blue dot or 4 natives with a blue dot. The rest know that either there are 4 natives with a blue dot or there are 5 natives with a blue dot. OK, that much is simple. Now, comes the harder part....the deductions made by the natives that have a blue dot about the knowledge of the other natives with a blue dot. The ones with a blue dot know that there are either 3 or 4 natives with a blue dot. But, they don't know what the other natives with a blue dot know. If they have a red dot, then there are only 3 natives with a blue dot, each of which think there are either 3 natives with a blue dot or two natives with a blue dot...depending on whether they have a blue dot themselves. If they themselves have a blue dot, then each of the other natives think there are either 3 or 4 natives with a blue dot. At this point, let me arbitrarily name the natives with the blue dots native A, B, C and D. Native A knows that natives B, C, & D will think there are 3 or 4 with a blue dot if A has a blue dot, and think there are 2 or three natives with a blue dot if A has a red dot. Now, native A does some deductions about the understanding of native B concerning the understanding of natives C & D if native A has a red dot. In this case, native B knows that, if he has a red dot, then native C thinks there is either one or two natives with a blue dot, depending on his dot. After D lives past the first midnight, he knows that the number must be 2, requiring him to have a blue dot. He would, if B had a red dot, kill himself (along with D who makes similar deductions) the second night. When this doesn't happen, if native A had a red dot, native B would know that he had a blue dot, and B, C, & D would kill themselves the third night. Since this doesn't happen, A, B, C, & D accurately deduce that they all have blue dots, and kill themselves on the 4th night. The key here is not what the natives know. In this case, it's a 4-fold regression...it's what the natives know about what the other natives know about what the other natives know about what the other natives know. It would not take an outsider to start the chain....any certain statement that allows each and every native to start the regression of knowledge of knowledge of knowledge would suffice. Does that help? Dan M. _______________________________________________ http://www.mccmedia.com/mailman/listinfo/brin-l
