JDG wrote: > > 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. > The problem is that you are not assuming that _all_ natives are omniintelligent.
> > *But*, if N(blue) > 2, then *every* native starts out with: > > Given: ~A > > Thus, the arrival of this anthropologist can't impart any additional > information, because the first step of the induction leads to a > conclusion that the natives have already reached anyways. > He does. > Maybe it will help to lay out different cases. It seems clear that if > a native can identify the value of N(blue), then mass suicide becomes > inevitable. > > ************************************************************************ > > Case I > > If N(blue) = 0, then every native exists in: > State 1: Sees only red dots, but doesn't know if N(blue) = 0 or N(blue) > =1 > > This case obviously doesn't apply to the given example. > Ok. In case, the anthropologist would say something like "why everybody has a red dot?" and mass suicide would happen in the first night. > ************************************************************************\ > * > > Case II > > If N(blue) = 1, then: > > State 1: One native sees only red dots, but doesn't know if N(blue) = 0 > or N(blue) =1 > > State 2: All other natives see one blue dot, but don't know if N(blue) > = 1 or N(blue) = 2 > > Moreover, in this case All Natives know that each and every Native knows > that each of them is either in State 1 or State 2. > > In this case, the anthropologist imparts information to the one native > in State 1, causing the cascade. > Yes. > ************************************************************************\ > * > > > Case III > > If N(blue) = 2 > > State 2: Two natives see one blue dot, but don't know if N(blue) = 1 or > N(blue) = 2 > > State 3: All other natives see two blue dots, but don't know if N(blue) > = 2 or N(blue) = 3 > > > Moreover, in this case All Natives know that each and every Native knows > that each of them is either in State 2 or State 3. > > In this case, the anthropologist doesn't impart any information to > anyone. Everyone knows that N(blue) >= 1. So, presuming that the > island existed in a steady state before the anthropologist's arrival, > then her arrival with the announcement that N(blue) >= 1 has no effect. > > Am I just missing something here? > Yes. Let's work it out in detail. Imagine that Gor and Kull are the two blue-dotted natives. When the anthropologist claims "I see a blue dotted native", Gor will reason this way: (G-B): I am blue dotted. Then, the anthropologist gave no new information to Kull. So, Kull will not know his color, and he will not suicide on the first night. (G-R): I am red dotted. Then, the anthropologist gave a new information to Kull - and he will know that he is blue dotted. So, he will suicide on the first night, poor bastard. On the first night, Kull does not suicide. Then, Gor will know that he is blue dotted, and he will suicide on the second night. /****************************************************************************/ Case IV: if (N(blue) = 3) Now we have Gor, Kull and Tar with blue dots. And _Tar_, being omniintelligent, will reason like this: (T-R): Suppose I am red dotted. But Gor is very intelligent, and he will reason as he did in Case III: if Kull does not suicide on the first night, then Gor must suicide on the second night. Gor does not suicide on the second night. So, Tar deduces that he is blue dotted, and suicides on the third night. /********************************************************************/ Case V: if (N(blue) = 4) Now we have Gor, Kull, Tar and Zira with blue dots. Zira will reason like this: (Z-R): Suppose I am red dotted. Since Tar is very intelligent, he will reason as he did in Case IV: if Kull does not suicide on the first night and Gor does not suicide on the second night, then Tar must suicide on the third night. Tar does not suicide on the third night. So, Zira deduces that she is blue-dotted, and suicides on the fourth night. Can you see that the induction process can only be started when the anthropologist gives the _absolute_ information that there is at least one blue dotted native? Alberto Monteiro _______________________________________________ http://www.mccmedia.com/mailman/listinfo/brin-l
