--- 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. > Induction Hypothesis: > > (b) Suppose that there are (N+1) blue dotted natives. Then, each > of these natives, noticing that the other (N) blue dotted natives > didn't commit suicide in the N-th night, will commit ritual > suicide in the (N+1)-th night > > The naturalist provides information because he starts the process, > by forcing step (a) of the induction. If I understand this correctly, here's how the "Induction Hypothesis" works, starting with step (a). Let A = There is one, and only one, blue dot native. Let B = One Native commites ritual suicide on the first night. The induction seems to be that: Given: If A then B. Given: ~B Then: ~A *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. 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. ************************************************************************\ * 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. ************************************************************************\ * 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? JDG _______________________________________________ http://www.mccmedia.com/mailman/listinfo/brin-l
