A professor of mine said that he stopped giving this problem, because a logician friend of his said there was something wrong with the solution. However, he never found out from his friend what the flaw was.
The problem is rather well-known. There are several different versions; I will give a cleaner version. There is an island of savages, some who have brown eyes and some who have blue eyes. It is tribal law that anyone who discovers that he has blue eyes must kill himself on the next sunrise. The blue-eyed islanders have managed to live so far, because to speak of eye color is taboo, and there are no mirrors or reflective surfaces of any kind on this island (assume the water is too murky). One day, the entire tribe is congregating about an infallible oracle, and the oracle announces, “There is at least one blue-eyed person on this island.” The question is, What happens?
The standard answer is that if there are n blue-eyed islanders, they all kill themselves on the nth day after the oracle’s announcement. The argument is as follows. Suppose there is 1 blue-eyed islander. After the oracle makes his announcement, this person knows that he must have blue eyes, since he sees that everyone else has brown eyes. He therefore kills himself the next day. Now, suppose there are n+1 blue-eyed islanders, and assume as the inductive hypothesis that if there are n blue-eyed islanders, they all kill themselves on the nth day. Each blue-eyed islander sees n people with blue eyes. He reasons that if he himself does not have blue eyes, these n people will kill themselves on the nth day. When this does not happen, he concludes that he must also have blue eyes, and kills himself the next day. Therefore, all n+1 blue-eyed islanders kill themselves on the (n+1)th day.
Is this argument flawed? If so, what is the flaw?
Here are some thoughts.
A. How do you know that on the (n+1)th day there are still n+1 blue-eyed islanders remaining? Maybe, some of them islanders killed themselves earlier, because they discovered they had blue eyes by an argument you have not thought of. The inductive hypothesis says that if there n blue-eyed islanders, they all kill themselves on the nth day. It does not say what happens if the number of blue-eyed islanders is more than n; it might be possible that some of them kill themselves earlier. Is this an essential flaw? Or can you get around it by inducing on two statements at once. For example,
(1) If there are n blue-eyed islanders, they will all kill themselves on the nth day.
(2) If there are more than n blue-eyed islanders, no one will commit suicide on any day up to and including the nth.
B. What new information does the oracle give? If there are 2 blue-eyed islanders, they each know that there is at least one blue-eyed person. You may say, “But they don’t know that the other knows that there is at least one.” OK, so consider the case where there are 3 blue-eyed islanders, Alice, Bob, and Charlie. Alice knows that there is at least one blue-eyed islander, and she knows that Bob and Charlie know too. Then, you reply, “But Alice doesn’t know that Bob knows that Charlie knows.” Continuing in this fashion, in the general case, the oracle’s announcement allows each blue-eyed islander to conclude chains of “I know that he knows that he knows . . . that there is at least one blue-eyed person.”
However, in and of itself, the oracle’s statment doesn’t tell anyone anything that he did not already know. It is the fact that he announced it publically that is important. So by publically announcing information that everyone already knows, he provides new information. Doesn’t that seem strange?
C. Perhaps, there is nothing wrong with the argument, but maybe it can’t be translated into first-order logic; maybe some other type of logic is needed.
Anyone with expert knowledge of logic care to comment on this?
