Hi
@andrewkirk:
I agree with you that there are ambiguities in the Wikipedia statement of the problem, which I quote below for convenience.
On an island, there are k people who have blue eyes, and the rest of the people have green eyes. At the start of the puzzle, no one on the island ever knows their own eye color. By rule, if a person on the island ever discovers they have blue eyes, that person must leave the island at dawn; anyone not making such a discovery always sleeps until after dawn. On the island, each person knows every other person's eye color, there are no reflective surfaces, and there is no discussion of eye color.
At some point, an outsider comes to the island, calls together all the people on the island, and makes the following public announcement: "At least one of you has blue eyes". The outsider, furthermore, is known by all to be truthful, and all know that all know this, and so on: it is common knowledge that he is truthful, and thus it becomes common knowledge that there is at least one islander who has blue eyes. The problem: assuming all persons on the island are completely logical and that this too is common knowledge, what is the eventual outcome?
In your post #84, you discuss the ambiguity in the text: "all persons on the island are completely logical". I want to discuss that ambiguity with you, but before I do that I thought it would be useful to mention another ambiguity:
By rule, if a person on the island ever discovers they have blue eyes, that person must leave the island at dawn
The ambiguity is due to the two underlined elements. There are two possible interpretations.
1. The rule specifies that a person has an obligation to leave the island at dawn the day following their ascertaining that they have blue eyes. However, there is no specification that a person who has the obligation will actually leave.
2. The "rule" is not just a rule, but rather it is some "wired in the brain programming", and "must" means "has an uncontrollable compulsion to".
Using (1) would produce the conclusion that no one has the knowledge that if someone deduced that they had blue eyes that person would actually leave the island. That means the answer to the puzzle is: No one leaves he island. (2) is the more interesting case, so perhaps the puzzle text would be improved by using some appropriate text to express (2) to replace the "By rule ..." text.
In #84 you deal with the ambiguity by deciding the puzzle had no valid answer. There are ways to improve the text of the puzzle to produce what I think is the most likely (but not the only possible) intention of the originator of the puzzle.
I would appreciate your help in coming up with such improved text. I am not completely happy with my attempt at it below:
All persons on the island are experts in and completely adept at using first order two-valued predicate logic, and they all know that what they deduce using this logic is TRUE regarding both hypothetical worlds as well as the world in which they live, and that all of this is also common knowledge.
With this change in the text the answer to the puzzle is the following:
Given that there are N blue eyed person on the island, All N of them will leave the island N days after the visitor makes his/her announcement.
I have no doubt that there are many ways that this result can be "proved", possibly including the one I outline in my post #238.
I would much appreciate any comments.
I plan to post another resolution of this ambiguity in another post at a later time.
Regards,
Buzz