Why does the announcement matter if everyone already sees blue eyes?

Question

On an island of perfect logicians, three people have blue eyes and the rest brown; nobody knows their own colour, and anyone who deduces it must leave that night. A visitor announces publicly, "I can see someone with blue eyes." The three blue-eyed islanders leave on the third night. Here is the objection everyone raises: each blue-eyed islander could already see two blue-eyed people before the visitor spoke. Everyone already knew that someone had blue eyes. The announcement seemed to say nothing new. Explain, concretely and without hand-waving, what information the announcement actually adde…

Accepted Answer

The announcement adds nothing to what anyone knows directly. It changes what everyone knows about what others know, and the induction argument runs on exactly that. The chain before the announcement. Call the blue-eyed islanders A, B and C. Each sees two blue-eyed people. - A knows that someone has blue eyes: she sees B and C. - A also knows that B knows someone has blue eyes: even if A herself were brown-eyed, B would still see C. - But now go one level deeper. A considers the possibility that she is brown-eyed. In that imagined world there are only two blue-eyed people, B and C. In that wor…

Why does the announcement matter if everyone already sees blue eyes?

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