I found this difficult to pinpoint, too, until I read about common knowledge. The example here is very relevant: http://en.wikipedia.org/wiki/Common_knowledge_(

I found this difficult to pinpoint, too, until I read about common knowledge. The example here is very relevant: http://en.wikipedia.org/wiki/Common_knowledge_(logic) http://en.wikipedia.org/wiki/Common_knowledge_(logic) At first, the islanders do not know that the other islanders are making the same logical judgments. The proof relies on the fact that each islander knows what the other islanders have deduced by each day. By making an announcement to everyone at once, the new information that the guru provides is the common knowledge that (i) there is a blue eyed person, (ii) the islanders know there is a blue eyed person, (iii) the islanders know the other islanders know that there is a blue eyed person, and so on. Note that if the guru went to each islander one by one and told them individually that there is a blue-eyed person, they do not learn anything new about the other islanders and nobody would leave.