"synchronize" is a bit vague. It can be made more precise. Simply put, it sets the initial condition in the recursion. Before someone says "there is a blue ey

"synchronize" is a bit vague. It can be made more precise. Simply put, it sets the initial condition in the recursion. Before someone says "there is a blue eyed person here", the initial condition in the recursion does not hold. The islanders are not allowed to say it so the visitor was the first one to make the statement. If indeed one of the islanders said it (whether blue or brown eyed) the condition would hold as well and the recursion would follow too. However none of them can talk about it.