Editing Common knowledge (logic) (section)

=== Puzzle ===

The idea of common knowledge is often introduced by some variant of [[induction puzzles]] (e.g. [[Induction puzzles#Muddy Children Puzzle|Muddy children puzzle]]):{{ref|Sevitan}}

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 communication of eye color.

At some point, an outsider comes to the island, calls together all the people on the island, and makes the following [[Dynamic epistemic logic#Public Events|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 they are truthful, and thus it becomes common knowledge that there is at least one islander who has blue eyes (<math>C_G[\exists x\! \in\! G ( Bl_{x})]</math>). The problem: finding the eventual outcome, assuming all persons on the island are completely logical (every participant's knowledge obeys the [[Epistemic modal logic#The properties of knowledge|axiom schemata for epistemic logic]]) and that this too is common knowledge.