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I give a quite detailed answer at the linked question, that tries to explain in detail how each event changes the state of (mental) affairs on the island, and how much of "common knowledge" is really required to eventually cause the individuals concerned to leave. In my notation there with $C$ for "it is common knowledge that" and $E$ for "everybody knows ...


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The blue-eyed people determine their eye colour by a proof-by-contradiction that creates hypothetical people each of whom uses a proof-by-contradiction based on hypothetical people etc. It assumes that every one of these hypothetical people is able to fully reason out the thinking of each of the hypothetical people they think of. In order for the proof to ...


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If $n=1$ and no one knows any more than they can see, then the one blue-eyed person does not know that there are blue-eyed people on the island. If $n = 2$, then blue-eyed person $a$ does not know that blue-eyed person $b$ knows that there are blue-eyed people on the island. For all $a$ knows, $b$ could be alone in his blue-eyed-ness, in which case $b$ ...


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The answer can be found in the Volume 3 of the Handbook of Philosophical Logic, 2nd Edition, 2001. See the "Correspondence Theory" chapter by Johan van Benthem, pp. 325-408. On page 333, Fact 5, he gives an example of an incomplete modal logic: The logic $L$ that extends $\mathbf{K}$ with axioms (of degree at most 2 and with only 1 variable) $$ ...



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