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Demystifier

Science Advisor

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## Main Question or Discussion Point

The blue-eye puzzle (or paradox, or riddle) is a well known logical puzzle, explained and discussed in many places, including

http://puzzling.stackexchange.com/questions/236/in-the-100-blue-eyes-problem-why-is-the-oracle-necessary

https://en.wikipedia.org/wiki/Common_knowledge_(logic)

http://math.stackexchange.com/questions/489308/blue-eyes-a-logic-puzzle

Since the puzzle is explained in those and many other places, I will assume that readers are familiar with the problem, so I will not explain what the problem is. I want to discuss the solution(s).

I have my own solution of the problem. (Perhaps someone already proposed that solution, but I am not aware of that.) In short, my solution is that the solution of the problem is

Let me explain. At the beginning of the puzzle it is said that all people are "

So, to get to the point, the two different solutions of the blue-eye problem correspond to an application of two different types of logic. It is not predefined which type of logic should be used (it is only said that "perfect logic" should be used, but that means nothing), so it is impossible to give a unique answer. In this sense, the problem is not well posed.

To conclude, the paradox stems from the false belief that there is such thing as "perfect logic", seducively suggesting that the solution should be unique. But there isn't. You must use one type of reasoning or the other, and neither of them is perfect or necessarily better than the other.

http://puzzling.stackexchange.com/questions/236/in-the-100-blue-eyes-problem-why-is-the-oracle-necessary

https://en.wikipedia.org/wiki/Common_knowledge_(logic)

http://math.stackexchange.com/questions/489308/blue-eyes-a-logic-puzzle

Since the puzzle is explained in those and many other places, I will assume that readers are familiar with the problem, so I will not explain what the problem is. I want to discuss the solution(s).

I have my own solution of the problem. (Perhaps someone already proposed that solution, but I am not aware of that.) In short, my solution is that the solution of the problem is

**not unique**. There are (at least) two solutions, and from the formulation of the problem it is impossible to eliminate one of them. One solution (the obvious one) is that nobody will do anything, and another solution (the standard one) is that they will all commit suicides after 100 days. In a sense, both solutions are "correct".Let me explain. At the beginning of the puzzle it is said that all people are "

**perfect logicians**". But that means absolutely nothing. There is no such thing as "perfect logic". If you open a logic textbook, you will find chapters such as Propositional logic, Predicate logic, Second order logic, Modal logic, etc. But you will not find chapter entitled "Perfect logic", simply because neither of those types of logic is "perfect". Each kind of logic has its own principles of inference, and in general there is no purely logical way to determine when to apply which kind of logic. The principles of inference for each kind of logic are defined by humans, not given by God. It is left to the human**intuition**(not to the human logic) to decide when to use which kind of logic.So, to get to the point, the two different solutions of the blue-eye problem correspond to an application of two different types of logic. It is not predefined which type of logic should be used (it is only said that "perfect logic" should be used, but that means nothing), so it is impossible to give a unique answer. In this sense, the problem is not well posed.

To conclude, the paradox stems from the false belief that there is such thing as "perfect logic", seducively suggesting that the solution should be unique. But there isn't. You must use one type of reasoning or the other, and neither of them is perfect or necessarily better than the other.