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蓝眼睛谜题

2018年12月16日 未分类 暂无评论 阅读 46 次

陶哲轩的博客上的一个puzzle。

地址在这里:https://terrytao.wordpress.com/2008/02/05/the-blue-eyed-islanders-puzzle/ 。你可以直接跳过去读原文。

这是发生在一个在虚构的岛上的故事,不过因为这里面的逻辑本身就非常微妙,所以故事描述的时候会稍显啰嗦,我直接摘下来:

There is an island upon which a tribe resides. The tribe consists of 1000 people, with various eye colours. Yet, their religion forbids them to know their own eye color, or even to discuss the topic; thus, each resident can (and does) see the eye colors of all other residents, but has no way of discovering his or her own (there are no reflective surfaces). If a tribesperson does discover his or her own eye color, then their religion compels them to commit ritual suicide at noon the following day in the village square for all to witness. All the tribespeople are highly logical and devout, and they all know that each other is also highly logical and devout (and they all know that they all know that each other is highly logical and devout, and so forth).

[Added, Feb 15: for the purposes of this logic puzzle, “highly logical” means that any conclusion that can logically deduced from the information and observations available to an islander, will automatically be known to that islander.]

Of the 1000 islanders, it turns out that 100 of them have blue eyes and 900 of them have brown eyes, although the islanders are not initially aware of these statistics (each of them can of course only see 999 of the 1000 tribespeople).

One day, a blue-eyed foreigner visits to the island and wins the complete trust of the tribe.

One evening, he addresses the entire tribe to thank them for their hospitality.

However, not knowing the customs, the foreigner makes the mistake of mentioning eye color in his address, remarking “how unusual it is to see another blue-eyed person like myself in this region of the world”.

 

所以,这个游客的这句话到底带来了什么影响了吗?

观点1:没有影响。因为游客不过说了大家都知道的事实(每个人都能看到其他人的眼睛颜色)。

观点2:100天后,所有蓝眼睛的人自杀了。原因可用数学归纳法证明(注:这个证明本身也应当先怀疑一下)。

 

到底哪个是正确的呢?

陶哲轩在评论区给了答案:观点2是正确的(归纳是正确的)。游客确实带来了一些新东西:common knowledge(查了下这个属于模态逻辑里面的概念,似乎被归到哲学系课程里面去了)。

 

真是一个悲伤的故事...

 

 

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