Sequim82 wrote:Let's say the three of us are standing together on the beach. I can see three of us, and I can see that you can see three of us, and I can see that he can three of us, and I can see that you can see that I can see three of us, and he can see that I can see that you can see that I can see three of us, etc, etc,etc. How many 'etc' are there before we come to the bifurcation between the knowledge about how many of us are standing on the beach that is common to the three of us, and statements about how many of us are standing on the beach that we all agree with? Just curious.
The difference between this and the blue eyes problem is that when the three of you can see eachother, it becomes common knowledge that there are three of you. In the blue eyes problem, this is like asking if it is common knowledge that there are 200 people on the island.
Here's where the common knowledge comes in the problem, and where the information that the guru says is clear.
Person1 is blue eyed, but doesn't know it. So, before the guru speaks, he sees that there are 99 blue eyed people and 100 brown eyed people. He also knows that every blue eyed person sees at least 100 brown eyed people and at least 98 blue eyed people. Now, there are two possible cases. Either Person1 has blue eyes, or he doesn't. If he does, then there are 100 brown eyes and 100 blue eyes on the island. If he does not, then what would the other blue eyed people be thinking right now?
Well, Person2, who is also blue eyed and doesn't know it, would see 98 blue eyed people, 100 brown eyed people, and Person1. This is what Person1 knows that Person2 knows. But since Person2 doesn't know his own eye color, then Person1 knows he has blue eyes, but does not know that Person2 knows this. That's easy to understand, right?
So the knowledge of Person2 from Person1's perspective is that there are 98 blue, 100 brown, and Person1. This is different from the actual knowledge of Person2 (which is identical to Person1's knowledge) because Person1 doesn't know everything that Person2 knows, only the information that he knows both absolutely must know.
Now, from Person1's perspective, Person2 could be using the same logic on Person3. "Well, I know Person3 sees 97 blues, 100 browns, myself and Person1." This is not what Person2 knows about Person3, it's what Person1 knows Person2 knows about Person3. Person2 actually knows that Person3 sees 98 blues, and Person1 knows that Person3 sees 98 blues, but Person1 only knows that Person2 knows that Person3 sees 97 blues.
Person4? Well, Person1 knows that Person4 sees 98 blues. And Person2 knows that Person4 sees 98 blues. And Person3 knows that Person4 sees 98 blues. But Person2 doesn't know that Person3 knows that; he only knows that Person3 knows that Person4 sees 97 blues. And Person1 doesn't know that; he only knows that Person2 knows that Person3 knows that Person4 sees 96 blues.
And so on and so forth, until you get to the bottom. Person1 knows that Person100 sees 98 blues, but Person1 only knows that Person99 knows that Person100 sees 97 blues. And Person1 only knows that Person2 knows that Person3 knows that... Person100 knows that 100 brown eyes and nothing about blue eyes at all.
So what happens when the guru speaks? Well, that last case changes. Now Person1 still only knows that Person100 sees 98 blue eyes, but now he knows that Person2 knows that Person3... knows that Person99 knows that Person100 knows that there are 100 browns and at least 1 blue.
Does that all make sense? That's why before the guru speaks, there is no induction. The sequence of knowledge of knowledge of knowledge gets cut off at the 100th person. Once the guru speaks, it's absurd even to imagine a 100th layer where someone doesn't know there is at least 1 blue eyed person, because that person must have been there when the guru spoke, and therefore knows there is at least 1.
This is why this statement is wrong:
3. There are three blue-eyes on the beach when the guru speaks. "Well DUH, guru", respond all the logicians in unison.
In the 3 person case, it's even easier to grasp why it's not common knowledge until the guru speaks. Say it's just John, Paul, and George on the island, all with blue eyes, following the same rules as the original problem. John sees that Paul and George have blue eyes, but doesn't know his own eye color. So, to John, Paul may see either 2 blues or 1 blue. If Paul only sees 1 blue eyed person (George), then he would know there was at least 1 blue eyed person, but what if Paul was not blue eyed? Then George would know nothing of blue eyes. So John knows that Paul knows there is at least 1 blue, and Paul knows that George knows there is at least 1 blue, but John does not know that Paul knows that George knows there is at least 1 blue. So when Ringo (with green eyes) shows up and says "There is one blue eyed person here." in front of all of them, then now John knows that Paul knows that George knows there is at least one blue eyed person. That's what changes.
If you still disagree with the 3 person case, consider this: What's the difference between 3 blue eyed people, and 2 blue eyed people with 1 brown eyed person? Would they still get no knowledge from the guru? If the 2 person case gets information but the 3 person case doesn't, then what goes on in the brown-eyed person's head in the 2/1 case? Does he still think "Well DUH, guru"?