Is there a limit to the sequence "I know that you know that I know..."?
Consider this hypothetical example. A and B are spies on opposing sides. B gets hold of some secret information about A’s side. B will be trying to deliver this information to his side.
1. B knows
A finds out that B has the information. A need not know the details of the information. A knows enough to try to stop B.
2. A knows that B knows.
B finds out that A is going after him, and will keep track of A’s whereabouts and try to stay away.
3. B knows that A knows that B knows.
It seems that each extra level of knowledge has diminishing impact. At this point, I am not sure if it makes much difference if A knows that B knows that A knows that B knows. Do you think this is the general case?
Consider the case where A tells something to B. Can we assume all levels of A knows that B knows that…?
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11 Answers
Who knows?
Now, I know this can be interpreted two ways and I know that you know that and I know that you know that I knew that you knew this when I typed it but did you know that you knew that I knew that you knew that I knew that?
just wondering
Yes. It’s called a period.
Reminds me of the old song:
_I was lookin’ back to see if she was lookin’ back to see
If I was lookin’ back to see if she was lookin’ back at me._
Perhaps it is because the extra knows are just details on the main issue, which is the first know. That’s why there’s a diminishing aspect. Knowing something about the main idea is less important than knowing the main idea, and knowing something about something about the main idea is even less important since without the main idea, that trivial excess knowledge is useless.
This is vaguely reminiscent of an episode of Friends.
“But they don’t’ know that we know they know we know!”
Of course there is.
Zeus.
We know, is the conclusion to this entire matter of posturing.
In practicality, there is no reason to go beyond a depth of 4. The reason is that, after a certain depth, it is impossible for one side to take preventative action.
Here is an example:
1) The German u-boat fleet is planning a fleet movement.
2) Let’s say that the British have cracked the enigma, and therefore know about it. They want to attack the fleet. They must cover up the fact that they have cracked the enigma, however, otherwise the Germans would increase encryption strength.
3) Let’s say that the Germans found out that the British had cracked the enigma. They then decide to increase encryption strength, which would severely hamper British intelligence gathering.
4) Let’s say that the British knew about #3 (that the increase in encryption strength was not coincidental). There is no major use for this information, because they can’t prevent it. Regardless of why the Germans decided to increase encryption strength, the end result is the same. They might be able to locate a German spy or two, but that is about it.
Theoretically speaking, there is no necessary end, but I suspect that most (almost all) applications run into a dead end where no preventative action is possible.
Linguistically, recursion is limitless. However, cognitively, the recursion only works up to a point because we don’t have sufficient working memory to handle it.
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