Knowledge About Knowledge
So certain kinds of knowledge have an eerie capacity to change us, and it feels like a violation of natural laws. Information shouldn't have so much power, being unreal and insubstantial. If all that has changed is your awareness of something, and the knowledge has made you worse off, then surely you can just forget it? It is a matter of rearranging a few molecules. But of course it isn't possible, and so we have aphorisms about rung bells and scrambled eggs, to express the irreversibility of things such as knowledge.
There are the germs of many good novels here (and a pretty good movie), but anyway what I want to say now is that the 2016 presidential election is perhaps dangerous as much for what it is teaching us about our country and our fellow countrymen as it is for its results (which, as I write, seem likely to be non-disastrous). We have always known a certain number of people to be cretinous, and others to be racist, and we are aware that the territory of overlap is especially well-populated. But the sheer volume of new evidence is disarming, and I have the feeling that bells are being rung all over that can never be unrung even when all the dust has settled. We don't only know that we are surrounded by these people, but that everyone knows, that we know it and they know it and they know that we know it, and so we are establishing common facts that will be very uncomfortable for a long time to come. The Republicans are desperately trying to retain as much ambiguity as possible, but the party's choices are out there for everyone to see.
I append a discussion that may be of interest to some of my readers.
There is a well-known thought experiment in game theory to illustrate the importance of common knowledge. Three people (who are all commonly known to be completely rational) sit facing each other, all of their faces visibly dirty, but each of them unable to see his own face. At the stroke of each hour, if a person stands up and states the number of people with dirty faces, he will either receive a reward (if he is correct) or a punishment (if he is wrong). Lacking a credible way to communicate information to each other, the three find themselves in a deadlock.
A fourth person enters the room, announces (with complete credibility) that at least one of the three has a dirty face, and exits. At this point, you may wish to predict what will happen before you read further.
To use up space between the puzzle and the answer, I will observe that these highly stylized puzzles are sometimes taken too literally. The point is not to be clever but to establish a conceptual possibility, and it is that conceptual possibility that should be the focus of the student's attention. To that end, I'll add a somewhat realistic example (in fairness to my own educators, I believe it is an example we discussed in the game theory class that I took).
And now the answer: As the clock strikes the hour for the third time after the announcement, all three stand up and declare that there are three dirty faces.
The logic is this: initially, each player must be agnostic about the status of his own face. Each player doesn't stand up initially because he is uncertain whether there are two dirty faces or three. But once everyone has learned that there is at least one dirty face, things proceed as follows. If only one player had a dirty face, he would stand up at the stroke of the first hour. This would be an easy win, because he would possess two pieces of information that, together, compel the conclusion that there is one dirty face: (1) there is at least one dirty face (as per the announcement), and (2) there are at least two clean faces (as per the evidence of his own eyes).
But no one stands up at the stroke of the first hour, because everyone can see two dirty faces. So after the first hour, everyone knows that everyone knows that there are at least two dirty faces. If a player could see one dirty face and one clean face, at the next opportunity he could stand up and confidently announce that there were two dirty faces. But no one can, because no one sees a clean face, and so no one can tell whether there are two dirty faces or three.
But since no one stands up, everyone knows that there are three dirty faces, and so at the third hour they all stand up.
The punchline here is that the fourth person announced a fact that was already known to all of the players. After all, each of them could see two dirty faces, so no one was surprised by the revelation. And in fact, not only did each player know that there was at least one dirty face, each player knew that every other player knew of at least one dirty face. (Again, this is a consequence of the fact that each player could see two dirty faces, and so at most any other player would see one clean face.)
You have to push back one more level before any new information was revealed. Let's call the players A, B, and C. A knew that there was at least one dirty face. A knew that B knew that there was at least one dirty face. But A didn't know that B knew that C knew that there was at least one dirty face. (From A's perspective, A might have a clean face. And if so, then from B's perspective, both A and B might have clean faces. And so A had no way of knowing that B knew that C knew that there was at least one dirty face.) Once the announcement was made, then of course every player knew that every other player knew, to any number of degrees you like.
I'm sorry if I've lost you. Here is a more concrete example. A man has cheated on his wife. She knows it (either because he confessed or because she found other evidence). The community also knows it. Moreover, the community knows that the wife knows. (Let us assume it is a traditional, backward, sex-negative society in which adultery is considered morally wrong. Moreover, it is considered humiliating for both spouses, including the wronged spouse.) Also, the wife knows that the community knows. So basically everyone knows everything, right?
Well, no. Let's walk through it. You are an acquaintance of the wife. You don't know whether she knows that you know about the adultery, and so you must pretend not to know. If in fact the wife doesn't know that you know, then maybe she won't feel ashamed. In fact, even if you know that she knows that you know about the adultery, you should pretend not to know unless you know that she knows that you know that she knows about the adultery. But anyway, the point is that outwardly everyone can pretend that nothing has happened, and her pain can remain a private affair.
This may explain the phenomenon in which a woman who has long known about her husband's adultery suddenly must divorce him when it becomes public, even if it was already widely known. Suddenly, something that was already known by everyone has become known to be known, and while it can still technically be ignored in public (as can almost anything), there is no avoiding the mutual awareness of the falseness of this stance.