Outer Limits of Reason by Noson S. Yanofsky

doors. You choose one of the doors and then Monty proceeds to open twenty-three other doors. Each door he opens reveals a goat, as in figure 3.8 .
    Figure 3.8
    An extended version of the Monty Hall problem
    Now there are two doors that Monty did not open: the one you chose and one other that he avoided. It could very well be that (a) the one you chose is the one with the car (a 1-out-of-25 chance) and that Monty is simply bluffing and hoping you change. Or it could be that (b) you picked a door with a goat behind it (a 24-out-of-25 chance) and since Monty knows where the car is, he is not going to open that door. It is obvious that you should switch. Here Monty is subtly giving you information about the whereabouts of the car by not telling you where the car is .
    Here is an interesting scenario to think about. Imagine that Monty himself does not know where the car is. Then he will be randomly opening doors. He might accidentally open the door with the car in it and the game is over. But if he does not accidentally open the door with the car, then should you switch? Answer: nope! There is nothing to gain. You should switch only when you know that Monty knows, and he is subtly giving you the information.
    This is just one of the strange aspects of knowledge and information that we explore in this section.
    Probably the simplest paradox about knowledge is the cousin of the famous liar paradox that we met in the last chapter. Simply hold the following idea in your head:
    This idea is false.
    As with the liar paradox, this idea is false if and only if it is true. This self-referential paradox also has many variants. For example, on Tuesday you can have the idea that while today you cannot think straight,
    Tomorrow, all my ideas will be clear and true.
    Then, on Wednesday, you can realize that
    All my thoughts of yesterday were false.
    Question: Was Tuesday’s thought true or false? A short argument following the implications will show that Tuesday’s idea was true if and only if it was false.
    One possible solution to this is that the human mind is full of contradictions. As I mentioned in chapter 1 , the human mind is not a perfect machine and has conflicting ideas. A little introspection will show that we all believe ideas that contradict each other.
    One of the more interesting paradoxes about knowledge is called the surprise-test paradox . A teacher announces that there will be a surprise test in the forthcoming week. The last day of class is Friday of that week. What day can the surprise test happen on? If the test is going to be on Friday, then after school on Thursday night the students will already know that the test is on Friday and it will not be a surprise test. So the test cannot happen on Friday. Since this was purely logical reasoning, everyone knows this. Can the test be on Thursday? After class on Wednesday night the students can deduce that since the test has not happened already and it cannot be on Friday, it must be on Thursday. But again, since they know that it must be on Thursday, it will no longer be a surprise test. So the test cannot occur on Thursday or Friday. We can continue reasoning in the same way and conclude that the test cannot happen on Wednesday, Tuesday, or Monday. When exactly will this surprise test occur?
    Logic has shown us that a teacher cannot give a surprise test within a given time interval. This is a paradox because it goes against the obvious fact that teachers have been torturing students with surprise tests for millennia.
    It is interesting to note that the paradox would not arise if the teacher just remained silent. The problems only arise because of the teacher announcing to the students that there will be a surprise test. The instant the students are told of the surprise test, they must hold the two contradictory thoughts simultaneously: there will be a surprise test and there cannot be a surprise test.
    Â 
    In 2006, Adam Brandenburger and Jerome Keisler published a