lew@ihuxr.UUCP (09/10/83)
Here's my explanation of the "surprise quiz" paradox. I think it is a paradox of the same ilk as "this statement is false". I would restate the teachers announcement as follows: --------------------------------- I am going to give a quiz, subject to two conditions: 1) It will be given during class one day this term. 2) At no time prior to the quiz will it be possible to infer from these conditions that the quiz will be given on a certain day. --------------------------------- I say that the paradox is now genuine. That is, it is impossible for the teacher to fulfill these conditions. The self-referential character of condition 2) is evident, and provides the key to the paradox. This paradox had a much bigger impact on me than others I've seen. I find it hard to extricate the logical structure from my judgement of what my real expectations would be if the teacher made the announcement. What if the teacher said, "I'm giving a quiz tomorrow, but you don't know this." ??? Lew Mammel, Jr. ihuxr!lew
FtG@rochester.UUCP (FtG) (09/12/83)
The surprise quiz and its variations, like "The Unexpected Hanging" (Gardner) are NOT really paradoxes. They are merely true statements. 1. It is not possible to make a RATIONAL choice that no other RATIONAL person can not anticipate. 2. People make IRRATIONAL choices all the time. Pick Wednesday. (Why? Why not.) The everyday world bears very little relation to the perfect world of logic. FtG
leichter@yale-com.UUCP (Jerry Leichter) (09/12/83)
The "surprise quiz" problem has often been stated in terms of a prisoner, who is told that he will be executed at dawn on some morning the following week, but will not be able to know the evening before that he will die the following morning; in fact, if had does know, and can prove it to the judge, he will be granted a pardon. W V Quine published a paper, the name of which I don't know, analyzing this problem quite nicely, calling it a pseudo-paradox. I think his disolution of the paradox is quite nice. It rests on the meaning of my "knowing" something. Analysis of what it means to know X (by Quine and others) shows that it has 3 components: X is true; I believe X to be true; I have a good reason for my belief that X is true. The classic example that shows that all three conditions are needed is the following: I look in a field and see sheep. Do I know there are sheep there? (a) suppose that there are actually pictures of sheep, but not sheep. Then it isn't reasonable for me to say that I know there are sheep because there aren't... One goes through a number of related examples, and ultimately looks at a case in which I see pictures of sheep, and behind the pictures are real sheep, which I can't see. Now, the first two conditions hold, but not the third - and in fact I'd rather not say I know there are sheep. How does this apply to the exam/prisoner problem? Suppose I am the teacher/judge, you the student/prisoner. The evening before the last day, you come to me and say: I know the exam/hanging is tomorrow. Well, do you KNOW this? I intend to test/hang you tomorrow, so what it is in fact true that you will be tested/die; apparently you believe it. But what is your REASON for this belief? Well, you took my first statement: You will be tested/be hanged this week. and assumed it to be true; from this, you concluded that tomorrow is the big day. However, I also made another statement: You won't know the day before that you will be tested/hanged. Your alleged "proof", the basis of your claim of knowledge, makes this statement false! What gives you any particular reason to prefer one statement over the other? If you are going to use my statements as starting points for a PROOF, you either take both as givens, or you make no assumptions at all about their veracity. You cannot give a meaningful proof by picking and choosing which hypotheses to accept and which to reject. Hence, your proof fails. You are in an interesting position: You will be tested/hanged tomorrow; you believe it; it's true; but you cannot, by the very nature of the situation, KNOW it to be true! Notice that if I had said: You won't BELIEVE the day before that you will be tested/hanged - you have an out: You just tell me you believe every single night, and you are quite safe. -- Jerry decvax!yale-comix!leichter leichter@yaele \\\ @yale