July 28, 2000/January 19, 2001 Eric Rasmusen, erasmuse@indiana.edu THE QUIZ ON FRIDAY PARADOX This is a paradox, seemingly about backwards induction, that I come across now and then. VERSION ONE. On Sunday, a teacher tells his students, "I will give a quiz on or before Friday and surprise you." If Friday came, and there had been no quiz, the students would know the quiz would be on Friday, and it wouldn't be a surprise the day they came in. So the quiz can't be on Friday. But then if Thursday came and there had been no quiz, the students would know it would have to be on Thursday, and it wouldn't be a surprise. So the quiz can't be on Thursday. By induction, it can't be on any day. The teacher was lying. Yet if the teacher gives a quiz on Monday, everyone will be surprised, and the teacher was not lying. VERSION TWO: The essentials of the paradox remain in a two-day version-- that is, the quiz will be on either Thursday or Friday. VERSION THREE: The essentials of the paradox remain even in a one-day version. On Sunday, the teacher says, "(a) I am going to give you a quiz tomorrow, and (b) you will be surprised." The students respond by thinking "The teacher must be lying-- there is only one possible day for the quiz, and so we can't be surprised." Then, if the teacher does give the quiz, they are surprised. THE PARADOX. The paradox in all three is that if the students believe the teacher's statement is true, then it is false in either part (a) or (b), either there is no surprise or no quiz is given. But if the student believe the statement is false, it is true in both parts, because the teacher will give the quiz and they will be surprised. A RESOLUTION: The resolution of all three paradoxes is that the students should conclude: "The teacher's statement is false. The teacher indeed might give us a quiz-- but maybe not. If she does give the quiz, we will therefore be surprised." THE FALSE STEP. The false step that creates the paradox is the implicit assumption that if the students believe the statement is false, they believe alternative Alpha: "There will not be a quiz" rather than alternative Beta: "There might be a quiz." The paradox would not arise if the teacher had said, "I might give a quiz and if I do I will surprise you, on or before the last of classes on Friday". Based on that, the students still wouldn't know on Friday if there was going to be a quiz. I think my answer is the same as the one the noted philosopher W. Quine gives in "On a So-Called Paradox," Mind, New Series, Vol. 62:65-67, (January, 1953), but I don't fully understand his article. He, too, eliminates the backwards induction part of the paradox, and emphasizes that if the students are willing to accept that the teacher might be lying, they should incorporate that into the original alternatives they might believe.