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.