7.3Ex V
The Surprise Exam Paradox

I've promised you a pop quiz -- sometime this year. But there are only four days left, call them 'a', 'b', 'c', and 'd', where 'a' stands for tomorrow and 'd' for a week from Thursday, the last day. I can't very well give it to you on, d, the last day of class is too late for a surprise. Then it wouldn't be a surprise: you'd know about it in advance. And so, if it's really to be a surprise, I need to give it to you on one of the next three days of class: a, b, or c.

But, once we know for certain that d is "too late for a surprise", then we see that the surprise exam is certainly going to be on one of a, b and c, we can use the same reasoning as above to show it's not going to be on c (else we'd not be surprised!)

But if c it too late for the surprise, and we know this, then you can figure out that it's got to be given on a or b. Then I can't very well wait until b, ... [put the same reasoning in here as before to prove that even b is too late] 

So, when can I give the exam? It looks like I can't surprise you, right? But once you've reasoned like this, to the conclusion that I can't give you a surprise exam, it becomes even easier to surprise you, right? 

That's the paradox.  

But our job for now is just to treat it in a formal way. 

The universe of discourse is the last four days of class
Bxy: x is the class before y
Kx: class x is too late for a surprise.  (In the sense made explicit above.)
a: tomorrow, b: etc. 

We can begin to symbolize the argument with the following premises:  

Bab&(Bbc&Bcd)     (a is the class before b, etc.)
Kd                         (d is too late for a surprise.) (^x)(^y)[(Kx&Byx)>Ky]  (if y is the class before one that is too late, then y is also too late)  

And we want the conclusion to be that there will be no surprise exam: (^x)Kx  

But we're missing a premise. We need to say something more.  

So, two study questions:

  1. What's the missing premise? Express it in English and then think about symbolizing.
  2. What's wrong with the argument? (Because surprise quizzes are possible in fact!)