Surprises in Logic (2016)

I don't think you need anything fancy to tackle the "surprise examination" or "unexpected hanging" paradox. This is my take on it, at least:

> The teacher says one day he'll give a quiz and it will be a surprise. So the kids think "well, it can't be on the last day then—we'd know it was coming." And then they think "well, so it can't be on the day before the last day, either!—we'd know it was coming." And so on... and they convince themselves it can't happen at all.

> But then the teacher gives it the very next day, and they're completely surprised.

The students convince themselves that it can't happen at all... and that's well and good, but once they admit that as an option, they have to include that in their argument—and if they do so, their entire argument falls apart immediate.

Consider the first time through: "It can't be on the last day, because we'd know it was coming, and so couldn't be a surprise." Fine.

Now compare the second time through: "If we get to the last day, then either it will be on that day, or it won't happen at all. We don't know which, so if it did happen on that day, it would count as a surprise." Now you can't exclude any day, the whole structure of the argument fell apart.

Basically, they start with a bunch of premises, arrive at a contradiction, and conclude some new possibility. But if you stop there, you just end up with a contradiction and can't conclude anything.

So you need to restart your argument, with your new possibility as one of the premises. And now you don't get to a contradiction at all.

I would encourage anyone who's intrigued by this paradox to read Timothy Chow's comprehensive paper on the subject (https://arxiv.org/abs/math/9903160).

In particular, he discusses what he calls the meta-paradox:

> The meta-paradox consists of two seemingly incompatible facts. The first is that the surprise exam paradox seems easy to resolve. Those seeing it for the first time typically have the instinctive reaction that the flaw in the students’ reasoning is obvious. Furthermore, most readers who have tried to think it through have had little difficulty resolving it to their own satisfaction.

> The second (astonishing) fact is that to date nearly a hundred papers on the paradox have been published, and still no consensus on its correct resolution has been reached. The paradox has even been called a “significant problem” for philosophy [30, chapter 7, section VII]. How can this be? Can such a ridiculous argument really be a major unsolved mystery? If not, why does paper after paper begin by brusquely dismissing all previous work and claiming that it alone presents the long-awaited simple solution that lays the paradox to rest once and for all?

> Some other paradoxes suffer from a similar meta-paradox, but the problem is especially acute in the case of the surprise examination paradox. For most other trivial-sounding paradoxes there is broad consensus on the proper resolution, whereas for the surprise exam paradox there is not even agreement on its proper formulation. Since one’s view of the meta-paradox influences the way one views the paradox itself, I must try to clear up the former before discussing the latter.

> In my view, most of the confusion has been caused by authors who have plunged into the process of “resolving” the paradox without first having a clear idea of what it means to “resolve” a paradox. The goal is poorly understood, so controversy over whether the goal has been attained is inevitable. Let me now suggest a way of thinking about the process of “resolving a paradox” that I believe dispels the meta-paradox.