This was written in 2009. I explain in this essay why I think the surprise exam paradox is just a subtle version of the paradox of the heap. This was not easy to figure out at all because this is one of the most confusing paradoxes on the planet.

Paradox Lost (1971), by Ian Stewart.

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The student claimed that the five-day announcement was just a more sophisticated form of the one-day announcement, but a blindspot all the same. A simple piece of induction now seems to bear out his claim.

As we just saw, if the

This argument is pleasingly simple and plausibly regarded as the essence of the student’s reasoning. It appears to confirm that the five-day announcement is a blindspot for the class, but, worse, if the relevant period was (say) 365 days, then the same conclusion would hold. Thus, should the teacher announce that a surprise exam will occur some time during the

Needless to say, this seems intolerable. Indeed, calling the five-day announcement a blindspot seems bad enough. But how should we respond?

We could resist the argument by rejecting one of its two premises:

But the first premise is indisputable and I have also explained why I think the second must be granted. In my opinion, there is nothing for it but to conclude that this is just an instance of the paradox of the heap!The one-day announcement is a blindspot.If then-day announcement is a blindspot, then so is the (n+1)-day announcement.

The paradox of the heap is that, from these two truisms:

it follows thatOne grain of sand is not a heap of sand.Ifngrains of sand is not a heap of sand, then neither isn+1 grains.

There is no consensus on how to resolve this paradox and we need not try to address it here. The point is only that

If the surprise exam is indeed a version of the heap, then it’s a fairly interesting one, because the statements that generate the paradox are not

Apart from this, we would also have a relatively

Thus, it would normally take many iterations of the inductive claim to yield a noticeable absurdity, e.g., thousands, for the grains of sand. But, in our case, just

But short heaps are possible as the following example shows.

No two adjacent squares are distinguishable to the naked eye in point of hue, so if you will call the first one

But it seems clear that, while the first one is cherry red, the last one is not. So short heaps do exist and the shortness of ours need not be a concern.

Finally, a feature deemed essential for generating such a paradox is the presence of a

Consider ‘know’ for example, which was used to introduce ‘blindspot.’ Some people mistakenly think of

For example, if you find yourself gradually dominating the field in a marathon race and end up winning by a mile, at which point did you

So, at various points, the question arises of whether the class knows this, or whether they know that, and the idea is that a determinate answer is not always needed, since it may be correct to think in terms of a slippery slope. Thus, the more days the teacher has at her disposal, the easier it is for her to surprise the class as intended, and the more absurd it would be to deny that they can

Likewise, we should not deny that the one-day announcement is unknowable to the class. But we might demur at the thought of drawing a sharp line at that point that would allow the two-day announcement (onwards) to be knowable without ado. Our analysis has also not borne this out. In contrast, we might sensibly expect the

Such a “knowledge leak” is easy to grasp and we can also see this from the point of view of the class. A one-day announcement would certainly befuddle them, whereas a two-day announcement, while still befuddling, would perhaps befuddle them a little less. A three-day announcement, in contrast, is not so much of a blindspot and a four-day announcement even less of one. And so on, until one has clearly eased out of a blindspot on the fifth day (say). The slippery slope approach does sit well with this paradox.

So the recommendation is to concede that the one-day announcement is a blindspot for the class, while accepting the student’s contention that if the

As for how we can blithely subscribe to three jointly inconsistent claims like that, that is of course the paradox of the heap! So while we have not solved the surprise exam paradox, we have at least managed to reduce it to another paradox and this seems like a good place to stop.

The American philosopher Roy Sorensen is one of the few people who has carefully examined whether the surprise exam may be reduced to the heap in any such way. His considered answer is no. I will end by explaining why I think he’s wrong.

Achilles & the tortoise

The surprise exam

Newcomb’s problem

Newcomb’s problem (sassy version)

Seeing and being

Logic test!

Philosophers say the strangest things

Favourite puzzles

Books on consciousness

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