Missives from a fly bottle
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Last revised 31 August 2017
The surprise exam


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.

Other Sites

Paradox Lost (1971), by Ian Stewart.
jaworski.co.uk
9. The heap

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 n-day announcement is a blindspot for the class, then so is the (n+1)-day one, for any positive integer n. But the one-day announcement is unquestionably a blindspot for the class. So the announcement appears to be a blindspot for the class for any value of n, in particular n = 5.

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 year, the class would still be unable to accept her words. They would be unable to take her seriously!

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:
The one-day announcement is a blindspot.
If the n-day announcement is a blindspot, then so is the (n+1)-day announcement.
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 paradox of the heap is that, from these two truisms:
One grain of sand is not a heap of sand.
If n grains of sand is not a heap of sand, then neither is n+1 grains.
it follows that any number of grains is not a heap of sand, i.e., there is no such thing as a heap of sand. Naturally, this is absurd, but the first claim is obviously true, while the second, though occasionally contested, is equally hard to dispute. (How can one grain of sand separate a non-heap from a heap?)

There is no consensus on how to resolve this paradox and we need not try to address it here. The point is only that our paradox has exactly the same form. So the question is whether the resemblance to the heap is just a coincidence. It’s hard to know how to decide this question but I can’t see any reason to resist the assimiliation!

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 obviously true in the way they normally are for the heap. Thus, it is relatively obvious that the two claims above for the grains of sand are true, but it is not likewise obvious that the corresponding two claims concerning blindspots are true – we have had to spell everything out. Of course, nothing in the heap requires that these things be obvious so long as they are true. In our case, it just means that the paradox is not obvious, but it is there all the same. Indeed, its unobviousness is what makes it an interesting case of the heap.

Apart from this, we would also have a relatively short heap.

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 five iterations seem to yield the requisite absurdity, assuming it absurd that the five-day announcement should be a blindspot.

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 cherry red, you are forced to say the same of the next, and eventually of the last one too.


 

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 vague expression like ‘heap’ or ‘red,’ which admits of grey (or borderline) cases where its application is indeterminate, even when all the relevant facts are known. Typically, we need a range of such cases, involving a gradual indeterminacy of application, as with the red squares. Other expressions which generate the paradox are ‘bald,’ ‘tall’ and ‘child.’ The paradox does not arise with relatively precise terms like ‘legally married’ or ‘right angle.’ In our case, the culprit vague expression is doubtless ‘blindspot’ or (at bottom) ‘know,’ ‘accept’ or ‘believe.’

Consider ‘know’ for example, which was used to introduce ‘blindspot.’ Some people mistakenly think of knowing as a black-or-white matter that admits of no grey cases. But, in the ordinary use of the term, there certainly are grey cases where it is indeterminate whether someone knows something, even when all the facts are plain.

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 know that you were going to win? There may be no specific moment marking the transition. “It dawned upon me,” you might say, using a well-known metaphor whose literal subject is often itself used to illustrate the paradox of the heap. Phrases like “don’t really know,” “sort of know,” and “know for sure” also abound in our language and the same is true of ‘accept’ and ‘believe.’

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 know that she will succeed. The talk is in degrees and it can make sense to leave it like that.

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 n-day announcement to “ease out” of being a blindspot as the value of n increases. As with the grains of sand, the exact point of transition will simply be indeterminate.

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 n-day announcement is a blindspot for the class, then so is the (n+1)-day announcement, for any value of n. At the same time, we may insist that the five-day announcement is not a blindspot for the class, siding with common sense.

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.