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.

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Paradox Lost (1971), by Ian Stewart.
10. Sorensen’s objection

In his intriguing book Blindspots, the American philosopher Roy Sorensen considers whether the surprise exam may be reduced to the paradox of the heap:

In light of the similarities between the two paradoxes, it is surprising that few commentators have tried to exploit the resemblance.
Indeed no one has simply asserted that the following is just another instance of the sorites:
The class knows that the exam won’t occur on the last day.
If the class knows that the exam won’t occur on day n, then they know that it won’t occur on day n-1.
Therefore, the class knows that the exam won’t occur.
Why not blame the whole puzzle on the vagueness of ‘know’? ... Despite its attractiveness, I have not found any clear examples of this strategy. (pp. 292-3, lightly modified)

Despite these encouraging words, Sorensen in fact goes on to deny that the surprise exam may be reduced to the paradox of the heap.

His main reason is that the first premise of the inductive argument shown above is questionable. This claims that the class may rule out Friday (the last day) as the exam day, e.g., on the familiar grounds that if the exam were held on Friday, the class would figure this out by Thursday’s end. But Sorensen rejects this claim for a reason that we have already seen. The reason is that, if the teacher were to hold the exam on Friday, then, at Thursday’s end, her words would reduce to a blindspot for the class, who would then stand to be surprised by an exam on the morrow!

But if the class cannot even rule out Friday, then there is no real comparison with the paradox of the heap, which rests on the immovable base that one grain of sand is not a heap of sand. As Sorensen says:

... it is clear that the analogy with the sorites is considerably weakened if the base step of the [surprise exam paradox] is uncompelling. (p. 325)

What should we make of this?

Well, relative to our purposes, it’s not difficult to see that Sorensen’s misgivings are misplaced. The trouble is that, like most other commentators, Sorensen takes the student to be arguing that the surprise exam cannot occur, which is why he lays out the inductive argument in the way he does above. But if the student is arguing only that the class cannot know (accept, believe) that the surprise exam will occur, then the inductive argument begins to look slightly different!

To my mind, the piece of induction that is meaningfully compared with the heap is this one, which we have already seen, and not the one Sorensen has above:

The one-day announcement is a blindspot for the class.
If the n-day announcement is a blindspot for the class, then so is the (n+1)-day one.
Therefore, the teacher’s announcement is a blindspot for the class, no matter how many days are at issue.

The difference is crucial because Sorensen’s objection does not touch this argument. Indeed, his complaint that the one-day announcement is a blindspot for the class actually affirms our initial premise.

The surprise exam paradox is so confoundingly slippery because it is easy to confuse two very different conclusions that the student may be driving at. One is that the surprise exam cannot occur, while the other is that the class cannot know (believe, accept, endorse) that it will occur. People almost always take the student to be driving at the first conclusion but this is not really the best interpretation of what he is up to!

In a sense, of course, there is no question of discovering which conclusion he is “really” driving at, since the student is an entirely fictitious character. Nevertheless, if he were driving at the first conclusion, then there is a very short way with his argument, which is that, from a structural point of view, it simply undermines itself in the manner explained at the start. (Much needless ink has been spilt on this paradox from failing to see this point.)

It makes more sense therefore to read him as driving at the second conclusion, and when his reasoning is spelt out clearly, it proves to be a fairly powerful piece of reasoning, but one which mercifully proves assimilable to the paradox of the heap.