THE CLEVER SCHOOLBOY

A Philosophical Conundrum

Sometimes philosophers take a break from the heavy lifting required by metaphysics, epistemology, value theory, and other arcane inquiries, in order to have a little fun with a puzzle. Here is one that baffled a number of professional philosophers several decades back, until British philosopher Gilbert Ryle revealed in a journal article that he had found a solution that had satisfied him for quite a long time. I offer my own version here for your entertainment and possible philosophical profit.

The setup

A math teacher announced to his class on a Friday, “There will be a surprise test next week. By that I mean that no student, while walking to school on the day of the test, will be able to predict the test will be given on that day.”

A clever student at the back of the class raised his hand and said, “Sir, that’s impossible. You can’t give a surprise test next week or any other week if you announce it beforehand. Here’s the thing: if you don’t give us the test by next Thursday, any student walking to school on Friday would know the test has to occur that day. But then it wouldn’t be a surprise. Therefore, your test can’t be given on Friday. Now, knowing that Friday is out, if the test has not been given by Wednesday, then any student walking to school on Thursday would realize the test would have to be given on that day. But then it wouldn’t be a surprise, so the test can’t be given on Thursday, either. The same line of thinking applies in turn to Wednesday, Tuesday, and Monday. There are no other possible test days. Therefore, you can’t give us a surprise test next week at all.”

The teacher praised the student for his ingenious objection and dismissed the class. The following week, he gave the test on Tuesday surprising, in the required sense, every member of the class.

The Conundrum: Where did the student’s argument go wrong?

Time out

You may wish to close your laptop at this point and adjourn to a coffee shop to think about this and possibly solve it on your own. However, be forewarned - it's not easy. If you find yourself after a while tempted to hurl your coffee mug at a wall, take a breath, return to your computer, and let the following restore your peace of mind.

The solution

Here is the argument again, this time set out in step-by-step logical form for ease of analysis:

The teacher decrees: "There will be a surprise test next week. By 'surprise' I mean that no student walking to school on any of the 5 days will know that the test will be given on that day."

The clever student argues to the contrary:

1. (a) If the teacher doesn’t give the test by next Thursday (i.e. by the end of the day’s class), any student walking to school on Friday would know the test would have to be given on that day. (b) But then it wouldn’t be a surprise, which contradicts the teacher's decree. (c) Therefore, the test can't be given on Friday.

2. (a) Now, knowing that Friday is out (can't be a surprise test day), if the test has not been given by Wednesday, then any student walking to school on Thursday would realize the test would have to be given on that day. (b) But then it wouldn’t be a surprise. (c) Therefore, the test can’t be given on Thursday either.

3. (a) Next, knowing that Friday and Thursday are out, any student walking to school on Wednesday would know the test has to be given on that day. (b) But then it wouldn't be a surprise. (c) Therefore, the test can't be given on Wednesday.

4. (a) Next, knowing that Friday, Thursday, and Wednesday are out, any student walking to school on Tuesday would know that the test has to be given on that day. (b) But then it wouldn't be a surprise. (c) Therefore, the test can't be given on Tuesday.

5. (a) Finally, knowing that Friday, Thursday, Wednesday, and Tuesday are out, any student walking to school on Monday would know the test has to be given on that day. (b) But then it wouldn't be a surprise. (c) Therefore, the test can't be given on Monday.

6. There are no other possible test days in the week.

Conclusion: Therefore, the teacher can’t give the students a surprise test next week at all.

Problem: The teacher administered the test on Tuesday the following week, surprising, in the required sense, everyone in the class. So we know something is wrong with the clever student's argument. But what exactly?

The student's reasoning consists of a chain of 5 deductive arguments, all of which have the same structure - premise a, premise b, conclusion c - only the names of the days are different in each argument. After Step 1, each argument depends on the one before. That means if the first argument is flawed, the entire chain collapses. And that is exactly what happens. Here's why:

A deductive argument can go wrong in two basic ways:

First, a deductive argument is invalid if the conclusion does not follow from the premises, even if the premises are true.

All men are mortal.

Some women are mortal.

Therefore, some women are men.

[By contrast, of course, an argument is valid if its conclusion follows necessarily from the premises. That is, if the premises are true, the conclusion has to be true.

All golden eagles are raptors.

All raptors are warm-blooded.

Therefore, all golden eagles are warm-blooded.]

Second, a deductive argument is unsound when one or more of the premises is false, even if the conclusion does follow from the premises.

All humans are mortal.

Aphrodite is a human. (False. Aphrodite is a goddess, not a human.)

Therefore, Aphrodite is mortal.

Of course, an argument may be both invalid and unsound, but that is not the case with the surprise test argument. It is valid but unsound.

Why valid? If you look carefully at the student's argument at Step 1, you can see that if the premises are true, the conclusion is necessarily true. That means it's valid. The logic is perfect. But is the argument sound? No. Consider the first premise: if the test is not given by Thursday, then it must be given on Friday. But that is true only if the student can be certain that there will be a test. Can he be certain of that?

The student assumes, as a matter of fact, that there will be a test next week. Is that assumption true? Is the student entitled to make that assumption? …. What does he know for sure? Only that the teacher said there will be a test. Could he logically infer from that that there will be a test?

The teacher said there will be a test next week.

Therefore, there will be a test next week.

No. An inference like this is sometimes warranted. For example, there is a scientific law that tells us that if we heat a piece of copper, it will expand - always; we know of no exceptions. So we can be certain that every time we heat our copper pan on the stove, it will expand. Now, is there a scientific law that states 'Every time a teacher says there will be a test next week, the test always takes place'? Of course not. Perhaps the student has never experienced an occasion when a teacher decreed a test but then the test didn't happen. Even so, does that justify his thinking that such an occasion can never happen? Certainly not.

With her decree - "There will be a surprise test next week" - the teacher was not making a prediction based on scientific law. She was stating her intention to give a test next week. As the clichÃ© has it, there is many a slip between cup and lip. Could the teacher guarantee that she would give the test next week? Of course not. She is not God. She can't control how the future will actually play out despite her best intentions.

I needn't bore you here with a list of the multitude of circumstances that might intervene to prevent the teacher from giving her test: she falls ill and can't teach on Friday, a blizzard forces a total school closure, the dog eats her test paper, etc. She might even change her mind at the last minute and announce to the class on Friday morning, "There won't be a test today, kids. I looked it over last night and decided it's not as good as it should be. Over the week-end I will revise it and give you a surprise test next week."

It should be clear now that a test, announced as a surprise in the previous week, can be administered even on Friday. So, if the test has not been given by Thursday, the surprised students would be well advised to study for the test anyway - just in case.

And what of our clever schoolboy? We give him an A for logical brilliance but an F for not realizing the difference between an intention and a scientific prediction.