Friday Paradox:

The Surprise Exam Paradox

Imagine that a professor provides the following blurb on her syllabus:

"In addition to the final paper and final exam, we will have one pop quiz (for 99% of your grade) on some class day between now and the end of the semester. (The topic will also be a surprise.) I won't tell you which day I am going to give the exam, but I will tell you this: I will definitely give an exam on one of the remaining class days, and on that day you will have no good reason to believe that it will be on that day, rather than some other. (This is just what it means for it be a surprise exam, of course.)"

You might object that what the professor has described is impossible. Consider how a sharp student might reply to the prof.

"Well, you can't give the exam on the last day of class, since then we would know that, there being no more classes remaining, you had to give the exam that day; and in that case it would not be a surprise. So we can safely eliminate the last day of class from the list of possible days on which you can give the exam. But then you can’t give it the second-to-last day of class either; for on that day we would know that you couldn’t wait until the last day-since then it would not be a surprise-and so we would know that you had to give it that day. But then, of course, it would not be a surprise. So we can also safely eliminate the second-to-last day of the semester as a possible date for the exam. But then you can't give it the third to last day either… (and so on, until all the remaining dates on which the professor could give such an exam are eliminated.) Therefore, you can't actually give a surprise exam!"

So here’s the paradox: Something is wrong with the student's reply. A professor can clearly give a surprise exam. What went wrong?

(Thanks to Jeff Speaks at Notre Dame for providing materials on paradoxes online. I've borrowed very heavily from this document.)

Friday Paradox:

This one is from a great paper by Saul Kripke (see page 392 for a fuller description):

Pierre is from France and so speaks French. He sees pictures and postcards of a city called, ‘Londres.’ When asked in French whether he finds that city pretty, Pierre whole-heartedly and enthusiastically assents. He earnestly says things like, “Londres est jolie” (“London is pretty” in French). As a consequence, it seems correct to say that Pierre believes that London is pretty. After some time, Pierre moves to a town he knows by the name ‘London’—but he is not aware that it is the same town that he and other Frenchmen call ‘Londres.’ In addition, Pierre was unfortunate enough to have moved into a rather shoddy and rundown neighborhood, and so finds the town that he understands to be ‘London’ to be an extremely ugly place. Having learned quite a bit of English since his move, he adamantly assents to the English sentence, “London is not pretty.” As a result, it seems natural to say that Pierre believes that London is not pretty. Moreover, since he has not learned that ‘London’ and ‘Londres’ are names for the same city, he remains willing to assent to the sentence “Londres est jolie.” He clearly thinks that ‘London’ and ‘Londres’ name two different places.

On the basis of his French utterances it would appear natural to say that Pierre believes that London is pretty. At the same time, on the basis of his English utterances, it also seems fair to say that Pierre believes that London is not pretty. It might appear, then, that Pierre has contradictory beliefs. He seemingly both has the belief that London is pretty and has the belief that London is not pretty. The paradox, or so Kripke suggests, is that Pierre does not seem guilty of any logical error. In this scenario, it would be rather perverse to accuse Pierre of logical inconsistency. After all, he is simply not aware that ‘Londres’ and ‘London’ name the same city. In regard to Pierre’s lamentable circumstance, we are faced with an intriguing philosophical question—namely: What exactly does Pierre believe about the attractiveness of London?

Friday Paradox:

Zeno put forward a number of paradoxes. The upshot of many of them is that motion is impossible. This is a pretty startling conclusion, but his reasons seem fairly plausible.

Suppose I want to go out to eat and I decide to walk to the restaurant. Let A be my house and B be the restaurant. In order for me to get from A to B, I must cover half the distance between them. So I must get to point C. But in order for me to get from A to C, I must cover half of that distance. So I must get to point D. But in order for me to get from A to D, I must cover half of that distance. So I must get to point E... you get the idea. The issue is that there are an infinite number of "halves" that I have to traverse and so I'll never be able to get the full distance. How can one ever traverse an infinite number of points? So I can never actually get to the restaurant. Of course, we can generalize this and so it looks like all movement is impossible. So, contrary to the way things appear, nothing is really moving!

This wonderful clip explains another of Zeno's paradoxes (sometimes called "Achilles and the Tortoise") quite well. It has quite a bit in common with the one I describe above and the visual representation of the issue might be helpful.

Now, of course, most of think motion is possible. So the trick is identifying where Zeno's argument went awry.

Good luck and enjoy the weekend!

Friday Paradox:

We often say that certain things don't exist. Santa, unicorns, Sherlock Holmes, and Atlantis are a few examples. But consider the sentence "Unicorns do not exist." This sentence seems to have a subject-predicate form in that there is a subject--unicorns--to which a predicate or property--non-existence--is being attributed. Consider other subject-predicate sentences: Sarah is tall. John doesn't like mushrooms. Liza does not have a car. By saying that these various people have these various properties, we are implying that these people exist. But aren't we then doing the same thing when we say of unicorns that they do not exist. Aren't we saying that unicorns have a certain property; namely, the property of non-existence? If so, then what are we talking about that have this property but unicorns? There has to be a subject of the sentence. Here's another way to put this: To be able to truly claim of an object that it doesn't exist, it seems that one has to presuppose that it exists, for doesn't a thing have to exist if we are to make a true claim about it?

Friday Paradox:

This one is from Bertrand Russell.

The Barber Paradox
A certain village has a barber named Saul. Saul is an affable fellow and is liked by everyone who knows him. But Saul has a peculiar practice from which he never waivers. He shaves all and only those adult male villagers who do not shave themselves. Now Saul is an adult male who lives in the village and he is clean-shaven. So an interesting question arises: Does he or does he not shave himself?

You might think that Saul does shave himself but this would contradict his practice of only shaving those individuals that do not shave themselves. Alternatively, you might think that Saul does not shave himself but this would contradict his practice of shaving all those individuals that do not shave themselves. Something has to give!?

* I should mention that Saul is a "normal" adult male and would have a beard if it were not for the fact that his face is regularly shaved.
(No cheating now... try to solve it on your own.)

Friday Paradox:

I'll occasionally post a paradox or puzzling case on a Friday. Today's is an old one--4th century B.C. It's from Eubulides.

The Horn Paradox:
(1) You don't have horns.
(2) If you have not lost something, you still have it.
(3) You have not lost any horns.
(4) Therefore, you (still) have horns. (This follows from (2) and (3))
(5) But (4) contradicts (1) and so we're saddled with a paradox.

If you think you've got a solution to the paradox, please share it with the rest of us by making a comment to this post.