Why should you study paradoxes? The easiest way to answer this question is with a story:

In 2002 I was attending a conference on self-reference in Copenhagen, Denmark. During one of the breaks I got a chance to chat with Raymond Smullyan, who is amongst other things an accomplished magician, a distinguished mathematical logician, and perhaps the most well-known popularizer of `Knight and Knave’ (K&K) puzzles.

K&K puzzles involve an imaginary island populated by two tribes: the Knights and the Knaves. Knights always tell the truth, and Knaves always lie (further, members of both tribes are forbidden to engage in activities that might lead to paradoxes or situations that break these rules). Other than their linguistic behavior, there is nothing that distinguishes Knights from Knaves.

Typically, K&K puzzles involve trying to answer questions based on assertions made by, or questions answered by, an inhabitant of the island. For example, a classic K&K puzzle involves meeting an islander at a fork in the road, where one path leads to riches and success and the other leads to pain and ruin. You are allowed to ask the islander one question, after which you must pick a path. Not knowing to which tribe the islander belongs, and hence whether she will lie or tell the truth, what question should you ask?

(Answer: You should ask “Which path would someone from the other tribe say was the one leading to riches and success?”, and then take the path not indicated by the islander).

Back to Copenhagen in 2002: Seizing my chance, I challenged Smullyan with the following K&K puzzle, of my own devising:

There is a nightclub on the island of Knights and Knaves, known as the Prime Club. The Prime Club has one strict rule: the number of occupants in the club must be a prime number at all times.

The Prime Club also has strict bouncers (who stand outside the doors and do not count as occupants) enforcing this rule. In addition, a strange tradition has become customary at the Prime Club: Every so often the occupants form a conga line, and sing a song. The first lyric of the song is:

“At least one of us in the club is a Knave.”

and is sung by the first person in the line. The second lyric of the song is:

“At least two of us in the club are Knaves.”

and is sung by the second person in the line. The third person (if there is one) sings:

“At least three of us in the club are Knaves.”

And so on down the line, until everyone has sung a verse.

One day you walk by the club, and hear the song being sung. How many people are in the club?

Smullyan’s immediate response to this puzzle was something like “That can’t be solved – there isn’t enough information”. But he then stood alone in the corner of the reception area for about five minutes, thinking, before returning to confidently (and correctly, of course) answer “Two!”

I won’t spoil things by giving away the solution – I’ll leave that mystery for interested readers to solve on their own. (Hint: if the song is sung with any other prime number of islanders in the club, a paradox results!) I will note that the song is equivalent to a more formal construction involving a list of sentences of the form:

At least one of sentences S 1 – S n is false.

At least two of sentences S 1 – S n is false.

⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮

At least n of sentences S 1 – S n is false.

The point of this story isn’t to brag about having stumped a famous logician (even for a mere five minutes), although I admit that this episode (not only stumping Smullyan, but meeting him in the first place) is still one of the highlights of my academic career.

Instead, the story, and the puzzle at the center of it, illustrates the reasons why I find paradoxes so fascinating and worthy of serious intellectual effort. The standard story regarding why paradoxes are so important is that, although they are sometimes silly in-and-of-themselves, paradoxes indicate that there is something deeply flawed in our understanding of some basic philosophical notion (truth, in the case of the semantic paradoxes linked to K&K puzzles).

Another reason for their popularity is that they are a lot of fun. Both of these are really good reasons for thinking deeply about paradoxes. But neither is the real reason why I find them so fascinating. The real reason I find paradoxes so captivating is that they are much more mathematically complicated, and as a result much more mathematically interesting, than standard accounts (which typically equate paradoxes with the presence of some sort of circularity) might have you believe.

The Prime Club puzzle demonstrates that whether a particular collection of sentences is or is not paradoxical can depend on all sorts of surprising mathematical properties, such as whether there is an even or odd number of sentences in the collection, or whether the number of sentences in the collection is prime or composite, or all sorts of even weirder and more surprising conditions.

Other examples demonstrate that whether a construction (or, equivalently, a K&K story) is paradoxical can depend on whether the referential relation involved in the construction (i.e. the relation that holds between two sentences if one refers to the other) is symmetric, or is transitive.

The paradoxicality of still another type of construction, involving infinitely many sentences, depends on whether cofinitely many of the sentences each refer to cofinitely many of the other sentences in the construction (a set is cofinite if its complement is finite). And this only scratches the surface!

The more I think about and work on paradoxes, the more I marvel at how complicated the mathematical conditions for generating paradoxes are: it takes a lot more than the mere presence of circularity to generate a mathematical or semantic paradox, and stating exactly what is minimally required is still too difficult a question to answer precisely. And that’s why I work on paradoxes: their surprising mathematical complexity and mathematical beauty. Fortunately for me, there is still a lot of work that remains to be done, and a lot of complexity and beauty remaining to be discovered.

Featured Image Credit: ‘Structure and Clarity’, by Dan Flavin, Photo uploaded by SuperCar-RoadTrip, CC by 2.0, via flickr.