A paradox arises over beer

Sat 04 Dec 2010 04:24 PM

A productive synergy of being a visiting fellow at the Center is that most of my social life consists of interacting with other fellows, and so philosophy gets done even in leisure time. Not all of it is serious philosophy, however, as evidenced by the following item that Bert Leuridan and I concocted over pizza and beer.

The Visiting Fellows Paradox

Step one: Note that there are things which are considered to be paradoxes, things which are commonly referred to as the such-and-so paradox, which nevertheless are not paradoxes. As one example, consider the Birthday Paradox: it only takes 23 people for the probability of at least two people having the same birthday to be greater than .5. As another, consider Simpson's Paradox: variables which are positively correlated in each subpopulation of a population p can be negatively correlated in p itself. A common thing to say about such things is that prima facie these are not really paradoxes at all. Rather, they are just surprising facts.

Step two: A paradox arises when a plausible line of reasoning leads to a contradictory conclusion. It is easy to show that the cases described in step one are secunda facie paradoxical. The proof goes like this:

Take one of these so-called paradoxes. Either it is after all a genuine paradox or it is not. If the former disjunct obtains, then the matter is shown. If the latter, its being called `the such-and-so paradox' gives us reason to believe that it is a paradox; yet it is not a paradox, by assumption. Letting P be `this is a paradox', we have a reason to believe P and also to believe not-P. So we have a paradox.

The latter disjunct, in which the commonly-called paradox's not being a paradox produces a paradox, shifts from using to mentioning the original alleged paradox. Rather than showing that the original statement about birthdays was a paradox, for example, it shows that the Birthday Paradox figures in a distinct but derivative paradox. Call this a second-order paradox.

Step three: In the proof above, we derived a paradox by cases. It was either an ordinary, first-order paradox or a meta-level, second-order paradox. In this step, we propose a paradox which does not require the disjunction, one which is exclusively a second-order paradox. We call this the Visiting Fellows Paradox.

We rule out the first disjunct - that it is paradoxical in the usual way - by refusing to explain to you the ordinary, first-order content of the Visiting Fellows Paradox. It is not, however, at all paradoxical on its own. We assure you of that.

The second-order paradox arises from the juxtaposition of reasons for thinking it is a paradox (because it is called one) and reasons for thinking it is not (because of our assurances). We provided our assurances in the previous paragraph, so all that remains is for it to be commonly called a paradox. One natural way to accomplish this is to publish a paper describing the new paradox so-called in a respected, scholarly journal. The success of our construction, and so the Visiting Fellows Paradox actually being a paradox, relies on the paper's being accepted by qualified referees of that scholarly journal. Yet, quite naturally, qualified referees will only accept the paper if it describes an actual paradox.

It follows from this that the authority of philosophers (referees, in this case) allows them to make something a paradox which would otherwise be merely a curiosity. Surely, this is a surprising fact. We decline to give a name to this fact, lest we unwittingly contribute to a further paradox.