Readings: Paradoxes, Barbers, Brexit, Groupon, etc.

How wonderful that we have met with a paradox. Now we have some hope of making progress.

Niels Bohr

The UK Brexit debates is a limitless source of the strange. Among my favorites this week was the claim that the Tory party might, in its ardor for a general election, turn against itself in a vote of no-confidence. Such a vote would take down the government, thus causing the government to then have to call an election, given its absence of confidence in itself.

Leaving aside whether this will happen — by which I mean, Please, please, please let it happen — the thing I appreciate most about this is its embrace of logical illogic. Governments don’t, however confidently, have votes of no-confidence in themselves. First, it means you’re no longer the government, which seems a bad idea, given that the main reason to be in government is to be in government. Second, and more fun, is that not having confidence in yourself forces an election that, one assumes, you think you will win, suggest you have confidence in your lack of confidence about your confidence to confidently govern. Or something.

The crazy thing, practically speaking, is that this is mostly logical-ish. If, as a minority party, you can’t convince the other parties to vote against you, thus forcing a general election, you are forced to vote against yourself. It makes perfect sense, even if it might seem mad.

This is, of course, a paradox. And I love paradoxes, arguments that, despite a sensible premise and logical argument, produce contradictory or illogical conclusions. Much of scientific progress can be connected to paradoxes — which is, in large part, why “Well, that’s weird” — is such a powerful observation when doing scientific research.

Much of the best comedy springs from paradoxes as well, which isn’t a coincidence.

But there are paradoxes and there are paradoxes, and analytic philosopher W. V. Quine argued that there are three kinds. The first kind Quine identified was a result that might seem nuts, but can be shown to be true anyway. There are many such paradoxes, but among the best known is the Monty Hall Problem, where a decision that seems a coin flip isn’t a coin flip. Quine called this a veridical paradox, where an absurd conclusion turns out to be true.

A second kind of paradox Quine identified was where something that appears false turns out to actually be false. Granted, this might not seem like much of a paradox — more like something better described as “stupid” — but it can be. An example: There many fairly compelling mathematical proofs that 1=2. These can be very convincing, even if they seem false, and we know they must be false, but it’s sometimes difficult to pin down why, exactly, that they are false.

The third kind of paradox Quine identified was where you reach an internally contradictory result when applying proper reasoning. This usually involves statements that reference themselves, producing bizarre conclusions. The Barber Paradox is an example: The barber is the one who shaves all those, and those only, who do not shave themselves, so, does the barber shave himself? (There are entire books of this sort of thing, like those by Raymond Smullyan.)

So, which kind of paradox is the British no-confidence vote? To my way of thinking, it is one of those statements that circles back and eats itself, a logical paradox created by language that implies the sentence is saying something that it can’t, which makes it — drumroll — an antimony. I’m doubtful, however, that this means we are making any progress. Sorry, Niels Bohr.


A few articles and papers worth reading, most with a paradox theme: