Re: Agenbites and Anti-Agenbites

Re your blogs on agenbites and anti-agenbites, Jody, I appreciate all the fun that can be had with such words, and I have much enjoyed many of the examples, but there’s another aspect to such words that—to me, at least—is even more fascinating. If we get strict with agenbites and require that they be literally true of themselves, then we straightaway run into the most profound logical difficulties. Consider, if you will, the following sentence:

1. “Anti-agenbite” is an anti-agenbite.

If this sentence (1) is true, then “anti-agenbite” is an anti-agenbite, and so not true of itself, which means that (1) is false. But if (1) is false, then “anti-agenbite” is not an anti-agenbite, which means that “anti-agenbite” is true of itself, and so (1) is true. In other words, sentence (1) is true if and only if it is false. This means that introducing a word like agenbite creates logical paradoxes in a language. That’s why such words are blocked by grammatical rules in the formalized languages of mathematics and theoretical physics.

I wish I had been smart enough to think this up on my own, but in fact the problem with (1) was first noted by German logician Kurt Grelling in 1910 and is known to logicians as the Grelling’s Paradox or the Heterological Paradox, heterologic al being the word Grelling used for “anti-agenbite.”

Grelling’s Paradox is a particular case of a more general logical principle: Nothing can bear a relation to all and only those things that do not bear that relation to themselves. Thus, we have Bertrand Russell’s example of the barber who shaves all and only the men who do not shave themselves. Does this barber shave himself or not? If he does, then he doesn’t, but if doesn’t, then he does. Hence, there is no such barber, his existence being logically impossible.

More seriously, Russell applied the principle to disprove the existence of a set of all sets not members of themselves. That observation, known today as Russell’s Paradox, refuted the supposed Axiom of Comprehension on which Frege had hoped to base his version of what we today would call set theory. The modern versions of set theory differ one from another by the expedients they adopt to avoid Russell’s Paradox.

Going back to agenbites and anti-agenbites, Grelling’s Paradox doesn’t mean that some words aren’t true of themselves, just as the barber paradox doesn’t mean that some men don’t shave themselves. Nor does Grelling’s Paradox threaten the fun we might have thinking up clever examples of agenbites or anti-agenbites. The paradox does mean, however, that words like agenbite and anti-agenbite cannot play a role in a fully vigorous theory of the world.

For an accessible account of such things, see W.V. Quine’s “The Ways of Paradox” in his The Ways of Paradox and Other Essays .