Sending copies of an xkcd cartoon on the plague of trochees, especially double trochees (Language Log discussion here), to friends, I reflected, not for the first time, that trochee is indeed a trochee (´ˇ), but so are dactyl and iamb, while anapest is a dactyl (´ˇˇ). Well, there’s no reason to expect that a predicative word (an adjective, or, in this case, noun) should be self-descriptive, but there’s a certain intellectual pleasure in playing with the fit or misfit between different properties of linguistic expressions when the semantics of their use vs. mention is involved.
Short is short, long is not long, polysyllabic is polysyllabic, monosyllabic is not monosyllabic. Eggcorn is an eggcorn, snowclone is not a snowclone. Bitchy is bitchy, vulgar is not vulgar. English is English, but French is not French (it’s English). And so on.
All we need to play the game is a property P of linguistic expressions, and then we can ask if an expression E that denotes P has the property P: is trochee a trochee? (yes) is vulgar vulgar? (no).
If E has the property P it denotes, then call it autologous; if not, heterologous. (These are not my labels; I’m using terms from the analysis of paradoxes in 20th-century philosophy.) Trochee is autologous, vulgar heterologous.
Ah, but heterologous and autologous are linguistic expressions denoting properties of linguistic expressions. So, is heterologous heterologous, is autologous autologous?
The first question leads us to the Grelling-Nelson Paradox (compact but excellent Wikipedia entry here). (Irrelevant but entertaining fact: Grelling-Nelson is a double trochee — as is Arnold Zwicky.) Heterologous is either autologous or heterologous. If it’s autologous, then it denotes the property of autologousness (or, if you will, autologicity); but, no, it denotes the property of heterologousness (or heterologicity). So it must be heterologous. But if so, it denotes the property of heterologousness — but that means it’s autologous. Either way, you seem to be screwed: if it’s autologous, then it’s heterologous, but if it’s heterologous, then it’s autologous, and it can’t be both at once.
There’s a fair literature on Grelling-Nelson. It bears a family resemblance to an even more famous logical antinomy, Russell’s Paradox (Wikipedia entry here), having to do with the set of all sets that are not members of themselves (it’s a member of itself if and only if it is not), the commonality having to do with the combination of self-reference and negation.
(By a pleasant coincidence, Ben Zimmer has just posted on Language Log about Saskia Hamilton, whose name is a double dactyl.)