A variant of Grelling's paradox

Consider the word "pentasyllabic." A nice sounding word, no? As many have noticed, the word "pentasyllabic" has five syllables. The word "pentasyllabic" is itself pentasyllabic. Not many words are like this. The word "tasty" is (presumably) not tasty. So, we have words that describe themselves and words that do not. Call the words that describe themselves the autological words, and call the words that do not describe themselves the heterological words.

Now, it seems like we may ask the following question: is the word "heterological" itself heterological? Well, if it is, then it does describe itself, and so it isn't. If it is not, then it does not describe itself, so it is! If it is, it isn't; if it isn't, it is. This is a logical contradiction known as Grelling's paradox.

Let's not worry about solutions to this paradox, though many have been proposed. Instead, let us consider a variant of this paradox. Consider the word "exiguous." It's not a widely used word. (It means something like "of small size, degree, amount" etc.) Our use of "exiguous" is itself exiguous. Let's call a word like "exiguous" that describes its own usage autochrestic. That is, a word "w" is autochrestic if and only if the use of "w" is w, and the use of a word "w" is autochrestic iff that use is w. Towards producing a variant of Grelling's paradox, we define "heterochrestic" for words and usages of words in the expected way.*

Many, many words are heterochrestic. It is hard for me to see how our usage of any word could be green or red, so "green" and "red" are easily heterochrestic. What about "purpureal"? One could fancifully argue that our use of this "chiefly poetic"-as the OED labels it-synonym for purple is purple, i.e. purpureal, in the sense of "excessively ornate"-again, the OED. This is probably playing too fast-and-loose with definitions/synonyms. But, I digress.

Now, going back to paradox, is "heterochrestic" itself heterochrestic? If it is, then our use of "heterochrestic" is not heterochrestic, but this would mean our use of "heterochrestic" is heterochrestic. So, we conclude that "heterochrestic" is not heterochrestic. But this means our use of "heterochrestic" is not heterochrestic, which means our use of "heterochrestic" is heterochrestic. We are led by our definitions to contradiction, and the paradox reappears. Really, that there is a viable variant of Grelling's paradox is no great discovery. Grelling's paradox itself exploits rather run-of-the-mill, self-referential shenanigans, like one finds perhaps most famously in Russell's paradox.

*I coined these words from the Greek "chresis," for "usage." Their derivation was modelled specifically on the English "catachrestic," meaning "pertaining to catachresis." Interestingly, "chresis" in "catachresis" has the same meaning as it does in "autochrestic" and "heterochrestic." For a catachresis is a misuse of a word.