Wednesday, June 13, 2007

The Grelling-Nelson Paradox

The meaning of ‘heterological’ is most easily shown by examples, e.g. since ‘long’ is not a long word, ‘long’ is heterological, whereas since ‘short’ is short, ‘short’ is not heterological, and similarly ‘monosyllabic’ is heterological because it is not monosyllabic, and ‘polysyllabic’ is not heterological—but what about ‘heterological’? If it is heterological then (by its intuitive definition) it is not, whereas if it is not then (similarly) it is.
......This paradox was designed to shed light on Russell’s paradox (of the collection of all the collections that are not members of themselves) and indeed, the best way to see its triviality is to consider the idea of a club for all and only those who don’t belong to any other club. Whilst such a club is perfectly feasible, if the word ‘other’ were missing from its definition it would become paradoxical unless nobody was not in some club.
......Similarly the intuitively heterological adjectives would just be the other adjectives that don’t describe themselves. After all, while it is quite clear what ‘ugly’ being an ugly word would amount to, we have no similarly intuitive grasp of how ‘heterological’ could be heterological. (Furthermore, how heterological are ‘ugly,’ ‘round,’ ‘necessary,’ etc.? Unlike ‘collection,’ ‘heterological’ does not even seem to name a definite concept.)

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