Grelling-Nelson paradox

The Grelling-Nelson paradox is a semantic paradox formulated in 1908 by Kurt Grelling and Leonard Nelson and sometimes mistakenly attributed to German philosopher and mathematician Hermann Weyl. It is thus occasionally called Weyl's paradox, as well as Grelling's paradox. Justice of attribution has increasingly encouraged the present name, however. It is closely analogous to several other well known paradoxes, in particular the Barber paradox and Russell's paradox.

Definition

Define the adjectives "autological" and "heterological" as follows:

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• An adjective is autological if and only if it describes itself. For example "short" is autological, since the word "short" is short. "Sophisticated" and "polysyllabic" are also autological.
• An adjective is heterological if and only if it does not describe itself. Hence "long" is a heterological word, as is "monosyllabic".

All adjectives, it would seem, must be either autological or heterological, for each adjective either describes itself, or it doesn't. The Grelling-Nelson paradox arises when we consider the adjective "heterological".

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There is no consistent answer to the question: is the word "heterological" heterological? On one hand, if the word "heterological" is heterological, then it does not describe itself. Since the fact of it not describing itself does, in fact, describe it, it is autological, which means it isn't heterological. On the other hand, if the word "heterological" is not heterological, then it must be autological, which means it describes itself, and therefore it must be heterological. Either case leads to the contradiction that the word "heterological" is both heterological and not heterological, which is impossible.

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