Grelling–Nelson paradox

The Grelling–Nelson paradox is a semantic self-referential paradox concerning the applicability to itself of the word "", meaning "inapplicable to itself." It was formulated in 1908 by Kurt Grelling and Leonard Nelson and sometimes mistakenly attributed to the German philosopher and mathematician Hermann Weyl. It is thus occasionally called Weyl's paradox as well as Grelling's paradox. It is closely analogous to several other well-known paradoxes, in particular the barber paradox and Russell's paradox.

Suppose one interprets the adjectives "autological" and "heterological" as follows:

All adjectives, it would seem, must be either autological or heterological, for each adjective either describes itself, or it doesn't. Problems arise in a number of instances, however:

The Grelling–Nelson paradox arises when we consider the adjective "heterological". One can ask: Is "heterological" a heterological word? If the answer is 'no', "heterological" is autological. This leads to a contradiction, for in this case "heterological" does not describe itself: it must be a heterological word. But if the answer is 'yes', "heterological" is heterological. This again leads to a contradiction, because if the word "heterological" describes itself, it is autological.

The paradox can be eliminated, without changing the meaning of "heterological" where it was previously well-defined, by modifying the definition of "heterological" slightly to hold all nonautological words except "heterological." But "nonautological" is subject to the same paradox, for which this evasion is not applicable because the rules of English uniquely determine its meaning from that of "autological." A similar slight modification to the definition of "autological" (such as declaring it false of "nonautological" and its synonyms) might seem to fix that, but the paradox still obtains for synonyms of "autological" and "heterological" such as "selfdescriptive" and "nonselfdescriptive," whose meanings also would need adjusting, and the consequences of those adjustments would then need to be pursued, and so on. Freeing English of the Grelling–Nelson paradox entails considerably more modification to the language than mere refinements of the definitions of "autological" and "heterological," which need not even be in the language for the paradox to arise. The scope of these obstacles for English is comparable to that of Russell's paradox for mathematics founded on sets.

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