Grelling-Nelson paradox

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The Grelling-Nelson paradox is a semantic paradox 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. 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.

1 Definition
2 Analysis
3 See also
4 External link

[edit] Definition

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

An adjective is autological if and only if it describes itself. For example "short" is autological, since the word "short" is short. "Sophisticated" and "pentasyllabic" 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".

To test if the word 'foo' is autological you can ask: Is 'foo' a foo word? If the answer is 'yes', 'foo' is autological. If the answer is 'no', 'foo' is heterological.

Is 'heterological' a heterological word? If the answer is 'yes', 'heterological' is autological (leading to a contradiction). If the answer is 'no', 'heterological' is heterological (again leading to a contradiction).

There is no consistent answer to the question: Is the word "heterological" heterological? On the 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.

[edit] Analysis

The Grelling-Nelson paradox can be translated into Bertrand Russell's famous paradox in the following way: identify each adjective with the set of objects to which that adjective applies. So, for example, the adjective "red" is equated with the set of all red objects. In this way, the adjective "pronounceable" is equated with the set of all pronounceable things, one of which is the word "pronounceable" itself. Thus, an autological word is understood as a set, one of whose elements is the set itself. The question of whether the word "heterological" is heterological becomes the question of whether the set of all sets not containing themselves contains itself as an element.