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 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.
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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 "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".
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.
Analysis
The Grelling-Nelson paradox is resolvable by simply admitting that the words "autological" and "heterological" do not in fact form two well-defined categories into which all adjectives fall. Thus the paradox can be taken as a demonstration that natural language does not necessarily partition all objects of thought into well-defined categories that will stand up to arbitrary logical scrutiny.
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.