Editing Grelling–Nelson paradox

{{Short description|Semantic self-referential paradox}}
{{refimprove|date=August 2018}}
The '''Grelling–Nelson paradox''' arises from the question of whether the term "non-self-descriptive" is self-descriptive. It was formulated in 1908 by [[Kurt Grelling]] and [[Leonard Nelson]], and is sometimes mistakenly attributed to the German philosopher and mathematician [[Hermann Weyl]]<ref>Weyl refers to it as a "well-known paradox" in ''[https://archive.org/details/daskontinuumkrit00weyluoft Das Kontinuum]'' (1918), p. 2, mentioning it only to dismiss it.  Its misattribution to him may stem from {{harvnb|Ramsey|1926}} (attested in {{harvnb|Peckhaus|2004}}).</ref> thus occasionally called '''Weyl's paradox''' or '''Grelling's paradox'''. It is closely related to several other well-known paradoxes, in particular, the [[barber paradox]] and [[Russell's paradox]]. It is an [[antinomy]], or a semantic self-referential paradox.

==Paradox==
[[File:Blueblue.svg|thumb|150px|The first instance of the word "blue" is autological, while the second is heterological.]]

Suppose one interprets the adjectives "autological" and "heterological" as follows:
# An [[adjective]] is ''[[autological word|autological]]'' (sometimes ''homological'') if it describes itself. For example, the English word "English" is autological, as are "unhyphenated" and "pentasyllabic".
# An adjective is ''heterological'' if it does not describe itself. Hence "long" is a heterological word (because it is not a long word), as are "hyphenated" (because it has no hyphen) and "monosyllabic" (because it has more than one syllable).

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

==={{anchor|Heterological}}Paradoxical cases===
{{unreferenced |section|date=August 2023}}
The Grelling–Nelson paradox arises when we consider the adjective "heterological". One can ask: Is "heterological" a heterological word? If the answer is "no", then "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", then "heterological" is heterological.  This again leads to a contradiction, because if the word "heterological" describes itself, it is autological.

* Is "heterological" a heterological word?
** no → "heterological" is autological → "heterological" describes itself → "heterological" is heterological, ''contradiction''
** yes → "heterological" is heterological → "heterological" does not describe itself → "heterological" is not heterological, ''contradiction''

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 correct that, but the paradox still remains for synonyms of "autological" and "heterological" such as "self-descriptive" and "non-self-descriptive", 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.

==={{anchor|Autological}}Arbitrary cases=== 
One may also ask whether "autological" is autological.
It can be chosen consistently to be either:
* if we say that "autological" is autological and then ask whether it applies to itself, then yes, it does, and thus is autological;
* if we say that "autological" is not autological and then ask whether it applies to itself, then no, it does not, and thus is not autological.

This is the opposite of the situation for heterological: while "heterological" logically cannot be autological or heterological, "autological" can be either. (It cannot be both, as the category of autological and heterological cannot overlap.)

In logical terms, the situation for "autological" is:
{{block indent|"autological" is autological if and only if "autological" is autological}}
{{block indent|A if and only if A, a [[tautology (logic)|tautology]]}}
while the situation for "heterological" is:
{{block indent|"heterological" is heterological if and only if "heterological" is autological}}
{{block indent|A if and only if not A, a [[contradiction]].}}

===Ambiguous cases===
One may also ask whether "loud" is autological or heterological. If said loudly, "loud" is autological; otherwise, it is heterological.  This shows that some adjectives cannot be unambiguously classified as autological or heterological. Newhard sought to eliminate this problem by taking Grelling's Paradox to deal specifically with [[type-token distinction|word types as opposed to word tokens]].<ref>{{cite journal |last=Newhard |first=Jay |title=Grelling's Paradox |journal=Philosophical Studies |date=October 2005 |volume=126 |issue=1 |pages=1–27 |doi=10.1007/s11098-004-7808-z|s2cid=170827394 }}</ref>

==Similarities with Russell's paradox==
The Grelling–Nelson paradox can be translated into [[Russell's paradox]] in the following way. First, one must identify each adjective with the [[set (mathematics)|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 which do not contain themselves contains itself.

==See also==
* [[Self-reference]]
* [[List of self–referential paradoxes]]
* ''[[Metamagical Themas]]''
* [[Use–mention distinction]]

==Notes==
{{reflist}}

==References==
*{{cite book |first1=K. |last1=Grelling |first2=L. |last2=Nelson |chapter=Bemerkungen zu den Paradoxien von Russell und Burali-Forti |title=Abhandlungen der Fries'schen Schule II |location=Göttingen |year=1908 |pages=301–334 }} Also in: {{cite book |first=Leonard |last=Nelson |title=Gesammelte Schriften III. Die kritische Methode in ihrer Bedeutung für die Wissenschaften |publisher=Felix Meiner Verlag |location=Hamburg |year=1974 |pages=95–127 |isbn=3787302220 }}
*{{cite journal |first=Frank P. |last=Ramsey |title=The Foundations of Mathematics |journal=[[Proceedings of the London Mathematical Society]] |series=2 |volume=25 |issue=1 |pages=338–384 |year=1926 |doi=10.1112/plms/s2-25.1.338 }}
*{{cite book |first=Volker |last=Peckhaus |chapter=Paradoxes in Göttingen |title=One hundred years of Russell's paradox: mathematics, logic, philosophy |editor-first=Godehard |editor-last=Link |publisher=Walter de Gruyter |location=Berlin |year=2004 |pages=501–516 |isbn=3110174383 }}

==External links==
*[http://www.segerman.org/autological.html Autological words]

{{Paradoxes}}

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[[Category:Self-referential paradoxes]]
[[Category:1908 introductions]]