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Grelling–Nelson paradox

February 21, 2023

The Grelling–Nelson paradox is an antinomy, or a semantic self-referential paradox, concerning the applicability to itself of the word "heterological", meaning "inapplicable to itself". It was formulated in 1908 by Kurt Grelling and Leonard Nelson, and is sometimes mistakenly attributed to the German philosopher and mathematician Hermann Weyl.[1] It is thus occasionally called Weyl's paradox and Grelling's paradox. It is closely related to several other well-known paradoxes, in particular, the barber paradox and Russell's paradox.

The paradox

 
The first instance of the word "blue" is autological, while the second is heterological.

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

  1. An adjective is autological (sometimes homological) if it describes itself. For example, the English word "English" is autological, as are "unhyphenated" and "pentasyllabic".
  2. 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.

Paradoxical cases

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 "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.

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:

"autological" is autological if and only if "autological" is autological
A if and only if A, a tautology

while the situation for "heterological" is:

"heterological" is heterological if and only if "heterological" is autological
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 word types as opposed to word tokens.[2]

Similarities with Russell's paradox

The Grelling–Nelson paradox can be translated into Bertrand Russell's famous paradox in the following way. First, one must 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 which do not contain themselves contains itself.

See also

Notes

  1. ^ Weyl refers to it as a "well-known paradox" in Das Kontinuum (1918), p. 2, mentioning it only to dismiss it. Its misattribution to him may stem from Ramsey 1926 (attested in Peckhaus 2004).
  2. ^ Newhard, Jay (October 2005). "Grelling's Paradox". Philosophical Studies. 126 (1): 1–27. doi:10.1007/s11098-004-7808-z. S2CID 170827394.

References

  • Grelling, K.; Nelson, L. (1908). "Bemerkungen zu den Paradoxien von Russell und Burali-Forti". Abhandlungen der Fries'schen Schule II. Göttingen. pp. 301–334. Also in: Nelson, Leonard (1974). Gesammelte Schriften III. Die kritische Methode in ihrer Bedeutung für die Wissenschaften. Hamburg: Felix Meiner Verlag. pp. 95–127. ISBN 3787302220.
  • Ramsey, Frank P. (1926). "The Foundations of Mathematics". Proceedings of the London Mathematical Society. 2. 25 (1): 338–384. doi:10.1112/plms/s2-25.1.338.
  • Peckhaus, Volker (2004). "Paradoxes in Göttingen". In Link, Godehard (ed.). One hundred years of Russell's paradox: mathematics, logic, philosophy. Berlin: Walter de Gruyter. pp. 501–516. ISBN 3110174383.

External links

  • Autological words

February 21, 2023
grelling, nelson, paradox, this, article, needs, additional, citations, verification, please, help, improve, this, article, adding, citations, reliable, sources, unsourced, material, challenged, removed, find, sources, news, newspapers, books, scholar, jstor, . This article needs additional citations for verification Please help improve this article by adding citations to reliable sources Unsourced material may be challenged and removed Find sources Grelling Nelson paradox news newspapers books scholar JSTOR August 2018 Learn how and when to remove this template message The Grelling Nelson paradox is an antinomy or a semantic self referential paradox concerning the applicability to itself of the word heterological meaning inapplicable to itself It was formulated in 1908 by Kurt Grelling and Leonard Nelson and is sometimes mistakenly attributed to the German philosopher and mathematician Hermann Weyl 1 It is thus occasionally called Weyl s paradox and Grelling s paradox It is closely related to several other well known paradoxes in particular the barber paradox and Russell s paradox Contents 1 The paradox 1 1 Paradoxical cases 1 2 Arbitrary cases 1 3 Ambiguous cases 2 Similarities with Russell s paradox 3 See also 4 Notes 5 References 6 External linksThe paradox Edit 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 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 Paradoxical cases Edit 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 contradictionThe 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 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 Arbitrary cases Edit 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 autological is autological if and only if autological is autological A if and only if A a tautology while the situation for heterological is heterological is heterological if and only if heterological is autological A if and only if not A a contradiction Ambiguous cases Edit 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 word types as opposed to word tokens 2 Similarities with Russell s paradox EditThe Grelling Nelson paradox can be translated into Bertrand Russell s famous paradox in the following way First one must 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 which do not contain themselves contains itself See also EditList of paradoxes Metamagical Themas Use mention distinction Liar paradoxNotes Edit Weyl refers to it as a well known paradox in Das Kontinuum 1918 p 2 mentioning it only to dismiss it Its misattribution to him may stem from Ramsey 1926 attested in Peckhaus 2004 Newhard Jay October 2005 Grelling s Paradox Philosophical Studies 126 1 1 27 doi 10 1007 s11098 004 7808 z S2CID 170827394 References EditGrelling K Nelson L 1908 Bemerkungen zu den Paradoxien von Russell und Burali Forti Abhandlungen der Fries schen Schule II Gottingen pp 301 334 Also in Nelson Leonard 1974 Gesammelte Schriften III Die kritische Methode in ihrer Bedeutung fur die Wissenschaften Hamburg Felix Meiner Verlag pp 95 127 ISBN 3787302220 Ramsey Frank P 1926 The Foundations of Mathematics Proceedings of the London Mathematical Society 2 25 1 338 384 doi 10 1112 plms s2 25 1 338 Peckhaus Volker 2004 Paradoxes in Gottingen In Link Godehard ed One hundred years of Russell s paradox mathematics logic philosophy Berlin Walter de Gruyter pp 501 516 ISBN 3110174383 External links EditAutological words Retrieved from https en wikipedia org w index php title Grelling Nelson paradox amp oldid 1096473433, wikipedia, wiki, book, books, library,

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