Grelling-Nelson paradox

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The Grelling-Nelson paradox is an antinomy, or a semantic self-referential paradox, of its own application of the word " heterological ", meaning "no apply to itself. " It was formulated in 1908 by Kurt Grelling and Leonard Nelson, and is sometimes mistakenly associated with the German philosopher and mathematician Hermann Weyl. It is sometimes called Weil's paradox and Grelling Paradox . This is closely related to some other notable paradoxes, particularly the paradox of the Russell barber and paradox.

Video Grelling-Nelson paradox

Paradoks

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

1. Autistic adjectives (sometimes homologous ) if it describes itself. For example, English words "English" are autologous, such as "unbroken" and "pentasyllabic".
2. Adjectives heterological if not self-described. Then "long" is a heterological word (because it is not a long word), as well as "hyphenated" and "syllable one".

All adjectives, it seems, must be autologous or heterological, because each adjective describes itself, or not. Problems appear in a number of examples, however:

The paradoxical case

The Grelling-Nelson paradox arises when we consider the "heterological" adjective. One can ask: Is "heterological" a heterological word? If the answer is 'no', 'heterological' is autologous. This leads to contradictions, because in this case "heterologists" do not describe themselves: it must be a heterological word. But if the answer is 'yes', 'heterological' is heterological. This again leads to contradictions, because if the word "heterologist" describes itself, it is autologous.

• Is "heterological" a heterological word?
  • no -> "heterological" is autologous -> "heterological" describes itself -> "heterological" is heterological, contradiction
  • yes -> "heterological" does not describe itself -> "heterological" is not heterological, contradiction

The paradox can be eliminated, without altering the "heterological" meaning which was previously well defined, by modifying the definition of "heterologists" slightly to hold all nonautological words except "heterologists." But "nonautological" is subject to the same paradox, which this avoidance can not be applied because English rules uniquely determine the meaning of "autolog". A few modifications similar to the definition of "autologic" (such as expressing them as "nonautological" and synonymous) may seem to improve, but the paradox still obtains "autolog" and "heterological" synonyms such as "selfdescriptive" and "unknown," whose meanings also need to be adjusted , and the consequences of that adjustment then need to be pursued, and so on. Liberating English from the Grelling-Nelson paradox requires more modifications to the language than just a refinement of the "autolog" and "heterological" definitions, which need not be in the language for the arising paradox. The scope of this obstacle to English is comparable to Russell's paradox for mathematics established on set.

An arbitrary case

One may also ask whether "autolog" is autologous. It can be consistently selected to be:

• if we say "autolog" is autologous, and then asks if it applies to itself, then yes, yes, and thus is autologous;
• if we say that "autolog" is not autologous, and then asks if it applies to itself, then no, it is not, and thus not autologous.

This is the opposite of situations for heterologies: while "heterologists" can not logically be autologous or heterological, "autologic" can be. (Can not be both, because the autologous and heterological categories can not overlap.)

Logically, the situation for "autolog" is:

"autological" is autologous if and only if "autolog" is autologous

A if and only if A, a tautology

while the situation for "heterologists" is:

"heterological" is heterogeneous if and only if "heterological" is autologous

A if and only if it is not A, a contradiction.

Causes ambiguous

One may also ask whether "hard" is autologous or heterological. If spoken aloud, "hard" is autologous; if not, it's heterological. This suggests that some adjectives can not be clearly classified as either autologous or heterologous. Newhard attempted to eliminate this problem by taking Grelling's Paradox to specifically handle the word type as opposed to the word token.

Maps Grelling-Nelson paradox

The equation with Russell's paradox

The Grelling-Nelson paradigm can be translated into the famous paradox of Bertrand Russell in the following way. The first must identify each adjective with the set of objects to which the adjective applies. So, for example, the adjective "red" is equal to the set of all red objects. In this way, the adjective "spoken" is equated with the set of all things that can be spoken, one of which is the word "spoken" itself. Thus, the word autologist is understood as a set, one element is the set itself . The question of whether the heterological "heterological" becomes the question of whether the set of all non-self-contained sets contains itself as an element.

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See also

• Paradox list
• Metamagical Themas
• Use-name differences

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Note

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References

• Grelling, K.; Nelson, L. (1908). "Bemerkungen zu den Paradoxien von Russell and Burali-Forti". Abhandlungen der Friesschen Schule II . GÃÆ'¶ttingen. pp.Ã, 301-334. Ã, Also in: Nelson, Leonard (1974). Gesammelte Schriften III. Die criticalche Methode in ihrer Bedeutung fÃÆ'¼r die Wissenschaften . Hamburg: Felix Meiner Verlag. pp. 95-127. ISBN: 3787302220. Ã,
• 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). "Paradox in GÃÆ'¶ttingen". At Link, Godehard. One hundred years Russell paradox: mathematics, logic, philosophy . Berlin: Walter de Gruyter. pp.Ã, 501-516. ISBN: 3110174383.

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External links

• Autological words

Source of the article : Wikipedia