# Grelling-Nelson Antinomy

The Grelling-Nelson antinomy is a semantic self-referential paradox published in 1908 by Kurt Grelling and Leonard Nelson as a variant of Russell's antinomy and is sometimes incorrectly ascribed to Hermann Weyl .

## description

In forming their antinomy, Grelling and Nelson assume that each class is defined by a feature that denotes a word. For example, the word "monosyllabic" denotes the characteristic of the class of all monosyllabic words. You then break the words down into two classes, which are defined as follows:

An autological word itself has the characteristic it designates, while a heterological word does not. The words “German” or “three-syllable” are autological, because “German” is a German word and “three-syllable” is a three-syllable word. Most words, however, are heterologous, for example “English” and “monosyllabic”, because “English” is not an English word and “monosyllabic” is not a monosyllabic word.

It seems that every word can be classified into one of these two classes without contradiction, but on closer inspection problems arise.

The actual Grelling-Nelson antinomy arises when trying to classify the word "heterological" in one of the two classes:

• Assuming “heterological” is an autological word, then by definition it describes itself and is, contrary to the assumption, a heterological word.
• Assuming the opposite is true and “heterological” is a heterological word, then according to the definition it does not describe itself and, contrary to the assumption, is not a heterological word (i.e. autological).

Nelson already formulated the paradox in a letter to Gerhard Hessenberg on May 28, 1907 , in which the word "heterological" was not yet used. This was coined by Otto Meyerhof , who used it in a letter to Nelson dated August 19, 1907. In Grelling and Nelson this was the fifth of a series of paradoxes, it is why it P 5 called.

It can be resolved by redefining something “heterological” so that it denotes the characteristic of all non-autological words except “heterological”. Then the paradox still exists with “non-autological”. Redefining the word “autological” seems to solve the problem, but the paradox then still persists with synonyms of “heterological” and “non-self-describing”. Liberating German from the antinomy requires considerably more changes than simple refinements of the definitions of “autological” and “heterological”. The extent of these obstacles in German is comparable to that of Russell's antinomy in set theory.

### Undefined cases

The word “autological” can be classified in each of the classes without contradiction. For "autological" the situation is:

“Autological” is autological if and only if “autological” is autological.
A if and only if A - a tautology .

In contrast, the situation of "heterological" is:

“Heterological” is heterological precisely when “heterological” is autological.
A if and only if not A - a contradiction .

### Ambiguous cases

Jay Newhard identified another problem in the classification of words into auto and heterology that is not already covered by Russell's antinomy: the word “loud” is autological when it is said out loud; otherwise, heterological. Newhard solved the problem by restricting the classification to types (word types) so that tokens (word occurrences) are excluded from the two categories.

## solutions

In their antinomy, Grelling and Nelson transferred Russell's antinomy to the level of language by assigning a word as a name to each class through a reversible unambiguous function ; The Russell class corresponds to the class of heterological words , so that the word means "heterological". Therefore the solution of the Grelling-Nelson antinomy is completely parallel to the solution of the Russell antinomy : One can prove that the class of all heterological words is not a set, but a so-called real class . ${\ displaystyle \ varphi}$${\ displaystyle H = \ {\ varphi (x) \ mid \ varphi (x) \ notin x \}}$${\ displaystyle \ varphi (H)}$${\ displaystyle \, H}$

The Grelling-Nelson antinomy thus has the following logical consequence: The predefined bijection , which gives the name of a word class, cannot be implemented logically. With a set of words over an alphabet, with which every common language is described, an intra-logical function which gives a name to all classes cannot be formed; real classes remain nameless here because they cannot be arguments in functions. This means that the linguistic requirements of the antinomy are not given. It is therefore one of the so-called semantic paradoxes, in which a metalinguistic fact is inadmissibly drawn onto the logical language level. Naming any class is only correct as a metalinguistic function that affects formula formation. But if one assumes an analogous logical function like Grelling-Nelson, then it can demonstrably not be a bijection, because the contradiction shows that this naive assumption is wrong. ${\ displaystyle \ varphi}$${\ displaystyle \ varphi}$${\ displaystyle \ varphi}$

With the solution in the less common branched type theory , the syntax is restricted in such a way that the statements and are no longer syntactically correct and the two word classes can no longer be formed and defined. This is because word classes have a higher type than their elements (words), and the function is still a higher type than word classes. Therefore function values ​​are not allowed as elements of . The type theory tries to eliminate the problems by introducing language levels and needs a complicated syntax which strongly restricts the language possibilities. The formulation in the first order predicate logic , which, as in Russell's antinomy, is completely sufficient for the solution, avoids this effort and allows said formulas; here the permitted conclusions are sufficient to demonstrate that the requirements of the Grelling-Nelson antinomy are inconsistent. ${\ displaystyle \ varphi (x) \ in x}$${\ displaystyle \ varphi (x) \ notin x}$${\ displaystyle \ varphi}$${\ displaystyle \ varphi (x)}$${\ displaystyle \, x}$

## Significance for entertainment linguistics

Because of their rarity, finding autological words is challenging, especially when excluding words with negatives such as "incombustible". In addition to adjectives, nouns, verbs (“end”, “contain”, “exist”), adverbs (English “polysyllabically” multi-syllable ) and other words (“es”, “here”) are mentioned, whereby there are two definitions for autological nouns gives. According to one definition, a noun is considered autological if it designates the characteristic it possesses, and according to another, if it designates what it is . According to the first definition, "four-syllable" (is four-syllable) and " antonymy " (is antonym, namely to synonymy ) are examples of autological nouns, after the second "three-syllable" (is a three-syllable) and "antonym" (is an antonym). The words "haplogy" (for haplology ) and " oxymoron " were formed to be autological according to the second definition.

The word “ Proparoxytonon ” is autological in the broader sense ( word stressed on the penultimate syllable , whether Greek or in another language). “ Neologism ” (new word creation) was once an autological word, but is no longer it. "Protologism" ( coined by Mikhail Epstein for proposed new words that are not yet widespread and thus have not yet achieved neologism status) is still autological, but could lose this status. “Unfinished” is unfinished, but does not correctly describe this property and is therefore not to be seen as an autological word. “Quote” is not autological, because the word “quote” is not a quote, but the quote “'quote'”.

## Individual evidence

1. Weyl mentions it in The Continuum . The miscalculation could be traced back to Ramsey 1926, as Peckhaus explains in 2004.
2. a b Grelling / Nelson p. 307: “Let φ ( M ) be the word that denotes the concept by which M is defined. This word is either an element of M or it is not. In the first case we want to call it “autological”, in the other “heterological”. ”According to Grelling / Nelson p. 306, φ is assumed here as a bijective, reversible one-to-one function.

## credentials

• Kurt Grelling, Leonard Nelson: Notes on the Paradoxes of Russell and Burali-Forti . In: Treatises of the Fries School II . Göttingen 1908, p. 301-334 . Reprinted in: Leonard Nelson: Gesammelte Schriften III. The critical method and its significance for the sciences . Felix Meiner Verlag, Hamburg 1974, p. 95-127 .
• Volker Peckhaus: The Genesis of Grelling's Paradox . , in: Ingolf Max, Werner Stelzner: Logic and Mathematics: Frege Colloquium Jena 1993 . Walter de Gruyter, Berlin 1995, p. 269-280 .
• Jay Newhard: Grelling's Paradox . In: Philosophical Studies . tape 126 , no. 1 .
• Dmitri A. Borgmann: Beyond Language . Adventures in Word and Thought. 1967.
• David L. Silverman: Kickshaws . In: Word Ways . tape 2 , no. 3 , 1969, p. 182-183 .
• David Morice: Kickshaws . In: Word Ways . tape 29 , no. 3 , 2012, p. 180-181 .
• Anthony Sebastian: On Reflexivity in Words . In: Word Ways . tape 21 , no. 3 , 2012.