Grelling–Nelson paradox

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

Description

Grelling and Nelson go in the formation of this antimony assume that each class is defined by a feature that is called a word. For example, the word " monosyllabic " the characteristic of the class of all one-syllable words. Then you break down the words into two classes, which are defined as follows:

An auto logical word does not itself own the feature called it, a hetero logical word against it. The words " German " or " three syllables " are autologically, for " German " is a German word and " three syllables " a word of three syllables. Most words are but heterological, for example, " English " and " monosyllabic ", because is "english " is not an English word and " monosyllabic " no monosyllable.

It seems that every word may be consistent classified into one of these two classes, on closer examination, however, problems arise.

Conflicting cases

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

• Suppose " heterological " is a car- logical word, it describes by definition itself and is in contradiction to the assumption a heterologous logical word.
• Suppose it the opposite is true and " heterological " is a straight logical word, it describes not by definition itself and is in contradiction with the assumption no heterosexual logical word (ie autologically ).

Nelson has already formulated the paradox on May 28, 1907 in a letter to Gerhard Hessenberg, in which the word was " heterological " not used yet. This was coined by Otto Meyerhof, who used it in a letter to Nelson on 19 August, 1907. In Grelling and Nelson, this was the fifth in a series of paradoxes, which is why they called P5.

It can be resolved by " heterological " somewhat redefined so that it is the characteristic of all non autological words called " hetero- logical" except. Then, the paradox is still "not autologically ". The word "auto logic" to redefine seems to fix the problem, but the paradox is then still with synonyms of " hetero- logical" as a "non- descriptive ". To liberate the German of the antinomy requires significantly more changes than simple refinements of the definitions of "auto logic" and " heterosexual logic". The extent of these obstacles in German is comparable to that of Russell's antinomy in set theory.

Undefined cases

The word "auto logic" can without contradiction be placed in each of the classes. "Auto logic" of the situation is:

In contrast, the situation is " heterological ":

Ambiguous cases

Jay Newhard presented in the arrangement of words to the car and heterology another issue which is not already covered by the Russell's antinomy: the word "loud" is autologically when it is said aloud, on the other hand heterological. Newhard solved the problem by restricting the classification to type (word types ) so that token ( Wortvorkommnisse ) are excluded from the two categories.

Solutions

Grelling and Nelson transferred in their antinomy Russell's antinomy on the language level, by each class a word as a name zuordneten by a reversible single-valued function; this corresponds to Russell's class, the class of hetero- logical words, so that the word refers to " heterological ". Therefore, the solution of Grelling -Nelson antinomy is completely parallel to the solution of Russell's antinomy: One can prove that the class of all heterological words is not a lot, but a so-called real class.

The Grelling -Nelson's antinomy thus has the following corollary: The bijection given that specifies the name of a word class is not within logically feasible. With a set of words over an alphabet with which any ordinary language is described, an internal logical function that returns all classes, a name that can not be formed; here real classes remain nameless because they can be no arguments in functions. This means that the language requirements of the antinomy are not given. Therefore, it is one of the so-called semantic paradoxes, in which a metalinguistic facts is inadmissible drawn to the logical language level. The designation of any class is in fact correct only as metalinguistic function, which relates to the formula formation. If, however, as analog logical function takes as Grelling - Nelson, then it is proven to be no bijection, because the contradiction shows that this naive assumption is wrong.

When dissolving in the ungebräuchlicheren branched type theory, the syntax is restricted so that the statements and syntax are no longer correct and the two word classes can no longer be formed and defined. Word classes have here that is a higher type than their elements (words), and the feature to a higher type than word classes. Therefore, the function values ​​are not allowed as elements of. Thus, the type theory attempts to eliminate the problems caused by the introduction of language levels and needs to a complicated syntax that restricts the language options greatly. The wording in the first order predicate logic, which is perfectly adequate as Russell's antinomy to the solution eliminates that hassle and allows said formulas; here the allowed range of inference from the evidence that the conditions of Grelling -Nelson's antinomy are inconsistent.

Importance for the entertainment Linguistics

Due to their rarity makes finding autological words is a challenge, especially when words with negations such as " non-flammable " excluded. Apart from adjectives and nouns, verbs ( " end ", " contain ", "exist" ), adverbs (English " polysyllabically " polysyllabic ) and other words ( " there ", " here " ), in which it autological Nouns two definitions there. According to a definition of a noun is considered autologically when it refers to the feature that it has, according to another if it denotes what it is. After the first definition are " Viersilbigkeit " ( is viersilbig ) and " antonym " ( is antonym, namely synonymy ) Examples of autological nouns, after the second " Dreisilbler " ( is a Dreisilbler ) and " antonym " (an antonym ). The words " Haplogie " ( for haplology ) and " oxymoron " were formed to be autologically according to the second definition.

The word " Proparoxytonon " in the broader sense ( accented on the antepenultimate syllable word, whether Greek or anderssprachig ) autologically. " Neologism " ( neologism ) was once a car- logical word, but it is not anymore. " Protologismus " ( marked by Mikhail Epstein for proposed new words that are not yet widespread and thus have not yet achieved the status as a neologism ) is still autologically, this status could also lose. " Unbeend " is indeed unfinished, but referred to this property is not correct, and thus not be seen as a self- logical word. "Quote " is not autologically, because not the word " quote" is a quote, but the quote " quote ' ".