13.4 Paradoxes of Grelling and Berry Autologous and Heterologic Words

Grelling Paradoxes and Berry

An interesting logical paradox was discovered by German logicians K. Grelling and L. Nelson (Grelling paradox). This paradox can be formulated very simply.

Autologous and Heterologic Words

Some words denoting properties have the very property they call. For example, the adjective “Russian” itself is Russian, and the “polysyllabic” is itself polysyllabic. Such self-related words are called self-identifying, or autologous. There are not so many similar words; overwhelmingly, adjectives do not possess the properties they call. "New" is not, of course, new, "hot" - hot, "one-syllable" - consisting of one syllable, and "English" - English. Words that do not have the properties denoted by them are called, or heterologous. Obviously, all adjectives denoting properties that are not applicable to words will be heterologous.

This division of adjectives into two groups seems clear and does not raise objections. It can be extended to nouns: “the word” is a word, “noun” is a noun, but “clock” is not a clock and “verb” is not a verb.

The paradox arises as soon as the question is asked: to which of the two groups does the adjective “heterologous” belong? If it is autologous, it has the property that it denotes and must be heterologous. If it is heterologous, it has no property called by it and must therefore be autologous. There is a paradox.

By analogs with this paradox, it is easy to formulate other paradoxes of the same structure. For example, is the one who kills every non-suicide and does not kill a single suicide, or is not a suicide?

It turned out that the Grelling paradox was known in the Middle Ages as an antinomy of expression that does not name itself. One can imagine the attitude to sophisms and paradoxes in modern times, if the problem requiring an answer and causing lively disputes was suddenly forgotten and was rediscovered only five hundred years later!

Another seemingly simple antinomy was indicated at the very beginning of the 20th century. D. Berry.

The set of natural numbers is infinite. The set of those names of these numbers, which are, for example, in Russian and contain less than, say, a hundred words, is finite. This means that there are natural numbers for which there are no names in the Russian language consisting of less than one hundred words. Among these numbers there is obviously the smallest number. It can not be called through the Russian expression, containing less than a hundred words. But the expression: “The smallest natural number for which its complex name does not exist in Russian, consisting of less than one hundred words” is just the name of this number! This name has just been formulated in Russian and contains only nineteen words. The obvious paradox: the number for which there is no name turned out to be named!