Self-Reference, Autograms and Autological Words

OuroborosWords and sentences that refer to themselves are fertile ground for word play and paradoxes. The topic was explored in depth in Douglas Hofstadter’s Metamagical Themas column in Scientific American and subsequent book by the same title.

Self-Referential Sentences

The original self-referential sentence is, “This sentence is false,” which also creates a paradox. Here are a few more examples of paradoxical self-referential sentences.

  • There are two errors in this sentance.
  • Fill in the blank with a word for a number, not a numeral:
    This sentence includes ________ e’s.
  • You are not finished reading this sentence.
  • Why is there no correct answer to this question?
  • If this sentence is true, then Bigfoot exists.

Autograms

MetamagicalThemasAn autogram is a sentence that provides a full inventory of its own characters. Lee Sallows composed this autogram, which was published in Metamagical Themas:

Only the fool would take trouble to verify that his sentence was composed of ten a’s, three b’s, four c’s, four d’s, forty-six e’s, sixteen f’s, four g’s, thirteen h’s, fifteen i’s, two k’s, nine l’s, four m’s, twenty-five n’s, twenty-four o’s, five p’s, sixteen r’s, forty-one s’s, thirty-seven t’s, ten u’s, eight v’s, eight w’s, four x’s, eleven y’s, twenty-seven commas, twenty-three apostrophes, seven hyphens and, last but not least, a single !

Autological and Heterological Words

Autological or homological words describe themselves. For example, “short” is a short word, “polysyllabic” has multiple syllables, and “noun” is a noun. Sometimes a word is only autological in context. For example, the word “legible” is legible if you are reading it, and the word “example” is being used as an example of an autological word in this sentence.

Heterological words do not describe themselves. For instance, “long” is not a long word, and the word “French” is not a word in French. Perhaps the most ironic case is that both “hyphenated” and “non-hyphenated” are heterological.

It seems that all adjectives must be either autological or heterological. However, this leads to a paradox.

The Grelling-Nelson Paradox

Is “heterological” a heterological word? If yes, then it does not describe itself; however, that means “heterological” is heterological, so in fact it does describe itself. If no, then it does describe itself; however, that means “heterological” is autological, so in fact it does not describe itself.

When we consider whether “autological” is an autological word, we find, not a paradox, but a tautology. The answer is that if it is, then it is, creating an infinite regress of truth values. However, it is just as easy to say that it is not because it is not.

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