Grelling-Nelson Paradox

Paradox
Define an adjective to be autological if it is true when applied to itself.

For instance, the word "English" is autological, as it is a word in English.

The word "multisyllabic" is also autological, as it contains multiple syllables.

Define an adjective to be heterological if it is not true when applied to itself.

For instance, the word "long" is heterological, as it is not a long word.

The word "monosyllabic" is also heterological, as it does not contain only one syllable.

All adjectives must either be autological or heterological, as they either apply to themselves or they don't.

"Autological" and "heterological" are thus each defined as the negation of the other.

The Grelling-Nelson paradox arises when trying to apply one of these adjectives to the word "heterological".

If "heterological" is autological, then it does apply to itself.

But then by the definition of "heterological", it does not apply to itself, making it heterological.

If "heterological" is heterological, then it does not apply to itself.

But then by the definition of "heterological," it is not the case that "heterological" does not apply to itself, and so it does apply to itself, making it autological.

We thus have:
 * heterological is autological $\iff$ heterological is heterological

for the contradictory predicates "heterological" and "autological".

This paradox is closely related to :
 * $\not \exists x: \forall y: \paren {\map \RR {x, y} \iff \neg \map \RR {y, y} }$

Define $x$ to be the predicate "heterological," and define the relation $\map \RR {x, y}$ to mean "the predicate $x$ applies to $y$".

The sentence:
 * $\map \RR {x, y} \iff \neg \map \RR {y, y}$

is thus interpreted:
 * $y$ is heterological $y$ does not apply to itself.

The assertion that the predicate "heterological" exists and is defined as such for all $y$ is then equivalent to:
 * $\forall y: \paren {\map \RR {x, y} \iff \neg \map \RR {y, y} }$

This leads to the same contradiction as -- when we instantiate $y$ as $x$, we obtain:
 * $\map \RR {x, x} \iff \neg \map \RR {x, x}$