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 Russell's Paradox: Corollary:

$\not \exists x: \forall y: \paren {\map \RR {x, y} \iff \neg \map \RR {y, y} }$

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 if and only if $y$ 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 Russell's Paradox: Corollary -- when we instantiate $y$ as $x$, we obtain:

$\map \RR {x, x} \iff \neg \map \RR {x, x}$

Also known as

The Grelling-Nelson Paradox is also known as:

Grelling's Paradox

Weyl's Paradox, as it is sometimes mistakenly attributed to Hermann Klaus Hugo Weyl.

Source of Name

This entry was named for Kurt Grelling and Leonard Nelson.

Historical Note

The Grelling-Nelson Paradox was stated by Kurt Grelling and Leonard Nelson in $1908$.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Grelling's paradox
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Grelling-Nelson paradox