# 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**