The Grelling-Nelson Paradox

Type theory as a resolution. Bertrand Russell's own solution to his paradox was the theory of types: divide all objects into a hierarchy of types, and restrict predicates to apply only to objects of lower type than themselves. On this view, a predicate cannot be applied to itself — the question "is 'heterological' heterological?" is not false but meaningless, because it attempts to apply a predicate at its own level rather than a lower one.

This is formally consistent, but it requires giving up a lot of intuitive reasoning. Many things we want to say — including many self-referential sentences that are perfectly well-behaved — become formally unutterable.

Tarski's hierarchy of languages. Alfred Tarski's approach divides language into levels: the object language and the metalanguage. Predicates in the object language can only be applied to objects; to say that a predicate applies to itself requires moving to the metalanguage. The hierarchy can be extended indefinitely, and no finite level contains its own truth predicate. This approach underlies formal theories of truth and avoids the Liar paradox and Grelling's paradox — at the cost of accepting that there is no single unified language in which all true statements can be expressed.

Is the paradox really different from Russell's? Some logicians regard Grelling-Nelson as simply a notational variant of Russell's paradox, since the same substitution — predicate for set, applies-to for membership — transforms one into the other. Others hold that they are genuinely different, because the Grelling version is about linguistic meaning and cannot be dissolved by formal restrictions on set membership. The question of whether the two paradoxes share a single root or have distinct sources is itself philosophically contested.

Grelling and Nelson published their paradox in 1908, the year after Russell published his theory of types as a response to his own paradox. They were working in the Friesian tradition of German philosophy, interested in the foundations of logic. The paradox appeared in a paper about Russell's and Burali-Forti's paradoxes, framed as a demonstration that the same problem could arise in predicate language.

The paradox is less famous than Russell's because set theory became the central arena for foundational work in mathematics, and Russell's version was directly relevant to the major mathematical programmes. Grelling's version is more relevant to the philosophy of language and linguistics.

Kurt Grelling was murdered at Auschwitz in 1942. Leonard Nelson had died in 1927. Neither lived to see the paradox become a standard teaching example.

It shows that the crisis in set theory's foundations was not just about sets. If you could replace all the set-theoretic language in Russell's paradox with predicate language and get the same contradiction, the problem was deeper than the axioms of set theory. It was about the logic of self-reference itself.

It drives the development of formal semantics. The need to avoid both the Liar paradox and Grelling-Nelson motivated Tarski's theory of truth and the hierarchy of metalanguages. That framework underlies the semantics of formal languages in logic and computer science.

It is a clean test case for theories of predication. Any theory of how predicates work must say something about whether a predicate can apply to itself. Grelling-Nelson shows that the most natural answer — that any predicate can be applied to any predicate — is inconsistent.

Type systems in programming. The design of type systems in programming languages is directly motivated by the need to prevent self-referential contradictions of exactly this kind. Languages that allow unrestricted self-application of types generate paradoxes at the level of the type system; stratified type systems avoid them at the cost of some expressiveness.

Natural language processing. Systems that reason about language must handle predicates that apply to predicates — "this sentence is true," "that description is accurate." Grelling-Nelson is a standing reminder that naive self-application of semantic predicates is inconsistent, and that any sufficiently expressive language must handle the resulting contradictions explicitly.

The formal structure

Let $P$ be the domain of predicates. Define the predicate $H$ ("heterological") by:

$$H(x) \iff \neg x(x)$$

(for any predicate $x$, $H(x)$ holds iff $x$ does not apply to itself).

Substituting $H$ for $x$:

$$H(H) \iff \neg H(H)$$

This is a direct contradiction. The derivation uses only the definition of $H$ and the universal generalisation that $H$ applies to all predicates satisfying its defining condition.

The parallel with Russell's paradox: define $R = {x : x \notin x}$. Then $R \in R \iff R \notin R$. The logical form is identical; only the domain (sets vs. predicates) and the membership relation (set membership vs. predicate application) differ.

What cannot fix it

Restricting self-application: If you simply rule that predicates cannot be applied to themselves, you lose the ability to say "'verb' is not a verb" and "'long' is not long," which are perfectly meaningful and true sentences. The restriction is too broad.

Declaring "heterological" undefined: This is Tarski's move. It is formally clean but requires accepting that the predicate "heterological" is not a legitimate predicate at any level of the hierarchy where it can be applied to itself. The cost is a stratified, non-unified language.

Paraconsistent logic: Some logicians accept that $H(H) \wedge \neg H(H)$ is simply true — that contradictions can be true without trivialising the system. This requires revising the principle of explosion (ex contradictione quodlibet: from a contradiction, anything follows), which is a significant departure from classical logic.