Grelling–Nelson Paradox

1. Introduction

The Grelling–Nelson paradox is a self-referential puzzle in the philosophy of language and logic that arises from sorting adjectives into two categories: those that describe themselves and those that do not. The paradox focuses on what happens when the adjective “heterological” (meaning: not describing itself) is applied to itself, and it appears to yield a contradiction under seemingly natural assumptions about meaning and classification.

This paradox is often presented as a particularly striking example of a semantic paradox formulated entirely in ordinary language, without explicit use of set-theoretic notation. It is therefore used to illustrate how problems similar to Russell’s paradox can emerge from everyday talk about words and their properties. Because the Grelling–Nelson construction involves an adjective classifying adjectives, it raises questions about self-reference, semantic categories, and the legitimacy of certain kinds of generalizations over linguistic expressions.

Philosophers and logicians have interpreted the paradox in different ways. Some regard it as a direct linguistic analogue of Russell’s paradox, revealing deep tensions in naive conceptions of meaning and predication. Others treat it as a pseudo-problem generated by category mistakes or by ignoring distinctions between language levels. Still others use it as a test case for formal approaches to semantics, including hierarchies of languages, partial truth predicates, or non-classical logics that tolerate certain contradictions.

Because of these multiple roles, the Grelling–Nelson paradox has become a standard example in discussions of the foundations of mathematics, formal semantics, and the theory of truth, even though its contemporary status is largely pedagogical rather than the focus of active technical research.

2. Origin and Attribution

The paradox is attributed to the German logician Kurt Grelling (1886–1942) and the philosopher Leonard Nelson (1882–1927). Their joint paper, usually cited as the first appearance of the paradox, is:

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Kurt Grelling and Leonard Nelson, “Bemerkungen zu den Paradoxien von Russell und Burali-Forti,” Abhandlungen der Fries’schen Schule, Neue Folge, Band 2 (1908), pp. 301–334.

In that article, Grelling and Nelson developed what they themselves described as an antinomy closely related to the then-recently discovered Russell and Burali-Forti paradoxes. Their central aim was to show that analogous contradictions could be formulated without the machinery of set theory, purely within the realm of ordinary language and logical predicates.

The specific classification into autological and heterological adjectives is widely credited to Grelling, although the exact distribution of contributions between Grelling and Nelson is not fully documented. Secondary literature generally follows the convention of naming the paradox after both authors, reflecting the joint context of the 1908 publication.

Later authors have sometimes abbreviated the label to “Grelling’s paradox,” or referred to it as the “paradox of heterological adjectives.” Despite this variation, there is broad agreement in the scholarly literature about the 1908 paper as the canonical source.

The following table summarizes the basic bibliographic facts:

Subsequent discussions by logicians such as Alfred Tarski, W.V.O. Quine, and Max Black have helped to standardize both the terminology and the reading of the original construction, even though they sometimes differ over how central the paradox is to foundational issues.

3. Historical Context

The Grelling–Nelson paradox emerged during the foundational crisis in mathematics at the turn of the 20th century. In the first decade of the century, logicians confronted a series of paradoxes—most notably Russell’s paradox (1901–1903) and the Burali-Forti paradox—that called into question the consistency of naive set theory and traditional logical assumptions.

Grelling and Nelson formulated their antinomy in 1908 against this backdrop, explicitly referencing Russell and Burali-Forti. Their work belongs to a broader effort to understand whether these paradoxes stem from set-theoretic notions alone or from more general features of language and logical predication.

Grelling and Nelson were associated with the Friesian school, which combined neo-Kantian and empiricist themes with a strong interest in logical rigor. In this milieu, paradoxes were seen not merely as curiosities but as pressure points for revising basic logical and semantic principles.

The paradox also predates, and in some cases foreshadows, later work by Tarski and others on formal semantics and truth. While Tarski’s influential papers on the concept of truth appeared in the 1930s and 1940s, his concerns about self-reference, semantic closure, and hierarchies of languages are closely aligned with the issues dramatized by the Grelling–Nelson construction.

Historically, then, the paradox occupies an intermediate position: it follows the first wave of set-theoretic antinomies but anticipates more systematic treatments of semantic paradoxes, contributing to the sense that foundational difficulties were not confined to mathematics but permeated language and meaning themselves.

4. The Paradox Stated

The Grelling–Nelson paradox arises from a seemingly simple classification of adjectives. First, two semantic categories are introduced:

• An adjective is autological if it applies to itself (for example, “English” is an English word).
• An adjective is heterological if it does not apply to itself (for example, “long” is not a long word).

Assuming that every adjective falls into exactly one of these two categories, one then considers the adjective “heterological” itself and asks whether it is autological or heterological.

The reasoning proceeds informally as follows:

1. Suppose “heterological” is heterological.
   Then, by the definition of “heterological,” it does not apply to itself. Hence, “heterological” is not heterological—a contradiction.

2. Suppose instead that “heterological” is not heterological.
   Then it must be autological. But if “heterological” is autological, it does apply to itself, which means that it is heterological after all—again yielding a contradiction.

Under the initial assumption that every adjective is either autological or heterological (and not both), the classification of “heterological” seems to lead to an impossibility, with each option forcing its negation.

A compact schematic representation is often given:

This internal tension is what is commonly referred to as the Grelling–Nelson paradox or the paradox of heterological adjectives.

5. Autological and Heterological Adjectives

The concepts of autological and heterological adjectives are central to the construction of the Grelling–Nelson paradox.

5.1 Autological adjectives

An adjective is autological (sometimes: “homological”) if it possesses the property it expresses. Typical examples used in the literature include:

• “English” (as used in English) is an English word.
• “Polysyllabic” is polysyllabic.
• “Short” is often claimed to be short (relative to ordinary word lengths).

The defining pattern is: an adjective A is autological if the sentence “A is A” is true when the first occurrence of A is used as a predicate and the second as a subject-designating word.

5.2 Heterological adjectives

An adjective is heterological if it does not possess the property it expresses. Common examples are:

• “Long” is not a long word.
• “German” (considered as an English word) is not German.
• “Blue” (as a word on the page) is arguably not blue.

Here the characteristic pattern is: an adjective B is heterological if the sentence “B is B” is false under the same use–mention shift.

5.3 Illustrative classification

A simple, often-cited classification runs as follows:

Philosophers note that this classification depends on background assumptions about spelling, pronunciation, context, and what counts as “possessing” a property (for instance, whether a word written in blue ink is “blue”). These complications have led some authors to question how well-defined the autological/heterological distinction is in natural language; however, the Grelling–Nelson paradox proceeds from the simplifying idealization that such a distinction is meaningful and exhaustive for the relevant adjectives.

6. Logical Structure and Reductio Form

The reasoning behind the Grelling–Nelson paradox is typically analyzed as a reductio ad absurdum: it assumes a plausible principle about adjectives and then derives a contradiction, suggesting that the principle must be rejected or modified.

6.1 Formal core of the argument

Let A(x) mean “x is an autological adjective” and H(x) mean “x is a heterological adjective.” Let h denote the adjective “heterological.”

The key assumptions are:

1. Exhaustiveness: For every adjective x, either A(x) or H(x).
2. Exclusivity: For no adjective x do both A(x) and H(x) hold.
3. Definitions:
   • A(x) ↔ x applies to itself.
   • H(x) ↔ ¬(x applies to itself).

One then considers h and reasons within classical logic:

• Assume H(h).
  By definition of heterological, ¬(h applies to h).
  But H(h) itself states that h does not apply to h, so if this is correct, “h is heterological” is true precisely when “h is not heterological,” yielding a contradiction.

• Assume ¬H(h).
  By exhaustiveness, A(h).
  By definition of autological, h applies to h.
  But if h applies to h, then it satisfies the condition for being heterological (not applying to itself) only by failing it, again producing inconsistency.

6.2 Reductio structure

The overall form is:

1. Assume a general principle P: every adjective is either autological or heterological, and not both.
2. Derive that P implies (H(h) ∨ ¬H(h)).
3. Show that each disjunct leads to a contradiction.
4. Conclude that P is untenable, at least in its naive, unrestricted form.

This pattern exemplifies how semantic paradoxes can be cast within standard logical frameworks: the reasoning uses classical logic, including the law of excluded middle and non-contradiction, together with definitional equivalences. Debates about the paradox often focus on which specific premise or inferential rule should be restricted, reinterpreted, or rejected.

7. Comparison with Russell’s Paradox

The Grelling–Nelson paradox is frequently compared to Russell’s paradox, and many authors treat it as a linguistic analogue of Russell’s set-theoretic construction.

7.1 Structural analogy

Russell’s paradox involves the set:

• R = {x | x ∉ x}, the set of all sets that are not members of themselves.

The central question is whether R ∈ R, and classical reasoning yields a contradiction:

• If R ∈ R, then by definition R ∉ R.
• If R ∉ R, then by definition R ∈ R.

The Grelling–Nelson paradox involves the adjective:

• “heterological” = the adjective that does not apply to itself.

The central question is whether “heterological” is heterological, and the dialectic mirrors Russell’s:

• If “heterological” is heterological, it does not apply to itself.
• If it is not heterological, it must be autological and apply to itself.

7.2 Interpretive significance

Many commentators hold that this analogy supports the view that both paradoxes stem from a common pattern: unrestricted self-reference combined with a negating condition (non-self-membership, non-self-application). On this reading, Grelling–Nelson is a semantic restatement of Russell’s insight, showing that similar difficulties arise even without explicit set-talk.

Others argue that the analogy is imperfect. They contend that Russell’s paradox concerns well-defined set-theoretic principles (like unrestricted comprehension), whereas the Grelling–Nelson construction depends on contentious assumptions about what it means for an adjective to apply to itself and about the domain of adjectives.

Nevertheless, the comparison has been influential in motivating type-theoretic and hierarchical responses: just as Russell introduced a hierarchy of sets to block self-membership, so later authors proposed hierarchies of languages or predicates to block self-application in the semantic case.

8. Semantic and Syntactic Issues

The Grelling–Nelson paradox hinges on subtle interactions between semantics (meaning, truth, application) and syntax (form, occurrence, category). Many proposed analyses focus on how these aspects are intertwined in the key definitions.

8.1 Use–mention and self-application

A central issue is the shift between using an adjective and mentioning it as a word. In “’English’ is English,” the first occurrence is mentioned (as a word), while the second is used as a predicate. The predicate “applies to itself” thus mixes syntactic facts about a word’s occurrence with semantic facts about what the predicate expresses.

Some authors argue that conflating these roles leads to serious ambiguity. On one reading, “heterological” is a word-type with certain typographic and phonetic properties; on another, it is a predicate expressing a property. The paradox appears to trade on sliding between these readings without explicit distinction.

8.2 Category and type distinctions

Another line of critique targets the category of things to which adjectives can apply. Ordinarily, adjectives modify nouns and noun phrases, not adjectives themselves. Saying that “’red’ is red” seems to treat a linguistic item as belonging to the same semantic category as the objects it describes, which some regard as a category mistake.

From a more formal perspective, this concern is related to the idea of logical types: expressions of one type (e.g., predicates of physical objects) are not, strictly speaking, applicable to items of another type (e.g., predicate-expressions themselves) without additional machinery.

8.3 Ambiguities in “applies to”

The phrase “applies to” can be read in multiple ways:

• Syntactic: an adjective occurs next to a noun in a grammatically correct way.
• Semantic: an adjective truly characterizes an object.
• Metalinguistic: a predicate in a metalanguage is satisfied by a linguistic item in an object language.

The paradox is often reconstructed as requiring the semantic or metalinguistic sense, but ordinary-language formulations may not clearly separate these aspects, leaving room for different reconstructions.

These semantic and syntactic issues motivate several responses: some treat the paradox as revealing deep semantic problems; others treat it as exposing a misuse of language where critical distinctions are elided.

9. Type-Theoretic and Hierarchical Responses

One influential family of responses to the Grelling–Nelson paradox adapts type-theoretic and hierarchical strategies originally developed to address set-theoretic and semantic paradoxes.

9.1 Russellian type theory

Inspired by Bertrand Russell’s theory of types, some approaches assign expressions to distinct logical levels. On such views, an adjective at level n can apply only to items of lower levels (e.g., objects or adjectives at level n−1), and no expression is permitted to apply, in a well-formed way, to itself at the same level.

Reconstructed in this framework, “heterological₁” at level 1 could be defined as applying to level-0 adjectives that do not apply to themselves as level-0 predicates. However, asking whether “heterological₁” is heterological₁ becomes ill-typed or meaningless, because a level-1 predicate cannot serve simultaneously as both subject and predicate at its own level.

9.2 Tarskian semantic hierarchies

Alfred Tarski’s semantic conception of truth introduced a hierarchy of object languages and metalanguages: a metalanguage can contain a truth predicate for sentences of the object language but not for sentences of the same level.

Analogously, some treatments impose a hierarchy on application predicates:

• An object language contains adjectives applying to objects.
• A metalanguage contains predicates (such as “heterological”) that speak about adjectives of the object language.
• A meta-metalanguage could then speak about predicates of the metalanguage, and so on.

On this view, the paradox depends on illicitly treating “heterological” as if it were both an object-language adjective and a metalanguage predicate. Once the hierarchy is enforced, “Is ‘heterological’ heterological?” is not a well-formed question at any single level.

9.3 Assessment of hierarchical strategies

Proponents argue that these strategies cleanly block the paradox by ruling out the crucial self-application. Critics contend that such hierarchies may be ad hoc or that natural language appears to allow cross-level talk more freely than rigid type systems permit. Nonetheless, type-theoretic and hierarchical responses remain a prominent and widely discussed way of handling the Grelling–Nelson antinomy.

10. Non-Classical and Partial Semantics Solutions

Another major family of responses to the Grelling–Nelson paradox modifies semantic assumptions or the underlying logic, rather than banning self-reference outright.

10.1 Partial semantics and undefinedness

Influenced by work such as Saul Kripke’s fixed-point theories of truth, some authors treat the autological/heterological classification as a partial predicate: for certain self-referential cases, including “heterological” itself, no truth value is assigned.

On this approach, there may be three semantic statuses:

Applied to the Grelling–Nelson paradox, the key step is to deny that “’heterological’ is heterological” (or its negation) must be either true or false. Because classical reasoning from “either H(h) or ¬H(h)” is blocked, the contradictory cycle cannot be completed.

10.2 Non-classical logics and dialetheism

Other responses alter the logic itself. Paraconsistent and dialetheist logics, as developed for example by Graham Priest, allow that some contradictions may be true without entailing triviality (i.e., without allowing one to prove every statement).

On a dialetheist reading, the sentence “’heterological’ is heterological” might be both true and false. The paradox is then interpreted not as a reductio of the autological/heterological classification but as evidence that certain semantic constructions generate true contradictions.

10.3 Comparative evaluation

Supporters of partial semantics emphasize alignment with intuitions that anomalous self-referential sentences lack a determinate truth value. Advocates of non-classical logics highlight the uniform treatment of paradoxes and the preservation of naive principles.

Critics of these approaches often question their cost: partial semantics can complicate standard logical laws, and paraconsistent systems revise entrenched inferential rules such as explosion (from a contradiction, anything follows). The Grelling–Nelson paradox thus serves as one of several test cases in broader debates about how far logic and semantics should be revised to accommodate self-reference.

11. Standard Objections and Critiques

A number of philosophers have argued that the Grelling–Nelson paradox does not reveal a deep inconsistency in language or logic but instead rests on questionable assumptions. Several widely discussed objections are the following.

11.1 Category mistake objection

Some critics maintain that the paradox arises from treating adjectives as if they were the right sort of things to which adjectives can apply. On this view, adjectives are linguistic items, not the objects that adjectives normally describe, so asking whether “long” is long or “German” is German is a category mistake. The predicates in question are designed for non-linguistic entities (words, perhaps, but not adjectives as such), and the self-application step is therefore illegitimate.

11.2 Ambiguity of “applies to itself”

Another objection focuses on the phrase “applies to itself.” This expression may blur distinctions between syntactic occurrence, semantic truth, and metalinguistic satisfaction. For instance, “’red’ occurs in a red context” is quite different from “’red’ is red” as a statement about its color. Critics suggest that the paradox trades on switching between readings of “applies to” without clear indication.

11.3 Hierarchy of languages objection

From a Tarskian point of view, it is argued that the paradox relies on using, within one and the same language, a predicate that speaks about all expressions of that language, including itself. If one insists on a hierarchy of languages, where semantic predicates apply only to a lower-level language, then the crucial question (“Is ‘heterological’ heterological?”) is simply not formulable without crossing language levels improperly.

11.4 Context-sensitivity and vagueness

Some philosophers emphasize the context-sensitivity and vagueness of many adjectives. Whether “short” is short, or “polite” is polite, can vary with conversational standards, and not all uses admit a sharp, context-independent classification as autological or heterological. On this reading, the paradox presupposes a degree of precision that ordinary language does not in fact support; the resulting difficulty reflects an idealization rather than a genuine contradiction in meaning.

These objections have prompted divergent views about the significance of the paradox. Some conclude that the autological/heterological distinction is too ill-formed to bear philosophical weight; others treat the objections as highlighting subtleties that any satisfactory semantic theory must address when reconstructing the paradox in more formal terms.

12. Impact on Theories of Truth and Meaning

The Grelling–Nelson paradox has played a notable—though often secondary—role in shaping modern theories of truth and meaning.

12.1 Motivation for semantic hierarchies

For theorists such as Alfred Tarski, the paradox reinforced concerns about self-referential semantic predicates. Although Tarski focused primarily on the liar paradox, the Grelling–Nelson construction provided an additional example in which seemingly natural semantic notions (here, “applies to itself”) lead to inconsistency when allowed to range over expressions of the same language. This helped motivate Tarski’s insistence on object-language/metalanguage distinctions and stratified definitions of truth and related semantic concepts.

12.2 Stress-testing ordinary-language semantics

In the philosophy of language, the paradox has been used as a test case for theories of predication, reference to words, and the semantics–pragmatics interface. It raises questions about:

• How predicates can apply to linguistic items.
• Whether a single, untyped language can express global semantic predicates about its own expressions.
• How context and speaker intentions affect the interpretation of self-referential descriptions.

Competing theories—such as truth-conditional semantics, deflationary accounts of truth, and pragmatic or use-based accounts—have each been examined for their ability to reproduce or dissolve the paradox.

12.3 Influence on non-classical and partial truth theories

In later work on partial and non-classical theories of truth (e.g., Kripke’s and Priest’s approaches), Grelling–Nelson is sometimes cited alongside the liar as an illustration of how self-referential constructions challenge classical, bivalent semantics. Whether one regards “heterological” as undefined, both true and false, or excluded by typing constraints, the paradox functions as a benchmark against which proposals for truth predicates, truth-value gaps, or truth-value gluts are evaluated.

Overall, while not as central as the liar paradox, the Grelling–Nelson antinomy has contributed to a broader realization that any comprehensive account of truth and meaning must confront the possibility of self-applicative, semantically loaded expressions within a single language.

13. Relation to Other Semantic Paradoxes

The Grelling–Nelson paradox is part of a wider family of semantic paradoxes that involve self-reference, negation, or circularity. Its relationships to these paradoxes illuminate common underlying patterns.

13.1 Relation to the liar paradox

The liar paradox involves a sentence that says of itself that it is not true (e.g., “This sentence is not true”), leading to a similar oscillation between truth and falsity. Both paradoxes:

• Use self-reference or quasi-self-reference.
• Involve a semantic predicate (truth, or “applies to itself”) with a negative condition.
• Generate contradictions under classical assumptions.

Some authors thus classify Grelling–Nelson as a liar-like paradox, differing mainly in surface form (adjectives instead of sentences) but sharing a common logical structure.

13.2 Relation to Berry, Richard, and other definability paradoxes

Paradoxes such as Berry’s paradox (“the smallest number not nameable in fewer than nineteen syllables”) and Richard’s paradox (concerning definable real numbers) also exploit self-reference, definability, and impredicative constructions. Grelling–Nelson is sometimes grouped with these as showing that:

• Talk about linguistic items using general classifications (“autological,” “definable,” “nameable”) can give rise to paradox.
• Problems arise when quantifying over “all” expressions or definitions within a language.

13.3 Relation to set-theoretic paradoxes

As noted earlier, there is a close analogy with Russell’s and Burali-Forti’s paradoxes. All of these can be viewed as manifestations of tensions between unrestricted comprehension principles and the possibility of self-involvement (self-membership, self-application).

Because of these parallels, many general strategies for handling semantic paradoxes—hierarchies, restrictions on self-reference, partiality, non-classical logics—are evaluated simultaneously against the liar, Grelling–Nelson, and related constructions, with success or failure in one case often taken to bear on the others.

14. Contemporary Assessments

In contemporary philosophy and logic, the Grelling–Nelson paradox is widely discussed but rarely regarded as the central focus of research. Its current status can be characterized along several dimensions.

14.1 Predominantly pedagogical role

Most textbooks and surveys in philosophical logic and philosophy of language mention the paradox as a clear and memorable illustration of:

• How self-reference can arise in ordinary language.
• How semantic notions (like “applying to itself”) may lead to inconsistency.
• How linguistic analogues of set-theoretic paradoxes can be constructed.

It thus serves mainly as a teaching tool, helping students grasp the broader landscape of semantic paradoxes.

14.2 Divergent views on depth and significance

Contemporary authors disagree about how philosophically deep the paradox is:

• Some treat it as a genuine challenge to naive semantics, comparable in importance to the liar paradox and supporting the need for hierarchies or restrictions.
• Others regard it as more fragile, heavily dependent on ambiguous or ill-formed notions such as “an adjective describing itself,” and therefore of limited diagnostic value for serious semantic theory.

Because of these disagreements, the paradox often plays a secondary role in technical developments, with researchers focusing first on more robust or canonical paradoxes (especially the liar) and then checking whether their solutions extend to the Grelling–Nelson case.

14.3 Ongoing relevance

Despite its largely pedagogical status, the paradox continues to be cited in discussions of:

• The expressive limits of languages with global semantic predicates.
• The interaction between use–mention distinctions and self-reference.
• Comparative evaluations of classical vs. non-classical semantic frameworks.

In this way, it functions as a standing test for the generality and explanatory power of theories developed primarily in response to other semantic or set-theoretic paradoxes.

15. Legacy and Historical Significance

The legacy of the Grelling–Nelson paradox lies less in ongoing technical research and more in its historical contribution to the understanding of self-reference and semantic paradoxes.

15.1 Place in the foundational crisis

Historically, the paradox contributed to a growing realization in the early 20th century that paradoxical phenomena were not confined to set theory. By presenting an antinomy couched entirely in terms of adjectives and their application, Grelling and Nelson helped demonstrate that analogous difficulties arise at the level of language and meaning. This broadened the scope of the foundational crisis from mathematics to semantics and the theory of logical form.

15.2 Influence on later semantic work

The paradox is frequently cited in historical reconstructions of the development of formal semantics and theories of truth, alongside the liar, Richard’s, and Berry’s paradoxes. It is part of the background that shaped the thinking of figures such as Tarski, Carnap, and later logicians who sought systematically to separate object language and metalanguage, define truth predicates, and handle self-reference in controlled ways.

15.3 Role in the canon of paradoxes

Within the “canon” of logical paradoxes, Grelling–Nelson occupies a stable but modest position. It is often mentioned in surveys of paradoxes, included in anthologies, and used in introductory courses, but it does not typically drive new specialized research in the way that some other paradoxes do.

Overall, the paradox’s enduring significance lies in its clear and memorable formulation of a self-referential semantic puzzle, which has helped shape discussions about how languages can, and cannot, coherently talk about their own expressions.