The Grelling-Nelson paradox

The Grelling–Nelson paradox is the paradox of the self-describing adjective. It is also sometimes called Weyl’s paradox or the Heterological paradox, or simply Grelling’s paradox. It first appeared in a publication in 1908. (Footnote: K. Grelling & L. Nelson. Bemerkungen zu den Paradoxien von Russell und Burali-Forti. in Abhandlungen der Fries’schen Schule 2, no. 3, pages 301-334, 1908. The English translation of the original wording of the paradox is:
Let φ(M) be the word that denotes the concept defining M. This word is either an element of M or not. In the first case we will call it “autological” in the other “heterological”. Now the word “heterological” is itself either autological or heterological. Suppose it to be autological; then it is an element of the set defined by the concept that is denoted by itself, hence it is heterological, contrary to the supposition. Suppose, however, that it is heterological; then it is not element of the set defined by the concept that is denoted by itself, hence it is not heterological, again against the supposition.)

The paradox is usually given in terms similar to the following:

Some adjectives can be said to describe themselves, such as multisyllabic, short, English, pronounceable, while others do not, such as monosyllabic, female, long, four-lettered, etc. So, we create a definition as follows:

• A word is self-descriptive if and only if it describes that word (if it describes itself). For example, ‘printable’, ‘finite’, and ‘polysyllabic’ are self-descriptive.
• A word is non-self-descriptive if and only if it does not describe that word (does not describe itself).

So, for example, ‘unprintable’, ‘incomplete’, ‘monosyllabic’, and ‘four-lettered’ are non-self-descriptive.

Then the question is asked:

“Is non-self-descriptive non-self-descriptive or self-descriptive?”

If it is self-descriptive, then non-self-descriptive is non-self-descriptive. But if non-self-descriptive is non-self-descriptive, then it does describe itself, so it must be self-descriptive.

That is the ‘paradox’.

Is unspellable unspellable?

Before going into an analysis of the Grelling-Nelson paradox, let’s consider another apparently paradoxical question, as follows. We define that:

• A word is unspellable if and only if it is not possible to spell the word.
• A word is spellable if and only if it is possible to spell the word.

Now we ask the question, ‘Is unspellable unspellable?’

Is this a paradox?

It isn’t. The difficulty in answering the posed question lies in the ambiguity of the question, since there are two quite valid interpretations of the question:

• It may be a question as to whether the word ‘unspellable’ is the word ‘unspellable’ - to which the answer is obviously ‘Yes’.
• Or it may be a question as to whether the word ‘unspellable’ has the property of being unspellable, that is, whether it is impossible to spell the word ‘unspellable’ - to which the answer is obviously ‘No’.

All that is happening here is that we are being befuddled by ambiguities in the English language. So the answer to the question depends on which interpretation one puts on the question.

This isn’t a paradox that is attributable to faulty logic, it is simply a case of confusion due to ambiguity in the question. The solution to the apparent paradox is to phrase the question less ambiguously. And it will be seen below that the Grelling-Nelson paradox is just the same sort of ill-defined question. The paradox operates by obfuscating the distinction between adjectives and nouns in its definition.

Adjectives and Nouns

Consider the expression ‘Joe is short.’

This has two interpretations/meanings:

• The word ‘Joe’ is short.
• Joe the person is short.

and what we are really intending is that the meaning is either:

• The symbol string ‘Joe’ is a short symbol string.

• The person that is represented by the symbol string ‘Joe’ is a short person.

In the above, we have assumed that the word short can only be an adjective. We could also have the possibly that Joe is also called ‘short’ as a nickname, and that it is spelt without an initial capital. Now we have more possibilities, since ‘short’ may be a noun or an adjective.

• The symbol string ‘Joe’ is a short symbol string.

• The person that is represented by the symbol string ‘Joe’ is a short person.
• The person that is represented by the symbol string ‘Joe’ is a person that is also represented by the symbol string ‘short’.

And we can say whether the statements are correct:

• Yes – it consists of only three letters
• No – Joe the person is in fact tall
• Yes – Joe is also called ‘short’

In other words, when we say in English that x is y, that does not necessarily mean that we intend the mathematical meaning that x equals y. For example, when we say that something is pink, we don’t mean that the thing and pink are identical; what we mean is that the thing has a property that we call pink.

In the Grelling-Nelson paradox, the word self-descriptive is ‘defined’ by a vague self-reference so that it is not actually clearly defined. The crucial point is that, in the Grelling-Nelson ‘paradox’, the words ‘self-descriptive’ and ‘non-self-descriptive’ may be nouns or adjectives depending on the context.

So, if we state the Grelling-Nelson paradox in a logical manner by removing the ambiguity of noun/adjective we get:

• A word is self-descriptive if, as an adjective, it describes that symbol combination that is that word.
• A word is non-self-descriptive if, as an adjective, it does not describe that symbol combination that is that word.

So now if we ask

‘Is non‑self-descriptive non‑self-descriptive?’

it is clear that the question is ambiguous, and can have different meanings. We can mean either:

• “Is the word ‘non-self-descriptive’ the word ‘non-self-descriptive’?” – in which case the answer is obviously “Yes”

  or

• “Does the adjective ‘non-self-descriptive’ describe the symbol string ‘non-self-descriptive’?” Here the latter ‘non-self-descriptive’ is a noun and not an adjective - and a noun does not describe anything - so the answer in this case is also “Yes”.
  Some people seem to have difficulty in understanding this. Here we are using the first instance of ‘non-self-descriptive’ as an adjective, and the second instance of ‘non-self-descriptive’ as an noun. Since nouns do not describe anything, the second instance of ‘non-self-descriptive’ does not describe anything, therefore we can say that the second instance of ‘non-self-descriptive’ does not describe itself, and so the adjective ‘non-self-descriptive’ (the first instance) applies to the second instance of ‘non-self-descriptive’.

Either way, the answer is “Yes”, and there is no paradox.

And if you ask:

“Is self-descriptive an adjective that applies to the symbol string ‘non-self-descriptive’?”

again, ‘non-self-descriptive’ is a noun and not an adjective, so that it cannot be descriptive at all, so again, it cannot describe itself. The answer is No, and there is no paradox.

It is apparent that many people find it difficult to shake off the compulsion to interpret the words ‘self-descriptive’ and ‘non-self-descriptive’ as adjectives where they actually occur as nouns. It is rather like those visual images that initially appear to contradict reality, such as those that show two lines that we see as being of different lengths, but when we measure them, we find that they are the same.

The apparent paradox arises simply because there is an ambiguity in how we are referring to the words of the language. In English, the meaning of a word in English may be dependent on its context. A word that is a describer is different to a thing that is described by a describer, even if the physical symbols that constitute the word can be both a describer and a thing that is described. In that case, the decision as to whether a word is a describer or a thing that is described is made according to context.

The Grelling-Nelson paradox is an expression that is ambiguously defined. Clarifying the Grelling-Nelson definition (assuming that it is intended to actually have a meaningful expression) removes the ambiguity from the expression.

Delimiters

In the normal syntax of a language, an adjective is not an object of the language. For logical languages, we want to avoid the ambiguities of natural language that can result from the noun/adjective confusion that can occur in cases such as Grelling-Nelson. A simple way to achieve this is by the use of what is called a delimiter. In natural language, quotation marks are often used for this purpose, as was done above. However, because natural language does not have any rigid rules for the application of delimiters, this does not eliminate the possibility of confusion.

Language

In terms of meta-language and sub-language, nouns are always objects of a language. In a logical language, adjectives are part of the syntax of the language and should not be objects within that language - but they could be objects as seen by a meta-language.

A reader has suggested that the paradox can be perpetuated by replacing the word ‘word’ by ‘string of characters’ in the definition, giving:

• x is self-descriptive if and only if x is a string of characters used to signify a property that x has
• x is non-self-descriptive if and only if x is a string of characters used to signify a property that x does not have

In fact, this only serves to show that the paradox relies on the confusion of meta-language and sub-language, since now it is obvious that the language for the ‘used to signify’ is not defined. If we were to attempt to define it as English, we now have a straightforward case of a language trying to be both an object language and a meta-language to that object language. (Footnote: Note: it might be thought that this can be obviated by defining that the English definition of ‘self-descriptive’ that the signification is defined as being in French, and that you have a similar definition in French where the definition is that the signification is defined as being in English. However, to obtain the full definition of ‘self-descriptive’ in English in this case, you need to reference the French definition which refers to English as an object language, so that the object/meta language confusion is not obviated at all.) The ambiguity of natural language such as English allows symbols strings to be at the same time, objects of the language, and valid syntax of the language. This sort of ambiguity cannot occur in a well formed well defined formal language.

Peter Smith’s ‘Proof’

It has come to my notice that, when asked about the demonstration of the flaw in his proof (see A Fundamental Flaw in an Incompleteness Proof by Peter Smith PDF), Smith refuses to engage in any logical discussion, and instead attempts to deflect attention away from any such discussion. If any other reader has tried to engage with Smith regarding my demonstration of the flaw, I would be interested to know what the outcome was.

There’s something about Gödel by Francesco Berto

There is a new addition to the page Yet another flawed incompleteness proof, where Berto’s proof of incompleteness in his book There’s something about Gödel comes under scrutiny.

Easy Footnotes

I found that making, adding or deleting footnotes in the traditional manner proved to be a major pain. So I developed a different system for footnotes which makes inserting or changing footnotes a doddle. You can check it out at Easy Footnotes for Web Pages (Accessibility friendly).

O’Connor’s “computer checked” proof

I have now added a new section to my paper on Russell O’Connor’s claim of a computer verified incompleteness proof. This shows that the flaw in the proof arises from a reliance on definitions that include unacceptable assumptions - assumptions that are not actually checked by the computer code. See also the new page Representability.

New page on Chaitin’s Constant

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