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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; the original wording is given below.

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

the word ?” – in which case the answer is obviously “Yes” “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 a 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 in the definition the word ‘word ’ by ‘string of characters ’ and replacing ‘describe’ by ‘used to signify a property ’ 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

This formulation raises the question as to what languages are involved, since the language for the ‘used to signify’ is not defined.

But another problem with this formulation is that the property is not defined, hence it can be any property. The problem is that defining that a string of characters does not have a property is meaningless unless that property is defined - no string of characters possesses every possible property, and hence every string of characters would be described by an adjective that states that for the string it describes, there is some property that it does not possess. Perhaps one could try reformulating it as:

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 that is not used to signify a property that x has

This still does not help, since in the first case ‘self-descriptive’ is an adjective, and in the second case ‘non-self-descriptive’ cannot, by the very definition itself, be an adjective. That means that if we apply the same disambiguation as before, and ask:

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

we note that in this case that since ‘non-self-descriptive’ is a noun the only valid meaning is:

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

The lesson should be that it is a waste of time and effort attempting to formulate “better” versions of a natural language “paradox”, since the supposed paradox can inevitably be shown to be dependent on the vagueness and ambiguity of natural language. 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.

Original Wording

The English translation of the original wording by Grelling and Nelson 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.”

Note that in Grelling and Nelson's original text there is no clear definition of the set M, since it is a circular definition. If we try to disambiguate it we might try:

Let φ(M) be the adjective that defines all words that are in the set M. Let MC be the set of all words that are not in the set M. If a word is an element of M it is a noun that may be described by the adjective ‘autological ’ (i.e: φ(M) and ‘autological ’ are synonymous adjectives). If a word is an element of MC it is either a noun, in which case it may be described by the adjective ‘heterological ’, or else it is not a noun.

Now ask is ‘heterological ’ an element of M or MC. Note that, of itself, the definition does not enable us to tell which set it is in.

If ‘heterological ’ is an element of M it is a noun described by the adjective ‘autological ’. There is no paradox.

If ‘heterological ’ is an element of MC, it is either a noun described by the adjective ‘heterological ’, or else it is not a noun.

Again there is no paradox, we simply note (in a meta-language) that depending on the context the word ‘heterological ’ could be an element of MC and be a noun, and could be an element of MC and be an adjective, but any particular instance of ‘heterological ’ cannot be a noun and also an adjective - it has to be one or the other.

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