Is " Heterological " heterological? (The Grelling- Nelson Paradox)

[michaellevenson]

A predicate expression is heterological if and only if it doesn't apply to itself, and autological only if it does.
For example, " is monosyllabic ", " is a French phrase", and " is three words long" are heterological since they don't apply to themselves. Whereas " is polysyllabic ", " is an English phrase", and " is four words long", are autological. It is either/ or, heterological or autological no middle ground possible. A word must either describe itself or it doesn't, the bivalency rule. Simple words like cat are heterological because the word cat isn't a cat. "Pronounceable" is autological because it describes itself ie it's pronounceable. "Unpronounceable" is heterological because it doesn't describe itself ie it is also perfectly pronounceable.
Now,
Q: is " heterological " heterological?
If " heterological " is heterological it doesn't apply to itself, so " heterological " isn't heterological, so it's autological, so it does apply to itself, so it is heterological, so it doesn't. If it does apply to itself, it doesn't, and if it doesn't it does!
We have a problem!
This video will make all this very clear.

 

[michaellevenson]

Spoiler

1a
One solution is to treat predicates about predicates as one level higher than their subjects, so that a statement about a predicate is only accepted as significant if its predicate is of a level one higher than that of its subject. So " short" cannot apply to itself since both subject and predicate in " Short is short" would be the same level.
Heterological is heterological will also be disqualified for the same reason, as will Heterological is autological.
On this view the question of whether Heterological is heterological or autological cannot properly be asked. But it is highly counterintuitive to dismiss as meaningless the statements that "Short" is short ,or "monosyllabic" is not monosyllabic.
This rule about predicate levels seems gerrymandered to avoid the paradox.
1 b
An alternative solution is to recognise a hierarchy of " heterologicals" and "autologicals."
Call " heterological ²" a second level predicate which is true of first level predicates which don't apply to themselves." Heterological ³" is true only of first and second level predicates and is a third level predicate. This hierarchy differs from 1a in allowing predicates to apply to themselves, except same level heterologicals and autologicals, this exception means there will be no " heterological " which applies to all levels so the paradox is avoided.
2.
A better solution is possibly to say Heterological is heterological is neither true or false since it is not a statement with any genuine content. When we consider whether it is true that " monosyllabic " is monosyllabic we look at the number of syllables and see it isn't. But we cannot tell from the expressions " heterological " or " autological" whether or not they apply to themselves. In order to know whether Heterological is heterological we need to know whether it applies to itself, that is, we need to already know the answer to our question before we then ask and then answer it ! But we can say " Short" is autological because it's short or monosyllabic is heterological because it's not monosyllabic.

Spoiler

Suppose one interprets the adjectives "autological" and "heterological" as follows:

1. An adjective is autological (sometimes homological) if it describes itself. For example, the English word "English" is autological, as are "unhyphenated" and "pentasyllabic".
2. An adjective is heterological if it does not describe itself. Hence "long" is a heterological word (because it is not a long word), as are "hyphenated" (because it has no hyphen) and "monosyllabic" (because it has more than one syllable).

All adjectives, it would seem, must be either autological or heterological, for each adjective either describes itself, or it does not. Problems arise in a number of instances, however.

The Grelling–Nelson paradox arises when we consider the adjective "heterological". One can ask: Is "heterological" a heterological word? If the answer is "no", then "heterological" is autological. This leads to a contradiction, for in this case "heterological" does not describe itself: it must be a heterological word. But if the answer is "yes", then "heterological" is heterological. This again leads to a contradiction, because if the word "heterological" describes itself, it is autological.

• Is "heterological" a heterological word?
  • no → "heterological" is autological → "heterological" describes itself → "heterological" is heterological, contradiction
  • yes → "heterological" is heterological → "heterological" does not describe itself → "heterological" is not heterological, contradiction

The paradox can be eliminated, without changing the meaning of "heterological" where it was previously well-defined, by modifying the definition of "heterological" slightly to hold all nonautological words except "heterological". But "nonautological" is subject to the same paradox, for which this evasion is not applicable because the rules of English uniquely determine its meaning from that of "autological". A similar slight modification to the definition of "autological" (such as declaring it false of "nonautological" and its synonyms) might seem to correct that, but the paradox still remains for synonyms of "autological" and "heterological" such as "self-descriptive" and "non–self-descriptive", whose meanings also would need adjusting, and the consequences of those adjustments would then need to be pursued, and so on. Freeing English of the Grelling–Nelson paradox entails considerably more modification to the language than mere refinements of the definitions of "autological" and "heterological", which need not even be in the language for the paradox to arise. The scope of these obstacles for English is comparable to that of Russell's paradox for mathematics founded on sets.

Arbitrary cases
One may also ask whether "autological" is autological. It can be chosen consistently to be either:

• if we say that "autological" is autological and then ask whether it applies to itself, then yes, it does, and thus is autological;
• if we say that "autological" is not autological and then ask whether it applies to itself, then no, it does not, and thus is not autological.

This is the opposite of the situation for heterological: while "heterological" logically cannot be autological or heterological, "autological" can be either. (It cannot be both, as the category of autological and heterological cannot overlap.)

In logical terms, the situation for "autological" is:
"autological" is autological if and only if "autological" is autological
A if and only if A, a tautology
while the situation for "heterological" is:
"heterological" is heterological if and only if "heterological" is autological
A if and only if not A, a contradiction.
Ambiguous case
One may also ask whether "loud" is autological or heterological. If said loudly, "loud" is autological; otherwise, it is heterological. This shows that some adjectives cannot be unambiguously classified as autological or heterological. Newhard sought to eliminate this problem by taking Grelling's Paradox to deal specifically with word types as opposed to word tokens.