Is " 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 " is heterological " heterological?
If " is heterological " is heterological it doesn't apply to itself, so " is 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.