About Grelling's Paradox

Question

Grelling's Paradox is about certain types of words. An adjective is heterological if it does not describe itself, like 'long', 'German', or 'monosyllabic'. An adjective is autological if it does describe itself, like 'short', 'English', or 'polysyllabic'. So the paradox is:

What is heterological: is it autological or heterological?

Assuming either leads to a contradiction, so a paradox exists if all adjectives can be categorized as autological or heterological.

But what if there are other categories? Could you say an adjective is 'semi-heterological' if its meaning cannot describe all of itself but does describe part of itself? If such a word exists, is it heterological or autological? Does this resolve the paradox?

Answer 1

There is no paradox as long as you realize that, when you say "all adjectives can be categorized as autological or heterological", the "or" is not exclusive.

To explain this a little bit better, let's try to describe this using math syntax. Let's call $f$ a function that returns the relevant description of a word. So $f($"short"$)$ is "short", and $f($"long$)$ is also "short". Then if $f(a)=a$ then $a$ is autological, while if $f(a)\ne a$ then $a$ is heterological.

Using "autological" in this formula works fine because $f($"autological"$)=$ "autological". However, what about $f($"heterological"$)$? If $f($"heterological"$)\ne$ "heterological", then it is heterological, and if $f($"heterological"$)=$ "heterological", then it is autological. This is where the apparent paradox comes in, but what this really suggests is that our function $f$ is poorly defined!

Obviously, words can be described by more than one description - for example, "polysyllabic" is both polysyllabic and long (or medium-sized, if you want). So why not have $f$ return a set of descriptions of the word? This would make an autological word one for which $a\in f(a)$. A heterological word could be defined either as $a\notin f(a)$, or one for which $\neg a\in f(a)$, where $\neg a$ is an antonym of the word. Assuming the first definition of heterological, if "heterological" $\notin f($"heterological"$)$, then "heterological" $\in f($"heterological"$)$, so we should use the second definition.

There's still something a little weird here, but it is no longer self-contradictory. Suppose "heterological" $\notin f($"heterological"$)$. Then it is not autological, and because "autological" $\notin f($"heterological"$)$, it is also not heterological. However, suppose "heterological" $\in f($"heterological"$)$. Then it is autological, and since "autological" $\in f($"heterological"$)$, it is also heterological. The logic in both cases is circular, but neither one actually results in a contradiction.

So while this does not answer the question of whether "heterological" is heterological or not, this does resolve the paradox - it is either both heterological and autological, or neither. I believe that we should consider it to be both, making it an auto-antonym. Also, it suggests that we refine the definition of "heterological" to be more like "it can describe something that it is not".