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?

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    $\begingroup$ "So a paradox exists if all adjectives can be categorized as autological or heterological." So then I guess not all adjectives can be categorized as autological or heterological! Maybe this is less of a paradox and more of a contradiction resulting from that assumption. To whatever degree it is a paradox, it seems like it's just a complex wrapper around "this sentence is a lie". $\endgroup$ Sep 18 '14 at 15:08
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    $\begingroup$ It's Russell's paradox. $\endgroup$
    – phs
    Sep 18 '14 at 22:55

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".


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