Neologism, Paleologisms, and Grelling’s Paradox

Self-proclaimed “Maverick Philosopher” William Vallicella brought to my attention the following paradox:

‘Neologism’ is not a new word, but an old word. Hence, ‘neologism’ is not a neologism. ‘Paleologism’ is not a word at all; or at least it is not listed in the Oxford English Dictionary. But it ought to be, so I hereby introduce it. Who is going to stop me? Having read it and understood it, you have willy-nilly validated its introduction and are complicit with me.

Now that we have ‘paleologism’ on the table, and an unvast conspiracy going, we are in a position to see that ‘neologism’ is a paleologism, while ‘paleologism’ is a neologism. Since the neologism/paleologism classification is both exclusive (every word is either one or the other)and exhaustive (no word is neither), it follows that ‘neologism’ is not a neologism, and ‘paleologism’ is not a paleologism. Such words are called heterological: they are not instances of the properties they express. ‘Useless’ and ‘monosyllabic’ are other examples of heterological expressions in that ‘useless’ is not useless and ‘monosyllabic’ is not monosyllabic. A term that is not heterological is called autological. Examples include ‘short’ and ‘polysyllabic.’ ‘Short’ is short and ‘polysyllabic’ is polysyllabic. Autological terms are instances of the properties they express.

Now ask yourself this question: Is ‘heterological’ heterological? Given that the heterological/autological classification is exhaustive, ‘heterological’ must be either heterological or else autological. Now if the former, then ‘heterological’ is not an instance of the property it expresses, namely, the property of not being an instance of the property it expresses. But this implies that ‘heterological’ is autological. On the other hand, if ‘heterological’ is autological, then it is an instance of the property it expresses, namely the property of not being an instance of the property it expresses. But this implies that ‘heterological’ is heterological.

Therefore, ‘heterological’ is heterological if and only if it is not. This contradiction is known in the trade as Grelling’s Paradox. It is named after Kurt Grelling, who presented it in 1908.

My thoughts on this are as follows:

(1) Heterological means not exemplifying the property it expresses. Autological means exemplifying the property it expresses. So defined, the two are mutually exclusive and jointly exhaustive. As such, every property expressing word must be one or the other.

(2) ‘Heterological’ and ‘autological’ are metaproperties–each is the second-order property of exemplifying some first-order property. Thus, ‘short”s property of being autological supervenes upon its property of being short.

(3) Thus, “exemplifying the property it expresses” doesn’t have any determinate meaning until the referent of “the property it expresses” is fixed. For example, is ‘short” short or not? It depends on what ‘short’ means, that is, upon which property it expresses. If ‘short’ means less than a inch, then ‘short’ is indeed short and autological to boot. But if ‘short’ means less than a millimeter, then ‘short’ is neither short nor autological, but heterological.

(4) As metaproperties, heterologicality and autologicality are properties of exemplifying a certain property, namely, the property that ‘heterological’ and ‘autological’, respectively, express. But in each case that property is also a metaproperty, specifically, itself. This leads to an infinite regress. For just as ‘short’ is neither determinately short or not short until we fix the meaning of the term, so neither are ‘heterological’ and ‘autological’ determinately heterological or autological until we fix the meanings of those terms. But when we try to do so, we find that the meaning is continually deferred. Thus, heterologicality is the property of exemplifying heterologicality, which is the property of exemplifying the property of exemplifying heterologicality, which is the property of exemplifying the property of exemplifying the property of exemplifying heterologicality, and so on. Every time we try to spell out exactly what the metaproperty is we wind up invoking a metaproperty, which just pushes the analysis back a step.

(5) So it looks like there’s something inherently problematic about ascribing metaproperties to metaproperties, for doing so generates an infinite regress. The only way that I can see to stop the regress is to restrict the application of metaproperties to lower-order properties. Thus, nth-order metaproperties cannot apply to nth-order metaproperties, but only to (n-1)th order properties, until eventually we come to some first-order, non-metaproperty.

Russell’s theory of types is lurking in the shadows.