Food for Thought: Paradoxes of Self-Reference

Paradox

Our English language, like all human languages, is full of surprises. One such is Grelling’s Paradox, named after the German logician Kurt Grelling (1886-1942). It begins with the innocuous observation that some English adjectives apply to themselves.

Standard examples include the words “short” (which is itself short), “polysyllabic” (which itself consists of several syllables), and “English” (which is itself English), but it’s easy to come up with further cases, such as “flamboyant,” “outré,” and (arguably) “awkward.” Grelling called such adjectives autological, from the Greek words “auto” for “self” and “logos” for “word.”

Most adjectives, however, do not apply to themselves: The word “French” is not French, “monosyllabic” does not consist of a single syllable, and “liquid” is not liquid. Following Grelling, let’s call such adjectives heterological (the Greek word “hetero” means “different”).

The paradox now arises by asking whether the word “heterological” is heterological. Let’s suppose it is and see what happens. An adjective is heterological just in case it doesn’t apply to itself. So if, as we’re assuming, “heterological” is heterological, it doesn’t apply to itself. But that means “heterological” is not heterological (otherwise it would apply to itself). Ouch! The supposition that “heterological” is heterological must therefore be false, for it has led to the contradictory conclusion that it is also not heterological. But if the supposition that “heterological” is heterological is false, that means the word is not heterological. Wait a minute! The adjectives that fail to be heterological are the autological ones, that is, the ones that apply to themselves. So if “heterological” is not heterological, it is autological and hence applies to itself. But if “heterological” applies to itself, that just means it is heterological after all.

This reasoning has led straight into a contradiction, for it now seems as though “heterological” cannot be heterological, yet at the same time it must be. Philosophers are still debating the sources of Grelling’s Paradox, but no consensus has been reached to date, despite the fact that the paradox is over a hundred years old. Many people have the suspicion that it is somehow illegitimate to let “heterological” loose on itself, since it’s been defined with reference to only those English adjectives that existed before the definition. However, it has proved surprisingly difficult to make that thought both precise and compelling.

Grelling’s Paradox is related to the Liar Paradox, which has perplexed philosophers and logicians to an even greater degree. Consider the sentence “This sentence is false.” Asking whether it is true or false yields catastrophic results. For suppose it is true. Then things are as the sentence says they are, that is, the sentence is false. But it cannot be both true and false, so the supposition of the sentence’s truth is untenable. Thus it must be false. But that’s exactly what the sentence says, and sentences that correctly express what is indeed the case are of course true. So the sentence, in addition to being false, is also true. But that’s absurd. Paradox!

It has been noted that both Grelling’s Paradox and the Liar Paradox crucially depend on what might be called self-reference: The former uses the notion of a word applying to itself, the latter revolves around a sentence that makes reference to itself. It therefore seems likely that the phenomenon of self-reference is somehow to blame for the paradoxical results, but again, it’s proved to be exceedingly difficult to make a rigorous case for this diagnosis. After all, many cases of linguistic self-reference are entirely benign: “This sentence is in English,” for instance, is a perfectly meaningful, true English sentence, and so is “This sentence consists of six words.”

In any case, the paradoxes of self-reference continue to provide philosophers with plenty of food for thought.

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