The n-Category Café

Gongsun Long

Gongsun Long was a Chinese philosopher who lived from around 325 to 250 BC. Besides the better-known Confucian and Taoist schools of Chinese philosophy, another important school at this time was the Mohists, who were more interested in science and logic. Gongsun Long is considered a member of the Mohist-influenced ‘School of Names’: a loose group of logicians, not really a school in any real sense. They are remembered largely for their paradoxes: for example, they independently invented a version of Zeno’s paradox.

As with Heraclitus, most of Gongsun Long’s writings are lost. Joseph Needham [N] has written that this is one of the worst losses of ancient Chinese texts, which in general have survived much better than the Greek ones. The Gongsun Longzi is a text that originally contained 14 of his essays. Now only six survive. The second essay discusses the question “when is a white horse not a horse?”

The White Horse Paradox

When I first heard this ‘paradox’ I didn’t get it: it just seemed strange and silly, not a real paradox. I’m still not sure I get it. But I’ve decided that’s what makes it interesting: it seems to rely on modes of thought, or speech, that are quite alien to me. What counts as a ‘paradox’ is more culturally specific than you might realize.

If a few weeks ago you’d asked me how the paradox goes, I might have said something like this:

A white horse is not a horse, because where there is whiteness, there cannot be horseness, and where there is horseness there cannot be whiteness.

However this is inaccurate because there was no word like ‘whiteness’ (let alone ‘horseness’) in classical Chinese.

Realizing that classical Chinese does not have nouns and adjectives as separate parts of speech may help explain what’s going on here. To get into the mood for this paradox, we shouldn’t think of a horse as a thing to which the predicate ‘whiteness’ applies. We shouldn’t think of the world as consisting of things $x$ and, separately, predicates $P$, which combine to form assertions $P(x)$. Instead, both ‘white’ and ‘horse’ are on more of an equal footing.

I like this idea because it suggests that predicate logic arose in the West thanks to peculiarities of Indo-European grammar that aren’t shared by all languages. This could free us up to have some new ideas.

Here’s how the dialog actually goes. I’ll use Angus Graham’s translation because it tries hard not to wash away the peculiar qualities of classical Chinese:

Is it admissible that white horse is not-horse?

S. It is admissible.

O. Why?

S. ‘Horse’ is used to name the shape; ‘white’ is used to name the color. What names the color is not what names the shape. Therefore I say white horse is not horse.

O. If we take horses having color as nonhorse, since there is no colorless horse in the world, can we say there is no horse in the world?

S. Horse obviously has color, which is why there is white horse. Suppose horse had no color, then there would just be horse, and where would you find white horse. The white is not horse. White horse is white and horse combined. Horse and white is horse, therefore I say white horse is non-horse.

(Chad Hansen writes: “Most commentaries have trouble with the sentence before the conclusion in F-8, “horse and white is horse,” since it appears to contradict the sophist’s intended conclusion. But recall the Mohists asserted that ox-horse both is and is not ox.” I’m not sure if that helps me, but anyway….)

O. If it is horse not yet combined with white which you deem horse, and white not yet combined with horse which you deem white, to compound the name ‘white horse’ for horse and white combined together is to give them when combined their names when uncombined, which is inadmissible. Therefore, I say, it is inadmissible that white horse is not horse.

S. ‘White’ does not fix anything as white; that may be left out of account. ‘White horse’ has ‘white’ fixing something as white; what fixes something as white is not ‘white’. ‘Horse’ neither selects nor excludes any colors, and therefore it can be answered with either yellow or black. ‘White horse’ selects some color and excludes others, and the yellow and the black are both excluded on grounds of color; therefore one may answer it only with white horse. What excludes none is not what excludes some. Therefore I say: white horse is not horse.

One possible anachronistic interpretation of the last passage is

The set of white horses is not equal to the set of horses, so “white horse” is not “horse”.

This makes sense, but it seems like a way of saying we can have $S \subseteq T$ while also $S \ne T$. That would be a worthwhile observation around 300 BC — and it would even be worth trying to get people upset about this, back then! But it doesn’t seem very interesting today.

A more interesting interpretation of the overall dialog is given by Chad Hansen [H]. He argues that to understand it, we should think of both ‘white’ and ‘horse’ as mass nouns or ‘kinds of stuff’.

The issue of how two kinds of stuff can be present in the same place at the same time is a bit challenging — we see Plato battling with it in the Parmenides — and in some sense western mathematics deals with it by switching to a different setup, where we have a universe of entities $x$ of which predicates $P$ can be asserted. If $x$ is a horse and $P$ is ‘being white’, then $P(x)$ says the horse is white.

However, then we get Leibniz’s principle of the ‘indistinguishability of indiscernibles’, which is a way of defining equality. This says that $x = y$ if and only if $P(x) \iff P(y)$ for all predicates $P$. By this account, an entity really amounts to nothing more than the predicates it satisfies!

This is where equality comes in — but as I said, all of this is seeming like too much of a distraction from my overall goals for this essay right now.

Notes

• [N] Joseph Needham, Science and Civilisation in China vol. 2: History of Scientific Thought, Cambridge U. Press, Cambridge, 1956, p. 185.

• [H] Chad Hansen, Mass nouns and “A white horse is not a horse”, Philosophy East and West 26 (1976), 189–209.