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This is embarrassing, but let's face it. It is a problem caused by Chinese grammar.
A white horse is not a horse. -- false, but no paradox.
White horses are not horses. -- false, but no paradox.
White horse is not horse -- English grammar forbids this expression.
This example illustrates how language can help clear thinking.
KEY
{philosophy} "a white horse is not a horse" (paradoxical claim by the philosopher Gōngsūn Lóng 公孫龍/公孙龙, 320-250 BCE, in his "White Horse Dialogue" where he tries to establish a conclusion contrary to common sense)
"a white horse is not a horse" is quite paradoxical.
When a white horse is not a horse
When A White Horse Is Not A Horse (Chinese: 白馬非馬; pinyin: Báimǎ fēi mǎ; Wade–Giles: Pai-ma fei ma; literally: "white horse not horse"), also known as the White Horse Dialogue (Chinese: 白馬論; pinyin: Báimǎ Lùn; Wade–Giles: Pai-ma Lun; literally: "white horse discourse"), is a famous paradox in Chinese philosophy. Gongsun Long wrote this circa 300 BCE dialectic analysis of the question Can it be that a white horse is not a horse?.
Wikipedia also contains the following, which is quite similar to the question that you pose about the difficulty of translation into English:
Difficulties of interpretation
The syntax, semantics, and logic of the White Horse Dialogue are ambiguous in the Classical Chinese original and thorny in English translation: for example, in Chinese, 白 (or bai, and so on) has the characteristic of a verb, in English. Thus, to translate, the term bai ma is more precisely rendered as the phrase "be white horse", or as referring to a "being white horse". In the Chinese context, color attributes are verbal rather than adjectival, whereas "horse" is nominal.
The Chinese bai ma fei ma 白馬非馬 syntactically hinges upon the negative fei 非 "not, is not; no, negative; oppose; wrong". The Classical construction "A fei B" "A非B" can ambiguously mean either "A is not a member of the class B" or "A is not identical to B".[2] Interpreting this equivocation fallacy, A.C. Graham says this "white horse" vs. "horse" paradox plays upon the ambiguity of whether "is" means:
"Is a member of the class (x)"
"Is identical to (x)"[2]
In other words, the expression "a white horse is not a horse" is ambiguous between "a white horse is not identical with a horse" (true, because "white horse" is more specific than "horse"), and "a white horse is not a member of the set of horses" (obviously false). The Advocate in the dialogue is asserting that "a white horse is not [identical with] a horse," while the Objector is interpreting the Advocate's statement as "a white horse is not [a member of the group of] horses."
An illustration of the alternative uses of "fei" may be found in the widely known "Happiness of Fish" dialogue in Zhuangzi (17, tr. Watson 1968:188-9). Huizi says "You're not a fish [子非魚] — how do you know what fish enjoy?" (denying that Zhuangzi is a member of the class of fish) and Zhuangzi replies "You're not I [子非我], so how do you know I don't know what fish enjoy?" (denying that the individuals Huizi and Zhuangzi are identical).
Beyond the inherent semantic ambiguities of Baima fei ma, the first line obscurely asks ke hu 可乎 "Can it be that …?". This dialogue could be an attempted proof that a white horse is not a horse, or a question if such a statement is possible, or both. Van Norden suggests "that the issue is not whether it is always true that 'a white horse is not a horse,' but whether it is possible for it to be true."
An alternative interpretation is offered in Feng Youlan's History of Chinese Philosophy (translated in 1952 by Derk Bodde):
Strictly speaking, names or terms are divided into those that are abstract and those that are concrete. The abstract term denotes the universal, the concrete term the particular. The particular is the denotation, and the universal the connotation, of the term. In western inflected languages there is no difficulty in distinguishing between the particular ('white' or 'horse') and the abstract ('whiteness' or 'horseness'). In Chinese, however, owing to the fact that the written characters are ideographic and pictorial and lack all inflection, there is no possible way, as far as the form of individual words is concerned, of distinguishing between abstract and concrete terms. Thus in Chinese the word designating a particular horse and that designating the universal, 'horseness,' are written and pronounced in the same way. Similarly with other terms, so that such words as 'horse' and 'white', being used to designate both the concrete particular and the abstract universal, thus hold two values.[3]
However, there are no contemporary histories of Chinese philosophy that subscribe to Fung's interpretation. Other philosophers and sinologists who have analyzed the dialogue include Hansen;[4] Graham;[5] Thompson;[6] Harbsmeier;[7] and Van Norden.1
Gongsun Long (3rd century BC) is a logician in the School of Names. His famous paradox highlights the difference between an entity ("name") and all collective attributes of that entity, effectively introducing the concept of "set". In logic, ancient or modern, a set and all members (attributes) of that set are not the same. Without distinguishing them, the ambiguity exists in any language -- Chinese or English
The simplest interpretation of 白馬非馬 (A white horse is not a horse) would be
A white horse is not a set of horse
The "set of horse" can of course be an empty set and the statement is clearly true as a white horse belong to a set of horse instead.
Of courses there are a number of different explicit interpretations based on the same idea.
A white horse set is (clearly) not a horse set
The two sets do not overlap completely.
But a set need not to be restricted to merely counting members vs. counting groups.
A (real) white horse (that you can ride on) is not an abstract name of a horse
The statement that "A white horse IS a horse" already assumes "a horse" as abstract. It goes without saying in every language, Chinese or English. Similarly, I can easily say "A horse does not exist" because clearly all that exist are real horses (that I can ride on), and I cannot ride on an abstract calling of a horse.
However, Gongsun Long went a little deeper. In the original of his thesis,
馬者，所以命形也；白者，所以命色也。命色者非名形也。故曰：「白馬非馬」。
"Horse, is a name of form; White, is a name of color. To name a form is not to name a color. Thus, a white horse is not horse"
Thus, besides "set" vs. "members" and "concrete" vs. "abstract", all names (abstract or not) and all their attributes can be of different set (of description) (i.e. of form or of color)... ad infinitum. With this in mind,
A white horse (who lives in the domain of form AND color) is (clearly) not the horse (who (merely) lives in the domain of form)
One can of course be ambiguous in any language. As Wikipedia states, the equivocation here comes from conflating "being an element of" and "being identical to".
Can one equivocate like this in English? I'll posit a few examples:
Could it be that being a white horse is not being a horse?
This turns it into verbal noun phrases, making the two concepts a bit fuzzier.
Could it be that the white horse is not the horse?
This uses a quirk of the English article the to produce the generic, abstract meaning. But I agree it's a little awkward. If you stress the second the, a new interpretation is formed (Is the white horse the "standard/ideal/best" horse?).
It should say 白马非驫, (biao1, all horses). Chinese does have a plural form for horses. Therefore this is not a paradox. It was a historical mistake. A paradox needs to have a clear definition in its language but contradict only in its logic.
We can see that, yes, 白马 is 马, a white horse is a horse.
马非驫, a horse is not a herd of horses.
"White horse" is not "horse".