White horses/all proofs

The old Chinese proof that white horses are not horses

You need to speak Chinese to understand this.

“ 熊猫非猫，章鱼非鱼，蜗牛非牛。所以，白马非马。 ”

— Ancient Chinese philosophers

The literal translation is: "Pandas are not cats, octopuses are not fish, snails are not cows. Hence, white horses are not horses." That would not make any sense to those who don't understand the Chinese language, because it is the uniquely stupid Chinese illogic.

Modern proofs that some white horses are horses

First, define an adjective oavdycous: something oavdycous is a horse, a white horse, and existing. By definition, something oavdycous exists. That thing is both a white horse and a horse. Therefore, there exists a white horse that is a horse. Call it A. Then define cawkremerous: something cawkremerous is a horse, a white horse, not oavdycous, and existing. By definition, something cawkremerous exists. Call it B. Then B is both a white horse and a horse. And it is not oavdycous, so it is not the A. Now there are two white horses that are horses. QED.

Alternative proof (Proof by Obviousness)

A white horses is obviously a horse. QED.

Improvement: All white horses are horses

There is a man whose penis is 19.1-inch-long - the largest in the world. His pseudonym is Daniel. He believes that all white horses are horses. Using Proof by Penis Size, we know that what he believes is true. But be careful. There is a considerable risk that some guy on Earth will have an even langer penis. If that happens, this proof will be in danger. Well, actually not. The record-breaker will probably spend all his time trying to stop Daniel from taking more penis enlargement pills or arguing against Daniel on who has the biggest penis. The record-breaker will not have time to think about white horses. So we are still quite safe.

Another proof

The sentence in this purple box is false.

Assume for contradiction that white horses are not horses. Now we need to divide into two cases. In the first case, the sentence in the purple box is true. Then the sentence in the purple box is false. That's a contradiction. In the second case, the sentence in the purple box is false. Then the sentence in the purple box must be true. Contradiction again. Therefore, the initial assumption that white horses are horses is wrong. So white horses are horses. QED.

Yet another

A white horse is an animal. A horse is an animal, so an animal is a horse. So a white horse is an animal which is a horse. QED.

Modern proofs that white horses are not horses

Assume for contradiction that white horses are horses. Then horses can be categorized into white and non-white. Obviously non-white horses exist, so the set of horses has more elements than the set of white horses. Therefore, the set of white horses is not equal to the set of horses. In other words, white horses are not horses. QED.

Alternative proof

In September 2020, an anonymous genius came up with a much simpler proof. You don't need to know set theory to understand this proof.

“ A white horse has eleven letters while a horse only has six. So they are not the same. ”

— Anonymous genius

People suspect that the "anonymous genius" might be Captain Яidiculous.

Another alternative proof

First we write sentences into two boxes. These sentences may be either true or false.

The sentence in the blue box below is false.

The sentence in the green box above is true.

Assume for contradiction that white horses are horses. Now we need to divide into two cases. In the first case, the sentence in the green box is true. Then the sentence in the blue box is false, so the one in the green box must also be false. That's a contradiction. In the second case, the sentence in the green box is false. Then the sentence in the blue box must be true, so the one in the green box must also be true. Contradiction again. Therefore, the initial assumption that white horses are horses is wrong. So white horses are not horses. QED.

Variant

The sentence in this orange box is false.

Assume for contradiction that white horses are horses. Now we need to divide into two cases. In the first case, the sentence in the orange box is true. Then the sentence in the orange box is false. That's a contradiction. In the second case, the sentence in the orange box is false. Then the sentence in the orange box must be true. Contradiction again. Therefore, the initial assumption that white horses are horses is wrong. So white horses are not horses. QED.

One more alternative proof (Proof by Obviousness)

Captain Obvious is a Captain instead of an Obvious. That's obviously true. Also, Lord Redund is a Lord instead of a Redund. In the exact same way, a white horse is a white instead of a horse. QED.

5th proof (Proof by Nonsense Letters)

"Some white horses are horses" can be paraphrased as "white horses that are horses exist". This phrasing is equivalent to the original, but contains a nonsense letter X. So it can't possibly be right. So no white horses are horses. QED. What? "QED" has a nonsense letter? Oh, SHIT!

6th proof

We already have 5 aforementioned proofs that white horses are not horses. So it has to be right. QED.

7th proof (Proof by Soviet Russia)

For the 7th proof, if you doubt whether there are Russian horses that are not white, there is one in this picture. The other horse is white, but nobody cares.

Some horses in Soviet Russia are not white. So, in Soviet Russia, some horses are not white horses. Now we un-reverse it and get: white horses are not some horses.

8th proof

This proof works by proving that a white horse is better than a horse so they can't be the same.

Eternal happiness is better than having just a horse. Nothing is better than eternal happiness. And, of course, having a white horse to ride is better than having nothing. So a white horse is better than nothing, which is better than eternal happiness, which is better than a horse. Therefore a white horse is better than a horse, so a white horse is not the same as a horse. QED.

9th proof (does not really work, but useful in China)

This proof isn't really true, but it can convince your Chinese and/or History teacher if you live in China.

The ancient Chinese philosopher Gongsun Long wrote the White Horse Dialog (Chinese: 白马论) talking about white horses that are not horses. Just cite it. Due to the ridiculous education and religion system in China that overly appreciates traditional Chinese culture, your teacher would have to admit.

10th proof (too long for everyday use)

This proof is too long and complex if you just want to explain to others why you believe that white horses are not horses. But if you want to show others how smart or good at math you are, this will be appropriate for your purpose.

We need two lemmas first.

Lemma 1: <latex>1=2</latex>
Proof: First, let <latex>x=1</latex>. Multiply both sides by <latex>x</latex> and we get <latex>x^2=x</latex>. Subtract both sides by 1 and we get <latex>x^2-1=x-1</latex>. We notice that the left hand side can be factored into <latex>(x-1)(x+1)</latex>. So <latex>(x-1)(x+1)=x-1</latex>. Divide both sides by <latex>(x-1)</latex> and get <latex>x+1=1</latex>. Substituting <latex>x=1</latex> gives the desired result.

Lemma 2: <latex>1=3</latex>
Proof: First, let <latex>x=1</latex>. Using lemma 1, we have <latex>x^2=x^1=1</latex> Multiply both sides by <latex>x</latex> and we get <latex>x^3=x</latex>. Subtract both sides by 1 and we get <latex>x^3-1=x-1</latex>. We notice that the left hand side can be factored into <latex>(x-1)(x^2+x+1)</latex>. So <latex>(x-1)(x^2+x+1)=x-1</latex>. Divide both sides by <latex>(x-1)</latex> and get <latex>x^2+x+1=1</latex>. Substituting <latex>x=1</latex> gives the desired result.

Now back to the problem.
Proof: By Lemma 1, 1=2. So 1=2 or white horses are not horses. (Here the "or" is the logical or.) By lemma 2, 1=3, so 1≠2. Since "1=2 or white horses are not horses" and 1≠2, white horses are not horses.