阿基里斯聽烏龜說


(洪偉) #1

新的一年開始,「康德散步」又開始連載了。不知道大家還記不記得,康德和阿草曾經討論過 Zeno 的「阿基里斯與烏龜」的問題(http://bit.ly/2lxmoBr),阿基里斯追不上烏龜,聽起來滿難過的。

這一次我們要來介紹一個叫作「阿基里斯聽烏龜說」的悖論,這是一個邏輯悖論,由 Lewis Carroll 提出,你可能沒聽過他的名字,但是你一定聽過《愛麗絲夢遊仙境》。

這個邏輯悖論從這個推論開始:

(1) 如果有兩個人都和愛彌爾一樣高,那這兩個人一樣高。
(2) 康德和阿草都跟愛彌爾一樣高。
© 康德和阿草一樣高。

烏龜問說,有沒有人會接受 (1) 和 (2) ,但是不接受 © 。阿基里斯想了想說,有可能有這樣的人。烏龜接著說,那你把我當成這種人,你有辦法說服我接受 © 嗎?

阿基里斯說,可以,我們可以加上這個:

(3) 如果 (1) 和 (2) 都成立,則 © 就會成立。

他問烏龜,你接受 (3) 嗎?烏龜說,我接受。阿基里斯說,那你就會接受 © 啦。但是烏龜又說,不,雖然我可以接受 (1) 、 (2) 、 (3) ,但我就是不接受 © 。

這時候如果阿基里斯為了說服烏龜,他就加上 (4) :

(4) 如果 (1) 、 (2) 、 (3) 都成立,則 © 就會成立。

但即使烏龜接受了 (1) 、(2) 、 (3) 、 (4) ,他好像還是必須要先接受 (5) 才能得出 © :

(5) 如果 (1) 、(2) 、 (3) 、 (4) 都成立,則 © 成立。

但是如果要從 (1) ~ (5) 推出 © 又好像需要 (6) … 這樣一來,烏龜真的有辦法被說服嗎?

這下子阿基里斯更沮喪了。

漫畫

還在想。


(香腸伯) #2

如果是和阿草這隻脖子可長可短的草尼馬比,答案就出來了。lol

另外,這卡羅先生是愛麗思夢遊仙境的作者?


(洪偉) #3

這哏好像有辦法用


(洪偉) #4

真的耶。。。


#5

這裡可以舉例嗎?接受1、2,不接受C的原因是?


(洪偉) #6

大概是單純無法接受 © ,譬如阿草一直相信自己比康德高。


(香腸伯) #7

稍微GOOGLE了一下

What the Tortoise Said to Achilles

http://www.ditext.com/carroll/tortoise.html

裡面用的例子是三角形。方法其實蠻好笑的。

烏龜說:“就算A和B是對的,是不是需要有C證明A和B是對的”

阿基里斯說:“對喔”

阿基里斯就寫說:“A和B和C是對的,假如D是A和B和C則Z對”

烏龜說:“是不是說需要有E證明…”

雖然每個命題都是可以需要用另一個命題保證其正確,但現實上把康德和阿草抓來一比就結案了。

先說我英文挺菜,有錯請指正一下。


(洪偉) #8

還是說我直接把這篇翻出來算了 XD


(洪偉) #9

Reprinted from Lewis Carroll, “What the Tortoise Said to Achilles,” Mind 6, No. 14 (April 1895): 278-280.


Achilles had overtaken the Tortoise, and had seated himself comfortably on its back.

“So you’ve got to the end of our race-course?” said the Tortoise. “Even though it does consist of an infinite series of distances? I thought some wiseacre or other had proved that the thing couldn’t be done?”

“It can be done,” said Achilles. “It has been done! Solvitur ambulando. You see the distances were constantly diminishing; and so --”

“But if they had been constantly increasing?” the Tortoise interrupted “How then?”

“Then I shouldn’t be here,” Achilles modestly replied; “and you would have got several times round the world, by this time!”

“You flatter me – flatten, I mean” said the Tortoise; “for you are a heavy weight, and no mistake! Well now, would you like to hear of a race-course, that most people fancy they can get to the end of in two or three steps, while it really consists of an infinite number of distances, each one longer than the previous one?”

“Very much indeed!” said the Grecian warrior, as he drew from his helmet (few Grecian warriors possessed pockets in those days) an enormous note-book and a pencil. “Proceed! And speak slowly, please! Shorthand isn’t invented yet!”

“That beautiful First Proposition of Euclid!” the Tortoise murmured dreamily. “You admire Euclid?”

“Passionately! So far, at least, as one can admire a treatise that won’t he published for some centuries to come!”

"Well, now, let’s take a little bit of the argument in that First Proposition – just two steps, and the conclusion drawn from them. Kindly enter them in your notebook. And in order to refer to them conveniently, let’s call them A, B, and Z: –

(A) Things that are equal to the same are equal to each other.

(B) The two sides of this Triangle are things that are equal to the same.

(Z) The two sides of this Triangle are equal to each other.

Readers of Euclid will grant, I suppose, that Z follows logically from A and B, so that any one who accepts A and B as true, must accept Z as true?"

“Undoubtedly! The youngest child in a High School – as soon as High Schools are invented, which will not be till some two thousand years later – will grant that.”

“And if some reader had not yet accepted A and B as true, he might still accept the sequence as a valid one, I suppose?”

“No doubt such a reader might exist. He might say ‘I accept as true the Hypothetical Proposition that, if A and B be true, Z must be true; but, I don’t accept A and B as true.’ Such a reader would do wisely in abandoning Euclid, and taking to football.”

“And might there not also he some reader who would say 'I accept A and B as true, but I don’t accept the Hypothetical '?”

“Certainly there might. He, also, had better take to football.”

“And neither of these readers,” the Tortoise continued, “is as yet under any logical necessity to accept Z as true?”

“Quite so,” Achilles assented.

“Well, now, I want you to consider me as a reader of the second kind, and to force me, logically, to accept Z as true.”

"A tortoise playing football would be – " Achilles was beginning

“-- an anomaly, of course,” the Tortoise hastily interrupted. “Don’t wander from the point. Let’s have Z first, and football afterwards!”

“I’m to force you to accept Z, am I?” Achilles said musingly. “And your present position is that you accept A and B, but you don’t accept the Hypothetical --”

“Let’s call it C,” said the Tortoise.

"-- but you don’t accept

© If A and B are true, Z must be true. "

“That is my present position,” said the Tortoise.

“Then I must ask you to accept C.”

“I’ll do so,” said the Tortoise, “as soon as you’ve entered it in that note-book of yours. What else have you got in it?”

“Only a few memoranda,” said Achilles, nervously fluttering the leaves: “a few memoranda of – of the battles in which I have distinguished myself!”

“Plenty of blank leaves, I see!” the Tortoise cheerily remarked. “We shall need them all!” (Achilles shuddered.) "Now write as I dictate: –

(A) Things that arc equal to the same are equal to each other.

(B) The two sides of this Triangle are things that are equal to the same.

© If A and B are true, Z must be true.

(Z) The two sides of this Triangle are equal to each other."

“You should call it D, not Z,” said Achilles. “It comes next to the other three. If you accept A and B and C, you must accept Z.”

“And why must I?”

“Because it follows logically from them. If A and B and C are true, Z must be true. You don’t dispute that, I imagine?”

“If A and B and C are true, Z must he true,” the Tortoise thoughtfully repeated. “That’s another Hypothetical, isn’t it? And, if I failed to see its truth, I might accept A and B and C’, and still not accept Z. mightn’t I?”

“You might,” the candid hero admitted; “though such obtuseness would certainly be phenomenal. Still, the event is possible. So I must ask you to grant one more Hypothetical.”

"Very good. I’m quite willing to grant it, as soon as you’ve written it down. We will call it

(D) If A and B and C are true, Z must be true.

“Have you entered that in your notebook?”

“I have!” Achilles joyfully exclaimed, as he ran the pencil into its sheath. “And at last we’ve got to the end of this ideal race-course! Now that you accept A and B and C and D, of course you accept Z.”

“Do I?” said the Tortoise innocently. “Let’s make that quite clear. I accept A and B and C and D. Suppose I still refused to accept Z?”

“Then Logic would force you to do it!” Achilles triumphantly replied. “Logic would tell you ‘You can’t help yourself. Now that you’ve accepted A and B and C and D, you must accept Z!’ So you’ve no choice, you see.”

“Whatever Logic is good enough to tell me is worth writing down,” said the Tortoise. "So enter it in your book, please. We will call it

(E) If A and B and C and D are true, Z must be true. Until I’ve granted that, of course I needn’t grant Z. So it’s quite a necessary step, you see?"

“I see,” said Achilles; and there was a touch of sadness in his tone.

Here narrator, having pressing business at the Bank, was obliged to leave the happy pair, and did not again pass the spot until some months afterwards. When he did so, Achilles was still seated on the back of the much-enduring Tortoise, and was writing in his note-book, which appeared to be nearly full. The Tortoise was saying, “Have you got that last step written down? Unless I’ve lost count, that makes a thousand and one. There are several millions more to come. And would you mind, as a personal favour, considering what a lot of instruction this colloquy of ours will provide for the Logicians of the Nineteenth Century – would you mind adopting a pun that my cousin the Mock-Turtle will then make, and allowing yourself to be re-named Taught-Us?”

“As you please!” replied the weary warrior, in the hollow tones of despair, as he buried his face in his hands. “Provided that you, for your part, will adopt a pun the Mock-Turtle never made, and allow yourself to be re-named A Kill-Ease!”


Transcribed into hypertext by Andrew Chrucky, July 10, 1997.