邏輯愛麗絲

8/24/2020 11:32:00 上午
《愛麗絲夢遊仙境》和《愛麗絲鏡中奇緣》的作者是路易斯卡羅(Lewis Carroll),這可能是眾所周知的事情。較少人知的是,卡羅其實是個厲害的邏輯學家,那兩本書有意無意用到大量邏輯技倆,令不少對話看起來吊詭,但實質內有玄奧。茲舉兩例。

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1. I say what I mean ≠ I mean what I say

在《愛麗絲夢遊仙境》的〈瘋狂茶會〉一章,愛麗絲認為 “I say what I mean” 和 “I mean what I say” 是同一回事,接著瘋帽子(Mad Hatter)、三月兔(March Hare)和睡鼠(Dormouse)就說了三組句子:
“You should say what you mean,” [said the March Hare, reproving Alice sharply.] “I do,” Alice hastily replied; “at least — at least I mean what I say — that’s the same thing, you know.”
“Not the same thing a bit!” said the Hatter. “Why, you might just as well say that ‘I see what I eat’ is the same thing as ‘I eat what I see’!”
“You might just as well say,” added the March Hare, “that ‘I like what I get’ is the same thing as ‘I get what I like’!”
“You might just as well say,” added the Dormouse, which seemed to be talking in its sleep, “that ‘I breathe when I sleep’ is the same thing as ‘I sleep when I breathe’!”
“It is the same thing with you,” said the Hatter, and here the conversation dropped.
卡羅在此處使用的邏輯技巧是「類比反駁法」(refutation by analogy)。愛麗絲最早宣稱:
Alice: I say what I mean = I mean what I say
瘋帽子、三月兔和睡鼠反對愛麗絲,用愛麗絲的說法類比另一個說法,用意是透過另一個說法荒謬,來彰顯愛麗絲的說法同樣荒謬:
Hatter: I see what I eat = I eat what I see
March Hare: I like what I get = I get what I like
Dormouse: I breathe when I sleep = I sleep when I breathe
這三組句子,左右兩句的意思都天差地別。我們雖然會見到自己吃的東西(I see what I eat),但卻不會吃光我們見到的東西(I eat what I see);有人或者能夠喜歡自己擁有的一切(I like what I get),但就連皇帝也無法擁有一切喜歡的事物(I get what I like);正常人睡覺時會呼吸(I breathe when I sleep),唯有植物人才連呼吸時都在睡覺(I sleep when I breathe)。瘋帽子、三月兔和睡鼠所舉的類比,皆用來反駁愛麗絲的說法,由此可見, I say what I mean ≠ I mean what I say 。

這裡的類比反駁法完全仰仗句子的結構。我見過有譯者將 “You should say what you mean” 譯成「那你就應該把話說清楚」、將 “at least I mean what I say” 譯成「至少我把要說的意思表達出來了」(高寶書版)。這個譯法丟失兩句在結構上的關連,連帶瘋帽子、三月兔和睡鼠的反駁都變得莫名其妙,原因要不是譯者粗心大意,就是連譯者也看不懂卡羅耍的技倆。

2. Nobody

“I see what I eat” 和 “I eat what I see” 在邏輯上稱為「換位」 (conversion) 。事實上,這兩句都是換了位的全稱肯定句:
  1. I see what I eat
    凡我吃的都是我見到的
    For every x, if I eat x, then I see x
    ∀x(Ex→Sx)
  2. I eat what I see
    凡我見到的都是我吃的(凡我見到的我都吃)
    For every x, if I see x, then I eat x
    ∀x(Sx→Ex)
卡羅的《符號邏輯I》有不少篇幅討論全稱語句,包括證明全稱肯定句不可換位,即是「所有 X 都是 Y」不可換成「所有 Y 都是 X」。

在邏輯學,「所有 X 都是 Y」是全稱肯定句,「沒有 X 是 Y」是全稱否定句,當中的「所有」和「沒有」都是量詞(quantifier),用來表示數量。卡羅的《鏡中奇緣》便利用量詞營造了一段奇怪的對話,那對話的開首是國王問信差:
“Who did you pass on the road?” the King went on, holding his hand out to the messenger for some hay.
Nobody,” said the messenger.
“Quite right,” said the King; “this young lady saw him too. So of course Nobody walks slower than you.”
信差說他在路上沒有遇到任何人(nobody),用的是量詞,意思是他在路上遇到的人數是零。國王將之曲解成信差在路上遇到 Nobody ,「Nobody」成了人名,信差的意思變成他在路上遇到一個叫「Nobody」的人。信差的量詞被國王扭曲成名詞,兩人驢唇不對馬嘴,不細想只會摸不著頭腦。

這種語言上的 twist 現在也有不少,例如將某個字詞當成名字:


或是利用量詞在自然語言的歧義,推導離奇的結論:
Nothing is better than money
Money is better than philosophy
Therefore, nothing is better than philosophy
這兩個例子我便不多解釋。

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我第一次讀《愛麗絲夢遊仙境》和《鏡中奇緣》,是為了找邏輯課用的例子,因為這兩本書的荒誕對話在邏輯上都有特殊意義。除了小說,卡羅連學術論文也喜歡透過荒誕對話呈現邏輯論點,例如他寫過一篇阿基里斯與龜續集,也寫過一篇討論質料條件句的論文,兩篇都是對話體。就連他那本專論三段論的《符號邏輯I》(例如 Part II, Book X, Chapter II),也不時蹦出一兩頁對話,用來論證某個哲學立場。在他的著作看到這些「怪談」,不妨停下來想想自己可以怎樣解釋趣怪之處,也許會有些意外收獲。
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