雙語暢銷書《艾倫圖靈傳》第5章:解謎接力賽(123)
In a less technical essay written in this period, 'The Reform of Mathematical Notation', he explained that despite all the efforts of Frege and Russell and Hilbert
... mathematics has profited very little from researches in Symbolic logic.
……數學從符號邏輯的研究中獲益甚少。The chief reason for this seems to be a lack of liaison between the logician and the mathematician-in-the-street.
主要的原因似乎就是，邏輯學家和數學家之間缺少一種默契。
Symbolic logic is a very alarming mouthful for most mathematicians, and the logicians are not very much interested in making it more palatable.
對於大部分數學家而言，符號邏輯是令人懼怕的，但對於邏輯學家而言，又沒興趣把它搞得讓人舒服些。His own effort to bridge the gap began with an attempt
艾倫自己進行了一項嘗試，試圖填平這條裂縫：... to put the theory of types into a form in which it can be used by the mathematician-in-the-street without having to study symbolic logic, much less use it.
……要把類型理論轉爲一種形式，使數學家不必學習符號邏輯就能使用。The statement of the type principle given below was suggested by lectures of Wittgenstein, but its shortcomings should not be laid at his door.
下面對類型原理的描述是由維特根斯坦在講座中給出的，但其中的缺陷並不是由他造成的。The type principle is effectively taken care of in ordinary language by the fact that there are nouns as well as adjectives.
自然語言中既有名詞也有謂詞，因此類型原理可以有效地用自然語言來描述。We can make the statement 'All horses are four-legged', which can be verified by examination of every horse, at any rate if there are only a finite number of them.
我們可以給出這樣的命題：所有的馬都有四條腿。對於有限數量的馬，我們可以通過逐個檢查每匹馬來證明。If however we try to use words like 'thing' or 'thing whatever' trouble begins.
但如果我們引入"東西"或"任何東西"這樣的詞，就會帶來麻煩。Suppose we understand 'thing' to include everything whatever, books, cats, men, women, thoughts, functions of men with cats as values, numbers, matrices, classes of classes, procedures, propositions...
假設"東西"包括所有的事物，書、貓、男人、女人、想法、數字、矩陣、過程、命題……Under these circumstances what can we make of the statement 'All things are not prime multiples of 6'.... What do we mean by it?
在這樣的情況下，我們考慮這個命題：任何東西都不是6的質因式。這個命題是什麼意思？我們能從中知道什麼？Under no circumstances is the number of things to be examined finite.
"東西"的數量是無限的，是無法通過檢驗來證明的。
