數學 1854

對思維規律的研究

喬治·布林

邏輯是一種代數——把推理化成服從定律的符號。

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In depth · the introduction

喬治·布林有一個既古怪又強大的念頭：邏輯的規則，可以像代數裡的算式一樣，被寫下來、被計算出來。

把這個想法拆開看

我們通常把邏輯——「所有 A 都是 B」「這個並且那個」「要麼……要麼……」——當作關於詞語和縝密思考的事。布林看出，它可以被做成數學。讓一個字母代表一個類別（比如 x =「白色的東西」），再讓類別的組合，遵循看上去就像代數的規則：「x 並且 y」成了一種乘法，x 乘以 y；「非 x」成了 1 減去 x；萬物構成的全域是 1，而空無一物是 0。

讓這一切成立的竅門是：他的字母永遠只取兩個值，0 或 1——關或開，假或真。在白色的東西裡再挑出白色的東西，挑來挑去還是那些白色的東西，所以 x 乘以 x 等於 x。這條不起眼的事實——在尋常算術裡除了 0 和 1 之外都不成立——正是一整套可以拿來計算的、完整而精確的邏輯的種子。

它從哪裡來

布林是一個幾乎全靠自學的英國人，鞋匠的兒子，從未拿過自己的大學學位，卻當上了愛爾蘭科克市皇后學院的數學教授。他在 1847 年一本薄薄的小冊子裡勾畫了這個想法；1854 年，他在這本書裡把它和盤托出，而它那宏大的書名——《對思維規律的研究》——宣告著他的信念：他找到了推理本身背後的數學。他的朋友奧古斯都·德摩根，當時也正追逐著相關的想法。布林英年早逝，1864 年去世，年僅 49 歲，從未看見他的代數會變成什麼。

它為何重要

兩千年來，邏輯一直是亞里斯多德的——縝密，卻是言辭的，也難以推廣。布林把它變成了可以用符號寫下、用計算求解的東西，讓邏輯向數學的全部力量敞開。這一步——把推理當作計算——是思想史上最深刻的步伐之一，它為後來的一切埋下了伏筆：從數理邏輯，到計算機的設計。

電燈開關

想像兩個開關，接到同一盞燈上。每個開關，關時記作 0，開時記作 1。把它們一前一後串成一排，只有兩個都開，燈才亮——這就是布林的「與」，x 乘以 y。把它們並排接上，任一個開，燈就亮——這就是他的「或」。一個會把信號反過來的開關，就是「非」。布林在 1854 年算出這套代數時，心裡並沒有任何機器；一個世紀後，它竟恰好描述了計算機內部的電路是怎麼「思考」的。撥動下面的開關，看著邏輯發生。

它落在哪裡

布林站在本館「推理」這條故事線的一處樞紐上。在他身後，是歐幾里得——證明了真理可以靠純粹的邏輯步驟推導出來；在他前方，是哥德爾——探問形式系統的極限，以及克勞德·香農——他的工作，把布林的二值代數徑直帶進了數位時代。當你用「與」和「或」做檢索，當任何一台計算機把兩個數相加，你用的，正是布林在這本書裡寫下的代數。

The original document

Original source text

George Boole · An Investigation of the Laws of Thought · London: Walton and Maberly, 1854

Chapter I — Nature and design of this work

The design of the following treatise is to investigate the fundamental laws of those operations of the mind by which reasoning is performed; to give expression to them in the symbolical language of a Calculus, and upon this foundation to establish the science of Logic and construct its method; to make that method itself the basis of a general method for the application of the mathematical doctrine of Probabilities; and, finally, to collect from the various elements of truth brought to view in the course of these inquiries some probable intimations concerning the nature and constitution of the human mind.

The universe of discourse: 0 and 1

Boole lets a literal symbol such as x stand for a class — say, all white things — and writes xy for the things that are at once x and y. He fixes the bounds of the discussion with two limiting symbols: 0 for Nothing, and 1 for the Universe — the class “comprehending every conceivable class of objects whether actually existing or not.” Every other class is some selection within that 1.

The fundamental law of thought

Because a class selected twice over is the same class — the white things among the white things are simply the white things — the symbols obey x·x = x, which Boole writes x² = x. In ordinary algebra only 0 and 1 satisfy that equation; that is the hinge of the whole book.

Let us conceive, then, of an Algebra in which the symbols x, y, z, &c. admit indifferently of the values 0 and 1, and of these values alone.

That axiom of Metaphysicians which is termed the principle of contradiction, and which affirms that it is impossible for anything to possess a quality, and in the same time not to possess it, is a consequence of the fundamental law of thought, whose expression is x² = x.

[ … ]

The second half of the treatise carries the same symbolic method into the theory of probabilities, treating a probability as a value between 0 and 1 and deriving the probability of compound events from the logical relations among them.

Queen's College, Cork · 1854