Logical Machines

In the "Voyage to Laputa" there is a description of a machine for evolving science automatically. "By this contrivance, the most ignorant person, at a reasonable charge, and with little bodily labor, might write books in philosophy, poetry, politics, laws, mathematics, and theology, without the least assistance from genius or study." The intention is to ridicule the Organon of Aristotle and the Organon of Bacon, by showing the absurdity of supposing that any "instrument" can do the work of the mind. Yet the logical machines of Jevons and Marquand are mills into which the premises are fed and which turn out the conclusions by the revolution of a crank. The numerous mathematical engines that have been found practically useful, from Webb's adder up to Babbage's analytical engine (which was designed though never constructed), are also machines that perform reasoning of no simple kind. Precisely how much of the business of thinking a machine could possibly be made to perform, and what part of it must be left for the living mind, is a question not without conceivable practical importance; the study of it can at any rate not fail to throw needed light on the nature of the reasoning process. Though the instruments of Jevons and of Marquand were designed chiefly to illustrate more elementary points, their utility lies mainly, as it seems to me, in the evidence they afford concerning this problem.

在《勒皮他游记》里，描写了一台能自动产生学问的机器。“靠这台装置，最无知的人只需花一笔合理的费用，出一点体力，就能写出哲学、诗歌、政治、法律、数学和神学方面的书，丝毫不需要天才或学习的帮助。”作者的用意是讥讽亚里士多德的《工具论》和培根的《新工具》，表明以为有什么“工具”能代替心灵工作是多么荒谬。可是，杰文斯和马昆德的逻辑机，正是一些磨坊：把前提喂进去，摇动曲柄，结论就转出来了。那许多已被证明切实有用的数学机械，从韦布的加法器，直到巴贝奇的分析机（它设计出来了，却从未造成），也都是执行着相当复杂的推理的机器。思维这件事，究竟有多少可以由机器来完成，又有哪一部分必须留给活的心灵，这个问题并非不可能有实际的重要性；无论如何，研究它一定能为推理过程的本质提供一些急需的启示。杰文斯和马昆德的仪器虽然主要是为说明一些更基本的论点而设计的，但在我看来，它们的用处主要在于它们为这个问题提供的证据。

we have only to have a connection which can be introduced at will, such that when one event A occurs in the machine, another event B must also occur. This connection being introduced between A and B, and also between B and C, it is necessarily virtually introduced between A and C. This is the same principle which lies at the foundation of every logical algebra; only in the algebra, instead of depending directly on the laws of nature, we establish conventional rules for the relations used. When we perform a reasoning in our unaided minds we do substantially the same thing, that is to say, we construct an image in our fancy under certain general conditions, and observe the result. In this point of view, too, every machine is a reasoning machine, in so much as there are certain relations between its parts, which relations involve other relations that were not expressly intended. A piece of apparatus for performing a physical or chemical experiment is also a reasoning machine, with this difference, that it does not depend on the laws of the human mind, but on the objective reason embodied in the laws of nature. Accordingly, it is no figure of speech to say that the alembics and cucurbits of the chemist are instruments of thought, or logical machines.

我们只需要有一种可以随意建立的联系，使得当机器中发生事件 A 时，事件 B 也必然发生。在 A 和 B 之间、B 和 C 之间都建立了这种联系，那么在 A 和 C 之间实际上也就必然建立了这种联系。这与一切逻辑代数的基础是同一个原理；只不过在代数中，我们不是直接依靠自然规律，而是为所用的关系制定约定的规则。当我们不借助任何工具、在自己心中进行推理时，我们做的基本上也是同样的事，也就是说，我们在想象中按照某些一般条件构造一个形象，然后观察结果。从这个观点看，每一台机器也都是一台推理机器，因为它的各个部件之间存在着某些关系，而这些关系又牵涉到另一些并非有意设计的关系。一套用来做物理或化学实验的仪器，也是一台推理机器，区别只在于它依靠的不是人类心灵的规律，而是体现在自然规律之中的客观理性。因此，说化学家的蒸馏器和葫芦瓶是思维的工具、是逻辑机器，并不是一种修辞。

Every reasoning machine, that is to say, every machine, has two inherent impotencies. In the first place, it is destitute of all originality, of all initiative. It cannot find its own problems; it cannot feed itself. It cannot direct itself between different possible procedures. For example, the simplest proposition of projective geometry, about the ten straight lines in a plane, is proved by von Staudt from a few premises and by reasoning of extreme simplicity, but so complicated is the mode of compounding these premises and forms of inference, that there are no less than 70 or 80 steps in the demonstration. How could we make a machine which would automatically thread its way through such a labyrinth as that? And even if we did succeed in doing so, it would still remain true that the machine would be utterly devoid of original initiative, and would only do the special kind of thing it had been calculated to do. This, however, is no defect in a machine; we do not want it to do its own business, but ours. The difficulty with the balloon, for instance, is that it has too much initiative, that it is not mechanical enough. We no more want an original machine, than a house-builder would want an original journeyman, or an American board of college trustees would hire an original professor. If, however, we will not surrender to the machine, the whole business of initiative is still thrown upon the mind; and this is the principal labor.

每一台推理机器，也就是说每一台机器，都有两种固有的无能。第一，它完全没有独创性，没有主动性。它不能找到自己的问题；它不能自己给自己喂料。它不能在不同的可能步骤之间为自己选择方向。例如，射影几何中最简单的命题，关于平面上十条直线的那个命题，冯·施陶特是从几个前提出发、用极其简单的推理证明的，可是这些前提和推理形式的组合方式如此复杂，证明竟有七八十步之多。我们怎样才能造出一台机器，让它自动地在这样一座迷宫里找到出路呢？而且即使我们成功了，事实仍然是：这台机器完全没有独创的主动性，只会做它被设计来做的那一类特定的事。不过，这对机器来说并不是缺点；我们并不要它做它自己的事，而是要它做我们的事。比如，气球的毛病，就在于它的主动性太多，不够机械。我们不想要一台有独创性的机器，正如一个盖房子的人不想要一个有独创性的帮工，美国的大学董事会不会聘请一位有独创性的教授一样。但是，如果我们不愿把主动权交给机器，那么主动的全部工作就仍然落在心灵身上；而这才是主要的劳动。

In the second place, the capacity of a machine has absolute limitations; it has been contrived to do a certain thing, and it can do nothing else. For instance, the logical machines that have thus far been devised can deal with but a limited number of different letters. The unaided mind is also limited in this as in other respects; but the mind working with a pencil and plenty of paper has no such limitation. It presses on and on, and whatever limits can be assigned to its capacity to-day, may be over-stepped to-morrow. This is what makes algebra the best of all instruments of thought; nothing is too complicated for it. And this great power it owes, above all, to one kind of symbol, the importance of which is frequently entirely overlooked—I mean the parenthesis. We can, of course, dispense with parentheses as such. Instead of (a+b) c=d, we can write a+b=t and tc=d. The letter t is here a transmogrified parenthesis. We see that the power of adding proposition to proposition is in some sort equivalent to the use of a parenthesis.

第二，机器的能力有绝对的限度；它被设计来做某一件事，别的什么也不能做。例如，迄今为止设计出来的逻辑机器，只能处理数目有限的几个不同字母。不借助工具的心灵在这一点上也和在其他方面一样，是有限度的；可是拿着铅笔、有足够多纸张的心灵，却没有这样的限度。它不断向前推进，无论今天能为它的能力划出什么界限，明天都可能被越过。正是这一点，使代数成为一切思维工具中最好的；没有什么东西复杂到它处理不了。而它的这种伟大力量，首先要归功于一种符号，它的重要性常常完全被忽视了——我指的是括号。当然，我们可以不用括号本身。我们可以不写 (a+b) c=d，而写 a+b=t 和 tc=d。这里的字母 t 就是一个变了形的括号。我们看到，把命题与命题相加的能力，在某种意义上就等于使用括号。

I do not think there would be any great difficulty in constructing a machine which should work the logic of relations with a large number of terms. But owing to the great variety of ways in which the same premises can be combined to produce different conclusions in that branch of logic, the machine, in its first state of development, would be no more mechanical than a hand-loom for weaving in many colors with many shuttles. The study of how to pass from such a machine as that to one corresponding to a Jacquard loom, would be likely to do very much for the improvement of logic.

我认为，造一台能用大量的项来处理关系逻辑的机器，不会有什么大的困难。可是在逻辑的这个分支里，同样的前提可以用许许多多不同的方式组合，得出不同的结论，因此这台机器在它发展的初期，并不会比一架用许多梭子织多种颜色的手工织机更机械。研究怎样从这样一台机器过渡到一台相当于雅卡尔织机的机器，很可能会对逻辑的改进大有帮助。