An Investigation of the Laws of Thought — Chapter I: Nature and Design of this Work

Chapter I. Nature and Design of this Work第一章　本书的性质与宗旨#

1. 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.

1．本书的宗旨在于：考察心智借以进行推理的那些运作的基本法则；用一种演算的符号语言来表达它们，并以此为基础确立逻辑科学、建构其方法；使这一方法本身成为应用数学概率论的一般方法的基础；最后，从这些探究过程中所揭示的种种真理要素里，汇集一些关于人类心智之本性与构造的或然的启示。

3. But although certain parts of the design of this work have been entertained by others, its general conception, its method, and, to a considerable extent, its results, are believed to be original. For this reason I shall offer, in the present chapter, some preparatory statements and explanations, in order that the real aim of this treatise may be understood, and the treatment of its subject facilitated.

3．不过，虽然本书宗旨的某些部分前人也曾抱有，但它的总体构想、它的方法，以及在相当程度上它的结果，我相信都是原创的。因此，我将在本章中先作一些预备性的陈述和说明，以便读者理解本书真正的目标，也便于对其主题的论述。

It is designed, in the first place, to investigate the fundamental laws of those operations of the mind by which reasoning is performed. It is unnecessary to enter here into any argument to prove that the operations of the mind are in a certain real sense subject to laws, and that a science of the mind is therefore possible. If these are questions which admit of doubt, that doubt is not to be met by an endeavour to settle the point of dispute à priori, but by directing the attention of the objector to the evidence of actual laws, by referring him to an actual science. And thus the solution of that doubt would belong not to the introduction to this treatise, but to the treatise itself. Let the assumption be granted, that a science of the intellectual powers is possible, and let us for a moment consider how the knowledge of it is to be obtained.

本书的第一项宗旨，是考察心智借以进行推理的那些运作的基本法则。这里无须展开任何论证来证明：心智的运作在某种真实的意义上服从法则，因而一门关于心智的科学是可能的。如果这些问题容许怀疑，那么回应这种怀疑的办法，不是试图先验地解决争执之点，而是让提出异议的人去注意实际法则的证据，把他引向一门实际存在的科学。这样一来，解决那个疑问的任务就不属于本书的导言，而属于本书本身了。让我们姑且承认这个假定，即一门关于理智能力的科学是可能的；再让我们略为考虑一下，这门科学的知识要如何获得。

4. Like all other sciences, that of the intellectual operations must primarily rest upon observation,—the subject of such observation being the very operations and processes of which we desire to determine the laws. But while the necessity of a foundation in experience is thus a condition common to all sciences, there are some special differences between the modes in which this principle becomes available for the determination of general truths when the subject of inquiry is the mind, and when the subject is external nature. To these it is necessary to direct attention.

4．同其他一切科学一样，关于理智运作的科学首先必须建立在观察之上——这种观察的对象，正是我们想要确定其法则的那些运作和过程本身。然而，以经验为基础虽是一切科学共同的条件，但当探究的对象是心智、与当探究的对象是外部自然时，这一原则被用来确定一般真理的方式之间，却存在某些特殊的差别。我们有必要注意这些差别。

5. But if the general truths of Logic are of such a nature that when presented to the mind they at once command assent, wherein consists the difficulty of constructing the Science of Logic? Not, it may be answered, in collecting the materials of knowledge, but in discriminating their nature, and determining their mutual place and relation. All sciences consist of general truths, but of those truths some only are primary and fundamental, others are secondary and derived. The laws of elliptic motion, discovered by Kepler, are general truths in astronomy, but they are not its fundamental truths. And it is so also in the purely mathematical sciences. An almost boundless diversity of theorems, which are known, and an infinite possibility of others, as yet unknown, rest together upon the foundation of a few simple axioms; and yet these are all general truths. It may be added, that they are truths which to an intelligence sufficiently refined would shine forth in their own unborrowed light, without the need of those connecting links of thought, those steps of wearisome and often painful deduction, by which the knowledge of them is actually acquired. Let us define as fundamental those laws and principles from which all other general truths of science may be deduced, and into which they may all be again resolved. Shall we then err in regarding that as the true science of Logic which, laying down certain elementary laws, confirmed by the very testimony of the mind, permits us thence to deduce, by uniform processes, the entire chain of its secondary consequences, and furnishes, for its practical applications, methods of perfect generality? Let it be considered whether in any science, viewed either as a system of truth or as the foundation of a practical art, there can properly be any other test of the completeness and the fundamental character of its laws, than the completeness of its system of derived truths, and the generality of the methods which it serves to establish. Other questions may indeed present themselves. Convenience, prescription, individual preference, may urge their claims and deserve attention. But as respects the question of what constitutes science in its abstract integrity, I apprehend that no other considerations than the above are properly of any value.

5．但如果逻辑的一般真理具有这样的性质——一旦呈现于心智，便立刻令人信服——那么，建构逻辑科学的困难究竟在哪里？可以这样回答：困难不在于收集知识的材料，而在于辨明它们的性质，确定它们相互之间的位置和关系。一切科学都由一般真理构成，但在这些真理中，只有一部分是首要的、根本的，其余的则是次要的、派生的。开普勒发现的椭圆运动定律，是天文学中的一般真理，却不是它的根本真理。在纯数学科学中也是如此。几乎无穷多样的已知定理，以及无限可能的、尚属未知的其他定理，共同建立在少数几条简单公理的基础之上；而它们全都是一般真理。还可以补充一句：对于一个足够精微的智能来说，这些真理会以它们自身的、并非借来的光芒照耀出来，而无需那些思想的连接环节，无需那些令人厌倦、往往令人痛苦的推演步骤——而我们实际上正是通过这些步骤才获得对它们的认识的。让我们把这样的法则和原理定义为根本的：科学的其他一切一般真理都能从中推导出来，也都能重新归结为它们。那么，如果我们把这样一种逻辑视为真正的逻辑科学——它确立某些为心智本身所证实的基本法则，允许我们由此以统一的程序推导出其次级推论的整个链条，并为其实际应用提供完全一般的方法——我们会错吗？请考虑一下：在任何一门科学中，无论把它看作一个真理体系，还是看作一门实用技艺的基础，除了其派生真理体系的完备性、以及它所确立的方法的一般性之外，还能有什么别的标准，来恰当地检验其法则是否完备、是否根本？当然，其他问题也可能出现。方便、惯例、个人偏好，也可能提出各自的要求，值得关注。但就“什么构成抽象完整意义上的科学”这一问题而言，我认为，除上述考虑之外，其他考虑都没有真正的价值。

6. It is designed, in the next place, to give expression in this treatise to the fundamental laws of reasoning in the symbolical language of a Calculus. Upon this head it will suffice to say, that those laws are such as to suggest this mode of expression, and to give to it a peculiar and exclusive fitness for the ends in view. There is not only a close analogy between the operations of the mind in general reasoning and its operations in the particular science of Algebra, but there is to a considerable extent an exact agreement in the laws by which the two classes of operations are conducted. Of course the laws must in both cases be determined independently; any formal agreement between them can only be established à posteriori by actual comparison. To borrow the notation of the science of Number, and then assume that in its new application the laws by which its use is governed will remain unchanged, would be mere hypothesis. There exist, indeed, certain general principles founded in the very nature of language, by which the use of symbols, which are but the elements of scientific language, is determined. To a certain extent these elements are arbitrary. Their interpretation is purely conventional: we are permitted to employ them in whatever sense we please. But this permission is limited by two indispensable conditions,—first, that from the sense once conventionally established we never, in the same process of reasoning, depart; secondly, that the laws by which the process is conducted be founded exclusively upon the above fixed sense or meaning of the symbols employed. In accordance with these principles, any agreement which may be established between the laws of the symbols of Logic and those of Algebra can but issue in an agreement of processes. The two provinces of interpretation remain apart and independent, each subject to its own laws and conditions.

6．其次，本书旨在用一种演算的符号语言来表达推理的基本法则。在这一点上只须说明：这些法则本身就提示了这种表达方式，并使它对于所要达到的目的具有一种特殊的、独一无二的适宜性。在一般推理中心智的运作，与它在代数这门特殊科学中的运作之间，不仅有着密切的类比，而且两类运作所遵循的法则在相当大的程度上是精确一致的。当然，两种情况下的法则都必须各自独立地确定；它们之间的任何形式上的一致，都只能通过实际比较，后验地加以确立。借用数的科学的记号，然后假定在新的应用中支配其使用的法则会保持不变，那就只是一种假设。确实存在某些以语言本身的性质为基础的一般原则，它们决定着符号——符号不过是科学语言的元素——的使用。在一定程度上，这些元素是任意的。对它们的解释纯属约定：我们可以按自己喜欢的任何意义使用它们。但这种许可受到两个必不可少的条件的限制：第一，在同一个推理过程中，我们绝不偏离一经约定的意义；第二，进行推理所依据的法则，必须完全建立在所用符号的上述固定意义之上。按照这些原则，在逻辑符号的法则与代数符号的法则之间所能确立的任何一致，都只能归结为程序上的一致。两个解释的领域仍然彼此分离、各自独立，各受其自身的法则和条件支配。

7. The principle has already been laid down (5) that the sufficiency and truly fundamental character of any assumed system of laws in the science of Logic must partly be seen in the perfection of the methods to which they conduct us. It remains, then, to consider what the requirements of a general method in Logic are, and how far they are fulfilled in the system of the present work.

7．前面（第 5 节）已经确立了这样一条原则：逻辑科学中任何一个假定的法则体系是否充分、是否真正根本，必须部分地从它引导我们达到的方法是否完善中看出来。因此，接下来要考虑的是：逻辑中的一般方法有哪些要求，以及本书的体系在多大程度上满足了这些要求。

Logic is conversant with two kinds of relations,—relations among things, and relations among facts. But as facts are expressed by propositions, the latter species of relation may, at least for the purposes of Logic, be resolved into a relation among propositions. The assertion that the fact or event A is an invariable consequent of the fact or event B may, to this extent at least, be regarded as equivalent to the assertion, that the truth of the proposition affirming the occurrence of the event B always implies the truth of the proposition affirming the occurrence of the event A. Instead, then, of saying that Logic is conversant with relations among things and relations among facts, we are permitted to say that it is concerned with relations among things and relations among propositions. Of the former kind of relations we have an example in the proposition—“All men are mortal;” of the latter kind in the proposition—“If the sun is totally eclipsed, the stars will become visible.” The one expresses a relation between “men” and “mortal beings,” the other between the elementary propositions—“The sun is totally eclipsed;” “The stars will become visible.” Among such relations I suppose to be included those which affirm or deny existence with respect to things, and those which affirm or deny truth with respect to propositions. Now let those things or those propositions among which relation is expressed be termed the elements of the propositions by which such relation is expressed. Proceeding from this definition, we may then say that the premises of any logical argument express given relations among certain elements, and that the conclusion must express an implied relation among those elements, or among a part of them, i.e. a relation implied by or inferentially involved in the premises.

逻辑处理两类关系：事物之间的关系，以及事实之间的关系。但由于事实是由命题来表达的，后一类关系至少就逻辑的目的而言，可以归结为命题之间的关系。断言事实或事件 A 是事实或事件 B 的恒常后果，至少在这个范围内，可以看作等同于这样的断言：肯定事件 B 发生的那个命题为真，总是蕴涵着肯定事件 A 发生的那个命题为真。因此，与其说逻辑处理事物之间的关系和事实之间的关系，我们不妨说它所关心的是事物之间的关系和命题之间的关系。前一类关系的例子，是命题“所有的人都会死”；后一类关系的例子，是命题“如果发生日全食，星星就会变得可见”。前者表达的是“人”与“会死的存在物”之间的关系，后者表达的是两个基本命题——“发生了日全食”与“星星会变得可见”——之间的关系。我认为，这类关系也包括那些就事物肯定或否定其存在、就命题肯定或否定其为真的关系。现在，让我们把那些被表达出关系的事物或命题，称为表达这种关系的命题的元素。从这个定义出发，我们就可以说：任何逻辑论证的前提，表达的是某些元素之间给定的关系；而结论必须表达这些元素、或其中一部分元素之间的一种蕴涵的关系，也就是由前提所蕴涵、或以推论的方式包含在前提之中的关系。

8. Now this being premised, the requirements of a general method in Logic seem to be the following:—

8．有了这些前提，逻辑中一般方法的要求似乎如下：——

1st. As the conclusion must express a relation among the whole or among a part of the elements involved in the premises, it is requisite that we should possess the means of eliminating those elements which we desire not to appear in the conclusion, and of determining the whole amount of relation implied by the premises among the elements which we wish to retain. Those elements which do not present themselves in the conclusion are, in the language of the common Logic, called middle terms; and the species of elimination exemplified in treatises on Logic consists in deducing from two propositions, containing a common element or middle term, a conclusion connecting the two remaining terms. But the problem of elimination, as contemplated in this work, possesses a much wider scope. It proposes not merely the elimination of one middle term from two propositions, but the elimination generally of middle terms from propositions, without regard to the number of either of them, or to the nature of their connexion. To this object neither the processes of Logic nor those of Algebra, in their actual state, present any strict parallel. In the latter science the problem of elimination is known to be limited in the following manner:—From two equations we can eliminate one symbol of quantity; from three equations two symbols; and, generally, from n equations n-1 symbols. But though this condition, necessary in Algebra, seems to prevail in the existing Logic also, it has no essential place in Logic as a science. There, no relation whatever can be proved to prevail between the number of terms to be eliminated and the number of propositions from which the elimination is to be effected. From the equation representing a single proposition, any number of symbols representing terms or elements in Logic may be eliminated; and from any number of equations representing propositions, one or any other number of symbols of this kind may be eliminated in a similar manner. For such elimination there exists one general process applicable to all cases. This is one of the many remarkable consequences of that distinguishing law of the symbols of Logic, to which attention has been already directed.

第一，由于结论必须表达前提所涉元素的全部或一部分之间的关系，我们就必须掌握这样的手段：既能消去那些我们不希望出现在结论中的元素，又能确定前提在我们希望保留的元素之间所蕴涵的全部关系。那些不出现在结论中的元素，用普通逻辑的语言来说，叫做中项；逻辑论著中所示范的那种消去，就是从两个含有一个共同元素即中项的命题中，推出一个把其余两项联结起来的结论。但本书所设想的消去问题，范围要广得多。它所提出的，不仅是从两个命题中消去一个中项，而是一般地从命题中消去中项，不论两者各有多少，也不论它们之间是何种联系。对于这一目标，无论是逻辑的程序还是代数的程序，就其现状而言，都没有提供任何严格的对应物。众所周知，在代数中，消去问题受到如下限制：从两个方程中，我们可以消去一个数量符号；从三个方程中，可以消去两个；一般地，从 n 个方程中，可以消去 n-1 个。然而，这一条件虽然在代数中是必要的，似乎也通行于现有的逻辑，在作为科学的逻辑中却没有本质的地位。在那里，要消去的项的数目与据以进行消去的命题的数目之间，不能证明存在任何关系。从代表单个命题的方程中，可以消去任意多个代表逻辑中的项或元素的符号；从代表命题的任意多个方程中，也可以同样地消去一个或任意其他数目的这类符号。对于这样的消去，存在一个适用于一切情形的一般程序。这是逻辑符号那条与众不同的法则——前面已经提请读者注意过它——所带来的许多显著后果之一。

2ndly. It should be within the province of a general method in Logic to express the final relation among the elements of the conclusion by any admissible kind of proposition, or in any selected order of terms. Among varieties of kind we may reckon those which logicians have designated by the terms categorical, hypothetical, disjunctive, &c. To a choice or selection in the order of the terms, we may refer whatsoever is dependent upon the appearance of particular elements in the subject or in the predicate, in the antecedent or in the consequent, of that proposition which forms the “conclusion.” But waiving the language of the schools, let us consider what really distinct species of problems may present themselves to our notice. We have seen that the elements of the final or inferred relation may either be things or propositions. Suppose the former case; then it might be required to deduce from the premises a definition or description of some one thing, or class of things, constituting an element of the conclusion in terms of the other things involved in it. Or we might form the conception of some thing or class of things, involving more than one of the elements of the conclusion, and require its expression in terms of the other elements. Again, suppose the elements retained in the conclusion to be propositions, we might desire to ascertain such points as the following, viz., Whether, in virtue of the premises, any of those propositions, taken singly, are true or false?—Whether particular combinations of them are true or false?—Whether, assuming a particular proposition to be true, any consequences will follow, and if so, what consequences, with respect to the other propositions?—Whether any particular condition being assumed with reference to certain of the propositions, any consequences, and what consequences, will follow with respect to the others? and so on. I say that these are general questions, which it should fall within the scope or province of a general method in Logic to solve. Perhaps we might include them all under this one statement of the final problem of practical Logic. Given a set of premises expressing relations among certain elements, whether things or propositions: required explicitly the whole relation consequent among any of those elements under any proposed conditions, and in any proposed form. That this problem, under all its aspects, is resolvable, will hereafter appear. But it is not for the sake of noticing this fact, that the above inquiry into the nature and the functions of a general method in Logic has been introduced. It is necessary that the reader should apprehend what are the specific ends of the investigation upon which we are entering, as well as the principles which are to guide us to the attainment of them.

第二，逻辑中的一般方法应当能够用任何可容许的种类的命题，或按照任何选定的项的次序，来表达结论诸元素之间的最终关系。在种类的差别中，我们可以算上逻辑学家们称为直言的、假言的、选言的等等那些命题。至于项的次序的选择，凡是取决于特定元素出现在构成“结论”的那个命题的主项还是谓项、前件还是后件之中的东西，都可以归入其中。但是，撇开经院的术语不谈，让我们来考虑一下，究竟有哪些真正不同的问题种类可能出现在我们面前。我们已经看到，最终的、即推出的关系，其元素可以是事物，也可以是命题。假定是前一种情形；那么，可能要求我们从前提中推出某一事物或某一类事物——它构成结论的一个元素——的定义或描述，并用结论中所涉及的其他事物来表达。或者，我们可以形成某个涉及结论中不止一个元素的事物或事物类的概念，并要求用其他元素来表达它。再假定结论中保留的元素是命题，那么我们可能想要弄清如下这些问题，即：凭借前提，这些命题中是否有哪一个单独来看是真的或假的？——它们的某些特定组合是真的还是假的？——假定某一特定命题为真，对于其他命题会不会推出什么后果？如果会，推出什么后果？——假定对其中某些命题设定某种特定条件，对于其余命题会不会推出什么后果？推出什么后果？如此等等。我要说，这些都是一般性的问题，应当属于逻辑中一般方法的范围或职分之内，由它来解决。也许我们可以把它们全都包括在对实用逻辑最终问题的如下一个陈述之中：给定一组表达某些元素——无论是事物还是命题——之间关系的前提；求：在任何提出的条件下、以任何提出的形式，明确表达出这些元素中任何元素之间由此推出的全部关系。这个问题在其所有方面都是可解的，这一点将在后文中看到。但上面之所以要考察逻辑中一般方法的性质和功能，并不是为了指出这一事实。读者有必要领会我们即将着手的这项研究的具体目的何在，以及引导我们达到这些目的的原则是什么。

9. Possibly it may here be said that the Logic of Aristotle, in its rules of syllogism and conversion, sets forth the elementary processes of which all reasoning consists, and that beyond these there is neither scope nor occasion for a general method. I have no desire to point out the defects of the common Logic, nor do I wish to refer to it any further than is necessary, in order to place in its true light the nature of the present treatise. With this end alone in view, I would remark:—1st. That syllogism, conversion, &c., are not the ultimate processes of Logic. It will be shown in this treatise that they are founded upon, and are resolvable into, ulterior and more simple processes which constitute the real elements of method in Logic. Nor is it true in fact that all inference is reducible to the particular forms of syllogism and conversion.—Vide Chap. xv. 2ndly. If all inference were reducible to these two processes (and it has been maintained that it is reducible to syllogism alone), there would still exist the same necessity for a general method. For it would still be requisite to determine in what order the processes should succeed each other, as well as their particular nature, in order that the desired relation should be obtained. By the desired relation I mean that full relation which, in virtue of the premises, connects any elements selected out of the premises at will, and which, moreover, expresses that relation in any desired form and order. If we may judge from the mathematical sciences, which are the most perfect examples of method known, this directive function of Method constitutes its chief office and distinction. The fundamental processes of arithmetic, for instance, are in themselves but the elements of a possible science. To assign their nature is the first business of its method, but to arrange their succession is its subsequent and higher function. In the more complex examples of logical deduction, and especially in those which form a basis for the solution of difficult questions in the theory of Probabilities, the aid of a directive method, such as a Calculus alone can supply, is indispensable.

9．这里或许有人会说：亚里士多德的逻辑，在其三段论规则和换位规则中，已经陈述了构成一切推理的基本程序，除此之外，一般方法既无余地，也无必要。我无意指摘普通逻辑的缺陷，也不想在必要的限度之外更多地提到它；提到它，只是为了让本书的性质得到真实的呈现。仅仅出于这个目的，我想指出：第一，三段论、换位等等并不是逻辑的终极程序。本书将会表明，它们建立在一些更深一层、更为简单的程序之上，也可以归结为这些程序，而后者才构成逻辑中方法的真正元素。事实上，也并非一切推论都可以归结为三段论和换位这些特定形式。——参见第十五章。第二，即使一切推论都可以归结为这两种程序（也有人主张，一切推论都可以只归结为三段论），一般方法也仍然同样必要。因为为了得到所要求的关系，我们仍然需要确定这些程序应当以怎样的次序相继进行，以及它们各自的具体性质。我所说的所要求的关系，是指这样一种完整的关系：它凭借前提，把从前提中随意选出的任何元素联结起来，并且还能以任何所要求的形式和次序来表达这种关系。如果我们可以从数学科学——已知方法最完善的典范——来判断，那么方法的这种指导性功能，正是它的主要职责和特征所在。例如，算术的基本程序本身，不过是一门可能的科学的元素。指明它们的性质，是这门科学的方法的首要任务；而安排它们的先后次序，则是它随后的、更高的职能。在逻辑推演较为复杂的例子中，尤其是在那些为解决概率论中的难题提供基础的例子中，一种只有演算才能提供的指导性方法的帮助，是不可或缺的。

10. Whence it is that the ultimate laws of Logic are mathematical in their form; why they are, except in a single point, identical with the general laws of Number; and why in that particular point they differ;—are questions upon which it might not be very remote from presumption to endeavour to pronounce a positive judgment. Probably they lie beyond the reach of our limited faculties. It may, perhaps, be permitted to the mind to attain a knowledge of the laws to which it is itself subject, without its being also given to it to understand their ground and origin, or even, except in a very limited degree, to comprehend their fitness for their end, as compared with other and conceivable systems of law. Such knowledge is, indeed, unnecessary for the ends of science, which properly concerns itself with what is, and seeks not for grounds of preference or reasons of appointment. These considerations furnish a sufficient answer to all protests against the exhibition of Logic in the form of a Calculus. It is not because we choose to assign to it such a mode of manifestation, but because the ultimate laws of thought render that mode possible, and prescribe its character, and forbid, as it would seem, the perfect manifestation of the science in any other form, that such a mode demands adoption. It is to be remembered that it is the business of science not to create laws, but to discover them. We do not originate the constitution of our own minds, greatly as it may be in our power to modify their character. And as the laws of the human intellect do not depend upon our will, so the forms of the science, of which they constitute the basis, are in all essential regards independent of individual choice.

10．逻辑的终极法则为什么在形式上是数学的；为什么它们除了一点之外，与数的一般法则完全相同；又为什么恰恰在这一点上与之不同——对这些问题，试图作出肯定的判断，恐怕离僭妄也不太远了。它们大概超出了我们有限的官能所能达到的范围。心智也许被允许认识它自身所服从的法则，却未必同时被赋予理解这些法则之根据与起源的能力，甚至——除非在非常有限的程度上——也未必能够领会：与其他可以设想的法则体系相比，这些法则为何适合于它们的目的。这样的知识对于科学的目的确实是不必要的，因为科学本来只关心“是什么”，而不去寻求偏爱的理由或安排的缘由。这些考虑，足以回答一切反对以演算形式展示逻辑的抗议。之所以必须采用这样一种方式，并不是因为我们自己选择赋予它这样一种显现的方式，而是因为思维的终极法则使这种方式成为可能，规定了它的性质，而且看来也不允许这门科学以任何其他形式得到完善的展现。应当记住，科学的任务不是创造法则，而是发现法则。我们并不创造自己心智的构造，尽管我们或许有很大的能力改变它的品性。正如人类理智的法则不取决于我们的意志，以这些法则为基础的这门科学的形式，在一切本质方面也都不取决于个人的选择。

11. Beside the general statement of the principles of the above method, this treatise will exhibit its application to the analysis of a considerable variety of propositions, and of trains of propositions constituting the premises of demonstrative arguments. These examples have been selected from various writers, they differ greatly in complexity, and they embrace a wide range of subjects. Though in this particular respect it may appear to some that too great a latitude of choice has been exercised, I do not deem it necessary to offer any apology upon this account. That Logic, as a science, is susceptible of very wide applications is admitted; but it is equally certain that its ultimate forms and processes are mathematical. Any objection à priori which may therefore be supposed to lie against the adoption of such forms and processes in the discussion of a problem of morals or of general philosophy must be founded upon misapprehension or false analogy. It is not of the essence of mathematics to be conversant with the ideas of number and quantity. Whether as a general habit of mind it would be desirable to apply symbolical processes to moral argument, is another question. Possibly, as I have elsewhere observed, the perfection of the method of Logic may be chiefly valuable as an evidence of the speculative truth of its principles. To supersede the employment of common reasoning, or to subject it to the rigour of technical forms, would be the last desire of one who knows the value of that intellectual toil and warfare which imparts to the mind an athletic vigour, and teaches it to contend with difficulties, and to rely upon itself in emergencies. Nevertheless, cases may arise in which the value of a scientific procedure, even in those things which fall confessedly under the ordinary dominion of the reason, may be felt and acknowledged. Some examples of this kind will be found in the present work.

11．除了对上述方法的原理作一般性的陈述之外，本书还将展示这一方法在分析相当多样的命题，以及构成证明性论证之前提的命题系列时的应用。这些例子选自不同的作者，在复杂程度上相差很大，涉及的题材范围也很广。在这一点上，有些人也许会觉得选择的范围放得太宽，但我认为不必为此作任何辩解。逻辑作为一门科学可以有非常广泛的应用，这是人们承认的；但同样确定的是，它的终极形式和程序是数学的。因此，如果有人以为在讨论道德问题或一般哲学问题时采用这样的形式和程序会有什么先验的反对理由，那么这种反对必定建立在误解或错误的类比之上。处理数和量的观念，并不是数学的本质。至于作为一种一般的心智习惯，把符号程序应用于道德论证是否可取，那是另一个问题。也许正如我在别处说过的，逻辑方法的完善，其主要价值或许在于为其原理在思辨上的真确性提供证据。要取代普通推理的运用，或者使它服从技术形式的严格约束，这将是任何一个懂得那种理智劳作与搏斗之价值的人最不愿意看到的事——正是这种劳作与搏斗赋予心智以运动员般的活力，教它与困难较量，在危急时依靠自己。然而，即使在那些公认属于理性日常统辖的事情上，也可能出现某些情形，使人感到并承认科学程序的价值。本书中会有一些这样的例子。

12. The general doctrine and method of Logic above explained form also the basis of a theory and corresponding method of Probabilities. Accordingly, the development of such a theory and method, upon the above principles, will constitute a distinct object of the present treatise. Of the nature of this application it may be desirable to give here some account, more especially as regards the character of the solutions to which it leads. In connexion with this object some further detail will be requisite concerning the forms in which the results of the logical analysis are presented.

12．上面所阐明的逻辑的一般学说和方法，也构成一种概率理论及其相应方法的基础。因此，依据上述原理来发展这样一种理论和方法，将是本书的又一个独立目标。关于这种应用的性质，这里不妨作一些说明，尤其是关于它所导致的解答具有何种特点。与这一目标相联系，还需要进一步详细说明逻辑分析的结果是以什么形式呈现的。

The ground of this necessity of a prior method in Logic, as the basis of a theory of Probabilities, may be stated in a few words. Before we can determine the mode in which the expected frequency of occurrence of a particular event is dependent upon the known frequency of occurrence of any other events, we must be acquainted with the mutual dependence of the events themselves. Speaking technically, we must be able to express the event whose probability is sought, as a function of the events whose probabilities are given. Now this explicit determination belongs in all instances to the department of Logic. Probability, however, in its mathematical acceptation, admits of numerical measurement. Hence the subject of Probabilities belongs equally to the science of Number and to that of Logic. In recognising the co-ordinate existence of both these elements, the present treatise differs from all previous ones; and as this difference not only affects the question of the possibility of the solution of problems in a large number of instances, but also introduces new and important elements into the solutions obtained, I deem it necessary to state here, at some length, the peculiar consequences of the theory developed in the following pages.

之所以必须先有一种逻辑方法作为概率理论的基础，理由可以用几句话说明。在我们能够确定某一特定事件预期发生的频率如何依赖于其他事件已知的发生频率之前，我们必须了解这些事件本身之间的相互依赖。用专门的话说，我们必须能够把所求概率的那个事件，表示为概率已知的那些事件的函数。而这种明确的确定，在一切情况下都属于逻辑的领域。然而，概率在其数学的意义上，是可以用数值度量的。因此，概率这门学科同等地属于数的科学和逻辑的科学。本书承认这两种要素并列共存，这是它与以往一切论著的不同之处；而由于这一不同不仅影响到大量情形中问题能否得到解答，还给所得的解答带来了新的、重要的要素，我认为有必要在这里较为详细地说明后文所发展的理论所特有的结论。

13. The measure of the probability of an event is usually defined as a fraction, of which the numerator represents the number of cases favourable to the event, and the denominator the whole number of cases favourable and unfavourable; all cases being supposed equally likely to happen. That definition is adopted in the present work. At the same time it is shown that there is another aspect of the subject (shortly to be referred to) which might equally be regarded as fundamental, and which would actually lead to the same system of methods and conclusions. It may be added, that so far as the received conclusions of the theory of Probabilities extend, and so far as they are consequences of its fundamental definitions, they do not differ from the results (supposed to be equally correct in inference) of the method of this work.

13．一个事件的概率的量度，通常被定义为一个分数：分子代表对该事件有利的情形的数目，分母代表有利和不利的情形的总数；所有情形都被假定为同等可能发生。本书采用这一定义。同时也将表明，这门学科还有另一个方面（下文即将提到），它同样可以被看作是根本的，并且实际上会导出同一套方法和结论。还可以补充说，在概率论公认的结论所及的范围内，就这些结论是其基本定义的推论而言，它们与本书方法所得的结果（假定在推理上同样正确）并无不同。

Again, although questions in the theory of Probabilities present themselves under various aspects, and may be variously modified by algebraical and other conditions, there seems to be one general type to which all such questions, or so much of each of them as truly belongs to the theory of Probabilities, may be referred. Considered with reference to the data and the quæsitum, that type may be described as follows:—1st. The data are the probabilities of one or more given events, each probability being either that of the absolute fulfilment of the event to which it relates, or the probability of its fulfilment under given supposed conditions. 2ndly. The quæsitum, or object sought, is the probability of the fulfilment, absolutely or conditionally, of some other event differing in expression from those in the data, but more or less involving the same elements. As concerns the data, they are either causally given,—as when the probability of a particular throw of a die is deduced from a knowledge of the constitution of the piece,—or they are derived from observation of repeated instances of the success or failure of events. In the latter case the probability of an event may be defined as the limit toward which the ratio of the favourable to the whole number of observed cases approaches (the uniformity of nature being presupposed) as the observations are indefinitely continued. Lastly, as concerns the nature or relation of the events in question, an important distinction remains. Those events are either simple or compound. By a compound event is meant one of which the expression in language, or the conception in thought, depends upon the expression or the conception of other events, which, in relation to it, may be regarded as simple events. To say “it rains,” or to say “it thunders,” is to express the occurrence of a simple event; but to say “it rains and thunders,” or to say “it either rains or thunders,” is to express that of a compound event. For the expression of that event depends upon the elementary expressions, “it rains,” “it thunders.” The criterion of simple events is not, therefore, any supposed simplicity in their nature. It is founded solely on the mode of their expression in language or conception in thought.

其次，概率论中的问题虽然以种种不同的面貌出现，也可能因代数的或其他的条件而有种种变化，但似乎存在一个普遍的类型，所有这类问题——或者说每个问题中真正属于概率论的那一部分——都可以归入其中。就已知与所求而言，这个类型可以描述如下：第一，已知的是一个或多个给定事件的概率，每一个概率或者是它所涉及的那个事件绝对实现的概率，或者是该事件在给定的假设条件下实现的概率。第二，所求，即所要寻求的对象，是另外某个事件绝对地或有条件地实现的概率；这个事件在表达上不同于已知中的那些事件，但或多或少涉及同样的元素。就已知而言，它们或者是由原因给定的——例如根据对一枚骰子构造的了解，推出掷出某一点数的概率——或者是从对事件成败的反复观察中得出的。在后一种情形下，一个事件的概率可以定义为：随着观察无限地继续下去，有利情形数与所观察的全部情形数之比所趋近的极限（以自然的齐一性为前提）。最后，就所涉事件的性质或关系而言，还有一个重要的区别：这些事件或者是简单的，或者是复合的。所谓复合事件，是指这样一种事件：它在语言中的表达，或在思想中的构想，依赖于其他事件的表达或构想，而相对于它而言，那些其他事件可以被看作简单事件。说“下雨了”，或说“打雷了”，是表达一个简单事件的发生；而说“又下雨又打雷”，或说“不是下雨就是打雷”，则是表达一个复合事件的发生。因为这一事件的表达依赖于“下雨了”“打雷了”这些基本的表达。所以，简单事件的标准，并不在于它们的本性有什么假定的简单性，而仅仅在于它们在语言中的表达方式，或在思想中的构想方式。

14. Now one general problem, which the existing theory of Probabilities enables us to solve, is the following, viz.:—Given the probabilities of any simple events: required the probability of a given compound event, i.e. of an event compounded in a given manner out of the given simple events. The problem can also be solved when the compound event, whose probability is required, is subjected to given conditions, i.e. to conditions dependent also in a given manner on the given simple events. Beside this general problem, there exist also particular problems of which the principle of solution is known. Various questions relating to causes and effects can be solved by known methods under the particular hypothesis that the causes are mutually exclusive, but apparently not otherwise. Beyond this it is not clear that any advance has been made toward the solution of what may be regarded as the general problem of the science, viz.: Given the probabilities of any events, simple or compound, conditioned or unconditioned: required the probability of any other event equally arbitrary in expression and conception. In the statement of this question it is not even postulated that the events whose probabilities are given, and the one whose probability is sought, should involve some common elements, because it is the office of a method to determine whether the data of a problem are sufficient for the end in view, and to indicate, when they are not so, wherein the deficiency consists.

14．现有的概率论使我们能够解决的一个一般问题如下：给定任何简单事件的概率；求一个给定的复合事件的概率，即由给定的简单事件以给定方式复合而成的事件的概率。当所求概率的复合事件受到给定条件——即同样以给定方式依赖于给定简单事件的条件——的限制时，这个问题也可以解决。除了这个一般问题之外，还有一些特殊问题，其解法的原理也是已知的。关于原因和结果的各种问题，在原因彼此互斥这一特殊假设下，可以用已知的方法解决，但在其他情况下显然不能。除此之外，对于可以看作这门科学之一般问题的那个问题，似乎还看不出已经取得了什么进展；这个问题就是：给定任何事件的概率，无论简单的还是复合的，有条件的还是无条件的；求任何其他事件的概率，这个事件在表达和构想上同样是任意的。在这个问题的陈述中，甚至没有要求概率已知的事件与所求概率的事件之间必须涉及某些共同的元素，因为确定一个问题的已知数据对于所要达到的目的是否充分，并在不充分时指出不足之处何在，这正是方法的职责。

This problem, in the most unrestricted form of its statement, is resolvable by the method of the present treatise; or, to speak more precisely, its theoretical solution is completely given, and its practical solution is brought to depend only upon processes purely mathematical, such as the resolution and analysis of equations. The order and character of the general solution may be thus described.

这个问题，即使以其最不受限制的形式来陈述，也可以用本书的方法解决；或者更确切地说，它的理论解答已经完全给出，而它的实际解答只取决于纯粹数学的程序，例如方程的求解和分析。这个一般解答的步骤和性质，可以描述如下。

15. In the first place it is always possible, by the preliminary method of the Calculus of Logic, to express the event whose probability is sought as a logical function of the events whose probabilities are given. The result is of the following character: Suppose that X represents the event whose probability is sought, A, B, C, &c. the events whose probabilities are given, those events being either simple or compound. Then the whole relation of the event X to the events A, B, C, &c. is deduced in the form of what mathematicians term a development, consisting, in the most general case, of four distinct classes of terms. By the first class are expressed those combinations of the events A, B, C, which both necessarily accompany and necessarily indicate the occurrence of the event X; by the second class, those combinations which necessarily accompany, but do not necessarily imply, the occurrence of the event X; by the third class, those combinations whose occurrence in connexion with the event X is impossible, but not otherwise impossible; by the fourth class, those combinations whose occurrence is impossible under any circumstances. I shall not dwell upon this statement of the result of the logical analysis of the problem, further than to remark that the elements which it presents are precisely those by which the expectation of the event X, as dependent upon our knowledge of the events A, B, C, is, or alone can be, affected. General reasoning would verify this conclusion; but general reasoning would not usually avail to disentangle the complicated web of events and circumstances from which the solution above described must be evolved. The attainment of this object constitutes the first step towards the complete solution of the question proposed. It is to be noted that thus far the process of solution is logical, i. e. conducted by symbols of logical significance, and resulting in an equation interpretable into a proposition. Let this result be termed the final logical equation.

15．首先，借助逻辑演算的预备方法，总是可以把所求概率的事件，表示为概率已知的那些事件的一个逻辑函数。结果具有如下性质：设 X 代表所求概率的事件，A、B、C 等代表概率已知的事件，这些事件或者是简单的，或者是复合的。那么，事件 X 与事件 A、B、C 等之间的全部关系，就以数学家所谓展开式的形式推导出来；在最一般的情形下，展开式由四类不同的项组成。第一类项表示事件 A、B、C 的这样一些组合：它们既必然伴随事件 X 的发生，又必然标示事件 X 的发生；第二类项表示这样一些组合：它们必然伴随事件 X 的发生，但并不必然蕴涵它；第三类项表示这样一些组合：它们与事件 X 一同发生是不可能的，但在其他情况下并非不可能；第四类项表示在任何情况下都不可能发生的组合。对于问题的逻辑分析所得的这一结果，我不再多加陈述，只想指出：它所呈现的这些元素，恰恰就是我们对事件 X 的预期——就其依赖于我们对事件 A、B、C 的知识而言——所受影响、或者说唯一能受影响的那些元素。一般推理可以验证这个结论；但一般推理通常无法解开那张由事件和情况交织而成的复杂之网，而上述解答正必须从这张网中推演出来。达到这一目标，构成了完整解决所提问题的第一步。应当注意，到此为止，求解的过程是逻辑的，也就是说，是用具有逻辑意义的符号进行的，其结果是一个可以解释为一个命题的方程。让我们把这个结果称为最终逻辑方程。

The second step of the process deserves attentive remark. From the final logical equation to which the previous step has conducted us, are deduced, by inspection, a series of algebraic equations implicitly involving the complete solution of the problem proposed. Of the mode in which this transition is effected let it suffice to say, that there exists a definite relation between the laws by which the probabilities of events are expressed as algebraic functions of the probabilities of other events upon which they depend, and the laws by which the logical connexion of the events is itself expressed. This relation, like the other coincidences of formal law which have been referred to, is not founded upon hypothesis, but is made known to us by observation (I.4), and reflection. If, however, its reality were assumed à priori as the basis of the very definition of Probability, strict deduction would thence lead us to the received numerical definition as a necessary consequence. The Theory of Probabilities stands, as it has already been remarked (I.12), in equally close relation to Logic and to Arithmetic; and it is indifferent, so far as results are concerned, whether we regard it as springing out of the latter of these sciences, or as founded in the mutual relations which connect the two together.

这个过程的第二步值得仔细注意。从前一步引导我们得到的最终逻辑方程出发，只须观察，就可以推出一系列代数方程，它们隐含着所提问题的完整解答。关于这一过渡是如何实现的，只须这样说：一方面是把事件的概率表示为它们所依赖的其他事件之概率的代数函数所依据的法则，另一方面是表达这些事件之间逻辑联系本身所依据的法则，两者之间存在着确定的关系。这一关系，同前面提到的其他形式法则上的吻合一样，并不是建立在假说之上的，而是通过观察（第一章第 4 节）和反思为我们所知的。然而，如果把它的实在性先验地假定下来，作为概率定义本身的基础，那么严格的演绎就会由此引导我们得出公认的数值定义，作为一个必然的推论。正如前面已经指出的（第一章第 12 节），概率论与逻辑和算术同样密切相关；就结果而言，无论我们把它看作出自后者，还是看作建立在联结这两门科学的相互关系之上，都无关紧要。

16. There are some circumstances, interesting perhaps to the mathematician, attending the general solutions deduced by the above method, which it may be desirable to notice.

16．用上述方法推出的一般解答，还伴有一些情况，或许会引起数学家的兴趣，这里不妨一提。

1st. As the method is independent of the number and the nature of the data, it continues to be applicable when the latter are insufficient to render determinate the value sought. When such is the case, the final expression of the solution will contain terms with arbitrary constant coefficients. To such terms there will correspond terms in the final logical equation (I. 15), the interpretation of which will inform us what new data are requisite in order to determine the values of those constants, and thus render the numerical solution complete. If such data are not to be obtained, we can still, by giving to the constants their limiting values 0 and 1, determine the limits within which the probability sought must lie independently of all further experience. When the event whose probability is sought is quite independent of those whose probabilities are given, the limits thus obtained for its value will be 0 and 1, as it is evident that they ought to be, and the interpretation of the constants will only lead to a re-statement of the original problem.

第一，由于这一方法不依赖于已知数据的数目和性质，当数据不足以使所求的值确定时，它仍然适用。在这种情况下，解答的最终表达式中将包含带有任意常数系数的项。与这些项相对应，最终逻辑方程（第一章第 15 节）中也会有一些项，对它们的解释将告诉我们，为了确定这些常数的值、从而使数值解答完整，还需要哪些新的数据。如果这样的数据无法获得，我们仍然可以通过给这些常数赋以其极限值 0 和 1，来确定所求概率必定落在其中的界限，而这些界限与一切进一步的经验无关。当所求概率的事件与概率已知的那些事件完全无关时，这样得到的界限将是 0 和 1——显然理应如此——而对常数的解释只会导致对原问题的重新陈述。

2ndly. The expression of the final solution will in all cases involve a particular element of quantity, determinable by the solution of an algebraic equation. Now when that equation is of an elevated degree, a difficulty may seem to arise as to the selection of the proper root. There are, indeed, cases in which both the elements given and the element sought are so obviously restricted by necessary conditions that no choice remains. But in complex instances the discovery of such conditions, by unassisted force of reasoning, would be hopeless. A distinct method is requisite for this end,—a method which might not inappropriately be termed the Calculus of Statistical Conditions. Into the nature of this method I shall not here further enter than to say, that, like the previous method, it is based upon the employment of the “final logical equation,” and that it definitely assigns, 1st, the conditions which must be fulfilled among the numerical elements of the data, in order that the problem may be real, i.e. derived from a possible experience; 2ndly, the numerical limits, within which the probability sought must have been confined, if, instead of being determined by theory, it had been deduced directly by observation from the same system of phænomena from which the data were derived. It is clear that these limits will be actual limits of the probability sought. Now, on supposing the data subject to the conditions above assigned to them, it appears in every instance which I have examined that there exists one root, and only one root, of the final algebraic equation which is subject to the required limitations. Every source of ambiguity is thus removed. It would even seem that new truths relating to the theory of algebraic equations are thus incidentally brought to light. It is remarkable that the special element of quantity, to which the previous discussion relates, depends only upon the data, and not at all upon the quæsitum of the problem proposed. Hence the solution of each particular problem unties the knot of difficulty for a system of problems, viz., for that system of problems which is marked by the possession of common data, independently of the nature of their quæsita. This circumstance is important whenever from a particular system of data it is required to deduce a series of connected conclusions. And it further gives to the solutions of particular problems that character of relationship, derived from their dependence upon a central and fundamental unity, which not unfrequently marks the application of general methods.

第二，最终解答的表达式在一切情况下都会涉及一个特定的数量元素，它可以通过解一个代数方程来确定。当这个方程次数较高时，在选择正确的根这一点上，似乎会出现困难。诚然，在有些情况下，已知元素和所求元素都受到必要条件如此明显的限制，以致没有选择的余地。但在复杂的情形中，单凭推理之力去发现这样的条件，是没有希望的。为此需要一种独特的方法——这种方法称之为“统计条件演算”，或许并无不当。对于这种方法的性质，我在这里不再深入，只想说明：同前一种方法一样，它的基础是对“最终逻辑方程”的运用；它明确地给出：第一，已知数据的数值元素之间必须满足的条件，以使问题是真实的，即来自一种可能的经验；第二，假如所求概率不是由理论确定，而是从提供已知数据的同一个现象系统中通过观察直接推出，那么它必定被限制在其中的数值界限。显然，这些界限将是所求概率的实际界限。现在，假定已知数据满足上面为它们规定的条件，那么在我所考察过的每一个实例中，最终代数方程都有一个根、而且只有一个根满足所要求的限制。这样，一切歧义的来源都被消除了。甚至看来，关于代数方程理论的一些新真理也因此附带地被揭示出来。值得注意的是，前面讨论所涉及的那个特殊数量元素，只取决于所提问题的已知，而完全不取决于它的所求。因此，每一个特殊问题的解答，都为一整个问题系统解开了难结，那就是：以拥有共同的已知数据为标志、而不论其所求为何的那个问题系统。每当需要从一个特定的已知数据系统推出一系列相互联系的结论时，这一情况就很重要。而且，它还赋予各个特殊问题的解答以一种亲缘关系的性质——这种性质来自它们对一个中心的、根本的统一体的依赖——而一般方法的应用，往往不乏这样的标志。

17. But though the above considerations, with others of a like nature, justify the assertion that the method of this treatise, for the solution of questions in the theory of Probabilities, is a general method, it does not thence follow that we are relieved in all cases from the necessity of recourse to hypothetical grounds. It has been observed that a solution may consist entirely of terms affected by arbitrary constant coefficients,—may, in fact, be wholly indefinite. The application of the method of this work to some of the most important questions within its range would—were the data of experience alone employed—present results of this character. To obtain a definite solution it is necessary, in such cases, to have recourse to hypotheses possessing more or less of independent probability, but incapable of exact verification. Generally speaking, such hypotheses will differ from the immediate results of experience in partaking of a logical rather than of a numerical character; in prescribing the conditions under which phænomena occur, rather than assigning the relative frequency of their occurrence. This circumstance is, however, unimportant. Whatever their nature may be, the hypotheses assumed must thenceforth be regarded as belonging to the actual data, although tending, as is obvious, to give to the solution itself somewhat of a hypothetical character. With this understanding as to the possible sources of the data actually employed, the method is perfectly general, but for the correctness of the hypothetical elements introduced it is of course no more responsible than for the correctness of the numerical data derived from experience.

17．不过，上述考虑以及其他类似的考虑，虽然证明了这样的断言是正当的，即本书解决概率论问题的方法是一种一般方法，但并不能由此推出：我们在一切情况下都免除了诉诸假说性根据的必要。前面已经指出，一个解答可能完全由带有任意常数系数的项组成——事实上，可能是完全不确定的。把本书的方法应用于其适用范围内一些最重要的问题时，如果只使用经验数据，就会得出这种性质的结果。在这种情况下，要得到一个确定的解答，就必须诉诸一些假说，它们或多或少具有独立的可能性，却无法得到精确的验证。一般说来，这样的假说与经验的直接结果的不同之处在于：它们带有逻辑的而非数值的性质；它们规定的是现象发生的条件，而不是给出现象发生的相对频率。然而，这一情况并不重要。无论这些假说的性质如何，一旦被采用，就必须从此被看作属于实际数据的一部分，尽管显然它们会使解答本身带有某种假说的性质。在对实际所用数据的可能来源作了这样的理解之后，这一方法是完全一般的；但对于所引入的假说性元素是否正确，它当然不负责任，正如它对于从经验得来的数值数据是否正确不负责任一样。

In illustration of these remarks we may observe that the theory of the reduction of astronomical observations rests, in part, upon hypothetical grounds. It assumes certain positions as to the nature of error, the equal probabilities of its occurrence in the form of excess or defect, &c., without which it would be impossible to obtain any definite conclusions from a system of conflicting observations. But granting such positions as the above, the residue of the investigation falls strictly within the province of the theory of Probabilities. Similar observations apply to the important problem which proposes to deduce from the records of the majorities of a deliberative assembly the mean probability of correct judgment in one of its members. If the method of this treatise be applied to the mere numerical data, the solution obtained is of that wholly indefinite kind above described. And to show in a more eminent degree the insufficiency of those data by themselves, the interpretation of the arbitrary constants (I. 16) which appear in the solution, merely produces a re-statement of the original problem. Admitting, however, the hypothesis of the independent formation of opinion in the individual mind, either absolutely, as in the speculations of Laplace and Poisson, or under limitations imposed by the actual data, as will be seen in this treatise, Chap. XXI., the problem assumes a far more definite character. It will be manifest that the ulterior value of the theory of Probabilities must depend very much upon the correct formation of such mediate hypotheses, where the purely experimental data are insufficient for definite solution, and where that further experience indicated by the interpretation of the final logical equation is unattainable. Upon the other hand, an undue readiness to form hypotheses in subjects which from their very nature are placed beyond human ken, must re-act upon the credit of the theory of Probabilities, and tend to throw doubt in the general mind over its most legitimate conclusions.

为说明这些论点，我们可以指出，天文观测的归算理论部分地建立在假说性的根据之上。它对误差的性质、误差以过量或不足的形式出现的概率相等等等，作了某些假定；没有这些假定，就不可能从一组相互冲突的观测中得出任何确定的结论。但是，一旦承认了上述这类假定，研究的其余部分就严格地落在概率论的范围之内了。类似的意见也适用于另一个重要问题：从一个议事会议的多数表决记录中，推出其成员之一作出正确判断的平均概率。如果把本书的方法应用于单纯的数值数据，所得的解答就属于上面所说的那种完全不确定的类型。而更为突出地显示这些数据本身之不足的是：对解答中出现的任意常数（第一章第 16 节）的解释，只会产生对原问题的重新陈述。然而，如果承认个人心中的意见是独立形成的这一假说——无论是像拉普拉斯和泊松的思辨中那样绝对地承认，还是像本书第二十一章将会看到的那样，在实际数据所施加的限制下承认——这个问题就会具有确定得多的性质。显然，在纯粹的实验数据不足以给出确定的解答，而对最终逻辑方程的解释所指示的那种进一步的经验又无法获得的地方，概率论更深一层的价值，必定在很大程度上取决于能否正确地构成这样的中介性假说。另一方面，在那些就其本性而言处于人类认识范围之外的题目上过于轻率地构造假说，必定会反过来损害概率论的信誉，并使一般人对它最正当的结论也产生怀疑。

18. It would, perhaps, be premature to speculate here upon the question whether the methods of abstract science are likely at any future day to render service in the investigation of social problems at all commensurate with those which they have rendered in various departments of physical inquiry. An attempt to resolve this question upon pure à priori grounds of reasoning would be very likely to mislead us. For example, the consideration of human free-agency would seem at first sight to preclude the idea that the movements of the social system should ever manifest that character of orderly evolution which we are prepared to expect under the reign of a physical necessity. Yet already do the researches of the statist reveal to us facts at variance with such an anticipation. Thus the records of crime and pauperism present a degree of regularity unknown in regions in which the disturbing influence of human wants and passions is unfelt. On the other hand, the distemperature of seasons, the eruption of volcanoes, the spread of blight in the vegetable, or of epidemic maladies in the animal kingdom, things apparently or chiefly the product of natural causes, refuse to be submitted to regular and apprehensible laws. “Fickle as the wind,” is a proverbial expression. Reflection upon these points teaches us in some degree to correct our earlier judgments. We learn that we are not to expect, under the dominion of necessity, an order perceptible to human observation, unless the play of its producing causes is sufficiently simple; nor, on the other hand, to deem that free agency in the individual is inconsistent with regularity in the motions of the system of which he forms a component unit. Human freedom stands out as an apparent fact of our consciousness, while it is also, I conceive, a highly probable deduction of analogy (Chap, XXII.) from the nature of that portion of the mind whose scientific constitution we are able to investigate. But whether accepted as a fact reposing on consciousness, or as a conclusion sanctioned by the reason, it must be so interpreted as not to conflict with an established result of observation, viz.: that phænomena, in the production of which large masses of men are concerned, do actually exhibit a very remarkable degree of regularity, enabling us to collect in each succeeding age the elements upon which the estimate of its state and progress, so far as manifested in outward results, must depend. There is thus no sound objection à priori against the possibility of that species of data which is requisite for the experimental foundation of a science of social statistics. Again, whatever other object this treatise may accomplish, it is presumed that it will leave no doubt as to the existence of a system of abstract principles and of methods founded upon those principles, by which any collective body of social data may be made to yield, in an explicit form, whatever information they implicitly involve. There may, where the data are exceedingly complex, be very great difficulty in obtaining this information,—difficulty due not to any imperfection of the theory, but to the laborious character of the analytical processes to which it points. It is quite conceivable that in many instances that difficulty may be such as only united effort could overcome. But that we possess theoretically in all cases, and practically, so far as the requisite labour of calculation may be supplied, the means of evolving from statistical records the seeds of general truths which lie buried amid the mass of figures, is a position which may, I conceive, with perfect safety be affirmed.

18．抽象科学的方法将来是否有一天会在社会问题的研究中，提供与它们在物理研究各部门中所提供的相当的服务——在这里对这个问题进行推测，也许为时过早。试图纯粹根据先验的推理来解决这个问题，很可能会把我们引入歧途。例如，对人的自由意志的考虑，乍看起来似乎排除了这样的想法：社会系统的运动竟会显示出我们在物理必然性的统治下所预期的那种有序演进的性质。然而，统计学家的研究已经向我们揭示了与这种预期相悖的事实。例如，犯罪和贫困的记录所显示的规则性，竟是那些感受不到人类欲求与激情之扰动影响的领域里所不曾有的。另一方面，季节的失调、火山的爆发、植物中枯萎病的蔓延或动物界中流行病的扩散——这些显然或主要是自然原因的产物——却拒绝服从有规律的、可以把握的法则。“变幻如风”，是一句谚语。对这些问题的反思，在某种程度上教会我们纠正较早的判断。我们认识到：在必然性的统治之下，除非产生秩序的原因的作用足够简单，否则我们不应期待一种人类观察所能察觉的秩序；另一方面，也不应认为个人的自由意志与他作为其组成单元的那个系统在运动中的规则性互不相容。人的自由作为我们意识中的一个显然事实而凸显出来；同时我认为，它也是从心智中我们能够研究其科学构造的那一部分的本性出发，通过类比（第二十二章）得出的一个极其可能的推论。但无论是作为一个依托于意识的事实而被接受，还是作为一个经理性认可的结论而被接受，对它的解释都必须不与一个已确立的观察结果相冲突，那就是：大群人参与其中所产生的现象，确实显示出非常显著的规则性，使我们能够在相继的每个时代收集起一些要素，而对那个时代的状况和进步的估计——就其表现于外在结果而言——必定依赖于这些要素。因此，对于为一门社会统计科学提供实验基础所必需的那类数据的可能性，并不存在站得住的先验反对理由。再者，无论本书还能完成什么别的目标，可以设想，它将使人们毫不怀疑地看到：存在一个抽象原理的体系，以及建立在这些原理之上的方法，凭借它们，任何一批社会数据，都可以以明确的形式给出它们所隐含的一切信息。在数据极其复杂的地方，获得这些信息可能会有非常大的困难——这种困难并非由于理论有什么缺陷，而是由于理论所指示的分析程序十分繁重。完全可以设想，在许多情形下，这种困难也许只有联合起来的努力才能克服。但是，我们在理论上于一切情形中、在实践上于所需的计算劳动能够提供的范围内，都拥有这样的手段：从统计记录中引出埋藏在大量数字之中的一般真理的种子——我认为，这一论点可以十分稳妥地加以肯定。

19. But beyond these general positions I do not venture to speak in terms of assurance. Whether the results which might be expected from the application of scientific methods to statistical records, over and above those the discovery of which requires no such aid, would so far compensate for the labour involved as to render it worth while to institute such investigations upon a proper scale of magnitude, is a point which could, perhaps, only be determined by experience. It is to be desired, and it might without great presumption be expected, that in this, as in other instances, the abstract doctrines of science should minister to more than intellectual gratification. Nor, viewing the apparent order in which the sciences have been evolved, and have successively contributed their aid to the service of mankind, does it seem very improbable that a day may arrive in which similar aid may accrue from departments of the field of knowledge yet more intimately allied with the elements of human welfare. Let the speculations of this treatise, however, rest at present simply upon their claim to be regarded as true.

19．但在这些一般的论点之外，我不敢用有把握的语气说话。把科学方法应用于统计记录，除了那些无须借助这种方法就能发现的结果之外，还可能期待得到哪些结果；这些结果能否足以补偿所需的劳动，以至于值得以适当的规模开展这样的研究——这一点也许只有经验才能确定。人们希望，而且不算太过僭越地期待，在这里也像在其他场合一样，科学的抽象学说所提供的不止是理智上的满足。而且，考虑到各门科学看来是按怎样的次序发展起来、又相继为人类服务的，那么，有一天，知识领域中与人类福祉的要素更为密切相关的部门也会带来类似的帮助，这似乎也并非很不可能。不过，目前且让本书的这些思辨只凭一点立足：它们有理由被看作是真的。

20. I design, in the last place, to endeavour to educe from the scientific results of the previous inquiries some general intimations respecting the nature and constitution of the human mind. Into the grounds of the possibility of this species of inference it is not necessary to enter here. One or two general observations may serve to indicate the track which I shall endeavour to follow. It cannot but be admitted that our views of the science of Logic must materially influence, perhaps mainly determine, our opinions upon the nature of the intellectual faculties. For example, the question whether reasoning consists merely in the application of certain first or necessary truths, with which the mind has been originally imprinted, or whether the mind is itself a seat of law, whose operation is as manifest and as conclusive in the particular as in the general formula, or whether, as some not undistinguished writers seem to maintain, all reasoning is of particulars; this question, I say, is one which not merely affects the science of Logic, but also concerns the formation of just views of the constitution of the intellectual faculties. Again, if it is concluded that the mind is by original constitution a seat of law, the question of the nature of its subjection to this law,—whether, for instance, it is an obedience founded upon necessity, like that which sustains the revolutions of the heavens, and preserves the order of Nature,—or whether it is a subjection of some quite distinct kind, is also a matter of deep speculative interest. Further, if the mind is truly determined to be a subject of law, and if its laws also are truly assigned, the question of their probable or necessary influence upon the course of human thought in different ages is one invested with great importance, and well deserving a patient investigation, as matter both of philosophy and of history. These and other questions I propose, however imperfectly, to discuss in the concluding portion of the present work. They belong, perhaps, to the domain of probable or conjectural, rather than to that of positive, knowledge. But it may happen that where there is not sufficient warrant for the certainties of science, there may be grounds of analogy adequate for the suggestion of highly probable opinions. It has seemed to me better that this discussion should be entirely reserved for the sequel of the main business of this treatise,—which is the investigation of scientific truths and laws. Experience sufficiently instructs us that the proper order of advancement in all inquiries after truth is to proceed from the known to the unknown. There are parts, even of the philosophy and constitution of the human mind, which have been placed fully within the reach of our investigation. To make a due acquaintance with those portions of our nature the basis of all endeavours to penetrate amid the shadows and uncertainties of that conjectural realm which lies beyond and above them, is the course most accordant with the limitations of our present condition.

20．最后，我打算尝试从前面各项研究的科学结果中，引出一些关于人类心智之本性与构造的一般启示。这类推论之所以可能，其根据这里不必深究。一两点一般性的说明，或许足以指明我将尽力遵循的路径。不能不承认，我们对逻辑科学的看法，必定会实质性地影响、甚至也许在很大程度上决定我们对理智官能之本性的意见。例如这样一个问题：推理是否仅仅在于应用某些原初就印在心智上的第一真理或必然真理；还是说心智本身就是法则的所在，其作用在个别公式中和在一般公式中同样明显、同样确定；抑或像一些颇有声望的作者似乎主张的那样，一切推理都是关于个别事物的——我要说，这个问题不仅影响逻辑科学，也关系到能否对理智官能的构造形成正确的看法。再者，如果得出结论说，心智就其原初构造而言是法则的所在，那么它服从这种法则的性质如何——例如，它是不是一种建立在必然性之上的服从，就像维持天体运行、保持自然秩序的那种服从；抑或是一种全然不同的服从——这也是一个具有深刻思辨意义的问题。此外，如果心智真的被确定为服从法则的主体，而它的法则也被正确地指明，那么，这些法则在不同时代对人类思想进程可能的或必然的影响如何，就是一个极为重要的问题，无论作为哲学问题还是作为历史问题，都值得耐心研究。这些以及其他一些问题，我打算在本书的结尾部分加以讨论，尽管讨论难免不够完善。它们也许属于或然的或猜测的知识领域，而不属于确实知识的领域。但可能会有这样的情形：在没有足够根据得出科学之确定性的地方，也许有充分的类比根据，可以提出极为可能的意见。在我看来，最好把这一讨论完全留到本书的主要任务——即对科学真理和法则的研究——之后。经验充分地告诉我们，一切追求真理的探究，正确的前进次序都是由已知到未知。即便在人类心智的哲学和构造之中，也有一些部分已经完全处在我们的研究所及之内。把对我们本性中这些部分的适当认识，作为一切试图深入那个位于它们之外、之上的猜测领域之阴影与不确定之中的努力的基础，这是最符合我们目前处境之局限的路径。