commonly called 'mathematical' symbols: as, moreover, I have devoted a separate treatise to its exposition, I will merely give a sketch here of the process by which the foundations of this treatment of the subject may be most readily reached.

Start then from the familiar account of Division as given by the later Aristotelians. This was, as we have seen, at every step bifurcate; branching into two contradictory subdivisions. Thus Substance was divided into corporeal and incorporeal : corporeal substance into what was complex and simple, and so on. There are two points which must be noticed here. In the first place there is not any very rigid adherence displayed to a truly formal dichotomy of the X and not-X kind. However accurately the two subdivisions may, as a matter of fact, be mutually exclusive of each other, they do not always show by their very form that they are so. And in the second place, as was pointed out in the last chapter, we do not undertake at every step to subdivide both the resultant classes so as to continually double their number. The process, as indicated by the technical expression 'the predicamental line', is directed towards the ultimate separation, by subdivision, of some predetermined class or individual.

The Division which we now contemplate is founded upon a thorough-going recognition of these two characteristics. It deals with nothing but formal contradictories, and it purposes to introduce every alternative of which the form will admit. Suppose that we start with a class S, and that we are concerned with three attributes which shall serve as the bases of division, viz. X, Y, and Z. We divide S into X, and not-X. then proceed to subdivide both of these by the introduction of Y, thus obtaining four classes. Introduce the third attribute Z, and make the same division again, and we get eight resultant classes. And this we might continue doing with any number of such dividing attributes.

We

So far nothing has been suggested except an improvement in respect of accuracy and completeness of method. But now attention must be directed to a decided alteration in the point of view, and one which gives nearly all its significance to the method in question. Instead of regarding X and not-X and their respective subdivisions as standing for actual classes, we shall regard them as standing for class-compartments. This

makes an enormous difference. It meets for instance the objection which has been raised by formal purists against the legitimacy of this kind of division, on the ground that unless we had reason to know the relevance of the attribute X to the objects included under S,—that is, unless we had the material information that some of them did possess X and some did not, --we might be led to the absurdity of a class which was without any members to compose it. Of course this would be absurd if what we were proposing to do was to make an arrangement or classification of existing objects of any kind;—in fact ‘the snakes of Iceland' have passed into a common joke;-but it is far from absurd if we are proposing an exhaustive arrangement of compartments. The emptiness of any one of these, as we shall see in a moment, gives all its significance to the practical employment of the scheme.

What we start with then, on this system of Logic, is a framework of class-compartments, the number of these being determined by the number of class-terms involved in the proposition or group of propositions which we are supposed to have before us. If the propositions involved, as is the case with the common syllogism, three terms, say X, Y, Z, we should have eight such compartments before us, viz. XYZ, XY. not-Z, X. not-Y. Z, X. not-Y. not-Z, and so on, till all the possible combinations were exhausted. Now this being so, it may be shown, and this forms the basis of the whole Theory of Symbolic Logic,-that every significant universal1 proposition must necessarily destroy some one or more of the possible classes; that is, it must cause some one or more of the compartments to be empty. And conversely, whatever may be the number and description of empty compartments, there must be some corresponding proposition which will unambiguously express the facts. Thus, for example, if I say, 'All X is Y', this is equivalent to saying that there is no X that is not Y', or in other words that there is no such class as X. not-Y. So if I say that 'All X is either Y or Z'; this is equivalent to saying that 'There is no X that is neither Y nor Z', that is, that there is no such class as X. not-Y. not-Z.

1 The case of particular propositions is somewhat different. It is found that they are best explained as declaring that such compartments are occupied.

A peculiar interpretation of the import of propositions is of course demanded on this system of Logic; but, granted this, the whole foundation of the system may be said to consist in this exhaustive scheme of compartments.

As this development of Logic has extremely little connection with Induction, being in fact nothing but a very broad generalization of Formal Logic, we must content ourselves here with this slight indication of its scope. The reader must however understand that this process of continued dichotomy, by the introduction of every relevant class-term, forms the entire basis of the subject. The elaborate apparatus of symbols, in great part identical with those employed in mathematics, which are introduced to express the resultant formulæ, are nothing more than convenient abbreviations of processes which are essentially logical and not mathematical in their character. It is necessary to enter this protest in the most decisive way at once, because the opposite opinion has become rather firmly rooted. So far from mathematical processes and conceptions being introduced into Logic, we only make use of a selection of its symbols to express logical processes and conceptions. We resort to these not from absolute necessity, but because we find them ready to hand and convenient for our purpose. We could work out our results without their aid at all, as was originally done to a considerable extent by Jevons in his Familiar Lessons, and has been yet more completely effected recently by Mr Keynes in his Deductive Logic. It is however, I think, convenient to help ourselves by symbolic notation if it were only for purposes of simplification. The manipulation of a multitude of class-elements, the introduction of eight terms, remember, will yield 512 such compartments,--is almost impossible in practice without some such aid. To this may be added another advantage. It is highly desirable that there should be some alternative mode open to the student, besides that of mathematics, for acquiring some knowledge of the nature and object of a symbolic language.

II. The extension of the old problem of Division which we have just discussed was a development of the formal side of the process. The one which we have now to consider proceeds upon the opposite plan, viz. that of developing to the utmost the material side. What this leads us to is nothing

1

else than the familiar problem of Classification, under which name we will now proceed to examine its nature and aims.

The main object of all Classification is simple enough. We are supposed to have before us a miscellaneous lot of objects, and we are directed to group them into classes, or rather into a sort of hierarchy of classes. The process may be commenced, so to say, from either end. On the old plan,—as the name Division implies, we were always supposed to start with the entire group or supreme class, and to proceed, as the phrase is, downwards; the process is one of continued subdivision. We might however just as well have proceeded in the reverse direction, arranging our classes in the upward order. Had we adopted this course, a more appropriate name would have been Aggregation rather than Division. In the physical process of sorting shot or gravel into a number of packets according to size, it would come to the same thing in the end whether we made use of the sieves by beginning with the finest or with the coarsest. We might again, if we liked, proceed irregularly, as indeed is probably the commonest practice when we are dealing with actual objects in the ordinary walks of life,-sometimes upwards and sometimes downwards, until the task is completed. The fact is that a system of Classification carried out entirely de novo is hardly anywhere to be found. Ages before the logician, or any one else who deals with systems, had a hand in the matter, the necessities of common life had been at work prompting. men to group the things which they observed. All names imply the recognition of groups, and a great number of names imply a subordination of groups, so that at the earliest stage to which we can transfer ourselves we find that we are already in possession of a rudimentary classification; and that we cannot even talk or think about the things without an appeal to this.

The remarks just made do not give the slightest suggestion in the way of guiding us to any principle of Classification; on the contrary they may serve to remind us that this principle must in some way be supplied by ourselves. The mere arrangement of one and the same lot of things into a hierarchy of successive classes can be carried out in a variety of ways which is for all practical purposes infinite. And even admitting the

restrictions imposed by common sense and inherited language; in other words, taking for granted the basement set of classes as they are popularly accepted, yet the number of ways in which the intermediate ones, between these and the top class, may be disposed is very great indeed.

We want then some principle to guide us. This can only be given in its details presently, after we have seen what a system of classification can effect,-for here, as in so many other cases, we can only clearly determine what to aim at after we have had some practice in shooting,-but the main outlines of what we should strive after can readily be sketched out. The general object of all classification is to keep our control over the facts by marshalling the objects in order: to know where to find a thing when it is wanted, and to economize our statements in the retention and communication of our knowledge. This is purposely phrased rather loosely, but it will serve as a starting point towards the distinction between one system and another.

There is a distinction which has been so frequently drawn in logical and other systematic treatises, that it has already begun to make its way into popular phraseology. It is that between Natural and Artificial systems of Classification. The particular phrase in which this distinction is conveyed does not seem to me a very satisfactory one. Every arrangement of the kind in question is artificial in so far as it is our own voluntary procedure and not a result offered to us from without it is also artificial in the sense that we are seldom or never proposing to group the actual objects themselves but only to make an ideal arrangement of them in our own minds. On the other hand every classification ought to aim at being natural in the sense of conforming to the facts and endeavouring to be as suitable as possible to the circumstances. 'Natural' is a word which it does not need the authority of Butler to condemn as rather ambiguous in philosophical discussion.

The distinction with which I prefer to start is that between Classification intended for special purposes and that which is intended for general purposes. There must be some purpose or aim presupposed in every arrangement of the kind in ques-' tion, just as there must be one for the shape and size of a tool; and the determination of this purpose at once puts its stamp upon the consequent classification. It is perfectly optional on