This principle of mediate comparison might be briefly (though obscurely) expressed by the formula "more than the more is more than the less"—the words more and less standing simply for degrees of increase along a constant direction of differences. Such a formula would cover all possible cases, as, earlier than early is earlier than late, worse than bad is worse than good, east of east is east of west; etc., etc., ad libitum.[538] Symbolically, we might write it as a < b < c < d.... and say that any number of intermediaries may be expunged without obliging us to alter anything in what remains written.
The principle of mediate comparison is only one form of a law which holds in many series of homogeneously related terms, the law that skipping intermediary terms leaves relations the same. This axiom of skipped intermediaries or of transferred relations occurs, as we soon shall see, in logic as the fundamental principle of inference, in arithmetic as the fundamental property of the number-series, in geometry as that of the straight line, the plane and the parallel. It seems to be on the whole the broadest and deepest law of man's thought.
In certain lists of terms the result of comparison may be to find no-difference, or equality in place of difference. Here also intermediaries may be skipped, and mediate comparison be carried on with the general result expressed by the axiom of mediate equality, "equals of equals are equal," which is the great principle of the mathematical sciences. This too as a result of the mind's mere acuteness, and in utter independence of the order in which experiences come associated together. Symbolically, again: a = b = c = d..., with the same consequence as regards expunging terms which we saw before.
CLASSIFICATORY SERIES.
Thus we have a rather intricate system of necessary and immutable ideal truths of comparison, a system applicable to terms experienced in any order of sequence or frequency, or even to terms never experienced or to be experienced, such as the mind's imaginary constructions would be. These truths of comparison result in Classifications. It is, for some unknown reason, a great æsthetic delight for the mind to break the order of experience, and class its materials in serial orders, proceeding from step to step of difference, and to contemplate untiringly the crossings and inosculations of the series among themselves. The first steps in most of the sciences are purely classificatory. Where facts fall easily into rich and intricate series (as plants and animals and chemical compounds do), the mere sight of the series fills the mind with a satisfaction sui generis; and a world whose real materials naturally lend themselves to serial classification is pro tanto a more rational world, a world with which the mind will feel more intimate, than with a world in which they do not. By the pre-evolutionary naturalists, whose generation has hardly passed away, classifications were supposed to be ultimate insights into God's mind, filling us with adoration of his ways. The fact that Nature lets us make them was a proof of the presence of his Thought in her bosom. So far as the facts of experience can not be serially classified, therefore, so far experience fails to be rational in one of the ways, at least, which we crave.
THE LOGIC-SERIES.
Closely akin to the function of comparison is that of judging, predicating, or subsuming. In fact, these elementary intellectual functions run into each other so, that it is often only a question of practical convenience whether we shall call a given mental operation by the name of one or of the other. Comparisons result in groups of like things; and presently (through discrimination and abstraction) in conceptions of the respects in which the likenesses obtain. The groups are genera or classes, the respects are characters or attributes. The attributes again may be compared, forming genera of higher orders, and their characters singled out; so that we have a new sort of series, that of predication, or of kind including kind. Thus horses are quadrupeds, quadrupeds animals, animals machines, machines liable to wear out, etc. In such a series as this the several couplings of terms may have been made out originally at widely different times and under different circumstances. But memory may bring them together afterwards; and whenever it does so, our faculty of apprehending serial increase makes us conscious of them as a single system of successive terms united by the same relation.[539]
Now whenever we become thus conscious, we may become aware of an additional relation which is of the highest intellectual importance, inasmuch as upon it the whole structure of logic is reared. The principle of mediate predication or subsumption is only the axiom of skipped intermediaries applied to a series of successive predications. It expresses the fact that any earlier term in the series stands to any later term in the same relation in which it stands to any intermediate term; in other words, that whatever has an attribute has all the attributes of that attribute; or more briefly still, that whatever is of a kind is of that kind's kind. A little explanation of this statement will bring out all that it involves.
We learned in the chapter on Reasoning what our great motive is for abstracting attributes and predicating them. It is that our varying practical purposes require us to lay hold of different angles of the reality at different times. But for these we should be satisfied to 'see it whole,' and always alike. The purpose, however, makes one aspect essential; so, to avoid dispersion of the attention, we treat the reality as if for the time being it were nothing but that aspect, and we let its supernumerary determinations go. In short, we substitute the aspect for the whole real thing. For our purpose the aspect can be substituted for the whole, and the two treated as the same; and the word is (which couples the whole with its aspect or attribute in the categoric judgment) expresses (among other things) the identifying operation performed. The predication-series a is b, b is c, c is d,... closely resembles for certain practical purposes the equation-series a = b, b = c, c = d, etc.
But what is our purpose in predicating? Ultimately, it may be anything we please; but proximately and immediately, it is always the gratification of a certain curiosity as to whether the object in hand is or is not of a kind connected with that ultimate purpose. Usually the connection is not obvious, and we only find that the object S is of a kind connected with P, after first finding that it is of a kind M, which itself is connected with P. Thus, to fix our ideas by an example, we have a curiosity (our ultimate purpose being conquest over nature) as to how Sirius may move. It is not obvious whether Sirius is a kind of thing which moves in the line of sight or not. When, however, we find it to be a kind of thing in whose spectrum the hydrogen-line is shifted, and when we reflect that that kind of thing is a kind of thing which moves in the line of sight; we conclude that Sirius does so move. Whatever Sirius's attribute is, Sirius is; its adjective's adjective can supersede its own adjective in our thinking, and this with no loss to our knowledge, so long as we stick to the definite purpose in view.