Chapter XIII

(p. 490) to be a cardinal mental fact. "A differs from B differs from C differs from D, etc.," makes a _series_ only when the differences are in the same direction. In any such difference-series all terms differ in just the same way from their predecessors. The numbers 1, 2, 3, 4, 5,... the notes of the chromatic scale in music, are familiar examples. As soon as the mind grasps such a series as a whole, it perceives that _two terms taken far apart differ more than two terms taken near together_, and that any one term differs more from a remote than from a near successor, and this no matter what the terms may be, or what the sort of difference may be, provided it is always the same sort.

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._

Now please note that this elimination of intermediary kinds and transfer of _is_'s along the line, results from our insight into the very meaning of the word _is_, and into the constitution of any series of terms connected by that relation. It has naught to do with what any particular thing is or is not; but, _whatever_ any given thing may be, we see that it also is whatever _that_ is, indefinitely. To grasp in one view a succession of _is_'s is to apprehend this relation between the terms which they connect; just as to grasp a list of successive equals is to apprehend their _mutual_ equality throughout. The principle of mediate subsumption thus expresses relations of ideal objects as such. It can be discovered by a mind left at leisure with any set of meanings (however originally obtained), of which some are predicable of others. The moment we string them in a serial line, that moment we see that we can drop intermediaries, treat remote terms just like near ones, and put a genus in the place of a species. This shows that _the principle of mediate subsumption has nothing to do with the

## particular order of our experiences, or with the outer coexistences and

sequences of terms._ Were it a mere outgrowth of habit and association, we should be forced to regard it as having no universal validity; for every hour of the day we meet things which we consider to be of this kind or of that, but later learn that they have none of the kind's properties, that they _do not_ belong to the kind's kind. Instead, however, of correcting the principle by these cases, we correct the cases by the principle. We say that if the thing we named an M has not M's properties, then we were either mistaken in calling it an M, or mistaken about M's properties; or else that it is no longer M, but has changed. But we never say that it is an M without M's properties; for by conceiving a thing as of the kind M I mean that it _shall_ have M's properties, be of M's kind, even though I should never be able to find in the real world anything which is an M. The principle emanates from my perception of what a lot of successive is's _mean_. This perception can no more be confirmed by one set, or weakened by another set, of outer facts, than the perception that black is not white can be confirmed by the fact that snow never blackens, or weakened by the fact that photographer's paper blackens as soon as you lay it in the sun.

The abstract scheme of successive predications, extended indefinitely, with all the possibilities of substitution which it involves, is thus an immutable system of truth which flows from the very structure and form of our thinking. _If_ any real terms ever do fit into such a scheme, they will obey its laws; _whether_ they do is a question as to nature's facts, the answer to which can only be empirically ascertained. _Formal logic_ is the name of the Science which traces in skeleton form all the remote relations of terms connected by successive _is_'s with each other, and enumerates their possibilities of mutual substitution. To our principle of mediate subsumption she has given various formulations, of which the best is perhaps this broad expression, that _the same can be substituted for the same in any mental operation._[540]

The ordinary logical series contains but three terms--"Socrates, man, mortal." But we also have 'Sorites'--Socrates, man, animal, machine, run down, mortal, etc.--and it violates psychology to represent these as syllogisms with terms suppressed. The ground of there being any logic at all is our power to grasp any series as a whole, and the more terms it holds the better. This synthetic consciousness of an uniform direction of advance through a multiplicity of terms is, apparently, what the brutes and lower men cannot accomplish, and what gives to us our extraordinary power of ratiocinative thought. The mind which can grasp a string of _is_'s as a whole--the objects linked by them may be ideal or real, physical, mental, or symbolic, indifferently--can also apply to it the principle of skipped intermediaries. _The logic-list is thus in its origin and essential nature just like those graded classificatory lists which we erewhile described._ The 'rational proposition' which lies at the basis of all reasoning, the _dictum de omni et nullo_ in all the various forms in which it may be expressed, the fundamental law of thought, is thus _only the result of the function of comparison_ in a mind which has come by some lucky variation to apprehend a series of more than two terms at once.[541] So far, then, _both Systematic Classification and Logic are seen to be incidental results of the mere capacity for discerning difference and likeness,_ which capacity is a thing with which the _order of experience_, properly so styled, has absolutely nothing to do.

* * * * *

But how comes it (it may next be asked) when systematic classifications have so little ultimate theoretic importance--for the conceiving of things according to their mere degrees of resemblance always yields to other modes of conceiving when these can be obtained--that the logical relations among things should form such a mighty engine for dealing with the facts of life?

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