Although there is no absolute summum genus, nevertheless relatively to any branch of knowledge or any particular argument, there is always some class or notion which bounds our horizon as it were. The chemist restricts his view to material substances and the forces manifested in them; the mathematician extends his view so as to comprehend all notions capable of numerical discrimination. The biologist, on the other hand, has a narrower sphere containing only organised bodies, and of these the botanist and the zoologist take parts. In other subjects there may be a still narrower summum genus, as when the lawyer regards only reasoning beings of his own country together with their property.
In the description of the Logical Alphabet it was pointed out (p. [93]) that every series of combinations is really the development of a single class, denoted by X, which letter was accordingly placed in the first column of the table on p. [94]. This is the formal acknowledgment of the principle clearly stated by De Morgan, that all reasoning proceeds within an assumed summum genus. But at the same time the fact that X as a logical term must have its negative x, shows that it cannot be an absolute summum genus.
There arises, again, the question whether there be any such thing as an infima species, which cannot be divided into minor species. The ancient logicians were of opinion that there always was some assignable class which could only be divided into individuals, but this doctrine appears to be theoretically incorrect, as Mr. George Bentham long ago stated.[575] We may put an arbitrary limit to the subdivision of our classes at any point convenient to our purpose. The crystallographer would not generally treat as different species crystalline forms which differ only in the degree of development of the faces. The naturalist overlooks innumerable slight differences between animals which he refers to the same species. But in a strictly logical point of view classification might be carried on as long as there is a difference, however minute, between two objects, and we might thus go on until we arrive at individual objects which are numerically distinct in the logical sense attributed to that expression in the chapter upon Number. Either, then, we must call the individual the infima species or allow that there is no such thing at all.
The Tree of Porphyry.
Both Aristotle and Plato were acquainted with the value of bifurcate classification, which they occasionally employed in an explicit manner. It is impossible too that Aristotle should state the laws of thought, and employ the predicables without implicitly recognising the logical necessity of that method. It is, however, in Porphyry’s remarkable and in many respects excellent Introduction to the Categories of Aristotle that we find the most distinct account of it. Porphyry not only fully and accurately describes the Predicables, but incidentally introduces an example for illustrating those predicables, which constitutes a good specimen of bifurcate classification. Translating his words[576] freely we may say that he takes Substance as the genus to be divided, under which are successively placed as Species—Body, Animated Body, Animal, Rational Animal, and Man. Under Man, again, come Socrates, Plato, and other particular men. Now of these notions Substance is the genus generalissimum, and is a genus only, not a species. Man, on the other hand, is the species specialissima (infima species), and is a species only, not a genus. Body is a species of substance, but a genus of animated body, which, again, is a species of body but a genus of animal. Animal is a species of animated body, but a genus of rational animal, which, again, is a species of animal, but a genus of man. Finally, man is a species of rational animal, but is a species merely and not a genus, being divisible only into particular men.
Porphyry proceeds at some length to employ his example in further illustration of the predicables. We do not find in Porphyry’s own work any scheme or diagram exhibiting this curious specimen of classification, but some of the earlier commentators and epitome writers drew what has long been called the Tree of Porphyry. This diagram, which may be found in most elementary works on Logic,[577] is also called the Ramean Tree, because Ramus insisted much upon the value of Dichotomy. With the exception of Jeremy Bentham[578] and George Bentham, hardly any modern logicians have shown an appreciation of the value of bifurcate classification. The latter author has treated the subject, both in his Outline of a New System of Logic (pp. 105–118), and in his earlier work entitled Essai sur la Nomenclature et la Classification des Principales Branches d’Art-et-Science (Paris, 1823), which consists of a free translation or improved version of his uncle’s Essay on Classification in the Chrestomathia. Some interest attaches to the history of the Tree of Porphyry and Ramus, because it is the prototype of the Logical Alphabet which lies at the basis of logical method. Jeremy Bentham speaks truly of “the matchless beauty of the Ramean Tree.” After fully showing its logical value as an exhaustive method of classification, and refuting the objections of Reid and Kames, on a wrong ground, as I think, he proceeds to inquire to what length it may be carried. He correctly points out two objections to the extensive use of bifid arrangements, (1) that they soon become impracticably extensive and unwieldy, and (2) that they are uneconomical. In his day the recorded number of different species of plants was 40,000, and he leaves the reader to estimate the immense number of branches and the enormous area of a bifurcate table which should exhibit all these species in one scheme. He also points out the apparent loss of labour in making any large bifurcate classification; but this he considers to be fully recompensed by the logical value of the result, and the logical training acquired in its execution. Jeremy Bentham, then, fully recognises the value of the Logical Alphabet under another name, though he apprehends also the limit to its use placed by the finiteness of our mental and manual powers.
Does Abstraction imply Generalisation?
Before we can acquire a sound comprehension of the subject of classification we must answer the very difficult question whether logical abstraction does or does not imply generalisation. It comes to exactly the same thing if we ask whether a species may be coextensive with its genus, or whether, on the other hand, the genus must contain more than the species. To abstract logically is (p. [27]), to overlook or withdraw our notice from some point of difference. Whenever we form a class we abstract, for the time being, the differences of the objects so united in respect of some common quality. If we class together a great number of objects as dwelling-houses, we overlook the fact that some dwelling-houses are constructed of stone, others of brick, wood, iron, &c. Often at least the abstraction of a circumstance increases the number of objects included under a class according to the law of the inverse relation of the quantities of extension and intension (p. [26]). Dwelling-house is a wider term than brick-dwelling-house. House is more general than dwelling-house. But the question before us is, whether abstraction always increases the number of objects included in a class, which amounts to asking whether the law of the inverse relation of logical quantities is always true. The interest of the question partly arises from the fact, that so high a philosophical authority as Mr. Herbert Spencer has denied that generalisation is implied in abstraction,[579] making this doctrine the ground for rejecting previous methods of classifying the sciences, and for forming an ingenious but peculiar method of his own. The question is also a fundamental one of the highest logical importance, and involves subtle difficulties which have made me long hesitate in forming a decisive opinion.
Let us attempt to answer the question by examination of a few examples. Compare the two classes gun and iron gun. It is certain that there are many guns which are not made of iron, so that abstraction of the circumstance “made of iron” increases the extent of the notion. Next compare gun and metallic gun. All guns made at the present day consist of metal, so that the two notions seem to be coextensive; but guns were at first made of pieces of wood bound together like a tub, and as the logical term gun takes no account of time, it must include all guns that have ever existed. Here again extension increases as intension decreases. Compare once more “steam-locomotive engine” and “locomotive engine.” In the present day, as far as I am aware, all locomotives are worked by steam, so that the omission of that qualification might seem not to widen the term; but it is quite possible that in some future age a different motive power may be used in locomotives; and as there is no limitation of time in the use of logical terms, we must certainly assume that there is a class of locomotives not worked by steam, as well as a class that is worked by steam. When the natural class of Euphorbiaceæ was originally formed, all the plants known to belong to it were devoid of corollas; it would have seemed therefore that the two classes “Euphorbiaceæ,” and “Euphorbiaceæ devoid of Corollas,” were of equal extent. Subsequently a number of plants plainly belonging to the same class were found in tropical countries, and they possessed bright coloured corollas. Naturalists believe with the utmost confidence that “Ruminants” and “Ruminants with cleft feet” are identical terms, because no ruminant has yet been discovered without cleft feet. But we can see no impossibility in the conjunction of rumination with uncleft feet, and it would be too great an assumption to say that we are certain that an example of it will never be met with. Instances can be quoted, without end, of objects being ultimately discovered combining properties which had never before been seen together. In the animal kingdom the Black Swan, the Ornithorhynchus Paradoxus, and more recently the singular fish called Ceratodus Forsteri, all discovered in Australia, have united characters never previously known to coexist. At the present time deep-sea dredging is bringing to light many animals of an unprecedented nature. Singular exceptional discoveries may certainly occur in other branches of science. When Davy first discovered metallic potassium, it was a well established empirical law that all metallic substances possessed a high specific gravity, the least dense of the metals then known being zinc, of which the specific gravity is 7·1. Yet to the surprise of chemists, potassium was found to be an undoubted metal of less density than water, its specific gravity being 0·865.
It is hardly requisite to prove by further examples that our knowledge of nature is incomplete, so that we cannot safely assume the non-existence of new combinations. Logically speaking, we ought to leave a place open for animals which ruminate but are without cleft feet, and for every possible intermediate form of animal, plant, or mineral. A purely logical classification must take account not only of what certainly does exist, but of what may in after ages be found to exist.