| ABCdE | aBCdE |
| ABcde | aBcde |
| AbCDE | abCDE |
| AbcDe | abcDe |
They are now classified, so that those containing A stand first, and those devoid of A second, but no other property seems to be correlated with A. Let us alter this arrangement and group the combinations thus:—
| ABCdE | AbCDE |
| ABcde | AbcDe |
| aBCdE | abCDE |
| aBcde | abcDe |
It requires little examination to discover that in the first group B is always present and D absent, whereas in the second group, B is always absent and D present. This is the result which follows from a law of the form B = d (p. [136]), so that in this mode of arrangement we readily discover correlation between two letters. Altering the groups again as follows:—
| ABCdE | ABcde |
| aBCdE | aBcde |
| AbCDE | AbcDe |
| abCDE | abcDe, |
we discover another evident correlation between C and E. Between A and the other letters, or between the two pairs of letters B, D and C, E, there is no logical connexion.
This example may seem tedious, but it will be found instructive in this way. We are classifying only eight objects or combinations, in each of which only five qualities are considered. There are only two laws of correlation between four of those five qualities, and those laws are of the simplest logical character. Yet the reader would hardly discover what those laws are, and confidently assign them by rapid contemplation of the combinations, as given in the first group. Several tentative classifications must probably be made before we can resolve the question. Let us now suppose that instead of eight objects and five qualities, we have, say, five hundred objects and fifty qualities. If we were to attempt the same method of exhaustive grouping which we before employed, we should have to arrange the five hundred objects in fifty different ways, before we could be sure that we had discovered even the simpler laws of correlation. But even the successive grouping of all those possessing each of the fifty properties would not necessarily give us all the laws. There might exist complicated relations between several properties simultaneously, for the detection of which no rule of procedure whatever can be given.
Bifurcate Classification.
Every system of classification ought to be formed on the principles of the Logical Alphabet. Each superior class should be divided into two inferior classes, distinguished by the possession and non-possession of a single specified difference. Each of these minor classes, again, is divisible by any other quality whatever which can be suggested, and thus every classification logically consists of an infinitely extended series of subaltern genera and species. The classifications which we form are in reality very small fragments of those which would correctly and fully represent the relations of existing things. But if we take more than four or five qualities into account, the number of subdivisions grows impracticably large. Our finite minds are unable to treat any complex group exhaustively, and we are obliged to simplify and generalise scientific problems, often at the risk of overlooking particular conditions and exceptions.
Every system of classes displayed in the manner of the Logical Alphabet may be called bifurcate, because every class branches out at each step into two minor classes, existent or imaginary. It would be a great mistake to regard this arrangement as in any way a peculiar or special method; it is not only a natural and important one, but it is the inevitable and only system which is logically perfect, according to the fundamental laws of thought. All other arrangements of classes correspond to the bifurcate arrangement, with the implication that some of the minor classes are not represented among existing things. If we take the genus A and divide it into the species AB and AC, we imply two propositions, namely that in the class A, the properties of B and C never occur together, and that they are never both absent; these propositions are logically equivalent to one, namely AB = Ac. Our classification is then identical with the following bifurcate one:—