Two statements are called complementary when they deny between them all the sixteen combinations without both denying any one; or, which is the same thing, when each denies just those combinations which the other permits to exist. It is obvious that when two statements are similar, the complementary statements will also be similar, and consequently for every type of n-fold statement, there is a complementary type of (16—n)-fold statement. It follows that we need only enumerate the types as far as the eighth order; for the types of more-than-eight-fold statement will already have been given as complementary to types of lower orders.
One combination, ABCD, may be converted into another AbCd by interchanging one or more of the classes with the complementary classes. The number of such changes is called the distance, which in the above case is 2. In two similar compound statements the distances of the combinations denied must be the same; but it does not follow that when all the distances are the same, the statements are similar. There is, however, only one example of two dissimilar statements having the same distances. When the distance is 4, the two combinations are said to be obverse to one another, and the statements denying them are called obverse statements, as in ABCD = 0 and abcd = 0 or again AbCd = 0 and aBcD = 0. When any one combination is given, called the origin, all the others may be grouped in respect of their relations to it as follows:—Four are at distance one from it, and may be called proximates; six are at distance two, and may be called mediates; four are at distance three, and may be called ultimates; finally the obverse is at distance four.
| Origin and four proximates. | Six mediates. | Obverse and four ultimates. | ||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| ab CD | ||||||||||||
| | | ||||||||||||
| a BCD | A bc D | | | A b C d | A bcd | ||||||||
| | | ╲ | | | ╱ | | | ||||||||
| ABC d | — | ABCD | — | A b CD | ╳ | abc D | — | abcd | — | a B cd | ||
| | | ╱ | | | ╲ | | | ||||||||
| AB c D | a B c D | | | a BC d | ab C d . | ||||||||
| | | ||||||||||||
| AB cd |
It will be seen that the four proximates are respectively obverse to the four ultimates, and that the mediates form three pairs of obverses. Every proximate or ultimate is distant 1 and 3 respectively from such a pair of mediates.
Aided by this system of nomenclature Professor Clifford proceeds to an exhaustive enumeration of types, in which it is impossible to follow him. The results are as follows:—