In the first edition (vol. i. p. 163), I asserted that some years of labour would be required to ascertain even the precise number of types of law governing the combinations of four classes of things. Though I still believe that some years’ labour would be required to work out the types themselves, it is clearly a mistake to suppose that the numbers of such types cannot be calculated with a reasonable amount of labour, Professor W. K. Clifford having actually accomplished the task. His solution of the numerical problem involves the use of a complete new system of nomenclature and is far too intricate to be fully described here. I can only give a brief abstract of the results, and refer readers, who wish to follow out the reasoning, to the Proceedings of the Literary and Philosophical Society of Manchester, for the 9th January, 1877, vol. xvi., p. 88, where Professor Clifford’s paper is printed in full.
By a simple statement Professor Clifford means the denial of the existence of any single combination or cross-division, of the classes, as in ABCD = 0, or AbCd = 0. The denial of two or more such combinations is called a compound statement, and is further said to be twofold, threefold, &c., according to the number denied. Thus ABC = 0 is a twofold compound statement in regard to four classes, because it involves both ABCD = 0 and ABCd = 0. When two compound statements can be converted into one another by interchange of the classes, A, B, C, D, with each other or with their complementary classes, a, b, c, d, they are called similar, and all similar statements are said to belong to the same type.
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. | ||||||||||||
| abCD | ||||||||||||||
| | | ||||||||||||||
| aBCD | AbcD | | | AbCd | Abcd | ||||||||||
| | | ╲ | | | ╱ | | | ||||||||||
| ABCd | — | ABCD | — | AbCD | ╳ | abcD | — | abcd | — | aBcd | ||||
| | | ╱ | | | ╲ | | | ||||||||||
| ABcD | aBcD | | | aBCd | abCd. | ||||||||||
| | | ||||||||||||||
| ABcd | ||||||||||||||
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:—
| 1-fold | statements | 1 | type |  | 159 | |
| 2 | " | " | 4 | types | ||
| 3 | " | " | 6 | " | ||
| 4 | " | " | 19 | " | ||
| 5 | " | " | 27 | " | ||
| 6 | " | " | 47 | " | ||
| 7 | " | " | 55 | " | ||
| 8-fold | statements | 78 | " | |||
Now as each seven-fold or less-than-seven-fold statement is complementary to a nine-fold or more-than-nine-fold statement, it follows that the complete number of types will be 159 × 2 + 78 = 396.
It appears then that the types of statement concerning four classes are only about 26 times as numerous as those concerning three classes, fifteen in number, although the number of possible combinations is 256 times as great.