Some years of continuous labour would be required to ascertain the types of laws which may govern the combinations of only four things, and but a small part of such laws would be exemplified or capable of practical application in science. The purely logical inverse problem, whereby we pass from combinations to their laws, is solved in the preceding pages, as far as it is likely to be for a long time to come; and it is almost impossible that it should ever be carried more than a single step further.
In the first edition, vol i. p. 158, I stated that I had not been able to discover any mode of calculating the number of cases in which inconsistency would be implied in the selection of combinations from the Logical Alphabet. The logical complexity of the problem appeared to be so great that the ordinary modes of calculating numbers of combinations failed, in my opinion, to give any aid, and exhaustive examination of the combinations in detail seemed to be the only method applicable. This opinion, however, was mistaken, for both Mr. R. B. Hayward, of Harrow, and Mr. W. H. Brewer have calculated the numbers of inconsistent cases both for three and for four terms, without much difficulty. In the case of four terms they find that there are 1761 inconsistent selections and 63,774 consistent, which with one case where no condition exists, make up the total of 65,536 possible selections.
The inconsistent cases are distributed in the manner shown in the following table:—
| Number of Combinations remaining. | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10, &c. |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Number of Inconsistent Cases. | 1 | 16 | 112 | 352 | 536 | 448 | 224 | 64 | 8 | 0 | 0, &c. |
When more than eight combinations of the Logical Alphabet (p. 94, column V.) remain unexcluded, there cannot be inconsistency. The whole numbers of ways of selecting 0, 1, 2, &c., combinations out of 16 are given in the 17th line of the Arithmetical Triangle given further on in the Chapter on Combinations and Permutations, the sum of the numbers in that line being 65,536.
Professor Clifford on the Types of Compound Statement involving Four Classes.
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.