a = bC ꖌ Bc, B = CA ꖌ ca, b = cA ꖌ Ca,
C = AB ꖌ ab, c = aB ꖌ Ab.
I do not think it needful to publish at present the complete table of 193 series of combinations and the premises corresponding to each. Such a table enables us by mere inspection to learn the laws obeyed by any set of combinations of three things, and is to logic what a table of factors and prime numbers is to the theory of numbers, or a table of integrals to the higher mathematics. The table already given (p. [140]) would enable a person with but little labour to discover the law of any combinations. If there be seven combinations (one contradicted) the law must be of the eighth type, and the proper variety will be apparent. If there be six combinations (two contradicted), either the second, eleventh, or twelfth type applies, and a certain number of trials will disclose the proper type and variety. If there be but two combinations the law must be of the third type, and so on.
The above investigations are complete as regards the possible logical relations of two or three terms. But when we attempt to apply the same kind of method to the relations of four or more terms, the labour becomes impracticably great. Four terms give sixteen combinations compatible with the laws of thought, and the number of possible selections of combinations is no less than 216 or 65,536. The following table shows the extraordinary manner in which the number of possible logical relations increases with the number of terms involved.
| Number of terms. | Number of possible combinations. | Number of possible selections of combinations corresponding to consistent or inconsistent logical relations. |
|---|---|---|
| 2 | 4 | 16 |
| 3 | 8 | 256 |
| 4 | 16 | 65,536 |
| 5 | 32 | 4,294,967,296 |
| 6 | 64 | 18,446,744,073,709,551,616 |
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.