series of combinations is the logical analogue, the chief points of difference being that there is a limit to the number of cases, and that prime numbers have no analogue in logic, since every series of combinations corresponds to a law or group of conditions. But the analogy is perfect in the point that they are both inverse processes. There is no mode of ascertaining that a number is prime but by showing that it is not the product of any assignable factors. So there is no mode of ascertaining what laws are embodied in any series of combinations but trying exhaustively the laws which would give them. Just as the results of Eratosthenes’ method have been worked out to a great extent and registered in tables for the convenience of other mathematicians, I have endeavoured to work out the inverse logical problem to the utmost extent which is at present practicable or useful.
I have thus found that there are altogether fifteen conditions or series of conditions which may govern the combinations of three terms, forming the premises of fifteen essentially different kinds of arguments. The following table contains a statement of these conditions, together with the numbers of combinations which are contradicted or destroyed by each, and the numbers of logically distinct variations of which the law is capable. There might be also added, as a sixteenth case, that case where no special logical condition exists, so that all the eight combinations remain.
| Reference Number. | Propositions expressing the general type of the logical conditions. | Number of distinct logical variations. | Number of combinations contradicted by each. |
|---|---|---|---|
| I. | A = B | 6 | 4 |
| II. | A = AB | 12 | 2 |
| III. | A = B, B = C | 4 | 6 |
| IV. | A = B, B = BC | 24 | 5 |
| V. | A = AB, B = BC | 24 | 4 |
| VI. | A = BC | 24 | 4 |
| VII. | A = ABC | 24 | 3 |
| VIII. | AB = ABC | 8 | 1 |
| IX. | A = AB, a B = a B c | 24 | 3 |
| X. | A = ABC, ab = ab C | 8 | 4 |
| XI. | AB = ABC, ab = abc | 4 | 2 |
| XII. | AB = AC | 12 | 2 |
| XIII. | A = BC ꖌ A bc | 8 | 3 |
| XIV. | A = BC ꖌ bc | 2 | 4 |
| XV. | A = ABC, a = B c ꖌ b C | 8 | 5 |
There are sixty-three series of combinations derived from self-contradictory premises, which with 192, the sum of the numbers of distinct logical variations stated in the third column of the table, and with the one case where there are no conditions or laws at all, make up the whole conceivable number of 256 series.
We learn from this table, for instance, that two propositions of the form A = AB, B = BC, which are such as constitute the premises of the old syllogism Barbara, exclude as impossible four of the eight combinations in which three terms may be united, and that these propositions are capable of taking twenty-four variations by