therefore every possible law which can exist concerning the relation of A and B must be marked by the exclusion of one or more of the above combinations. The number of possible laws then cannot exceed the number of selections which we can make from these four combinations. Since each combination may be present or absent, the number of cases to be considered is 2 × 2 × 2 × 2, or sixteen; and these cases are all shown in the following table, in which the sign 0 indicates absence or non-existence of the combination shown at the left-hand column in the same line, and the mark 1 its presence:—
| 1 | 2 | 3 | 4 | 5 | 6 | 7 * | 8 * | 9 | 10 * | 11 | 12 * | 13 | 14 * | 15 * | 16 * | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| AB | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| Ab | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
| aB | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
| ab | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |
Thus in column sixteen we find that all the conceivable combinations are present, which means that there are no special laws in existence in such a case, and that the combinations are governed only by the universal Laws of Identity and Difference. The example of metals and conductors of electricity would be represented by the twelfth column; and every other mode in which two things or qualities might present themselves is shown in one or other of the columns. More than half the cases may indeed be at once rejected, because they involve the entire absence of a term or its negative. It has been shown to be a logical principle that every term must have its negative (p. [111]), and when this is not the case, inconsistency between the conditions of combination must exist. Thus if we laid down the two following propositions, “Graphite conducts electricity,” and “Graphite does not conduct electricity,” it would amount to asserting the impossibility of graphite existing at all; or in general terms, A is B and A is not B result in destroying altogether the combinations containing A, a case shown in the fourth column of the above table. We therefore restrict our attention to those cases which may be represented in natural phenomena when at least two combinations are present, and which correspond to those columns of the table in which each of A, a, B, b appears. These cases are shown in the columns marked with an asterisk.
We find that seven cases remain for examination, thus characterised—
Four cases exhibiting three combinations,
Two cases exhibiting two combinations,
One case exhibiting four combinations.
It has already been pointed out that a proposition of the form A = AB destroys one combination, Ab, so that this is the form of law applying to the twelfth column. But by changing one or more of the terms in A = AB into its negative, or by interchanging A and B, a and b, we obtain no less than eight different varieties of the one form; thus—
| 12th case. | 8th case. | 15th case. | 14th case. | ||||
A = | AB | A = | Ab | a = | aB | a = | ab |
b = | ab | B = | aB | b = | Ab | B = | AB |
The reader of the preceding sections will see that each proposition in the lower line is logically equivalent to, and is in fact the contrapositive of, that above it (p. [83]). Thus the propositions A = Ab and B = aB both give the same combinations, shown in the eighth column of the table, and trial shows that the twelfth, eighth, fifteenth and fourteenth columns are thus accounted for. We come to this conclusion then—The general form of proposition A = AB admits of four logically distinct varieties, each capable of expression in two modes.
In two columns of the table, namely the seventh and tenth, we observe that two combinations are missing. Now a simple identity A = B renders impossible both Ab and aB, accounting for the tenth case; and if we change B into b the identity A = b accounts for the seventh case. There may indeed be two other varieties of the simple identity, namely a = b and a = B; but it has already been shown repeatedly that these are equivalent respectively to A = B and A = b (p. [115]). As the sixteenth column has already been accounted for as governed by no special conditions, we come to the following general conclusion:—The laws governing the combinations of two terms must be capable of expression either in a partial identity or a simple identity; the partial identity is capable of only four logically distinct varieties, and the simple identity of two. Every logical relation between two terms must be expressed in one of these six forms of law, or must be logically equivalent to one of them.
In short, we may conclude that in treating of partial and complete identity, we have exhaustively treated the modes in which two terms or classes of objects can be related. Of any two classes it can be said that one must either be included in the other, or must be identical with it, or a like relation must exist between one class and the negative of the other. We have thus completely solved the inverse logical problem concerning two terms.[85]