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—
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 = | A b | a = | a B | a = | ab |
| b = | ab | B = | a B | b = | A b | 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 A b and a B, 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