“In the Indirect process of Inference we found that from certain propositions we could infallibly determine the combinations of terms agreeing with those premisses. The inductive problem is just the inverse. Having given certain combinations of terms, we need to ascertain the propositions with which they are consistent, and from which they may have proceeded. Now if the reader contemplates the following combinations,—
| ABC | abC | |
| aBC | abc, |
he will probably remember at once that they belong to the premisses A = AB, B = BC. If not, he will require a few trials before he meets with the right answer, and every trial will consist in assuming certain laws and observing whether the deduced results agree with the data. To test the facility with which he can solve this inductive problem, let him casually strike out any of the possible combinations involving three terms, and say what laws the remaining combinations obey. Let him say, for instance, what laws are embodied in the combinations,—
| ABC | aBC | |
| Abc | abC, |
“The difficulty becomes much greater when more terms enter 526 into the combinations. It would be no easy matter to point out the complete conditions fulfilled in the combinations,—
| ACe |
| aBCe |
| aBcdE |
| abCe |
| abcE. |
After some trouble the reader may discover that the principal laws are C = e, and A = Ae ; but he would hardly discover the remaining law, namely that BD = BDe” (Principles of Science, 1st ed., vol. I., p. 144; 2nd ed., p. 125).
“The inverse problem is always tentative, and consists in inventing laws, and trying whether their results agree with those before us” (Studies in Deductive Logic, p. 252).
The problem may preferably be stated as follows:—
Given a complex proposition of the form
Everything is P1P2 … or Q1Q2 … or …,