There is a simple but highly important process of inference which enables us to abstract, eliminate or disregard all circumstances indifferently present and absent. Thus if I were to state that “a triangle is a three-sided rectilinear figure, either large or not large,” these two alternatives would be superfluous, because, by the Law of Duality, I know that everything must be either large or not large. To add the qualification gives no new knowledge, since the existence of the two alternatives will be understood in the absence of any information to the contrary. Accordingly, when two alternatives differ only as regards a single component term which is positive in one and negative in the other, we may reduce them to one term by striking out their indifferent part. It is really a process of substitution which enables us to do this; for having any proposition of the form A = ABC ꖌ AB*c*, (1) we know by the Law of Duality that AB = ABC ꖌ AB*c*. (2) As the second member of this is identical with the second member of (1) we may substitute, obtaining A = AB. This process of reducing useless alternatives may be applied again and again; for it is plain that A = AB (CD ꖌ C*d* ꖌ *c*D ꖌ *cd*) communicates no more information than that A is B. Abstraction of indifferent terms is in fact the converse process to that of development described in p. 89; and it is one of the most important operations in the whole sphere of reasoning. The reader should observe that in the proposition AC = BC we cannot abstract C and infer A = B; but from AC ꖌ A*c* = BC ꖌ B*c* we may abstract all reference to the term C. It ought to be carefully remarked, however, that alternatives which seem to be without meaning often imply important knowledge. Thus if I say that “a triangle is a three-sided rectilinear figure, with or without three equal angles,” the last alternatives really express a property of triangles, namely, that some triangles have three equal angles, and some do not have them. If we put P = “Some,” meaning by the indefinite adjective “Some,” one or more of the undefined properties of triangles with three equal angles, and take A = triangle B = three-sided rectilinear figure C = with three equal angles, then the knowledge implied is expressed in the two propositions PA = PBC *p*A = *p*B*c*. These may also be thrown into the form of one proposition, namely, A = PBC ꖌ *p*B*c*; but these alternatives cannot be reduced, and the proposition is quite different from A = BC ꖌ B*c*. *Illustrations of the Indirect Method.* A great variety of arguments and logical problems might be introduced here to show the comprehensive character and powers of the Indirect Method. We can treat either a single premise or a series of premises. Take in the first place a simple definition, such as “a triangle is a three-sided rectilinear figure.” Let A = triangle B = three-sided C = rectilinear figure,