the proposition (1) is of the form A = B ꖌ C hence AB = B ꖌ BC and AC = BC ꖌ C; but these inferences are not equivalent to the false ones (2) and (3). We can readily represent disjunctive reasoning by the *modus ponendo tollens*, when it is valid, by expressing the inconsistency of the alternatives explicitly. Thus if we resort to our instance of Water is either salt or fresh, and take A = Water B = salt C = fresh, then the premise is apparently of the form A = AB ꖌ AC; but in reality there is an unexpressed condition that “what is salt is not fresh,” from which follows, by a process of inference to be afterwards described, that “what is fresh is not salt.” We have then, in letter-terms, the two propositions B = B*c* C = *b*C. If we substitute these descriptions in the original proposition, we obtain /* A = AB*c* ꖌ A*b*C; */ uniting B to each side we infer AB = AB*c* ꖌ AB*b*C or AB = AB*c*; that is, Water which is salt is water salt and not fresh. I should weary the reader if I attempted to illustrate the multitude of forms which disjunctive reasoning may take; and as in the next chapter we shall be constantly treating the subject, I must here restrict myself to a single instance. A very common process of reasoning consists in the determination of the name of a thing by the successive exclusion of alternatives, a process called by the old name *abscissio infiniti*. Take the case: Red-coloured metal is either copper or gold (1) Copper is dissolved by nitric acid (2) This specimen is red-coloured metal (3) This specimen is not dissolved by nitric acid (4) Therefore, this specimen consists of gold (5) Let us assign the letter-symbols thus-- A = this specimen B = red-coloured metal C = copper D = gold E = dissolved by nitric acid. Assuming that the alternatives copper or gold are intended to be exclusive, as just explained in the case of fresh and salt water, the premises may be stated in the forms B = BC*d* ꖌ B*c*D (1) C = CE (2) A = AB (3) A = A*e* (4) Substituting for C in (1) by means of (2) we get B = BC*d*E ꖌ B*c*D From (3) and (4) we may infer likewise A = AB*e* and if in this we substitute for B its equivalent just stated, it follows that A = ABC*d*E*e* ꖌ AB*c*D*e* The first of the alternatives being contradictory the result is A = AB*c*D*e* which contains a full description of “this specimen,” as furnished in the premises, but by ellipsis asserts that it is gold. It will be observed that in the symbolic expression (1) I have explicitly stated what is certainly implied, that copper is not gold, and gold not copper, without which condition the inference would not hold good. CHAPTER VI. THE INDIRECT METHOD OF INFERENCE.