It is hardly necessary to point out that, however numerous the terms combined, or the alternatives in those terms, we may effect the combination, provided each alternative is combined with each alternative of the other terms, as in the algebraic process of multiplication. Some processes of deduction may be at once exhibited. We may always, for instance, unite the same qualifying term to each side of an identity even though one or both members of the identity be disjunctive. Thus let A = B ꖌ C. Now it is self-evident that AD = AD, and in one side of this identity we may for A substitute its equivalent B ꖌ C, obtaining AD = BD ꖌ CD. Since “a gaseous element is either hydrogen, or oxygen, or nitrogen, or chlorine, or fluorine,” it follows that “a free gaseous element is either free hydrogen, or free oxygen, or free nitrogen, or free chlorine, or free fluorine.” This process of combination will lead to most useful inferences when the qualifying adjective combined with both sides of the proposition is a negative of one or more alternatives. Since chlorine is a coloured gas, we may infer that “a colourless gaseous element is either (colourless) hydrogen, oxygen, nitrogen, or fluorine.” The alternative chlorine disappears because colourless chlorine does not exist. Again, since “a tooth is either an incisor, canine, bicuspid, or molar,” it follows that “a not-incisor tooth is either canine, bicuspid, or molar.” The general rule is that from the denial of any of the alternatives the affirmation of the remainder can be inferred. Now this result clearly follows from our process of substitution; for if we have the proposition A = B ꖌ C ꖌ D, and we insert this expression for A on one side of the self-evident identity A*b* = A*b*, we obtain A*b* = AB*b* ꖌ A*b*C ꖌ A*b*D; and, as the first of the three alternatives is self-contradictory, we strike it out according to the law of contradiction: there remains A*b* = A*b*C ꖌ A*b*D. Thus our system fully includes and explains that mood of the Disjunctive Syllogism technically called the *modus tollendo ponens*. But the reader must carefully observe that the Disjunctive Syllogism of the mood *ponendo tollens*, which affirms one alternative, and thence infers the denial of the rest, cannot be held true in this system. If I say, indeed, that Water is either salt or fresh water, it seems evident that “water which is salt is not fresh.” But this inference really proceeds from our knowledge that water cannot be at once salt and fresh. This inconsistency of the alternatives, as I have fully shown, will not always hold. Thus, if I say Gems are either rare stones or beautiful stones, (1) it will obviously not follow that A rare gem is not a beautiful stone, (2) nor that A beautiful gem is not a rare stone. (3) Our symbolic method gives only true conclusions; for if we take A = gem B = rare stone C = beautiful stone,