and, as the first of the three alternatives is self-contradictory, we strike it out according to the law of contradiction: there remains
Ab = AbC ꖌ AbD.
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