A = ABC ꖌ ABcDEF ꖌ ABcDEf ꖌ ABcDeF.
The above gives the strict logical interpretation of the sentence, and the first alternative ABC is capable of development into eight cases, according as D, E and F are or are not present. Although from our knowledge of the matter, we may infer that weakness of character cannot be asserted of a person absolutely mad, there is no explicit statement to this effect.
Inference by Disjunctive Propositions.
Before we can make a free use of disjunctive propositions in the processes of inference we must consider how disjunctive terms can be combined together or with simple terms. In the first place, to combine a simple term with a disjunctive one, we must combine it with every alternative of the disjunctive term. A vegetable, for instance, is either a herb, a shrub, or a tree. Hence an exogenous vegetable is either an exogenous herb, or an exogenous shrub, or an exogenous tree. Symbolically stated, this process of combination is as follows,
A(B ꖌ C) = AB ꖌ AC.
Secondly, to combine two disjunctive terms with each other, combine each alternative of one with each alternative of the other. Since flowering plants are either exogens or endogens, and are at the same time either herbs, shrubs or trees, it follows that there are altogether six alternatives—namely, exogenous herbs, exogenous shrubs, exogenous trees, endogenous herbs, endogenous shrubs, endogenous trees. This process of combination is shown in the general form
(A ꖌ B) (C ꖌ D ꖌ E) = AC ꖌ AD ꖌ AE ꖌ BC ꖌ BD ꖌ BE.
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.