76 THE PRINCIPLES OF SCIENCE. [chap. 



F = subjected to most unfavourable circumstances, 

 the proposition will take the form 



A = ABjO-l-D (E-l-F)}, 

 and if we develop the alternatives, expressing some of 

 the different cases which uuiy happen, we obtain 



A = ABC -I- ABrDEF •!• ABcDE/-]- ABcDcF. 

 The above gives the strict logical interpretation of the 

 sentence, and the first alternative ABC is capable of de- 

 velopment 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, hy Disjunctive Propositions. 



Before we can make a free use of disjunctive proposi- 

 tions in the i^rocesses of inference w^e 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 nmst 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.|.C) = AB-l-AC. 



Secondly, to combine two disjunctive terms with each 

 other, combine each alternative of one with each alterna- 

 tive of the other. Since flowering plants are either 

 exogens or endoirens, 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 -I- B) (C -I- D -I- E) = AC •!• AD •!• AE -I- BC -I- BD -I- 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 alter- 

 native is combined with each alternative of the other 

 terms, as in the algebraic process of multiplication. 



