447. The Resolution of Particular Complex Propositions into Equivalent Compound Propositions.—Particular complex propositions cannot be resolved into compound conjunctives, but they may under certain conditions be immediately resolved into equivalent compound alternative propositions in which the alternants are relatively simple. This is the case so far as there is alternative combination in the subject or conjunctive combination in the predicate of a particular negative, or alternative combination either in the subject or in the predicate of a particular affirmative. Thus,
(1) Some P or Q is not R = Some P is not R or some Q is not R ;
(2) Some P is not QR = Some P is not Q or some P is not R ;
(3) Some P or Q is R = Some P is R or some Q is R ;
(4) Some P is Q or R = Some P is Q or some P is R.
Particular complex propositions cannot be immediately resolved into compound propositions (either conjunctive or alternative) so far as there is conjunctive combination in the subject or alternative combination in the predicate if the proposition is negative, or so far as there is conjunctive combination either in the subject or in the predicate if the proposition is affirmative. In these cases, however, the complex proposition implies a compound conjunctive proposition, though we cannot pass back from the latter to the former. Thus,
(i) Some PQ is not R implies Some P is not R and Some Q is not R ;
(ii) Some P is not either Q or R implies Some P is not Q and some P is not R ;
(iii) Some PQ is R implies Some P is R and some Q is R ;
(iv) Some P is QR implies Some P is Q and some P is R.
It must be particularly noticed that, although in these cases the 485 compound proposition can be inferred from the complex proposition, still the two are not equivalent. For example, from Some P is Q and some P is R it does not follow that Some P is QR, for we cannot be sure that the same P’s are referred to in the two cases.
All the results of this section follow from those of the preceding section by the application of the rule of contradiction to the propositions themselves and the rule of contraposition to the relations of implication between them.
448. The Omission of Terms from a Complex Proposition.—From the two preceding sections we may obtain immediately the following rules for inferring from a given proposition another proposition in which certain terms contained in the original proposition are omitted:
(1) Any determinant may be omitted from an undistributed term ;[494]
(2) Any alternant may be omitted from a distributed term.[495]
[494] The subject of a particular or the predicate of an affirmative proposition.
[495] The subject of a universal or the predicate of a negative proposition.
For example,—
Whatever is A or B is CD, therefore, All A is C ;
Some AB is CD, therefore, Some A is C ;
Nothing that is A or B is C or D, therefore, No A is C ;
Some AB is not either C or D, therefore, Some A is not C.
The above rules may also be justified independently, as will be shewn in the following section. The results which they yield must be distinguished from those obtained in section [445]. In the cases discussed in that section, the terms omitted were superfluous in the sense that their omission left us with propositions equivalent to our original propositions; but in the above inferences we cannot pass back from conclusion to premiss. From Some A is C, for example, we cannot infer that Some AB is C.
449. The Introduction of Terms into a Complex Proposition.—Corresponding to the rules laid down in the preceding section we have also the following:
(1) Any determinant may be introduced into a distributed term ;
(2) Any alternant may be introduced into an undistributed term.