479. The Conjunctive Combination of Particulars with Particulars.—Particulars cannot to any purpose be conjunctively combined with particulars so as to yield a new complex proposition. It is true that from Some X is P1 or P2 and some X is Q1 or Q2, we can pass to Some X is P1 or P2 or Q1 or Q2. But this is a mere weakening of the information given by either of the premisses singly; and by the rule that an alternant may at any time be introduced into an undistributed term (section [449]), it could equally well be inferred 502 from either premiss taken by itself. Again from Some X is not either P1 or P2 and some X is not either Q1 or Q2 we can pass to Some X is not either P1Q1 or P1Q2 or P2Q1 or P2Q2. But similar remarks again apply, since we have already found that a determinant may at any time be introduced into a distributed term.
480. The Alternative Combination of Universal Propositions.—Given a number of universal propositions as alternants in a compound alternative proposition we cannot obtain a single equivalent complex proposition. From the compound proposition Either all A is P1 or P2 or all A is Q1 or Q2 we can indeed infer All A is P1 or P2 or Q1 or Q2; but we cannot pass back from this to the original proposition.[499]
481. The Alternative Combination of Particular Propositions.—It follows from the equivalences shewn in section [447] that a compound alternative proposition in which all the alternants are particular can be reduced to the form of a single complex proposition. If all the alternants of the compound proposition have the same subject and are all affirmative, their predicates must be alternatively combined in the complex proposition; if they all have the same subject and are all negative, their predicates must be conjunctively combined in the complex proposition. If the alternants have different subjects, they must all be reduced to the form Something is … before their predicates are combined; if they differ in quality, recourse must be had to the process of obversion. It is unnecessary to discuss these different cases in detail, but the following may be taken as examples:
| (i) | Some X is P or some X is Q = Some X is P or Q ; |
| (ii) | Some X is not P or some X is not Q = Some X is not PQ ; |
| (iii) | Some X is P or some Y is Q = Something is XP or YQ ; |
| (iv) | Some X is P or some Y is not Q = Something is XP or Yq. |
482. The Alternative Combination of Particulars with Universals.—From a compound alternative proposition in which some of the alternants are particular and some universal, we can infer a particular complex proposition; but in this case we cannot pass back from the complex proposition to the compound proposition. The following are examples:
| (1) | All A is P or some A is Q, therefore, Something is a or P or Q ;[500] |
| (2) | All A is P or some B is not Q, therefore, Something is a or Bq or P. |
[500] We cannot infer Some A is P or Q, since this implies the existence of A, whereas the non-existence of A is compatible with the premiss.