CHAPTER IV.
THE COMBINATION OF COMPLEX PROPOSITIONS.
474. The Problem of combining Complex Propositions.—Two or more complex propositions given in simple combination, either conjunctive or alternative, constitute a compound proposition. Hence the problem of dealing with a combination of complex propositions so as to obtain from them a single equivalent complex proposition, which is the problem to be considered in the present chapter, is identical with that of passing from a compound proposition to an equivalent complex proposition; and it is, therefore, the converse of the problem which was partially discussed in sections [446], 447. The latter problem, namely, that of passing from a complex to an equivalent compound proposition, will be further discussed in [chapter 6].
475. The Conjunctive Combination of Universal Affirmatives.—We may here distinguish two cases according as the propositions to be combined have or have not the same subject.
(1) Universal affirmatives having the same subject.
All X is P1 or P2 …… or Pm,
All X is Q1 or Q2 …… or Qn,
may for our present purpose be taken as types of universal affirmative propositions having the same subject. By conjunctively combining their predicates, thus,
All X is (P1 or P2 … or Pm) and also (Q1 or Q2 … or Qn),
that is, All X is P1Q1 or P1Q2 … or P1Qn
or P2Q1 or P2Q2 … or P2Qn
or ……
……
or PmQ1 or PmQ2 … or PmQn, 499
we may obtain a new proposition which is equivalent to the conjunctive combination of the two original propositions; it sums up all the information which they jointly contain, and we can pass back from it to them.
In almost all cases of the conjunctive combination of terms there are numerous opportunities of simplification; and, after a little practice, the student will find it unnecessary to write out all the alternants of the new predicate in full. The following are examples:—
| (i) | All X is AB or bce, |
| All X is aBC or DE ; | |
| therefore, | All X is ABDE. |
It will be found that all the other combinations in the predicate contain contradictories.
| (ii) | All X is A or Bc or D, |
| All X is aB or Bc or Cd ; | |
| therefore, | All X is ACd or aBD or Bc. |
| (iii) | Everything is A or bd or cE, |
| Everything is AC or aBe or d ; | |
| therefore, | Everything is AC or Ad or bd or cdE. |
(2) Universal affirmatives having different subjects.
Given the conjunctive combination of two universal affirmative propositions with different subjects, a new complex proposition may be obtained by conjunctively combining both their subjects and their predicates. Thus, if All X is P1 or P2 and All Y is Q1 or Q2, it follows that All XY is P1Q1 or P1Q2 or P2Q1 or P2Q2. But in this case the new proposition obtained is not equivalent to the conjunctive combination of the original propositions; and we cannot pass back from it to them.
A single complex proposition which sums up all the information contained in the original propositions may, however, be obtained by first reducing each of them to the form Everything is X1 or X2 … or Xn, and then conjunctively combining their predicates.
476. The Conjunctive Combination of Universal Negatives.—Here again we may distinguish two cases according as the propositions to be combined have or have not the same subject.
(1) Universal negatives having the same subject
No X is P1 or P2 …… or Pm,
No X is Q1 or Q2 …… or Qn,
may for our present purpose be taken as types of universal negative propositions having the same subject. Given these two propositions 500 in conjunctive combination, a new complex proposition may be obtained by alternatively combining their predicates. Thus,
No X is P1 or P2 …… or Pm or Q1 or Q2 …… or Qn.
This new proposition is equivalent to the two original propositions taken together, so that we can pass back from it to them. The process of combining the predicates is again likely to give opportunities of simplification. The following are examples:
| (i) | No X is either aB or aC or aE or bC or bE, |
| No X is either Ad or Ae or bd or be or cd or ce ; | |
| therefore, No X is either a or b or d or e.[498] | |
| (ii) | Nothing is aBC or aBe or aCD or aDe, |
| Nothing is AcD or abD or aDE or bcD or cDE ; | |
| therefore, Nothing is aBC or aBe or aD or cD. |
(2) Universal negatives having different subjects.
Given the conjunctive combination of two universal negative propositions with different subjects a new complex proposition may be obtained by conjunctively combining their subjects and alternatively combining their predicates. Thus, if No X is P1 or P2 and No Y is Q1, or Q2, it follows that No XY is P1 or P2 or Q1 or Q2. In this case the inferred proposition is not equivalent to the premisses; and we cannot pass back from it to them.
A single complex proposition which sums up all the information contained in the original propositions may, however, be obtained by first reducing each of them to the form Nothing is X1, or X2 … or Xn, and then alternatively combining their predicates.
477. The Conjunctive Combination of Universals with Particulars of the same Quality.—We may here consider, first, affirmatives, and then, negatives.
(1) Affirmatives. From the conjunctive combination of a universal affirmative and a particular affirmative having the same subject, a new particular affirmative proposition may be obtained by conjunctively combining their predicates. If All X is P1 or P2 and Some X is Q1 or Q2, it follows that Some X is P1Q1 or P1Q2 or P2Q1 or P2Q2. Here the particular premiss affirms the existence of X and of either XQ1 or XQ2; and the universal premiss implies that if X exists then either XP1 or XP2 exists.
We can pass back from the conclusion to the particular premiss, but not to the universal premiss. The conclusion is, therefore, not equivalent to the two premisses taken together.
501 A new complex proposition cannot be directly obtained from the conjunctive combination of a universal affirmative and a particular affirmative having different subjects. The propositions may, however, be reduced respectively to the forms Everything is P1 or P2 … or Pm, Something is Q1 or Q2 … or Qn, and their predicates may then be conjunctively combined in accordance with the above rule.
(2) Negatives. From the conjunctive combination of a universal negative and a particular negative having the same subject, a new particular negative proposition may be obtained by the alternative combination of their predicates. If No X is either P1 or P2 and Some X is not either Q1 or Q2 it follows that Some X is not either P1 or P2 or Q1 or Q2. The validity of this process is obvious since the particular premiss affirms the existence of X. By obversion it can also be exhibited as a corollary from the rule given above in regard to affirmatives. We can again pass back from the conclusion to the particular premiss, but not to the universal premiss.
With regard to the conjunctive combination of universal negatives and particular negatives having different subjects, the remarks made concerning affirmatives apply mutatis mutandis.
478. The Conjunctive Combination of Affirmatives with Negatives.—By first obverting one of the propositions, the conjunctive combination of an affirmative with a negative may be made to yield a new complex proposition in accordance with the rules given in the preceding sections. For example,
| (1) | All X is A or B, |
| No X is aC, | |
| therefore, All X is A or Bc ; | |
| (2) | Everything is P or Q, |
| Nothing is Pq or pR, | |
| therefore, Nothing is pR or q ; | |
| (3) | All X is AB or bce, |
| Some X is not either aBC or DE, | |
| therefore, Some X is ABd or ABe or bce. |
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.
EXERCISES.
483. Reduce the propositions All P is Q, No Q is R to such a form that the universe of discourse appears as the subject of each of them; and then combine the propositions into a single complex proposition. How is your result related to the ordinary syllogistic conclusion No P is R? [K.]
484. Combine the following propositions into a single equivalent complex proposition: All X is either A or b ; No X is either AC or acD or CD ; All a is B or x. [K.]
485. Every voter is both a ratepayer and an occupier, or not a ratepayer at all; If any voter who pays rates is an occupier, then he is on the list; No voter on the list is both a ratepayer and an occupier.
Examine the results of combining these three statements. [V.]
486. Every A is BC except when it is D ; everything which is not A is D ; what is both C and D is B ; and every D is C. What can be determined from these premisses as to the contents of our universe of discourse? [M.]