437. Give the contradictories of the following terms in their simplest forms as series of alternants:—AB or BC or CD ; AB or bC or cD ; ABC or aBc ; ABcD or Abcde or aBCDe or BCde. [K.]
438. Simplify the following terms:
(1) Ab or aC or BCd or Bc or bD or CD ;
(2) ACD or Ac or Ad or aB or bCD ;
(3) aBC or aBe or aCD or aDe or AcD or abD or bcD or aDE or cDE ;
(4) (A or b) (A or c) (a or B) (a or C) (b or C). [K.]
439. Prove the following equivalences:
(1) AB or AC or BC or aB or abc or C = a or B or C ;
(2) aBC or aBd or acd or ABd or Acd or abd or aCd or BCd or bcd = aBC or ad or Bd or cd ;
(3) Pqr or pQs or pq or prs or qrs or pS or qR = p or q. [K.]
CHAPTER II.
COMPLEX PROPOSITIONS AND COMPOUND PROPOSITIONS.
440. Complex Propositions.—A complex proposition may be defined as a proposition which has a complex term either for its subject or its predicate. The ordinary distinctions of quantity and quality may be applied to complex propositions; thus All AB is C or D is a universal affirmative complex proposition. Some AB is not EF is a particular negative complex proposition. In the following pages propositions written in the indefinite form will be interpreted as universal, so that AB is CD will be understood to mean that all AB is CD. It is to be added that in dealing with complex propositions we interpret particulars as implying, but universals as not implying, the existence of their subjects in the universe of discourse.
441. The Opposition of Complex Propositions.—The opposition of complex terms has been already dealt with, and the opposition of complex propositions in itself presents no special difficulty. It must, however, be borne in mind that as we interpret particulars as implying the existence of their subjects, but universals as not doing so, we have the following divergences from the ordinary doctrine of opposition: (1) we cannot infer I from A, or O from E; (2) A and E are not necessarily inconsistent with each other; (3) I and O may both be false at the same time. The ordinary doctrine of contradictory opposition remains unaffected. The following are examples of contradictory propositions: All X is both A and B, Some X is not both A and B ; Some X is Y and at the same time either P or Q or R, No X is Y and at the same time either P or Q or R.
442. Compound Propositions.[485]—A compound proposition may be defined as a proposition which consists in a combination of other propositions. The combination may be either conjunctive (i.e., when 479 two or more propositions are affirmed to be true together) or alternative (i.e., when an alternative is given between two or more propositions); for example, All AB is C and some P is not either Q or R is a compound conjunctive proposition; Either all AB is C or some P is not either Q or R is a compound alternative proposition. Propositions conjunctively combined may be spoken of as determinants of the resulting compound proposition; and propositions alternatively combined may be spoken of as alternants of the resulting compound proposition. In what follows, both conjunctive and alternative propositions are interpreted as being assertoric.