453. Simplify the following propositions:—
(1) All AB is BC or be or CD or cE or DE ;
(2) Nothing that is either PQ or PR is Pqr or pQs or pq or prs or qrs or pS or qR. [K.]
CHAPTER III.
IMMEDIATE INFERENCES FROM COMPLEX PROPOSITIONS.
454. The Obversion of Complex Propositions—The doctrine of obversion is immediately applicable to complex propositions; and no modification of the definition of obversion already given is necessary. From any given proposition we may infer a new one by changing its quality and taking as a new predicate the contradictory of the original predicate. The proposition thus obtained is called the obverse of the original proposition.
The only difficulty connected with the obversion of complex propositions consists in finding the contradictory of a complex term; but a simple rule for performing this process has been given in section [426]:—Replace all the simple terms invoked by their contradictories, and throughout substitute alternative combination for conjunctive and vice versâ.
Applying this rule to AB or ab, we have (a or b) and (A or B), that is, Aa or Ab or aB or Bb ; but since the alternants Aa and Bb involve self-contradiction, they may by rule (5) of section [433] be omitted. The obverse, therefore, of All X is AB or ab is No X is Ab or aB.
As additional examples we may find the obverse of the following propositions: (1) All A is BC or DE ; (2) No A is BcE or BCF ; (3) Some A is not either B or bcDEf or bcdEF.
(1) All A is BC or DE yields No A is (b or c) and at the same time (d or e), or, by the reduction of the predicate to a series of alternants, No A is bd or be or cd or ce.
(2) No A is BcE or BCF. Here the contradictory of the 489 predicate is (b or C or e) and (b or c or f), which yields b or Cc or Cf or ce or ef. Cc may be omitted by rule (5) of section [433]; also ef by rule (7), since ef is either Cef or cef. Hence the required obverse is All A is b or Cf or ce.