(5) In a universal negative or a particular affirmative proposition the contradictory of any determinant of the subject may be indifferently introduced or omitted as an alternant of the predicate.
By this rule the two following propositions are affirmed to be equivalent to one another: No AB is a or C ; No AB is C ; and also the two following: Some AB is a or C ; Some AB is C.
The rule follows directly from rule (2) by obversion.
(6) In a universal negative or a particular affirmative proposition the contradictory of any determinant of the predicate may be indifferently introduced or omitted as an alternant of the subject.
This rule follows from rule (3) by obversion.
446. The Resolution of Universal Complex Propositions into Equivalent Compound Propositions.—We may enquire how far complex propositions are immediately resolvable into a conjunctive or alternative combination of relatively simple propositions. Universal propositions will be considered in this section, and particulars in the next.
Universal Affirmatives. Universal affirmative complex propositions may be immediately resolved into a conjunction of relatively simple ones, so far as there is alternative combination in the subject or conjunctive combination in the predicate. Thus,
(1) Whatever is P or Q is R = All P is R and all Q is R ;
(2) All P is QR = All P is Q and all P is R.
Universal Negatives. Universal negative complex propositions may be immediately resolved into a conjunction of relatively simple ones, so far as there is alternative combination either in the subject or in the predicate. Thus,
(3) Nothing that is P or Q is R = No P is R and no Q is R ;
(4) No P is either Q or R = No P is Q and no P is R.
So far as there is conjunctive combination in the subject or alternative combination in the predicate of universal affirmative propositions, or conjunctive combination either in the subject or in the predicate of universal negative propositions, they cannot be 484 immediately[493] resolved into either a conjunctive or an alternative combination of simpler propositions. It may, however, be added that propositions falling into this latter category are immediately implied by certain compound alternatives. Thus,
(i) All PQ is R is implied by All P is R or all Q is R ;
(ii) All P is Q or R is implied by All P is Q or all P is R ;
(iii) No PQ is R is implied by No P is R or no Q is R ;
(iv) No P is QR is implied by No P is Q or no P is R.
[493] It will be shewn subsequently that even in these cases universal complex propositions may be resolved into a conjunction of relatively simpler ones by the aid of certain immediate inferences.