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.