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.