Hence our proposition is resolvable into the following:
| (i) | All AbC is DE ; |
| (ii) | All ade is BC ; |
| (iii) | All BE is ACD ; |
| (iv) | AcD is non-existent ; |
| (v) | All DE is AC. |
But by (v) All BE is AC or d ; therefore, (iii) may be reduced to All BE is D. Again by (iv), All DE is a or C ; therefore, (v) may be reduced to All DE is A.
Hence we have the following as our final solution:—
| (1) All AbC is DE ; |
| (2) All ade is BC ; |
| (3) All BE is D ; |
| (4) All cD is a ; |
| (5) All DE is A. |
536. Another Method of Solution of the Inverse Problem.—Another method of solving the inverse problem, suggested to me by Dr Venn, is to write down the original complex proposition in the negative form, i.e., to obvert it, before resolving it. It has been already shewn that a negative proposition with an alternative predicate may be immediately broken up into a set of simpler propositions.
In some cases, especially where the number of destroyed combinations as compared with those that are saved is small this plan is of easier application than that given in the preceding section.
531 To illustrate this method we may take two or three of the examples already discussed.
I. Everything is ABC or Abc or aBC or abC ;
therefore, by obversion, Nothing is AbC or ac or Bc ;
and this proposition is at once resolvable into
| ⎰ | All Ab is c, |
| ⎱ | All c is Ab.[528] |