By contraposition one or more of these complex terms may be brought over from the predicate into the subject, so that we have
Whatever is not either P or S or &c. is Q or T or &c.
The selection of certain terms for transposition in this way is arbitrary (and it is here that the indeterminateness of the problem becomes apparent); but it will generally be found best to take two or three which have as many common determinants as possible.
What is not either P or S or &c. is Q or T or &c.
will, when the subject is written in the affirmative form, be immediately resolvable into a series of propositions, which taken together give all the information originally given.[522] Any of these propositions which still involve alternative combination may be dealt with in the same way, until no alternative combination remains.
We shall now be left with a set of propositions which will satisfy the required conditions. The possibility of various simplifications has, however, to be considered. Thus, it will be necessary to make sure that each of the propositions is itself expressed in its simplest form;[523] and to observe whether any two or more of the propositions 528 admit of a simple recombination.[524] It may also be found that some of the propositions can be altogether omitted, inasmuch as they add nothing to the information jointly afforded by the remainder; or that, considered in their relation to the remaining propositions, they may, at any rate, be simplified by the omission of one or more of the terms which they contain.[525] When these simplifications have been carried as far as is possible we shall have our final solution.[526]
[523] For example, All AB is BC may be reduced to All AB is C.
[524] For example, All ac is d and All Bc is d may be combined into All cD is Ab.
[525] Thus, for the propositions All AB is CD and All Ab is C we may substitute the propositions All AB is D and All A is C.