[526] It may be observed that it is no part of our object to obtain a set of propositions which are mutually independent. As a matter of fact, it will generally be found that the maximum simplification involves the repetition of some items of information. Thus, in the example given in the preceding note the propositions All AB is CD and All Ab is C are quite independent of one another; but the proposition All A is C renders superfluous part of the information given by the proposition All AB is D.
The solution may, if we wish, be verified by recombining into a single complex proposition the propositions that have been obtained, an operation by which we shall arrive again at a series of alternants substantially identical with those originally given us. Such verification is, however, not essential to the validity of our process, which, if it has been correctly performed, contains no possible source of error.
The following examples will serve to illustrate the above method.
I. For our first example we may take one of those chosen by Jevons in the extract quoted in the preceding section.
Given the proposition, Everything is either ABC or Abc or aBC or abC, we are to find a set of propositions not involving alternative combination which shall be equivalent to it.
By the reduction of aBC or abC to aC, followed by contraposition, we have What is neither ABC nor Abc is aC ; therefore, What is a or Bc or bC is aC ; and this may be resolved into the three propositions:—
| ⎧ | All a is C, |
| ⎨ | Bc is non-existent, |
| ⎩ | All bC is a. |
Bc is non-existent is reducible to All B is C ; and this proposition and All a is C may be combined into All c is Ab.
529 Hence we have for our solution the two propositions:—
| ⎰ | All c is Ab, |
| ⎱ | All bC is a. |
It will be found that by the recombination of these propositions we regain the original proposition.