Therefore, by contraposition, Whatever is not ABC or AbD or aBc is ABcd or aBCD or abCd.
But Whatever is not ABC or AbD or aBc is equivalent to Whatever is ABc or aBC or ab or bd. Hence we have for our solution the following set of propositions:
| (1) All ABc is d, | (2) All aBC is D, |
| (3) All ab is Cd, | (4) All bd is a.[527] |
This is equivalent to the solution given by Jevons, Studies, p. 256.
[527] We first obtain All bd is aC ; but since by (3) All abd is C, this may be reduced to All bd is a.
530 IV. The following example is also from Jevons, Principles of Science, 2nd edition, p. 127 (Problem viii). In his Studies, p. 256, he speaks of the solution as unknown. A fairly simple solution may, however, be obtained by the application of the general rule formulated in this section.
The given alternants are ABCDE, ABCDe, ABCde, ABcde, AbCDE, AbcdE, Abcde, aBCDe, aBCde, aBcDe, abCDe, abCdE, abcDe, abcdE.
By the reduction of duals these alternants may be written: ABCe or ABcde or Abcd or ACDE or aBCde or abdE or aDe.
Therefore, by contraposition, Whatever is not either ABCe or ABcde or Abcd or abdE or aDe is ACDE or aBCde.
But it will be found that, by the application of the ordinary rule for obtaining the contradictory of a given term, Whatever is not either ABCe or ABcde or Abcd or abdE or aDe is equivalent to Whatever is AbC or ade or BE or AcD or DE.