All Ab is C or X ; (i)
with the five corresponding propositions;
All Ac is B or X ; (ii)
All Ax is B or C ; (iii)
All bc is a or X ; (iv)
All bx is a or C ; (v)
All cx is a or B. (vi)
By a repetition of the same process, we have
| All Abc is X (which is the original proposition over again); | (α) | |
| and corresponding to this: | All Abx is C ; | (β) |
| All Acx is B ; | (γ) | |
| All bcx is a. | (δ) | |
It will be observed that the following are pairs of full contrapositives;—(1) (δ), (2) (γ), (3) (β), (4) (α), (i) (vi), (ii) (v), (iii) (iv).
A further series of propositions may be obtained by obverting all the above; and as there has been no loss of logical power in any of the processes employed we have in all thirty propositions that are equivalent to one another.
461. If AB is either Cd or cDe, and also either eF or H, and if the same is true of BH, what do we know of that which is E? [K.]
Whatever is AB or BH is (Cd or cDe) and (eF or H);
therefore, Whatever is AB or BH is CdeF or cDeF or CdH or cDeH ;
therefore, Whatever is ABE or BHE is CdH ;
therefore, All E is ah or b or CdH.
462. Given A is BC or BDE or BDF, infer descriptions of the terms Ace, Acf, ABcD. [Jevons, Studies, pp. 237, 238.]