And we should find this Rule to hold good with any Pair of Premisses which fulfil the given conditions.
[The Reader had better satisfy himself of this, by working out, on Diagrams, several varieties, such as
x′m0 † m1y0 (which ¶ xy′1)
m′1x0 † m′y′0 (which ¶ x′y1)
m1x′0 † m1y′0 (which ¶ xy1).]
The Formula, to be remembered, is
xm0 † ym0 † m1 ¶ x′y′1
with the following Rule (which is merely the Formula expressed in words):—
Two Nullities, with Like Eliminands asserted to exist, yield an Entity, in which both Retinends change their Signs.
In order to help the Reader to remember the peculiarities and Formulæ of these three Figures, I will put them all together in one Table.
| [pg078]TABLE IX. [Fig. I.] xm0 † ym′0 ¶ xy0 Two Nullities, with Unlike Eliminands, yield a Nullity, in which both Retinends keep their Signs. A Retinend, asserted in the Premisses to exist, may be so asserted in the Conclusion. [Fig. II.] xm0 † ym1 ¶ x′y1 A Nullity and an Entity, with Like Eliminands, yield an Entity, in which the Nullity-Retinend changes its Sign. [Fig. III.] xm0 † ym0 † m1 ¶ x′y′1 Two Nullities, with Like Eliminands asserted to exist, yield an Entity, in which both Retinends change their Signs. |
| [Fig. I.] xm0 † ym′0 ¶ xy0 Two Nullities, with Unlike Eliminands, yield a Nullity, in which both Retinends keep their Signs. A Retinend, asserted in the Premisses to exist, may be so asserted in the Conclusion. |
| [Fig. II.] xm0 † ym1 ¶ x′y1 A Nullity and an Entity, with Like Eliminands, yield an Entity, in which the Nullity-Retinend changes its Sign. |
| [Fig. III.] xm0 † ym0 † m1 ¶ x′y′1 Two Nullities, with Like Eliminands asserted to exist, yield an Entity, in which both Retinends change their Signs. |
I will now work out, by these Formulæ, as models for the Reader to imitate, some Problems in Syllogisms which have been already worked, by Diagrams, in [Book V., Chap. II.]