When once we have found, by Diagrams, the Conclusion to a given Pair of Premisses, and have represented the Syllogism in subscript form, we have a Formula, by which we can at once find, without having to use Diagrams again, the Conclusion to any other Pair of Premisses having the same subscript forms.
[Thus, the expression
xm0 † ym′0 ¶ xy0
is a Formula, by which we can find the Conclusion to any Pair of Premisses whose subscript forms are
xm0 † ym′0
For example, suppose we had the Pair of Propositions
“No gluttons are healthy;
No unhealthy men are strong”.proposed as Premisses. Taking “men” as our ‘Universe’, and making m = healthy; x = gluttons; y = strong; we might translate the Pair into abstract form, thus:—
“No x are m;
No m′ are y”.These, in subscript form, would be
xm0 † m′y0
which are identical with those in our Formula. Hence we at once know the Conclusion to be
xy0
that is, in abstract form,
“No x are y”;
that is, in concrete form,
“No gluttons are strong”.]
I shall now take three different forms of Pairs of Premisses, and work out their Conclusions, once for all, by Diagrams; and thus obtain some useful Formulæ. I shall call them “Fig. I”, “Fig. II”, and “Fig. III”.
[pg075]Fig. I.
This includes any Pair of Premisses which are both of them Nullities, and which contain Unlike Eliminands.
The simplest case is
| xm0 † ym′0 | |
 |  ∴ xy0 |
In this case we see that the Conclusion is a Nullity, and that the Retinends have kept their Signs.
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
m1x0 † ym′0 (which ¶ xy0)
xm′0 † m1y0 (which ¶ xy0)
x′m0 † ym′0 (which ¶ x′y0)
m′1x′0 † m1y′0 (which ¶ x′y′0).]