_Formulæ for solving Problems in Syllogisms._ 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 xm_{0} + ym'_{0} ¶ xy_{0} is a Formula, by which we can find the Conclusion to any Pair of Premisses whose subscript forms are xm_{0} + 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 xm_{0} + m'y_{0} which are identical with those in our _Formula_. Hence we at once know the Conclusion to be xy_{0} 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 ·---------------· ·-------· xm_{0} + ym'_{0} |(O) | | |(O)| | | ·---|---· | |---|---| | |(O)|(O)| | | | | |---|---|---|---| ·-------· | | | | | | ·---|---· | .'. xy_{0} |(O) | | ·---------------· 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 m_{1}x_{0} + ym'_{0} (which ¶ xy_{0}) xm'_{0} + m_{1}y_{0} (which ¶ xy_{0}) x'm_{0} + ym'_{0} (which ¶ x'y_{0}) m'_{1}x'_{0} + m_{1}y'_{0} (which ¶ x'y'_{0}).] If either Retinend is asserted in the _Premisses_ to exist, of course it may be so asserted in the _Conclusion_. Hence we get two _Variants_ of Fig. I, viz. (a) where _one_ Retinend is so asserted; (b) where _both_ are so asserted. [The Reader had better work out, on Diagrams, examples of these two Variants, such as