2.  Syllogisms

Now suppose we divide our Universe of Things in three ways, with regard to three different Attributes.  Out of these three Attributes, we may make up three different couples (for instance, if they were a, b, c, we might make up the three couples ab, ac, bc).  Also suppose we have two Propositions given us, containing two of these three couples, and that from them we can prove a third Proposition containing the third couple. (For example, if we divide our Universe for m, x, and y; and if we have the two Propositions given us, “no m are x’” and “all m’ are y”, containing the two couples mx and my, it might be possible to prove from them a third Proposition, containing x and y.)

In such a case we call the given Propositions ‘the Premisses’, the third one ‘the conclusion’ and the whole set ‘A syllogism’.

Evidently, one of the Attributes must occur in both Premisses; or else one must occur in one Premiss, and its contradictory in the other.

In the first case (when, for example, the Premisses are “some m are x” and “no m are y’”) the Term, which occurs twice, is called ‘the middle term’, because it serves as a sort of link between the other two Terms.

In the second case (when, for example, the Premisses are “no m are x’” and “all m’ are y”) the two Terms, which contain these contradictory Attributes, may be called ‘the middle terms’.

Thus, in the first case, the class of “m-Things” is the Middle Term; and, in the second case, the two classes of “m-Things” and “m’-Things” are the Middle Terms.

The Attribute, which occurs in the Middle Term or Terms, disappears in the Conclusion, and is said to be “eliminated”, which literally means “turned out of doors”.

Now let us try to draw a Conclusion from the two Premisses—­

“Some new Cakes are unwholesome;
No nice Cakes are unwholesome.”

In order to express them with counters, we need to divide Cakes in three different ways, with regard to newness, to niceness, and to wholesomeness.  For this we must use the larger Diagram, making x mean “new”, y “nice”, and m “wholesome”. (Everything inside the central Square is supposed to have the attribute m, and everything outside it the attribute m’, i.e. “not-m".)

You had better adopt the rule to make m mean the Attribute which occurs in the middle Term or Terms. (I have chosen m as the symbol, because ‘middle’ begins with ’m’.)

Now, in representing the two Premisses, I prefer to begin with the negative one (the one beginning with “no"), because grey counters can always be placed with certainty, and will then help to fix the position of the red counters, which are sometimes a little uncertain where they will be most welcome.

Let us express, the “no nice Cakes are unwholesome (Cakes)”, i.e. “no y-Cakes are m’-(Cakes)”.  This tells us that none of the Cakes belonging to the y-half of the cupboard are in its m’-compartments (i.e. the ones outside the central Square).  Hence the two compartments, No. 9 and No. 15, are both ‘empty’; and we must place a grey counter in each of them, thus:—­