And now how are we to represent the contradictory Proposition "SOME x are m"? This is a difficulty I have already considered. I think the best way is to place a red counter ON THE DIVISION-LINE between No. 11 and No. 12, and to understand this to mean that ONE of the two compartments is 'occupied,' but that we do not at present know WHICH. This I shall represent thus:--

------------------- | | | | _____|_____ | | | | | | | | -1- | | | | | | | -------------------

Now let us express "all x are m."

This consists, we know, of TWO Propositions,

"Some x are m," and "No x are m'."

Let us express the negative part first. This tells us that none of the Cakes, belonging to the upper half of the cupboard, are to be found OUTSIDE the central Square: that is, the two compartments, No. 9 and No. 10, are EMPTY. This, of course, is represented by

------------------- | 0 | 0 | | _____|_____ | | | | | | | | | | | | | | | | -------------------

But we have yet to represent "Some x are m." This tells us that there are SOME Cakes in the oblong consisting of No. 11 and No. 12: so we place our red counter, as in the previous example, on the division-line between No. 11 and No. 12, and the result is

------------------- | 0 | 0 | | _____|_____ | | | | | | | | -1- | | | | | | | -------------------

Now let us try one or two interpretations.

What are we to make of this, with regard to x and y?

------------------- | | 0 | | _____|_____ | | | | | | | | 1 | 0 | | | | | | | -------------------

This tells us, with regard to the xy'-Square, that it is wholly 'empty', since BOTH compartments are so marked. With regard to the xy-Square, it tells us that it is 'occupied'. True, it is only ONE compartment of it that is so marked; but that is quite enough, whether the other be 'occupied' or 'empty', to settle the fact that there is SOMETHING in the Square.

If, then, we transfer our marks to the smaller Diagram, so as to get rid of the m-subdivisions, we have a right to mark it

----------- | | | | 1 | 0 | | | | -----------

which means, you know, "all x are y."

The result would have been exactly the same, if the given oblong had been marked thus:--

------------------- | 1 | 0 | | _____|_____ | | | | | | | | | 0 | | | | | | | -------------------

Once more: how shall we interpret this, with regard to x and y?

------------------- | 0 | 1 | | _____|_____ | | | | | | | | | | | | | | | | -------------------

This tells us, as to the xy-Square, that ONE of its compartments is 'empty'. But this information is quite useless, as there is no mark in the OTHER compartment. If the other compartment happened to be 'empty' too, the Square would be 'empty': and, if it happened to be 'occupied', the Square would be 'occupied'. So, as we do not know WHICH is the case, we can say nothing about THIS Square.

The other Square, the xy'-Square, we know (as in the previous example) to be 'occupied'.

If, then, we transfer our marks to the smaller Diagram, we get merely this:--

----------- | | | | | 1 | | | | -----------

which means, you know, "some x are y'."

These principles may be applied to all the other oblongs. For instance, to represent "all y' are m'" we should mark the ------- RIGHT-HAND UPRIGHT OBLONG (the one | | that has the attribute y') thus:-- |--- | | 0 | | |---|-1-| | 0 | | |--- | | | -------

and, if we were told to interpret the lower half of the cupboard, marked as follows, with regard to x and y,

------------------- | | | | | | | | 0 | | | | | | | | -----|----- | | 1 | 0 | -------------------