The question is a very thoughtful one, and does you GREAT credit, dear Reader! It DOES mean both. If you choose to take x (that is, "new Cakes") as your Subject, and to regard No. 5 as part of a HORIZONTAL oblong, you may read it "some x are y", that is, "some new Cakes are nice": but, if you choose to take y (that is, "nice Cake") as your Subject, and to regard No. 5 as part of an UPRIGHT oblong, THEN you may read it "some y are x", that is, "some nice Cakes are new". They are merely two different ways of expressing the very same truth.

Without more words, I will simply set down the other ways in which this upright oblong might be marked, adding the meaning in each case. By comparing them with the various cases of the horizontal oblong, you will, I hope, be able to understand them clearly.

You will find it a good plan to examine yourself on this table, by covering up first one column and then the other, and 'dodging about', as the children say.

Also you will do well to write out for yourself two other tables--one for the LOWER half of the cupboard, and the other for its RIGHT-HAND half.

And now I think we have said all we need to say about the smaller Diagram, and may go on to the larger one.

_________________________________________________
               |
  Symbols.     |          Meanings.
_______________|_________________________________
    -----      |
   |     |     |
   |     |     |  Some y are x';
   |     |     |     i.e. Some nice are not-new.
    -----      |
   |     |     |
   |  1  |     |
   |     |     |
    -----      |
               |
    -----      |
   |     |     |  No y are x;
   |  0  |     |      i.e. No nice are new.
   |     |     |
    -----      |  [Observe that this is merely another way of
   |     |     |      expressing "No new are nice."]
   |     |     |
   |     |     |
    -----      |
               |
    -----      |
   |     |     |
   |     |     |  No y are x';
   |     |     |      i.e. No nice are not-new.
    -----      |
   |     |     |
   |  0  |     |
   |     |     |
    -----      |
               |
    -----      |
   |     |     |
   |  1  |     |  Some y are x, and some are x';
   |     |     |     i.e. Some nice are new, and some are
    -----      |            not-new.
   |     |     |
   |  1  |     |
   |     |     |
    -----      |
               |
    -----      |
   |     |     |
   |  0  |     |  No y are x, and none are x'; i.e. No y
   |     |     |       exist;
    -----      |     i.e. No Cakes are nice.
   |     |     |
   |  0  |     |
   |     |     |
    -----      |
               |
    -----      |
   |     |     |
   |  1  |     |  All y are x;
   |     |     |       i.e. All nice are new.
    -----      |
   |     |     |
   |  0  |     |
   |     |     |
    -----      |
               |
    -----      |
   |     |     |
   |  0  |     |  All y are x';
   |     |     |       i.e. All nice are not-new.
    -----      |
   |     |     |
   |  1  |     |
   |     |     |
    -----      |
_______________|_________________________________

This may be taken to be a cupboard divided in the same way as the last, but ALSO divided into two portions, for the Attribute m. Let us give to m the meaning "wholesome": and let us suppose that all WHOLESOME Cakes are placed INSIDE the central Square, and all the UNWHOLESOME ones OUTSIDE it, that is, in one or other of the four queer-shaped OUTER compartments.

We see that, just as, in the smaller Diagram, the Cakes in each compartment had TWO Attributes, so, here, the Cakes in each compartment have THREE Attributes: and, just as the letters, representing the TWO Attributes, were written on the EDGES of the compartment, so, here, they are written at the CORNERS. (Observe that m' is supposed to be written at each of the four outer corners.) So that we can tell in a moment, by looking at a compartment, what three Attributes belong to the Things in it. For instance, take No. 12. Here we find x, y', m, at the corners: so we know that the Cakes in it, if there are any, have the triple Attribute, 'xy'm', that is, "new, not-nice, and wholesome." Again, take No. 16. Here we find, at the corners, x', y', m': so the Cakes in it are "not-new, not-nice, and unwholesome." (Remarkably untempting Cakes!)

It would take far too long to go through all the Propositions, containing x and y, x and m, and y and m which can be represented on this diagram (there are ninety-six altogether, so I am sure you will excuse me!) and I must content myself with doing two or three, as specimens. You will do well to work out a lot more for yourself.

Taking the upper half by itself, so that our Subject is "new Cakes", how are we to represent "no new Cakes are wholesome"?

This is, writing letters for words, "no x are m." Now this tells us that none of the Cakes, belonging to the upper half of the cupboard, are to be found INSIDE the central Square: that is, the two compartments, No. 11 and No. 12, are EMPTY. And this, of course, is represented by


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

The Game of Logic Page 06

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