The only two compartments, available for Things which are xm', are No. 9 and No. 10. Of these, No. 9 is already marked as 'empty'; so our red counter must go into No. 10.

Similarly, the only two, available for ym, are No. 11 and No. 13. Of these, No. 11 is already marked as 'empty'; so our red counter MUST go into No. 13.

The final result is


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

And now how much of this information can usefully be transferred to the smaller Diagram?

Let us take its four compartments, one by one.

As to No. 5? This, we see, is wholly 'empty'. (So mark it with a grey counter.)

As to No. 6? This, we see, is 'occupied'. (So mark it with a red counter.)

As to No. 7? Ditto, ditto.

As to No. 8? No information.

The smaller Diagram is now pretty liberally marked:--


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

And now what Conclusion can we read off from this? Well, it is impossible to pack such abundant information into ONE Proposition: we shall have to indulge in TWO, this time.

First, by taking x as Subject, we get "all x are y'", that is,

"All Dragons are not-Scotchmen":

secondly, by taking y as Subject, we get "all y are x'", that is,

"All Scotchmen are not-Dragons".

Let us now write out, all together, our two Premisses and our brace of Conclusions.

"All Dragons are uncanny; All Scotchmen are canny. &there4 All Dragons are not-Scotchmen; All Scotchmen are not-Dragons."

Let me mention, in conclusion, that you may perhaps meet with logical treatises in which it is not assumed that any Thing EXISTS at all, by "some x are y" is understood to mean "the Attributes x, y are COMPATIBLE, so that a Thing can have both at once", and "no x are y" to mean "the Attributes x, y are INCOMPATIBLE, so that nothing can have both at once".

In such treatises, Propositions have quite different meanings from what they have in our 'Game of Logic', and it will be well to understand exactly what the difference is.

First take "some x are y". Here WE understand "are" to mean "are, as an actual FACT"--which of course implies that some x-Things EXIST. But THEY (the writers of these other treatises) only understand "are" to mean "CAN be", which does not at all imply that any EXIST. So they mean LESS than we do: our meaning includes theirs (for of course "some x ARE y" includes "some x CAN BE y"), but theirs does NOT include ours. For example, "some Welsh hippopotami are heavy" would be TRUE, according to these writers (since the Attributes "Welsh" and "heavy" are quite COMPATIBLE in a hippopotamus), but it would be FALSE in our Game (since there are no Welsh hippopotami to BE heavy).

Secondly, take "no x are y". Here WE only understand "are" to mean "are, as an actual FACT"--which does not at all imply that no x CAN be y. But THEY understand the Proposition to mean, not only that none ARE y, but that none CAN POSSIBLY be y. So they mean more than we do: their meaning includes ours (for of course "no x CAN be y" includes "no x ARE y"), but ours does NOT include theirs. For example, "no Policemen are eight feet high" would be TRUE in our Game (since, as an actual fact, no such splendid specimens are ever found), but it would be FALSE, according to these writers (since the Attributes "belonging to the Police Force" and "eight feet high" are quite COMPATIBLE: there is nothing to PREVENT a Policeman from growing to that height, if sufficiently rubbed with Rowland's Macassar Oil--which said to make HAIR grow, when rubbed on hair, and so of course will make a POLICEMAN grow, when rubbed on a Policeman).

Thirdly, take "all x are y", which consists of the two partial Propositions "some x are y" and "no x are y'". Here, of course, the treatises mean LESS than we do in the FIRST part, and more than we do in the SECOND. But the two operations don't balance each other--any more than you can console a man, for having knocked down one of his chimneys, by giving him an extra door-step.

If you meet with Syllogisms of this kind, you may work them, quite easily, by the system I have given you: you have only to make 'are' mean 'are CAPABLE of being', and all will go smoothly. For "some x are y" will become "some x are capable of being y", that is, "the Attributes x, y are COMPATIBLE". And "no x are y" will become "no x are capable of being y", that is, "the Attributes x, y are INCOMPATIBLE". And, of course, "all x are y" will become "some x are capable of being y, and none are capable of being y'", that is, "the Attributes x, y are COMPATIBLE, and the Attributes x, y' are INCOMPATIBLE." In using the Diagrams for this system, you must understand a red counter to mean "there may POSSIBLY be something in this compartment," and a grey one to mean "there cannot POSSIBLY be anything in this compartment."

The Game of Logic Page 10

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