Thus a simple four-dimensional poiograph has enabled us to detect a mistake in the mnemonic lines which have been handed down unchallenged from mediæval times. To discuss the subject of these lines more fully a logician defending them would probably say that a particular statement cannot be a major premiss; and so deny the existence of the fourth figure in the combination of moods.
To take our instance: some Americans are of African stock; no Aryans are of African stock. He would say that the conclusion is some Americans are not Aryans; and that the second statement is the major. He would refuse to say anything about Aryans, condemning us to an eternal silence about them, as far as these premisses are concerned! But, if there is a statement involving the relation of two classes, it must be expressible as a statement about either of them.
To bar the conclusion, “Aryans do not include the whole of Americans,” is purely a makeshift in favour of a false classification.
And the argument drawn from the universality of the major premiss cannot be consistently maintained. It would preclude such combinations as major O, minor A, conclusion O—i.e., such as some mountains (M) are not permanent (P); all mountains (M) are scenery (S); some scenery (S) is not permanent (P).
This is allowed in “Jevon’s Logic,” and his omission to discuss I, E, O, in the fourth figure, is inexplicable. A satisfactory poiograph of the logical scheme can be made by admitting the use of the words some, none, or all, about the predicate as well as about the subject. Then we can express the statement, “Aryans do not include the whole of Americans,” clumsily, but, when its obscurity is fathomed, correctly, as “Some Aryans are not all Americans.” And this method is what is called the “quantification of the predicate.”
The laws of formal logic are coincident with the conclusions which can be drawn about regions of space, which overlap one another in the various possible ways. It is not difficult so to state the relations or to obtain a symmetrical poiograph. But to enter into this branch of geometry is beside our present purpose, which is to show the application of the poiograph in a finite and limited region, without any of those complexities which attend its use in regard to natural objects.
If we take the latter—plants, for instance—and, without assuming fixed directions in space as representative of definite variations, arrange the representative points in such a manner as to correspond to the similarities of the objects, we obtain configuration of singular interest; and perhaps in this way, in the making of shapes of shapes, bodies with bodies omitted, some insight into the structure of the species and genera might be obtained.
CHAPTER IX
APPLICATION TO KANT’S THEORY OF EXPERIENCE
When we observe the heavenly bodies we become aware that they all participate in one universal motion—a diurnal revolution round the polar axis.
In the case of fixed stars this is most unqualifiedly true, but in the case of the sun, and the planets also, the single motion of revolution can be discerned, modified, and slightly altered by other and secondary motions.