In order to draw a conclusion about S we have to admit the statement, “S does not contain the whole of P,” as a valid logical form—it is a statement about S which can be made. The logic which gives us the form, “some P is not S,” and which does not allow us to give the exactly equivalent and equally primary form, “S does not contain the whole of P,” is artificial. And I wish to point out that this artificiality leads to an error. If one trusted to the mnemonic lines given above, one would conclude that no logical conclusion about S can be drawn from the statement, “some P are M, no M are S.” But a conclusion can be drawn: S does not contain the whole of P. It is not that the result is given expressed in another form. The mnemonic lines deny that any conclusion can be drawn from premisses in the moods I, E, respectively. 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.”