First notions of logic (preparatory to the study of geometry)

by Augustus De Morgan

	Every	A is X	All the △ is in the ○	A
	No	B is X	None of the □ is in the ○	E
∴	No	A is B	None of the △ is in the □	E

is the only way in which a universal negative conclusion can be drawn.

III. In what manner can a particular affirmative conclusion be drawn; namely, that part or all of one figure is contained in the other? Only when we are able to assert that the whole circle is part of one of the figures, and that the whole, or part of the circle, is part of the other figure. We have then two forms.

	Every	X is A		All the ○ is in the △	A
	Every	X is B		All the ○ is in the □	A
∴	Some	A is B	∴	Some of the △ is in the □	I
	Every	X is A		All the ○ is in the △	A
	Some	X is B		Some of the ○ is in the □	I
	Some	A is B		Some of the △ is in the □	I

The second of these contains all that is strictly necessary to the conclusion, and the first may be omitted. That which follows when an assertion can be made as to some, must follow when the same assertion can be made of all.

IV. How can a particular negative proposition be inferred; namely, that part, or all of one figure, is not contained in the other? It would seem at first sight, whenever we are able to assert that part or all of one figure is in the circle, and that part or all of the other figure is not. The weakest syllogism from which such an inference can be drawn would then seem to be as follows.

	Some A is X		Some of the △ is in the ○
	Some B is not X		Some of the □ is not in the ○
∴	Some B is not A	∴	Some of the △ is not in the □

But here it will appear, on a little consideration, that the conclusion is only thus far true; that those As which are Xs cannot be those Bs which are not Xs; but they may be other Bs, about which nothing is asserted when we say that some Bs are not Xs. And further consideration will make it evident, that a conclusion of this form can only be arrived at when one of the figures is entirely within the circle, and the whole or part of the other without; or else when the whole of one of the figures is without the circle, and the whole or part of the other within; or lastly, when the circle lies entirely within one of the figures, and not entirely within the other. That is, the following are the distinct forms which allow of a particular negative conclusion, in which it should be remembered that a particular proposition in the premises may always be changed into a universal one, without affecting the conclusion. For that which necessarily follows from “some,” follows from “all.”

	Every A is X		All the △ is in the ○	A
	Some B is not X		Some of the □ is not in the ○	O
∴	Some B is not A		Some of the □ is not in the △	O
	No A Is X		None of the △ is in the ○	E
	Some B is X		Some of the □ is in the ○	I
∴	Some B is not A		Some of the □ is not in the △	O
	Every X is A		All the ○ is in the △	A
	Some X is not B		Some of the ○ is not in the □	O
∴	Some A is not B		Some of the △ is not in the □	O

It appears, then, that there are but six distinct syllogisms. All others are made from them by strengthening one of the premises, or converting one or both of the premises, where such conversion is allowable; or else by first making the conversion, and then strengthening one of the premises. And the following arrangement will shew that two of them are universal, three of the others being derived from them by weakening one of the premises in a manner which does not destroy, but only weakens, the conclusion.