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

by Augustus De Morgan

1.	Every A is X		3.	Every A is X
	Every X is B			No B is X			.........
	Every A is B			No A Is B
	│			│
	│
2.	Some A is X	4.	Some A is X	5.	Every A is X	6.	Every X is A
	Every X is B		No B is X		Some B is not X		Some X is not B
	Some A is B		Some A is not B		Some B is not A		Some A is not B

We may see how it arises that one of the partial syllogisms is not immediately derived, like the others, from a universal one. In the preceding, AEE may be considered as derived from AAA, by changing the term in which X enters universally into its contrary. If this be done with the other term instead, we have

No	A is X	from which universal premises we cannot deduce a universal conclusion, but only Some B is not A.
Every	X is B

If we weaken one and the other of these premises, as they stand, we obtain

Some A is not X		No A is X
Every X is B	and	Some X is B
No conclusion		Some B is not A

equivalent to the fourth of the preceding: but if we convert the first premiss, and proceed in the same manner,

	No X is A		Some X is not A
From	Every X is B	we obtain	Every X is B
	Some B is not A		Some B is not A

which is legitimate, and is the same as the last of the preceding list, with A and B interchanged.

Before proceeding to shew that all the usual forms are contained in the preceding, let the reader remark the following rules, which may be proved either by collecting them from the preceding cases, or by independent reasoning.

1. The middle term must enter universally into one or the other premiss. If it were not so, the one premiss might speak of one part of the middle term, and the other of the other; so that there would, in fact, be no middle term. Thus, ‘Every A is X, Every B is X,’ gives no conclusion: it may be thus stated;