Part II, Chapter 4: Of the Opposition between Propositions having the same Subject and Attribute
We have said there are four sorts of propositions—A, E, I, O. We inquire now what agreement or disagreement they have together, when we make from the same subject, and the same attribute, different kinds of propositions. This is what is called opposition.
And it is easy to see that this opposition can be only of three kinds, though one of the three is subdivided into two others.
For if propositions are opposed, both in quantity and quality, they are called contradictories, as A O, and E I, Every man is an animal, Some man is no animal,—No man is sinless, Some man is sinless.
If they differ in quantity alone, and agree in quality, they are called subalterns, as A I, and E O, Every man is an animal, Some man is an animal,—No man is sinless, Some man is not sinless.
And if they differ in quality, and agree in quantity, they are then called contraries, or sub-contraries. Contraries when they are universal, as Every man is an animal, No man is an animal. Sub-contraries when they are particular, as Some man is an animal, Some man is not an animal. In considering these opposed propositions, according to their truth or falsehood, we may easily determine—
1°. That contradictories are never both either true or false together, but if one is true the other is false; and if one is false the other is true. For if it be true that every man is an animal, it cannot be true that some man is not an animal; and if, on the contrary, it be true that some man is not an animal, it is consequently not true that every man is an animal. This is so clear that it would only be obscured by further explanation.
2°. Contraries can never be both true, but they may be often both false. They cannot be true because the contradictories would be true. For if it be true that Every man is an animal, it is false that Some man is not an animal, which is the contradictory; and, by consequence, still more false that No man is an animal, which is the contrary. But the falsehood of one does not imply the truth of the other, for it may be false that all men are just, without its being true, on that account, that no man is just, since there may be just men, though all are not just.
3°. Sub-contraries, by a rule directly opposed to that of contraries, may be both true, as these, Some man is just. Some man is not just, because justice may belong to one part of men, and not to another; and thus the affirmation and negation do not regard the same subject, since some men is taken for one part of men, in one of the propositions, and for another in the other. But they cannot be both false, since otherwise the contradictories would be both false; for if it were false that Some men were just, it would therefore be true that No man was just, which is the contradictory, and with much more reason that Some man is not just, which is the sub-contrary.
4°. With regard to the Subalterns, there is no true opposition, since the particulars are consequents of the general, for if All men are animals, Some man is an animal; if No man is an ape, Some man is not an ape. Hence the truth of the universals involves that of the particulars; but the truth of the particulars does not involve that of the universals, for it does not follow, because it is true that some man is just, that it should be also true that every man is just; and, on the contrary, the falsehood of particulars involves the falsehood of universals, for if it be true that some man is sinless, it is still more false that every man is sinless. But the falsity of the universals does not involve the falsity of the particulars, for although it may be false that every man is just, it does not follow that it is false to say that some man is just. Hence it follows that there are many cases in which these subalternate propositions are both true, and others in which they are both false.
We have said nothing touching the reducing of opposed propositions to the same sense, because this is altogether useless, and because the rules which are given for it are for the most part true only in Latin.