§ 8. Sub-contrariety is the relation of two propositions, concerning the same matter that may both be true but are never both false. This is the case with I. and O. If it be true that Some men are wise, it may also be true that Some (other) men are not wise. This follows from the maxim in [chap. vi. § 6], not to go beyond the evidence.

For if it be true that Some men are wise, it may indeed be true that All are (this being the subalternans): and if All are, it is (by contradiction) false that Some are not; but as we are only told that Some men are, it is illicit to infer the falsity of Some are not, which could only be justified by evidence concerning All men.

But if it be false that Some men are wise, it is true that Some men are not wise; for, by contradiction, if Some men are wise is false, No men are wise is true; and, therefore, by subalternation, Some men are not wise is true.

§ 9. The Square of Opposition.—By their relations of Subalternation, Contrariety, Contradiction, and Sub-contrariety, the forms A. I. E. O. (having the same matter) are said to stand in Opposition: and Logicians represent these relations by a square having A. I. E. O. at its corners:

As an aid to the memory, this diagram is useful; but as an attempt to represent the logical relations of propositions, it is misleading. For, standing at corners of the same square, A. and E., A. and I., E. and O., and I. and O., seem to be couples bearing the same relation to one another; whereas we have seen that their relations are entirely different. The following traditional summary of their relations in respect of truth and falsity is much more to the purpose:

(1)If A. is true,I. is true,E. is false,O. is false.
(2)If A. is false,I. is unknown,E. is unknown,O. is true.
(3)If I. is true,A. is unknown,E. is false,O. is unknown.
(4)If I. is false,A. is false,E. is true,O. is true.
(5)If E. is true,A. is false,I. is false,O. is true.
(6)If E. is false,A. is unknown,I. is true,O. is unknown.
(7)If O. is true,A. is false,I. is unknown,E. is unknown.
(8)If O. is false,A. is true,I. is true,E. is false.

Where, however, as in cases 2, 3, 6, 7, alleging either the falsity of universals or the truth of particulars, it follows that two of the three Opposites are unknown, we may conclude further that one of them must be true and the other false, because the two unknown are always Contradictories.

§ 10. Secondary modes of Immediate Inference are obtained by applying the process of Conversion or Obversion to the results already obtained by the other process. The best known secondary form of Immediate Inference is the Contrapositive, and this is the converse of the obverse of a given proposition. Thus:

DATUM.OBVERSE.CONTRAPOSITIVE.
A. All S is P∴ No S is not-P∴ No not-P is S
I. Some S is P∴ Some S is not not-P∴ (none)
E. No S is P∴ All S is not-P∴ Some not-P is S
O. Some S is not P∴ Some S is not-P∴ Some not-P is S