A is the cause of B ∴ B is the effect of A.
The rule assumes that the reciprocal of a given relation is definitely known; and so far as this is true it may be extended to more concrete relations—
A is a genus of B ∴ B is a species of A
A is the father of B ∴ B is a child of A.
But not every relational expression has only one definite reciprocal. If we are told that A is the brother of B, we can only infer that B is either the brother or the sister of A. A list of all reciprocal relations is a desideratum of Logic.
§ 5. Obversion (otherwise called Permutation or Æquipollence) is Immediate Inference by changing the quality of the given proposition and substituting for its predicate the contradictory term. The given proposition is called the 'obvertend,' and the inference from it the 'obverse.' Thus the obvertend being—Some philosophers are consistent reasoners, the obverse will be—Some philosophers are not inconsistent reasoners.
The legitimacy of this mode of reasoning follows, in the case of affirmative propositions, from the principle of Contradiction, that if any term be affirmed of a subject, the contradictory term may be denied (chap. vi. § 3). To obvert affirmative propositions, then, the rule is—Insert the negative sign, and for the predicate substitute its contradictory term.
In agreement with this mode of inference, we have the rule of modern English grammar, that 'two negatives make an affirmative.'
Again, by the principle of Excluded Middle, if any term be denied of a subject, its contradictory may be affirmed: to obvert negative propositions, then, the rule is—Remove the negative sign, and for the predicate substitute its contradictory term.
Thus, by obversion, each of the four propositions retains its quantity but changes its quality: A. to E., I. to O., E. to A., O. to I. And all the obverses are infinite propositions, the affirmative infinites having the sense of negatives, and the negative infinites having the sense of affirmatives.
Again, having obtained the obverse of a given proposition, it may be desirable to recover the obvertend; or it may at any time be requisite to change a given infinite proposition into the corresponding direct affirmative or negative; and in such cases the process is still obversion. Thus, if No S is not-P be given us to recover the obvertend or to find the corresponding affirmative; the proposition being formally negative, we apply the rule for obverting negatives: 'Remove the negative sign, and for the predicate substitute its contradictory.' This yields the affirmative All S is P. Similarly, to obtain the obvertend of All S is not-P, apply the rule for obverting Affirmatives; and this yields No S is P.
§ 6. Contrariety.—We have seen in chap. iv. § 8, that contrary terms are such that no two of them are predicable in the same way of the same subject, whilst perhaps neither may be predicable of it. Similarly, Contrary Propositions may be defined as those of which no two are ever both true together, whilst perhaps neither may be true; or, in other words, both may be false. This is the relation between A. and E. when concerned with the same matter: as A.—All men are wise; E.—No men are wise. Such propositions cannot both be true; but they may both be false, for some men may be wise and some not. They cannot both be true; for, by the principle of Contradiction, if wise may be affirmed of All men, not-wise must be denied; but All men are not-wise is the obverse of No men are wise, which therefore may also be denied.
At the same time we cannot apply to A. and E. the principle of Excluded Middle, so as to show that one of them must be true of the same matter. For if we deny that All men are wise, we do not necessarily deny the attribute 'wise' of each and every man: to say that Not all are wise may mean no more than that Some are not. This gives a proposition in the form of O.; which, as we have seen, does not imply its subalternans, E.
If, however, two Singular Propositions, having the same matter, but differing in quality, are to be treated as universals, and therefore as A. and E., they are, nevertheless, contradictory and not merely contrary; for one of them must be false and the other true.
§ 7. Contradiction is a relation between two propositions analogous to that between contradictory terms (one of which being affirmed of a subject the other is denied)—such, namely, that one of them is false and the other true. This is the case with the forms A. and O., and E. and I., in the same matter. If it be true that All men are wise, it is false that Some men are not wise (equivalent by obversion to Some men are not-wise); or else, since the 'Some men' are included in the 'All men,' we should be predicating of the same men that they are both 'wise' and 'not-wise'; which would violate the principle of Contradiction. Similarly, No men are wise, being by obversion equivalent to All men are not-wise, is incompatible with Some men are wise, by the same principle of Contradiction.
But, again, if it be false that All men are wise, it is always true that Some are not wise; for though in denying that 'wise' is a predicate of 'All men' we do not deny it of each and every man, yet we deny it of 'Some men.' Of 'Some men,' therefore, by the principle of Excluded Middle, 'not-wise' is to be affirmed; and Some men are not-wise, is by obversion equivalent to Some men are not wise. Similarly, if it be false that No men are wise, which by obversion is equivalent to All men are not-wise, then it is true at least that Some men are wise.
By extending and enforcing the doctrine of relative terms, certain other inferences are implied in the contrary and contradictory relations of propositions. We have seen in chap. iv. that the contradictory of a given term includes all its contraries: 'not-blue,' for example, includes red and yellow. Hence, since The sky is blue becomes by obversion, The sky is not not-blue, we may also infer The sky is not red, etc. From the truth, then, of any proposition predicating a given term, we may infer the falsity of all propositions predicating the contrary terms in the same relation. But, on the other hand, from the falsity of a proposition predicating a given term, we cannot infer the truth of the predication of any particular contrary term. If it be false that The sky is red, we cannot formally infer, that The sky is blue (cf. chap. iv. § 8).
§ 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