[The diagram has diagonal lines, not represented here, from corner A to corner O, and from corner E to corner I, each labelled “Contradictory Opposition”. Tr.]
A E
Contrary Opposition.
Sub-contrary Opposition.
I O
A = Universal Affirmative. All men are mortal.
E = Universal Negative. No men are mortal.
I = Particular Affirmative. Some men are mortal.
O = Particular Negative. Some men are not mortal.
Sub-contrary Opposition has no real meaning; the judgments so opposed are compatible.
It is not true both that “All M.P.’s are wise,” and that “No M.P.’s are wise,” but both may be false; while Contradictory Negation implies the Law of Excluded Third or excluded Middle, “X is either A or not A,” the principle of disjunction, or rather, the simplest case of it. It is not {128} false both that “All M.P.’s are wise” and that “Some M.P.’s are not wise.” The point is, then, on the one hand, that in Contradiction you can go from falsehood to truth, [1] while in Contrariety you can only go from truth to falsehood; but also that in Contradiction the Affirmative and Negative are not at all on a level in meaning, while in Contrariety they are much more nearly so. Then if we leave out the relations of mere plurality, of All and Some, which enable you to get contrary negation in pure negative form in the common Logic, we may say generally that in contrary negation something is asserted, and in contradictory negation taken quite literally nothing is asserted, but we have a “bare denial,” a predicate is merely removed. In actual thought this cannot be quite realised, because a bare denial is really meaningless, and we always have in our mind some subject or universe of discourse within which the denial is construed definitely. But this definite construing is not justified by the bare form of contradiction, which consists simply in destroying a predication and not replacing it by another. In as far as you replace it by another, defined or undefined, you are going forward towards contrary negation.
[1] I.e. Contradictory alternatives are exhaustive.
Contrary Negation
2. Thus, Contrary Negation in its essence is affirmation with a negative intention, and we may take as a type of it in this wider sense the affirmation of a positive character with the intention of denying another positive character. E.g. when you deny “This is a right-angled triangle” by asserting “This is an equilateral triangle,” you have typical contrary negation. It is not really safe to speak of contraries except with reference to judgments, intended to deny each {129} other; but it is common to speak of species of the same genus as contraries or opposites, because the same thing cannot be both. [1]
[1] Bain, p. 55 ff.