Cursus Philosophicus Thomisticus, Prima Pars Artis Logicae

Equipollence is nothing other than “the equivalence of opposed propositions through variation of negation, and the signification of the same sense.”

There are three rules for this:

FIRST: Given two contradictories, if negation is prefixed to one of them, they become equipollent. This is understood provided that negation is taken negatively and not as an infinitizing negation. For example, suppose you say, “Every man disputes,” and “Some man does not dispute.” If you prefix negation to one, saying, “Not every man disputes,” it is the same as saying, “Some man does not dispute.” And if you say, “Not some man does not dispute,” it is the same as saying, “Every man disputes.” From this you can also infer how any given proposition is made contradictory merely by prefixing negation, since negation is of a malignant nature, destroying whatever it finds after it. Thus, just as prefixed to a contradictory it makes that proposition equipollent with it, so prefixed to the very same proposition it makes it contradictory to itself.

SECOND RULE: Given two contrary propositions, they become equipollent by placing negation after the subject of the proposition and before the copula. Thus, “Every man runs” and “No man runs” become equipollent by saying, “Every man does not run,” or “No man does not run.” This rule is understood where the distribution is complete; otherwise a counterexample is possible, as with “Each of the two men disputes” and “No man disputes.” They do not become equipollent by merely placing negation after the subject, and yet by definition they are contraries, since they can both be false in the case where only one man disputes and not both; there is, however, no case in which they can both be true.

THIRD RULE concerns subalterns, such as “Every man disputes” and “Some man disputes.” They become equipollent by prefixing and postposing negation. Thus, if you say, “Not every man does not dispute,” it is equipollent to “Some man disputes”; and “Not some man does not dispute” is equipollent to “Every man disputes.” Hence the verse:

Before the Contradictories, after the Contraries, before-and-after the Subalterns.

No equipollences are given for subcontraries, because they are not capable of them. For if you prefix negation, a universal proposition results and consequently it is not equipollent to a particular, as with “Some man is white” and “Some man is not white.” If you say, “Not some man is white,” it is the same as “No man is.” If you postpose negation to the subject in an affirmative proposition, it becomes not equipollent but formally negative: for example, from “Some man is white,” if you postpose negation, it becomes the negative “Some man is not white,” but it is not equipollent to the negative. If, however, negation is postposed to the negative proposition itself, an unhelpful repetition results: “Some man is not not white.”