# Contraries as duals

Contrariety is a property of pairs of propositions, but it also applies to pairs of terms or concepts. “Two general terms are contraries if and only if, by virtue of their meaning alone, they apply to possible cases on opposite ends of a scale. Both terms cannot apply to the same possible case, but neither may apply.” (Aristotelian Logic, Parry and Hacker, p. 216) Opposite ends of a scale are also called extremes, which are contrasted with means between the extremes.

Every pair of contraries forms a duality by inverting the scale of which they are opposites. For example, quantitative contraries such as rich and poor become poor and rich when the scale is inverted. Every measurement scale can be inverted so in this sense a measurement and its inverse are a contrary pair that forms a self-duality. Every ratio or mapping of two variables, f(x, y), can be interchanged and form a duality, f(y, x). For example, the equation v = Δst can be interchanged to become u = v−1 = Δts.

The scale may be qualitative, too. For example, the qualitative contraries up and down become down and up, respectively, by looking upside-down. The contraries left and right become right and left when looked at facing the other way. Extension and intension are opposites that may be inverted by interchanging them with each other. Compare the duality of top-down and bottom-up perspectives.

“A pair of terms is contradictory if and only if by virtue of their meaning alone each and every entity in the universe must be named by one or the other but not both.” (Aristotelian Logic, Parry and Hacker, p. 216) May the terms X and not-X be made into duals? That depends. If not-X is the contradictory of X and means everything other than X, that includes things that are non-dual. But in some cases, not-X means the opposite of X, so that contraries are indicated.