Duality Principle (Order Theory)/Global Duality

Theorem
Let $\Sigma$ be a statement about ordered sets (whether in natural or a formal language).

Let $\Sigma^*$ be the dual statement of $\Sigma$.

The following are equivalent:


 * $(1): \quad \Sigma$ is true for all ordered sets
 * $(2): \quad \Sigma^*$ is true for all ordered sets

$(1)$ implies $(2)$
Let $\left({S, \preceq}\right)$ be an ordered set, and let $\left({S, \succeq}\right)$ be its dual,

By assumption, $\Sigma$ is true for $\left({S, \succeq}\right)$.

By Local Duality, this implies $\Sigma^*$ is true for $\left({S, \preceq}\right)$.

Since $\left({S, \preceq}\right)$ was arbitrary, the result follows.

$(2)$ implies $(1)$
From Dual of Dual Statement (Order Theory), $\left({\Sigma^*}\right)^* = \Sigma$.

The result thus follows from applying the other implication to $\Sigma^*$.

Also see

 * Dual Statement
 * Duality Principle (Category Theory), a more general duality principle.