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 $\struct {S, \preceq}$ be an ordered set.

Let $\struct {S, \succeq}$ be its dual.

By assumption, $\Sigma$ is true for $\struct {S, \succeq}$.

By Local Duality, this implies $\Sigma^*$ is true for $\struct {S, \preceq}$.

Since $\struct {S, \preceq}$ was arbitrary, the result follows.

$(2)$ implies $(1)$
From Dual of Dual Statement (Order Theory):
 * $\paren {\Sigma^*}^* = \Sigma$

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

Also see

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