\begin{definition}[Definition:Dual Statement (Order Theory)]
Let $\struct {S, \preceq}$ be an ordered set.
Let $\succeq$ be the dual ordering to $\preceq$.
Let $\Sigma$ be any statement pertaining to $\struct {S, \preceq}$ (be it in natural language or a formal language).
The '''dual statement of $\Sigma$''', denoted $\Sigma^*$, is the statement obtained from replacing every reference to $\preceq$ in $\Sigma$ with a reference to its dual $\succeq$.
This '''dual statement''' may then be turned into a statement about $\preceq$ again by applying the equivalences on Dual Pairs (Order Theory).
\end{definition}