Two-dimensional partially ordered set

A partially ordered set is two-dimensional when there are two total orders ${<}_1$ and ${<}_2$ on its ground set such that

A partially ordered set is two-dimensional if every one of its finite induced suborders is two-dimensional. Encode two candidate total orders by propositional variables; every finite collection of the order, extension, and intersection clauses concerns a finite induced suborder, so the propositional compactness theorem supplies two global realizing orders.