Is there a dual to term "vacuously true" for a universal set? For an empty set, any statement that claims "for all ... is true/false" are considered "vacuously true". So, can we construct a universal set in which any statement that claims "there exists ... is false/true" as a dual to the vacuously true statements?
 A: $\forall x\in X.P(x)$ is usually1 viewed as shorthand for $\forall x.x\in X\to P(x)$. With this definition, when $X=\varnothing$ then $x\in X$ is equivalent to $\bot$, the constantly false proposition. We then have (arguably by definition of $\bot$) that $\bot\to Q$ is always true. To this end, the universal quantifier isn't particularly relevant. $\exists x.x\in\varnothing\to P(x)$ is also always true.
If $\forall x\in\varnothing.P(x)$ is always true then $\neg\forall x\in\varnothing.P(x)$ is always false, which is to say (verify this) $\exists x\in\varnothing.Q(x)$ is always false. Again, this becomes $\exists x.x\in\varnothing\land Q(x)$ and again $\bot\land Q$ is always false.
There is no "universal set" in most set theories, but $\forall x$/$\exists x$ already runs over all possible sets.
1 In type theory the story would be quite different.
A: Not without loosing consistancy.  Consider $\exists x : x \ne x $ a mathematical system like this would have bugs. 
The idea of vacourious truth comes from the implication $ \bot \to \top $ whoes contrapostive is itself. so in a way the dual of vacorious truth is vacorious truth. 
