TITLE: What precisely is a vacuous truth?
QUESTION [33 upvotes]: Is there a proper and precise definition that goes something like this?
Definition. A statement $S$ is a vacuous truth if ... ...

REPLY [2 votes]: Contrary to some of the existing answers, I don't have the impression that one typically speaks of a vacuous truth if the statement is a pure implication whose antecedent happens to be false; the usual use of "vacuous truth" occurs in the context of universal claims where the antecedent is always (i.e. for every object) false.
A universally quantified implication is vacuously true iff it is true just because there are no objects to satisfy the antecedent. Formally,

A statement of the form $\forall u (\phi \to \psi)$ (where $u$ is a varible and $\phi$ and $\psi$ are arbitrary formulas) is vacuously true
in a structure $\mathcal{S} = \langle \mathcal{D}, \mathcal{I} \rangle$
iff
for all $a \in \mathcal{D}$ and arbitarary $v$: $\mathcal{S},
 v^{[u \mapsto a]} \not \models \phi$.