The ambigous definition of vacuous truth It is no doubt that the vacuous truth is related to material implication  "$P\Rightarrow Q$". We say the material implication statement is true when $P$ is false. However, seems that this is not the definition for vacuously truth. Do we call it vacuously true only when $P$ can never be true? (Seems that all examples are in this way).
More clearly, suppose we have a statement "$P(x)\Rightarrow Q(x)$", and at some domains of $x$ (we may denote the domain by $T_x$), $P(x)$ is true;  at some other domains of $x$ (denote the domain by $F_x$), $P(x)$ is false. Can we say the statement is vacuously true in the domain $F_x$? Is there any real example?
To make the problem more clear, we may check the statement "For all $x$, $P(x)\Rightarrow Q(x)$". Can we say this statement is vacuously true in the domain $F_x$ ($F_x$ is defined as above)?

*

*I use this inclusive definition:


*

*A vacuous truth is an implication or universally-quantified implication whose antecedent is true or universally true, respectively.
As such, both $$P{\implies}Q$$ and $$\forall x\;[P(x){\implies}Q(x)]$$ are vacuously true. (Wikipedia gives both types of examples.)


*In particular, $$(P\land\lnot P){\implies}Q$$ is a vacuous tautology, while $$\forall x\;[x\ne x{\implies}Q(x)]$$ is a vacuous validity.

