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)?
 A: I would frame the concept as pertaining to first order sentences of the form $$\forall x(P(x)\to Q(x)).$$ Vacous truth is the particular situation when there are no $x$ in the domain satisfying $P(x),$ in which case standard semantics says the statement is true. So yes, whether a first order sentence is vacuously true or not depends on the interpretation, and hence the domain, much as regular truth does. (However this doesn’t usually matter: when we’re talking about math, we generally are making statements relative to some fixed interpretation.)
A: In my understanding if we want to use quantifier, the domain of discourse has to be specified first. This let us focus on those things we care the most. E.g. let $S$ be the set containing "all people not taller than 180cm", then $\bar S$ may contain a tree taller than 180, or a monkey who likes tomato, etc. Since the definition/description from Wikipedia uses quantifier:

["]In mathematics and logic, a vacuous truth is a statement that asserts that all members of the empty set have a certain property.["]

It depends on the domain of discourse, or the universal set, to say something is vacuously true. Using the just example following the definition
$$\textrm{all cell phones in the room are turned off.}$$
will be vacuously true whenever there are no cell phones in the room: Here a room with no cell phones is the domain of discourse, and in this case it's vacuously true. The adjective "vacuously" is omitted in the text.
But as the example you mentioned in the comment, when quantifier is not used, it can also be said vacuously true, and it's more formal (which is what Wiki says, not me)

More formally, a relatively well-defined usage refers to a conditional statement with a false antecedent.

In short, when


*

*Quantifier is used, and the target $S$ is an empty (sub)set. (which leads to antecedent be true, because no counter-example can be found), more specifically
$$[\forall x\in S, P(x)]\equiv T, \textrm{when }S=\emptyset$$
And this means the truth of 
$$[\forall x\in S, P(x)\implies Q]\equiv Q$$
is depends on $Q$.

*the antecedent (a single target) is false.


both are called vacuously true, because we don't care this case so much, we interested in "if $P$ happen, $Q$ will happen".

To make the problem more clear, we may check the statement "For all x, P(x)⇒Q(x)". Can we say this statement is vacuously true in the domain Fx (Fx is defined as above)?

Yes, only in the domain $F_x$.
A: Logicians typically define concepts such as these purely in terms of truth values rather than what's possible; this isn't a concept of modal logic. I think the examples you've seen (you didn't mention any specifically) probably just chose impossible $p$ so it's obvious to you that they're false.
A: *

*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.

A: 
Do we call it vacuously true only when P can never be true?

No. In classical logic, for any logical propositions P and Q (it doesn't matter if they are true or false), we have:
$P\implies [\neg P \implies Q]$
This is a tautology. See truth table here.
