Meaning of vacously true statement I'm stuck with the concept of vacously true statement. For example, I know that the statement "Every element of the empty set is a zebra" is a vacously true statement because we can't find an element of the empty set which IS NOT a zebra.
However, there is one point that I can't understand. That is we also can't find an element of the empty set which IS a zebra.
So in some sense, the statement is both "true" and "false" (which is quite disturbing).  So why we claim that the vacously true statement TRUE even when we can't prove it is true ?
I have heard something like "In mathematics, everything is true unless proven false" as an excuse for the vacously true statement.  But i think it is quite false, for example in the case of many hypothesis (for example Riemann hypothesis) we wait and we try to prove or disprove it without automatically supposing it is true (until someone prove it is false).
Could you please explain me the philosophy of the "vacuously true" statement and the logic of the stament "in mathematics, everything is true unless proven false" ?
 A: Both

$(A)$ "Every element of the emptyset is a zebra"

and

$(B)$ "Every element of the emptyset is not a zebra"

are true. However, they are not the negations of each other - the negation of $(A)$, for example, is

$(\neg A)$ "Some element of the emptyset is not a zebra."

We may reflexively expect $(\neg A)$ to be a consequence of $(B)$ but that's based on a glitch of intuition. The intuitively-obvious implication $$\forall x\in U(P(x))\rightarrow \exists x\in U(P(x))$$ only holds if the scope of quantification $U$ is nonempty, which in this case it isn't.
So in fact there is no simultaneous truth and falsity going on here. $(A)$ and $(B)$ are each true, and their negations $(\neg A)$ and $(\neg B)$ are each false.
A: Statements like the ones above are true because we require that all statements be true or false. So, if "for all  in the empty set,  is a zebra" is false, then we are saying that there exists an  in the empty set such that  is not a zebra. That can never happen.
In general, all statements of the form $$\forall x P(x)$$ are true for the empty set because their negations are $$\exists x\lnot P(x)$$
A: I think to understand how this comes about you need to appreciate two things.

*

*Sets are conceived as abstractions of properties.  Two properties might have quite different meanings, but if every object with each property also has the other, then the properties correspond to the same set.  Properties are a complicated idea, and sets are intended to simplify this complicated idea: if two sets contain the same elements, they are the same set, even if they were defined in very different ways.
One important consequence of this is that there is only one empty set.  The property of being an even prime number greater than 10 is nothing at all like the property of being a living crown prince of the Ottoman Empire.  But the two sets are the same set.


*Consider the following common-sense claim:

If all rubies are red, then all rubies belonging to MJD are red.

As it happens, I don't own any rubies.  The set of rubies belonging to MJD is empty. If you don't agree that all the rubies in the empty set are red, then you should deny the common-sense claim above.   You should want me to qualify it:

If all rubies are red, then all rubies belonging to MJD are red, if there are any.

We would have to to include this unimportant qualification every time we made a claim about elements of any subset of anything.  This would be unhelpful.  So we agree that such claims can be vacuously true: if all rubies are red, then all my rubies are red, whether I have any rubies and even if I don't.
When these two ideas come together, one gets the odd-sounding claim that every element of the empty set is red.  (I own no rubies, so the set of my rubies is empty, and every ruby in that set is red, because every ruby is red.)  But if we want sets to work as abstractions of properties, we have to accept that every element of the empty set is red.
By the same reasoning, every element of the empty set is blue, because all my blue diamonds are blue.
If you don't like that every element of the empty set is both red and blue, you need to find a way to say that the set of all my rubies is different from the set of all my blue diamonds.  But this is exactly what what we don't want to do; it's exactly the complicated situation that sets were invented to simplify.
When you simplify anything, you necessarily lose some of its nuance.  There is a philosophical distinction between the property of being one of my rubies and the property of being one of my blue diamonds. Set theory intentionally discards this distinction.
And anyway what is wrong with the idea that every element of the empty set is both a red ruby  and a blue diamond?  There aren't any things that are both red rubies and blue diamonds, so the set of all such things ought to be exactly the empty set.
