MJD's "rubies in my vault" discussion is an excellent example of a vacuously true statement, but I don't think it confronts the source of your confusion head on.
This question comes from the notion that truth is defined where something cannot be proved false - which to me has always felt insufficient.
Being unable to prove something false is indeed insufficient to say it's true. But that's not the meaning of the term "vacuously true".
Only statements of a certain type are said to be vacuously true. The two most common cases (already mentioned in the comments, and in MJD's example):
- "For all $x$ with property $P$, the property $Q$ also holds", where there are no $x$'s with property $P$.
- $A\to B$, where $A$ is false.
According to the standard conventions used in mathematics, these are both vacuously true. You can see how (1) and (2) are closely related: if "$x$ has property $P$" is false, then "if $x$ has property $P$ then it also has property $Q$" is vacuously true.
MJD's discussion explains clearly why we use this convention in mathematics and logic. But for a statement like, say, "there are no odd perfect numbers", just because we haven't been able (so far) to prove it false, doesn't mean we regard it as vacuously true. It doesn't fit the pattern. Likewise for the negation, "there exists an odd perfect number".
Let me dig a little deeper into the form "all $x$ with property $P$ have property $Q$". It will be vacuously true if there are no $x$'s with property $P$. But to prove that it's vacuously true, you have to prove that there are, in fact, no $x$'s with property $P$. The statement "all odd perfect numbers are the product of at least 3000 primes" might be vacuously true, but we don't know. If tomorrow someone proved that there are no odd perfect numbers, then the vacuous truth of that "3000 primes" assertion would follow.
You asked why statements of the form (1) are vacuously true rather than vacuously false.
The answer is "convenience", but I mean that in a very strong sense---in the same sense, for example, that it's convenient to have a number 0 with the property that $x+0=x$ for all $x$, or that negative times negative is positive. These are mathematical conventions, and you can construct consistent mathematical systems with different conventions, but they're usually awkward. You could say they offend against some indefinable sense of mathematical "rightness".
Consider the statement "all odd perfect numbers have at least 3000 prime factors". If it turns out that there are no odd perfect numbers, what truth-value should we assign to this? Three options:
- It doesn't have a truth-value.
Option (3) would be a so-called "truth-value gap" (a term coined by Strawson). Classical logic relies heavily on the law of the excluded middle, which forbids truth-value gaps. (In everyday language, we're not so rigid.)
Option (2) would be very awkward. It's been proven that an odd perfect number must have at least 101 prime factors. I haven't seen the proof, but I'm sure it looks something like this: "Assume $n$ is an odd-perfect number. Then $n=2m+1$ for some $m$, and $\sigma(n)=2n$...". Then some standard reasoning, concluding with at least 101 prime factors. We can check the reasoning for validity without knowing if there are any odd perfects, but only because we regard the assertion as true if there aren't any. With option (2), the reasoning becomes somehow invalid in the absence of odd perfect numbers, even though it would be fine if thare is one.
And that's a general phenomenon. We want to be able to prove conditional statements "If $P$ then $Q$" without knowing if $P$ is true. The classical truth table for $P\to Q$ has F only on the "TF" line ($P$ true and $Q$ false). If you make the "FT" and "FF" lines also F, then you get the truth table for $P\wedge Q$. In other words, to prove that $P$ implies $Q$ you'd have to prove that $P$ is true. That would be most inconvenient.