Vacuous falsehood – does it exist, and are there examples

Question

I've ben struggling with the concept of vacuous truth, as used (1) in proving implications, (2) as base cases for induction proofs.

To help me understand, it would be useful to understand if the concept of vacuous falsehood exists, and if so, what simple examples might be.

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Discussion

This question comes from the notion that truth is defined where something cannot be proved false – which to me has always felt insufficient.

So I ask myself, if something can't be proved true, is it vacuously false?

I expect the answer is that there is no such thing as vacuously false. Explaining this will shed light on my original confusion.

Answer 1

Here's my favorite example of vacuous truth. First let's agree that

all rubies are red.

This is true, by definition; the same gemstone, when not colored red, is called a sapphire.

Since all rubies are red, we can conclude

all the rubies in my vault are red.

This makes sense because if all rubies are red, then the ones in the vault are certainly red.

"Vacuous truth" just means that we agree that we will consider "all the rubies in my vault are red" to be true even if there are no rubies in the vault. This is mostly for convenience. Otherwise we find ourselves saying a lot of things like

all the rubies in my vault are red, if there are any rubies in my vault.

which makes mathematical discussion more confusing and complicated for no real benefit.

Now to your question. What would a vacuous falsity be? Tian Vlašić's suggestion in the comments is just what we need:

If a statement is vacuously true, then its negation is vacuously false.

The negation of

all the rubies in my vault are red

is

there is a ruby in my vault that is not red.

One way this could be false is if there are some rubies in my vault and they are all red.

But another way this could be false is that there might not be any rubies in my vault at all. This is the vacuously false case.

And in the vacuously false case we don't even need to read the second half of the sentence:

there is a ruby in my vault that …

It doesn't matter what “…” is. We know immediately this is vacuously false, because there is no ruby of any sort in my vault.

The irrelevance of the “…is not red” part is mirrored exactly in the irrelevance of the “… are red” part in the vacuously true statement:

all the rubies in my vault are …

Again, if my vault has no rubies, then it doesn't matter what “…” is. We know the statement is vacuously true, because there is no ruby of any sort in my vault.