Why is a statement "vacuously true" if the hypothesis is false, or not satisfied?

Why isn't a conditional statement said to "not apply" instead of be "vacuously true" if the hypothesis is not satisfied? That would seem more appropriate.

It's not appropriate because it doesn't reflect the fact that the statement is actually vacuously true. More useful might be understanding what vacuous actually means: without contents, empty, lacking in ideas or intelligence, meaningless, etc.

Thus, something like the following is vacuously true:

If the moon is made of barbecue and spare ribs, then I'm smarter than Ramanujan.

This is vacuously true; that is, logically it is true, but it really doesn't mean anything. The moon is obviously not made of barbecue and spare ribs; thus, whatever statement I have is true in no meaningful way (i.e., the statement is vacuously true).

Saying the proposition "does not apply"...what would be the use in that? Doesn't apply to what? Saying something is vacuously true communicates what ultimately needs to be communicated--that a statement is true, but its truth is meaningless.

We would often say that a statement of the form $P\rightarrow Q$ doesn't apply if $P$ cannot be assumed. This is not a definition or anything formal though - it is merely a statement that knowing that $P\rightarrow Q$ holds gives us no information about where $Q$ holds if $P$ does not. This is, in fact, an interpretation of what vacuous truths mean. That is to say, we have two cases. If we're lucky, we get:

Suppose $P\rightarrow Q$ and $P$. Then, if we look at a truth table, the only way this could be would be if $Q$ were true. Thus, we can apply the statement $P\rightarrow Q$ to the situation.

If our truth is vacuous, however, we end up with:

Suppose $P\rightarrow Q$ and $\neg P$. If we look at a truth table, we see that this could be true if $Q$ is false or if $Q$ is true. So, knowing these two facts tells us nothing about $Q$ - so, our statement $P\rightarrow Q$ cannot be advantageously applied.

The point here is that the truth-table definition of $P\rightarrow Q$ gives rise to the fact that vacuously true statements aren't very helpful. So, while the definition is counterintuitive, it behaves exactly as we expect when we apply it.

In an intuitive sense, if we wanted to think about a statement like "if a bird is a crow, it is black," we would find that consistent with the following observations

• We see a crow that is black.

• We see a raven, which is also black.

• We see a blue jay, which is not black.

which correspond to the three "true" cases of $P\rightarrow Q$ on a truth table. We would be surprised, however, if we saw a crow that was not black - which is the only "false" case. So, the way we interpret $P\rightarrow Q$ is only about "Is it consistent to believe this?" not "Is it useful?"

The phrase "vacuously true" is commonly applied to statements like $\forall x \in X : P(x) $ because when $X$ is the empty set, the statement is always true, regardless of what $P$ represents. Notice the connection between the empty set and the phrase vacuous.

The reasoning here is because the sentence $\forall x \in X : P(x) $ is really a shorthand for $\forall x : x \in X \rightarrow P(x)$. When $X = \emptyset$, the premise of the if then statement is always false -- and false implies everything.

For example, if I have an empty bag and I make the claim "everything in this bag is red!", the only way to demonstrate that my claim is false is by finding a counterexample -- something in the bag that is not red. Since there is nothing in the bag, there is no counter example, and therefore the claim must be true. More specifically, my claim is vacuously true.

The reason this same phrase "vacuously true" applies in general statements like $P \rightarrow Q$ when $P$ is false is due to the notion that properties can represent sets (classes really, but this is not important here). When $P$ is false, the collection of all objects with the property $P$ is the empty set.