TITLE: Why is a statement "vacuously true" if the hypothesis is false, or not satisfied?
QUESTION [24 upvotes]: 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.

REPLY [4 votes]: 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.