Definition:Vacuous Truth

Definition
Let $$P \implies Q$$ be a conditional statement.

Suppose that $$P$$ is false.

Then the statement $$P \implies Q$$ is a vacuous truth, or is vacuously true.

It is frequently encountered in the form:
 * $$\forall x: P \left({x}\right) \implies Q \left({x}\right)$$

when the propositional function $$P \left({x}\right)$$ is false for all $$x$$.

Such a statement is also a vacuous truth.

For example, the statement:
 * "All cats who are expert chess-players are also fluent in ancient Sanskrit"

is (vacuously) true, because (as far as the author knows) there are no cats who are expert chess-players.