Definition:Vacuous Truth

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

Suppose that $$P$$ is false.

Then the statement $$P \implies Q$$ 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 vacuously true.

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.

The word "vacuous" derives from the Latin word "vacuum", meaning "empty space".