Definition:Vacuous Truth

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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:

$\forall x: \map P x \implies \map Q x$

when the propositional function $\map P x$ 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.

Linguistic Note

The word vacuous literally means empty.

It derives from the Latin word vacuum, meaning empty space.