# Definition:Vacuous Truth

From ProofWiki

## 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.

## Sources

- James R. Munkres:
*Topology*(2nd ed., 2000)... (previous)... (next): $1$: Set Theory and Logic: $\S 1$: Fundamental Concepts