# Definition:Vacuous Truth

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## 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:

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

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**.

## Sources

- 2000: James R. Munkres:
*Topology*(2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 1$: Fundamental Concepts - 2010: Raymond M. Smullyan and Melvin Fitting:
*Set Theory and the Continuum Problem*(revised ed.) ... (previous) ... (next): Chapter $2$: Some Basics of Class-Set Theory: $\S 3$ Axiom of the empty set