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:

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

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.

The word vacuous literally means empty.

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

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