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:

$\forall x: P \left({x}\right) \implies Q \left({x}\right)$

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