Universal Statement has no Existential Import

Theorem

A universal statement of the form:

All $A$ are $B$

has no existential import.

Proof

If there exist no $A$, then:

All $A$ are $B$

is vacuously true, and hence remains true.

If there exist no $B$, then:

All $A$ are $B$

is vacuously true when there exist no $A$.

Hence the result by definition of existential import.

$\blacksquare$

Examples

French Kings

The statement:

All French kings are bald

has no existential import, as there are no French kings.

Hence All French kings are bald is vacuously true.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): existential import
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): existential import