Universal Statement has no Existential Import
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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