Definition:Square of Opposition/Categorical Statements/Vacuous Terms

Definition
The traditional treatment of the categorical syllogism makes the assumption that no term is vacuous.

However, from the point of view of the full predicate logic, this assumption may not be valid.

Note that if $S$ is empty, then the square of opposition no longer holds.

Although All $S$ are $P$ is vacuously true for such an empty universe, Some $S$ are $P$ is not.

Thus Some $S$ are $P$ is no longer subimplicant to All $S$ are $P$.

Similarly, as Some $S$ are not $P$ is also false, it follows that All $S$ are $P$ and Some $S$ are not $P$ are no longer subcontrary.

Also see

 * Universal Affirmative and Universal Negative are Contrary iff First Predicate is not Vacuous
 * Particular Affirmative and Particular Negative are Subcontrary iff First Predicate is not Vacuous
 * Universal Affirmative implies Particular Affirmative iff First Predicate is not Vacuous
 * Universal Negative implies Particular Negative iff First Predicate is not Vacuous