Some Things that are Right with the Traditional Square of Opposition

The doctrines encoded in the traditional square of opposition were invented by Aristotle.  These doctrines differ from modern views principally in that the Aform (Every P is Q) and the O form (No P is Q) are contraries.  This results in the A form having existential import: if every P is Q then there must be P's. More importantly, the O form (Some P is not Q) turns out to be true when there are no P's.  In late medieval times this was taken to be the correct view.  The idea that the O form is true when its subject term is empty may be defended onthe grounds that these forms are pieces of canonical  notation in a theory of logic, and that they do not necessarily reflect ordinary usage.  On this view, the doctrine is coherent; it leads to theory in which all main terms of affirmative propositions have existential import, and the main terms of negative propositions are the opposite: a negative proposition is automatically true whenever any of its main terms are empty.

The question remains whether a semantics can  be given which  agrees with these results, and in which the O form (Some P is not Q) is treated as an existentially quantified proposition with a negation inside.  This becomes important when the notation is expanded - as it was in late medieval times - by quantifying the predicate term, and allowing negation signs to occur more widely, so as to yield forms such as "not every P is not no Q".  I argue that the doctrine works smoothly, and preserves the generalization about affirmative and negative forms given above.