Initially, I suspected no because categorical propositions that have contradictory relations according to the modern square of opposition will always have opposing truth values. For instance, if we have the following propositions
A: "All S are P"
O: "Some S are not P"
then both cannot be true simultaneously since what they claim oppose or contradict each other, so proposition O being true would necessitate proposition A is false, and vice versa. But I'm confused when my textbook (Patrick Hurley's A Concise Introduction to Logic, 12 e., pg. 216) says the statement ~A: "it is false that all S are P" makes the claim that "All S are P" is false and shows that its Venn diagram matches proposition O's Venn diagram.
The author then concludes
as the diagram shows, "it is false that all S are P" is actually a particular proposition.
Which I interpret to mean ~A is equivalent to O since their diagrams match.
Since ~A is equivalent to saying "All S are P" is false and ~A's diagram matches O's diagram, then doesn't that mean proposition A being false and proposition O being true have the same Venn diagram and thus are logically equivalent? Or expressed symbolically, since ~A = A(F) and ~A = O(T) then A(F) = O(T)?
