The classical square of opposition presupposes that the terms of any categorical proposition have referents. If we adopt the modern doctrine about existential import, then some of the relationships in the traditional square no longer hold.
Subalternation and the Modern Square of Opposition
If the I proposition, "Some S are P" implies the existence of Ss, while the corresponding A proposition does not, then the truth of the A proposition does not imply the truth of the corresponding I. The same is true for E and O. In other words, subalternation must be removed from the square.
Subcontrary and the Modern Square of Opposition
Two statements are subcontraries if, by virtue of their logical form, they could both be true but could not both be false. However, if no Ss exist, then both particular statements are false. I and O no longer fit the definition of subcontraries. Thus, subcontrary must be removed from the square.
Contrary and the Modern Square of Opposition
Since neither of the universal statements have existential import, then they are both true in the case where there are no Ss. Thus, A and E are no longer contraries. Contrary must, then, be removed from the square.
Contradiction and the Modern Square of Opposition
The only relationship that survives in the modern square of opposition is that of contradictories. If there exists a single thing that is both S and P, then the I proposition is true and the E false. However, if nothing is both S and P, then the I proposition is false and the E is true--even if the absence of things that are both S and P is due to the fact that there aren't any Ss at all.
The same reasoning applies to the A and O propositions. So E is true if and only if I is false, and A is true if and only if O is false.