Another Square of Opposition

In my former post, I showed Terrence Parsons' theory of Aristotle's Square of Opposition in symbolic form.  I also noted that the Square of Opposition holds up under the modern interpretation of the four forms of Term Logic if it is assumed a priori that the subject terms of the forms is non-empty (I am not the first to note this; see section 2.2.2 of iLogic).  So there are two interpretations of the four forms which affirm the Square of Opposition.  But are there others?

I found another interpretation of the four forms of Term Logic that affirms the Square of Opposition.  It's a parallel to Parsons' Square.  Parson constructs the square by taking the modern interpretation of the four forms, bestowing existential import upon Form A and denying existential import to the form on the opposite corner - Form O.  The interpretation I found is constructed by by taking the modern interpretation of the four forms, bestowing existential import upon Form E, and denying existential import to Form I on the opposite corner.  Here is a statement of it in symbolic form:

// Another Square of Opposition

// "All S are P", with no existential import
A <=> (x,Sx->Px)

// "No S are P", with existential import
E <=> ((x,Sx->~Px)&(3x,Sx))

// "Some S are P", with no existential import
// "If there are any S, some of them are P" might be a better way to state it.
I <=> ((3x,Sx)->(3x,Sx&Px))

// "Some S are not P" under the modern interpretation with existential import
// Since it has existential import, there's no need to state is as "Not all S are P".
O <=> ~(x,Sx->Px)

->

// Contraries
~(A&E)

// Contradictories
A ^ O
I ^ E

// Subcontraries
I | O

// Subalterns
A -> I
E -> O

I'd like know if anyone else has thought of it before.

This interpretation, like Parsons' interpretation and the modern interpretation combined with a non-empty subject term, affirms the Logical Hexagon:

// The Logical Hexagon:

// "All S are P", with no existential import
A <=> (x,Sx->Px)

// "No S are P", with existential import
E <=> ((x,Sx->~Px)&(3x,Sx))

// "Some S are P", with no existential import
// "Some S are P, if any S exist" is a better way to state it.
I <=> ((3x,Sx)->(3x,Sx&Px))

// "Some S are not P", existential import
// Since it has existential import, there's no need to state is as "Not all S are P".
O <=> ~(x,Sx->Px)

// The statement U may be interpreted as "Either all S are P or all S are not P."
U <=> ((x,Sx->Px)|(x,Sx->~Px))

// The statement Y may be interpreted as "Some S is P and some S is not P"
Y <=> ((3x,Sx&Px)&(3x,Sx&~Px))

->

// Subalterns: AI, AU, EU, EO, YI, YE
A->I
A->U
E->U
E->O
Y->I
Y->O

// Contraries: AE, EY, YA
~(A&E)
~(E&Y)
~(Y&A)

// Subcontraries: IU, UO, OI
I|U
U|O
O|I

// Contradictories: AO, UY, EI
A^O
U^Y
E^I