*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

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