...intuitionistic logic by taking ∃X.F = ∀O(∀X(F → O) → O). It is also known that the converse is not true in intuitionistic logic, but ∀ is definable in the second-order propositional subtractive logic =-=[Cro01]-=- from ∃ and − . Indeed, by applying the duality, ∀ is definable as follows: ∀X.F = ∃O((O − ∃X(O − F)). The second-order propositional subtractive logic is conservative over the second-order propositio...

...rmula built from ⊤, ⊥, ∨ , ∧ , → , ∃ has the same semantics as the formula ∀X(X ∨ ¬X). 2 Kripke Models Let us recall the primary interpretation of second order propositional formulas in Kripke models =-=[Kre97]-=-. This definition includes the semantics of subtraction [Cro01]. Definition 1. A second-order Kripke structure is a triple (X, O, P) where (X, O) is a bi-topological space (i.e. the upper-closed sets ...