A Note on Partial Postulate Sets for Propositional Logic

Assume modus ponens and the rule of substitution for propositional variables as rules of inference. Then does a system F of n axioms exist that satisfies the following six conditions: (i) each axiom of F is independent of the remaining axioms, (ii) F yields the theorems of the classical sentential calculus with ⊃, &, ⋁ and ¬ as primitive connectives, (iii) there is a subsystem H of F with n-1 axioms that yields the theorems of Hey ting’s intuitionistic sentential calculus, (iv) there is a subsystem J of H with n-2 axioms that yields the theorems of Johansson’s minimal calculus, (v) each axiom of F contains ⊃ and no axiom contains more than two different connectives, (vi) for each selection γ of connectives (from ⊃, &, ⋁ and ¬) that includes implication, and each choice S from F, H and J, there is a subsystem S γ of S, the axioms of which contain no other connectives than those of γ and which yields all theorems derivable from S that contain connectives of γ Only?1