Would propositional calculus be sound and valid if it only had the natural deductions MPP and ^I and nothing else?

Question

If we made a PC system that only allows for the two rules MPP (Modus Ponens) and ^I (Or Introduction) would the resulting system be sound and would it be complete? In other words is it possible to rewrite all the natural deduction rules using only MPP and ^I

Answer 1

Hilbert-style Axiom systems are sound and complete systems that use Modus Ponens as their only inference rule. But, that is only because in these systems you can put down instantiations of certain axioms schemas as lines in a proof. Without those axioms, they would be sound, but not complete. Indeed, one could regard these axioms as special inference rules that allow one to derive a statement from nothing, and as such they would have several inference rules other than just Modus Ponens.

Without the use of such axioms, Modus Ponens together with Or Introduction is not going to be complete. You couldn't, for example, do something as simple as deriving $P$ from $P \land Q$. But since Modus Ponens and Or Introduction are both valid inferences, the system is sound.