How quantum mechanics might be when measuring commuting observables

We consider how quantum mechanics might be when measuring commuting observables if we rely on the property of the Kronecker delta. Quantum mechanics is reduced to a classical theory when we consider only commuting observables. Using this fact, we discuss an inconsistency within quantum mechanics when accepting we use the property of the Kronecker delta without extra assumptions about the reality of observables. One of the objectives of this paper is for us to remain wondering the extension of quantum mechanical axiom to concrete commuting observables themselves.