Natural deduction for belief at most

We present a new approach to the logic of at most, introducing the notion of parametric propositions to modal natural deduction proofs. We apply the method with a natural deduction formulation of the doxastic logic N, a new system in the “only knowing” family. Using parametric proof rules, we give introduction and elimination rules for belief at most that directly match the natural second-order formulation of the concept. N is sound and complete with respect to the class of intended models. We conjecture that it weakly normalizes and that it satisfies the subformula property.