A Kalmár-style completeness proof for the logics of the hierarchy ${\mathbb{I}}^n {\mathbb{P}}^k$

The logics of the family ${\mathbb{I}}^n {\mathbb{P}}^k$:=$\{{ I^n P^k}\}_{(n,k) \in \omega^2}$ are formally defined by means of finite matrices, as a simultaneous generalization of the weakly-intuitionistic logic $I^1$ and of the paraconsistent logic $P^1$. It is proved that this family can be naturally ordered, and it is shown a sound and complete axiomatics for each logic of the form $I^n P^k$. The involved completeness proof showed here is obtained by means of a generalization of the well-known Kalmár's method, usually applied for many-valued logics.