# A General Axiomatization for the logics of the Hierarchy ${\mathbb{I}}^n {\mathbb{P}}^k$

## TL;DR

This paper introduces a unified axiomatization for a family of logical systems that generalize weakly-intuitionistic and paraconsistent logics, providing a formal framework and natural ordering.

## Contribution

It presents a general axiomatization for the hierarchy ${\mathbb{I}}^n {\mathbb{P}}^k$ using finite matrices, unifying and extending previous logics.

## Key findings

- The family ${\mathbb{I}}^n {\mathbb{P}}^k$ can be ordered naturally.
- An adequate axiomatics for each logic $I^n P^k$ is established.
- The logics are formally defined and characterized.

## Abstract

In this paper, 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 an adequate axiomatics for each logic of the form $I^n P^k$.