Abstract: 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.
Keywords: mathematical logic; Kalmár's completeness proof; many-valued logic
DOI: DOI 10.14712/1213-7243.2024.009
AMS Subject Classification: 03B50 03B53