The logics of the family ${𝕀}^n{\mathbb{P}}^k$:=${\left\{I^n{P}^k\right\}}_{(n,k)\in {\omega }^2}$ 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.

In this paper we present a topological duality for a certain subclass of the $F_{\omega }$-structures defined by M. M. Fidel, which conform to a non-standard semantics for the paraconsistent N. C. A. da Costa logic $C_{\omega }$. Actually, the duality introduced here is focused on $F_{\omega }$-structures whose supports are chains. For our purposes, we characterize every $F_{\omega }$-chain by means of a new structure that we will call down-covered chain (DCC) here. This characterization will allow us to prove the dual equivalence between the...