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...

In [6] da Costa has introduced a new hierarchy $N_i$, $1\le i\le w$ of logics that are both paraconsistent and paracomplete. Such logics are now known as non-alethic logics. In this article we present an algebraic version of the logics $N_i$ and study some of their properties.

The Author describes new systems of logic (called "nonalethic") which are both paraconsistent and paracomplete. These systems are connected with the logic of vagueness and with certain philosophical problems (e.g. with some aspects of Hegel's logic).