Restricted swap structures for da Costa's $C_n$ and their category

TL;DR

This paper introduces restricted swap-structures for da Costa's $C_n$ logics, extending their semantic characterization via RNmatrices over Boolean algebras, and shows the category of these structures is isomorphic to Boolean algebras.

Contribution

It extends RNmatrices for $C_n$ using restricted swap-structures over Boolean algebras, providing a new semantic framework and categorical equivalence.

Findings

RNmatrices characterize $C_n$ logic with decision procedures.

Restricted swap-structures generalize previous semantics.

Category of RNmatrices is isomorphic to Boolean algebras.

Abstract

In a previous article we introduced the concept of restricted Nmatrices (in short, RNmatrices), which generalize Nmatrices in the following sense: a RNmatrix is a Nmatrix together with a {\em subset} of valuations over it, from which the consequence relation is defined. Within this semantical framework we have characterized each paraconsistent logic $C_{n}$ in the hierarchy of da Costa by means of a $(n+2)$-valued RNmatrix, which also provides a relatively simple decision procedure for each calculus (recalling that $C_1$ cannot be characterized by a single finite Nmatrix). In this paper we extend such RNmatrices for $C_{n}$ by means of what we call {\em restricted swap-structures} over arbitrary Boolean algebras, obtaining so a class of non-deterministic semantical structures which characterizes da Costa's systems. We give a brief algebraic and combinatorial description of the elements of the underlying RNmatrices. Finally, by presenting a notion of category of RNmatrices, we show that the category of RNmatrices for $C_{n}$ is in fact isomorphic to the category of non-trivial Boolean algebras.