A taxonomy for controlling (in)consistency

TL;DR

This paper introduces a hierarchy of Logics of Controlled Consistency (LCC) inspired by da Costa's systems, providing a structured way to represent varying degrees of paraconsistency and related notions of negation and consistency.

Contribution

It develops a hierarchy of LFIs with swap structure semantics, introduces a new 5-valued LFI3, and offers a comprehensive semantic and axiomatic framework for these logics.

Findings

Well-known LFIs are particular cases of LCCs.

The hierarchy allows representation of different philosophical positions from skepticism to dogmatism.

LFI3 exemplifies a many-valued LFI with more than three values.

Abstract

In this article, the hierarchy of LFIs L$_n^k$, Logics of Controlled Consistency (LCC), is introduced. Inspired by da Costa's original C$_n$ systems, this hierarchy can represent different degrees of paraconsistent commitment and different related notions of consistency, inconsistency, and negation associated with each two-dimensional level of these logics. In one dimension, the logics become increasingly more paraconsistent by allowing the consistency operator to behave inconsistently up to a fixed iteration. In another dimension, the negation is increasingly strengthened. Initially, we present these logics with a swap structure semantics, showing their soundness and completeness. Some well-known LFIs are shown to be particular cases of LCCs. With some examples, we show how these different logics represent different types of paraconsistent commitment: from skepticism to dogmastism, these logics have the multiplicity to represent these different philosophical positions. Furthermore, the development of the hierarchy in a general manner allows pragmatism to take place when considering the different types of paraconsistent commitment. Each level we go up in this direction we get a stronger family of logics. Furthermore, we also present an extension of an LCC, a 5-valued LFI called LFI3, a sublogic of LFI1. LFI3 presents a paradigmatic case for the development of many-valued LFIs that have more than three values. Using a technique that combines Karnaugh Maps and Twist Structures, we give an axiomatization and a semantical account of LFI3. Finally, using RNmatrices, we give a general semantical account of the L$_n^k$ family of logics, and we also prove its soundness and completeness.