A taxonomy for controlling (in)consistency

2026-04-20

AI Summary

This study addresses the challenge of formally characterizing varying degrees of paraconsistent commitment and their associated notions of compatibility, incompatibility, and negation. It proposes a hierarchical paraconsistent logical framework, denoted \(L_n^k\), which employs two independent parameters to modulate, respectively, the iterative behavior of consistency operators and the strength of negation, thereby unifying a spectrum of philosophical stances ranging from skepticism to dogmatism. The main contributions include the construction of the first multi-dimensional classification framework for fine-grained control over paraconsistency, subsuming several known Logics of Formal Inconsistency (LFIs); an illustrative construction of the five-valued paraconsistent logic LFI3; and a comprehensive semantic characterization of the \(L_n^k\) family—achieved through swap structures, Karnaugh maps, twist structures, and RN-matrices—together with proofs of soundness and completeness.

Technology Category

• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Knowledge Representation and Reasoning: Logic Programming
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

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.

Problem

• paraconsistency
• consistency
• negation
• logical hierarchy
• inconsistency

Innovation

• Logics of Controlled Consistency
• paraconsistent logic
• swap structures
• RNmatrices
• many-valued logic