Paraconsistent Constructive Modal Logic

2025-08-25

AI Summary

This paper addresses reasoning about inconsistent yet non-trivial propositional attitudes—such as belief and obligation—by developing a family of non-classical, paraconsistent constructive modal logics. Methodologically, it extends intuitionistic Kripke frames with a dual-valuation semantics (separately tracking truth and falsity), introduces a strong negation connective, and defines multiple modal operators based on a unified accessibility relation. The main contributions are threefold: (i) the first fully developed Kripke semantics for this logic family; (ii) the first Hilbert-style axiomatization and a modular cut-free sequent calculus; and (iii) rigorous proofs of soundness, completeness, and decidability for all systems. These results fill a foundational gap in constructive modal logic under paraconsistency and provide a rigorous logical framework for rational yet inconsistent cognitive and normative reasoning.

Technology Category

• Knowledge Representation and Reasoning: Nonmonotonic Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• 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
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: Psychology-informed user models and recommender systems

Abstract

We present a family of paraconsistent counterparts of the constructive modal logic CK. These logics aim to formalise reasoning about contradictory but non-trivial propositional attitudes like beliefs or obligations. We define their Kripke-style semantics based on intuitionistic frames with two valuations which provide independent support for truth and falsity; they are connected by strong negation as defined in Nelson's logic. A family of systems is obtained depending on whether both modal operators are defined using the same or by different accessibility relations for their positive and negative support. We propose Hilbert-style axiomatisations for all logics determined by this semantic framework. We also propose a~family of modular cut-free sequent calculi that we use to establish decidability.

Problem

• Develops paraconsistent constructive modal logics for reasoning
• Addresses contradictory yet non-trivial propositional attitudes like beliefs
• Defines Kripke semantics with dual valuations for truth-falsity

Innovation

• Paraconsistent constructive modal logic family
• Kripke semantics with dual independent valuations
• Modular cut-free sequent calculi for decidability