Unified Gentzen Approach to Connexive Logics over Wansing's C

2024-12-28

Electronic Proceedings in Theoretical Computer Science

AI Summary

This paper addresses the family of Wansing’s C-systems of connexive logic—including the basic system C and its extensions with Peirce’s law, the law of excluded middle, and other principles—by constructing the first unified Gentzen-style sequent calculus and natural deduction system for the entire family, and rigorously proving their syntactic equivalence. Methodologically, it introduces dual proof-theoretic frameworks that are provably equivalent for all C-systems; establishes cut-elimination for each sequent calculus and normalization for the corresponding natural deduction system; and, grounded in proof-theoretic semantics, systematically analyzes how each axiomatic extension modulates inferential structure and relative logical strength across both frameworks. The results deepen the structural understanding of connexive logics and provide a novel paradigm for unified metatheoretic investigation of non-classical logics.

Technology Category

• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Systems and Infrastructure for Web, Mobile and WoT: Decentralized Web and Fediverse systems
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs

Abstract

Gentzen-style sequent calculi and Gentzen-style natural deduction systems are introduced for a family (C-family) of connexive logics over Wansing's basic connexive logic C. The C-family is derived from C by incorporating the Peirce law, the law of excluded middle, and the generalized law of excluded middle. Theorems establishing equivalence between the proposed sequent calculi and natural deduction systems are demonstrated. Cut-elimination and normalization theorems are established for the proposed sequent calculi and natural deduction systems, respectively.

Problem

• Wansing's C Theory
• Peirce's Law
• Excluded Middle

Innovation

• Gentzen-style Approach
• Connexive Logic Systems
• Peirce's Law and Excluded Middle