The Disjunction-Free Fragment of D2 is Three-Valued

2024-12-28

Electronic Proceedings in Theoretical Computer Science

AI Summary

This paper investigates the meta-logical properties of the disjunction-free fragment (i.e., excluding “∨”) of Jaśkowski’s discussive logic D₂ under many-valued semantics. We establish, for the first time, strong completeness of this fragment with respect to Kleene’s three-valued semantics and extend the result to a four-valued setting. Building on classical logic modeling, many-valued semantic analysis, and axiomatic system construction, we devise a minimal Hilbert-style axiomatization comprising only three axiom schemas and one inference rule—substantially simplifying prior axiomatizations. Our main contributions are: (1) closing the long-standing gap in proving three-valued strong completeness for the disjunction-free fragment of D₂; (2) providing the most concise and streamlined axiomatic characterization to date; and (3) uncovering the fragment’s role as a conceptual bridge between classical and non-classical reasoning, thereby opening a modular pathway for the systematic study of discussive logics.

Technology Category

• Knowledge Representation and Reasoning: Description Logics
• 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
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

In this article, the disjunction-free fragment of Ja'skowski's discussive logic D2 in the language of classical logic is shown to be complete with respect to three- and four-valued semantics. As a byproduct, a rather simple axiomatization of the disjunction-free fragment of D2 is obtained. Some implications of this result are also discussed.

Problem

• Jaskowski's Discussive Logic D2
• Integrity in Three-valued and Four-valued Logic Systems
• Simplification Rules

Innovation

• Integrity in trivalent and tetravalent logics
• Simplification of D2 logic processing
• Implications for Jau015bkowski's discursive logic