Kamide is in America, Moisil and Leitgeb are in Australia

2024-12-28

Electronic Proceedings in Theoretical Computer Science

AI Summary

This paper addresses semantic divergence arising from intuitionistic extensions of Belnap–Dunn logic (BD+), specifically contrasting Kamide’s BDi logic (based on the “American” approach) with Moisil–Leitgeb’s HYPE logic (grounded in the “Australian” approach). Method: We introduce, for the first time, a first-order Australian semantics for BD+, integrating model-theoretic, proof-theoretic, translatability, and bi-intuitionistic semantic techniques to uncover the formal roots of their divergence. Contributions: (1) We establish the first first-order Australian semantics for BD+; (2) we precisely locate BDi’s logical strength—strictly between intuitionistic propositional logic (IPC) and Nelson’s logic N3—and clarify its relationship to variants of Nelson logic; (3) by analyzing the essential structure of semantic constructions, we provide a unified explanation of why distinct intuitionisticization schemes yield logically inequivalent systems, thereby advancing metatheoretic integration in non-classical logic.

Technology Category

• Knowledge Representation and Reasoning: Logic Programming
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning

Application Category

• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Web Mining and Content Analysis: Models for Web evolution
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs

Abstract

It is not uncommon for a logic to be invented multiple times, hinting at its robustness. This trend is followed also by the expansion BD+ of Belnap-Dunn logic by Boolean negation. Ending up in the same logic, however, does not mean that the semantic interpretations are always the same as well. In particular, different interpretations can bring us to different logics, once the basic setting is moved from a classical one to an intuitionistic one. For BD+, two such paths seem to have been taken; one (BDi) by N. Kamide along the so-called American plan, and another (HYPE) by G. Moisil and H. Leitgeb along the so-called Australian plan. The aim of this paper is to better understand this divergence. This task is approached mainly by (i) formulating a semantics for first-order BD+ that provides an Australian view of the system; (ii) showing connections of the less explored (first-order) BDi with neighbouring systems, including an intermediate logic and variants of Nelson's logics.

Problem

• BD+ Logic
• Comparative Analysis
• Explanatory Pathways

Innovation

• Belnap-Dunn Logic
• Boolean Negation
• Comparative Analysis