Modal Extensions of Generalised Nelson Logics

2026-10-07

AI Summary

This study addresses the absence of modal extension semantics for generalized Nelson logics by constructing a Kripke-style modal extension for the semi-relevant logic RM based on Dunn’s binary relational semantics, which is subsequently generalized to the entire family of generalized Nelson logics. Methodologically, this work proposes a unified modal axiomatization framework that systematically integrates Kripke semantics with axiomatic techniques. The principal contribution lies in establishing soundness and completeness proofs for the modal extensions of all generalized Nelson logics. By bridging the modal gap in relevant epistemic logics, this research provides a general paradigm for the modalization of non-classical logics.

Abstract

Motivated by relevant epistemic logic, we extend Dunn's simple Kripke-style binary relational semantics for the semi-relevant logic RM to fit a modal extension of RM. It is shown that the modal axioms and rules that need to be added to RM to obtain a sound and complete axiomatisation of its modal extension give an axiomatisation of modal extensions of all logics in the family of generalised Nelson logics, also studied by Dunn.

Problem

• Modal Extensions
• Generalised Nelson Logics
• Semi-relevant Logic RM
• Kripke Semantics
• Axiomatisation

Innovation

• Generalised Nelson Logics
• Modal Extensions
• Kripke Semantics
• Relevant Logic RM
• Axiomatisation