Modal Extensions of CLoN with Bi-neighborhood Semantics

2026-06-29

AI Summary

This study addresses the problem of system collapse in the non-normal modal logic CLoN when weak negation coexists with deontic principles. To resolve this, the paper proposes a dual neighborhood semantic framework, equipping each modal operator with two neighborhood functions and integrating rejection-set techniques to adequately capture modal semantics under weak negation. This approach constitutes the first application of dual neighborhood semantics to CLoN, successfully validating non-trivial modal axioms involving weak negation. The resulting logical system accommodates standard deontic principles while formally representing moral dilemmas without trivialization, thereby establishing a robust semantic and axiomatic foundation for deontic logics capable of handling moral conflicts.

Abstract

In this paper we will present neighborhood semantics for non-normal modal extensions of $\clon$, which is a sublogic of {\sf FDE}. Our framework is built upon earlier work on {\sf FDE}-based non-normal modal logics and employs two different neighborhood functions for each modal operator. Despite being a logic with a very weak negation operator, we will show that with the right definition of the rejection sets of the modal operators, we can validate non-trivial axioms that contain the weak negation operator. The philosophical aim of our approach is to construct the basis for deontic logics that are able to accommodate both the usual deontic principles and moral dilemmas, without resulting in trivialization of the system.

Problem

• deontic logics
• moral dilemmas
• non-normal modal logics
• weak negation
• trivialization

Innovation

• bi-neighborhood semantics
• non-normal modal logic
• weak negation
• deontic logic
• moral dilemmas