Swap Kripke models for deontic LFIs

TL;DR

This paper introduces a novel semantic framework combining swap structures and Kripke models for deontic logics based on LFIs, providing new axiomatizations and semantics for the $C^{D}_n$ hierarchy, including the first deontic paraconsistent system.

Contribution

It presents the first combination of swap structures with Kripke models via swap Kripe models and offers a full axiomatization and semantics for the $C^{D}_n$ hierarchy.

Findings

Developed swap Kripe models for deontic LFIs

Provided axiomatization for $C^{D}_n$ hierarchy

Extended semantics to include the $C^{D}_1$ system

Abstract

We present a construction of nondeterministic semantics for some deontic logics based on the class of paraconsistent logics known as Logics of Formal Inconsistency (LFIs), for the first time combining swap structures and Kripke models through the novel notion of swap Kripe models. We start by making use of Nmatrices to characterize systems based on LFIs that do not satisfy axiom (cl), while turning to RNmatrices when the latter is considered in the underlying LFIs. This paper also presents, for the first time, a full axiomatization and a semantics for the $C^{D}_n$ hierarchy, by use of the aforementioned mixed semantics with RNmatrices. This includes the historical system $C^{D}_1$ of da Costa-Carnielli (1986), the first deontic paraconsistent system proposed in the literature.