Towards Propositional KLM-Style Defeasible Standpoint Logics

TL;DR

The paper addresses integrating standpoints with KLM-style defeasible propositional logic to model multiple, potentially conflicting beliefs within a single ontology. It introduces Defeasible Restricted Standpoint Logic (DRSL), combining ranked interpretations for the base logic with ranked standpoint structures and standpoint modal operators $\Box_s$ and $\Diamond_s$. A non-monotonic entailment, $\mathcal{K}\mid \approx_{RC}\phi$, is defined by a constructive algorithm (StandpointSplit and RCStandpoint) that reduces DRSL to propositional KLM entailment and is proven to be equivalent to a single representative model $M_{RC}^{\mathcal{K}}$. Complexity results show RCStandpoint lies in $P^{NP}$ (polynomial with an NP oracle), with a Horn-fragment in P, mirroring the propositional rational closure, and the framework is illustrated on a canonical example involving tomatoes.

Abstract

The KLM approach to defeasible reasoning introduces a weakened form of implication into classical logic. This allows one to incorporate exceptions to general rules into a logical system, and for old conclusions to be withdrawn upon learning new contradictory information. Standpoint logics are a group of logics, introduced to the field of Knowledge Representation in the last 5 years, which allow for multiple viewpoints to be integrated into the same ontology, even when certain viewpoints may hold contradicting beliefs. In this paper, we aim to integrate standpoints into KLM propositional logic in a restricted setting. We introduce the logical system of Defeasible Restricted Standpoint Logic (DRSL) and define its syntax and semantics. Specifically, we integrate ranked interpretations and standpoint structures, which provide the semantics for propositional KLM and propositional standpoint logic respectively, in order to introduce ranked standpoint structures for DRSL. Moreover, we extend the non-monotonic entailment relation of rational closure from the propositional KLM case to the DRSL case. The main contribution of this paper is to characterize rational closure for DRSL both algorithmically and semantically, showing that rational closure can be characterized through a single representative ranked standpoint structure. Finally, we conclude that the semantic and algorithmic characterizations of rational closure are equivalent, and that entailment-checking for DRSL under rational closure is in the same complexity class as entailment-checking for propositional KLM.