Authors: Nicholas Leisegang, Thomas Meyer, Sebastian Rudolph
Organizations: University of Cape Town and CAIR, South Africa · Technische Universität Dresden, Dresden, Germany · ScaDS.AI – Center for Scalable Data Analytics and Artificial Intelligence Dresden/Leipzig, Germany
Abstract
In this paper, we integrate the defeasible logic of Kraus, Lehmann and Magidor (KLM) with the standpoint logic framework of Gómez Álvarez and Rudolph. This is done with the goal of formally expressing knowledge taking into account multiple (possibly contradicting) viewpoints, which in turn may hold defeasible beliefs. In doing so, we utilise Defeasible Restricted Standpoint Logics (DRSL), introduced by Leisegang et al. Our work expands on previous work by providing a foundational representation result for DRSL semantics and systematically lifting several well-known entailment relations from the propositional case to the standpoint-enhanced setting. In particular, we characterise the semantics for DRSL through a set of KLM-style postulates adapted for the standpoints case. We furthermore provide a means to lift preferential entailment, and the class of entailment relations based on single ranking functions from the purely propositional to the standpoint-enhanced context, including rational and lexicographic closure. We show this can be done equivalently through semantic and algorithmic means. Furthermore, we show that, for each considered form of entailment, the complexity class of entailment checking does not change when moving from propositional KLM to DRSL.
Recent work in defeasible reasoning has seen notions of preferential semantics and entailment in the style of Kraus et al. applied to modal logics. However, work in this field has focussed primarily on satisfiability checking, and monotonic notions of entailment, which may be inferentially weak. One particular modal logic where this has been introduced is propositional standpoint logics, where modalities can express the views of different viewpoints. This has resulted in the formalisation of propositional defeasible standpoint logic (PDSL). In this paper, we propose a means of lifting the class of (non-monotonic) rational entailment relations from traditional KLM-style reasoning to a fragment of PDSL. In order to do so, we extend the expressivity of PDSL via situated standpoint conditionals, allowing us to talk about a defeasible conditional holding in the context of a given standpoint. This allows us to re-characterise the syntax of PDSL in terms of situated conditionals, and shows that a large fragment of PDSL is expressible as a set of situated conditionals. We then focus on characterising non-monotonic entailment in this fragment, defining a method to transport any ranking-based entailment relation from the propositional case into the PDSL case. This is first described in the general case and then considered in the specific cases of rational and lexicographic closures, providing a faithful translation of each inference into PDSL. We also show that entailment-checking in this fragment of PDSL can be done largely using algorithms from the propositional case, while preserving complexity bounds.
Defeasible statements are statements that are likely, or probable, or usually true, but may occasionally be false. Plausible reasoning makes conclusions from statements that are either facts or defeasible statements without using numbers. So there are no probabilities or suchlike involved. Seventeen principles of logics that do plausible reasoning are suggested and several important plausible reasoning examples are considered. There are 14 necessary principles and 3 desirable principles, one of which is not formally stated. A first-order logic, called Plausible Logic (PL), is defined that satisfies all but two of the desirable principles and reasons correctly with all the examples. As far as we are aware, this is the only such logic. PL has 8 reasoning algorithms because, from a given plausible reasoning situation, there are different sensible conclusions. This article is a condensation of my book `Plausible Reasoning and Plausible Logic' (PRPL), which is to be submitted. Each section of this article corresponds to a chapter in PRPL, and vice versa. The proofs of all the results are in PRPL, so they are omitted in this article.
This paper establishes and proves complexity results for entailment for cumulative propositional dependence logic and for cumulative propositional logic with team semantics. As recently shown, cumulative logics are famously characterised by System~C and exactly captured by the cumulative models of Kraus, Lehmann and Magidor. This gives rise to the entailment problem via relational models, which is specifically considered here.