Defeasible Conditional Obligation in a Two-tiered Preference-based Semantics (Extended Version)

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.

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.

In Dung-style abstract argumentation, various semantics capture notions of acceptability of arguments. The admissibility semantics capture the notion that an argument can be consistently defended from any potential counterargument. Weak semantics often relax the demands of admissibility by restricting which counterarguments must be taken seriously (e.g., discounting self-defeating or otherwise incoherent attacks). Many prominent proposals for weak semantics remain extension-based in a stronger sense. While these semantics discount attacks from arguments which are considered unreasonable, they still require a uniform defense against all reasonable arguments, even if they are collectively inconsistent. This uniformity can be too demanding when defensibility is inherently strategic, and thus the appropriate reply depends on the opponent's line of attack. We introduce tenability, a family of dialogue-based semantics that formalize when a designated argument (or a set of arguments) can be maintained in debate by a proponent against any conflict-free attack which the opponent may present. The approach is motivated by three natural benchmark patterns: self-defeating attack, floating assignment, and disjunctive reinstatement, on which tenability behaves differently from all weak semantics previously considered in the literature. We define three variants -- static tenability, tenability, and strong tenability -- via monotone commitment games over finite conflict-free moves, differing in the obligations imposed on the disputants. We establish the relative strength of these notions, prove implications and separations with previously studied weak semantics, and we analyze computational complexity on finite frameworks: deciding static tenability is $Π^P_2$-complete, while deciding tenability and strong tenability is PSPACE-complete.