# Some new papers on 1st order nonmonotonic logic, formal dialogues of explanation, and using inconsistency measures for normative reasoning

We got a few new paper accepted.

Christian Straßer and Sanderson Molick,

*Minimal inconsistency-tolerant logics: a quantitative approach*forthcoming in the Australasian Journal of Logic. Here’s the abstract:In order to reason in a non-trivializing way with contradictions, para- consistent logics reject some classically valid inferences. As a way of re- covering some of these inferences, Graham Priest (1991) proposed to nonmonotonically strengthen the Logic of Paradox by allowing the se- lection of “less inconsistent” models via a comparison of their respective inconsistent parts. This move recaptures a good portion of classical logic in that it does not block, e.g., disjunctive syllogism, unless it is applied to contradictory assumptions. In Priest’s approach the inconsistent parts of models are compared in an extensional way by considering their inconsis- tent objects. This distinguishes his system from the standard format of (inconsistency-)adaptive logics pioneered by Diderik Batens, according to which (atomic) contradictions validated in models form the basis of their comparison. A well-known problem for Priest’s extensional approach is its lack of the Strong Reassurance property, i.e., for specific settings there may be infinitely descending chains of less and less inconsistent models, thus never reaching a minimally inconsistent model. In this paper, we demonstrate that Strong Reassurance holds for the extensional approach under a cardinality-based comparison of the incon- sistent parts of models. Furthermore, we introduce and investigate the metatheory of the class of first-order nonmonotonic inconsistency-tolerant construct over the extensional or quantitative comparisons of their respective models. Core model-theoretic properties for these logics, such as the Löwenheim-Skolem theorems, along with other nonmonotonic properties, are further studied.

Kees van Berkel and Christian Straßer,

*Towards Deontic Explanations Through Dialogue*in the proceedings of ArgXAI. Here’s the abstract:Deontic explanations answer why-questions concerning agents’ obligations and permissions. Normative systems are notoriously conflict sensitive, making contrastive explanations pressing: “Why am I obliged to do 𝜙, despite my (seemingly) conflicting obligation to do 𝜓?” In this paper, we develop a model of contrastive explanatory dialogues for the well-established defeasible reasoning formalism Input/Output logic. Our model distinguishes between successful, semi-successful, and unsuccessful deontic dialogues. We prove that the credulous and skeptical (under shared reasons) entailment relation of Input/Output logic, can be characterized in formal argumentation using preferred and grounded semantics. This result allows us to leverage known results for dialogue models of the latter two semantics. Since this work is the first of its kind, we discuss 5 key challenges for deontic explanations through dialogue.

Kees van Berkel and Christian Straßer,

*A Tutorial in Proof-Theoretic Approaches to Logical Argumentation*forthcoming in Proceedings of the Reasoning Web Summer School 2023 (Springer Lecture Notes in Computer Science). Here’s the abstract:This article provides a tutorial in proof-theoretic approaches to logical argumentation. We ﬁrst introduce and discuss defeasible rea- soning and nonmonotonic logic. This naturally paves the way to formal argumentation. An argumentation framework structures a given knowl- edge base by tracking conﬂicts. Argumentation semantics oﬀer ways of selecting arguments. Logical argumentation provides structure both to arguments and to attacks. By adopting a proof-theoretic perspective on logical argumentation, we generate arguments and derive new ones from those derived previously. We study some meta-theoretic properties of the resulting systems. Finally, we apply these formal methods to the prob- lem of reasoning with norms and obligations with a special emphasis on explanations. In closing we, discuss an enhanced Argumentative Knowl- edge Representation and Reasoning pipeline, including the construction of explanations.

Ofer Arieli, Kees van Berkel, Badran Raddaoui, and Christian Straßer,

*Deontic Reasoning based on Inconsistency Measures*forthcoming in the proceedings of KR 2024. Here’s our abstract:Conflicts are inherent to normative systems. In this paper, we explore a novel approach to normative reasoning by quanti- fying the amount of conflicts within normative systems. We refine the idea from classical logic, according to which a formula is a consequence of a knowledge base in case its negation renders the knowledge base inconsistent. In our ap- proach, whether a formula is a logical consequence depends, for instance, on its negation’s marginal contribution to the in- consistency of the given knowledge base. Accordingly, var- ious inconsistency measures and corresponding (nonmono- tonic and paraconsistent) normative entailment relations are analyzed relative to a number of logical properties. To illus- trate our approach, we adopt Input/Output logic, a renowned formalism in deontic logic, specifically designed for defea- sible normative reasoning. As an application, the resulting entailment relations provide recommendations to agents for minimizing norm conflicts, and may be incorporated in a number of implementations (like the Tweety libraries and the LogiKey framework) by involving inconsistency measure- ments in normative reasoning.