Syntactic Reasoning with Conditional Probabilities in Deductive Argumentation

Evidence from studies, such as in science or medicine, often corresponds to conditional probability statements. Furthermore, evidence can conflict, in particular when coming from multiple studies. Whilst it is natural to make sense of such evidence using arguments, there is a lack of a systematic formalism for representing and reasoning with conditional probability statements in computational argumentation. We address this shortcoming by providing a formalization of conditional probabilistic argumentation based on probabilistic conditional logic. We provide a semantics and a collection of comprehensible inference rules that give different insights into evidence. We show how arguments constructed from proofs and attacks between them can be analyzed as arguments graphs using dialectical semantics and via the epistemic approach to probabilistic argumentation. Our approach allows for a transparent and systematic way of handling uncertainty that often arises in evidence

On the equivalence between assumption-based argumentation and logic programming

Assumption-Based Argumentation (ABA) has been shown to subsume various other non-monotonic reasoning formalisms, among them normal logic programming (LP). We re-examine the relationship between ABA and LP and show that normal LP also subsumes (flat) ABA. More precisely, we specify a procedure that given a (flat) ABA framework yields an associated logic program with almost the same syntax whose semantics coincide with those of the ABA framework. That is, the 3-valued stable (respectively well-founded, regular, 2-valued stable, and ideal) models of the associated logic program coincide with the complete (respectively grounded, preferred, stable, and ideal) assumption labellings and extensions of the ABA framework. Moreover, we show how our results on the translation from ABA to LP can be reapplied for a reverse translation from LP to ABA, and observe that some of the existing results in the literature are in fact special cases of our work. Overall, we show that (flat) ABA frameworks can be seen as normal logic programs with a slightly different syntax. This implies that methods developed for one of these formalisms can be equivalently applied to the other by simply modifying the syntax

Syntactic reasoning with conditional probabilities in deductive argumentation

Evidence from studies, such as in science or medicine, often corresponds to conditional probability statements. Furthermore, evidence can conflict, in particular when coming from multiple studies. Whilst it is natural to make sense of such evidence using arguments, there is a lack of a systematic formalism for representing and reasoning with conditional probability statements in computational argumentation. We address this shortcoming by providing a formalization of conditional probabilistic argumentation based on probabilistic conditional logic. We provide a semantics and a collection of comprehensible inference rules that give different insights into evidence. We show how arguments constructed from proofs and attacks between them can be analyzed as arguments graphs using dialectical semantics and via the epistemic approach to probabilistic argumentation. Our approach allows for a transparent and systematic way of handling uncertainty that often arises in evidence

A postulate-driven study of logical argumentation

Logical argumentation is a well-known approach to modeling non-monotonic reasoning with conflicting information. In this paper we provide a comprehensive postulate-based study of properties of logical argumentation frameworks and a full characterization of their semantics and inference relations. In this way we identify well-behaved formal argumentative models of drawing logically justified inferences from a given set of possibly conflicting defeasible, as well as strict assumptions. Given some desiderata in terms of rationality postulates, we consider the conditions that an argumentation framework should fulfill for the desiderata to hold. One purpose of this approach is to assist designers to “plug-in” pre-defined formalisms according to actual needs. To this end, we present a classification of argumentation frameworks relative to the types of attacks they implement. In turn, for each class we determine which desiderata are satisfied. Our study is highly abstract, supposing only a minimal set of requirements on the considered underlying deductive systems, and in this way covering a broad range of formalisms, including classical, intuitionistic and modal logics

Identifying the Class of Maxi-Consistent Operators in Argumentation

Dung’s abstract argumentation theory can be seen as a general framework for non-monotonic reasoning. An important question is then: what is the class of logics that can be subsumed as instantiations of this theory? The goal of this paper is to identify and study the large class of logic-based instantiations of Dung’s theory which correspond to the maxi-consistent operator, i.e. to the function which returns maximal consistent subsets of an inconsistent knowledge base. In other words, we study the class of instantiations where every extension of the argumentation system corresponds to exactly one maximal consistent subset of the knowledge base. We show that an attack relation belonging to this class must be conflict-dependent, must not be valid, must not be conflict-complete, must not be symmetric etc. Then, we show that some attack relations serve as lower or upper bounds of the class (e.g. if an attack relation contains canonical undercut then it is not a member of this class). By using our results, we show for all existing attack relations whether or not they belong to this class. We also define new attack relations which are members of this class. Finally, we interpret our results and discuss more general questions, like: what is the added value of argumentation in such a setting? We believe that this work is a first step towards achieving our long-term goal, which is to better understand the role of argumentation and, particularly, the expressivity of logic-based instantiations of Dung-style argumentation frameworks. 1