Abstract The paper develops a formal theory of the degree of justification of arguments, which relies solely on the structure of an argumentation framework, and which can be successfully interfaced with approaches to instantiated argumentation. The theory is developed in three steps. First, the paper introduces a graded generalization of the two key notions underpinning Dung's semantics: self-defense and conflict-freeness. This leads to a natural generalization of Dung's semantics, whereby standard extensions are weakened or strengthened depending on the level of self-defense and conflict-freeness they meet. The paper investigates the fixpoint theory of these semantics, establishing existence results for them. Second, the paper shows how graded semantics readily provide an approach to argument rankings, offering a novel contribution to the recently growing research programme on ranking-based semantics. Third, this novel approach to argument ranking is applied and studied in the context of instantiated argumentation frameworks, and in so doing is shown to account for a simple form of accrual of arguments within the Dung paradigm. Finally, the theory is compared in detail with existing approaches.
@cite_41: Abstract We explore a framework for argumentation (based on classical logic) in which an argument is a pair where the first item in the pair is a minimal consistent set of formulae that proves the second item (which is a formula). We provide some basic definitions for arguments, and various kinds of counter-arguments (defeaters). This leads us to the definition of canonical undercuts which we argue are the only defeaters that we need to take into account. We then motivate and formalise the notion of argument trees and argument structures which provide a way of exhaustively collating arguments and counter-arguments. We use argument structures as the basis of our general proposal for argument aggregation. There are a number of frameworks for modelling argumentation in logic. They incorporate formal representation of individual arguments and techniques for comparing conflicting arguments. In these frameworks, if there are a number of arguments for and against a particular conclusion, an aggregation function determines whether the conclusion is taken to hold. We propose a generalisation of these frameworks. In particular, our new framework makes it possible to define aggregation functions that are sensitive to the number of arguments for or against. We compare our framework with a number of other types of argument systems, and finally discuss an application in reasoning with structured news reports.
@cite_1: An argumentation system consists of a set of interacting arguments and a semantics for evaluating them. This paper proposes a new family of semantics which rank-orders arguments from the most acceptable to the weakest one(s). The new semantics enjoy two other main features: i) an attack weakens its target but does not kill it, ii) the number of attackers has a great impact on the acceptability of an argument. We start by proposing a set of rational postulates that such semantics could satisfy, then construct various semantics that enjoy them.
@cite_22: Argumentation is based on the exchange and valuation of interacting arguments, followed by the selection of the most acceptable of them (for example, in order to take a decision, to make a choice). Starting from the framework proposed by Dung in 1995, our purpose is to introduce "graduality" in the selection of the best arguments, i.e. to be able to partition the set of the arguments in more than the two usual subsets of "selected" and "non-selected" arguments in order to represent different levels of selection. Our basic idea is that an argument is all the more acceptable if it can be preferred to its attackers. First, we discuss general principles underlying a "gradual" valuation of arguments based on their interactions. Following these principles, we define several valuation models for an abstract argumentation system. Then, we introduce "graduality" in the concept of acceptability of arguments. We propose new acceptability classes and a refinement of existing classes taking advantage of an available "gradual" valuation.
Other graph propagation approaches that do not use an equational fixed point approach, typically account for the attack and defense paths terminating in the argument being ranked, where these paths are sequences of, respectively even and odd, numbers of attacking arguments. Besnard and Hunter () @cite_41 rank classical logic arguments in acyclic graphs, through a function -- @math defined as @math -- that assigns high values to arguments with low-valued attackers (and the maximum value to un-attacked arguments) and low values to arguments with high-valued attackers. Cayrol and Lagasquie-Schiex () @cite_22 then generalises use of this function to Dung @math s to develop (in their terms) a local approach' to valuation of arguments, and then formalise a global approach' that they argue gives more intuitive outcomes. Their approach requires a highly involved transformation of cyclic graphs to infinite acyclic graphs. More recently, Amgoud and Ben-Naim @cite_1 propose two ranking-based semantics, which they call discussion-based () and burden-based (). These semantics are also based on the processing of attack paths and their general applicability relies on conjectures concerning the processing of cyclic paths.