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_30: 
@cite_5: In almost all existing semantics in argumentation, a strong attack has a lethal effect on its target that a set of several weak attacks may not have. This paper investigates the case where several weak attacks may compensate one strong attack. It defines a broad class of ranking semantics, called α—BBS, which satisfy compensation. α—BBS assign a burden number to each argument and order the arguments with respect to those numbers. We study formal properties of α—BBS, implement an algorithm that calculates the ranking, and perform experiments that show that the approach computes the ranking very quickly. Moreover, an approximation of the ranking can be provided at any time.
@cite_33: In this paper we take a step towards using Argumentation in Social Networks and introduce Social Abstract Argumentation Frameworks, an extension of Dung's Abstract Argumentation Frameworks that incorporates social voting. We propose a class of semantics for these new Social Abstract Argumentation Frameworks and prove some important non-trivial properties which are crucial for their applicability in Social Networks.
Amongst the graph propagation approaches, we mention the equational approaches that assign a more fine grained ranking to arguments by evaluating fixed points of functions that assign a numerical value to any given argument based on the values of its attackers. In particular, the equational approach of Gabbay and Rodrigues @cite_30 who conjecture that their approach yields a unique solution for cyclic graphs, the compensation based semantics of Amgoud et.al. @cite_5 , which assigns the same ranking to all arguments in cycles and yield a unique solution for cyclic graphs, and the social argumentation approach of Leite and Martin () @cite_33 (who are concerned with propagating user votes on arguments, but also yield fine grained rankings without recourse to exogenous information).