Dung's abstract argumentation framework consists of a set of interacting arguments and a series of semantics for evaluating them. Those semantics partition the powerset of the set of arguments into two classes: extensions and nonextensions. In order to reason with a specific semantics, one needs to take a credulous or skeptical approach, i.e. an argument is eventually accepted, if it is accepted in one or all extensions, respectively. In our previous work [1], we have proposed a novel semantics, called counting semantics, which allows for a more fine-grained assessment to arguments by counting the number of their respective attackers and defenders based on argument graph and argument game. In this paper, we continue our previous work by presenting some supplementaries about how to choose the damaging factor for the counting semantics, and what relationships with some existing approaches, such as Dung's classical semantics, generic gradual valuations. Lastly, an axiomatic perspective on the ranking semantics induced by our counting semantics are presented.Definition 1 (Abstract Argumentation Framework). An argumentation framework is a pair AF = X , R where X is a finite set of arguments and R ⊆ X × X is a binary relation on X , also called attack relation. (a, b) ∈ R means that a attacks b, or a is an attacker of b. Often, we write (a, b) ∈ R as aRb.We denote by R − (x) (respectively, R + (x)) the subset of X containing those arguments that attack (respectively, are attacked by) the argument x ∈ X , extending this notation in the natural way to sets of arguments, so that for S ⊆ X , R − (S) {x ∈ X : ∃y ∈ S such that xRy} and R + (S) {x ∈ X : ∃y ∈ S such that yRx}. Now, let us characterise two fundamental notions of conflict-free and defence.Definition 2 (Conflict-free, Defense). Let AF = X , R be an argumentation framework, let S ⊆ X and x ∈ X .
