-   if $|SCCS_{AF}| = 1$, $S$ is a stage extension of $AF$;

-   otherwise $\forall Args \in SCCS_{AF} (S \cap Args) \in \sigma_{stage2}(AF\downarrow_{UP_{AF}(Args, S)}) \; otherwise$, with\
    $AF\downarrow_{UP_{AF}(Args,S)} = (UP, S \cap UP \times UP)$, the *restriction of $AF$ to $UP$* [@BARONI2005162] and for any $S, Args \subset AR, UP_{AF}(Args, S) = \{a \in Args | \nexists b \in S \\ Args: (b, a) \in Attacks \}$.

In words, CF2 and stage2 semantics decomposition an argumentation framework into SCCS and recursively resolve the framework component by component, starting with the SCCS that are not attacked by any other SCC. Note that a detailed explanation of CF2 and stage2 semantics is beyond the scope of the paper.

Finally, we provide the established definitions for *credulous* and *skeptical* acceptance of arguments [@10.1007/978-3-540-30227-8_39], given a specific argumentation semantics and in addition introduce the concept of *lenient* acceptance to allow for more concise formal work.

 [@10.1007/978-3-540-30227-8_39]

-   **Credulous Acceptance.** Given an argumentation framework\
    $AF = (AR, Attacks)$ and an argumentation semantics $\sigma_x$, an argument $a \in AR$ is credulously accepted w.r.t. $\sigma_x$ iff it is contained in *at least one* of the extensions of $\sigma_x(AF)$.

-   **Skeptical Acceptance.** Given an argumentation framework\
    $AF =(AR, 