$R \subseteq A \times A$ represents (binary) **attacks**, and $N \subseteq (2^A \setminus \emptyset) \times A$ is the **necessity relation**.

Given a set $B\subseteq A$ and an argument $a$, $B N a$ should be read as “at least one element of $B$ needs to be present in order to accept $a$”. The AFN semantics are built around the notions of coherence:

We say that a set of arguments $B$ is **coherent** iff every $b\in B$ is powerful, i.e. there exists a sequence $a_0,..,a_n$ of some elements of $B$ s.t

$a_n = b$,

$\nexists C\subseteq A$ s.t. $CN a_0$, and

for $1\leq i \leq n$ it holds that for every set $C\subseteq A$ if $CN a_i$, then $C \cap \{a_0,...,a_{i-1}\} \neq \emptyset$.

A coherent set $B$ is **strongly coherent** iff it is conflict–free.

Although it may look a bit complicated at first, the definition of coherence grasps the intuition that we need to provide sufficient acyclic support for the arguments we want to accept. Defense in AFNs is understood as the ability to provide support and to counter the attacks from any coherent set.

We say that a set $B\subseteq A$ **defends** $a$, if $B\cup \{a\}$ is coherent and for every 