interpretation. $v \sqcap w$ gives us a new interpretation $w' = \{a:{\mathbf{u}}, b:{\mathbf{u}}, c:{\mathbf{f}}, d:{\mathbf{u}})$: as the assignments of $a,b$ and $d$ differ between $v$ and $w$, the resulting value is ${\mathbf{u}}$. On the other hand, $c$ is in both cases ${\mathbf{f}}$ and thus retains its value.

We will use $v^x$ to denote a set of arguments mapped to $x$ by $v$, where $x$ is some truth–value. Given an acceptance condition $C_s$ for some argument $s \in S$ and an interpretation $v$, we define a shorthand $v(C_s)$ as $C_s(v^{\mathbf{t}}\cap par(s))$. For a given propositional formula $\varphi$ and an interpretation $v$ defined over all of the atoms of the formula, $v(\varphi)$ will just stand for the value of the formula under $v$. However, apart from knowing the “current” value of a given acceptance condition for some interpretation, we would also like to know if this interpretation is “final”. By this we understand that no new information will cause the value to change. This is expressed by the notion of decisive interpretations, which are at the core of the extension–based ADF semantics.

Given an interpretation $v$ defined over a set $A$, **completion** of $v$ to a set $Z$ where $A\subseteq Z$ is 