The Attack as Intuitionistic Negation

We translate the argumentation networks $\mathcal{A}=(S,R)$ into a theory $D$ of intuitionistic logic, retaining $S$ as the domain and using intuitionistic negation to model the attack $R$ in $\mathcal{A}$: the attack $xRy$ is translated to $x\to \mathrm{\neg }y$. The intuitionistic models of $D$ characterise the complete extensions of $\mathcal{A}$. The reduction of argumentation networks to intuitionistic logic yields, in addition to a representation theorem, some additional benefits: it allows us to give semantics to higher level attacks, where an attack “$xRy$” can itself attack another attack “$uRv$”; one can make higher level meta-statements $W$ on $(S,R)$ and such meta-statements can attack and be attacked in the domain.