Reinstatement

So far, we have only considered relations between pairs of arguments. However, this is insufficient to determine the status of an argument, namely, whether we should accept it or not. More precisely, this is insufficient to establish whether an argument is justified, such that we should accept its conclusion; overruled, such that we should not pay attention to it; or merely defensible, such that we should remain uncertain as to whether to accept it or not (on justified, overruled, and defensible arguments, see Prakken and Sartor 1997). This is because an argument A that is defeated by a counterargument B can still be acceptable when B is in turn defeated by a further argument C : we would have rejected A if we had accepted B , but since we do not accept B (given that it is defeated by C ), then A remains acceptable.
To clarify this point it is useful to specify the conditions that an argument should meet to be IN (acceptable) or OUT (inacceptable). The basic idea is that only a defeater which is IN can turn OUT the argument it attacks; a defeater which is OUT is not relevant to the status of the argument it attacks. Thus, we can state the following rules:
  1. An argument \(\mathcal{A}\) is IN iff no argument which defeats \(\mathcal{A}\) is IN.
  2. An argument \(\mathcal{A}\) is OUT iff an argument which defeats \(\mathcal{A}\) is IN.
To clarify our analysis let us consider the legal example in Figure 11, which extends Figure 8 with labels denoting the statuses of the corresponding arguments: