Clearly, we cannot endorse both arguments A and B at the same time (their conclusions are contradictory), and so we must either choose between them or remain uncertain as to which one we should choose. When two arguments conflict in such a way that the (final or intermediate) conclusions of one of them contradicts a (final or intermediate) conclusion of the other, we have a rebutting conflict between two arguments. To determine the outcome of a rebutting conflict we must consider the comparative strength of the two arguments. If one argument is stronger than the other (at the juncture at which the conflict takes place), then it prevails, i.e., it defeats its opponent without being defeated by it. In this case, the prevailing argument is said to strictly defeat its opponent. If neither of the conflicting arguments is stronger than the other, they each weakly defeat the other, i.e., their conflict remains undecided (for a logical analysis of these notions, see Prakken and Sartor 1997; Prakken 2010). To compare arguments, we adopt the so-called “last-link” principle, which affirms that when two defeasible arguments contradict each other, to determine the comparative strength of the two argument, at the point of where they clash, we must compare only the defaults that directly deliver the conflicting conclusions (possibly with the help of deductive inferences). We do not consider the defaults eventually used, in multi-step arguments, to establish the preconditions of the directly conflicting defaults (for a discussion of the last-link principle and a formal definition, see Prakken 2010, Section 6).
In our example, let us assume that we consider that the argument on the left in Figure 5 (Fido presumably is aggressive, given than it is a Doberman) is stronger that the argument on the right (Fido presumably is not aggressive, being a pet dog). According to this priority relation between the two arguments, the first can be said to strictly defeat the second: we should accept the conclusion that Fido is indeed aggressive (and be careful in approaching him).
A second kind of attack against defeasible arguments consists in contesting the support link between the premises and the conclusion of the argument, namely, in denying that in the case at hand, these premises can provide sufficient support for the conclusion (on undercutting, see Pollock 2008). Let us assume that we are dealing with another dog—let us call him Tommy—and let us assume that we know Tommy to be a pet dog, but we also that he has been reared in an isolated mountain hut, having had contact only with his owner, and that we believe that the nonaggressiveness of pet dog toward strangers is mainly due to their experience in previous interactions with a large enough set of humans. We can then reasonably claim that, under these particular circumstances, the fact that Tommy is a pet dog does not adequately support the conclusion that he is friendly toward strangers. This kind of conflict is called undercutting (see Figure 6).