In such a case, which arguments are justified depends on where we start from: if arguments two argument \(\mathcal{A}\) and \(\mathcal{B}\) attack each other (and neither of them is OUT on other grounds), then if we assume that \(\mathcal{A}\) is IN and \(\mathcal{B}\) will be OUT, and if we assume that \(\mathcal{B}\) is IN, then \(\mathcal{A}\) will be OUT. We can deal with this situation by considering all possible assignments of IN and OUT labels to the arguments at stake, consistently with rules (1) and (2) above: an argument is justified if it is IN according to every assignment; it is overruled if it is OUT according every assignment; it is defensible if it is IN according to some assignment and OUT according to some other assignment (Pollock 2008). An equivalent approach by which to assess the status of arguments consists in constructing alternative extensions, namely, maximal sets of consistent arguments: justified, defensible, and overruled arguments are contained in all, some, or no extensions (Dung 1995).
Unresolved conflicts concerning legal and factual issues are addressed in different ways in the adversarial context of legal disputes, where the judge is assumed to know the applicable law, while the parties should bring evidence on the facts of the case. If an unresolved conflict between competing arguments concerns a legal issue (e.g., there are arguments supporting alternative interpretations of the same source of law), the decision-maker (the judge) should resolve the conflict by assigning priority to one of the conflicting arguments. If the unresolved conflict concerns a factual premise that is needed to construct an argument, it will be assumed that the factual premises have not been legally substantiated, i.e., the factual argument will be OUT.
The dialectical interaction between arguments and counterarguments is reflected in the allocation of burdens of proof and, more generally, of burdens of argumentation. The idea of the burden of proof applies to many dialectical interaction, in context-dependent ways (see Walton 2008, 59), but it acquires a specific significance in the law (see Sartor 1993, Prakken and Sartor 2009). In a legal case, the party that is interested in establishing a legal outcome bears the burden of presenting and substantiating an argument supporting that outcome. For instance, in the example in Figure 13, plaintiff Mary must provide argument A , establishing John’s liability for negligence. She must substantiate the argument’s normative premises (the general rule of civil liability) by referring to sources of law, and its factual premise (John culpably damaged Mary) by bringing in appropriate evidence. This argument will be sufficient for Mary to win the case if its premises are accepted and no counterarguments defeating it are provided by John (at least with regard to factual premises, since the judge may independently bring in legal information).
Thus, Mary can be said to bear the burden of proving that John did damage to her, since without establishing this fact she will not be able to construct argument A , which supports the outcome she favours. She does not bear the burden of proving that John was not incapable, since to build argument A , she does not need to establish that fact. On the contrary, John bears the burden of proving that he was incapable, since without establishing this fact, he will not be able to substantiate argument B , which could defeat Mary’s argument A (switching A ’s status from IN to OUT, in the absence of further interfering arguments).
In general, when a party \(\pi_{1}\) fails to construct a certain legally acceptable argument \(\mathcal{A}\) supporting her side, we say that, unless evidence is provided for premise \(P\), \(\pi_{1}\)has the burden of proof regarding P. This does not mean that the counterparty \(\pi_{2}\) has no interest in \(P\). It is true that \(\pi_{1}\) will fail to build the argument based on \(P\) if \(\pi_{1}\) fails to provide evidence for \(P\), even if \(\pi_{2}\ \)remains inactive. However, if \(\pi_{1}\) provides sufficient evidence for \(P\) (and the other premises of \(\mathcal{A}\) have been established), then \(\pi_{2}\) must provide evidence against \(\varphi\), or counterarguments against \(\mathcal{A}\), if he does not want to lose on the basis of \(\mathcal{A}\) (on the logic of the burden of proof, and for further refinements, including the distinction among the burden of production, the burden of persuasion, tactical burden, and standards of proof, see Prakken and Sartor 2009, on the connection between defeasibility and proof, see also Sartor 1993, Duarte de Almeida 2012, Brewer 2011). Figure 13 below exemplifies the context of the burden of proof. Let us assume that the plaintiff (the alleged victim) has the burden of showing that his cancer was caused by smoke and that the standard of preponderance of the evidence applies. Then, even if the two arguments have equal weight, the plaintiff’s argument would be strictly defeated by the defendant’s argument (to defeat the defendant’s argument, the plaintiff’s argument must meet the required standard of proof). Thus, in an adversarial legal context governed by the burden of proof, the status assignment of Figure 12 (no justified arguments, two defensible one) would be transformed into the assignment of Figure 13 (one justified and one overruled argument).