The distinction between linked arguments and convergent argument structures enables us to address more clearly two important concepts: the concept of a contributory condition and the concept of a contributory reason.
A contributory condition , relatively to a conclusion, is a necessary element of a (presumably or conclusively) sufficient condition of conclusion. This concept applies to the (unconditional) premises that concur in matching the conjunctive antecedent of the warrant in a linked argument, such as premises \(P_{1}\) and \(P_{2}\) in Figure 2
A contributory reason , relatively to a conclusion, is a presumably sufficient condition of that conclusion. This concept applies to each (unconditional) conjunction of premises that contributes, by triggering the corresponding warrant, to one of the defeasible arguments that are merged in a convergent argument. This is the case for premises \(P_{1}\) and \(P_{2}\) in Figure 3 and Figure 4.
Thus, premises \(P_{1}\) and \(P_{2}\)in Figure 2 are contributory conditions, but fail to qualify as contributory reasons, since neither of them separately supports the conclusion of the argument including both of them. Therefore, neither of them can be properly characterised as a reason for that conclusion. On the other hand, premises \(P_{1}\) and \(P_{2}\) in Figure 3 and Figure 4 do qualify as (contributory) reasons for their common conclusion, since each of them (together with the corresponding default warrant) independently supports that conclusions, besides contributing to provide a stronger joint support to that conclusion.
In the legal domain, the idea of a contributory reasons applies to the domain of principles, understood as optimisation requirements (Alexy 2002, Ch. 4) or value-norms (Sartor 2013, Section D). The fact that a choice advances a principle (a legal value) is a contributory reason for adopting that choice (of for its constitutional legitimacy). When the same choice advances multiple principles, this originates multiple convergent arguments —the advancement of each principle being a contributory reason— that join to provide a stronger support to that choice. Similarly, the fact that a choice negatively affects the realisation of a principle, is a contributory reason for not adopting the choice or against its legitimacy. When multiple principles are negatively affected this originates multiple convergent arguments against that choice.
The idea of a contributory reason also applies to the antecedent conditions of legal rules. As I shall argue in the following, the antecedent of a legal rules usually only provides a presumably sufficient condition for the conclusion of that rule to hold. For instance, the driver exceeding the speed limit may not be subject to the sanction in case his behaviour is justified by self-defence (he was trying to escape from a killer) or state of necessity (he was running to the hospital for an emergency). Rule-warranted arguments and principle-warranted arguments, while sharing the same basic logical structure, present some relevant differences. Firstly, rule-warranted arguments may “exclude” (undercut, in our terminology, see Section 5), rather than oppose (rebut), certain contrary arguments warranted by principles (if we follow the idea of Raz 1985, also adopted by Hage 2000). Secondly, convergent rule-based argument structures usually do not provide a stronger support to their conclusion than the constituting arguments do. For instance, assume that a person has committed a violation that triggers his or her liability both in contract and in torts. This provides for a converging argument structure for the liability of this person. However, this convergent argument structure arguably does not provide a stronger support to the liability conclusion than the strongest of the two separate arguments for that conclusion. I have preferred to speak of a convergent argument structure, rather than or a convergent argument, to denote the combination of arguments leading to the same conclusions, to maintain the concept of an argument I introduced above, that requires a single warrant linking premises and conclusion.