Conclusive and defeasible arguments

Different approaches to defeasible (nonmonotonic) reasoning and its formalisation have been developed (see Ginzberg 1987, Horty 2001, Prakken and Vreeswijk 2002). Here I shall approach defeasible reasoning as argumentation, namely, as the derivation of a provisionally justified conclusion through the dialectical opposition of competing arguments (on argumentation, see Walton 2013). This is indeed the perspective that better fits the argumentative and dialectical nature of legal reasoning, as it emerges in analysis, advocacy, and decision-making.
A valid argument can be said to consist of three elements: a set of premises, a conclusion, and a support relation between premises and conclusion. In a deductively valid argument, the premises provide conclusive support for the conclusion: if we accept the premises we must necessarily accept the conclusion. In a defeasibly valid argument , the premises only provide presumptive support for the conclusion: if we accept the premises we should also accept the conclusion, but only so long as we do not have prevailing arguments to the contrary. We extend the notion of an argument also to unsupported claim: in this case the argument will only consist in a conclusion. The unsupported claim of a proposition will be sufficient to substantiate it, when the truth of the proposition is evident or is anyway agreed upon.
Defaults usually have a general form and consequently have to be mapped onto or instantiated to the specific propositions to which they are applied. For instance, to apply the general default “pet dogs are presumably unaggressive,” i.e., in a conditional form, “if something is a pet dot, then presumably it is nonaggressive,” to Fido, we must specify or “instantiate” the default to the case of Fido, i.e., to its specification: “if Fido is a pet dog, then presumably Fido is not aggressive.” The specification, in combination with the premise that Fido is a pet dog, leads us to the presumable conclusion that Fido is not aggressive, through defeasible modus ponens. In the examples that follow, I will omit the specification step, presenting the conclusion as directly resulting from the general default and the specific conditions matching its general antecedent. In fact, a general default can be seen as the set of all of its specific instances, which include the one applied to the case at hand.
I shall use a diagrammatic representation for arguments, as exemplified below, where the boxes include premises or conclusions, and combinations of premises are linked to the conclusion they conjunctively support. In the diagram of Figure 1, we can see a deductive argument (A ) supporting the conclusion that Fido, being a dog, is a mammal and a defeasible argument (B ) supporting the conclusion that Fido, being a pet dog, is presumably not aggressive. I have represented the premises both in natural language and in the usual formalism of predicate logic, and have labelled the connection between premises and conclusion by the letters C and D to distinguish conclusive from defeasible arguments.