Figure 1
Arguments in natural language usually have an enthymematic form, meaning that they may omit some of the premises that are needed to support their conclusions. Here I shall present all arguments in their complete form, that is, as including all premises that are sufficient to conclusively or defeasibly establish their conclusion.
In particular, I assume that each defeasible argument includes (a ) a set of antecedent conditions, and (b ) a defeasible conditional, called a default , according to which the (conjunction of the) conditions presumably determines the argument’s conclusion. I represent defaults in the form \("\)if \(P_{1}\ \)and … and\(\ P_{n}\)then presumably \(Q\)”, in formula \(P_{1}\ \land\cdots\land P_{\text{n\ }}\Rightarrow Q\), where the arrow \(\Rightarrow\) denotes defeasible conditionality (I will use the arrows \(\Rightarrow\), →, and \(\twoheadrightarrow\) to denot defeasible, material and strict conditional respectively, see Section ). Thus a single-step defeasible argument has the following form:
  1. \(P_{1},\ \cdots,P_{n}\ \)(the antecedent conditions), and
  2. if \(P_{1}\ \)and … and\(\ P_{n}\ \)then presumably \(Q\) (the default, in formula: \(\text{\ P}_{1}\ \land\cdots\land P_{\text{n\ }}\Rightarrow Q)\).
therefore
  1. \(Q\).
This inference is called defeasible modus ponens to distinguish it from the conclusive modus ponens inference of deductive logic. We can represent a defeasible argument by providing the set of its premises (conditions and default): \(\left\{{P_{1},\ \cdots,P_{n},\ P}_{1}\ \land\cdots\land P_{\text{n\ }}\Rightarrow Q\right\}\), the conclusion of the argument being conclusion of the default. Given a defeasible modus ponens inference (argument) \(\mathcal{A=\ }\left\{{P_{1},\ \cdots,P_{n},\ P}_{1}\ \land\cdots\land P_{\text{n\ }}\Rightarrow Q\right\}\), I will say that the conjunction of the \(P_{1},\ \cdots,P_{n}\) conditions is the reason for (concluding that) \(Q\) and that the default \(P_{1}\ \land\cdots\land P_{\text{n\ }}\Rightarrow Q\) is the warrant for \(Q\). I will also say that \(Q\) is warranted by that default.
For instance, given argument \(B\) in Figure 1, we can say that the fact that Fido is a pet dog is a reason for concluding that he is not aggressive and that that this conclusion is warranted by the default that pet dogs are not aggressive. As example of conjunctive reason, consider the argument in Figure 2, where the conjunction of the two premises \(P_{1}\) and \(P_{2}\) provides the reason for the conclusion warranted by the default \(D\). Note that, I freely use symbols \(P_{1},\ \cdots,P_{n}\ \)as names for propositions and \(D_{1},\ \cdots,D_{n}\) as names for defaults, whenever needed.
The notion of a defeasible argument can be generalised to multi-step defeasible arguments, which will consist of the set of the arguments providing the conditions of the top default, plus that default. For instance, if \(\left\{P,\ P\Rightarrow Q\right\}\) is a defeasible argument, so is also \(\left\{\left\{P,\ P\Rightarrow Q\right\},\ Q\Rightarrow R\right\}:\ \)(for an example of multistep defeasible argument, see Figure 16, for a formal definition of the general notion of an argument, possibly including both defeasible and deductive steps, see Prakken 2010, Section 3.2).