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.