In this paper, we have introduced graduality in the two main related
issues of argumentation systems:
- the valuation of the arguments,
- the acceptability of the arguments.
Regarding the first issue, we have defined two formalisms introducing
an interaction-based gradual valuation of arguments.
- First, a generic gradual valuation which covers existing
proposals (for example  and ). This approach
is essentially ``local'' since it computes the value of the
argument only from the value of its direct attackers.
- Then, an approach based on a labelling which takes the form of
a pair of tuples; this labelling memorises the structure of the
graph representing the interactions (the ``attack graph''),
associating each branch with its length (number of the edges from
the leaf to the current node) in the attack graph (if the length
of the branch is an even integer, the branch is a defence branch
for the current node, otherwise the branch is an attack branch for
the current node). This approach is said to be ``global'' since it
computes the value of the argument using the whole attack graph
influencing the argument.
We have shown that each of these valuations induces a preordering
on the set of the arguments, and we have brought to light the main
differences between these two approaches.
Regarding the second issue, two distinct approaches have been
- First, in the context of the collective acceptability
of : three levels of acceptability (uni-accepted,
exi-accepted, not-accepted) were already defined. More graduality
can be introduced in the collective acceptability using the notion
of cleanly-accepted arguments (those whose direct attackers
- Then, in the context of individual acceptability: using the
previously defined gradual valuations, the new notion of
well-defended arguments has been introduced (those which are
preferred to their direct attackers in the sense of a given gradual
The first concept induces a refinement of the level of exi-accepted in
two sublevels (cleanly-accepted arguments and only-exi-accepted
arguments). The gradual valuation allows graduality inside each level
of this collective acceptability.
The second concept induces two new levels of acceptability
(well-defended arguments and not-well-defended arguments). The
gradual valuation also allows graduality inside each level of this
Regarding our initial purpose of introducing graduality in the
definition of acceptability, we have adopted a basic principle:
- acceptability is strongly related to the interactions between
arguments (represented on the graph of interactions),
- and an argument is all the more acceptable if it is preferred to
its direct attackers.
Then, we have followed two different directions. One is based on a
refinement of an existing partition and remains in the framework of
Dung's work. The other one is based on the original concept of ``being
well-defended'', and deserves further investigation, in particular
from a computational point of view.