In this work we show a possible form of classification of comparison methods in argumentation systems [AG95,BV,Dung93,PRAK,Sim92]. The main idea in these systems is that any proposition will be accepted as true if there exists an argument that supports it, and this argument is acceptable according to an analysis between it and its counterarguments. This analysis requires a process of comparison of conflicting arguments, in order to decide which one is preferable. After this dialectical analysis in the set of arguments of the system, some of them will be acceptable or justified arguments, while others not. In this classification, the argument comparison method plays a very important role.Eje: Inteligencia Artificial Distribuida, Aspectos Teóricos de la Inteligencia Artificial y Teoría de la ComputaciónRed de Universidades con Carreras en Informática (RedUNCI

García, Alejandro Javier

Martínez, Diego C.

English

El Servicio de Difusión de la Creación Intelectual

Argument Comparison Criteria Analysis
Diego C. Martínez1
dcm@cs.uns.edu.ar
Alejandro J. García
ajg@cs.uns.edu.ar
Laboratorio de Investigación y Desarrollo en Inteligencia Artificial
Departamento de Ciencias de la Computación – Universidad Nacional del Sur
Avenida Alem 1253 – (0291) 4595135 – Bahía Blanca – Argentina
1. Introduction
In this work we show a possible form of classification of comparison methods in argumentation
systems [AG95,BV,Dung93,PRAK,Sim92]. The main idea in these systems is that any proposition
will be accepted as true if there exists an argument that supports it, and this argument is acceptable
according to an analysis between it and its counterarguments.  This analysis requires a process of
comparison of conflicting arguments, in order to decide which one is preferable. After this
dialectical analysis in the set of arguments of the system, some of them will be acceptable or
justified arguments, while others not. In this classification, the argument comparison method plays
a very  important role.
2. Comparison methods
Arguments support conclusions with different strength. The target of any comparison method is to
determine the difference of conclusive force in conflicting arguments. It determines an order of
conclusive force between arguments, which is partial, because some pairs of arguments may be
incomparable.
There exists a lot of methods for comparing arguments, such as the principle of specificity,
introduced by Poole in 1985, directness by Loui, preemption by Horty et. al., combined defeat by
Prakken, and accumulation of numerical strength, by Pollock. On the other hand, some authors do
not commit to a specific method to compare arguments [Vree97,DungLP]. This attitude saves them
from the responsibility of telling how and why a particular argument should overrule any other
argument.
When two arguments A and B are in conflict, they are compared so the strongest argument is
identified, say A and it is noted B<A. Then, a defeat relation is established between A and B. In
this case, the argument A defeats the argument B, because it is stronger according to the
comparison method.
However, it is possible for two conflicting arguments to be incomparable, according to the selected
comparison criterion. In this case, the conflict can not be solved, and it is usually translated to
reciprocal defeat relations: A defeats B and viceversa. This situation is not good, because the status
of A and B as accepted or rejected can not be determined. Therefore, it is said that A and B are
undecided arguments [DCM2k].
The outcome of a system with a set of arguments Arg depends on the set of all defeat relations
defined in Arg. The defeat relation is based on the outcome of the comparison method that is being
used, so it is clear the importance of this method in the system. Changing the method will probably
change the outcome, so the chosen criterion will always have a strong influence in argument
classification.
The worst case is to have a comparison criterion that makes  all the arguments incomparable, so the
system fall in a general indecision state.
                                                          
1
 Becario de la Comisión de Investigaciones Científicas de la Provincia de Buenos Aires.
However, it is not possible to construct a complete argument hierarchy. And this is not a real
disadvantage, because there exists by nature incomparable arguments in the real world. Any
criterion that always adopt a preference for any pair of arguments tends to be fictitious, because of
the arbitrary conflict resolution, and it hides de possibility of indecision in the system.
2.1 Levels of strength
A comparison criterion can be analyzed in function of its indecision. The indecision appears when
two arguments are incomparable (the information available is not enough to find the strongest
argument) or they have the same conclusive force (the information available can not distinguish
any relative difference in strength). The partial order induces a relation of equivalence between
arguments. We say that two arguments A and B are equivalent in strength if A≤B and A≥B. The
relation R≡ =“A is equivalent in strength to B” is a relation of equivalence, because it is a reflexive,
symmetrical and transitive and therefore it determines a partition of the set of arguments. For any
set of arguments Arg and any argument A, we define
A≡ =df { X : X ∈ Arg , X is equivalent to A}
which leads to the quotient set Arg/R≡.
If two arguments A1 and A2 are in different partition sets, then A1 is stronger than A2 , or A2 is
stronger than A1 or they are incomparable. If an argument of a set C1 is stonger than an argument of
the set C2, then we say that C1 is preferable to C2.
For this reason, a hierarchy between partition sets can be established, in the following manner:
• Sets not preferable to any other sets are in level 0.
• Any set  C is in level n, if it is preferable to at least one set C1 of level n-1, and there is no other
set C’ such that C is preferable to C’ and  C’ is preferable to C1
An argument A is an argument of level k, if A belongs to a set in level k. Two arguments have equal
relative force if the are in the same level. Arguments of the same level, but in different sets are
incomparable.
This structure of partition sets can be used to make a classification of methods. For example, a
comparison criterion is determinant if there is only one partition set in each level. It is not strictly
ranked, if all the strict arguments are in the same level.
The best scenario is a structure that contains a lot of levels, with only one partition set in each level
(so the method should be determinant) and only one argument in each set, which corresponds to a
complete order between arguments. If unary sets of arguments is not possible then the method
should produce the highest possible number of sets.  The main question arises: is it possible to
build a comparison method with this structure? If not, then the criterion should produce at least
small partition sets and levels with few sets. We should focus us in the precision of the methods,
and its refinements. The structure of levels leads us to a definition of accuracy between methods.
Definition [Accuracy of methods]. A comparison criterion θ1 is more accurate than θ2, if it
induces more strength levels than θ2 under the same scenario of arguments.   
 
A method θ1  is a refinement of θ2  if at least one set C of θ2 is equal to the union of sets of
arguments C1, C2,…Cn of θ1 .  That is to say, if θ1  has almost the same sets as θ2, but it establish a
hierarchy between arguments in at least one set of θ2 . Any refinement of a method is better
because reduces the amount of incomparable arguments and therefore is more accurate.
In order to minimize the possibility of indecision in the system, the following considerations must
be taken into account:
• The partition sets should not contain a lot of arguments, in order to minimize the possibility of
finding conflicts between arguments of equal strength, because these conflicts can not be
solved.
• Few partition sets in each level, in order to reduce the amount of incomparable arguments in
the system, and therefore maximizing the capability of solving conflicts.
• The method should induce a high number of levels so it is easy to find a difference of
conclusive force between arguments.
In order to fulfill these requirements, the following it should be noted that:
• The criterion must be very specific, and not based only in general aspects of the structure of
arguments. For example, the criterion based in the number of defeasible rules of the arguments
does not obey this rule, because it is very easy to find two arguments structurally equal. A
refinement of the criterion could provoke a shrink in the size of the sets.
• For each level, there is a trade-off between producing arguments of equal force and producing
incomparable arguments. Of these two possibilities, the former is preferable. If there are a lot
of sets in each level, then the criterion is not very specific, and it is based on properties not
existent in every argument. A refinement is needed.
• To produce a lot of levels the criterion should be based in different properties of arguments, not
only structural properties, in order to make easy the task of finding a difference of conclusive
force. This is the best way of avoiding indecision.
3. Actual work and next step.
It is possible to identify more properties of methods based on the structure of levels of strength.
These properties are being gathered in order to build a good and complete taxonomy of the
comparison criteria. We can therefore “compare”  these comparison methods, so we can define the
concept of “better methods”, use them,  and therefore improve the performance of the system.
In the future, a very important target is the construction of a criterion based, not only in the
composition of the arguments but also in the structure of the dialogue in which the arguments take
part. In this way, the criterion could obtain different results in argument comparison, so we can
avoid indecision when it is due to structural equality of the pair of arguments being compared.
Bibliography
[AG97] Alejandro García. La programación en lógica rebatible. Su definición teórica y computacional. Tesis
de Magister en Cs. de la Computación. Universidad Nacional del Sur. 1997.
[BV] Bart Verheij. Rules, reasons and arguments: formal studies of argumentation and defeat. PhD Thesis.
ISBN 90-9010071-7
[DCM2k] Diego C. Martínez, Alejandro J. García. Una metodología para el tratamiento de la indecisión en
los sistemas argumentativos abstractos.. Congreso Argentino de Ciencias de la Computacion CACIC 2000.
Ushuaia, Octubre de 2000.
[DUNG93] P. M Dung. On the Acceptability of Arguments and its Fundamental Role in Nonmotonic
reasoning and Logic Programming and N-persons games, IJCAI 93 852-857, 1993.
[PRAK] Henry Prakken y Giovanni Sartor. Argument-based extended logic programming with defeasible
priorities Journal of applied non classical logic 7, 25-75.
[PVREE] Henry Prakken y Gerard Vreeswijk. Logics for Defeasible Argumentation Survey in defeasible
argumentation.
[VREE97] Gerard A.W. Vreeswijk. Abstract argumentation systems. Artificial Intelligence, 90:225--279,
1997.


Argument comparison criteria analysis

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