The influence of defeated arguments in defeasible argumentation

Transcription

1 The influene of efeate arguments in efeasible argumentation Bart Verheij University of Limburg, Department of Metajuriia P.O. Box 616, 6200 MD Maastriht, The Netherlans fax: Abstrat Formal efeasible argumentation is urrently the subjet of ative researh. Formalisms of efeasible argumentation are haraterize by a notion of efeasible argument. The influene of arguments on whih onlusions an be rawn istinguishes formalisms of efeasible argumentation from nonmonotoni logis. This influene ours for two reasons: by the struture of an argument, an by interation with other arguments. In the proess of argumentation not all arguments are available at one. At eah stage of argumentation new arguments are taken into aount. In efeasible argumentation, where arguments an be efeate by other arguments, this results in the possible hange of the status of arguments, epening on whih arguments have been onsiere. Existing formalisms of efeasible argumentation o not provie a proess view on argumentation, or overlook the influene on this proess of the efeate arguments that have been taken into aount. In this paper we argue that a moel of the proess of argumentation requires that the arguments that are efeate at some stage of argumentation annot simply be ignore. Otherwise, ifferent stages of argumentation annot be istinguishe, an orers of argumentation an isappear. Keywors: efeasible argumentation, nonmonotoni logi 1 Introution Currently the formal stuy of efeasible argumentation gets muh attention [BoToKo93, Du93, HaVe94, Li93, Po94, Pr93a, Pr93b, Vr93, Ve95]. Formalisms of efeasible argumentation an be istinguishe from nonmonotoni logis in general. The prinipal istintion is that a notion of efeasible argument is entral. An argument is like a proof: It represents the argumentation from premises to onlusion, inluing the intermeiate steps. Unlike proofs, however, efeasible arguments are not strit, an an beome efeate by other arguments. There are two main reasons to take arguments into aount in efeasible argumentation. First, the struture of an argument influenes whether it is efeate or not. Seon, whether an argument is efeate is influene by other arguments. The struture of an argument The quality of an argument is influene by its struture. For instane, an argument with a number of weak steps is worse than an argument with fewer weak steps, an an argument that uses more information is better than (or as goo as) an argument that uses less. When one only looks at onlusions an reasons (the iret

2 preeessors of a onlusion in an argument) the influene of the struture of arguments is overlooke. Therefore, it is natural to etermine whih arguments an be base on a given set of information first, an then whih onlusions are justifie by those arguments. Other arguments By the interation of arguments some of the arguments taken into aount are unefeate, others efeate. For instane, arguments an impair other arguments, resulting in the efeat of the latter. Arguments an also reinfore eah other, so that they remain unefeate. Beause only unefeate arguments justify their onlusions, an the interation of arguments influenes whih arguments are efeate an whih unefeate, arguments have to be onsiere when one etermines the onlusions that follow from given information. Not all formalisms mentione use arguments for both reasons. Together these reasons an be use to istinguish formalisms of efeasible argumentation from others (see also setion 6). In the next setion we explain why efeate arguments annot be ignore at a stage in the proess of argumentation. In the setions thereafter we esribe a formalism appropriate for moeling the influene of efeate arguments. After a brief omparison with some other formalisms of efeasible argumentation, we en with the onlusion of this paper. 2 How efeate arguments influene argumentation An overlooke aspet of efeasible argumentation is the influene of the efeate arguments that have been taken into aount. A goo example an be given in ase of arual of arguments. 1 Arguments for a onlusion arue, if they reinfore eah other, an therefore together give better support to their onlusion than on their own. We give an example. Suppose we have the situation that there are three arguments, enote α 1, α 2 an β, available to a reasoner. It an be the ase that the arguments α 1 an α 2 are both on their own efeate by β, but together remain unefeate, an even efeat β. So, we have the following situation: The argument β efeats the argument α 1, if α 1 an β are the arguments onsiere. The argument β efeats the argument α 2, if α 2 an β are the arguments onsiere. The arguments α 1 an α 2 efeat the argument β, if α 1, α 2 an β are the arguments onsiere. 2 1 The term is use by Pollok [Po91]. Even though Pollok fins it a natural supposition that arguments arue, he surprisingly rejets it. Verheij [Ve95] argues that arguments o arue, an provies a formal moel for it. This paper extens that moel. 2 The following natural language example is taken from Verheij [Ve94]. Assume that John has robbe someone, so that he shoul be punishe (α 1 ). Nevertheless, a juge eies that he shoul not be punishe, beause he is a first offener (β). Or, assume that John has injure someone, an shoul therefore be punishe (α 2 ). Again, the juge eies he shoul not be punishe, being a first offener (β). Now assume John has robbe an injure someone at the same time, so that there are two arguments for punishing him (α 1, α 2 ). In this ase, the juge might eie that John shoul be punishe, even though he is a first offener (β).

3 There are six orers in whih the arguments an be taken into aount by a reasoner, suh as α 1, α 2, β or α 2, β, α 1. In figure 1, the six orers are shown in one iagram. Eah orner of the blok represents a stage in the proess of argumentation, an has a label representing whih arguments have been onsiere then. The 0 represents that no argument has been onsiere yet. An argument in brakets is efeate. Eah arrow enotes that an argument is being taken into aount. For instane, the arrow from α 2 to β (α 2 ) means that the argument α 2 beomes efeate after β has been taken into aount. 0 α 1 β α 2 β (α 1 ) α 1 α 2 β (α 2 ) α 1 α 2 (β) Figure 1: Consiering arguments in ifferent orers Now we ome to the influene of the efeate arguments. If we woul forget about the efeate arguments, as in other work on efeasible argumentation, we are left with the following (wrong) piture. Again, eah orner represents a stage of argumentation, but this time only the unefeate arguments at that stage are enote. Just as in the previous figure, eah arrow means that one argument is being taken into aount. 0 α 1 α 2 β α 1 α 2 Figure 2: The wrong piture This piture is wrong for two reasons: Different stages of argumentation have ollapse into one. For instane β, β (α 1 ) an β (α 2 ) an no longer be istinguishe. An orer of argumentation has isappeare. There is no iret arrow from β to α 1 α 2, beause it has beome unlear what it means to go from β to α 1 α 2 by taking a single extra argument into aount. In the orret piture the intermeiate stages β (α 1 ) an β (α 2 ) issolve this problem.

4 We summarize the influene of efeate arguments: If the efeate arguments taken into aount are overlooke, ifferent stages of argumentation annot be istinguishe, an orers of argumentation an isappear. 3 Arguments We start with the formal efinition of an argument. Our notion of an argument is relate to that of Lin an Shoham [LiSh89] an Vreeswijk [Vr91, Vr93], an is basially a tree of sentenes in some language. Our approah to efeasible argumentation is inepenent of the hoie of a language. Therefore, we treat a language as a set without any struture. A language oes not even ontain an element to enote negation or ontraition. This is not require, beause in our formalism ontraition is not the trigger for efeat. We briefly ome bak to this in the next setion. 3 Definition 3.1 A language is a set, whose elements are the sentenes of the language. An argument is like a proof, possibly with onitions. An argument supports its onlusion (relative to its onitions), but unlike a proof, an argument is efeasible. Any argument an be efeate by other arguments. Eah argument has a onlusion an onitions (possibly zero). An argument an ontain arguments for its onlusion. Arguments ontain sentenes, an have initial an final parts. A speial kin of argument is a rule. Definition 3.2 Let L be a language. An argument in the language L is reursively efine as follows: 1. Any element s of L is an argument in L. In this ase we efine Con(s) = s Cons(s) = Sents(s) = Initials(s) = Finals(s) = {s} 2. If Α is a set of arguments in L, s an element of L, an s Sents[Α], 4 then Α s is an argument in L. In this ase we efine Con(Α s) = s Cons(Α s) = Cons[Α] Sents(Α s) = {s} Sents[Α] Initials(Α s) = {Α s} Initials[Α] Finals(Α s) = {s} {Β s f: f is a surjetive funtion from Α onto Β, suh that α: f(α) Finals(α)} 5 Con(α) is the onlusion of α. An element of Cons(α), Sents(α), Initials(α), an Finals(α) is a onition, a sentene, an initial argument, an a final argument of α, respetively. The onlusion of an initial argument of α, other than the argument α itself, is an intermeiate onlusion of α. An argument in L is a rule, if it has the form S s, where S L an s L. For eah argument α we efine the set of arguments Subs(α), whose elements are the subarguments of α: Subs(α) = Initials[Finals(α)] 3 Lin an Shoham [LiSh89], Vreeswijk [Vr91, Vr93] an Dung [Du93] o more or less the same. Lin an Shoham use a language with negation, an Vreeswijk one with ontraition. Dung even goes a step further, an uses ompletely unstruture arguments. 4 If f: V W is a funtion an U V, then f[u] enotes the image of U uner f. 5 This means that the set Β arises by replaing eah argument in the set Α by one of its final arguments.

5 A proper subargument of an argument α is a subargument other than α. If α is a subargument of β, then β is a superargument of α. A subargument of an argument α that is a rule is a subrule of α. Notation If Α is finite, i.e. Α = {α 1, α 2,..., α n }, we write α 1, α 2,..., α n s for an argument Α s = {α 1, α 2,..., α n } s, if no onfusion an arise. Intuitively, if Α s is an argument (in some language L), the elements of Α are the arguments supporting the onlusion s. It may seem strange that also sentenes are onsiere to be arguments. An argument of the form s, where s is a sentene in the language L, represents the egenerate (but in pratie most ommon) kin of argument that a sentene is put forwar without any arguments supporting it. Some examples of arguments in the language L = {a, b,, } are {{a} b} an {{a}, {b} }. They are graphially represente in figure 3. a b a b Figure 3: Examples of arguments The onitions of the argument {{a}, {b} } are a an b. It has as its onlusion. Some of its initial arguments are b, {a} an the argument itself. Some of its final arguments are, {}, an {, {b} }. Among its subarguments are an {b}. The formal struture of our arguments iffers from those of Lin an Shoham [LiSh89] an Vreeswijk [Vr91, Vr93]. In these formalisms, eah onition of an argument an only be supporte by a single argument. Beause of our belief that arguments an arue, in our formalism onitions an be supporte by several arguments. 6 As a result, we an make weakenings (an strengthenings) of an argument expliit. Intuitively, an argument beomes weaker if less arguments support its onlusion an intermeiate onlusions. For instane, the argument {{b} } is a weakening of the argument {{a}, {b} }. The latter ontains {a} an {b} to support the intermeiate onlusion, while the former only ontains {b}. Definition 3.3 Let L be a language. For any argument α in the language L we reursively efine a set of arguments Weaks(α): 1. For α = s, s L, Weaks(s) = {s}. 2. For α = Α s, Α Args(L), s L, Weaks(Α s) = {Β s Β Weaks[Α] an Con[Β] = Con[Α]} An element of Weaks(α) is a weakening of α. A weakening of α, other than α, is a proper weakening of α. If α is a weakening of β, then β is a strengthening of α. Weakenings are in general not subarguments. For instane, {{a} } is not a subargument of {{a}, {b} }. 6 This is formally aomplishe by making the arguments supporting a onlusion a set of arguments, instea of a sequene.

6 Different arguments an have the same subrules. The arguments in figure 4 all have the subrules {a}, {b}, {} an {} e. There is some reunany in the last argument, beause the arguments {{a} } an {{b} } are weakenings of the argument {{{a}, {b} } } e. In ase one of the three arguments supporting is efeate suh reunany an beome meaningful. a b a b e e a b a b e Figure 4: Different arguments with the same subrules In other formalisms ifferent arguments with the same subrules are not istinguishe. For instane, the formal arguments of Pollok [Po87-Po94], Simari an Loui [SiLo92], Prakken [Pr93a, Pr93b], an Bonarenko et al. [BoToKo93] are (more or less) sets of rules, so that the istintion is oneale. In the formalisms of Lin an Shoham [LiSh89] an Vreeswijk [Vr91, Vr93], the arguments in figure 4 are not base on the same subrules, beause the onitions of a rule are a sequene, an not a set. For example, they treat an, as ifferent rules. 4 Defeaters As sai before, in efeasible argumentation, arguments are efeasible. In our formalism, all arguments are efeasible. Exept for Dung [Du93], other authors have separate lasses of strit an efeasible arguments. Arguments remain unefeate, if there is no information that makes them efeate. So, if one wants a lass of strit arguments, for instane, to moel eutive argumentation, it an be efine, by not allowing information that leas to the efeat of the arguments in that lass. In our formalism this is easy, beause the efeat of arguments is the result of efeat information that is expliit an iret. Expliit efeat information Pollok's efeaters [Po87-Po94], Prakken's kins of efeat [Pr93a, Pr93b], Vreeswijk's onlusive fore [Vr91, Vr93], an Dung's attaks [Du93] are examples of expliit efeat information. Instea of hiing the information in a general proeure, for instane base on speifiity, expliit information etermines whih arguments beome efeate an whih remain unefeate. Expliit efeat information is require beause no general proeure an be flexible enough to be universally vali. Diret efeat information By iret efeat information, we mean information speifying onitions that iretly imply the efeat of one or more arguments. Pollok's efeaters an Dung's attaks are examples of iret efeat information. Examples of iniret efeat information are

7 Prakken's kins of efeat an Vreeswijk's onlusive fore. In their formalisms efeat of arguments is triggere by a onflit of arguments. If there is a onflit, one of the arguments involve is selete using the efeat information. The selete argument beomes efeate, an the onflit is resolve. We think that iniret efeat information is not suffiient. An important kin of efeat requiring iret efeat information is efeat by an unerutting argument [Po97]. An unerutting argument only efeats another argument, without ontraiting the onlusion. In our formalism the efeat information is speifie by expliit an iret efeaters. In ontrast with Pollok's efeaters [Po87-Po94], an Dung's attaks [Du93], our efeaters an expliitly represent ompoun efeat: a set of arguments for a onlusion an efeat another set of arguments, instea of only a single argument another single argument. This is neee for the aequate moeling of arual of arguments [Ve95]. A efeater onsists of two sets of arguments: The arguments in one set beome efeate if the arguments in the other set are unefeate. Definition 4.4 Let L be a language. A efeater of L has the form Α (Β), where Α an Β are sets of arguments of L, suh that 1. All arguments in Α have the same onlusion. 2. All arguments in Β have the same onlusion. 3. No argument in Α has a subargument or weakening that is an element of Β. The arguments in Α are the ativating arguments of the efeater. The arguments in Β are its efeate arguments. Α Β is the range of the efeater. 7 Notation A efeater Α (Β) with finite range, i.e. Α = {α 1, α 2,..., α n } an Β = {β 1, β 2,..., β m }, is written α 1 α 2... α n (β 1 β 2... β m ), if no onfusion an arise. The meaning of a efeater Α (Β) is that if the arguments in Α are unefeate, the arguments in Β must be efeate. For instane, the efeater a (b ) efeats the rule b, if the argument a is unefeate. By the thir requirement in the efinition a efeater annot efeat a subargument or strengthening of one of its ativating arguments. For instane, if the argument a b is ativating in a efeater, it annot efeat the argument b. If the argument a is ativating in a efeater, it annot efeat the argument {a, b }. As sai, our efeaters an represent ompoun efeat whih is neee in ase of arual of arguments. For instane, the example in setion 2 requires not only the regular efeaters β (α 1 ) an β (α 2 ), but also a efeater that represents ompoun efeat, namely α 1 α 2 (β). 5 Argumentation stages We are about to efine an argumentation theory. It formally represents whih arguments are available to a reasoner, an when arguments an beome efeate. Our notion of an argumentation theory is relate to that of an argument system [Vr91, Vr93] an of an argumentation framework [Du93]. A theory onsists of a language, arguments, an efeaters. The language of a theory speifies the sentenes that an be use in 7 Beause arguments are their own subarguments an strengthenings, the sets Α an Β an have no elements in ommon.

8 arguments. The arguments of a theory are the arguments that are available. The efeaters of a theory represent the situations in whih arguments efeat other arguments. Definition 5.1 An argumentation theory is a triple (L, Args, Defs), where 1. L is a language, 2. Args is a set of arguments in L, lose uner initial arguments, 8 an 3. Defs is a set of efeaters of L, with their ranges in Args. 9 For instane, a theory that represents the example in setion 2 is efine as follows: L = {a 1, a 2, a, b}, Args = {a 1, a 1 a, a 2, a 2 a, b}, Defs = {β (α 1 ), β (α 2 ), α 1 α 2 (β)}, where α 1 = a 1 a, α 2 = a 2 a, β = b. So we have two separate arguments α 1 an α 2 that support the onlusion a, an an argument β that supports b. The efeaters say that α 1 an α 2 are on their own efeate by β, but together efeat β. We use this theory as an illustration of the oming efinitions. It is hosen, beause it is a key example of arual of arguments, an therefore suitable to show the influene of efeate arguments. It is however too simple to illustrate all aspets of the efinitions. The next efinition is that of an argumentation stage. It an represent the arguments that at a ertain stage in the proess of argumentation have been taken into aount, an whih of them are then efeate. (Later we efine argumentation stages that are aeptable with respet to a theory. These are the atual stages of argumentation that are mae possible by an argumentation theory.) Our efinition of an argumentation stage is relate to the argumentation strutures of Lin an Shoham, an Vreeswijk. They require however that it is a set without ontraiting arguments (see the isussion of iret efeat information in setion 4). Eah of the requirements in our efinition orrespons to a simple intuition on stages in argumentation. For instane, one requirement is that an argument an only be taken into aount if all its initial arguments alreay have been. The range of an argumentation stage onsists of the arguments taken into aount at that stage. Definition 5.2 Let (L, Args, Defs) be an argumentation theory. An argumentation stage of (L, Args, Defs) has the form Σ (Τ), where Σ an Τ are subsets of Args, suh that: 1. Σ is lose uner initial arguments. 2. No argument an be an element of both Σ an Τ. 3. No proper subargument of an element of Σ an be an element of Τ. 4. Not all proper weakenings of an element of Τ that has proper weakenings an be elements of Σ. The arguments in Σ are unefeate, an those in Τ efeate. The set Σ Τ is the range of Σ (Τ). Remark: efeaters an argumentation stages are the same in form. 8 The set of arguments is not lose uner 'rule appliation', as in other formalisms. Although the arguments of an ieal reasoner woul be, this is in general an unreasonable assumption. 9 The efeaters of a theory o not neessarily agree with eah other. For instane, both α (β) an β (α) an be efeaters of a theory. A lassi example of suh a situation is the Nixon iamon.

9 Some argumentation stages of the example theory are a 1 α 1 (β), a 2 β (a 1 α 2 ), an a 1 α 1 a 2 α 2 (β). Definition 5.2 is ruial: it reflets that there an be efeate arguments taken into aount at a stage of argumentation. Whih arguments of a theory beome efeate an whih not is etermine by its efeaters. Arguments are normally unefeate, but an at some stage of argumentation be efeate beause of relevant efeaters. A efeater is relevant at some argumentation stage if all its arguments have been taken into aount at that stage, or are parts of suh arguments. Formally, this means that its range is a subset of the final parts of the arguments taken into aount. Definition 5.3 Let (L, Args, Defs) be an argumentation theory, Α (Β) a efeater in Defs, an Σ (Τ) an argumentation stage of (Args, Defs). Α (Β) is relevant for Σ (Τ), if Α Β Finals[Σ Τ]. So, in the example theory, β (α 2 ) is relevant for a 2 β (a 1 α 2 ), an all three efeaters of the theory are relevant for a 1 α 1 a 2 α 2 (β). This notion of relevane of efeaters has no analogue in other formalisms. Normally, all efeaters are onsiere relevant. We an o better, beause our argumentation stages represent whih arguments are taken into aount. A efeater only justifies the efeat of its efeate arguments, if its ativating arguments are parts of unefeate arguments, i.e., if they are subarguments of unefeate arguments. The efeater is then ativate. Definition 5.4 Let (L, Args, Defs) be an argumentation theory, Α (Β) a efeater in Defs, an Σ (Τ) an argumentation stage of (Args, Defs). Α (Β) is ativate in Σ (Τ), if it is relevant an Α Finals[Σ]. In the argumentation stage a 2 β (a 1 α 2 ) the efeater β (α 2 ) is ativate. In the stage a 1 α 1 a 2 α 2 (β) all three efeaters are ativate. Argumentation stages only represent atual stages of the proess of argumentation, if they are aeptable with respet to an argumentation theory. An argumentation stage is aeptable, if The efeat of eah of the efeate arguments is justifie. The efault is namely that an argument is not efeate. Defeat is justifie by ativate efeaters. If the efeat of an argument is justifie, it must atually be efeate. Otherwise put, efeaters that are ativate in the stage must be obeye. No relevant efeater is unjustly ignore. This requirement nees yet another efinition. Relevant efeaters must be eativate, for instane, beause one of its ativating arguments is efeate. A efeater is eativate, if two onitions hol. First, there must be another efeater that justifies the efeat of one of its ativating arguments. (It is of ourse even suffiient that the efeat of a subargument or a strengthening of one of the ativating arguments is justifie.) However, α 1 α 2 (β) an β (α 1 ) o not eativate eah other. Only the former an eativate the latter. The reason for this is the arual of the arguments α 1 an α 2. The efeater α 1 α 2 (β) overrules β (α 1 ), an therefore annot be eativate by it. This leas to the seon onition: a efeater an only be eativate by a efeater it oes not overrule. Formally, this is apture in the following efinition.

10 Definition 5.5 Let (L, Args, Defs) be an argumentation theory, Α (Β) an Γ ( ) efeaters in Defs, an Σ (Τ) an argumentation stage of (Args, Defs). Α (Β) eativates Γ ( ), if both are relevant for Σ (Τ), an the following hol: 1. There is an element of Β that is a subargument or a strengthening of an element of Γ. 2. Α is not a subset of, or Β is not a proper subset of Γ. We an finally efine when an argumentation stage is aeptable with respet to an argumentation theory. 10 The requirements in the efinition have alreay been briefly explaine just after efinition 5.4. Definition 5.6 Let (L, Args, Defs) be an argumentation theory, an Σ (Τ) an argumentation stage of (L, Args, Defs). Σ (Τ) is aeptable with respet to (Args, Defs), if the following hol: 1. If τ Τ, there is an ativate Α (Β) Defs, suh that τ Β. 2. If Α (Β) Defs is ativate, then Β Τ. 3. If Α (Β) Defs is relevant, but not ativate, then there is an ativate Γ ( ) Defs that eativates Α (Β). An aeptable argumentation stage of our example theory is β (a 1 α 1 ). The stage a 1 a 2 α 2 (β) is not aeptable, beause the efeat of β is not justifie, an beause β (α 2 ) is not eativate. An extension of an argumentation theory is an aeptable stage of argumentation that annot be ontinue. It must therefore be maximal with respet to set inlusion. 11 Definition 5.7 Let (L, Args, Defs) be an argumentation theory, an Σ (Τ) an argumentation stage of (L, Args, Defs). Σ (Τ) is an extension of (L, Args, Defs), if the following hol: 1. Σ (Τ) is aeptable with respet to (L, Args, Defs), an 2. There is no argumentation stage Σ' (Τ'), aeptable with respet to (L, Args, Defs), suh that Σ Τ is a proper subset of Σ' Τ'. The unique 12 extension of our example theory is a 1 α 1 a 2 α 2 (β). Even though the theory ontains the efeaters β (α 1 ), β (α 2 ) that an efeat α 1 an α 2 separately, they remain unefeate by supporting eah other. This is a real ase of the arual of the arguments α 1 an α 2, as an be seen by looking at other aeptable argumentation stages: β a 1 (α 1 ) an β a 2 (α 2 ). Here α 1 an α 2 are on their own efeate by β. The arguments α 1 an α 2 only remain unefeate if they reinfore eah other. 6 Other formalisms In orer to ompare our formalism with others, we en with an overview of some of its harateristis. 10 The way we efine aeptable efeasible argumentation stages is relate to the way Dung efines his amissible sets of arguments [Du93], an Pollok his partial status assignments [Po94]. However, these o not represent stages in the proess of argumentation. 11 Dung's preferre extensions [Du93] an Pollok's status assignments [Po94, p. 393] are efine similarly. 12 A theory an have any number of extensions: zero, one, or several.

11 The struture of arguments an influene whih arguments are efeate an whih are not (setion 1). Other arguments an influene whih arguments are efeate an whih are not (setion 1). The formalism is inepenent of the hoie of a language (setion 3). Defeat is the result of expliit efeat information (setion 4). Defeat information represents iret efeat (setion 4). Defeat is not triggere by a onflit. Stages in the proess of argumentation are represente (setion 5). Arguments an arue (setion 2, 3, 4, 5). Defeate arguments influene the proess of argumentation (setion 2, 4, 5). Table 1 relates these harateristis to some other formalisms. An entry in the table is rosse if the formalism in that olumn has the harateristi iniate in the row. BoToKo93 Du93 Pr93a, Pr93b HaVe94 LiSh89, Li93 Po87- Po94 Vr91, Vr93 Struture of argument Other arguments Language inepenent Expliit efeat information Diret efeat information Argumentation stages Arual of arguments Influene of efeate arguments Table 1: Formalisms of efeasible argumentation an their harateristis 7 Conlusion In this paper we have argue that the proess of argumentation is better moele if the arguments that are efeate at a ertain stage of argumentation are not ignore. This is espeially lear if several arguments for a onlusion arue: They an at early stages of argumentation be taken into aount, but be efeate by other arguments, an later exert their influene, an beome unefeate. We have provie a formalism that aptures these ieas. Key efinitions are those of argumentation stages (efinition 5.2), in whih the efeate arguments are expliitly taken into aount, an of relevane of efeaters (efinition 5.3), whih assures that only efeaters are use that ontain arguments atually onsiere. Aknowlegments This researh was partly finane by the Founation for Knowlege-base Systems (SKBS) as part of the B3.A projet. SKBS is a founation with the goal to improve the level of expertise in the Netherlans in the fiel of knowlege-base systems an to promote the transfer of knowlege in this fiel between universities an business ompanies.

12 I thank Jaap Hage, Arno Loer an Gerar Vreeswijk for our stimulating an lively arguments about argumentation. Referenes [BoToKo93] A. Bonarenko, F. Toni an R. A. Kowalski, An assumption-base framework for non-monotoni reasoning, in Logi programming an non-monotoni reasoning. Proeeings of the seon international workshop, L. M. Pereira an A. Neroe (es.), MIT Press, Cambrige (Massahusetts), 1993, pp [Du93] P. M. Dung, On the aeptability of arguments an its funamental role in nonmonotoni reasoning, logi programming, an human s soial an eonomial affairs, manusript, [HaVe94] J. Hage an B. Verheij, Reason-Base Logi: a logi for reasoning with rules an reasons, to appear in Law, Computers an Artifiial Intelligene, Vol. 3 (1994), No. 2, report SKBS/B3.A/ [LiSh89] F. Lin an Y. Shoham, Argument Systems: a uniform basis for nonmonotoni reasoning, in Proeeings of the First International Conferene on Priniples of Knowlege Representation an Reasoning, R. J. Brahman, H. J. Levesque an R. Reiter (es.), Morgan Kaufmann Publishers, San Mateo (California), 1989, pp [Li93] F. Lin, An argument-base approah to nonmonotoni reasoning. Computational Intelligene, Vol. 9 (1993), No. 3, pp [Po87] J. L. Pollok, Defeasible reasoning, Cognitive Siene 11 (1987), pp [Po90] J. L. Pollok, A theory of efeasible reasoning, International Journal of Intelligent Systems, Vol. 6 (1990), pp [Po91] J. L. Pollok, Self-efeating arguments, Mins an Mahines 1 (1991), pp [Po92] J. L. Pollok, How to reason efeasibly, Artifiial Intelligene 57 (1992), pp [Po94] J. L. Pollok, Justifiation an efeat, Artifiial Intelligene 67 (1994), pp [Pr93a] [Pr93b] [SiLo92] [Ve94] H. Prakken, Logial tools for moelling legal argument, H. Prakken, Amsteram, H. Prakken, A logial framework for moelling legal argument, in The Fourth International Conferene on Artifiial Intelligene an Law. Proeeings of the Conferene, ACM, New York, 1993, pp G. R. Simari an R. P. Loui, A mathematial treatment of efeasible reasoning an its appliations, Artifiial Intelligene 53 (1992), pp H. B. Verheij, Reason Base Logi an legal knowlege representation, in Proeeings of the Fourth National Conferene on Law, Computers an Artifiial Intelligene, I. Carr an A. Narayanan (es.), University of Exeter, 1994, pp , report SKBS/B3.A/94-5. [Ve95] B. Verheij, Arual of arguments in efeasible argumentation, in Duth/German Workshop on Nonmonotoni Reasoning. Proeeings of the Seon Workshop, Delft University of Tehnology, Utreht University, 1995, pp , report SKBS/B3.A/ [Vr91] G. Vreeswijk, Abstrat argumentation systems: preliminary report, in Proeeings of the First Worl Conferene on the Funamentals of Artifiial Intelligene, D. M. Gabbay an M. De Glas (es.), Angkor, Paris, 1991, pp

13 [Vr93] G. Vreeswijk, Stuies in efeasible argumentation, G. A. W. Vreeswijk, Amsteram, 1993.