The recently proposed ASPIC, standing for ASPIC+ revisited is an evolution of the prominent ASPIC+ formalism for structured argumentation, which overcomes some limitations of ASPIC+ by resorting in particular to an alternative notion of attack referring to sets of arguments. While this notion of attack is a key element for achieving the technical advantages ofered by ASPIC, it also gives rise to some peculiar situations, where some attacks are, in a sense, polymorphic since there are multiple reasons by which a given set of arguments can attack an argument. After pointing out that polymorphic attacks are also possible in ASPIC+, we provide some examples of polymorphic attacks in ASPIC+ and ASPIC, discuss the underlying technical and conceptual issues and provide a preliminary discussion about how to revise the formalisms in order to encompass alternative options for their management.
ASPIC+ [
In recent years, however, it has also been pointed out that ASPIC+ sufers from some specific technical
problems concerning, in particular, the behavior in the presence of multiple contradictories [
While this idea is a key factor in enabling the advantageous technical properties of ASPIC, it also makes some peculiar situations possible. In particular, there may be multiple reasons for the same set to attack an argument , and each diferent reason corresponds to a diferent type of attack. In such a case, we say that the attack from to is polymorphic, since it potentially can be ascribed to multiple classes. We point out, however, that, though this has never been evidenced before in the literature, polymorphic attacks may also occur in the context of ASPIC+.
This peculiar feature has not been evidenced in previous works on ASPIC+ and ASPIC and, while not afecting, as we will discuss, the technical correctness of the formalisms, poses both technical and conceptual questions and opens, in particular, the way to the study of alternative options in the definition of attacks in ASPIC. The present work aims at providing the basis for this investigation line by carrying out a first analysis of the phenomenon of polymorphic attacks and discussing their conceptual and technical status in perspective. The paper is organised as follows. Section 2 recalls the necessary background notions on argumentation frameworks and ASPIC+, while Section 3 reviews ASPIC. Sections 4 and 5 discuss polymorphic attacks in ASPIC+ and ASPIC respectively. Section 6 provides some perspectives on their treatment, and Section 7 concludes the paper.
Definition 1. An argumentation framework (AF) is a pair ℱ = ⟨A, ↠ ⟩, where A is a set of arguments and ↠ ⊆ (A × A) is a binary relation on A.
When (, ) ∈↠ (also denoted as ↠ ) we say that attacks . For a set ⊆ A and an
argument ∈ A we write ↠ if ∃ ∈ : ↠ and ↠ if ∃ ∈ : ↠ , and we
denote the arguments attacking as − ≜ { ∈ A | ↠ } and the arguments attacked by as
+ ≜ { ∈ A | ↠ }. An extension-based argumentation semantics specifies the criteria for
identifying, for a generic AF, a set of extensions, where each extension is a set of arguments considered
to be acceptable together. Given a generic argumentation semantics , the set of extensions prescribed
by for a given AF ℱ is denoted as ℰ (ℱ ). Several argumentation semantics are recalled in Definition
2, along with some basic underlying notions. For more details, the reader is referred to [
Definition 3. Given a set a justification labeling of is a function : → Σ , where Σ = {, , }. Given an AF ℱ = ⟨A, ↠ ⟩ and a semantics , the justification labeling of A according to is defined as follows 1: ( ) = if ∈ ⋂︀E∈ℰ (ℱ) E; ( ) = if ∈ ⋃︀E∈ℰ (ℱ) E and /∈ ⋂︀E∈ℰ (ℱ) E; ( ) = if /∈ ⋃︀E∈ℰ (ℱ) E.
In words, we will say respectively that is skeptically justified, credulously justified, and not justified. We now recall the essential notions of the ASPIC+ formalism.
Definition 4.
An argumentation system is a tuple = (ℒ,¯, ℛ, ) where: 1. ℒ is a logical language 1We avoid reference to ℱ and in for ease of notation. Moreover, with respect to the traditional notion of justification we keep skeptical and credulous justification disjoint for reasons which will be clear later. base using rules.
Definition 6. theory.
Definition 7.
1. if undefined. wf in
ℒ), and ℛ ∩ ℛ = ∅ 4. : ℛ → ℒ is a naming convention for ℛ.
contradictory 2. ¯ is a contrariness function from ℒ to 2ℒ such that: (i) is a contrary of if ∈ , /∈ ; (ii) is a contradictory of (denoted by = − ) if ∈ , ∈ ; (iii) each ∈ ℒ has at least one 3. ℛ = (ℛ , ℛ) is a pair of sets of strict (ℛ ) and defeasible (ℛ) inference rules of the form 1, . . . , → and 1, . . . , ⇒ respectively (where , are meta-variables ranging over In the following, given a set ⊆ ℒ contraries and contradictories as = ⋃︀ with a little abuse of notation, we will denote the set of its ∈ { | ∈ }. Given a rule = 1, . . . , → (⇒) , we antecedents of the rule denoted as ant (). will say that is the consequent of the rule, denoted as cons() and that { 1, . . . , } is the set of the
Closure under transposition of strict rules is a desirable property as it ensures (together with other
conditions) that an argumentation system satisfies some rationality postulates [
Given = (ℒ,¯, ℛ, ), the set of strict rules ℛ is closed under transposition if if ∈ ℛ then, for = 1 . . . , 1, . . . , − 1, − , +1, . . . , → − ∈ ℛ . Given a ⋃︀ ∈ℛ (), where
A knowledge base is a subset of ℒ including certain (called axioms) and defeasible (called ordinary) premises. It gives rise to the notion of argumentation theory. Arguments are built from a knowledge (the axioms) and (the ordinary premises). The tuple = (, ) is called an argumentation A knowledge base in = (ℒ,¯, ℛ, ) is a set ⊆ ℒ consisting of two disjoint subsets
An argument on the basis of a knowledge base in = (ℒ,¯, ℛ, ) is: ∈ with: Prem( ) = { }; Conc( ) = ; Sub( ) = { }; Rules( ) = ∅; Top( ) = 2. 1, . . . , → (⇒) if 1, . . . , are arguments such that there exists a strict (defeasible) rule Conc( 1), . . . , Conc( ) → (⇒) in ℛ (ℛ) with: Prem( ) = Prem( 1) ∪ . . . ∪ Prem( ); Conc( ) = ; Sub( ) = Sub( 1) ∪ . . . ∪ Sub( ) ∪ { }; Rules( ) = Rules( 1) ∪ . . . ∪ Rules( ) ∪ {Conc( 1), . . . , Conc( ) → (⇒) }; Top( ) = Conc( 1), . . . , Conc( ) → (⇒) ; DefRules( ) = { | ∈ Rules( ) ∩ ℛ}; StRules( ) = { | ∈ Rules( ) ∩ ℛ }. ifnite if Rules( ) is finite.
For any argument , Prem( ) = Prem( ) ∩ ; Prem( ) = Prem( ) ∩ . is: strict if DefRules( ) = ∅, defeasible if DefRules( ) ̸= ∅; firm if Prem( ) ⊆ ; plausible if Prem( ) ⊈ ; ⊢ . Given a set of arguments , Prem() ≜ ⋃︀ Rules(), Top(), DefRules(), StRules().
Notation 1. Some further notations are useful. Given ⊆ ℒ , ⊢ denotes that there exists a strict argument such that Conc( ) = , with Prem( ) ⊆ . ⊢ denotes that ⊢ and ∄ ⊊ : ∈ Prem( ), and similarly for Conc(), Sub(),
Definition 8.
An argument attacks an argument if undercuts, rebuts, or undermines where: undercuts (on ′) if Conc( ) ∈ () for some ′ ∈ Sub( ) such that = Top( ′) is defeasible. contrary-rebuts if Conc( ) is a contrary of . undermines (on ′) if Conc( ) ∈ for some ′ = , ∈ Prem( ). In such a case contrary-undermines if Conc( ) is a contrary of .
In some cases, attack efectiveness depends on a preference ordering ⪯ over arguments (assumed to
be a preorder as in [
Definition 9. Let attack on ′. If undercuts, contrary-rebuts, or contrary-undermines on ′, then preference-independent attacks on ′, otherwise preference-dependent attacks on ′. Then, defeats if for some ′ either preference-independent attacks on ′ or preference-dependent attacks on ′ and ⊀ ′.
Then a structured argumentation framework ( ) can be defined from an argumentation theory, 2 using the attack relation. Using the defeat relation, an AF is then derived from a . Definition 10. Let = (, ) be an argumentation theory. A structured argumentation framework ( ), defined by is a triple (S, C, ⪯ ) where S is the set of all finite arguments constructed from in (called the set of arguments on the basis of ), C ⊆ S × S is such that (, ) ∈ C if attacks , and ⪯ is an ordering on S.
Definition 11. Let Δ = (S, C, ⪯ ) be a , and D ⊆ S × S be the defeat relation according to Definition 9. The corresponding to Δ is defined as ℱΔ = (S, D).
Given an argumentation semantics , the justification status of arguments in S according to is determined by the set of extensions ℰ (ℱΔ) according to Definition 3.
In this section, we present ASPIC+ revisited (in the following ASPIC), introduced in [
The main diferences of ASPIC with respect to ASPIC+ can be summarized in two points: • an original form of closure, not involving any structural constraint on the set of rules, but rather resorting to a sort of extension of the contrariness function based on the strict rules; • an alternative notion of attack referring to sets of arguments, which leads to the construction of a substantially diferent argumentation framework.
In addition, ASPIC relaxes some requirements and adjusts some hypotheses, as illustrated step by step in the following.
As to Definition 4, ASPIC adopts the same basic elements of ASPIC+ but does not require that every element of the language has a contradictory, while adding the constraint that the name of a rule cannot be the contrary of anything, i.e. for every rule ∄ ∈ ℒ such that () ∈ . Similarly, the names of the rules cannot be included in the antecedent of a rule or in the knowledge base3.
Thus, in Definition 4 items 1 and 4 are unmodified, while item 2 is reduced to: ¯ is a function from ℒ to 2ℒ∖ℛ where ℛ = {() | ∈ ℛ}, and in item 3 are meta-variables ranging over wf in ℒ ∖ ℛ while is a meta-variable ranging over wf in ℒ.
In Definition 6 it is prescribed ⊆ ℒ ∖ ℛ while Definition 7 and Notation 1 are left unmodified.
Among the properties considered in Definition 12 of [
Definition 12. Let = (, ) be an argumentation theory, where = (ℒ,¯, ℛ, ). is weakly
well-formed if whenever ∈ and ∈ or can be obtained from by applying strict rules, it also
holds that ∈ .
2We do not consider here the alternative notion of c-structured argumentation framework in [
One of the main standpoints of ASPIC is avoiding the requirement that strict rules are closed under
transposition. Rather, the satisfaction of the rationality postulates is ensured (as shown in [
Definition 13. Given an argumentation system = (ℒ,¯, ℛ, ) the strict knowledge base * for is given by = ℒ, = ∅ and the corresponding argumentation theory is defined as * = (, * ). ⊢* and ⊢* denote respectively that ⊢ and ⊢ in * .
The extended contrariness relation EC() refers to sets of language elements and is obtained by applying a sort of closure with respect to strict derivability, together with a requirement of minimality. Definition 14. Given an argumentation theory = (, ) with = (ℒ,¯, ℛ, ), let EC* () ⊆ 2ℒ × 2ℒ be defined as EC* () = {(, ) | ⊢* , ⊢* and ∈ }. Letting for ⊆ ℒ , ̂︀ = ∖, the extended contrariness relation is defined as EC() = {(̂︀, ̂︀) | (, ) ∈ EC* () and ∀(′, ′) ∈ EC* () s.t. ̂︀′ ⊆ ̂︀ and ̂︀′ ⊆ ̂︀, ̂︀′ = ̂︀ and ̂︀′ = ̂︀} ⊆ 2ℒ × 2ℒ. is a contrary of if (, ) ∈ EC() and (, ) ∈/ EC(); is a contradictory of if (, ) ∈ EC() and (, ) ∈ EC().
In words, EC* () corresponds to the completion of the contrariness relation ¯ on the basis of the set of strict rules, i.e. a set is regarded as ‘being against’ a set if a strict consequence of ‘is against’ a strict consequence of . Then for each pair (, ) ∈ EC* () the corresponding pair (̂︀, ̂︀) where axioms are excluded is considered. Among these pairs, only those which respect a minimality condition, and hence are not redundant, belong to EC().
Definition 14 can be equivalently expressed in the form encompassed by the following more constructive definition.
Definition 15. Given an argumentation theory = (, ) with = (ℒ,¯, ℛ, ), let EC1(), EC2(), EC3() be the following subsets of 2ℒ × 2ℒ • EC1() = {({ }, { }) | ∈ }; • EC2() = {(, { }) | ⊢* and ∈ }; • EC3() = {(, ) | ⊢* and (, { }) ∈ EC1() ∪ EC2()}.
Letting EC* () = EC1() ∪ EC2() ∪ EC3(), and, for ⊆ ℒ , ̂︀ = ∖ , the extended contrariness relation is defined as EC() = {(̂︀, ̂︀) | (, ) ∈ EC* () and ∀(′, ′) ∈ EC* () s.t. ̂︀′ ⊆ ̂︀ and ̂︀′ ⊆ ̂︀, ̂︀′ = ̂︀ and ̂︀′ = ̂︀} ⊆ 2ℒ × 2ℒ. is a contrary of if (, ) ∈ EC() and (, ) ∈/ EC(); is a contradictory of if (, ) ∈ EC() and (, ) ∈ EC().
Let us comment on the above definition. First, EC* () is defined in a constructive incremental way starting from EC1() which mirrors, at level of singletons, the contrariness relation ¯. Then EC2() is defined taking into account EC1(), i.e. the contrariness relation ¯, and the strict rules: basically a set is put in relation to { }, i.e. is regarded as being against { }, if it is possible to strictly derive a contradiction to from . Finally EC3() is defined taking into account EC1() and EC2(): a set is put in relation to , i.e. is regarded as being against , if is against some { } according to EC1() or EC2() and strictly follows from . Then, as in Definition 14, for each pair (, ) ∈ EC* () the corresponding pair (̂︀, ̂︀) where axioms are excluded is considered. Among these pairs, only those that are not redundant belong to EC().
The equivalence between the two definitions is proved in [
Proposition 1 (Prop. 24 of [
The various forms of attack are then revised, replacing Definition 8 with the following definition concerning attacks at the level of sets of arguments. ment if undercuts, rebuts, or undermines where: Definition 16.
Given an argumentation theory = (, ), a set of arguments attacks an argucontrary-rebuts if (, ) ∈/ EC().
undermines if (, ) ∈/ EC().
• undercuts (on ′) if for some ′ ∈ Sub( ) such that = Top( ′) ∈ ℛ, the following
condition holds: ∃, such that ∪ = Conc() ∪ {()}, (, ) ∈ EC() and () ∈ .
• rebuts (on ′) if for some ′ ∈ Sub( ) of the form 1′′, . . . , ′′ ⇒ the following condition
holds: ∃, such that ∪ = Conc() ∪ { }, (, ) ∈ EC() and ∈ . In this case
• undermines (on ′) if for some ′ = , ∈ Prem( ) the following condition holds: ∃,
such that ∪ = Conc() ∪ { }, (, ) ∈ EC() and ∈ . In this case
contraryAs the efectiveness of some attacks depends on the preference relation, the notion of preference
ordering needs to be generalized to sets of arguments. Note, however, that in ASPIC there are no
requirements on the preference relation (in [
Definition 17.
Given a preorder ⪯ on a set of arguments , we extend ⪯ to 2 × as follows. An argument is at least as preferred as a set of arguments , denoted ⪯ , if ∃ ∈ such that ⪯ . is strictly preferred to , denoted ≺ , if ∃ ∈ such that ≺ , not strictly preferred to , denoted
Definition 18.
Let the set of arguments attack an argument on ′ according to Definition 16. If undercuts, contrary-rebuts, or contrary-undermines on ′, then preference-independent attacks on ′, otherwise preference-dependent attacks on ′. Then, defeats if either preferenceindependent attacks on ′ or preference-dependent attacks on ′ and ⊀ ′. minimally defeats , denoted as ⇝ , if defeats and ∄ ′ ⊊ such that ′ defeats .
An argumentation framework based on the notion of defeat provided in Definition 18 is then defined to evaluate the justification status of arguments. The idea is that the framework nodes represent relevant sets of arguments. In particular, we need a node for each singleton corresponding to a produced argument, and a node for each set of ultimately fallible arguments (as per Definition 19) that minimally defeats some produced argument.
S | Top( ) ∈ ℛ}.
Definition 19.
Given an argumentation theory = (, ), let S be the set of the arguments produced in on the basis of . The set of ultimately fallible arguments of is defined as UF(S) ≜ ∪ { ∈ Definition 20.
Given an argumentation theory = (, ) with ordering ⪯ , let S be the set of the RS( ), is defined as RS( ) = {{ } | ∈ S} ∪ { | ⊆ UF(S) and ∃ ∈ S : ⇝ }. arguments produced in on the basis of . The set of relevant sets of arguments of , denoted as
Then, a relevant set of ultimately fallible arguments attacks another one simply if it minimally defeats one of its members. and ∃ ∈ : ⇝ .
Definition 21.
Let , ∈ RS( ) for an argumentation theory = (, ) and S be the set of the arguments produced in on the basis of . D-attacks , denoted as ‖‖ ↠ ‖ ‖, if ⊆ UF(S)
The relevant set based argumentation framework is defined accordingly.
Definition 22. Given an argumentation theory = (, ), the RS-based argumentation framework induced by is defined as RS− F( ) = ({‖‖ | ∈ RS( )}, ↠ ).
Finally, given an argumentation semantics , the justification status of an argument corresponds to the one of ‖{ }‖ in RS− F( ) according to Definition 3.
It is proved in [
In ASPIC+ the potential occurrence of an attack betwen two arguments (Definition 8) depends essentially on a relation of contrariness between the conclusion of the attacking argument and an attackable element (either the conclusion or the name of a defeasible rule or a defeasible premise) of (a subargument of) the attacked argument.
Three types of attack (undercut, rebut, and undermining) are identified, depending on the attackable element involved. Moreover, rebutting and undermining attacks are partitioned into two subclasses, depending on whether the relevant contrariness relation is symmetric or asymmetric (in the latter case the attack is called a contrary-attack).
It is worth noting5, that given two arguments and , Definition 8 does not per se guarantee that the three types of attack are mutually exclusive. In fact, it is formally possible, in specific situations, that more than one of the conditions establishing the occurrence of an attack hold at the same time for the same pair of arguments and .
In more detail, in order to have that undercuts and rebuts at the same time, the conclusion of should be a contrary both of a conclusion of a subargument of and of the name of a defeasible rule used in a subargument of , as shown in Example 1.
Example 1. Consider an argumentation system 1 = (ℒ,¯, ℛ, ) such that: ℒ = {, , , }; ∈ ; ∈ ; ℛ = ℛ = {1} where 1 = ⇒ ; (1) = . Consider then the following knowledge base in 1: = = {, }. We have then the following arguments: 1 = , 2 = , 3 = 1 ⇒ . It turns out that 2 both rebuts and undercuts 3
Similarly, in order to have that undercuts and undermines at the same time, the conclusion of should be a contrary both of an ordinary premise of and of the name of a defeasible rule used in a subargument of , as shown in Example 2 Example 2. Consider an argumentation system 2 = (ℒ,¯, ℛ, ) which is a variation of 1 with the only diference that ∈ instead of ∈ . Consider then the same knowledge base in 2: = = {, }. We have, similarly to above, the following arguments: 1 = , 2 = , 3 = 1 ⇒ . It turns out that 2 both undermines and undercuts 3.
Turning to a further case, in order to have that rebuts and undermines at the same time, the
conclusion of should be a contrary both of an ordinary premise of and of the conclusion of a
subargument of , as shown in Example 3
Example 3. Consider an argumentation system 3 = (ℒ,¯, ℛ, ) which is a variation of 1 with
the only diference that ∈ instead of ∈ . Consider then the same knowledge base in 3:
= = {, }. We have, similarly to above, the following arguments: 1 = , 2 = , 3 = 1 ⇒ .
It turns out that 2 both undermines and rebuts 3.
4More precisely, in [
Finally, note that, as is probably already evident, it is also possible to have a situation where an argument attacks another argument in all three possible ways.
Example 4. Consider an argumentation system 3 = (ℒ,¯, ℛ, ) which is a variation of 1 with the only diference that ∈ in addition to ∈ and ∈ . Consider then the same knowledge base in 4: = = {, }. We have, similarly to above, the following arguments: 1 = , 2 = , 3 = 1 ⇒ . It turns out that 2 undermines, rebuts, and undercuts 3.
In all the examples above, there is a language element that belongs to the contrariness function of at least two other elements. One may wonder whether this is a necessary condition for ℎ attacks to occur in ASPIC+, i.e., whether ℎ attacks may occur even under the constraint that each language element belongs to at most one contrariness function. The answer is positive, though rather peculiar situations must be considered, as shown by the following examples. Example 5. Consider an argumentation system 5 = (ℒ,¯, ℛ, ) such that: ℒ = {, , , }; ∈ ; ℛ = ℛ = {1} where 1 = , ⇒ ; (1) = . Consider then the following knowledge base in 1: = = {, , }. We have then the following arguments: 1 = , 2 = , 3 = , 4 = 1, 3 ⇒ . It turns out that 2 both undermines and undercuts 4.
Example 6. Consider an argumentation system 6 = (ℒ,¯, ℛ, ) such that: ℒ = {, , , , }; ∈ ; ℛ = ℛ = {1, 2} where 1 = , ⇒ and 2 = ⇒ ; (1) = . Consider then the following knowledge base in 1: = = {, , }. We have then the following arguments: 1 = , 2 = , 3 = , 4 = 3 ⇒ , 5 = 1, 4 ⇒ . It turns out that 2 both rebuts and undercuts 5.
For the sake of brevity and simplicity, the examples presented above are rather compact. More articulated examples, in particular involving longer rule chains, can easily be generated. Thus, it emerges that a variety of polymorphic attacks are actually possible in the context of ASPIC+.
Two main questions then arise. On the application side, one may wonder whether polymorphic attacks may occur in practice, i.e., if there are realistic reasoning contexts giving rise to them. On the technical side, one has, in any case, to consider whether they are problematic in any sense and how they are managed by the current definition of the formalism.
As to the first question, consider the following situation. Suppose you have a defeasible rule 1 stating that people wearing overalls are usually workers, and another defeasible rule 2 stating that workers are typically adults. Moreover you know that 1 is not applicable to children (since they may wear overalls in masquerade or for other fun reasons) and that children are not adults. Now you have a picture including a person who seems to be wearing overalls and to be a child: being a child is both a rebut and an undercut for the argument that the person is an adult, as shown in the example below. Example 7. Consider an argumentation system = (ℒ,¯, ℛ, ) defined as follows. ℒ = {Wo, Wk , Ch, Ad , N1 , N2 } where Wo means wears overalls, Wk means is a worker, Ch means is a child, Ad means is an adult, and N1 and N2 are symbols used for rule names. Then we assume Ch ∈ Ad , Ad ∈ Ch and Ch ∈ N1 . We consider a set of two defeasible rules: ℛ = ℛ = {1, 2} where 1 = Wo ⇒ Wk with (1) = 1 and 2 = Wk ⇒ Ad with (2) = N2 . We assume the following knowledge base in : = = {Wo, Ch}. We have then the following arguments: 1 = Wo, 2 = Ch, 3 = 1 ⇒ Wk , 4 = 3 ⇒ Ad . It emerges that 2 undercuts 3 and then 2 both rebuts 4 (on 4) and undercuts 4 (on 3). Moreover 4 undermines 2.
Example 7 provides an instance of polymorphic attack (rebut and undercut) arising from a realistic reasoning situation. While a systematic analysis of realistic examples (if any) giving rise to other combinations of types of attack is left to future work, we suggest that Example 7 is suficient to show that polymorphic attacks are more than a formal curiosity and should not be overlooked.
We turn, therefore, to consider the questions raised by polymorphic attacks from a technical and conceptual perspective.
Starting from the technical side, we remark first that Definition 8 is agnostic with respect to polymorphic attacks: it says that an argument attacks an argument if undercuts, rebuts, or undermines , and does not explicitly assume nor implicitly relies on the fact that the disjunction is exclusive.
Definition 9 then distinguishes preference-dependent and preference-independent attacks from an argument to an argument with reference to the subargument ′ to which the attack is directed. Thus, it turns out that an argument can attack another argument in a preference-dependent and in a preference-independent way at the same time (on diferent subarguments). For instance, in Example 7, 2 preference-dependent attacks 4 on 4 and preference-independent attack 4 on 3.
While this is a bit peculiar, it is, per se, not problematic for the notion of defeat introduced in Definition 9. It states that an argument defeats an argument if there is some ′ ∈ Sub( ) on which the attack is successful. As previously discussed, it may happen that: (i) there can be several such subarguments ′ and (ii) preferences may or may not play a role for the success of attacks concerning diferent subarguments. However the facts (i) and (ii) do not represent a problem: there will in any case a single defeat from to (possibly corresponding to more than one attack).
The defeat relation is then the basis for the construction of the argumentation framework in Definition 11, on which the assessment of the acceptability of arguments is based. Thus, technically speaking, polymorphic attacks do not pose problems nor require a revision of the relevant definitions in ASPIC+.
We argue, however, that from the conceptual point of view, some issues arise. As remarked above, Definition 9 admits that an argument can both preference-independent and preference-dependent attack another argument since the quality of being preference-(in)dependent depends on the pair (, ′), where ′ is the subargument of which is attacked, rather than on the pair (, ).
Let us refer to Example 7. The undercutting attacks from 2 to 3 and from 2 to 4 are, of course, preference-independent, while the rebutting attacks from 2 to 4 and vice versa are preferencedependent. According to Definition 9, 2 then defeats both 3 and 4 independently of any preference while 4 defeats 2 only if 4 ⊀ 2. Assuming 4 ⊀ 2, we get that in the argumentation framework ℱ prescribed by Definition 11 for Example 7 (see Figure 1) there is a formally symmetric defeat between 4 and 2 whose branches are however conceptually asymmetric: one branch has a polymorphic origin, while the other arises from a rebutting attack.
Let us examine the outcomes of the evaluation of the arguments in ℱ according to standard semantics. The grounded extension consists of the unique unattacked argument 1 (GR(ℱ) = { 1}), while in the case of preferred, stable, and semi-stable semantics, we have two extensions: ℰPR(ℱ) = ℰST(ℱ) = ℰSST(ℱ) = {{ 1, 2}, { 1, 3, 4}}.
In words, 1 is the only argument skeptically accepted with any semantics, while 2 (namely the evidence that the person is a child) is regarded as dubious, due to the mutual attack with 4. It is interesting to note that 4 defends its own subargument 3 from the undercutting attack coming from 2. This is a bit peculiar: a consequence that the undercutting attack is meant to prevent to draw is used to rebut the source of the undercutting attack itself.
Whether this is conceptually legitimate or problematic can be regarded as a matter of discussion, which we leave to future work. In this paper, we limit ourselves to suggesting that further investigation on polymorphic attacks in ASPIC+ is worth pursuing and that it may stimulate the study of alternative notions of defeat in ASPIC+. In particular, one might consider a revision of Definition 9 where the fact that an attack whose source is an argument gives rise to a successful defeat takes into account the possible existence of polymorphic attacks involving proper subarguments of , which, in a sense, might override the self-defence capability of against them.
In the context of ASPIC, the notions of undercutting, rebutting, and undermining attacks refer to the relation between a set of arguments, in the role of attacker, and an individual argument, in the role of attackee, as specified in Definition 16. Essentially, the starting point for the identification of attacks is the extended contrariness relation EC(), which consists of pairs of sets of language elements, say pairs (, ) with , ⊆ ℒ . Intuitively, each such pair indicates that the elements in ∪ cannot stay together and, more precisely, indicates that if all the elements in hold, then an least one element of cannot hold. It follows that if there are arguments such that all their conclusion or names of top rule coincide with the set ∪ , one of the arguments whose conclusion or name of top rule is included in cannot be accepted if all the others are. The three kinds of attacks realize this idea in diferent ways depending on the role of the language element involved and on the structure of the attacked argument. Thus: • a set 6 undercuts an argument on a subargument ′ when ′ has a defeasible top rule whose name is included in ; • a set rebuts an argument on a subargument ′ when ′ has a defeasible top rule and its conclusion is included in ; • a set undermines an argument on a subargument ′ when ′ is an ordinary premise which is included in .
Concerning the occurrence of polymorphic attacks in ASPIC, it is first of all interesting to examine what happens in the examples introduced in Section 4 for ASPIC+.
As a general observation, with reference to Definition 15 we can remark that since the examples do not contain any strict rule in all cases EC2() and EC3() do not contain any additional element with respect to EC1(). i.e. EC* () = EC1(). Moreover, since the examples do not contain any axiom, for any set of arguments it follows that ̂︀ = . We get then that EC() = EC* () = EC1() and by Proposition 1 EC() = EC1(). In words this amounts to the fact that, in the examples, the extended contrariness function EC() of ASPIC essentially coincides with the contrariness function of ASPIC+, the only diference being that the pair of language elements are replaced by the pairs of the corresponding singletons. For instance, in Example 1 ∈ becomes ({}, { }) ∈ EC() (and similarly for ∈ and for all the pairs in the other examples).
Since every pair in EC() consists of two singletons, in turn the attacks prescribed by Definition 16 have necessarily a singleton as a source, and skipping some relatively straightforward steps, we reach the conclusion that all the examples introduced in Section 4 are treated by ASPIC essentially in the same way as in ASPIC+, since (modulo the replacement of individual arguments with the corresponding singletons) the attacks are the same and hence the argumentation frameworks prescribed by Definitions 11 and 22 are, in these examples, isomorphic.
In summary, all the considerations on polymorphic attacks drawn in Section 4 are still valid for ASPIC.
In particular, it is worth remarking that Definition 16, like Definition 8 is agnostic with respect to polymorphic attacks and does not explicitly assume nor implicitly relies on the fact that the disjunction of the kinds of attacks is exclusive.
Definition 18, similarly to Definition 9, is compatible with the fact that a set of arguments attacks an argument both in a preference-independent and in a preferemce-dependent way and the notion of defeat simply requires that at least one successful attack exists, not excluding that there are several. Subsequent definitions from 19 to 22 are not afected by polymorphic attacks.
While the discussion above suggests a substantial similarity between ASPIC+ and ASPIC concerning polymorphic attacks, it is worth noting that extending the consideration to examples where strict rules are present, ASPIC raises further questions on polymorphic attacks. To see this, consider the following example. 6For brevity, we do not repeat here the condition of coverage of ∪ that the set , together with the attacked argument, must satisfy.
Example 8. Consider an argumentation system = (ℒ,¯, ℛ, ) such that: ℒ = {, ¬, , , , , }; ∈ ¬ ; ¬ ∈ ; ∈ ; ℛ = ℛ = {1, 2, 3, 4} where 1 = → ; 2 = , → ¬ ; 3 = → ; 4 = , → . Consider then the following knowledge base in : = = {, , }. We have then the following arguments: 1 = ; 2 = ; 3 = ; 4 = 1 → ; 5 = 2, 3 → ¬ ; 6 = 2 → ; 7 = 1, 3 → . According to Definition 15 we get: • EC1() = {({ }, {¬ }), ({¬ }, { }), ({ }, { })}; • EC2() ∖ EC1() = {({}, {¬ }), ({, }, { }), ({ }, { })}; • EC3() ∖ (EC1() ∪ EC2()) = {({¬ }, {}), ({, }, {}), ({ }, {, }), ({}, {, }), ({ }, {, }), ({ }, {, })}.
Given that = ∅ and that all pairs listed above respect the minimality condition specified in Definition 15 we get that the set EC() consists exactly of these pairs.
Let us now focus on the attacks involving ultimately fallible arguments (namely 1, 2, 3) which, in virtue of Definitions 16, 20, 21, are the basis for the construction of the argumentation framework RS− F( ) (Definition 22).
In particular, we note that there are two distinct conditions entailing that the set { 2, 3} undermines argument 1 (on 1) according to Definition 16. The first condition is the presence in RS− F( ) of the pair ({, }, {}), the second condition is the presence of the pair ({ }, {, }).
The two conditions difer concerning the kind of undermining: in the first case we have that also ({}, {, }) ∈ RS− F( ), while in the second case ({ }, {, }) ∈/ RS− F( ). It follows that the first condition corresponds to a preference-dependent undermining attack, while the second condition corresponds to a preference-independent contrary-undermining attack.
Definition 16 does not consider explicitly the possible occurrence of situations like the one presented in Example 8, however it implicitly gives the priority to contrary-undermining (and contrary-rebutting) attacks: if there is a pair (, ) ∈ RS− F( ) satisfying the specified conditions and (, ) ∈/ RS− F( ) then the attack is of the contrary kind, and hence is preference-independent. This is not afected by the possible existence of other pairs ( ′, ′) ∈ RS− F( ) still satisfying the required conditions and such that also ( ′, ′) ∈ RS− F( ).
At a general level, this appears a sort of conservative choice: it prevents the possible preferencedependent removal of the attacks whose nature is not univocal. In turn, given that attacks are motivated by the contrariness relation, this choice can be regarded as favoring the respect of consistency properties.
There are however also potential downsides of conservativeness: on one hand, it seems to demote the role of preferences, on the other hand, it tends to increase undecidedness, since it can preserve some mutual attacks (which intuitively correspond to an undetermined conflict) in situations where they could become a unidirectional attack (corresponding to a more determined conflict situation, where one element dominates the other).
In general, it emerges that polymorphic attacks pose more questions, but possibly also more opportunities, in ASPIC than in ASPIC+, for the reasons discussed in the following.
We have already remarked that all the examples of polymorphic attacks presented in Section 4 for ASPIC+, not involving strict rules nor axioms, have a direct correspondence in ASPIC. In the presence of strict rules, the significant diferences between ASPIC+ and ASPIC emerge.
First of all, it must be recalled that, in order to ensure the satisfaction of the rationality postulates,
ASPIC+ imposes several constraints limiting its expressiveness. In particular, Example 8, which is fully
legitimate and manageable in ASPIC, cannot be correctly handled and is simply forbidden in ASPIC+.
The reason is that the language element has a contrary (namely ) and is the consequent of a strict
rule (namely 4). This is not allowed in ASPIC+ since it violates the condition of well formedness (fifth
item of Definition 12 in [
Thus, it turns out that the larger variety of polymorphic attacks in ASPIC is also related to its greater expressiveness.
Another point to be remarked, concerns the diferent nature of attacks in ASPIC, with respect to ASPIC+, when strict rules are involved. Both in ASPIC+ and in ASPIC the so-called restricted rebut is adopted, namely attacks can be directed only towards ordinary premises or arguments whose last rule is defeasible and then on their superarguments (see Definitions 8 and 16).
The use of restricted rebut is crucial to ensure some desirable properties (see [
The, so to speak, ‘recovery strategy’ in ASPIC+ is based on ‘inverting’ strict rules so as to obtain additional arguments which can attack other arguments as needed. For this reason, in ASPIC+ the requirement of closure under transposition of the set of strict rules, recalled in Definition 5, has been introduced8. This approach is based on the traditional notion of binary attack between arguments.
ASPIC adopts a radically diferent ‘recovery strategy’, which, in a nutshell, aims at identifying the sets of ultimately fallible arguments which are the roots of the occurring inconsistencies. The fact that these arguments are incompatible is captured through an innovative notion of attack at the level of sets.
A side efect of the strategy illustrated above is that the same ultimately fallible arguments can be identified as the roots of distinct inconsistencies. This is exactly what happens in Example 8: the arguments 1, 2, and 3 are at the basis both of the inconsistency between 4 and 5 and of the one between 6 and 7. The diferent nature of these inconsistencies (roughly speaking, one is symmetric while the other is not) leads to the existence of a polymorphic attack.
It is then evident that, at least in principle, the cases of polymorphic attacks one can conceive are virtually unlimited, given that nothing prevents having very diferent derivations based on the same premises. It is also rather evident that most of these cases will hardly have a meaningful counterpart in practice: while, as already discussed, the notion of polymorphic attacks is not just a technical curiosity, it has also to be acknowledged that the practical interest of its instances is likely to decrease with the increase of their complexity.
After having evidenced and discussed the phenomenon of polymorphic attacks, in this section we carry out a preliminary discussion on possible variations of the ASPIC+ and ASPIC formalisms oriented to the treatment of polymorphic attacks.
Starting from ASPIC+, the first point of possible revision would be Definition 8. Here the key
point is that the notion of attack is equated to the disjunction of three conditions (undercut, rebut
or undermining). This implicitly prevents to distinguish the occurrence of one condition from the
occurrence of more of them, i.e. somehow ‘hides’ or ‘flattens’ the possible existence of polymorphic
attacks. To avoid this flattening, one could simply avoid, at the beginning, the reference to the common
notion of attack and simply introduce all the types of attack relation, with a more explicit emphasis
on the role of the subargument ′, i.e., as ternary relations. In the following (as in [
Definition 23. Let S be the set of arguments on the basis of an argumentation theory . We define the
following relations on S × S × S. Given arguments , , ′ ∈ S :
is an axiom while is an ordinary premise, according to ASPIC+ there is no attack between them and both would be
accepted, leading to an inconsistency.
8In alternative, the requirement of closure under contraposition has been considered. It has not been recalled here as it is not
constructive and not necessary for the present paper. The interested reader may refer to Definition 12 of [
Then, to explicitly take into account the possible occurrence of polymorphic attacks, one can introduce the notion of attack set for a pair of arguments.
Definition 24. The set ℛ of attack relations for ASPIC+ is defined as ℛ = {, ℛ, ℳ, ℛ, ℳ}. Given a pair of arguments (, ) the set of attacks from to denoted as (, ) is defined as (, ) = {(, , ′, ) | ∈ ℛ and (, , ′) ∈ }. Given a set of arguments A, the set of all the attacks relevant to A is defined as Ω(A) = ⋃︀, ∈A (, ).
For instance, in Example 7, ( 2, 4) = {( 2, 4, 4, ℛ), ( 2, 4, 3, )}, while ( 4, 2) = {( 4, 2, 2, ℳ)}. Moreover, ( 2, 3) = {( 2, 3, 3, )} and, as an example of the absence of any attack, ( 1, 3) = ∅.
Then, in place of Definition 9, one can introduce the notion of a function for deriving the defeat relation from the set of attack tuples.
Definition 25. A defeat generation function for ASPIC+ ℱ is a function which, given a set of arguments A, its relevant set of attacks Ω(A) and a preorder on A, returns a binary relation on A.
Note that we do not make any assumption on the defeat generation function, leaving open, at this stage, all possible alternatives for its design.
Definitions 10 and 11 can then be modified as follows.
Definition 26. Let = (, ) be an argumentation theory. A structured argumentation framework ( ), defined by is a triple (S, Ω(S), ⪯ ) where S is the set of all finite arguments constructed from in (called the set of arguments on the basis of ), Ω(A) is the set of all relevant attacks according to Definition 24, and ⪯ is an ordering on S.
Definition 27. Let Δ = (S, Ω(A), ⪯ ) be a and ℱ be a defeat generation function. The corresponding to Δ is defined as ℱΔ = (S, ℱ (S, Ω(S), ⪯ )).
In particular, the defeat generation function corresponding to the traditional version of ASPIC+, denoted as ℱ A+ , can be characterized as follows.
Definition 28. Given a set of arguments A, a relevant set of attacks Ω(A) and a preorder ⪯ on A, the defeat generation function ℱ A+ is defined as follows: (, ) ∈ ℱ A+ (A, Ω(A), ⪯ ) if ∃(, , ′, ) ∈ Ω(A) such that ∈ {, ℛ, ℳ} or ∃(, , ′, ) ∈ Ω(A) such that ∈ {ℛ, ℳ} and ⊀ ′.
The parametric scheme introduced above easily accommodates alternative treatments of polymorphic attacks, by varying the function ℱ . We leave this investigation to future work, remarking that, in particular, it would be interesting to identify properties of the defeat generation function which provide necessary or suficient conditions for the satisfaction of the rationality postulates. Ideally, one could achieve the characterization of a family of defeat generation functions, such that rationality postulates are satisfied for every choice of a function ℱ in that family.
Turning to ASPIC, a modification along the same lines can be devised.
First, we revise Definition 16 analogously to Definition 23.
Definition 29. Given an argumentation theory = (, ), let S be the set of arguments on the basis of . We define the following relations on 2S × S × S. Given a set of arguments ⊆ S and arguments , ′ ∈ S: • undercuts on ′ if ′ ∈ Sub( ) is such that = Top( ′) ∈ ℛ and the following condition holds: ∃, such that ∪ = Conc() ∪ {()}, (, ) ∈ EC() and () ∈ . This is denoted as (, , ′) ∈ . • (plain-)rebuts on ′ if ′ ∈ Sub( ) has the form 1′′, . . . , ′′ ⇒ and the following condition holds: ∃, such that ∪ = Conc() ∪ { }, (, ) ∈ EC() and ∈ and (, ) ∈ EC(). This is denoted as (, , ′) ∈ ℛ. • contrary-rebuts on ′ if ′ ∈ Sub( ) has the form 1′′, . . . , ′′ ⇒ and the following condition holds: ∃, such that ∪ = Conc() ∪ { }, (, ) ∈ EC() and ∈ and (, ) ∈/ EC(). This is denoted as (, , ′) ∈ ℛ. • (plain-)undermines on ′ if ′ = for some ∈ Prem( ) and the following condition holds: ∃, such that ∪ = Conc() ∪ { }, (, ) ∈ EC() and ∈ and (, ) ∈ EC().
This is denoted as (, , ′) ∈ ℳ. • contrary-undermines on ′ if ′ = for some ∈ Prem( ) and the following condition holds: ∃, such that ∪ = Conc() ∪ { }, (, ) ∈ EC() and ∈ and (, ) ∈/ EC(). This is denoted as (, , ′) ∈ ℳ.
On this basis, the notion of attack set for ASPIC can be introduced.
Definition 30. The set ℛ of attack relations for ASPIC is defined as ℛ = {, ℛ, ℳ, ℛ, ℳ}. Given a pair (, ) where ⊆ S and ∈ S the set of attacks from to denoted as (, ) is defined as (, ) = {(, , ′, ) | ∈ ℛ and (, , ′) ∈ }. Given a set of arguments A, the set of all the attacks relevant to A is defined as Ω(A) = ⋃︀⊆ A, ∈A (, ).
Then, in place of Definition 18, one can introduce the notion of a function for deriving the defeat relation from the set of attack tuples.
Definition 31. A defeat generation function for ASPIC ℱ is a function which, given a set of arguments A, its relevant set of attacks Ω(A) and a preorder on A, returns a binary relation on 2A × A, denoted as ⇝ .
Definitions 20, 21, and 22 are then unchanged, under the assumption that a defeat generation function is used to generate the defeat relation ⇝ .
Similarly to the case of ASPIC+, this generalization paves the way to several possible developments which are left to future work. In particular, also for ASPIC, it will be very interesting to investigate general properties of the defeat generation function ensuring the satisfaction of rationality postulates.
In this paper we have carried out a preliminary analysis concerning polymorphic attacks, namely the occurrence of multiple attacks of diferent kinds between the same pair of entities in the context of the ASPIC+ and ASPIC formalisms for structured argumentation. As to our knowledge, this peculiar phenomenon has not been considered in previous literature. We presented and discussed several examples of its occurrence, showing in particular that there are realistic reasoning cases where it may occur and that the increased expressiveness of ASPIC with respect to ASPIC+ also gives rise to a richer variety of polymorphic attacks.
From a technical viewpoint, we have commented how polymorphic attacks are (in a sense, implicitly) treated in the current versions of ASPIC+ and ASPIC and suggested that dealing with them explicitly may lead to considering alternative options for handling them. Towards this investigation direction, we presented a preliminary proposal of suitable revisions of some definitions of ASPIC+ and ASPIC in order to make the notion of defeat relation parametric with respect to a defeat generation function.
In addition to those already discussed in previous sections, among the directions of future work we mention a broader analysis of the potential importance of polymorphic attacks in practical reasoning contexts and the study of the possible relationships with formalisms where attacks are associated with a numerical weight, like for instance in weighted argument systems [15].
This work was supported by MUR project PRIN 2022 EPICA ‘Enhancing Public Interest Communication with Argumentation’ (CUP D53D23008860006) funded by the European Union - Next Generation EU, mission 4, component 2, investment 1.1.
[15] P. E. Dunne, A. Hunter, P. McBurney, S. Parsons, M. J. Wooldridge, Weighted argument systems: Basic definitions, algorithms, and complexity results, Artif. Intell. 175 (2011) 457–486.