The phenomenon of spurious preferences in argumentation can be described as an unjustified unequal treatment of some arguments emerging in a context where equal treatment would be expected. Using the ASPIC+ formalism as a basis, we provide an emblematic example of spurious preference and introduce a basic requirement of spurious preference avoidance for a suitable family of argumentation theories. We then show that a variant of ASPIC+, introduced to deal with problems concerning multiple contradictories, satisfies this requirement.
and ↠ ⊆ (A ×
An argumentation framework (AF) is a pair F = ⟨A, ↠ ⟩, where A is a set of arguments A) is a binary relation on A.
When (α, β ) ∈↠ (also denoted as α ↠ β ) we say that α attacks β . For a set X ⊆ A and an
argument α ∈ A we write α ↠ X if ∃β ∈ X : α ↠ β and X ↠ α if ∃β ∈ X : β ↠ α , and we
denote the arguments attacking X as X − ≜ {α ∈ A | α ↠ X } and the arguments attacked by X as
X + ≜ {α ∈ A | X ↠ α }. An extension-based argumentation semantics σ 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 σ , the set of extensions prescribed
by σ for a given AF F is denoted as Eσ (F ). Several argumentation semantics are recalled in Definition
2, along with some basic underlying notions. For more details, the reader is referred to [
Definition 2. Let F = ⟨A, ↠ ⟩ be an AF, α ∈ A and X ⊆ A. X is conflict-free , denoted as X ∈ ECF(F ), if X ∩ X− = ∅. α is acceptable with respect to X (or α is defended by X) if {α }− ⊆ X+. The function FF : 2A → 2A which, given a set X ⊆ A, returns the set of the acceptable arguments with respect to X, is called the characteristic function of F . X is admissible (denoted as X ∈ EAD(F )) if X ∈ ECF(F ) and X ⊆ FF (X). X is a complete extension (denoted as X ∈ ECO(F )) if X ∈ ECF(F ) and X = FF (X). X is the grounded extension (denoted as X = GR(F ) or X ∈ EGR(F )) if X is the least fixed point of FF (equivalently, the least complete extension). X is a preferred extension (denoted as X ∈ EPR(F )) if X is a maximal (with respect to set inclusion) admissible set. X is a stable extension (denoted as X ∈ EST(F )) if X+ = A \ X. X is a semi-stable extension (denoted as X ∈ ESST(F )) if it is a complete extension such that X ∪ X+ is maximal (wrt ⊆ ) among all complete extensions.
Argument justification status is defined on the basis of extension membership.
Definition 3. Given a set S a justification labeling of S is a function J : S → ΣJ , where ΣJ = {Sk, Cr, N o}. Given an AF F = ⟨A, ↠ ⟩ and a semantics σ , the justification labeling of A according to σ is defined as follows 1: J (α ) = Sk if α ∈ TE∈Eσ (F) E; J (α ) = Cr if α ∈ SE∈Eσ (F) E and α /∈ TE∈Eσ (F) E; J (α ) = N o if α /∈ SE∈Eσ (F) E.
In words, we will say respectively that α is skeptically justified, credulously justified, and not justified. We now recall the essential notions of the ASPIC+ formalism.
An argumentation system is a tuple AS = (L,¯, R, n) where: 1. L is a logical language 2. ¯ is a contrariness function from L to 2L such that: (i) φ is a contrary of ψ if φ ∈ ψ , ψ /∈ φ; (ii) φ is a contradictory of ψ (denoted by φ = − ψ ) if φ ∈ ψ , ψ ∈ φ; (iii) each φ ∈ L has at least one contradictory 3. R = (RS , RD) is a pair of sets of strict (RS ) and defeasible (RD) inference rules of the form φ1, . . . , φn → φ and φ1, . . . , φn ⇒ φ respectively (where φi, φ are meta-variables ranging over wf in L), and RS ∩ RD = ∅ 4. n : RD → L is a naming convention for RD.
In the following, given a set S ⊆ L with a little abuse of notation we will denote the set of its contraries and contradictories as S = Sφ∈S{ψ | ψ ∈ φ}. Given a rule r = φ1, . . . , φn → (⇒)φ, we will say that φ is the consequent of the rule, denoted as cons(r) and that {φ1, . . . , φn} is the set of the antecedents of the rule denoted as ant (r).
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 [
Definition 5. Given an argumentation system AS = (L,¯, R, n), the set of strict rules RS is closed under transposition if if φ1, . . . , φn → ψ ∈ RS then, for i = 1 . . . n, φ1, . . . , φi− 1, − ψ, φ i+1, . . . , φn → − φi ∈ RS . Given a set of strict rules RS , its closure under transposition is defined as Cltr(RS ) ≜ RS ∪ Sr∈RS tr(r), where for any r = φ1, . . . , φn → ψ , tr(r) = Si=1...n{φ1, . . . , φi− 1, − ψ, φ i+1, . . . , φn → − φi}.
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 base using rules. 1We avoid reference to F and σ in J 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.
Definition 6. A knowledge base in an argumentation system AS = (L,¯, R, n) is a set K ⊆ L consisting of two disjoint subsets Kn (the axioms) and Kp (the ordinary premises). The tuple AT = (AS, K) is called an argumentation theory.
is:
An argument α on the basis of a knowledge base K in an argumentation system (L,¯, R, n) 1. φ if φ ∈ K with: Prem(α ) = {φ}; Conc(α ) = φ; Sub(α ) = {φ}; Rules(α ) = ∅; Top(α ) = undefined. 2. α 1, . . . , α n → (⇒) ψ if α 1, . . . , α n are arguments such that there exists a strict (defeasible) rule Conc(α 1), . . . , Conc(α n) → (⇒) ψ in RS (RD) with: Prem(α ) = Prem(α 1) ∪ . . . ∪ Prem(α n); Conc(α ) = ψ ; Sub(α ) = Sub(α 1) ∪ . . . ∪ Sub(α n) ∪ {α }; Rules(α ) = Rules(α 1) ∪ . . . ∪ Rules(α n) ∪ {Conc(α 1), . . . , Conc(α n) → (⇒) ψ }; Top(α ) = Conc(α 1), . . . , Conc(α n) → (⇒) ψ ; DefRules(α ) = {r | r ∈ Rules(α ) ∩ RD}; StRules(α ) = {r | r ∈ Rules(α ) ∩ RS }.
For any argument α , Premn(α ) = Prem(α ) ∩ Kn; Premp(α ) = Prem(α ) ∩ Kp. α is: strict if DefRules(α ) = ∅, defeasible if DefRules(α ) ̸= ∅; firm if Prem(α ) ⊆ K n; plausible if Prem(α ) ⊈ Kn; ifnite if Rules(α ) is finite.
Notation 1. Some further notations are useful. Given S ⊆ L , S ⊢ φ denotes that there exists a strict argument α such that Conc(α ) = φ, with Prem(α ) ⊆ S. S ⊢min φ denotes that S ⊢ φ and ∄T ⊊ S : T ⊢ φ. Given a set of arguments X, Prem(X) ≜ Sα ∈X Prem(α ), and similarly for Conc(X), Sub(X), Rules(X), Top(X), DefRules(X), StRules(X).
Three kinds of attack between arguments are considered.
Definition 8. An argument α attacks an argument β if α undercuts, rebuts, or undermines β where: α undercuts β (on β ′) if Conc(α ) ∈ n(r) for some β ′ ∈ Sub(β ) such that r = Top(β ′) is defeasible. α rebuts β (on β ′) if Conc(α ) ∈ φ for some β ′ ∈ Sub(β ) of the form β 1′′, . . . , β n′′ ⇒ φ. In such a case α contrary-rebuts β if Conc(α ) is a contrary of φ. α undermines β (on β ′) if Conc(α ) ∈ φ for some β ′ = φ, φ ∈ Premp(β ). In such a case α contrary-undermines β if Conc(α ) is a contrary of φ.
In some cases, attack efectiveness depends on a preference ordering ⪯ over arguments (assumed to
be a preorder as in [
Definition 9. Let α attack β on β ′. If α undercuts, contrary-rebuts, or contrary-undermines β on β ′, then α preference-independent attacks β on β ′, otherwise α preference-dependent attacks β on β ′. Then, α defeats β if for some β ′ either α preference-independent attacks β on β ′ or α preference-dependent attacks β on β ′ and α ⊀ β ′.
Then a structured argumentation framework (SAF ) can be defined from an argumentation theory, 2 using the attack relation. Using the defeat relation, an argumentation framework is then derived from a SAF .
Definition 10. Let AT = (AS, K) be an argumentation theory. A structured argumentation framework (SAF ), defined by AT is a triple (S, C, ⪯ ) where S is the set of all finite arguments constructed from K in AS (called the set of arguments on the basis of AT ), C ⊆ S × S is such that (α, β ) ∈ C if α attacks β , and ⪯ is an ordering on S.
Definition 11. Let Δ = (S, C, ⪯ ) be a SAF , and D ⊆ S × S be the defeat relation according to Definition 9. The AF corresponding to Δ is defined as FΔ = (S, D).
Given an argumentation semantics σ , the justification status of arguments in S according to σ is
determined by the set of extensions Eσ (FΔ) according to Definition 3.
2We do not consider here the alternative notion of c-structured argumentation framework in [
A8 A1
A10
A11 A12 A6 A9
A7
A2 A5
To illustrate the phenomenon of spurious preferences, let us consider the following example (where we assume a simple language consisting just of a set of symbols and their negation). You have some uncertain evidence about the birthdate (b), birthplace (p) and domicile (d) of a person. From the known birthdate, it can be derived with certainty that the person is over 18 (m), and from the known birthplace it can be derived with certainty that the person is a citizen of a given country (c) (because ius soli holds in the country where the birthplace is located). Finally, from the domicile, age majority, and citizenship, it can be derived that the person must be included in the taxpayers’ list (ω). However, it turns out with certainty that the person is not included in the list (¬ω). Thus, some of the uncertain evidence has to be rejected. In the absence of any preference among them, the three pieces of evidence are equal candidates to be retracted, and this is, in fact, what happens with the formalization of the example that we describe below.
The three evidences are represented by ordinary premises Kp = {d, b, p} while the certain fact is represented by an axiom Kn = {¬ω} and the domain knowledge is represented through strict rules RSb = {b → m; p → c; d, c, m → ω}.
Closure under transposition (which is required to ensure the satisfaction of rationality postulates) leads to consider the following additional set of strict rules: RSt = {¬m → ¬b; ¬c → ¬p; d, c, ¬ω → ¬m; d, m, ¬ω → ¬c; c, m, ¬ω → ¬d}, thus RS = RSb ∪ RSt.
The following arguments are then built: A1 = d; A2 = b; A3 = p; A4 = A3 → c; A5 = A2 → m; A6 = A1, A4, A5 → ω; A7 = ¬ω; A8 = A1, A4, A7 → ¬m; A9 = A1, A5, A7 → ¬c; A10 = A4, A5, A7 → ¬d; A11 = A8 → ¬b; A12 = A9 → ¬p.
Since no defeasible rules are involved, all attacks have the form of undermining. Focusing on ordinary premises first, we have that: A1 is undermined by A10, A2 is undermined by A11 and A3 is undermined by A12. Then, considering the use of ordinary premises in the construction of other arguments, the following attacks also occur: A10 undermines A6, A8, A9, A11, A12; A11 undermines A5, A6, A9, A10, A12; A12 undermines A4, A6, A8, A10, A11. The relevant argumentation framework is shown in Figure 1.
It can be then observed that the core of the framework consists of the three arguments A10, A11, A12, which are the only sources of attacks and mutually attack each other. As a consequence, it can be seen that in the case of stable, preferred, or semi-stable semantics, we get the same three extensions: E1 = A12 A10 A1 A6 A8
A13 A11
A5
A2 {A2, A3, A4, A5, A7, A10}; E2 = {A1, A3, A4, A7, A8, A11}; and E3 = {A1, A2, A5, A7, A9, A12}.
Each of these extensions corresponds to the rejection of one of the three ordinary premises (A1, A2, and A3, respectively) corresponding to the three uncertain pieces of evidence, which reflects the absence of any preference among them: according to Definition 3 any of them is credulously justified, while none of them is skeptically justified.
Consider now a slight variation of the above example, namely a country where ius soli has been introduced at a certain date (which we assume to be before the known birthdate of the person). In this case, the rule that derives citizenship will use both the birthplace and the birthdate as premises. Intuitively, this small variation in the structure of the strict rules should not afect the result: the three uncertain pieces of evidence should still be regarded to be equal candidates for rejection, given that, together, they give rise to a contradiction with a certain fact and that there is no preference among them. Somehow surprisingly, this is not the case, as illustrated below.
With respect to the representation of the original example, everything remains the same with the exception of the strict rule p → c, which becomes b, p → c. This gives rise to RSb = {b → m; b, p → ′ c; d, c, m → ω}.
Closure under transposition of R′Sb gives rise to R′St = {¬m → ¬b; p, ¬c → ¬b; b, ¬c → ¬p; d, c, ¬ω → ¬m; d, m, ¬ω → ¬c; c, m, ¬ω → ¬d}, and the overall set of strict rules is given by R′S = RSb ∪ RSt
′ ′ .
The following arguments are then built (for simplicity of notation, we use the same names as above since the distinction is clear from the context): A1 = d; A2 = b; A3 = p; A4 = A2, A3 → c; A5 = A2 → m; A6 = A1, A4, A5 → ω; A7 = ¬ω; A8 = A1, A4, A7 → ¬m; A9 = A1, A5, A7 → ¬c; A10 = A4, A5, A7 → ¬d; A11 = A8 → ¬b; A12 = A2, A9 → ¬p; A13 = A3, A9 → ¬b.
The attacks occurring directly on ordinary premises are the following: A1 is undermined by A10; A2 is undermined by A11 and A13; A3 is undermined by A12. Then, according to the use of the premises, the following attacks also occur: A10 undermines A6, A8, A9, A11, A12, A13; A11 and A13 undermine A4, A5, A6, A8, A9, A10, A11, A12, A13; finally A12 undermines A4, A6, A8,A10, A11, A13.
The relevant argumentation framework is shown in Figure 2.
The core of the resulting argumentation framework consists of the four arguments A10, A11, A12, and A13 which are the only sources of attacks. The four arguments are all mutually attacking each other, moreover A11 and A13 are self defeating. It turns out that according to the preferred, stable and semi-stable semantics, there are two extensions: one where A10 is accepted while A11, A12, and A13 are rejected, and one where A12 is accepted while A10, A11, and A13 are rejected. Taking into account the attack relations involving other arguments it turns out that the two extensions are as follows: E1 = {A2, A3, A4, A5, A7, A10} and E2 = {A1, A2, A5, A7, A9, A12}.
It turns out that, focusing on the ordinary premises, A1 and A3 are alternatively rejected, while A2 is always accepted, which corresponds to a sort of implicit preference for argument A2, which turns out to be skeptically justified, with respect to arguments A1 and A3, which are credulously justified. This implicit preference can be regarded as an accidental side efect of the structure of the set of strict rules, and has therefore a rather dubious conceptual status. For this reason, we call this implicit preference spurious.
Indeed, the fact that the information on birthdate is involved in two intermediate reasoning steps turns out to give it a privileged status that appears to be accidental and unjustified. We regard spurious preferences as an interesting but undesired behavior, and we develop a relevant preliminary investigation in the next sections.
A broad analysis of spurious preferences and the cases where they may arise appears to be a challenging task that is beyond the limits of this paper. As a preliminary contribution in this direction, in this section we define a simple reference context, which includes the motivating example as an instance, where it is possible to provide a formal notion of spurious preference. Accordingly, we will formulate a requirement of spurious preference avoidance, which is meant to support the comparison of diferent formalisms in situations like the one described in Section 3.
To this purpose, we resort to a notion of core argumentation theory, namely an argumentation theory which contains the essential elements for the representation of a reasoning case but may need to be completed with other elements, derivable from the core, before being used for the construction of arguments and their assessment. The idea is that diferent completions of the same core may behave diferently with respect to spurious preference avoidance. In particular, we refer to a family of core argumentation theories called SSDOP (Simple Strict Derivation from Ordinary Premises), defined as follows.
Definition 12. Let AS = (L,¯, R, n) be an argumentation system and K a knowledge base in AS. An argumentation theory AT = (AS, K) is said to be an instance of the SSDOP family if the following conditions hold: • the language L consists of the closure of a given set Σ of symbols and their negation, namely
L = Σ ∪ {¬s | s ∈ Σ}; • the contrariness function coincides with the classical notion of negation: for every s ∈ Σ, s = {¬s} and ¬s = { }
s ; • R = (RSb, ∅), namely the set of defeasible rules is empty; • ∄r, r′ ∈ RSb : cons(r) ∈ cons(r′), namely no contradiction can be derived using the strict rules only; • Kn = {¬ω} for some ω ∈ Σ that will be called contradiction focus; • ∀r ∈ RSb, ant (r) ∩ {ω, ¬ω} = ∅; • the set of ordinary premises Kp satisfies the following conditions – |Kp| ≥ 2; – Kp ∩ {ω, ¬ω} = ∅; – ∄p, p′ ∈ Kp : p ∈ p′; – ∄r ∈ RSb : cons(r) ∈ Kp ∪ Kp – there is an argument α such that Prem(α ) = Kp, Conc(α ) = ω, and there is no argument α ′ such that Prem(α ′) ⊊ Kp, Conc(α ′) = ω; – for every p1, p2 ∈ Kp, p1 ≃ p2.
It is easy to see that the examples presented in Section 3 are SSDOP instances. The idea is to encompass cases where a set Kp of ordinary premises gives rise, through strict rules, to a contradiction with a certain fact, and all ordinary premises are equal candidates to be rejected to avoid this contradiction. The assumptions on Kp are meant to ensure multiple choices (|Kp| ≥ 2) and that there are no other possible reasons to discard some elements of Kp.
In a SSDOP instance, the main focus of the evaluation is represented by the ordinary premises, as they are the only defeasible elements. To capture this central aspect and to abstract away formal aspects which are specific of diferent variants of ASPIC+ to be compared, we introduce a generic notion of evaluation mechanism, incorporating all steps leading from a SSDOP instance to the evaluation of the justification status of the premises.
Definition 13. An SSDOP evaluation mechanism E is a function which, given a SSDOP instance AT with ordinary premises Kp, returns a justification labeling of Kp, denoted as EAT .
The evaluation mechanism is semantics-dependent in ASPIC+. It consists of applying closure under transposition to the set of strict rules3, then carrying out the reasoning steps corresponding to Definitions 7-11, and finally applying a chosen semantics σ to derive the justification labeling of ordinary premises from Eσ (FΔ).
We can now introduce a basic requirement concerning spurious preference avoidance. First, we need to formalize the notion of modifying an argumentation theory by adding a premise to the antecedents of a strict rule.
Definition 14. Given an argumentation system AS = (L,¯, R, n), where R = (RSb, ∅) and a knowledge base K such that AT = (AS, K) belongs to the SSDOP family, we say that R′Sb is a P -addition of RSb if ∃r ∈ RSb such that R′Sb = (RSb \ {r}) ∪ {r′} where cons(r′) = cons(r) and ant (r′) = ant (r) ∪ {p} for some p ∈ Kp.
With a little abuse of language, we will also say that AS′ is a P -addition of AS when AS′ is obtained from AS by replacing RSb with a P -addition R′Sb, and we will say that AT ′ is a P -addition of AT with the same meaning. It is easy to see that for any SSDOP instance AT , every P -addition of AT also belongs to the SSDOP family since the addition of a premise to any strict rule does not afect any of the conditions in Definition 12.
It is also easy to see that in Section 3, the first example corresponds to an instance of the SSDOP family, and the second example is a P -addition of the first one.
We can now specify a basic spurious preference avoidance requirement with reference to an evaluation mechanism. This requirement refers to the adoption of a multiple-status semantics and adheres to a credulous perspective: in a SSDOP instance each premise should be accepted in some scenario, but rejected in some other. Under a single-status semantics, like the grounded semantics, a skeptical treatment, where all the premises are rejected altogether, would be appropriate and a complementary analysis would need to be developed, which is left to future work.
Definition 15. An argumentation theory AT = (AS, K) which belongs to the SSDOP family is Crpremise-fair with respect to an evaluation mechanism E if for every ordinary premise p ∈ Kp, EAT (p) = Cr.
Definition 16. An evaluation mechanism E satisfies the requirement of basic spurious preference
avoidance if given any argumentation theory AT which is Cr-premise-fair with respect to E, it holds that every
P -addition of AT is Cr-premise-fair too.
3In [
In plain words, if the premises of a SSDOP instance AT are treated equally (with a credulous outcome), the equal treatment should be preserved in every P -addition of AT .
From the example in Section 3, it emerges then that the evaluation mechanism corresponding to
ASPIC+ does not guarantee basic spurious preference avoidance with preferred, stable, and semi-stable
semantics. In Section 5, we show that a variant of ASPIC+, introduced in [
One of the main standpoints of ASPICR is avoiding the requirement that strict rules are closed under
transposition. Rather, the satisfaction of the rationality postulates is ensured (as shown in [
Definition 17. Given an argumentation system AS = (L,¯, R, n) the strict knowledge base KA∗S for AS is given by Kn = L, Kp = ∅ and the corresponding argumentation theory is defined as AT A∗S = (AS, KA∗S). S ⊢∗ φ and S ⊢∗min φ denote respectively that S ⊢ φ and S ⊢min φ in AT A∗S. Definition 18. Given an argumentation theory AT = (AS, K) with AS = (L,¯, R, n), let EC1(AS), EC2(AS), EC3(AS) be the following subsets of 2L × 2L • EC1(AS) = {({φ}, {ψ }) | φ ∈ ψ }; • EC2(AS) = {(S, {ψ }) | S ⊢∗min φ and φ ∈ ψ }; • EC3(AS) = {(S, T ) | T ⊢∗min ψ and (S, {ψ }) ∈ EC1(AS) ∪ EC2(AS)}.
Letting EC∗ m(AS) = EC1(AS) ∪ EC2(AS) ∪ EC3(AS), and, for S ⊆ L , Sb = S \ Kn, the extended contrariness relation is defined as EC(AS) = {(Sb, Tb) | (S, T ) ∈ EC∗ m(AS) and ∀(S′, T ′) ∈ EC∗ m(AS) s.t. Sb′ ⊆ Sb and Tb′ ⊆ Tb, Sb′ = Sb and Tb′ = Tb} ⊆ 2L × 2L. U is a contrary of V if (U, V ) ∈ EC(AS) and (V, U ) ∈/ EC(AS); U is a contradictory of V if (U, V ) ∈ EC(AS) and (V, U ) ∈ EC(AS).
The various forms of attack are then revised, replacing Definition 8 with the following definition concerning attacks between a set of arguments and an argument.
Definition 19. Given an argumentation theory AT = (AS, K), a set of arguments X attacks an argument β if X undercuts, rebuts, or undermines β where: • X undercuts β (on β ′) if for some β ′ ∈ Sub(β ) such that r = Top(β ′) ∈ RD, the following condition holds: ∃T, U such that T ∪ U = Conc(X) ∪ {n(r)}, (T, U ) ∈ EC(AS) and n(r) ∈ U . • X rebuts β (on β ′) if for some β ′ ∈ Sub(β ) of the form β 1′′, . . . , β n′′ ⇒ φ the following condition holds: ∃T, U such that T ∪ U = Conc(X) ∪ {φ}, (T, U ) ∈ EC(AS) and φ ∈ U . In this case X contrary-rebuts β if (U, T ) ∈/ EC(AS). • X undermines β (on β ′) if for some β ′ = φ, φ ∈ Premp(β ) the following condition holds: ∃T, U such that T ∪ U = Conc(X) ∪ {φ}, (T, U ) ∈ EC(AS) and φ ∈ U . In this case X contraryundermines β if (U, T ) ∈/ EC(AS).
As the efectiveness of some attacks depends on the preference relation, the notion of preference ordering needs to be generalized to sets of arguments.
Definition 20. Given a preorder ⪯ on a set of arguments X, we extend ⪯ to 2X × X as follows. An argument α is at least as preferred as a set of arguments Y , denoted Y ⪯ α , if ∃β ∈ Y such that β ⪯ α . α is strictly preferred to Y , denoted Y ≺ α , if ∃β ∈ Y such that β ≺ α , not strictly preferred to Y , denoted Y ⊀ α if ∄β ∈ Y such that β ≺ α .
On this basis, an extended notion of defeat is introduced, replacing Definition 9.
Definition 21. Let the set of arguments Y attack an argument β on β ′ according to Definition 19. If Y undercuts, contrary-rebuts, or contrary-undermines β on β ′, then Y preference-independent attacks β on β ′, otherwise Y preference-dependent attacks β on β ′. Then, Y defeats β if either Y preferenceindependent attacks β on β ′ or Y preference-dependent attacks β on β ′ and Y ⊀ β ′. Y minimally defeats β , denoted as Y ⇝ β , if Y defeats β and ∄Y ′ ⊊ Y such that Y ′ defeats β .
An AF based on the notion of defeat provided in Definition 21 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 22) that minimally defeats some produced argument.
Definition 22. Given an argumentation theory AT = (AS, K) let S be the set of the arguments produced in AS on the basis of K. The set of ultimately fallible arguments of AT is defined as UF(S) ≜ Kp ∪ {α ∈ S | Top(α ) ∈ RD}.
Definition 23. Given an argumentation theory AT = (AS, K) with ordering ⪯ , let S be the set of the arguments produced in AS on the basis of K. The set of relevant sets of arguments of AT , denoted as RS(AT ), is defined as RS(AT ) = {{α } | α ∈ S} ∪ {X | X ⊆ UF(S) and ∃β ∈ S : X ⇝ β }.
Then, a relevant set of ultimately fallible arguments attacks another one simply if it minimally defeats one of its members.
Definition 24. Let X, Y ∈ RS(AT ) for an argumentation theory AT = (AS, K) and S be the set of the arguments produced in AS on the basis of K. X D-attacks Y , denoted as ∥X∥ ↠ ∥Y ∥, if X ⊆ UF(S) and ∃α ∈ Y : X ⇝ α .
Definition 25. Given an argumentation theory AT = (AS, K), the RS-based argumentation framework induced by AT is defined as RS− F(AT ) = ({∥X∥ | X ∈ RS(AT )}, ↠ ).
Finally, given an argumentation semantics σ , the justification status of an argument α corresponds to the one of ∥{α }∥ in RS− F(AT ) according to Definition 3.
ASPICR provides a diferent evaluation mechanism for SSDOP argumentation theories, consisting in deriving the extended contrariness relation and then carrying out the reasoning steps established by Definitions 17-25 till the assignment of a justification status, as described above. As in the case of ASPIC+, the evaluation mechanism is parametric with respect to the choice of an argumentation semantics σ . We now prove that ASPICR satisfies the requirement of basic spurious preference avoidance for preferred, stable and semi-stable semantics.
Proposition 1. Any argumentation theory AT = (AS, K) which belongs to the SSDOP family is Crpremise-fair with respect to the evaluation mechanism provided by ASPICR under the choice of preferred semantics.
Proof: Let AS = (L,¯, R, n) be an argumentation system and K a knowledge base in AS, such that the argumentation theory AT = (AS, K) belongs to the SSDOP family. Assume Kp = P and for every x ∈ P , let us denote Px = P \ {x}. From the conditions in Definition 12, we have that P ⊢∗min ω, from which, with reference to Definition 18, it follows that (P, {¬ω}) ∈ EC2(AS) and ({¬ω}, P ) ∈ EC3(AS). Since ¬ω ∈ Kn, we get {(P, ∅), (∅, P )} ∈ EC(AS). From (∅, P ) ∈ EC(AS), it follows that for every x ∈ P , Px undermines x according to Definition 19 (with T = ∅ and U = P ). By the conditions in Definition 12, Px ⊀ x and to conclude that Px defeats x according to Definition 21 we have to show that ∄P ′ ⊊ Px such that P ′ defeats x. To this purpose, let us refer to the elements of the extended contrariness relation of the form (S, T ) such that x ∈ T , taking into account the conditions specified in Definition 12. First, ({¬x}, {x}) is the only such pair in EC1(AS) and we note ¬x is not a premise nor an axiom and that there is no rule r such that ¬x = cons(r). Therefore there cannot be any argument with conclusion ¬x. For the same reason, there is no pair (S, {x}) with S ̸= {¬x} in EC2(AS). Turning to EC3(AS), it contains any pair (S, T ) with x ∈ T satisfying the third bullet of Definition 18. This means that S ⊢∗min φ and T ⊢∗min ψ with φ ∈ ψ . However, Definition 12 prevents that a contradiction is derived through strict rules only, which means that S or T must be a singleton, i.e. S = {φ} or T = {ψ }, and that, to have an attack according to Definition 19, φ or ψ must be an ordinary premise or an axiom. The fact that φ or ψ is an ordinary premise is prevented by the conditions in Definition 12 (no ordinary premise is a contrary of another premise, no rule can conclude an ordinary premise or a contrary of an ordinary premise). Since by Definition 19 arguments cannot be undermined on axioms, the remaining case is where φ is an axiom, i.e. φ = ¬ω. We have already shown above that it cannot be the case that the pair ({¬ω}, {x}) belongs to EC1(AS) ∪ EC2(AS). The only possible case is ({¬ω}, T ) ∈ EC3(AS), for some T such that T ⊢min ω and x ∈ T . From the conditions in Definition 12 the only set T satisfying these constraints is P , from which it follows, as desired, that ∄P ′ ⊊ Px such that P ′ defeats x. It follows that Px defeats x according to Definition 21 (recall that all premises are equally preferred, hence Px ⊀ x). Note now that, by the conditions in Definition 12, UF(AT ) = Kp = P . It follows that, for every x ∈ P , Px ∈ RS(AT ) according to Definition 23. Then, according to Definition 24, ∥Px∥ ↠ ∥X∥ for every X such that X ∈ RS(AT ) and x ∈ X. This means in particular that for every x, y ∈ P with x ̸= y, there is a mutual attack between ∥Px∥ and ∥Py∥ in the argumentation framework RS− F(AT ).
We want now to show that a generic ∥Px∥ has no other attackers in RS− F(AT ). Suppose there is X ∈ RS(AT ) such that ∥X∥ ↠ ∥Px∥. By Definition 24, this means that ∃y ∈ Px such that X ⇝ y. Now, since y ∈ P , by the same reasoning carried out above, the only possibility is that X = Py.
It follows that for every x ∈ P , ∥Px∥ is admissible and hence there is a preferred extension E
including ∥Px∥. By Lemma 3 of [
Proof: We use the same notation introduced in the proof of Proposition 1 and we show that preferred
extensions coincide with stable extensions (and hence with semi-stable extensions) in the argumentation
framework RS− F(AT ). The conclusion then follows from Proposition 1. From the proof of Proposition
1 we know that for every x ∈ P it holds that ∥Px∥ is attacked by all and only the elements ∥Py∥ with
y ∈ P and attacks these elements in turn. It follows that every preferred extension E must include
exactly one element ∥Px∥ for some x ∈ P . We want now to show that given a generic preferred
extension E, for every argument α , ∥α ∥ is either defended by (and hence included in) E or attacked by
E. By the properties of RS− F(AT ) proved in [
The property of basic spurious preference avoidance follows from Propositions 1 and 2. Theorem 1. The evaluation mechanism provided by ASPICR under the choice of preferred, stable, and semi-stable semantics satisfies the basic spurious preference avoidance requirement.
Proof: The conclusion follows from the fact that given an argumentation theory AT which belongs to the SSDOP family every P -addition of AT belongs to the SSDOP family too. From Propositions 1 and 2 it follows then that both AT and any P -addition of AT are Cr-premise-fair, thus complying with the requirement of basic spurious preference avoidance. □
As an illustration of the above result, we describe the behavior of ASPICR in the second version of the taxpayers’ list example without closure under transposition, giving rise to arguments: A1 = d; A2 = b; A3 = p; A4 = A2, A3 → c; A5 = A2 → m; A6 = A1, A4, A5 → ω; A7 = ¬ω.
Concerning the ⊢min relation we have that {φ} ⊢∗min φ for every φ ∈ L; {b} ⊢∗min m; {b, p} ⊢∗min c; {d, c, m} ⊢∗min ω; {d, c, b} ⊢∗min ω; {d, p, b} ⊢∗min ω. As to EC(AS) we get: • EC1(AS) = {({φ}, {¬φ}), ({¬φ}, {φ}) | φ ∈ L}; • EC2(AS) \ EC1(AS) = {({b}, {¬m}), ({b, p}, {¬c}), ({d, c, m}, {¬ω}), ({d, c, b}, {¬ω}), ({d, p, b}, {¬ω})}; • EC3(AS)\(EC1(AS)∪EC2(AS)) = {({¬m}, {b}), ({¬c}, {b, p}), ({¬ω}, {d, c, m}), ({¬ω}, {d, c, b}), ({¬ω}, {d, p, b})}; • EC(AS) = {({φ}, {¬φ}), ({¬φ}, {φ}) | φ ∈ L \ {ω}} ∪ {(∅, {ω}), ({ω}, ∅), ({b}, {¬m}), ({b, p}, {¬c}), ({¬m}, {b}), ({¬c}, {b, p}), ({d, c, m}, ∅), ({d, c, b}, ∅), ({d, p, b}, ∅), (∅, {d, c, m}), (∅, {d, c, b}), (∅, {d, p, b})}.
According to Definition 19, from (∅, {d, p, b}) ∈ EC(AS) we get that {A1, A2} undermines A3, {A1, A3} undermines A2, and {A2, A3} undermines A1. Taking into account the subargument relations, {A1, A2} undermines also A4 and A6, {A1, A3} undermines also A4, A5 and A6, {A2, A3} undermines also A6. Definition 19 encompasses also the following attacks: {A4, A5} undermines A1 and A6; {A1, A4} undermines A2, A4, A5 and A6; {A2, A4} undermines A1 and A6. They however are “filtered out” by Definition 23 since they involve arguments which are not ultimately fallible. All the attack relations listed above are also minimal defeats.
It follows that RS(AT ) = {{A1}, {A2}, {A3}, {A4}, {A5}, {A6}, {A7}, {A1, A2}, {A1, A3}, {A2, A3}}.
Then, according to Definition 24, ∥{A1, A2}∥ D-attacks the singletons ∥{A3}∥, ∥{A4}∥, ∥{A6}∥, and the relevant sets including them, namely ∥{A1, A3}∥ and ∥{A2, A3}∥. Similarly ∥{A1, A3}∥ D-attacks ∥{A2}∥, ∥{A4}∥, ∥{A5}∥, ∥{A6}∥, and ∥{A2, A3}∥ and ∥{A1, A2}∥, while ∥{A2, A3}∥ D-attacks∥{A1}∥, ∥{A6}∥, ∥{A1, A2}∥ and ∥{A1, A3}∥. The resulting argumentation framework is shown in Figure 3. It has three preferred, stable and semi-stable extensions, namely E1 = {∥{A1, A2}∥, ∥{A1}∥, ∥{A2}∥, ∥{A5}∥, ∥{A7}∥}, E2 = {∥{A2, A3}∥, ∥{A2}∥, ∥{A3}∥, ∥{A4}∥, ∥{A5}∥, ∥{A7}∥}, E3 = {∥{A1, A3}∥, ∥{A1}∥, ∥{A3}∥, ∥{A7}∥}. As expected, each of the three defeasible premises is credulously justified.
∥{A7}∥ ∥{A3}∥
∥{A4}∥ ∥{A1, A2}∥
∥{A6}∥ ∥{A1}∥ ∥{A2, A3}∥ ∥{A1, A3}∥ ∥{A2}∥ ∥{A5}∥
Preferences have been considered in various ways both in structured and abstract argumentation [
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.