There is an extensive body of work in logic-based argumentation, which links logic and argumentation, which is a potential solution to address inconsistencies or conflicting information in knowledge bases (KBs) by ofering dialogue games as proof procedures to determine and explain the acceptance of propositions. Most existing work, though, focuses on specific logics (such as description logics, existential rules, defeasible and propositional logics), has limitations of representational aspects, for selected semantics and binary conflicts. In this paper, we generalise this work by introducing G-SAF, which generalises the notions of arguments, dialogues and dialogue trees for more general logical reasoning with inconsistencies, including the most common semantics and to facilitate reasoning with non-binary conflicts using argumentation with collective attacks (SAFs).
attacks are introduced for defeasible logic programming. However, [
To compute and explain answers w.r.t the semantics of logical reasoning, argumentation
provides dialogue models as proof procedures, but, there is no unifying approach for the most
common inconsistency-tolerant semantics. In the Datalog± context, the framework of [
Previous proposed dialogue models are used in LAFs with binary attacks, which are not
generic enough to deal with collective attacks. In [
Main contributions. The existing approaches are mostly restricted to: (1) specific logics or limitations of representational aspects; (2) specific inconsistency-tolerant semantics; (3) (dialectical) proof procedures used in AFs with binary attacks. This paper closes this gap: (1) we propose a unifying framework, G-LAF, that allows a translation of KBs in a broad family of logics into argumentation frameworks; (2) our unified approach includes inconsistency-tolerant semantics considered in previous literature (e.g. IAR and Brave in the case of Datalog± ) as well as new ones (AR in the case of Datalog± ); (3) we propose an extended version of the dialogue model that applies to AFs/LAFs with collective attacks, generalizing the dialogue model used in AFs/LAFs with binary attacks.
We only sketch the proofs for our main results in this paper, more details can be found in the appendix.
Most of our discussion applies to arbitrary logics (monotonic and non-monotonic) consisting of a set ℒ (of sentences) and a function CN from 2ℒ to 2ℒ returning the logical consequences of a 1The notion of repair corresponds to "maximal consistent subsets" in our work set of sentences (the consequence operator), that satisfies the axiom: Expansion ⊆ CN(), Idempotence CN(CN()) = CN(). Fix a logic (ℒ, CN) and a set of sentences ⊆ ℒ . We say that: • is consistent wrt (ℒ, CN) if CN() ̸= ℒ. It is inconsistent otherwise; • A knowledge base (KB) is any subset of ℒ. A knowledge base may be inconsistent. Reasoning in inconsistent KBs ⊆ ℒ
amounts to: 1. Constructing maximal consistent subsets, 2. Applying classical entailment mechanism on a choice of the maximal consistent subsets. Motivated by this idea, we give the following definition.
Definition 1. Let be a KB and ⊆ be a set of sentences. is a maximal (for set-inclusion) consistent subset of if is consistent, there is no ′ such that ⊂ ′ and ′ is consistent. We denote the set of all maximal consistent subsets by MCS().
Inconsistency-tolerant reasoning in KB has semantics to determine diferent types of answers. Definition 2. Let be a KB and ⊢: 2MCS() ↦→ MCS() be an entailment mechanism. A sentence ∈ ℒ is entailed in • all maximal consistent subsets, written ⊢⋃︀ MCS() , if for all Δ ∈ MCS(), ∈ CN(Δ); • some maximal consistent subset, written ⊢MCS() , if for some Δ ∈ MCS(), ∈
CN(Δ); • the intersection of all maximal consistent subsets, written ⊢⋂︀ MCS() , if for Ψ = ⋂︀{Δ | Δ ∈ MCS()}, ∈ CN(Ψ).
In this section, we introduce: (1) A general logic-based argumentation with collective
attacks (G-SAF) translated from a KB. (2) General formal model of dialogue (for G-SAF)
considered as a dialectical proof procedure to determine and explain reasoning outcomes in KBs.
This dialogue model is inspired by previous works [
Arguments are built from a KB. They represent proofs for conclusions. Thus, an argument has two parts: a support (also called a set of premisses) and a conclusion.
Definition 3 (G-SAF arguments). An argument induced from a KB is the pair = (Γ, ) such that Γ ⊆ and ∈ CN(Γ). For such argument, we denote by Sup() = Γ the support set of and by Con() = the consequence of . We use Arg to denote the set of arguments constructed from . Fix ⊆ Arg, Cons() = ⋃︀ {Con()} are the conclusions in . ∈
Some proposals for logic-based argumentation stipulate additionally that the argument’s
support is consistent and/or that none of its subsets entails the argument’s conclusion (see [
Diferent attack relations have been considered in the literature for logic-based argumentation.
However, some definitions of attack are not suitable to capture non-binary conflicts [
Definition 4 (Collective Attacks). arguments.
Let = (Γ, ) be an argument and ⊆
Arg be a set of • undercut-attacks if there is Γ′ ⊆ Γ s.t ⋃︀∈ {Con( )} ∪ Γ′ is inconsistent. • rebuttal-attacks if ⋃︀∈ {Con( )} ∪ { } is inconsistent.
We can say that attacks for short. We use Att ⊆ induced from . 2Arg × Arg to denote the set of attacks
For KBs with only binary conflicts (e.g. core DL-Lite dialects), we have that | | = 1. We refer this case to (Dung) AFs where they consider an attack between two arguments (i.e. a binary attack).
Definition 5. Let be a KB, the corresponding general logic-based argumentation (G-SAF) ℱ is the pair (Arg, Att), where Arg is the set of arguments induced from and Att is the set of attacks.
Its semantics are now defined as in the definition of argumentation semantics for SAFs [
Given a G-SAF ℱ = (Arg, Att) and ⊆ Arg. attacks if ∃ ∈ s.t. attacks . defends if for each ⊆ Arg s.t. attacks , some ′ ⊆ attacks . An extension is called • conflict-free if it does not attack itself. • admissible (ad) if it is conflict-free and defends itself. • complete (cm) if it is an admissible extension containing all arguments that it defends. • preferred (pr) if it is an inclusion-maximal admissible extension. • stable (st) if it is conflict-free and attacks every argument which is not in it. • grounded (gr) if it is an inclusion-minimal complete extension.
Let Exts(ℱ ) denote the set of all extensions of ℱ under the semantics s {ad, cm, st, pr, gr}. Let us define acceptability in our G-SAF: ∈ Definition 6. ∈ ℒ is
Let ℱ be the corresponding G-SAF of a KB and s ∈ {ad, st, pr}. A sentence • credulously accepted under s if for some ℰ ∈ Exts(ℱ ), ∈ Cons(ℰ ). • sceptically accepted under s if for all ℰ ∈ Exts(ℱ ), ∈ Cons(ℰ ). • groundedly accepted under gr if for some
ℰ ∈ Extgr(ℱ ), ∈ Cons(ℰ ).
Proposition 1 shows a relation between extensions of G-SAFs and maximal consistent subsets of KBs.
Proposition 1. Let ℱ be the corresponding G-SAF of a KB . Then,
of ℱ . • the maximal consistent subset of coincides with the stable/ preferred extension of ℱ ; • the intersection of the maximal consistent subsets of coincides with the grounded extension Proof 1. The idea of the proof is to show that every preferred extension is the set of arguments generated from a MCS, that every such set of arguments is a stable extension, and that every stable extension is a preferred extension.
The following is the main result of this section, which generalises related results of previous works.
Theorem 1. Let ℱ be the corresponding G-SAF of a KB , ∈ ℒ a sentence and s ∈ {st, ad, pr}. Then, • ⊢MCS() if is credulously accepted under s. • ⊢⋃︀ MCS() if is sceptically accepted under s.
• ⊢⋂︀ MCS() if is groundedly accepted under gr.
The proof of Theorem 1 follows Proposition 1.
To argue the quality of G-SAF, it can be shown that it satisfies the rationality postulates
introduced in [
Let ℱ be the corresponding G-SAF of a KB . Wrt. s ∈ {ad, st, pr, gr}, Definition 7. ℱ is 1. closed under CN if for all ℰ ∈ Exts(ℱ ), Cons(ℰ ) = CN(Cons(ℰ )); 2. consistent if for all ℰ ∈ Exts(ℱ ), Cons(ℰ ) is consistent.
Proposition 2. Wrt. to any semantics in {ad, st, pr, gr}, ℱ satisfies consistency, closure.
To further illustrate the generality of our framework for monotonic and nonmonotonic logics, let us consider the following example.
Example 1. Let be a set of propositional atoms. Any atoms ∈ is a well-formed formula wrt. . If and are well-formed formulas wrt. then ¬, ∧ , ∨ are well-formulas wrt. (we also assume that the usual abbreviations →, ↔ are defined accordingly). Then ℒ is the set of well-formed formulas wrt. . Let |= be the entailment relation, i.e., |= if all models of are models of in the propositional semantics. A consequence operator CN : 2ℒ → 2ℒ is defined for each ⊆ ℒ by CN( ) = { ∈ ℒ | |= }. The propositional logic can be defined as (ℒ, CN).
Consider the propositional atoms 1 = {, } and the knowledge base ′ = {, , → ¬} ⊆ ℒ. The following set of arguments: 1 = ({}, ); 2 = ({}, ); 3 = ({ → ¬}, → ¬); 4 = ({, → ¬}, ¬); 5 = ({, → ¬}, ¬); 6 = ({, }, ∧ ).
The attack relations:{(1, 5), (5, 1), (2, 4), (4, 2), (3, 6), (6, 3)}.
The preferred extensions: {1, 2, 3}, {4, 5, 6}.
Example 2. Let (ℒ, CN) be a defeasible logic such as used in defeasible logic programming [
The consequence operator CN is a function from 2ℒ to 2ℒ such that for all ⊆ ℒ , ∈ CN( ) if there exists a sequence 1, . . . , such that 1. is , and 2. for each ∈ {1, . . . , }, • there is 1, . . . , → ∈ , or 1, . . . , → ∈ , such that {1, . . . , } ⊆ {1, . . . , − 1}, • or ∈
Consider ′′ = {, ¬, → ¬, ¬ ∧ ¬ → , → , ∧ → }, the following is an argument in defeasible logic = ({, ¬, → ¬, ¬ ∧ ¬ → }, ).
Example 3. Let (ℒ, CN) be Datalog± [
Let Nt be a set of terms that contain variables, constants and function terms. An atom is of the form (⃗), with a predicate name and ⃗ a vector of terms, which is ground if it contains no variables. A database is a finite instance. A tuple-generating dependency (TGD) is a firstorder formula of the form ∀⃗∀⃗(⃗, ⃗) → ∃⃗ (⃗, ⃗), where (⃗, ⃗) and (⃗, ⃗) are non-empty conjunctions of atoms. 2 A negative constraint (NC) is a rule of the form ∀⃗ (⃗) → ⊥ where (⃗) is a conjunction of atoms. The language ℒ contain a set of facts, TGDs, and NCs.
Assume that |=ℒ is an entailment of first-order formulas, i.e., Δ |=ℒ holds if every model of all elements in Δ is also a model of . The consequence operator CN is defined for ⊆ ℒ by CN( ) = { ∈ ℒ | |=ℒ } in the first-order semantics.
A knowledge base is a tuple (ℱ , ℛ, ) where a database ℱ , a set ℛ of TGDs and a set of NCs.
Consider KB = {ℛ, , ℱ } ⊆ ℒ , where:
ℛ ={1 : Le() → Em(), 2 : Re() → Em(), 2We usually leave out the universal quantification, and refer to (⃗, ⃗) and (⃗, ⃗) as the body ad head of . Argument
Sup() Table 1 shows the supports and conclusions of arguments induced from .
The KB admits six MCSs (called repairs) ℬ1, . . . , ℬ6, e.g., ℬ1 = {ta(v, KD), UC(KD)}. The corresponding G-SAF admits extensions: Extst/pr(ℱ ) = {ℰ1, . . . , ℰ6}, where ℰ1 = Args({ta(v, KD), UC(KD)}) 3 = {5, 6, 7}, and ℰ2, . . . , ℰ6 are obtained in an analogous way. By Theorem 1, the extensions correspond to the repairs of the KBs, for example, ℰ1 corresponds to ℬ1.
Consider 1 = Re(v). We have that "Re(v) is a possible answer" since 1 is entailed in ℬ2, ℬ3, ℬ5. In other words, 1 is credulously accepted under st (pr) extensions.
We have presented the translation of KBs into G-SAFs as an instantiation of SAFs. Subsequently, we shift the problem of determining the entailment of a sentence to computing and explaining the acceptance of arguments for . This section shows how to determine the acceptance of an argument, in various argumentation semantics from an explanatory dialogue (or "dialogue" for short).
Dialogues can be viewed as dialectical proof procedures of moving arguments as a 2-person
dialogue game. Informally, a dialectical proof procedure is formalised by a dialogue between
two players a proponent (P) and an opponent (O). The dialogue begins with P moving an initial
argument that it wants to put to the test. O and P take turns in moving arguments that attack
their counterpart’s last move. We adapt the notion of dialogue model4 from [
For such utterance , we set pl() = PL. Let denote the set of utterances. • A dialogue G is a finite sequence 1, . . . , of utterances such that the first utterance 1 is played by P, i.e. pl(1) = P. We denote by < ∞ the set of all dialogues. • A protocol Θ on specifying the legal moves at each stage of a dialogue. Formally, a protocol on is a function Θ : ↦→ 2 such that ⊆ < ∞.
The elements of are called the legal finite dialogues . For G ∈ , the elements of Θ(G) are called the moves allowed after G. If G is a legal dialogue and Θ(G) = ∅, then G is said to be a terminated dialogue. Θ must satisfy the following condition: for all finite dialogues G and moves , G ∈ and ∈ Θ(G) if G, ∈ .
The dialogue represents a compact representation of a tree where nodes are arguments played by both parties. To display all disclosed information from the dialogues, we introduce the notion of dialogue trees in terms of a meta-level, which includes arguments and counter-arguments. Definition 8. Given a dialogue G = 1, . . . , (for an argument ∈ Arg). The general dialogue tree drawn from G is a labelled tree G = (, ℰ ), where is the set of nodes and ℰ is the set of hyperedges that are directed from nodes to their parent node. G is defined as follows: • The set of all nodes is defined as = {arg() | ∈ G} with arg(1) as the root node of the tree s.t. arg(1) = . • ℰ is recursively defined as: – ℰ1 = ∅ if the player P utters 1, – ℰ− 1 ∪ {(arg(− 1), arg())} if the player PL ∈ {P, O} utters .
4Following [
A branch in a dialogue tree may be finite or infinite. A finite branch represents a winning debate (i.e., P wins) that ends with an argument by P against which O cannot attack. An infinite branch represents a winning debate in which P counterattacks every attack of O, ad infinitum. The requirement that "the proponent must counterattack every attack" does not guarantee that the proponent does not attack itself. This further requirement is incorporated in the definition of admissible dialogue tree: A dialogue tree is said to be admissible if P wins and no argument labels both a proponent and an opponent node.
Notice that, in admissible dialogue trees, it is not required that the opponent and proponent nodes have no arguments in common. This is because the opponent can use the proponent’s arguments against the proponent. If the opponent can attack the proponent using only the proponent’s arguments, then the proponent loses. To win, the proponent must identify and counter-attack each opponent’s attack with some culprit not part of their own defence.
To ensure credulous soundness, all possible O nodes must be considered. But if such a parent
node is already in the dialogue tree, then deploying it will not help O win the dialogues. For
this reason we call an admissible tree non-redundant [
The following definition defines successful dialogues.
Definition 9. Let G be the dialogue tree drawn from a dialogue G for an argument ∈ Arg. G is called • admissible-successful if G is admissible; • preferred-successful if it is admissible-successful; • grounded-successful if G is admissible and finite; • sceptical-successful if G is admissible and for no opponent node O in it there exists an admissible dialogue tree for the argument labelling O.
We now determine the acceptance of an argument from its dialogues and dialogue trees.
Theorem 2. Let G be a dialogue for an argument ∈ ArgG. Then is
• credulously accepted in some admissible extension of ℱ G if G is admissible-succesful;
• credulously accepted in some preferred extension of ℱ G if G is preferred-successful;
• groundedly accepted in ℱ G if G grounded-successful;
• sceptical accepted in all preferred extensions of ℱ G if G is sceptical-successful.
Proof 2. Theorem 2 generalizes known results about the abstract dispute trees of [
The following is a direct corollary of Theorem 1, 2 and Definition 9, which shows how to determine the entailment of a sentence wrt dialogues. The acceptance of a sentence corresponds to the acceptance of a set of arguments for .
Corollary 1. Let be a KB and be a sentence in ℒ. Let ℱ be the corresponding G-SAF and ⊆ Arg be the set of arguments for . Then, is P : claim(3) O : contrary(7) P : contrary(9) P : contrary(10) 3 : ⟨{t(v, KR), GC(KR)}, Re(v)⟩ 7 : ⟨{t(v, KD), UC(KR)}, ta(v)⟩ 9 : ⟨{t(v, KR)}, Le(v)⟩ 10 : ⟨{t(v, KD)}, Le(v)⟩
• credulously accepted, and ⊢MCS() if there is a dialogue G for some ∈ s.t. G is preferred-successful; • groundedly accepted, and ⊢⋂︀ MCS() if there is a dialogue G for all ∈ s.t. G is grounded-successful; • sceptically accepted, and ⊢⋃︀ MCS() if 1. there is a dialogue G for some ∈ such that G is sceptical-successful; 2. or there is a dialogue G for all ∈ s.t. G is preferred-successful and is contained in all preferred extensions of ℱ .
Example 4 (Continue Example 3). Assume that the user wants to understand why "v is a possible researcher". The deliver system considers a persuasion dialogue: P is persuading O to agree that v is a researcher. The agreement can be reached through a dialogue (3). Figure 1 shows the dialogue tree drawn from the dialogue (3).
In this paper, we generalized the translation of arbitrary logic into SAFs. We introduced an argumentation dialogue framework and investigated the relation between its outcome and inconsistency-tolerant semantics in KBs. Our dialogue framework functions as proof procedures to compute and explain the acceptability of propositions. It can be applied to formalisms with collective attacks, thereby we show that this framework can be a generalization of the dialogue model used in AFs/LAFs with binary attacks. However, our framework has limitations: (1) Our dialogue model is still defined abstractly, only considering arguments and attacks and ignoring the internal structure of arguments; (2) Space complexity wrt the large input size, since the procedure considers all arguments translated from KBs as input to compute attacks. In future, we would address the limitations.
This work is partially supported by the Hybrid Intelligence programme (https://www. hybrid-intelligence-centre.nl/), funded by a 10 year Zwaartekracht grant from the Dutch Ministry of Education, Culture and Science. 41–55
Notation 1. Let be a KB, ⊆ Then,
be a set of formulas and ⊆ Arg be a set of arguments. • Args( ) = { ∈ Arg | Sup() ⊆ } are the set of arguments generated by , • Base() = ⋃︀ Sup() are the set of supports of arguments in ,
∈ • An argument is a subargument of argument if Sup() ⊆ Sup(). We denote the set of subarguments of as Subs(). • is a minimal conflict of if ′ ⊊ implies ′ is consistent. We denote the set of minimal conflicts of by Conflicts().
We prove Proposition 1 through Lemma 1 and Lemma 2.
Lemma 1. Let be a KB, the corresponding G-LAF ℱ . Then, the maximal consistent subset of coincides with the stable/ preferred extension of ℱ ; Proof 3. To prove the lemma, we first prove it with the undercut-attacks. For the rebuttal-attacks, the proof is similar to that of the undercut-attacks. We separate the proof of the lemma into two parts: 1. We prove {Args() | ∈ MCS()} ⊆ Extst(ℱ ) 2. We prove Extpr(ℱ ) ⊆ { Args() | ∈ MCS()} 1. We prove {Args() | ∈ MCS()} ⊆ Extst(ℱ ). Suppose that ∈ MCS() (i.e., is consistent) and ℰ = Args(), we show that ℰ is a stable extension of ℱ , i.e., ℰ ∈ Extst(ℱ ). Which means that, by definition, we prove that ℰ is conflict-free and attacks all arguments not belonging to ℰ .
First, we prove that ℰ is conflict-free. Suppose for a contradiction that ℰ is not conflict-free. There exists ℰ ′ ⊆ ℰ and an argument ∈ ℰ such that ℰ ′ attacks . By Definition 4, there exists ∈ Sup() such that ⋃︀∈ℰ′ {Con()} ∪ { } is inconsistent. Thus ⋃︀∈ℰ′ Sup() ∪ { } is inconsistent. Since ℰ ′ ⊆ ℰ , it follows that ⋃︀∈ℰ Sup() ∪ { } is inconsistent. Thus is inconsistent which is a contradiction to is consistent. Thus ℰ must be conflict-free.
Second, we prove that ℰ attacks all arguments not belonging to ℰ . Let ∈ Arg ∖ ℰ and ∈ Sup() ∖ . Since /∈ and is a maximal consistent subset of , then for every ⊆ ℰ = Args() such that ⋃︀∈ Sup() ∪ { } is inconsistent, there follows that ⋃︀∈ {Con()} ∪ { } is inconsistent. By Definition 4, attacks . Since ℰ = Args(), this proves that ℰ attacks all arguments not belonging to ℰ . Since ℰ is conflict-free and attacks all arguments not belonging to ℰ , it follows that ℰ is a stable extension.
2. We next prove Extpr(ℱ ) ⊆ { Args() | ∈ MCS()}. Suppose that ℰ ∈ Extpr(ℱ ), we prove that there exists a maximal consistent subset such that ℰ = Args(). Which means that, by definition, we prove that is maximal consistent.
First, suppose that = Base(ℰ ) and we prove is consistent. Assume for a contradiction that is inconsistent. Let { 1, . . . , } = ′ ⊆ be an inconsistent set and every proper subset of it is consistent. Let ∈ ℰ be an argument such that ∈ Sup() and be the set of arguments generated from ′∖{ } (i.e., = { ∈ Arg | Sup() ⊆ ′∖{ }}) such that ⋃︀∈ Sup() ∪ { } is inconsistent. It follows that ⋃︀∈ {Con()} ∪ { } is inconsistent, then attacks (by Definition 4). Since Sup() ⊆ ′, attacks . Since ℰ is conflict-free, we have ⊈ ℰ ; and since ℰ is also an admissible set, there is a set of arguments ⊆ ℰ such that attacks . Then, there exist some arguments ∈ and ∈ Sup(), ∈ {1, . . . , − 1} such that ⋃︀∈{Con()} ∪ { } is inconsistent. Next, we will show attacks an argument in ℰ . Since ∈ Base(ℰ ), there exists an argument ∈ ℰ s.t. ∈ Sup( ), which implies attacks . Clearly, is attacked by but cannot be defended, which is in contradiction with the fact that ℰ is admissible. Hence is consistent.
Second, we show that is maximal consistent, i.e., there is no maximal consistent subset ′ ∈ MCS() such that ′ ⊆ and ′ is inconsistent. Assume the contrary. Since ℰ is a preferred extension, it follows that Args()∖Args(′) ̸= ∅, and let Ω ⊆ ∖ ′ and ℳ be the set of arguments generated by Ω, i.e., ℳ = { ∈ Arg | Sup( ) ⊆ Ω}. Since Ω ⊆ ∖ ′, it follows that ℳ ⊆ Args()∖Args(′). By the first part of the proof, Args(′) is a stable extension of ℱ , there must be Φ ⊆ ′ such that the set of all arguments generated by Φ, that is = { ∈ Args(′) | Sup() ⊆ Φ}, attacks ℳ. Since ℰ is the preferred extension, there must be a set of arguments ℋ ⊆ ℰ and ℋ attacks . Since ℋ ⊆ ℰ and = Base(ℰ ), it follows that Base(ℋ) ⊆ . By assumption, i.e., ′ ⊆ , it follows that ′ is inconsistent, which contradicts the assumption. Hence is maximal consistent. We conclude that ∈ MCS().
Since every stable extension is a preferred one [
Lemma 2. Let be a KB, the corresponding G-LAF ℱ . Then, the intersection of the maximal consistent subsets of coincides with the grounded extension of ℱ .
Proof 4. Let be the grounded extension of ℱ and be the intersection of the maximal consistent subsets of . Let ′ ∈ MCS() and = ⋂︀′∈MCS() ′. By Lemma 1, ′ is a stable extension of ℱ . Since every stable extension is also a complete extension, and is the minimal (w.r.t. set inclusion) complete extension of ℱ , it follows that ⊆ Args(′). Thus ⊆ Args().
In the other direction, let Δ ⊊ . Then there is no ∈ Conflicts() such that Δ ⊊ (If Δ ⊊ , ∖ Δ would be consistent and could be extended to a maximal consistent subset of that would not contain Δ). Hence, there are no arguments that attack the argument ∈ Args() where Sup() = Δ. Since has no attackers, then any extension trivially defends it, so must be in . It follows that Args() ⊆ .
We prove each postulate:
1. Closure: Let us show that for every ℰ ∈ Exts(ℱ ), Cons(ℰ ) = CN(Cons(ℰ )). From Expansion axiom, it follows that Cons(ℰ ) ⊆ CN(Cons(ℰ )).
In the other direction, we prove CN(Cons(ℰ )) ⊆ Cons(ℰ ). Let ∈ CN(Cons(ℰ )). Since ℰ is a preferred or stable extension, there is a set of sentences ⊆ such that ℰ = Args() (by Proposition 1). Thus ℰ = Args(Base(ℰ )). Since the supports of the arguments of ℰ include the sentences in , it follows that ∈ CN(). Hence, there is an argument ∈ ℰ such that Con() = .
3. Consistency:
We prove that for ℰ ∈ Exts(ℱ ), Cons(ℰ ) is consistent. First, we consider the case of stable or preferred extensions. Let ℰ be a stable or preferred extension of ℱ . By Proposition 1, there is a maximal consistent subset ∈ MCS() such that ℰ = Args(). Since is a preferred maximal consistent subset, we see that CN() is consistent. Since Cons(ℰ ) = CN() and CN() is consistent, it follows that Cons(ℰ ) is consistent.
We consider the case of grounded semantics. We denote as the grounded extension of ℱ . We prove that for every ℰ ∈ Extgr(ℱ ), Cons(ℰ ) is consistent. Since the grounded extension is a subset of the intersection of the preferred extensions, and since there is at least one preferred extension ℰ1, then ⊆ ℰ 1. From Proposition 1, there is a maximal consistent subset ∈ MCS() s.t. ℰ1 = Args(). By the first part of the proof, Cons(ℰ1) is consistent. It follows that Cons() is consistent.
• ⊢MCS() if is credulously accepted under s semantics. • ⊢⋃︀ MCS() if is sceptically accepted under s semantics.
• ⊢⋂︀ MCS() if is groundedly accepted under gr semantics.
By Proposition 1, there exists a link between the set of maximal consistent subsets on and the set of extensions on ℱ . Obviously, for every sentence ∈ ℒ and maximal consistent ∈ MCS(), it is the case that ∈ CN() if ∈ Cons(Args()). The theorem is now proven as follows: 1. ⊢⋃︀ MCS() if for every ∈ MCS(), ∈ CN() if for every Args() ∈ Ext(ℱ ), ∈ Cons(Args()) if is sceptically accepted. 2. ⊢MCS() if for some ∈ MCS(), ∈ CN() if for some Args() ∈ Ext(ℱ ), ∈ Cons(Args()) if is credulously accepted. 3. ⊢⋂︀ MCS() if for every ∈ MCS(), ∈ CN(⋂︀ ) if for every Args() ∈
Ext(ℱ ), ∈ Cons(⋂︀ Args()) if is accepted under grounded semantics. This ends the proof of Theorem 1.
Proof 5. It is clear that is accepted in some admissible extension of ℱ G if G is admissible.
Let G be the dialogue tree drawn from G. The argument given in Definition 2 and Lemma 1
of [
G is preferred-successful if G is admissible-successful. This result directly follows from the
results of [
The other statement follows in a similar way as a straightforward generalization of Theorem
1 of [