A Study of Argumentative Characterisations of Preferred Subtheories

1 Introduction

The impact of Dung's seminal theory of argumentation [Dung, 1995] is largely due to its dialectical characterisation of non-monotonic inference. Given a base B of logical formulae, one defines the arguments related by binary attacks. Preferences over arguments can then be used to determine which attacks succeed as defeats. Given the resulting argument framework (AF) consisting of arguments related by attacks or defeats, the winning sets (extensions) of arguments under various semantics are evaluated, and the conclusions of arguments in these extensions identify the inferences defined by the 'instantiating' B. These are then shown to correspond to the inferences defined directly over B by various non-monotonic logics.

A number of works [Amgoud and Vesic, 2010; Modgil and Prakken, 2013; Thang and Luong, 2014] provide argumentative characterisations of non-monotonic inference in Brewka's Preferred Subtheories [Brewka, 1989]. The latter defines credulous and sceptical inference relations over a totally ordered set B of classical wff. These are shown to correspond to the conclusions of arguments in at least one, respectively all, stable extensions of the AFs instantiated by B in the aforementioned works. However, these works formalising standard accounts of classical logic argumentation (Cl-Arg) do not fully account for practical applications, as the correspondences and satisfaction of rationality postulates [Caminada and Amgoud, 2007; Caminada et al., 2012] assume construction of all classical logic arguments (which are in general infinite) defined by B. Moreover, they enforce checking that the arguments' premises are consistent and 'subset minimal' (no proper subset of the premises entail the conclusion). Hence, [D'Agostino and Modgil, 2016; 2018] provide a fully rational dialectical account of Cl-Arg that makes minimal assumptions as to the arguments that need to be constructed from B, and eschews the need to check for subset minimality and consistency of premises. They then give an argumentative characterisation (again by reference to stable extensions) of Preferred Subtheories inference defined under the assumption that agents have limited inferential capabilities.

This paper makes two key contributions to the study of these argumentative characterisations of Preferred Subtheories inference. Firstly, we show that for both the standard approach to Cl-Arg (as formalised in [Modgil and Prakken, 2013]), and the dialectical approach for use by resource bounded agents, the preferred and stable extensions coincide. Hence the above correspondences with Preferred Subtheories inferences are shown to hold under the preferred semantics. The significance of these results is that argumentative characterisations of non-monotonic inference pave the way for individual agent non-monotonic reasoning via the use of argument game proof theories for establishing membership of arguments in admissible extensions (e.g., [Modgil and Caminada, 2009]), and generalisation of these proof theories to dialogical models of distributed non-monotonic reasoning (e.g., [Prakken, 2005]). However, each admissible extension of an AF is not necessarily a subset of a stable extension, but is a subset of a preferred extension. Hence, correspondences for Preferred Subtheories inferences under the preferred semantics enables use of these argument games and dialogues for establishing Preferred Subtheories inference. Secondly, we discuss how use of the standard approach to Cl-Arg results in: 1) the 'foreign commitment problem' when applying argumentation in dialogical contexts [Caminada et al., 2014], and; 2) overly sceptical results when defining argumentation based inferences under the grounded semantics. We show how both these problems are avoided in the dialectical approach. Avoiding the latter problem then means that the grounded semantics applied to the dialectical characterisation more closely 'approximates' sceptical Preferred Subtheories inference (as explained in Section 4).

This paper is organised as follows. Section 2 reviews Preferred Subtheories and its argumentative characterisations in the standard and dialectical approaches to Cl-Arg. Sections 3 and Section 4 respectively establish the first and second contributions described above. We conclude in Section 5.

2 Background

2.1 Preferred Subtheories

Preferred Subtheories [Brewka, 1989; 1991] resolves inconsistencies amongst an ordered set of classical formulae so as to formalise both non-monotonic reasoning and belief revision. Henceforth, ⊢c denotes the classical consequence relation, lower and upper case greek letters respectively denote individual, and sets of, first order formulae. For simplicity, examples will use a propositional language (lower case roman letters denote propositions).

Definition 1 (Γ,≤) is a default theory where ≤ is a total ordering over the set Γ of first order classical wffs (well formed formulas). (Γ,≤) can be represented as (Γ1, . . . , Γn), where Γ = ⋃ni=1 Γi and ∀α ∈ Γi, ∀β ∈ Γj : i < j iff α > β. Hence α, β ∈ Γi iff α ≃ β1 .

A preferred subtheory (ps) is a set Σ = Σ1 ∪ · · · ∪ Σn such that for i = 1, . . . , n, Σ1 ∪ · · · ∪ Σi is a maximal (under set inclusion) consistent subset of Γ1 ∪ · · · ∪ Γi.

φ is a sceptical (credulous) ps-inference from (Γ, ≤), denoted (Γ,≤) |∼sc ps φ ((Γ,≤) |∼cr ps φ) iff φ is classically entailed by all (respectively at least one) ps of (Γ,≤)

Intuitively, a ps is obtained by taking a maximal under set inclusion consistent subset (mcs) of Γ1, extending this to a mcs of Γ1 ∪ Γ2, and so on.

Example 1 The default theory (Γ1 = {¬a ∨ ¬b}, Γ2 = {a, b, a ∧ b → ¬c}, Γ3 = {c}) has two pss: Σ = {¬a ∨ ¬b, a, a ∧ b → ¬c, c} and Σ′ = {¬a ∨ ¬b, b, a ∧ b → ¬c, c}.

2.2 Preferred Subtheories as an instance of Classical Logic Argumentation

Given a set of first order wffs Γ, classical logic arguments are tuples (∆, α) such that ∆ ⊆ Γ, ∆ ⊢c α, ∆ is 'subset minimal' (no proper subset of ∆ entails α) and ∆ ⊬c ⊥ [Besnard and Hunter, 2008]. Henceforth, conc(X) = α, prem(X) = ∆, and prem(S) = ⋃X∈S prem(X). [Modgil and Prakken, 2013] then use the total ordering ≤ over Γ to define strict preferences over arguments, and hence defeats:

Definition 2 Let A be all arguments defined by Γ in the default theory (Γ,≤). ∀X, Y ∈ A:

• (X, Y ) ∈ C (i.e., X attacks Y ) iff conc(X) = φ, -φ ∈ prem(Y ), where for any φ, if φ is of the form ¬α then -φ = α, else -φ = ¬φ.

X is said to attack Y on the argument Y ′ = ({-φ},-φ) (abusing notation we may write 'X attacks Y on -φ').

• X ≺ Y iff ∃α ∈ prem(X), ∀β ∈ prem(Y ): α < β
• (X, Y ) ∈ ⇒ (i.e., X defeats Y , denoted X ⇒ Y ) iff X attacks Y on Y ′ , and X ⊀ Y ′ . X ⇏ Y denotes (X, Y ) /∈⇒.
• The extensions of the argument framework (AF) (A,⇒) defined by (Γ, ≤) are defined in the usual way [Dung, 1995]:

1x < y iff x ≤ y, y ≰ x, and x ≃ y iff x ≤ y and y ≤ x.

E ⊆ A is conflict free iff ∀X, Y ∈ E,X ⇏ Y . A conflict free E is an admissible extension iff ∀X ∈ E, X is acceptable w.r.t. E. An admissible E is a complete extension if X is acceptable w.r.t. E implies X ∈ E. E is a preferred extension if E is a maximal (under set inclusion) complete extension. E is the grounded extension if E is the mimimal (under set inclusion) complete extension. E is a stable extension if E is conflict free and ∀Y /∈ E, ∃X ∈ E s.t. X ⇒ Y .

• For S ∈ {preferred, grounded, stable}, the sceptical and credulous inferences are respectively defined as:

(A,⇒) |∼sc S φ if every S extension contains an argument concluding φ ;

(A,⇒) |∼cr S φ if at least one S extension contains an argument concluding φ.

Example 2 Let (A,⇒) be the AF defined by the default theory (Γ,≤) in Example 1. A consists of all classical inferences from consistent subsets of Γ, including:

• A1 = ({b, ¬a ∨ ¬b}, ¬a), A2 = ({b, a ∧ b → ¬c, c}, ¬a), B1 = ({a, ¬a ∨ ¬b}, ¬b), B2 = ({a, a ∧ b → ¬c, c}, ¬b), C = ({a, b}, ¬(¬a ∨ ¬b)), D = ({a, b, c}, ¬(a ∧ b → ¬c)), E = ({a, b, a ∧ b → ¬c}, ¬c), and; Y1 = ({¬a ∨ ¬b}, ¬a ∨ ¬b), Y2 = ({a}, a), Y3 = ({b}, b), Y4 = ({a ∧ b → ¬c}, a ∧ b → ¬c), Y5 = ({c}, c)
• A1 and A2 attack Y2, B1, B2, C, D, E on Y2. B1 and B2 attack Y3, A1, A2, C, D, E on Y3. C attacks Y1, A1, B1 on Y1. D attacks Y4, A2, B2, E on Y4. E attacks Y5, D on Y5.
• Only the attacks from A1, B1 and E succeed as defeats, since A2 ≺ Y2, B2 ≺ Y3, C ≺ Y1 and D ≺ Y4. There are two stable extensions:

We recall Theorem 34 in [Modgil and Prakken, 2013], employing the notation args(Σ) = {X ∈ A|prem(X) ⊆ Σ}:

Theorem 3 Let (A,⇒) be the AF defined by (Γ,≤). Then: If E is a stable extension of (A,⇒) then prem(E) is a preferred subtheory of (Γ, ≤);

If Σ is a preferred subtheory of (Γ, ≤) then args(Σ) is a stable extension of (A,⇒).

Hence there is a ps Σ of (Γ,≤) such that α is classically entailed from Σ iff there is an argument in a stable extension of (A,⇒) that concludes α. Hence:

2.3 Dialectical Characterisation of Preferred Subtheories

The consistency and closure rationality postulates [Caminada and Amgoud, 2007], and results in Theorem 3, are satisfied assuming that all arguments (which are in general infinite) that can be constructed from a set of formulae ('base') Γ, are included in an AF , and that the premises of each constructed argument is checked for consistency. These assumptions are clearly not feasible for real-world agents with bounded resources. Moreover, the inconsistency of arguments' premises is typically demonstrated dialectically (as in the classic Socratic move whereby an interlocutor is shown to commit to inconsistent premises). Hence [D'Agostino and Modgil, 2016; 2018] develop a dialectical account of Cl-Arg that is provably rational under minimal assumptions as to the arguments that need be constructed, and that eschews the need to check for consistency of premises; inconsistency is demonstrated dialectically. The key insight is to account for a distinction ubiquitous in dialectical practice, whereby an argument entails a conclusion from assumptions regarded as premises assumed to be true, and assumptions that are supposed true for the 'sake of argument' (i.e., those premises that an interlocutor commits to).

Definition 3 X = (∆, Φ, α) is a dialectical argument defined by Γ, if (∆∪Φ) ⊆ Γ, ∆∩Φ = ∅, and ∆∪Φ ⊢c α.

• ∆, Φ and α are respectively X's premises, suppositions and conclusion, denoted prem(X), supp(X) and conc(X). X is unconditional if supp(X) = ∅. Also, assum(X) = prem(X) ∪ supp(X) denotes X's assumptions. The preceding notation is generalised to sets of arguments in the obvious way.
• Let [X] = {(∆′ , Φ′ , α)|∆′ ∪ Φ′ = ∆ ∪ Φ}.Then ∀Y, Z ∈ [X], Y and Z are said to be logically equivalent. That is to say, [X] denotes all 'epistemic variants' in which X's assumptions are differentially labelled according to whether they are referenced as premises or suppositions.
• Any strict preference relation ≺ over arguments is then assumed to be invariant modulo logical equivalence (imle): ∀X′ ∈ [X], ∀Y ′ ∈ [Y ], X ≺ Y iff X′ ≺ Y ′ . ( i.e., the strength of an argument is invariant with respect to the distinction between premises and suppositions).

Any Z challenging the acceptability of X w.r.t. a set of arguments S, can suppose the truth of premises of any arguments in S ∪ {X}. In turn, some Y ∈ S that defends X, can counter Z by supposing the truth of any premises in Z.

For example, suppose S contains X = ({a}, ∅, a) and Z = ({b, ¬a ∨ ¬b}, ∅, ¬a) defeats X (assuming Z ⊀ X). Then Y = ({a}, {¬a ∨ ¬b}, ¬b) ∈ S can, without committing to the truth of ¬a ∨ ¬b as a premise, suppose the truth of Z's premise ¬a ∨ ¬b to defeat Z on b, and so defend X. Also, suppose some S containing arguments X1 and X2 that respectively have premises ¬a ∨ ¬b and a, b. Then Z = (∅, {¬a ∨ ¬b, a, b}, ⊥) dialectically demonstrates that X1 and X2 in S make use of inconsistent premises, by attacking X1 on ¬a ∨ ¬b and X2 on a and b. These attacks succeed as defeats independently of preferences since such a demonstration should clearly not be contingent on preferences. One cannot then defend against Z's defeats as Z cannot be defeated by virtue of its empty premises. Now, suppose Z ′ = ({¬a ∨ ¬b, a, b}, ∅, ¬c) defeats ({c}, ∅, c) ∈ S. The 'contaminating' effect of Z ′ , exploiting the explosivity of its inconsistent premises to entail the syntactically disjoint

¬c, is avoided by Z ∈ S defeating Z ′ (independently of preferences given that Z demonstrates the inconsistency of Z ′ 's premises). Note that Z can be included in any admissible extension as it cannot be defeated given its empty premises.

Definition 4 Let X = (∆, Φ, α), Y = (Π, Σ, β) ∈ A. Then:

• if α = ⊥, X attacks Y on any φ ∈ Φ ∩ Π;
• if α ̸= ⊥, X attacks Y on -α ∈ Π.
• The defeat relation D ⊆ 2A is defined as follows: (X, Y ) ∈ D if X attacks Y on φ, and either X is of the form (∅, Φ, ⊥), or X ⊀ ({φ}, ∅, φ).
• Then X defeats Y with respect to S, denoted X ⇒S Y , if (X, Y ) ∈ D and supp(X) ⊆ prem(S ∪ {Y }).
• Y is acceptable w.r.t. S if ∀X s.t. X ⇒S Y , ∃Z ∈ S s.t. Z ⇒{X} X. One then defines conflict free sets (with '⇒E' replacing '⇒'), and admissible, complete, preferred and stable (with '⇒{Y }' replacing '⇒') extensions of a dialectical AF (A, D) in the same way as in Definition 2. Then:
• (A,D) |∼sc S φ ((A,D) |∼cr S φ) if every (at least one) S extension contains an unconditional argument concluding φ. 2

Suppose A is any set of dialectical arguments defined by Γ, that satisfies the following:

• P1 If α ∈ Γ then ({α}, ∅, α) ∈ A;
• P2 If X ∈ A then ∀X′ ∈ [X], X′ ∈ A (i.e, if one can construct X from Γ, then one has access to all logical equivalents of X that differentially label the assumptions);
• P3 If (∆, Γ, α), (Π,Φ,-α) ∈ A then (∆∪Π, Γ∪Φ,⊥) ∈ A (i.e, if one can construct arguments with conflicting conclusions then one can combine their assumptions to yield an argument concluding ⊥).

Then [D'Agostino and Modgil, 2018] show satisfaction of the consistency and closure postulates assuming any preference relation. The crash resistance and non-interference rationality postulates [Caminada et al., 2012] are satisfied if: 1) whenever X in A includes premises redundantly used to entail the conclusion (since subset minimality is not enforced) by virtue of these premises being syntactically disjoint from the remaining premises and the conclusion, then resources suffice to construct the non-redundant argument excluding these premises; 2) ≺ is such that adding redundant premises does not strengthen an argument (this is satisfied by ≺ as defined in Definition 2).

One can then use dialectical Cl-Arg to give a fully rational characterisation of Preferred Subtheories inference for agents who do not have unlimited inferential capabilities:

Definition 5 Let ⊢r ⊆ ⊢c be any resource-bounded approximation of classical consequence such that ∀∆: 1) if β ∈ ∆

then ∆ ⊢r β; 2) if ∆ ⊢r α and ∆ ⊢r ¬α then ∆ ⊢r ⊥.

∆ is said to be r-inconsistent iff ∆ ⊢r ⊥; else r-consistent. The r-preferred subtheories of (Γ,≤) are then defined as in Definition 1, with 'r-consistent' replacing 'consistent'.

Theorem 4 Let (A,D) be the dialectical AF defined by (Γ,≤), where X ∈ A iff assum(X) ⊢r conc(X), and D

2 The inferences are conclusions of unconditional arguments that commit to the truth of all their assumptions as premises.

is defined based on the strict preference relation: X ≺ Y if ∃α ∈ assum(X), ∀β ∈ assum(Y ), α < β.

Then (letting Args(Σ) denote {X ∈ A|prem(X) ⊆ Σ}):

If E is a stable extension of (A,D), then Σ = ⋃X∈E Prem(X) is an r-preferred subtheory of (Γ, ≤).

If Σ is an r-preferred subtheory of (Γ, ≤), then E = Args(Σ) is a stable extension of (A, D).

Theorem 4 implies correspondences between resourcebounded Preferred Subtheories and argumentation based inference as formulated at the end of Section 2.2 (with '(A, D)' substituting for '(A,⇒)').

Example 5 Assuming (Γ, ≤) from Example 1, let ⊢r⊆⊢c be such that A is the following set of dialectical arguments:

A = ({b, ¬a ∨ ¬b}, ∅, ¬a), B = ({a, ¬a ∨ ¬b}, ∅, ¬b),

F = ({a, b, a∧b → ¬c, c}, ∅, ⊥)}, Y1 = ({¬a∨¬b}, ∅, ¬a∨¬b),

Y2 = ({a}, ∅, a), Y3 = ({b}, ∅, b), Y4 = ({a ∧ b → ¬c}, ∅, a ∧

b → ¬c), Y5 = ({c}, ∅, c) and their logical equivalents (e.g.,

Z = (∅, {a, b, ¬a ∨ ¬b}, ⊥) ∈ [D]).

(A, Y2) ∈ D and (B, Y3) ∈ D, and so both (A, X), (B, X) ∈ D for X ∈ {C,D,Z ′ ,E, F}. Also, (E, Y5), (Z ′ , Y5) ∈ D and (Z, D), (Z, Z ′ ) ∈ D. There are two stable extensions:

Note that each stable extension includes Z.

3 Preferred Equals Stable

The above correspondences allow one to identify credulous ps inferences with the conclusions of (unconditional) arguments in stable extensions of a (dialectical) AF defined by (Γ,≤). However, it would ideally suffice to identify these inferences with conclusions of arguments in admissible extensions, since: 1) there are numerous works establishing argument game proof theories and algorithms for deciding membership of admissible extensions (e.g., [Modgil and Caminada, 2009]), which moreover need not consider all arguments and attacks/defeats in an AF (unlike when deciding membership of a stable extension; see footnote 3); 2) argumentation based dialogues formalising distributed nonmonotonic reasoning typically involve showing membership of an argument in an admissible extension (of the AF incrementally defined by contents of locutions, e.g., [Prakken, 2005]). However, it is not always the case that an admissible extension is a subset of a stable extension 3 . Hence, for practical purposes it would be useful to also establish the previous section's correspondences for the preferred semantics. That is, if we can show that credulous ps inferences correspond to the conclusions of (unconditional) arguments in preferred extensions of a (dialectical) AF, we can then utilise the above argument game proof theories and dialogues for individual and distributed credulous ps reasoning, since any admissible extension is a subset of a preferred extension 4 . We now establish these correspondences for the standard and dialectical approaches, by showing that in both cases the preferred and stable extensions coincide.

3 E.g., consider the AF consisting of X and Y where Y defeats itself. Then {X} is admissible (and preferred) but there is no stable extension. This also illustrates the need to account for the whole AF when deciding membership of a stable extension.

3.1 Result for Standard Cl-Arg Characterisation

We now show that given an AF defined by (Γ, ≤) in Section 2.2, the preferred and stable extensions coincide. Straightforwardly, any stable extension of a Dung AF is preferred [Dung, 1995]. It therefore suffices to show that each preferred extension is stable. We make use of the following lemma.

Lemma 6 Let E be a complete extension of an AF (A,⇒) defined by (Γ,≤). Then ∀X ∈ A such that prem(X) ⊆ prem(E), X ∈ E.

Proposition 7 Let (A,⇒) be defined by (Γ,≤). If E is a preferred extension of (A,⇒) then E is a stable extension of (A, ⇒).

PROOF. Let E be a preferred extension of (A,⇒). Given Theorem 3, it suffices to show that Σ = prem(E) is a preferred subtheory (ps). Firstly, Σ must be consistent, since if Σ ⊢c ⊥, then ∀α: Σ ⊢c α, ¬α. Hence ∃Σ′ ,Σ′′ ⊆ Σ such that (by Lemma 6) (Σ′ , α), (Σ′′ , ¬α) ∈ E, contradicting direct consistency (Theorem 14 in [Modgil and Prakken, 2013]). Now, let Σ = Σ1, . . . ,Σn where for i = 1, . . . , n, Σi = prem(E) ∩ Γi. Suppose for contradiction that Σ is not a ps. Without loss of generality, assume:

i-1 ⋃ j=1 Σj )∪Σi is consistent but not maximally consistent

(2) and so let Λ ̸= ∅ be any maximal subset of Γi \ Σi s.t.:

We show that E ∪ E′ is admissible, contradicting E is a preferred extension. We first show that E ∪ E′ is conflict free.

i) Suppose X ∈ E, Y ∈ E′ , X ⇒ Y on α. It must be that α ∈ Λ else X defeats some Z ∈ E on α, contradicting E is conflict free. By Eq. 3, ⋃ij=1 Σj ⊬ -α. Hence, ∃β ∈ prem(X) s.t. β ∈ Σj>i. But then X ≺ ({α}, α), contradicting X ⇒ Y .

ii) Suppose X ∈ E, Y ∈ E′ , Y ⇒ X on α. Since E is admissible, ∃Z ∈ E, Z ⇒ Y . But then this leads to a contradiction as shown in i).

iii) Suppose X ∈ E′ , Y ∈ E′ , X ⇒ Y on α ∈ Λ. By Eq. 3, ⋃ij=1 Σj ∪ Λ ⊬ -α. Hence ∃β ∈ prem(X) s.t. β ∈ Σj>i.

4 This result is shown for Dung AFs in [Dung, 1995], and for dialectical AFs in [D'Agostino and Modgil, 2018].

But then X ≺ ({α}, α), contradicting X ⇒ Y . iv) Suppose X ∈ E′ , Y ∈ E′ , X ⇒ Y on α ∈ prem(E). Then ∃Z ∈ E, X ⇒ Z and we are in case ii).

We show that all arguments Y ∈ E′ are acceptable w.r.t. E ∪ E′ . Consider the following mutually exclusive cases.

1) X ⇒ Y on α, where at least one premise of X is in Γj>i. If α ∈ Λ, then X ≺ ({α}, α), contradicting X ⇒ Y . Hence α ∈ prem(E). Hence ∃Z ∈ E, X ⇒ Z on α. Since E is admissible, ∃W ∈ E s.t. W ⇒ X. Hence Y is acceptable w.r.t. E ∪ E′ .

1. X ⇒ Y on α, where all of X's premises are in Γj , j ≤ i: 2.1) Suppose prem(X) ⊆ prem(E) ∪ Λ. By Eq. 4, X ∈ E ∪E′ , contradicting E ∪E′ is conflict free.
2. 2.2) Hence ∃β ∈ prem(X) s.t. β in ⋃ij=1 Γj-(⋃ij=1 Σj∪Λ). 2.2.1) Let β ∈ Γj - Σj for some j = 1, . . . , i - 1. By Eq.1:

Hence ⋃jk=1 Σk ⊢ -β and ∃Z s.t. prem(Z) ⊆ ⋃jk=1 Σk, conc(Z) = -β, and Z ⇒ ({β}, β) (it cannot be that Z ≺ ({β}, β) given that β ∈ Γj - Σj). Hence Z ⇒ X. Moreover, by Lemma 6, Z ∈ E. Hence Y is acceptable w.r.t. E ∪ E′ .

2.2.2) Let β ∈ Γi - (Σi ∪ Λ). Since ⋃ij=1 Σj ∪ Λ is maximally consistent (Eq. 3), ⋃ij=1 Σj ∪ Λ ⊢ -β. By Eq. 4, ∃Z ∈ E ∪ E′ , prem(Z) ⊆ ⋃ij=1 Σj ∪ Λ, conc(Z) = -β, and Z ⇒ ({β}, β) (it cannot be that Z ≺ ({β}, β) since β ∈ Γi). Hence Z ⇒ X, and Y is acceptable w.r.t. E ∪ E′ . QED

3.2 Result for Dialectical Characterisation

We now show that given a dialectical AF (A, D) defined by (Γ, ≤) in Section 2.3, where A satisfies P1-P3, then the preferred and stable extensions coincide (it is straightforward to show that stable implies preferred for dialectical AFs). We make use of Lemma 9 in [D'Agostino and Modgil, 2018]:

Lemma 8 Let E be a complete extension of a dialectical AF (A,D) where A satisfies P1. Then ∀X ∈ A such that prem(X) ⊆ prem(E), X ∈ E.

Proposition 9 Let (A,D) be defined by (Γ,≤), where (∆,Φ, α) ∈ A iff ∆ ∪ Φ ⊢r α. If E is a preferred extension of (A, D) then E is a stable extension of (A, D).

PROOF. Let E be a preferred extension of (A,D). Given Theorem 4, it suffices to show that Σ = prem(E) is a rpreferred subtheory (r-ps). Firstly, Σ is r-consistent, since if Σ ⊢r ⊥, then ∃Σ′ ⊆ Σ such that (by Lemma 8), (Σ′ , ∅,⊥) ∈ E, contradicting direct consistency (Theorem 11 in [D'Agostino and Modgil, 2018]).

Now, let Σ = Σ1, . . . ,Σn where for i = 1, . . . , n, Σi = prem(E)∩Γi. Suppose for contradiction that Σ is not a r-ps. Without loss of generality assume that:

( i-1 ⋃ j=1 Σj )∪Σi is r-consistent but not maximally r-consistent

(6)

and so let Λ ̸= ∅ be any maximal subset of Γi \ Σi s.t.:

We show that there exists an admissible E∗ ⊃ E ∪ {({α}, ∅, α)|α ∈ Λ}, contradicting E is a preferred extension of (A,D). Let

We first show that E∗ is conflict free. Suppose for contradiction that X, Y ∈ E∗ , X ⇒E∗ Y on ({β}, ∅, β). By construction of E∗ , and supp(X) ⊆ prem(E∗ ) (given Y ∈ E∗ and X ⇒E∗ Y ), assum(X) ∪ {β} ⊆ prem(E′ ).

• Suppose β ∈ ⋃ij=1 Σj ∪ Λ (recall that Λ ⊆ Γi). Then assum(X) ⊆ ⋃ij=1 Σj ∪ Λ (else X ⇏E∗ Y ). If conc(X) = ⊥ then this contradicts Eq. 7. If conc(X) = -β, then β, -β and so ⊥ are all ⊢r entailed from ⋃ij=1 Σj ∪ Λ, contradicting Eq. 7.
• Suppose β ∈ ⋃nj=i+1 Σi and so β ∈ prem(E), and by P1 B = ({β}, ∅, β) ∈ A. Let X = (Π, Ω, φ).

Suppose that (Π∪Ω) ∩Λ = ∅. Then Π∪Ω ⊆ prem(E), and by P2, X′ = (Π ∪ Ω, ∅, φ) ∈ A. By Lemma 8, X′ , B ∈ E. But then X′ ⇒E B, contradicting E is conflict free. Hence (Π ∪ Ω) ∩ Λ ̸= ∅.

Let Θ = (Π∪Ω)∩Λ. By P2, X′ = (Θ, (Π∪Ω)\Θ), φ) ∈ A. Since (Π ∪ Ω) \ Θ ⊆ prem(E) and β ∈ prem(E), we have that X′ ⇒E Y ′ on β, for some Y ′ ∈ E. Since E is admissible, ∃Z ∈ E s.t. Z ⇒{X′ } X′ on some α ∈ Θ ⊆ Λ. Hence Z ⊀ ({α}, ∅, α) and so assum(Z) ⊆ ⋃ij=1 Σj ∪ Λ. If conc(Z) = ⊥ then this contradicts Eq. 7. If conc(Z) = -α, then -α, α and so ⊥ are ⊢r entailed by ⋃ij=1 Σj ∪ Λ, contradicting Eq. 7.

We now show that if Y ∈ E∗ then Y is acceptable w.r.t. E∗ :

1. Suppose X = (∆, Ψ, φ) ⇒E∗ Y on δ ∈ prem(E). By P1 and Lemma 8: Y ′ = ({δ}, ∅, δ) ∈ E.
2. 1.1) Suppose X ⇒E Y ′ . By the admissibility of E, ∃Z ∈ E (and so Z ∈ E∗ ) such that Z ⇒{X} X.

1.2) Suppose X ⇏E Y ′ . Since Λ = prem(E∗ ) \ prem(E), then (ΨΛ = Ψ ∩ Λ) ̸= ∅, and:

By admissibility of E, ∃Z = (Π, Φ, γ) ∈ E, Z ⇒{X′ } X′ .

1.2.1) Suppose Z ⇒{X′ } X′ on α ∈ ΨΛ. Hence Z ⊀ ({α}, ∅, α). Since α ∈ Γi, it must be that Π ⊆ ⋃ij=1 Σj .

Moreover, if Φ ⊆ ⋃ij=1 Σj ∪ Λ, then either ⊥ is ⊢r entailed from ⋃ij=1 Σi ∪ Λ (in the case that γ = ⊥), contradicting Eq. 7, or (in the case that γ = -α), -α, α and so ⊥ are ⊢r entailed by ⋃ij=1 Σj ∪ Λ, contradicting Eq. 7.

Hence, ∃β ∈ Φ s.t. β ∈ ⋃ij=1 (Γj \ Σj) \ Λ (if β ∈ Γj>i then this would mean Z ≺ ({α}, ∅, α) and so Z ⇏{X′ } X′ ). Moreover, since β /∈ Λ, then β /∈ ΨΛ, and so β ∈ (Φ ∩∆).

If 1.2.1.1) β /∈ Γi, then by Eq. 5, ⋃i-1 j=1 Σj ∪ {β} ⊢r ⊥, else if 1.2.1.2) β ∈ Γi, then by Eq. 7: ⋃ij=1 Σj ∪ Λ ∪ {β} ⊢r ⊥. In either case, ∃Ω ⊆ prem(E∗ ) s.t. Ω ∪ {β} ⊢r ⊥, where Ω ⊆ ⋃ij=1 Σj ∪Λ. By construction of E∗ , (Ω, {β}, ⊥) ∈ E∗ . Moreover, (Ω, {β},⊥) ⇒{X′ } X′ on β ∈ ∆. Hence (Ω, {β}, ⊥) ⇒{X} X on β ∈ (prem(X) = ∆).

1.2.2) Suppose Z ⇒{X′ } X′ on some α ∈ ∆. By P2, Z ′ = (Π ∪ ΨΛ,Φ \ ΨΛ, γ) ∈ A. By construction of E∗ , Z ′ ∈ E∗ . Since (Φ \ ΨΛ) ⊆ ∆, Z ′ ⇒{X} X on α.

2) Suppose X = (∆,Ψ, φ) ⇒E∗ Y ∈ E∗ , on some δ /∈ prem(E). Hence δ ∈ Λ, and so ∆ ∪ Ψ ⊆ ⋃ij=1 Γj (else X ≺ ({δ}, ∅, δ)). Then either ∆∪Ψ∪ {δ} ⊢r entails ⊥ (in the case that φ = ⊥) or ∆ ∪ Ψ ∪ {δ} ⊢r entails -δ, δ and so ⊥ (in the case that φ = -δ). Hence, by Eq. 7, it must be that ∃β ∈ ∆∪Ψ s.t. β ∈ (⋃i-1 j=1 Γj \ Σj) ∪ (Γi \ (Σi ∪ Λ)). Hence, since Ψ ⊆ prem(E∗ ) ⊆ ⋃ Σni=1Σi ∪ Λ, then β ∈ ∆. Then, as in 1.2.1.1 and 1.2.1.2 above, ∃(Ω, {β},⊥) ∈ E∗ such that (Ω, {β}, ⊥) ⇒{X} X on β ∈ (prem(X) = ∆).

We have shown that if Σ = prem(E) is not a r-preferred subtheory, then this contradicts the assumption that E is a preferred extension QED

4 Foreign Commitment, Sceptical Inference and the Grounded Semantics

The distinction between premises and suppositions adopted in dialectical argumentation, solves the so called 'foreign commitment' problem which arises in dialogues in which distinct agents make commitments when exchanging arguments. Adapting an example in [Caminada et al., 2014], suppose the default theory (Γ1 = {g, g → (¬a ∨ ¬b)}, Γ2 = {a, b}), where g denotes 'budget', and a and b respectively denote 'bob will attend conference A' and 'bob will attend conference B'. Using the standard approach to Cl-Arg, Ag1 submits X = ({a, b}, a ∧ b). However, given that the budget does not support attendance at both conferences, Ag2 would ideally be able to defeat X on its conclusion with ({g, g → (¬a ∨ ¬b)}, ¬(a ∧ b)). However, it is well known that attacks and defeats that target conclusions of classical logic arguments results in violation of the rationality postulates [Caminada and Amgoud, 2007; Modgil and Prakken, 2013]. Hence, Ag2 is forced to commit to one of Ag1's premises, by defeating X on its premises with either Y1 = ({a, g, g → (¬a ∨ ¬b)}, ¬b) or Y2 = ({b, g, g → (¬a ∨ ¬b)}, ¬a). In either case Ag2 is forced to commit to the truth of a proposition that she does not believe in, which is unnatural from the perspective of commitment in dialogue [Walton and Krabbe, 1995]. However, in dialectical Cl-Arg, Ag2 can defeat X = ({a, b}, ∅, a∧b) with either Y1 = ({g, g → (¬a ∨ ¬b)}, {a}, ¬b) or Y2 = ({g, g → (¬a∨¬b)}, {b}, ¬a), neither of which commit to the truth of Ag1's premises, by deploying them as suppositions.

Moreover, notice that the sceptical ps inferences in Example 1 include c, corresponding to c being an argumentation defined sceptical inference under the stable/preferred semantics in Examples 2 and 5. Now, ideally one would identify the sceptical ps inferences of (Γ,≤) with the conclusions of arguments in the single grounded extension of the (dialectical) AF defined by (Γ,≤), since computing the grounded extension 5 is computationally far less demanding than computing sceptical inference under preferred/stable semantics [Dunne and Wooldridge, 2009]. However, in Example 2 Y5 = ({c}, c) is not in the grounded extension EG. To see why, observe that although Y1 = ({¬a∨¬b}, ¬a∨¬b) ∈ EG, Y1 does not defend against E's defeat on Y5, since Y1 does not individually target either a or b in E's premises. If one were to then allow attacks targeting multiple premises (i.e., a and b), then this leads to violation of the consistency postulates [Gorogiannis and Hunter, 2011]. Hence one can only defeat E and so defend Y5, with A1 or B1, neither of which are in EG. On the other hand, in the dialectical approach one can effectively simulate attacks on multiple premises. In Example 5, the grounded extension includes Y1 = ({¬a ∨ ¬b}, ∅, ¬a ∨ ¬b) and the logical equivalents A′ = ({¬a ∨ ¬b}, {b}, ¬a), B′ = ({¬a ∨ ¬b}, {a}, ¬b) (which only commit to the truth of ¬a ∨ ¬b) of A and B respectively. A′ and B′ defeat E = ({a, b, a ∧ b → ¬c}, ∅, ¬c) on a and b respectively, so defending Y5 = ({c}, ∅, c) which is therefore in the grounded extension. Hence the grounded semantics for dialectical Cl-Arg are less sceptical than for the standard approach. That is to say, assuming (dialectical) AFs (A,⇒) ((A,D)) defined by default theories (Γ,≤) (as in Sections 2.2 and 2.3), and writing 'g' to denote grounded: ∀(Γ,≤) (A,⇒) |∼sc g φ implies (A,D) |∼sc g φ;

Unfortunately, (Γ,≤) |∼sc ps φ iff (A,D) |∼sc g φ does not in general hold for dialectical Cl-Arg because of the well known syntax sensitivity of syntactic approaches to non-monotonic inference.

Example 10 Suppose: Γ1 = {(¬a∨¬b)∧d, (¬a∨¬b)∧¬d}; Γ2 = {a, b, (a ∧ b) → ¬c, c}. Then (Γ,≤) |∼sc ps c (there are four preferred subtheories, none of which contain both a and b and so entail ¬c). However, the grounded extension of the dialectical AF no longer contains arguments with premise ¬a ∨ ¬b (and suppositions {a} or {b}) that then defeat ({a, b, a∧b → ¬c}, ∅, ¬c) and so defend ({c}, ∅, c). This is because any argument that commits to the truth of ¬a∨¬b in the premises, must also commit to d or ¬d, and the ordering (preferences) do not arbitrate between the latter. It is this somewhat unnatural coupling of the 'fate' of ¬a ∨ ¬b with d (¬d) through use of the conjunction connective, that disrupts the correspondence between sceptical ps inference and sceptical inference under the grounded semantics.

5 Which in both standard and dialectical approaches can (starting with the empty set) be computed by iteratively adding arguments acceptable w.r.t. the previous iteration, until reaching a fixed point that is the grounded extension.

5 Conclusions

Argumentative characterisations of Preferred Subtheories have been developed using both standard approaches to ClArg, and dialectical approaches more suited for real-world resource-bounded agents. This paper is the first to establish these characterisations for the preferred semantics, by showing that the preferred and stable extensions of (dialectical) AFs instantiated by default theories coincide.These results imply that argument game proof theories and algorithms for deciding membership of arguments in admissible extensions, suffice for realising the practical use of argumentation in deciding credulous Preferred Subtheories inference by single agents, and in distributed reasoning amongst multiple agents. However, it should be noted that argument game proof theories for Dung AFs cannot straightforwardly be applied to dialectical AFs. In particular, argument games involve a proponent submitting arguments that define an admissible extension S (containing the argument whose membership in an admissible extension is being decided), so as to defend against arguments moved by an opponent that defeat proponent's submitted arguments (e.g., [Modgil and Caminada, 2009]). Future work is therefore required to adapt such proof theories so that defeats moved by the opponent are parameterised so as to reference (the premises of arguments in) the set S that the proponent has thus far committed to.

Moreover, we have shown that under the grounded semantics, the dialectical approach more closely approximates sceptical Preferred Subtheories inference, as compared with the standard approach. We also conjecture that a full correspondence fails only in 'pathological' cases in which syntactically unrelated formulae are conjoined. Future work will more precisely articulate and verify this conjecture. Finally, we note that [Thang and Luong, 2014] provide argumentative characterisations of Preferred Subtheories inference using assumption based argumentation [Bondarenko et al., 1997], and do so for the more general case in which the preferred subtheories are defined by all total orderings that complete a given partial ordering over a set Γ of classical wff. It remains to study characterisations of this more general case using standard and dialectical approaches to Cl-Arg.