A Postulate-Driven Study of Logical Argumentation

Logical argumentation is a well-known approach to modeling non-monotonic reasoning with conflicting information. In this paper we provide a comprehensive postulate-based study of properties of logical argumentation frameworks and a full characterization of their semantics and inference relations. In this way we identify well-behaved formal argumentative models of drawing logically justified inferences from a given set of possibly conflicting defeasible, as well as strict assumptions. Given some desiderata in terms of rationality postulates, we consider the conditions that an argumentation framework should fulfill for the desiderata to hold. One purpose of this approach is to assist designers to “plug-in” pre-defined formalisms according to actual needs. To this end, we present a classification of argumentation frameworks relative to the types of attacks they implement. In turn, for each class we determine which desiderata are satisfied. Our study is highly abstract, supposing only a minimal set of requirements on the considered underlying deductive systems, and in this way covering a broad range of formalisms, including classical, intuitionistic and modal logics.


Introduction
Logical argumentation is a common AI-based method for making inferences in the presence of arguments and counter-arguments (see [8] for an overview). Its setting, called an argumentation framework, consists of two ingredients: • arguments, which are pairs ⟨Γ, ψ⟩ of a set of formulas (the argument's support, Γ) and a formula (the argument's conclusion, ψ) in some propositional language, such that ψ follows from Γ according to some underlying logic, and • attacks, which are instances of a binary relation on the set of arguments, relating arguments and counter-arguments.
Given such a framework, an argumentation semantics [40] determines what arguments can be mutually accepted, and so what conclusions can be drawn from this setting.
The nature of an argumentation framework thus depends on several factors. This includes, among others, the following elements: • The underlying languages and logics (consequence relations), on top of which the arguments are specified: For instance, the works in [24,25,45,48] are based on classical logic as the underlying logic, while those in [4,13,15] consider argumentation frameworks that are based on any Tarskian logics and arbitrary propositional languages (see Definition 1 below).
• The nature of the attack relations, that is: what triggers conflicts between arguments (see, e.g., [24,48,74] for some proposals of attack relations in different contexts).
• The semantics of the framework, namely: criteria for selecting extensions [19,40] of the argumentation framework, which are sets of arguments that can be collectively accepted (without, e.g., attacking each other).
• The induced entailment relation that indicates what conclusions can be drawn, based on the arguments at hand, the attacks among them, and the chosen semantics.
The fact that there are so many possibilities to define logical argumentation frameworks raises the question how to choose the most appropriate framework for specific needs. The purpose of this work is to put some order in this 'jungle' of argumentation frameworks and to provide some guidelines on how to construct robust frameworks for particular purposes. For this, we first specify some criteria for choosing among the candidate frameworks. A common way to do so is by checking the satisfiability of rationality postulates, that is, to consider formal properties that the intended framework should satisfy. In the sequel we consider several types of such postulates, like those that refer to the properties of the extensions of the logical argumentation framework at hand (see, e.g., [29]), and postulates that refer to the properties of the induced entailment relations (see [30,54,55]).
The essence of this work is, therefore, to investigate the interplay between the basic ingredients of logical argumentation frameworks on one hand, and the properties of the frameworks and their entailment relations on the other hand. This allows us to assemble logical argumentation frameworks according to the desired properties that they and their entailment relations should have. As a result, we form an argumentative basis for what Prakken and Vreeswijk [66,77] (inspired by Rescher [70]) call plausible reasoning, namely: "sound (i.e., deductive) reasoning from uncertain premises" [66, p. 286]. 1 In our study, logical argumentation is investigated in the context of sequent-based argumentation [13], a simple and modular deductive argumentation setting, borrowing the proof-theoretic notion of sequents [47] for representing arguments. The incorporation of such notions provides a solid abstract representation of logical argumentation (in the sense that will be described in Section 2, see e.g., Note 1), and furthermore allows to properly construct and reason with arguments (see also [15]). This paper is a revised and largely extended version of the conference papers in [11,12]. It provides a different perspective to earlier works on the subject (e.g., [2,5,48]) in several senses.
1. More postulates are considered 2 and their compatibility (i.e., their mutual satisfaction) is shown.
2. We cover many known Dung-based semantics for argumentation frameworks, including those (like stage or eager semantics) whose postulate-based behavior with respect to some attack relations has not been investigated before. 3 3. Unlike some previous studies on logical argumentation (e.g., [11,48]), we do make a distinction between two types of premises. Strict premises are considered to be certain and therefore they cannot be attacked. Defeasible premises, on the other hand, are plausible yet uncertain assumptions. As such, they can be attacked. Distinguishing between these two types of assumptions has a significant effect on the form of the attack rules and their consequences. 1 This is contrasted with defeasible reasoning (in the technical sense) as "unsound (i.e., defeasible) reasoning from firm premises" [65, p. 262] that is for instance modeled in Reiter's default logic [69] or in instantiations of ASPIC + , in which genuinely defeasible rules are used (see [57,64] for a comparative study of default logic and ASPIC + ). 2 Exhaustiveness (Definition 12), for example, was only defined and justified so far, but it was not characterized elsewhere, in the sense that the precise conditions for its satisfaction were not known (cf. . Moreover, we define variants of some postulates (dubbed "limited" versions), fine-tuned to our generalized setting with strict and defeasible assumptions. 3 The number of Dung-style semantics for argumentation frameworks is continuously growing, thus some semantics of lesser centrality, or those that are heavily based on graph-theoretic intuitions, such as the recursive CF2 [21], are left out. 4. We provide new results on how the nature of the attack rules (subset attacks versus direct attacks) affects the properties of the framework. This is done while avoiding some conditions that are used elsewhere, but which are problematic since they are hard to verify or too restrictive (see Notes 12 and 15). 5. We provide some new characterizations of the form of extensions for different Dung-style semantics (Theorems 1 and 3), as well as representations of the induced entailment relations in terms of inferences by maximally consistent subsets of the premises (Theorems 4 and 5). 6. Several assumptions that are made elsewhere are lifted in our case. For instance, in [2,5] it is assumed that the supports of the arguments are minimal and consistent, (that is, ⟨Γ, ψ⟩ is an argument only if Γ is consistent and there is no set Γ ′ ⊊ Γ such that ⟨Γ ′ , ψ⟩ is a valid argument), and in [48] it is further assumed that the base logic is classical logic. None of these assumptions are made here. See [13,16,39] for a discussion on the advantages of avoiding these assumptions.
It is important to note that each one of the above 6 items is already (partially) addressed in the literature. This paper fills several gaps in the known results. Furthermore, against the background of a uniform setting, it provides an in-depth coverage of properties (in terms of different rationality postulates) and characterizations (of extensions and entailment relations) of logical argumentation frameworks. This is done with respect to a variety of base logics, different kinds of assumptions (strict or defeasible), arguments (whose supports need not be minimal or consistent), several types of attack rules gathered according to their nature (direct attack, subset attacks, etc.), and all the main Dung-style semantics, where the corresponding extensions may be considered as individuals (credulous approach) or as a collective (skeptical or weakly skeptical approach). This allows us to compare logic-based argumentation frameworks with respect to different contexts and settings, demonstrating the advantages and shortcomings in each case.
The rest of the paper is organized as follows. The next section contains some basic notions and notations that are related to (sequent-based) logical argumentation, as well as further definitions concerning these frameworks, needed to configure the setting of this work and obtain its results. We then turn to the heart of the paper, which is divided to three parts: 1. Section 3 is an evaluation of logical argumentation frameworks from a postulate-based perspective. We distinguish between postulates concerning individual extensions (Section 3.1), and those that are concerned with sets of extensions (Section 3.2). At the end of each section we provide a table that summarizes the results in the section and shows what postulates are satisfied by which frameworks. These tables clearly indicate the large diversity among different argumentation frameworks and the crucial role of their ingredients in determining their properties.
Section 4 provides some further examples that demonstrate the results of the above-mentioned study in case of argumentation frameworks based on non-classical logics.
2. Section 5 contains a sequence of characterization theorems that are obtained from the propositions in Section 3. These theorems are of two kinds: those that characterize the extensions according to different semantics of the logical argumentation frameworks (Section 5.1), and those that characterize the entailment relations that are induced by the frameworks (Section 5.2).
3. Section 6 further relies on insights gained in the previous sections for providing some results on the behavior of the argumentation-based entailments. Again, we conduct a postulate-driven study that mainly considers two types of properties: those that are related to the non-monotonic nature of the inferences (Section 6.1) and those that are related to the way inconsistency is tolerated (Section 6.2). As a by-product, we also provide in this section some new results on reasoning with maximal consistency with strict and defeasible premises and abstract base logics.
In Section 7 we discuss related work, and in Section 8 we conclude.

Preliminaries
In this section we describe the setting of our work and its context. In Section 2.1 we review some basic notions concerning logical argumentation frameworks, and in Section 2.2 we express some minimal assumptions on these frameworks that are needed for our results.

Logical Argumentation
In what follows we shall assume that the underlying language L is propositional. Atomic formulas in L are denoted by lower-case letters (p, q, r), formulas are denoted by lower-case Greek letters (ϕ, ψ, δ, γ), sets of formulas are denoted by upper-case (calligraphic) letters (S, T , X ), and finite sets of formulas are denoted by upper-case Greek letters (Γ, ∆, Π, Θ), all of which can be primed or indexed. The set of atomic formulas of L (respectively, the set of atomic formulas appearing in the formulas of S) is denoted Atoms(L) (respectively, Atoms(S)). The set of the (well-formed) formulas of L is denoted WFF(L). The power set of a set S is denoted ℘(S).
In what follows, we shall assume that L contains at least a ⊢-negation operator (¬), satisfying p ̸ ⊢ ¬p and ¬p ̸ ⊢ p (for atomic p), 5 and a ⊢-conjunction operator (∧), for which S ⊢ ψ∧ϕ iff S ⊢ ψ and S ⊢ ϕ. 6 Also, we denote by Γ the conjunction of all the formulas in Γ. Besides ¬ and ∧, L may contain other logical operators, such as disjunction, implication, modal operators, logical primitives, etc.
Definition 2 (closure and consistency). Let L = ⟨L, ⊢⟩ be a logic and let S be a set of L-formulas. The ⊢-closure of S is the set CN L (S) = {ϕ | S ⊢ ϕ}. We say that S is ⊢-consistent, if it does not have any subset whose negation follows from an empty set of assumptions, namely: there are no formulas ϕ 1 , . . . , ϕ n ∈ S for which ⊢ ¬(ϕ 1 ∧ · · · ∧ ϕ n ). 7 Having a logic L = ⟨L, ⊢⟩ at our disposal, we can now define the notion of an argument, based on that logic. In its most general form, such an argument is just a pair ⟨Γ, ψ⟩, the first component of which (Γ) is a finite set of L-formulas which are the argument's premises (or supports), and the other component (ψ) is an L-formula, called the argument's conclusion. A minimal requirement from an argument is that it should be logically valid, that is: its conclusion should follow, according to L, from the premises. In the above notations, this means that Γ ⊢ ψ. Using a well-known terminology from proof theory, this is what is called there a sequent [47]. In our case, the set of premises of a sequent may consist of two kinds of formulas: strict and defeasible ones. Intuitively, the former are formulas that are 'taken for granted' while the latter are formulas that may be challenged (i.e., attacked by other arguments).
Definition 3 (arguments). Let L = ⟨L, ⊢⟩ be a logic, X a ⊢-consistent set of L-formulas (the strict assumptions), and S an arbitrary set of L-formulas (the defeasible assumptions), such that X ∩ S = ∅.
• An L-sequent (sequent for short) is an expression of the form Γ ⇒ ∆, where Γ and ∆ are finite sets of formulas in L and ⇒ is a symbol that does not appear in L.
• An L-argument based on S and X is an L-argument Γ ⇒ ψ, where Γ ⊆ S ∪ X . We denote by Arg X L (S) the set of all the L-arguments based on S and X .
Note 1. Sequents are a general representation of arguments in the sense that the only requirement on these expressions is that their conclusions will logically follow from their supports (according to the underlying logic). In addition to this requirement, it is sometimes also assumed that the argument's support is ⊢-consistent and/or that none of its proper subsets ⊢-entails the arguments' conclusion (see, e.g., [5,25]). As our goal here is to keep the discussion as general as possible, we do not make such restrictions. We refer to [13,16] and the discussion below for further justifications of this choice. Other works on sequent-based argumentation without these restrictions appear, e.g., in [10,15,26,74].
Formal systems for constructing sequents (and so arguments) for a logic L = ⟨L, ⊢⟩ are called sequent calculi [47], denoted here by C. The construction of arguments from simpler arguments is done by means of derivations via the inference rules of the sequent calculus. A sequent is provable (or derivable) in C if there is a derivation for it in C. 9 In what follows we shall assume that the calculus C is sound and complete for its logic (i.e., Γ ⇒ ψ is provable in C iff Γ ⊢ ψ). Note that this implies, in particular, that for given sets S and X , all the elements in Arg X L (S) are C-provable. Just as arguments are constructed by inference rules in C, conflicts (attacks) between arguments are represented in a rule-like manner. Such an attack (or, sequent-elimination) rule consists of an attacking argument (the first condition of the rule), an attacked argument (the last condition of the rule), conditions for the attack (the other conditions of the rule) and a conclusion (the eliminated attacked sequent). The outcome of an application of such a rule is that the attacked sequent is 'eliminated' (or 'invalidated', see below the exact meaning of this). The elimination of a sequent Γ ⇒ ϕ is denoted by Γ ̸ ⇒ ϕ. In sum, the general scheme for attack rules is as follows: We now define the attack rules that are central to our study.
Definition 4 (attack rules). Given a set X of strict (non-attacked) formulas, we consider the following attack rules: where γ ̸ ∈ X • Undercut (Ucut X ): where γ ̸ ∈ X • Consistency Ucut (ConUcut X ): The rules above indicate different cases in which the attacker challenges the attacked argument. For instance, when {p, ¬p} ⊆ S and classical logic (CL) is the core logic, the sequents p ⇒ p and ¬p ⇒ ¬p attack each other according to (Direct) Defeat and (Direct) Undercut. In contrast, the tautological sequent ⇒ ψ ∨ ¬ψ is not (direct) defeated or (direct) undercut by any argument in Arg X CL (S), since it has an empty support set. We also note that the attack rules differentiate between defeasible and strict assumptions by only allowing attacks in the former. E.g., when {¬p} ⊆ S and X = {p} the attack is uni-directional from p ⇒ p to ¬p ⇒ ¬p, since an argument cannot be attacked in its strict premises. Note 2. In the particular case where Γ 1 is empty, Consistency Undercut indicates that arguments with an inconsistent set of premises are attacked by tautological arguments. In relation to Note 1, then, support inconsistency may be handled by attack rules of the frameworks rather than simply ruled out (we refer to [16] for a further discussion on this). Note 3. When X = ∅, the rules in Definition 4 coincide with those of [13,74]. There, the reader can also find many other attack rules. In [26] sequent-based argumentation has been generalized along similar lines, where the left side of a sequent is a pair Π | Γ consisting of a set of defeasible premises Π and a set of strict premises Γ.
An argumentation framework is now defined as follows: Definition 5 (argumentation frameworks). A (sequent-based) argumentation framework (AF), based on a logic L and a set A of attack rules, for a set of defeasible premises S and a ⊢-consistent set of strict premises X such that S ∩X = ∅, is a pair AF X L,A (S) = Arg X L (S), A , where A ⊆ Arg X L (S)×Arg X L (S) and (a 1 , a 2 ) ∈ A iff there is an R ∈ A such that a 1 R-attacks a 2 (that is, (a 1 , a 2 ) is an instance of the relation R). 10 A pair (a 1 , a 2 ) is an instance of an attack rule R in case that R can be instantiated in such a way that a 1 is the attacking sequent, a 2 is the attacked sequent, and the sequents in the condition of R (if any) can be derived in the underlying calculus C (in particular, they need not be in Arg X L (S)).
The subscripts L, A and/or the superscript X will be omitted when they are clear from the context or arbitrary. Note that the attacking and the attacked arguments must be elements of Arg X L (S), to prevent "irrelevant attacks", in which, e.g., ¬p ⇒ ¬p attacks p ⇒ p although p ∈ S and ¬p ̸ ∈ S ∪ X . Example 1. Let AF X CL,A (S) = Arg X CL (S), A be an argumentation framework for S = {q, ¬p ∨ ¬q, r} and X = {p}, classical logic (CL) as the base logic, and A is obtained from the attack rules in A, where {ConUcut} ⊊ A ⊆ {DDef, DUcut, ConUcut}. Among others, the following sequents are in Arg X CL (S): The left part of Figure 1 is a graphical representation of the argumentation framework AF X CL,A (S) restricted to the arguments above, and where direct defeat and consistency undercut are the attack rules. The right part of the figure represents the framework with the same attack rules, but when p is a defeasible assumption rather than a strict assumption (i.e., when S = {p, q, ¬p ∨ ¬q, r} and X = ∅). In the figure, nodes represent arguments, and directed edges represent attacks from the attacking to the attacked arguments. Dashed arrows designate the differences between the two parts of the figure. Given an argumentation framework AF, Dung-style semantics [19,40] can be applied to it, to determine what combinations of arguments (called extensions) can collectively be accepted from AF.
Definition 6 (extensions). Let AF = AF X L,A (S) = Arg X L (S), A be an argumentation framework and let S ⊆ Arg X L (S) be a set of arguments. It is said that: • S attacks a if there is an a ′ ∈ S such that (a ′ , a) ∈ A. The set of all arguments attacked by S is denoted by S + . The set S ∪ S + is called the range of S.
• S defends a if S attacks every a ′ such that (a ′ , a) ∈ A.
• S is conflict-free if for no a 1 , a 2 ∈ S it holds that (a 1 , a 2 ) ∈ A.
• S is naive (nav) if it is a ⊆-maximal conflict-free set.
• S is a stage (stg) extension of AF if it is conflict-free and S ∪ S + is ⊆-maximal among the ranges of the conflict-free sets.
• S is admissible in AF if it is conflict-free and defends all of its elements.
• S is a complete (cmp) extension of AF if it is an admissible set that contains all the arguments that it defends.
• S is the grounded (grd) extension of AF if it is the ⊆-minimal complete extension of Arg X L (S).
• S is a preferred (prf) extension of AF if it is a ⊆-maximal complete extension of Arg X L (S).
• S is the ideal (idl) extension of AF if it is the ⊆-maximal admissible set that is included in each preferred extension.
• S is a stable (stb) extension of AF if it is a conflict-free set in Arg X L (S) that attacks every argument not in it (that is, the range of S is the whole set of arguments, Arg X L (S)).
• S is a semi-stable (sstb) extension of AF if it is a complete extension whose range is ⊆-maximal.
• S is the eager (egr) extension of AF if it is the ⊆-maximal admissible set that is included in every semi-stable extension.
We denote by Ext sem (AF) the set of all the extensions of AF of type sem for some sem ∈ {nav, stg, cmp, grd, prf, idl, stb, sstb, egr}.
Note 4. As shown in [40], the grounded extension is unique for a given framework, and as shown in [19], the ideal and the eager extensions are unique complete extensions for a framework. Furthermore, a stable extension is also a semi-stable extension, which in turn is a preferred extension. Other extensions and their properties are discussed, e.g., in [18,19,20]. A graphical representation of the relations among alternative extension-based semantics can be found, e.g., in [19, Figure 13].
On the basis of argumentation frameworks and their semantics we can define two general types of entailment relations: skeptical and credulous ones.
Definition 7 (entailments). Given an argumentation framework AF X L,A (S) and a semantics sem for it, the following entailment relations are induced from them: • Skeptical entailment: S |∼ ∩ L,A,X ,sem ϕ if there is an argument a ∈ Ext sem (AF X L,A (S)) such that Conc(a) = ϕ.
• Weakly skeptical entailment: S |∼ ⋒ L,A,X ,sem ϕ if for every E ∈ Ext sem (AF X L,A (S)) there is an argument a ∈ E such that Conc(a) = ϕ.
For fixed L, A, X and sem, we clearly have that |∼ ∩ L,A,X ,sem ⊆ |∼ ⋒ L,A,X ,sem ⊆ |∼ ∪ L,A,X ,sem . The subscripts L, A, X and sem (or some of them) are omitted when they are clear from the context or arbitrary. Since the grounded, ideal and eager extensions are unique, |∼ ∩ sem , |∼ ⋒ sem and |∼ ∪ sem coincide for a fixed sem ∈ {grd, idl, egr}, so all three are denoted by |∼ sem .
CL,{Ucut} (S) be an argumentation framework for S = {p, ¬p, q}, based on CL and Undercut as the sole attack rule. As noted in the paragraph below Definition 4, the sequent ⇒ p ∨ ¬p belongs to every complete extension of AF ∅ CL,{Ucut} (S), since it cannot be Undercut-attacked. 11 Similarly, q ⇒ q belongs to every complete extension of the framework, since ⇒ p ∨ ¬p counter-attacks

The Scope of the Study
Interestingly, despite the diversity of logics and their sequent calculi covered in this work, only a few assumptions on the sequent calculi are necessary for our results. In fact, we only need to assume that the rules of the basic calculus from Figure 2 are part of (or admissible in) C. Example 4. Gentzen's calculus LK for classical logic, its single-conclusion variation LJ for intuitionistic logic, as well as their extensions to modal logics, are some well-known calculi for base logics that are covered by our study.
We start with some simple lemmas that will be needed for the proofs in what follows. Lemma 1. For a formula ϕ and a finite set of formulas Γ, the sequents ϕ ⇒ ¬¬ϕ and Γ ⇒ Γ are C-derivable.
By Lemma 2, we also have the following result: The proof of (ii) is left to the reader.
The rules of Figure 2 also imply the following necessary requirement for an attack between two arguments: Lemma 5. For any attack rule in Definition 4 it holds that Γ ⇒ γ attacks ∆ ⇒ δ only if ∆ ∪ Γ is inconsistent.
Proof. Suppose that Γ ⇒ γ attacks ∆ ⇒ δ. By [Cut] together with the condition of any of the attack rules in Definition 4, it is easy to see that there is a ∆ ′ ⊆ ∆ for which Γ ⇒ ¬ ∆ ′ is C-derivable. By Lemma 2, ⇒ ¬ (Γ ∪ ∆ ′ ) is also C-derivable. Hence, by the soundness of C, Γ ∪ ∆ is inconsistent.
In what follows we distinguish between three types of attacks in argumentation frameworks, denoted set, dir and con.
Definition 8 (types of attack rules). Attack rules are categorized in the sequel as follows: set: A ∩ {Def, Ucut} ̸ = ∅ (i.e., attack rules in which an argument is attacked on a subset of its support, and where at least one of the rules is Undercut or Defeat), Note 6. In the presence of the inference rules in Table 2, Consistency Undercut is admissible if Defeat or Undercut are part of the attack rules. To see this, consider the following two derivations, where we assume that ConUcut can be applied and therefore that Γ 1 ⇒ ¬ Γ 2 and Γ 2 , Γ ′ 2 ⇒ ψ can be derived for some Γ 1 ⊆ X .

ConUcut Assumption
Concerning the categorization of the semantics in our setting, many of the results in what follows will apply to completeness-based semantics, as defined next: Definition 9 (completeness-based semantics). A semantics sem (e.g., one of those in Definition 8) is completeness-based relative to attack type †, iff for every argumentation framework AF with attack rules of type †, it holds that Ext sem (AF) ⊆ Ext cmp (AF).
Note 7. By their definitions, complete, grounded, and preferred semantics are all completeness-based relative to any type of attacks. Since it can be shown that every stable, semi-stable, ideal, and eager extension is always complete (see, e.g., [19]), the same holds for these semantics. In general, naive and stage extensions need not be complete. We will show below (Proposition 1) that for the three types of attacks in Definition 8 (i.e, for every † ∈ {dir, con, set}) also stage semantics is completeness-based. 13 Definition 11 (CS X L (S), MCS X L (S), Free X L (S)). For a logic L = ⟨L, ⊢⟩ a set S of L-formulas, and a ⊢-consistent set X of L-formulas, we say that S is ⊢ X -consistent if S ∪ X is ⊢-consistent. We denote: • CS X L (S): the set of the ⊢ X -consistent subsets of S, • MCS X L (S): the ⊆-maximal sets in CS X L (S), • Free X L (S): the formulas in S that are not part of any ⊆-minimal ⊢ X -inconsistent subset of S. 14 Lemma 6. Let AF X L,A (S) be a framework with attack rules of one of the types in Definition 8 and let T ∈ MCS X L (S). Then Arg X L (T ) ∈ Ext stb (AF X L,A (S)).
Proof. The conflict-freeness of Arg X L (T ) follows from Lemma 5. Consider now an argument a ∈ Arg X L (S) \ Arg X L (T ). If there is no such argument then Arg X L (T ) is clearly stable. We show that a is attacked by Arg X L (T ). Indeed, by the choice of a, there is a formula ϕ ∈ Supp(a) \ T . By the maximal ⊢ X -consistency of T , by the compactness of L and by Lemma 4 (ii), there is a Γ ⊆ T ∪ X for which Γ ⇒ ¬ϕ ∈ Arg X L (T ). The latter attacks a according to (D)Ucut and DDef (note that ¬ϕ ⇒ ¬ϕ is derivable by Reflexivity). 12 The exclusion of cmp from ME is explained in Note 22. 13 We note that naive semantics is not completeness-based (see, for instance, Footnote 11 for a counter-example). 14 It is well-known (and easily verified) that Free X L (S) consists exactly of those formulas in S that are contained in every maximal ⊢ X -consistent subset of S, namely: Free X L (S) = MCS X L (S) (see, for instance, [71, p. 186], [23, p. 24]).

Proposition 1.
If AF is a framework with attacks of one of the types in Definition 8, then Ext stb (AF) = Ext stg (AF) = Ext sstb (AF).
Proof. The directions Ext stb (AF) ⊆ Ext stg (AF) and Ext stb (AF) ⊆ Ext sstb (AF) are trivial. For the converse, let sem ∈ {stg, sstb} and suppose that E ∈ Ext sem (AF). By Lemma 6, there is a E ′ ∈ Ext stb (AF). 15 Since E ′ is conflict-free and E ′ ∪ E ′ + = Arg(AF), also E ∪ E + = Arg(AF), and so E ∈ Ext stb (AF). Figure 3 summarizes the relations between Dung-style semantics (Definition 6) and their classes (Definition 10). Strict lines indicate inclusion relations that hold in general (see, e.g., [19,40]). Dashed arrows indicate additional relations that hold in our logic-based setting (as shown in Theorem 1 below). As we shall see in the sequel, the main insight concerning extension-based semantics in logicbased argumentation, is that the standard multiple-extension semantics (respectively, the standard singleextension semantics) collapses to a single type.  To summarize, in what follows we consider argumentation frameworks based on any propositional logic L with a sound and complete sequent calculus C, in which the rules in Figure 2 are admissible, the set of premises may contain both strict (X ) and defeasible (S) formulas, the set of attack rules A may be of any of the three types given in Definition 8, and the semantics sem may be any completeness-based ones. To the best of our knowledge, this variety has not been considered previously in systematic studies of meta-theoretic properties of logical argumentation. In the next sections we evaluate these frameworks with respect to different rationality postulates, divided according to their nature.

Evaluation of Logical Argumentation Frameworks
As indicated previously, the definitions of sequent-based argumentation frameworks and the entailment relations induced by them leave plenty of choices to be made in their construction, as the base logic L, the attack rules A, the underlying semantics sem, and the inference method (skeptical, weakly skeptical, and credulous) may vary from one case to another. In this section we check how these choices affect the properties of the frameworks that are obtained and their entailment relations. For this, we consider several desirable properties and then check in what setting they can be warranted. In this section, we consider rationality postulates for the extensions of the frameworks, in Section 5 we show some characterization results, and in Section 6 the properties of the induced entailment relations are investigated.

Postulates Concerning Individual Extensions
First, we consider rationality postulates for completeness-based semantics that are concerned with the properties of individual extensions.
Definition 12 (rationality postulates I). Let AF X L,A (S) = Arg X L (S), A be an argumentation framework, sem a semantics for it, E ∈ Ext sem (AF X L,A (S)), and a ∈ Arg X L (S). In Table 1 we list properties that AF X L,A (S) may have. 16 17 Property Definition  Note 8. The postulates in Table 1 indicate some desirable properties of a framework's extensions.
• The first three postulates state different closure properties. For instance, the first postulate expresses that the set of conclusions of the arguments in an extension is logically closed.
• Support inclusion is a kind of reflexivity condition, demanding that anything that is assumed can be inferred.
• The next three postulates are consistency requirements regarding the supports and the conclusions of an extension. Violations of these postulates mean that contradictory conclusions may be inferred. Pairwise support consistency is a weaker version of support consistency, assuring that the support sets of each pair of arguments in an extension are mutually consistent. We shall see that only this form of consistency can be guaranteed in all circumstances.
• Exhaustiveness and strong exhaustiveness, like the closure postulates, state that what is expected to be in the extension is indeed there and is not left out. In [2], exhaustiveness is justified by the aspiration that 'if each step in an argument [namely, each sub-argument] 18 is good enough to be in a given extension, then so is the argument itself'.
• Free precedence and strong free precedence refer to the containments of the 'safe arguments', that is: those that are based on the formulas in (the intersection of) the consistent subsets of the premises. In other words, these postulates state that any argument that is supported only by formulas that are not involved in any minimal conflict, is included in the extension. The stronger postulate, requiring an absolute identity between the elements of an extension and the safe arguments, might be regarded as too strong, but when it is satisfied all the other postulates are guaranteed (see Figure 4).
• Finally, we also consider limited versions of some of the postulates, restricted to 'non-tautological' extensions.
The postulates in Table 1 are particularly useful in credulous reasoning, which is based on the content of specific extensions. In such cases these postulates assure the plausibility of the conclusion making process (being consistent, closed under logical inferences, etc.). For further descriptions of the postulates in Table 1 and discussions on the intuition behind them, we refer e.g. to [2,28,29].
Note 9. Clearly, some postulates in Definition 12 are related to each other. For instance, sub-argument closure follows from closure under support, pairwise support consistency follows from support consistency, exhaustiveness follows from strong exhaustiveness, free precedence follows from strong free precedence, and the limited version of a postulate follows from the non-limited version of the same postulate. These relations, as well as some other relations between the postulates are presented in Figure 4. Further conditions for relating some of the postulates in Definition 12 are given in [2].
Let us now check the satisfiability of the postulates in Definition 12 (Table 1), in relation to the underlying semantics and the type of the attack rules. In what follows E denotes a sem-extension, for some sem ∈ CMP. When the results are restricted to specific types of attacks or semantics, we assume that E is a sem-extension of frameworks satisfying such restrictions.
Proposition 2 (support consistency). Frameworks with attacks of type dir or con satisfy support consistency for all the completeness-based semantics: Supps(E) is ⊢ X -consistent for every complete extension E ∈ Ext cmp (AF).
Proof. Assume for a contradiction that there is a ⊆-minimal set Θ = {ϕ 1 , . . . , ϕ n } ⊆ Supps(E) for which X ⊢ ¬ Θ. Note that Θ \ X ̸ = ∅, since X is assumed to be ⊢-consistent. By the completeness of C, X ⇒ ¬ Θ is C-provable and by Lemma 2, a = X , ϕ 1 , . . . , ϕ n−1 ⇒ ¬ϕ n is C-provable, where without loss of generality, ϕ n / ∈ X . By the minimality of Θ and the soundness of C, a is not ConUcutattacked. Since for each ϕ i ∈ Θ there is an a i ∈ E for which ϕ i ∈ Supp(a i ), any attacker of a is an attacker of some a i ∈ E. By the admissibility of E, a is defended by E and by the completeness of E, a ∈ E. This contradicts the conflict-freeness of E since a attacks a n (according to both DDef and DUcut).
Note 10. Proposition 2 does not hold for the naive semantics. To see this, note that in the argumentation framework from Example 1, part of which is shown in Figure 1 (left), the set of arguments S = {a 2 , a 3 , a 4 , a 5 , a 6 } is a conflict-free set (and is therefore also part of a ⊆-maximal conflict-free set). However, Supps(S) = {p, q, ¬p ∨ ¬q} is not consistent. We will see that the same counterexample also applies to Corollary 2, Proposition 12, Corollary 13, and Proposition 19.
Note 11. A variation of Example 1 can also be used to show that support consistency does not hold for frameworks with attack rules of type set, thus in Proposition 2 it is essential to consider direct attacks. Indeed, let CL is the base logic, S = {p, q, ¬p ∨ ¬q}, X = ∅, and Defeat or Undercut rather than their direct versions is the attack rule. A graphical representation of the framework with the attack rule Defeat (using the same notations for the arguments as in Example 1) is presented in Figure 5.  Figure 4: Some relations between the postulates for individual extensions in our setting. If a property at the origin of an arrow in the diagram holds relative to classes of argumentation frameworks for a given semantics sem and a given attack type † ∈ {dir, con, set}, then also the property at the end of the same arrow holds (for the specified semantics and attack type). For the dashed arrows sub-argument closure, resp. closure under support, is supposed. In each node label, the lower part indicates for which combinations of attack types † and semantics sem the property holds (as is proved in the propositions in this section).
is a stable (and hence also semi-stable, stage, and preferred) extension, yet Concs(E) and Supps(E) are not consistent, neither is For frameworks with attack rules of type set we get a weaker version of support consistency.
Proposition 3 (pairwise support consistency). Frameworks with set-type attacks satisfy pairwise support consistency for all complete extensions: Note that a ′ has the same attackers as a, and is therefore defended by E. By the completeness of E, a ′ ∈ E as well. Since a ′ Def-attacks and Ucut-attacks b, this contradicts the conflict-freeness of E.
Next we show that with respect to single-extension semantics, support consistency holds for any framework in our setting (including those with attack rules of type set).

Lemma 7.
Let AF X L,A (S) be an S-based framework with any type † ∈ {dir, con, set} of attacks considered in Definition 8, and let Ext sem (AF X L,A (S)) = {E} for sem ∈ SE. Then E ⊆ Arg X L (Free X L (S)).
Proof. Assume for a contradiction that Supps(E) contains a minimally ⊢ X -inconsistent subset {ϕ 1 , . . . , ϕ n } of S (by ⊢-compactness, this set is finite). Let T be a maximal ⊢ X -consistent subset of S. Then there is an Proposition 4 (support consistency II). Any framework AF with set-type attacks satisfies support consistency for every sem ∈ SE: Proof. Follows immediately from Lemma 7.
Note 12. The condition Arg X L (Supps(E)) = E (for every sem-extension E) is frequently assumed in, e.g., [2] for assuring some rationality postulates (in the case where X = ∅). Yet, this condition is not easily verified, and, as Note 11 indicates, it is rather strict, since, in general, frameworks with set-type attacks do not satisfy it for multi-extension semantics. 19 We thus do not assume it for our results (see also Note 15 below).
For showing further results concerning the consistency postulates, we next relate some postulates.
Proposition 5 (support consistency → consistency). Any framework AF with any type † ∈ {dir, con, set} of attacks and any complete semantics sem ∈ CMP satisfies consistency if it satisfies support consistency.
Proof. We show the contraposition. Suppose that consistency does not hold for AF. Then there is a E ∈ Ext sem (AF) for which Concs(E) is ⊢ X -inconsistent. By compactness, there are a 1 , . . . , a n ∈ E for which Θ = {Conc(a 1 ), . . . , Conc(a n )} ⊆ Concs(E), and ⊢ ¬ Θ. By the completeness of C, the sequent ⇒ ¬ Θ is provable, and by Lemma 4, so is the sequent Θ ⇒. By [Cut], we have that Supp(a 1 ), . . . , Supp(a n ) ⇒ is also provable in C. By Lemma 4 and the soundness of C, Supps(E) is ⊢-inconsistent.
Proposition 6 (consistency and closure under support → support consistency). Any framework AF with any type † ∈ {dir, con, set} of attacks and any complete semantics sem ∈ CMP satisfies support consistency if it satisfies consistency and closure under support.
Proof. We show a contraposition of the statement. Suppose that closure under support is satisfied for AF and that for some E ∈ Ext sem (AF), Supps(E) is ⊢ X -inconsistent. By ⊢-compactness, there are a 1 , . . . , a n ∈ E for which {Supp(a 1 ), . . . , Supp(a n )} is ⊢ X -inconsistent. By closure under support and Lemma 1, The following corollaries are a consequence of Propositions 2, 4 and 5.
Corollary 2 (consistency). Frameworks with attack rules of type dir or con satisfy consistency for any complete semantics sem ∈ CMP: For every E ∈ Ext sem (AF), Concs(E) is ⊢ X -consistent.
As the example in Note 10 shows, Corollary 2 does not hold for naive semantics.
Corollary 3 (consistency II). Frameworks with attack rules of type set satisfy consistency for every semantics sem ∈ SE: Concs(E) is ⊢ X -consistent for every E ∈ Ext sem (AF).
As shown in Note 11, consistency does not hold for frameworks with attack rules of type set under multiple extension semantics. By Note 10, it does not hold for naive semantics as well.
We turn now to (strong) free precedence. First, we need the following lemma.
The converse holds by ⊢-monotonicity.
We first show that a limited version of free precedence is satisfied by frameworks with direct attacks under any complete semantics: Proposition 7 (limited free precedence). Frameworks with attack rules of type dir or con satisfy limited free precedence for any complete semantics sem ∈ CMP: If E is an extension of such a framework and . Suppose that either b = ∆ ⇒ ψ attacks a or a attacks b. Then ∆ ∪ Γ is ⊢ X -inconsistent by Lemma 5 and so ∆ is ⊢ X -inconsistent by Lemma 8. By Proposition 2, b / ∈ E. Thus, E ∪ {a} is conflict-free. Also, since ∆ is ⊢ X -inconsistent, by Lemma 2 and [RMon], c = ∆ ⇒ ¬σ ∈ Arg X L (S), and by its definition, c attacks a ′ . Since a ′ ∈ E and E is admissible, E defends a ′ by attacking c, and so E also attacks b. As a consequence, E defends a. Thus, a ∈ E by the completeness of E.
To check (unlimited) free-precedence, the next lemma is useful.
A be an argumentation framework with attack rules of any type † ∈ {dir, con, set} considered in Definition 8, and let E ∈ Ext prf (AF). If Supps(E) ⊆ X then E = Arg X L (∅) and all the formulas in S are ⊢ X -inconsistent.
Proof. By Lemma 6, for any maximal ⊢ X -consistent subset T of S, Arg X L (T ) ∈ Ext stb (AF) and thus Arg X L (T ) ∈ Ext prf (AF). Hence, since Supps(E) ⊆ X , by the ⊆-maximality of E, the only maximal ⊢ X -consistent subset T of S is empty, and so E = Arg X L (∅).
Proposition 8 (free precedence). Any framework AF X L,A (S) with diror con-type attacks satisfies free precedence with respect to every sem ∈ ME: for every Proof. Let E ∈ Ext prf (AF). By Proposition 7, we only need to consider the case that Supps(E) ⊆ X . By Lemma 9, E = Arg X L (∅) and Free X L (S) = ∅. Thus, Arg X L (Free X L (S)) = E. We have shown, then, that Ext prf (AF) = {E} and Arg X L (Free X L (S)) = E. Hence, we also have that Arg X L (Free X L (S)) ⊆ E ′ for every E ′ ∈ Ext sem (AF) (where sem ∈ ME), since by Proposition 1 it holds that Ext stg (AF) = Ext stb (AF) ⊆ Ext sstb (AF) ⊆ Ext prf (AF).
Note 13. For frameworks with only dir-type attack rules (that is, when ConUcut is not one of the attack rules), free precedence does not hold for grounded, ideal and eager semantics (thus, strong free precedence does not hold for these semantics either in this case). To see this, let S = {p ∨ ¬p, p, ¬p}, X = ∅, A = {DDef}, and suppose that the underlying logic is CL. In that case, the grounded (which is also the ideal and eager) extension is Arg X CL (∅). Note that p ∨ ¬p ⇒ p ∨ ¬p is attacked by p, ¬p ⇒ ¬(p ∨ ¬p). In the absence of ConUcut the latter is not attacked by the grounded argument ⇒ ¬(p ∧ ¬p). Yet, Free ∅ CL (S) = {p ∨ ¬p}. We now show that (unlike the case where there are only dir-type attack rules) frameworks with set or con attack rules satisfy unlimited free precedence with respect to every complete semantics.
Proposition 9 (free precedence II). Any framework AF X L,A (S) whose attack rules are of type set or con satisfies free precedence for every complete semantics sem ∈ CMP, that is: Proof. Consider an argument a = Γ ⇒ ϕ in Arg X L (Free X L (AF)) and an argument b = ∆ ⇒ ψ in Arg X L (S) that attacks a or is attacked by a. By Lemmas 5 and 8, ∆ is ⊢ X -inconsistent and so by Lemma 2 (and [RMon] and ⊢-compactness), c = Θ ⇒ ¬ ∆ is in Arg X L (∅) for some Θ ⊆ X . Since c has no attackers it is defended by E and by the completeness of E, c ∈ E. Also, c ConUcut-attacks and Def-attacks b. Thus, E defends a and again by the completeness of E, a ∈ E.
By Lemma 7 and Proposition 9, we obtain the following corollary: Corollary 4 (strong free precedence). Any framework AF X L,A (S) whose attack rules are of type set or con satisfies strong free precedence with respect to every single extension semantics sem ∈ SE. That is:

Corollary 5 (SE collapse I).
For any S-based framework AF whose attack rules are of type set or con, we have: Ext grd (AF) = Ext idl (AF) = Ext egr (AF).
We now consider a special case under which strong free precedence holds for all complete semantics and any attack-types.
Lemma 10. Let AF X L,A (S) be a framework with attack rules of type † ∈ {dir, con, set}. If there are no ⊆-minimal ⊢ X -inconsistent subsets of S of size greater than 1 (namely, every formula in S is either free or inconsistent with X ), then Ext sem (AF X L,A (S)) = {Arg X L (Free X L (S))} for all sem ∈ CMP. Proof. By the supposition we have two types of arguments: (1) those that contain ⊢ X -inconsistent formulas in their support, and (2) those whose support is a subset of Free X L (S) ∪ X . Note that where ∆, ϕ ⇒ ψ is an argument of type (1) with ϕ being a ⊢ X -inconsistent formula, it is attacked by Θ ⇒ ¬ϕ of type (2) for some Θ ⊆ X . We also note that Θ ⇒ ¬ϕ has no attackers. That means that all arguments of type (2) are defended by arguments without attackers, while arguments of type (1) cannot be defended. That means every completeness-based semantics has a unique extension consisting of all arguments of type (2). The rest follows immediately.
Corollary 6 (strong free precedence II). Let AF be an S-based framework with attack rules of type † ∈ {dir, con, set}. AF satisfies strong free precedence for any complete semantics, if there are no ⊆-minimal ⊢ X -inconsistent subsets of S of size greater than 1.
When strong free precedence is satisfied, several other postulates are met.
Proposition 10 (strong free precedence → closure under support, support consistency, free precedence). Any framework AF with attack type † ∈ {dir, con, set} and any complete semantics sem ∈ CMP satisfies closure under support, support consistency and free precedence, if it satisfies strong free precedence.
Proof. Immediate from the definitions of the postulates.
Proposition 11 (strong free precedence → strong exhaustiveness). Any framework AF with attack type † ∈ {dir, con, set} and any complete semantics sem ∈ CMP satisfies strong exhaustiveness if it satisfies strong free precedence.
Proof. Let E ∈ Ext sem (AF) and suppose that Supp(a) ⊆ Concs(E). By strong free precedence, Supp(a) ∈ CN L (Free X L (S)). We note that CN L (Free X L (S)) ∩ S = Free X L (S) and therefore a ∈ E.
Next, we consider the closure properties.
Proposition 12 (closure under support). Any framework AF = Arg X L (S), A with attack rules of type dir or con is closed under supports for all the complete semantics sem ∈ CMP: If E ∈ Ext sem (AF), a ∈ Arg X L (S), and Supp(a) ⊆ Supps(E), then a ∈ E.
Proof. Assume that for a ∈ Arg X L (S), Supp(a) ⊆ Supps(E). If a is not attacked then obviously a ∈ E. Suppose that some b ∈ Arg X L (S) attacks a. By Proposition 2, a is not ConUcut-attacked. Thus, b either DUcut-or DDef-attacks a, and so there is a ϕ ∈ Supp(a) for which there is a c ∈ E for which ϕ ∈ Supp(c) and so b also attacks c. Since E is complete, it defends c, thus E must attack b. It follows that a is also defended by E, and by the completeness of E, a ∈ E.
Note 14. Note 10 provides a counterexample to Proposition 12 in case of naive semantics.
By Corollary 4 and Proposition 10 we have: Two other immediate corollaries are the following: Corollary 8. For any framework AF = Arg X L (S), A with attack rules of type dir or con and any complete semantics sem ∈ CMP, we have: If E ∈ Ext sem (AF) then E = Arg X L (Supps(E)). Corollary 9. For any framework AF = Arg X L (S), A with attack rules of type set and any sem ∈ SE, we have: Closure under support implies some other postulates: Proposition 13 (closure under support → sub-argument closure). Any framework AF for any attack type † ∈ {dir, con, set} and any complete semantics sem ∈ CMP satisfies sub-argument closure, if it satisfies closure under support.
Proof. By the definitions of closure under support and sub-argument closure.
Proposition 14 (closure under support → closure of extensions). Any framework AF for any attack type † ∈ {dir, con, set} and any complete semantics sem ∈ CMP satisfies closure of extensions, if it satisfies closure under support.
For the converse, suppose that ϕ ∈ CN L (Concs(E)). Then there are a 1 , . . . , a n ∈ E with Γ i = Supp(a i ) and ϕ i = Conc(a i ) (1 ≤ i ≤ n) such that ϕ 1 , . . . , ϕ n ⊢ ϕ. (Note that n is finite by the compactness of L (Definition 1)). By the completeness of C, ϕ 1 , . . . , ϕ n ⇒ ϕ is C-derivable, and by Corollary 10 (sub-argument closure). Any framework AF = Arg X L (S), A with attack rules of type dir or con is closed under sub-arguments for any complete semantics sem ∈ CMP: For every E ∈ Ext sem (AF) and a ∈ E it holds that Sub(a) ⊆ E.
Proposition 15 (sub-argument closure II). Any framework AF = Arg X L (S), A with attack rules of type set is closed under sub-arguments for any complete semantics sem ∈ CMP: For every E ∈ Ext sem (AF) and a ∈ E it holds that Sub(a) ⊆ E.
Proof. Let a ∈ E and b ∈ Sub(a). Suppose that c attacks b. Note that every attacker of b is an attacker of a. (Indeed, the only non-trivial case is Defeat. In this case, let a = ∆, Thus, b is defended by E and by the completeness of E, b ∈ E. Corollary 11. Any framework AF = Arg X L (S), A with attack rules of type set or dir or con is closed under sub-arguments for any complete semantics sem ∈ CMP. For every E ∈ Ext sem (AF) and a ∈ E it holds that Sub(a) ⊆ E.
Proof. Follows from Corollary 10 and Proposition 15.
Note 15. Sub-argument closure does not hold for sem = nav and any of the types of attacks in Definition 8. To see this, consider an argumentation framework constructed from classical logic and the set S = {p ∧ ¬q, q}. Some of the arguments in this case are: One possible conflict-free set of arguments is {a 1 , a 4 }. Note that a 2 , a 3 ∈ Sub(a 1 ). However, a 2 and a 3 (Direct) Defeat a 4 , thus {a 1 , a 2 , a 3 , a 4 } would no longer be conflict-free. By similar arguments, support inclusion (Proposition 12) does not hold either for maximal conflict-free semantics.
In [2] sub-argument closure for naive semantics is shown, under the assumption that the support set of each extension is consistent (note that S = Supps({a 1 , a 4 }) is not consistent). To keep our results as general as possible, we do not make that assumption here (see also Note 12).
Proposition 16 (sub-argument closure → support inclusion). Any framework with any attack type † ∈ {dir, con, set} and any complete semantics sem ∈ CMP satisfies support inclusion if it satisfies sub-argument closure.
Corollary 12 (support inclusion). Let AF be an argumentation framework with attack rules of any type † ∈ {dir, con, set} considered in Definition 8. Then AF is closed under support inclusion for every complete semantics sem ∈ CMP: If E ∈ Ext sem (AF) then Supps(E) ⊆ Concs(E).
Corollary 13 (closure of extensions). Any framework AF with attack rules of type dir or con satisfies closure of extensions for any complete semantics sem ∈ CMP: If E ∈ Ext sem (AF) then Concs(E) = CN L (Concs(E)).
Note 16. Again, by Note 10, the last corollary does not hold for naive semantics.
Corollary 14 (closure of extensions II). Any framework AF with attack rules of type set satisfies closure of extensions for every sem ∈ SE: If E ∈ Ext sem (AF) then Concs(E) = CN L (Concs(E)).
Note 17. As shown in Note 11, closure of extensions does not hold for complete semantics and frameworks with attack rules of type set. By Note 10, it also does not hold for naive semantics.
Next, we consider (strong) exhaustiveness. Like free precedence, for assuring strong exhaustiveness for frameworks with dir-type attack rules and completeness-based extensions, one has to consider the limited version of the postulate: Proposition 17 (limited strong exhaustiveness). Any framework AF = Arg X L (S), A with attack rules of type dir or con satisfies limited strong exhaustiveness for every complete semantics sem ∈ CMP: For every E ∈ Ext sem (AF) and a ∈ Arg X L (S), if Supp(a) ⊆ Concs(E), and Supps(E) \ X ̸ = ∅, then a ∈ E.
In both cases E defends a, and by completeness, a ∈ E.
Proposition 18 (strong exhaustiveness). Any framework AF = Arg X L (S), A with attack rules of type dir satisfies strong exhaustiveness with respect to any semantics sem ∈ ME: For every E ∈ Ext sem (AF) and a ∈ Arg X L (S), if Supp(a) ⊆ Concs(E) then a ∈ E.
Proof. Since Ext stb (AF) = Ext stg (AF) ⊆ Ext sstb (AF) ⊆ Ext prf (AF), we need to show the proposition only for sem = prf. Let E ∈ Ext prf (AF) and let a ∈ Arg X L (S) with Supp(a) ⊆ Concs(E). By Proposition 17 we only need to consider the case in which Supps(E) \ X = ∅. By Lemma 9, E = Arg X L (∅) and all ϕ ∈ S are ⊢ X -inconsistent. Since by Corollary 2, Concs(E) is ⊢-consistent, Supp(a) ⊆ X , and hence a ∈ E.
By Proposition 18 and Note 9, we get: Corollary 15 (exhaustiveness). Frameworks with attack relations of type dir satisfy exhaustiveness with respect to any multiple-extension semantics. Note 18. The example in Note 13 illustrates also the failure of exhaustiveness (thus also the failure of strong exhaustiveness) for grounded, complete, ideal and eager extensions for frameworks with dirtype attack relations. Indeed, let AF be such a framework with the base logic CL. ). However, the argument p ∨ ¬p ⇒ p ∨ ¬p is not defended against the argument p, ¬p ⇒ ¬(p ∨ ¬p) in Arg ∅ CL ({p ∨ ¬p}).
Proposition 19 (strong exhaustiveness II). Frameworks with attack rules of type con satisfy strong exhaustiveness for any complete semantics sem ∈ CMP: If E ∈ Ext sem (AF) and Supp(a) ⊆ Concs(E) then a ∈ E.
Proof. Let E ∈ Ext sem (AF) and a ∈ Arg X L (S) with Supp(a) ⊆ Concs(E). By Proposition 17 we only need to consider the case in which Supps(E) ⊆ X . Consider an argument b = Γ ⇒ γ ∈ Arg X L (S) that attacks a. Since Supp(a) ⊆ Concs(E) and by Corollary 2, b does not ConUCut-attack a. By [Cut], Γ ⇒ ¬ϕ is derivable for a ϕ ∈ (Supp(a) \ X ). Since Supp(a) ⊆ Concs(E), we have that ϕ ∈ Concs(E), and so there is a c = Θ ⇒ ϕ ∈ E for some finite Θ ⊆ X . By [Cut] and Lemma 2, Θ, Γ ⇒ and d = Θ, Γ ∩ X ⇒ ¬ (Γ \ X ) are derivable, and since X is ⊢-consistent, we get that (Γ \ X ) ̸ = ∅. Since d has no attackers, by the completeness of E, d ∈ E. Since d ConUcut-attacks b, it defends a from b. Altogether, E defends a and by completeness, a ∈ E.
Note 19. Note 10 provides a counterexample to Proposition 19 in case of naive semantics.
By Proposition 19 and Note 9 we get: Corollary 16 (exhaustiveness II). Frameworks with attack rules of type con satisfy exhaustiveness with respect to every complete semantics sem ∈ CMP: If Supp(a) ∪ {Conc(a)} ⊆ Concs(E) for some E ∈ Ext sem (AF) and a ∈ Arg X L (S), then a ∈ E.

By Corollary 4 and Proposition 11 we have:
Proposition 20 (strong exhaustiveness III). Frameworks with attack rules of type set satisfy strong exhaustiveness for every semantics sem ∈ SE: If E ∈ Ext sem (AF) and Supp(a) ⊆ Concs(E) then a ∈ E.
Corollary 17 (exhaustiveness III). Frameworks with attack rules of type set satisfy exhaustiveness with respect to every semantics sem ∈ SE: If Supp(a) ∪ {Conc(a)} ⊆ Concs(E) for some E ∈ Ext sem (AF) and a ∈ Arg X L (S), then a ∈ E.
As shown in Note 11 above, (limited) exhaustiveness does not hold for argumentation frameworks with non-direct attacks. Table 2 summarizes the results in this section. It indicates the properties of individual extensions of argumentation frameworks in relation to the type of the attack rules and the semantics classes. For instance, it shows that (strong) exhaustiveness of extensions is satisfied with respect to every completeness-based extension when using con-type attack rules, but for dir-type (respectively, set-type) attack rules, this property is met only with respect to multiple-extension (respectively, single-extension) semantics.
The results in Table 2 show that in general the set-based attack rules are inferior to the direct attack rules (with or without ConUcut), since for frameworks with set-based attack rules some important properties, like consistency, closure, and exhaustiveness of the extensions, cannot be guaranteed unless  the semantics is a single extension one. In contrast, all the properties of Definition 12 (excluding strong free precedence) can be assured for frameworks with direct attacks under either the stable, semi-stable, preferred, or stage semantics. When ConUcut is also part of the attack rules, these properties can be guaranteed for all the complete semantics.
The analysis above should be taken with care, though, for several reasons. First, as we shall see in what follows, the entailment relations induced by direct attacks are preferential and rational (Definition 18) for a larger set of semantics than the entailments induced by subset attacks (see Section 6.1), while for some postulates (e.g., non-interference) subset-based attacks have some advantages over direct attacks (see Section 6.2). Second, this analysis cannot necessarily be applied to base logics that are not explosive or not contrapositive (recall Corollary 1). For such logics we note the following: 1. Not all the attack rules considered here (in Definition 4) may be applicable. For instance, while reflexivity, [⇒ ¬] and [⇒ ∧] assure that ConUcut is applicable in case that all the rules in Figure 2 are admissible, this is not guaranteed in other cases. For instance, in Dunn-Belnap's logic of firstdegree entailments (FDE) [22,42] no sequent with an empty left-hand side is valid (i.e, derivable in a sound and complete sequent calculus for FDE), thus ConUcut is not applicable in a setting with FDE as the base logic and X = ∅.
2. Even in cases that the attack rules are applicable, they may yield results that are considerably different than those that are obtained with logics that meet the conditions in Section 2.2. For instance, in Example 1, using the same set S and the same attack rule, but when Priest's 3-valued logic LP [67,68] is the base logic (instead of CL), both a 2 and a 3 are not attacked, simply because a 8 and a 9 (respectively) are not derivable (the Disjunctive Syllogism does not hold in LP).

Postulates Concerning Sets of Extensions and the Collapse of the Classes
We now turn to properties that refer to sets of extensions of the same type (that is, sets of the same sem ∈ {nav, stg, cmp, grd, prf, idl, stb, sstb, egr}). As a byproduct of establishing these properties we obtain the following theorem, which expresses that all multiple-extension, respectively all single-extension semantics, coincide in the logic-based setting studied in this paper (recall Fig. 3).

Proof. Item 1 follows by Corollary 19 (below) and Item 2 by Corollaries 5 and 20 (below).
Definition 13 (rationality postulates II). Let AF X L,A (S) = Arg X L (S), A be an argumentation framework and sem a semantics for it. In the remainder of this section we consider the postulates in Table 3. 21 Property Definition maximal consistency

strong stability
Ext sem (AF X L,A (S)) = Ext stb (AF X L,A (S)). Note 20. The postulates in Table 3 refer to some properties of the extensions of a framework, given some semantics.
• Maximal consistency requires that the set of the sem-extensions coincides with the set of the maximal consistent subsets of the premises. Although the link between logical argumentation and reasoning with maximal consistency is well-known (see, e.g. [6,7,10,32,76]) we note that this requirement is a rather strong demand (see a discussion on this in [7, Section 7.1], pointing out that this requirement is violated in several settings). This is one of the reasons for introducing in Definition 13 a weaker postulate, weak maximal consistency.
• Stability and strong stability refer to the existence of stable extensions, a property that does not hold for every argumentation framework. 22 These properties will be useful for the crash-resistance result in Section 6.
• When strong stability is satisfied for preferred semantics, we have that
Stability follows immediately from Lemma 6. However, we will show that the considered frameworks also satisfy strong stability. For this we need the following lemma and proposition. 21 For these postulates we need some notions from Definition 11. 22 When sem = prf, strong stability is sometimes called coherence; see [44,Def. 31].
Lemma 11. Any argumentation framework AF, based on any attack type † ∈ {dir, con, set} and any complete semantics sem ∈ CMP, satisfies Ext sem (AF) ⊆ {Arg X L (T ) | T ∈ CS X L (S)}, if it satisfies support consistency and closure under support.
Proposition 21 (support consistency & closure under support → strong stability, maximal consistency). Any AF based on any attack type † ∈ {dir, con, set} and any semantics sem ∈ ME satisfies strong stability and maximal consistency, if it satisfies support consistency and closure under support.
Corollary 18 (strong stability and maximal consistency). If AF = Arg X L (S), A is an argumentation framework whose attack rules are of type dir or con, then for every semantics sem ∈ ME it holds that Ext sem (AF) = {Arg X L (T ) | T ∈ MCS X L (S)}.
Concerning frameworks with set-type attacks, we have strong stability (but not maximal consistency -see Note 11): Proposition 22 (strong stability II). Frameworks with attack rules of type set satisfy strong stability for any sem ∈ ME: Ext sem (AF) = Ext stb (AF).
Proof. By [40,Lemma 15], every stable extension is preferred. Moreover, every stable extension is semi-stable (trivially) and by Proposition 1 every stable extension is also stage. Thus, we only need to show that every preferred extension is stable. For this, let E ∈ Ext prf (AF). If E ̸ ∈ Ext stb (AF) then there is an a = ∆ ⇒ δ ∈ Arg X L (S) \ E and E does not attack a. We show towards a contradiction that E ′ = E ∪ Sub(a) is admissible. Note first that since E does not attack a it also does not attack any argument in Sub(a) and no argument in Sub(a) attacks an argument in E. So E ′ is conflict-free. Suppose that some argument b attacks some a ′ ∈ Sub(a). Thus, is C-derivable and so E ′ attacks b. But then E ′ is admissible which is a contradiction to our assumption. Thus, E ∈ Ext stb (AF).
In view of the strong stability shown in Corollary 18 and Proposition 22, we get: Corollary 19 (ME collapse). If AF is an argumentation framework whose attack rules are of type dir, con or set, then Ext prf (AF) = E sstb (AF) = E stg (AF) = E stb (AF).
We now generalize SE-collapse (Corollary 5) to all attack-forms (completing Fig. 3 and the proof of Theorem 1). For this, we first need the next lemma.
Lemma 12. Let AF be an S-based framework with attack rules of type dir. If there is a ⊆-minimal ⊢ X -inconsistent subset of S of size greater than 1, then Ext sem (AF) = {Arg X L (∅)} for all sem ∈ SE.
Proof. If there is a ⊆-minimal ⊢ X -inconsistent subset of S of size greater than 1, the corollary follows by Lemma 12. Otherwise it follows by Lemma 10.
Note 21. Maximal consistency does not hold for set-type attack rules (recall Note 11) and also it is violated in frameworks with the other types of attack rules when the semantics is not preferred, stage, stable, or semi-stable: • For sem = nav this follows from the example in Note 10, where the set of argument S considered there is contained in a naive extension, and Supps(S) = {p, q, ¬p ∨ ¬q}.
• For sem ∈ SE this follows immediately since these semantics result in a single extension, while MCS L (S) may contain several maximally consistent subsets.
In the next proposition we therefore consider a weaker version of maximal consistency.
Proposition 23 (weak maximal consistency). Frameworks with attack rules of type † ∈ {dir, con, set} satisfy weak maximal consistency with respect to any semantics sem ∈ ME ∪ {cmp}: It holds that {Arg X L (T ) | T ∈ MCS X L (S)} ⊆ Ext sem (AF).
The results of this section are summarized in Table 4 (The sign ✓ indicates that the property always holds).

Properties of extensions sets
dir-attacks con-attacks set-attacks maximal consistency ME ME − weak maximal consistency ME ∪ {cmp} ME ∪ {cmp} ME ∪ {cmp} stability ✓ ✓ ✓ strong stability ME ME ME We conclude this section by an interesting corollary of the results in Sections 3.1 and 3.2: Theorem 2. All the postulates in Definition 12 (Table 1) and Definition 13 (Table 3), except for strong free precedence, are compatible (i.e., they can be mutually satisfied). Tables 2 and 4, when sem ∈ ME, all of these postulates, excluding strong free precedence, are satisfied by every framework whose attack rules are of type con.

Some Illustrations
In this section we consider two further examples, based on non-classical core logics, that illustrate some of the results in the previous section.
Example 5. Consider again the argumentation framework in Example 2, but this time where the base logic is intuitionistic logic (IL). For this, one has to replace the sequent calculus accordingly, e.g., trade LK by its single-conclusion counterpart LJ (see [47, page 192] , thus properties such as strong stability remain valid despite the change of the base logic. The next example (a variation of [74, Example 3]) demonstrates the use of the modal logic S4 as the core logic of a framework, and -more generally -the use of modal languages, which allow to qualify statements with modal operators for expressing alethic arguments (concerning necessity and possibility), epistemic ones (concerning knowledge and belief ) and deontic assertions (concerning obligation and permission). 23 A sequent calculus for S4 may be obtained by adding the following rules to LK: Example 6. Let S = {q, p ⊃ □r, q ⊃ □¬r} and X = {p}. Some of the arguments in Arg X S4 (S) are the following: a 1 = p ⇒ p a 4 = q, q ⊃ □¬r ⇒ □¬r a 2 = q ⇒ q a 5 = p, p ⊃ □r, q ⊃ □¬r ⇒ ¬q a 3 = p, p ⊃ □r ⇒ □r a 6 = p, q, p ⊃ □r ⇒ ¬(q ⊃ □¬r) a 7 = p, q, q ⊃ □¬r ⇒ ¬(p ⊃ □r) a 8 = q, p ⊃ □r, q ⊃ □¬r ⇒ ¬p Figure 6 is a graphical representation of (part of) the argumentation framework for the above setting with direct defeat as the sole attack rule.
The preferred extensions in this case are: These extensions are also the stable extensions. Also, This corresponds to Proposition 2, Corollary 2, and Proposition 18. 23 In all of these contexts there are various application in which distinguishing between strict and defeasible assumptions is useful. For instance, prima facia norms in deontic logic are often modeled as defeasible assumptions (e.g., in constrained inputoutput logic [60]), alethic arguments may be based on ceteris paribus regularities, beliefs on default assumptions, etc.  Figure 6: A representation of the framework of Example 6. Note that the argument a8 with conclusion ¬p does not directly defeat arguments that contain p as a support (such as a1, a3, etc.), since p is a strict assumption.

Characterization Results
We now use the results in Section 3 for characterizing the semantic extensions of the argumentation frameworks from Definition 5 (Section 5.1) and the entailment relations induced by them (Section 5.2).

Characterizations of the Semantic Extensions
During the investigations of rationality postulates for different argumentation frameworks, we have characterized the extensions of some types of attacks and particular semantics (see Corollaries 5,19 and 20). In this section we recall these results and extend them to all the attack types in Definition 8 and all the semantics in Definition 6.
The next definition is required for characterizing all the extensions of a logic-based argumentation framework and the entailment relations that are induced from it, which are not represented by maximally consistent subsets (MCS) of the premises (see Proposition 24 and Theorem 5 below).
Definition 14 (Ω X L (S)). Let L = ⟨L, ⊢⟩ be a logic and let S and X be two disjoint sets of L-formulas, such that X is ⊢-consistent. We denote by Ω X L (S) ⊆ ℘(℘(S)) the set of subsets of ℘(S), where for every ω ∈ Ω X L (S) the following two requirements are satisfied: 1. the elements of ω are pairwise ⊢ X -consistent: T i ∪ T j is ⊢ X -consistent for every T i , T j ∈ ω.
2. for every finite set Θ ∈ ℘(S) there is a set T ∈ ω such that either Θ ⊆ T or Θ ∪ T is ⊢ Xinconsistent.
Intuitively, Ω X L (S) contains all subsets ω of ℘(℘(S)) which only contain pairwise consistent subsets of S and that satisfy the following closure property: for any finite subset Θ of S for which no superset is contained in ω, there is a set Θ ′ in ω that is inconsistent with Θ.
We are ready to characterize extension-based semantics in logical argumentation frameworks. The following theorem states that for nearly all combinations of multiple and single extension semantics with attack rules in one of the three classes set, dir and con, the resulting extensions of an argumentation framework can be characterized by a set Arg X L (T ) where T is an appropriate consistent subset of the given defeasible assumptions. The exception is the combination of multiple extension semantics with attack rules in set. In this case, extensions are characterized by sets T ∈ω Arg X L (T ) where for each T , T ′ ∈ ω, T ∪ T ′ is a consistent set of defeasible assumptions.
Theorem 3 (characterization of extensions). Given an argumentation framework AF X L,A (S) in which attack rules that are of type † ∈ {dir, con, set}. It holds that: where † ∈ {dir, con} and sem ∈ ME.
Proof. Item 1 is maximal consistency (Corollary 18), Item 2 is shown in Proposition 24 (below), and Item 3 is strong free precedence (Corollary 4). If there is a ⊆-minimal ⊢ X -inconsistent subset of S of size greater than 1, Item 4 follows by Lemma 12. Otherwise it follows by Lemma 10.
Proposition 24. Let AF X L,A (S) be an argumentation framework in which the attack rules are of type set. Then, for every sem ∈ ME it holds that Ext sem (AF X L,A (S)) = {Arg X L (ω) | ω ∈ Ω X L (S)}.
Proof. By strong stability (Proposition 22) we only need to show this for stable extensions.
Let ω ∈ Ω X L (S). We first show that Arg X L (ω) is conflict-free. Suppose for a contradiction that a = Γ ⇒ γ, b = ∆ ⇒ δ ∈ Arg X L (ω), and a attacks b. By Lemma 5, Γ ∪ ∆ is ⊢-inconsistent. Since there are T i , T j ∈ ω for which Γ ⊆ T i and ∆ ⊆ T j , this implies that T i ∪ T j is ⊢-inconsistent, which contradicts the first condition in Definition 14. Thus, Arg X L (ω) is conflict-free. Let now a = Γ ⇒ γ ∈ Arg X L (S)) \ Arg X L (ω). Then there is some T ∈ ω and some Θ ⊆ T such that ⊢ X ¬ (Γ ∪ Θ), and hence b = Θ ′ , Θ ⇒ ¬ Γ ∈ Arg X L (ω) for some Θ ′ ⊆ X . Thus b Def-attacks or Ucut-attacks a. This shows that Arg X L (ω) ∈ Ext stb (AF). Let now E ∈ Ext stb (AF). By sub-argument closure (Corollary 11), there is a set . We show Items 1 and 2 of Definition 14.
1. Assume for a contradiction that there are i, j ∈ I for which T i ∪T j is ⊢ X -inconsistent. Thus, there are Θ i ⊆ T i and Θ j ⊆ T j such that ⊢ X ¬ (Θ i ∪ Θ j ). By Lemma 2, a = Θ i , Θ ⇒ Θ j ∈ E for some Θ ⊆ X , in contradiction to pairwise support consistency (Proposition 3).
2. Suppose for a contradiction that there is a set Θ ∈ ℘(S) such that Θ ̸ ⊆ T i and Θ ∪ T i is consistent for all i ∈ I. Let a = Θ ⇒ Θ ∈ Arg X L (S). Note that a / ∈ E. By the stability of E, there is an argument b = ∆ ⇒ δ ∈ E that attacks a. Since b ∈ i∈I Arg X L (T i ), there is an i ∈ I for which ∆ ⊆ T i . But then Θ ∪ T i is ⊢ X -inconsistent by Lemma 5, a contradiction to our assumption.

Characterizations of the Induced Entailments
We now turn to characterizations of the entailment relations that are induced by logical argumentation frameworks according to Definition 7. Like before, we consider the three types of argumentation frameworks from Definition 8, based on a logic L = ⟨L, ⊢⟩ with a corresponding sound and complete sequent calculus C, in which the rules from Figure 2 are admissible.
Since the results in this section refer to families of argumentation frameworks and not to specific frameworks, in what follows we shall somewhat modify the notations of the entailments in Definition 7 and sometimes use the abbreviation |∼ ⋆ L, †,X ,sem , where † ∈ {set, dir, con}, sem ∈ CMP = {cmp, grd, prf, idl, stb, sstb, egr, stg} and ⋆ ∈ {∩, ⋒, ∪}. Thus, for instance, a result that is applied to |∼ ∩ L,dir,X ,sem (that is, when ⋆ = ∩ and † = dir) covers, in terms of Definition 7, any skeptical entailment relation of the form |∼ ∩ L,A,X ,sem in which ∅ ̸ = A ⊆ {DDef, DUcut} (recall Definition 8). The latter, in turn, covers any instance of sem ∈ CMP.
In these notations, Theorem 1 shows that for every fixed † ∈ {dir, con, set} and ⋆ ∈ {∩, ⋒, ∪}, we actually have two different argumentative entailments of the form |∼ ⋆ L, †,X ,sem : one for sem ∈ ME and the other one for sem ∈ SE. This is formalized in the next corollary.

Note 22.
To simplify the presentation, i.e., in order to avoid too many case distinctions, we have not listed cmp among the semantics in SE and in ME. However, we note that |∼ ⋆ L, †,X ,cmp = |∼ ⋆ L, †,X ,sem either when ⋆ = ∪ and sem ∈ ME, or when ⋆ ∈ {∩, ⋒} and sem ∈ SE. Thus, implicitly, reasoning with sem = cmp is covered (here and in what follows) as well.
Our next result (Theorem 4) shows a correspondence between many argumentative entailments and inference by maximally consistent sets of the premises ( [71], see also [7,10]). For this, we recall the following entailments known from the area of inconsistency-tolerant (non-monotonic) logics.
Definition 15 (MCS-based entailments). Let L = ⟨L, ⊢⟩ be a logic, S a set of L-formulas and X a ⊢-consistent set of L-formulas such that S ∩ X = ∅. Recall that CN L (S) denotes the ⊢-closure of S (Definition 2) and that MCS X L (S) denotes the set of the ⊆-maximally ⊢ X -consistent subsets of S (Definition 11). The following entailments are defined in a way similar to those in Definition 7: We now relate the entailments in Definitions 7 and 15. A similar result to the next proposition was shown in [7] (see also [8,Section 2.3.1]), but only for direct undercut with consistency undercut as attack rules and for sem ∈ {grd, prf, stb} and without the distinction between strict and defeasible assumptions. Here we generalize the setting to sequent calculi in which the rules from Figure 2 are admissible, every complete semantics sem ∈ CMP = {cmp, grd, prf, idl, stb, sstb, egr, stg}, and for the three different types of attack relations.
Proof. We first show Item 1. Let sem ∈ ME.
[⇐] Suppose that S |∼ ∪ L,X ,mcs ψ. Thus, there is a maximal ⊢ X -consistent subset T of S for which T ∪X ⊢ ψ. By Lemma 6, Arg X L (T ) ∈ Ext stb (AF). By Theorem 1, Arg X L (T ) ∈ Ext sem (AF). Since L is finitary and by the completeness of C, there is an argument Γ ⇒ ψ ∈ Arg X L (T ). Thus, S |∼ ∪ L, †,X ,sem ψ.
[⇒] Suppose that S |∼ ∪ L, †,X ,sem ψ. Thus, there is a set E ∈ Ext sem (AF) and an argument Γ ⇒ ψ ∈ E. By (pairwise) support consistency (Proposition 2 resp. Proposition 3),Γ \ X is a ⊢ X -consistent subset of S. By the soundness of C, Γ ⊢ X ψ. Thus, there is a maximal ⊢ X -consistent subset T of S for which T ⊢ X ψ, and so S |∼ ∪ L,X ,mcs ψ. The other items directly follow from previous results: Item 2 for sem ∈ SE follows by strong free precedence (Corollary 4). Item 3 follows by maximal consistency (Corollary 18) and since Free X L (S) = MCS X L (S). Item 4 follows by maximal consistency (Corollary 18). Item 5 follows from Item 2 and since the entailments |∼ ∩ L, †,X ,sem and |∼ ⋒ L, †,X ,sem coincide for single extension semantics sem.
To complete the characterization in Theorem 4 to the other argumentative entailments considered here, we need the following notations (see also Definition 14): Definition 16 (Ω-based entailments). Let L = ⟨L, ⊢⟩ be a logic, S a set of L-formulas and X a ⊢-consistent set of L-formulas such that S ∩ X = ∅. We denote: The following result immediately follows from Definition 16 and Proposition 24.
Theorem 5 (characterization of entailments II). For † = set and every sem ∈ ME and ⋆ ∈ {∩, ⋒, ∪}, it holds that S |∼ ⋆ L, †,X ,sem ψ iff S |∼ ⋆ L,X ,Ω ψ Note 23. The only case that is not covered by Theorems 4 and 5 is when sem = SE and † = dir. The corresponding entailment relation is directly obtained by Item 4 in Theorem 3.
Note 24. By Theorem 4, Theorem 5, and Note 23, it follows that all the entailment relations that are obtained in this paper can be described in terms of consistent subsets of the premises (albeit not necessarily the maximal ones).

Evaluation of the Reasoning Process
We now examine the properties of the entailment relations which were characterized in the previous section with respect to the setting described in Section 2.2. In what follows we divide the entailment properties to two kinds: those that are concerned with non-monotonic reasoning, and those for inconsistency handling.

Inference Principles for Non-Monotonic Reasoning
By Definition 1, consequence relations are monotonic, that is: the set of conclusions cannot decrease when the set of assumption is increased. Human reasoning, on the other hand, is often non-monotonic, as previously drawn conclusions are sometimes retracted in light of new data. In this section we examine the argumentation-based entailment relations in Definition 7 relative to the monotonicity property. As it turns out, credulous entailments |∼ ∪ L, †,X ,sem with respect to multi-extension semantics (i.e., when sem ∈ {cmp, prf, stb, sstb, stg}) for † ∈ {dir, con, set}, are monotonic (Proposition 25), while skeptical entailments are non-monotonic. The latter are further divided according to the general patterns for nonmonotonic reasoning introduced in [46,54,55,59,72].
We start with some results concerning the (non-)monotonicity of argumentation-based entailments. For this, we need the next lemma that generalizes the result in [23] from classical logic to any compact Tarksian logic L.
Lemma 13. Let L = ⟨L, ⊢⟩ be a compact Tarskian logic. Then |∼ ∪ L,X ,mcs is monotonic. Proof. Suppose that S |∼ ∪ L,X ,mcs ψ. Then, there is a maximal ⊢ X -consistent subset T of S for which T ∪ X ⊢ ψ. Consider now S ∪ S ′ . Clearly, T is a ⊢ X -consistent subset of S ∪ S ′ , and so there is a maximal ⊢ X -consistent subset T ′ of S ∪ S ′ such that T ⊆ T ′ . By the monotonicity of ⊢, we have that T ′ ∪ X ⊢ ψ, thus S, S ′ |∼ ∪ L,X ,mcs ψ as well. Proposition 25 (monotonicity). Every entailment of the form |∼ ∪ L, †,X ,sem , where sem ∈ ME and † ∈ {dir, con, set}, is monotonic.
Logical consequences of entailed formulas are entailed.
We refer to [54,59] for a detailed discussion on ⊢ X -cREF, ⊢ X -RW, ⊢ X -LLE, CM and OR, and to [46] for a discussion on CC. The postulate RM was introduced in [55]. 25 All of these properties are well-known and have been extensively examined in different contexts and for different purposes involving inference in a non-monotonic way.
The properties in Definition 17 are often gathered for defining systems for non-monotonic inference. 24 The reason for excluding the strict assumptions from the premises is to keep our supposition that the strict and the defeasible premises are disjoint, and to preserve the similarity to Definition 11, where consistency is taken with respect to subsets of S. 25 RM is sometimes considered more controversial than other postulates. For instance, in [73] Stalnaker claims by a counterexample that RM is not a desirable property in some application contexts of defeasible reasoning. Nevertheless, also other reasoning principles have been criticized (e.g., CC in [63]), and some are not valid in central approaches of non-monotonic inference (e.g., CM does not hold in Reiter's default logic, see [59,Observation 3.2
• preferential (w.r.t. ⊢ X ), if it is ⊢ X -cumulative and satisfies OR.
We first give results concerning cumulativity and preferentiality. Given our characterization of the argumentation-induced entailments in Theorem 4, the following two lemmas will be useful.
The following example, from [23], shows that the entailment |∼ ∪ L,X ,mcs does not satisfy CC and is therefore neither cumulative, preferential nor rational. In view of this counterexample and Theorem 4 (Item 1), it follows that |∼ ∪ L, †,X ,sem is not cumulative, and therefore not preferential nor rational, for every † ∈ {con, set, dir} and sem ∈ ME.
Proof. We divide the proof according to the different items: 1. For † ∈ {con, set} this item follows from Lemma 14 and Item 3 of Theorem 4. For † = dir and sem ∈ ME it follows from Lemma 14 and Item 4 of Theorem 4, while for sem ∈ SE it follows from Lemma 15.
2. This item follows from Lemma 14 and Item 2 of Theorem 4 for sem ∈ ME. For † = con and sem ∈ SE it follows from Lemma 14 and Item 5 of Theorem 4. For † = dir and sem ∈ SE it follows from Lemma 15.
3. This item follows from Lemma 14 and Item 5 of Theorem 4.
While Item 2 of Proposition 26 holds for frameworks with attack rules of type con or dir, it fails for frameworks with attack rules of type set. Before giving a counter-example, we present a partially positive result concerning CM, for which the following lemma will be useful.
Lemma 16. Let S and X be disjoint sets of L-formulas and suppose that X is ⊢-consistent.
We now show that CC is violated for |∼ ⋒ L,set,X ,sem when sem ∈ ME.
We now turn to checking preferentiality. For this we need to consider the property OR, and thus a disjunction connective ∨. In what follows we suppose that, in addition to the rules in Figure 2, the calculus C also has two admissible rules characterizing ∨, as shown in Figure 7. Figure 7: Rules for ∨ that are part of (or admissible in) the calculus C (where ∆ is empty or a singleton and ∆ ′ is empty, in case that C is a single-conclusion calculus).
Lemma 18 (mcs preferentiality). For every propositional logic L = ⟨L, ⊢⟩ for which the rules in Figures 2 and 7 hold, 26 the entailment |∼ ⋒ L,X ,mcs is ⊢ X -preferential with respect to every ⊢-consistent X .
Proposition 28 (preferentiality). Let L = ⟨L, ⊢⟩ be a logic satisfying the conditions in Section 2.2, as well as the disjunction rules in Figure 7.
We now show that OR fails for the cases not mentioned in Proposition 28 (see Table 5 for an overview). In all the examples here we consider classical logic as the base logic L. In Example 10 we give counter-examples for those entailments that are with Theorem 4 identical to |∼ ∩ L,X ,mcs , namely |∼ ∩ L, †,X ,sem where † ∈ {con, set} and sem ∈ CMP (Item 3), |∼ ∩ L,dir,X ,sem where sem ∈ ME (Item 4), and |∼ ⋒ L, †,X ,sem for † ∈ {set, con} and sem ∈ SE (Item 5). In Example 11, we consider |∼ ⋒ L,set,X ,sem where sem ∈ ME.
We now also have the stable (preferred, semi-table and stage) extension We now turn to rationality. First, we show that RM fails for |∼ ⋒ L,X ,mcs and in view of Theorem 4 it also fails for all the entailments |∼ ⋒ L, †,X ,sem where † ∈ {con, dir} and sem ∈ ME. Finally, we give a counter-example for |∼ ⋒ L,set,∅,sem where sem ∈ ME.
Example 14. Let S = {p ∧ s, q ∧ s, ¬(p ∧ q) ∧ (s ⊃ t)}. We have the following stable (which are also preferred, semi-stable and stage) extensions: . Note that S|̸ ∼ ⋒ L,set,∅,sem ¬¬t and S |∼ ⋒ L,set,∅,sem s. Let now S ′ = S ∪{¬t}. Note that we now also have the stable (which is also a preferred, semi-stable and stage) extension Yet, some entailments are rational. The next proposition lists them.
Proposition 29 (rationality). Let L = ⟨L, ⊢⟩ be a logic satisfying the conditions in Section 2.2 and in Figure 7, and let sem ∈ SE. Then the entailments |∼ ⋒ L,dir,X ,sem and |∼ ∩ L,dir,X ,sem are rational.
Let AF X L,A (S) be an S and X -based framework whose core logic is L and the set A of attack rules consists only of dir-type attacks. Based on Theorem 3 (Item 4), and using its notations, we consider two cases: (a) S ⊤ is ⊢ X -consistent, (b) S ⊤ is ⊢ X -inconsistent. Consider (a). Then Ext sem (AF X L,A (S)) = {Arg X L (S ⊤ )}. Thus, S ⊤ ∪ X ̸ ⊢ ¬ψ and so S ⊤ ∪ {ψ} = (S ∪ {ψ}) ⊤ is also ⊢ X -consistent. Also, by the main supposition, there is an argument Γ ⇒ ϕ ∈ Arg X L (S ⊤ ). Again by Theorem 3 (Item 4), . Then, by Theorem 3 (Item 4), Ext sem (AF X L,A (S)) = {Arg X L (∅)}. Thus, also (S ∪ {ϕ}) ⊤ is ⊢ X -inconsistent, and so Ext sem (AF X L,A (S ∪{ψ})) = {Arg X L (∅)} = Ext sem (AF X L,A (S)). Thus, by the main supposition, S ∪ {ψ} |∼ ϕ.  Recall from Table 2 that, in almost all the cases, properties of extensions are satisfied by settings with con-attacks. In that respect, con-attacks may be preferred. However, as Table 5 indicates, as far as principles of non-monotonic reasoning are concerned, settings with dir-attacks may be superior.

Inference Principles for Inconsistency Handling
We now consider properties that are related to reasoning with inconsistent assumptions.
• We say that a set S such that Atoms(S ∪ X ) ⊊ Atoms(L) is contaminating (w.r.t. |∼ X ), if for every S ′ such that S ∪ X ∥ S ′ and for every L-formula ϕ, it holds that S |∼ X ϕ iff S, S ′ |∼ X ϕ.
A contaminating set of formulas renders every syntactically disjoint set uninformative. For instance, in the context of classical logic the singleton {ϕ ∧ ¬ϕ} is a contaminating set.
We say that |∼ X satisfies: • conservative ⊢-consistency: for every ⊢ X -consistent set S of L-formulas and every L-formula ψ it holds that S |∼ X ψ iff X , S ⊢ ψ.
• crash-resistance: there is no |∼ X -contaminating set of L-formulas.
Conservative ⊢-consistency means that |∼ X is a conservative extension of ⊢ that coincides with the latter with respect to ⊢ X -consistent premises. Paraconsistency [35] is a well-investigated logical property, disallowing the inference of every conclusion whatsoever from a single contradiction (see, e.g., [17,31] for some surveys on this subjects with many references). Non-interference and the related notion of crash-resistance were introduced in [30] for similar purposes, and are mainly investigated in the context of logical argumentation frameworks. Non-interference expresses that a formula ϕ should be |∼ X -entailed by S 1 if and only if it is entailed by S 1 ∪ S 2 where S 2 contains only information irrelevant to S 1 ∪ X and ϕ (in the sense that S 2 is syntactically disjoint from S 1 ∪ X ∪ {ϕ}).
Proof. Suppose that S is a ⊢ X -consistent set of L-formulas. Suppose first that there is some a ∈ Arg X L (S) such that b ∈ Arg X L (S) attacks a. Thus Conc(b) ⇒ ¬ϕ is derivable, where ϕ ∈ Supp(a) for frameworks with dir or con attacks, and ϕ = Supp(a) for frameworks with set attacks (note that b cannot be a ConUcut-attacker, since S is supposed to be ⊢ X -consistent , ⇒ ¬( Supp(b)∧ϕ) is derivable, a contradiction to the consistency of S. Thus Arg X L (S) is conflict-free. Therefore, Ext sem (AF X L,A (S)) = {Arg X L (S)} for every sem ∈ CMP. Let now Γ ⇒ ϕ ∈ Arg X L (S). Then Γ ⊢ ϕ by the soundness of C, and by the monotonicity of ⊢ (Definition 1), S, X ⊢ ϕ. Hence, |∼ ⋆ L, †,X ,sem ⊆ ⊢. Now, suppose that S, X ⊢ ϕ. Thus, by the completeness of C for L, and since L is finitary, there is a Γ ⊆ S ∪ X , such that Γ ⇒ ϕ ∈ Arg X L (S). Since for every sem ∈ {nav, stg, cmp, grd, prf, idl, stb, sstb, egr} Ext sem (AF X L,A (S)) = {Arg X L (S)}, we have that S |∼ ⋆ L, †,X ,sem ϕ. Thus ⊢ ⊆ |∼ ⋆ L, †,X ,sem . Altogether, ⊢ = |∼ ⋆ L, †,X ,sem .
We now turn to paraconsistency. We will show that it holds for uniform logics.
Definition 20 (uniformity). A logic L = ⟨L, ⊢⟩ is said to be uniform [58,75], if for every two sets of L-formulas S 1 , S 2 and a formula ϕ such that S 2 is both ⊢-consistent and syntactically disjoint from Note 26. By Łos-Suzsko Theorem [58], a compact propositional logic is uniform and structural, iff, it has a single characteristic matrix. Thus, classical logic, as well as many other logics, are uniform.
Proof. Let p, q ∈ Atoms(L), p ̸ = q, q / ∈ Atoms(X ). Let sem be a complete semantics and E a semextension of AF X L,A ({p, ¬p}). Suppose that there is an argument a = Γ ⇒ q ∈ Arg X L ({p, ¬p}). Thus, Γ \ X ⊢ X q. By the uniformity of L and since q / ∈ Atoms(Γ), Γ \ X is ⊢ X -inconsistent or ⊢ q. The latter is excluded by the structurality and non-triviality of L. By Proposition 2, a / ∈ E. This shows that p, ¬p ̸|∼ ⋆ L,dir,X ,sem q.
We turn now to non-interference and crash-resistance. In the remainder of this section, we suppose that L = ⟨L, ⊢⟩ is a uniform logic (as before, with a corresponding sound and complete calculus C in which the rules of the basic calculus from Figure 2 are admissible). We also suppose that S 1 and S 2 are syntactically disjoint sets of L-formulas (S 1 ∥ S 2 ), and that both of them are disjoint from the set of strict assumptions (i.e., (S 1 ∪ S 2 ) ∥ X ). In what follows we denote S = S 1 ∪ S 2 .
Proof. We show inclusions in both directions.
[⊆]: Suppose that T ∈ CS X L (S), and let T i = T ∩ S i for i ∈ {1, 2}. By ⊢-monotonicity, if T i were ⊢ X -inconsistent then also T would be ⊢ X -inconsistent. Thus, T i ∈ CS X L (S i ).
Proof. We show inclusions in both directions.
[⊇]: Suppose that T i ∈ MCS X L (S i ) for i ∈ {1, 2}. By Lemma 22, T = T 1 ∪ T 2 ∈ CS X L (S). Assume for a contradiction that there is a T ′ ⊋ T such that T ′ ∈ CS X L (S). By Lemma 23, which contradicts the ⊆-maximal ⊢ X -consistency of T 1 and T 2 .
[⊆]: Suppose that T ∈ MCS X L (S), and let T i = T ∩ S i for i ∈ {1, 2}. By Lemma 23, By Lemma 24 and Item 1 of Theorem 3, we get: Let AF X L,A (S) be an argumentation framework with attack rules of type † ∈ {dir, con}, and let sem ∈ ME. Then Ext sem (AF X L, Proof. We show containments in both directions. [⊇]: We show that ω satisfies the two requirements from Definition 14. For Item 1, let (i 1 , j 1 ), (i 2 , j 2 ) ∈ I 1 × I 2 . Then T i1 , T i2 ∈ CS X L (S 1 ) and T j1 , T j2 ∈ CS X L (S 2 ). By Item 1 in Definition 14, For Item 2 let Θ be a finite subset of S and let . Without loss of generality, j = 1. We again show that the two conditions of Definition 14 hold for ω 1 .
For Item 1 let i, k ∈ I. Then ( For Item 2 let Θ ⊆ S 1 . Suppose that Θ ̸ ⊆ (T i ∩ S 1 ) for all i ∈ I. Since S 1 ∩ S 2 = ∅, also Θ ̸ ⊆ T i for all i ∈ I. Thus, Θ ∪ T i is ⊢ X -inconsistent for some i ∈ I. By Lemma 21, Θ ∪ (T i ∩ S 1 ) is ⊢ X -inconsistent.
Now we are ready to show non-interference. First, we show this property for MCS-based and Ωbased entailments, from which we can conclude non-interference with respect to argumentative entailments.
Proof. For fixed L and X , we distinguish between the three cases where ⋆ ∈ {∪, ∩, ⋒}.
Proof. We show the claim for ⋆ = ⋒. The other two cases (⋆ ∈ {∩, ∪}) are similar (using Lemmas 21 and 25) and are left to the reader.
Proof. The items are immediate consequences of the results listed in the following Proof. Let |∼ X be any of the entailments relations in Corollary 24 (given a logic L = ⟨L, ⊢⟩ as in the corollary and a ⊢-consistent set X ), and let S ′ be any set of L-formulas for which there is a atom p ∈ Atoms(L) \ (Atoms(S ′ ) ∪ Atoms(X )). By the non-triviality of ⊢, necessarily ⊬ p. Suppose first that S ′ is ⊢ X -consistent. Then by Lemma 21, S ′ ⊬ X p and therefore S ′ |̸ ∼ X p. Suppose now that S ′ is ⊢ X -inconsistent. Then, by Lemma 21, if there is an argument a = Γ ⇒ p ∈ Arg X L (S ′ ), Γ is ⊢ X -inconsistent. By Proposition 2 and 3, S ′ |̸ ∼ X p.
On the other hand, since p |∼ X p (by Proposition 26), by non-interference (shown in Corollary 24), p, S ′ |∼ X p. Therefore, S ′ cannot be contaminating. Since S ′ was arbitrary this completes the proof.
Note 27. The only case missing in Corollary 24 and in Proposition 36, which involves skeptical entailments (i.e., when ⋆ ∈ {∩, ⋒}), is for † = dir and sem ∈ SE. Indeed, we give a counter-example for this case. Let X = ∅, S 1 = {p} and S 2 = {q, ¬q}. It is easy to see that S 1 |∼ ⋆ L, †,∅,sem p while S 1 , S 2 |̸ ∼ ⋆ L, †,∅,sem p. 27 Note also that {q, ¬q} is a contaminating set in this case. We are now ready to summarize the results in this section.
Note 28. Throughout this section we have shown the results first for mcs-based entailments and then applied them to argumentation-based entailments. Some of the mcs-based results are novel (also due to the distinction between strict and defeasible premises). An overview of the results concerning mcsbased reasoning provided in this paper is given in  Table 6: Summary of the results on mcs-based entailments. For paraconsistency, non-interference and crashresistance L is assumed to be uniform. Table 7 summarizes the results on inconsistency-tolerance by argumentative entailments.

Related Work
In the context of formal argumentation, postulate-based investigations of argumentation frameworks play a primary role, allowing not only to indicate how the ingredients of the framework affect its prop- 27  crash-resistance, † = dir ME ME ME ∪ {cmp} Table 7: Summary of the results in Section 6.2. For paraconsistency, non-interference and crash-resistance L is also assumed to be uniform. (The results concerning sem = cmp follow from Note 22.) erties, but also to compare related approaches to argumentation-based reasoning. In [48], for instance, Gorogiannis and Hunter study the properties of attack relations in logic-based argumentation frameworks. In particular, they consider various necessary and sufficient conditions on attack relations similar to those in Definition 4. Unlike our case, however, the discussion in [48] is concentrated on classical logic as the base logic of the frameworks, where the supports of the arguments are assumed to be classically consistent and the minimal ones that entail the argument's conclusion (recall Note 1).
Studies on requirements on the attack relations to fulfill rationality postulates are also presented in [3,76], where the conditions are somewhat different than the ones presented here, and include, among others, Again, the discussion is limited to classical logic as the base logic and to arguments whose support sets are both consistent and minimal in the sense discussed previously.
The interplay between logical principles about argumentation, on the one hand, and inference principles as studied in proof theory, on the other hand, is also considered in [33]. In that paper a series of logical principles of attack relations in argumentation frameworks is stated, and their collection leads to a characterization of classical logical consequence relations that only involves argumentation frameworks. We refer to [33] and [34] for further details.
Properties of extensions of logic-based argumentation frameworks are studied in, e.g., [2,3,29,48], again with respect to restricted supports of arguments. For checking the postulates, the following two properties of attack relations are assumed in [2]: • if Supp(a) ⊆ Supp(b), then (c, a) ∈ A implies (c, b) ∈ A.
Postulates that are related to reasoning with maximally consistent subsets [71] play a primary role in several works and may be traced back to Cayrol [32]. For detailed discussions and surveys on this 28 Where F is the propositional constant for falsity, satisfying F ⊢ ψ for every formula ψ.
subject we refer to [7,10]. Rationality postulates for other forms of structured argumentation, such as ASPIC + and ABA systems, can be found, e.g., in [38,61,62] (for ASPIC + systems), in [37,53] (for ABA systems), and in [41] (for logic-associated abstract argumentation frameworks). We refer to the survey in [8] for some comparisons of the rationality postulates that these approaches to structured argumentation satisfy. Moreover, a variety of ABA frameworks (such as the simple contrapositive ABAs in [51] and (non-prioritized) ASPIC systems) can be embedded in sequent-based argumentation frameworks, 29 thus the results provided here may be carried on to those systems.
Our study involves some ideas and notions from proof theory. 30 The main contribution of this work in relation to related works such as the ones mentioned above is that it provides a comprehensive presentation of the semantical as well as the inferential properties of logic-based argumentation frameworks, where only minimal (proof-theoretical) requirements are made on the base logic and very little is assumed on the form of the arguments. This allows to capture a wide range of core logics and to base arguments only on deducibility in the core logic. In our study, we avoid the use of further conditions (such as conflict dependence, conflict sensitivity and the condition discussed in Notes 12 and 15) that are computationally demanding, and so are difficult to verify.
Studies of inferential behavior of logical argumentation, and in particular its relation to non-monotonic reasoning, can be found in [15,Section 5], in the context of dynamic proof systems. Similar studies for ABA and ASPIC systems appear, respectively, in [36,52,53] and [56]. A comprehensive survey that relates these disciplines and mentions further results regarding logic-based approaches to formal argumentation appears in [8]. This survey also cites (without proofs) some results from our paper. The main results that were formulated after the writing of the survey (and therefore do not appear in it) are related to the extensions characterization according semantics classes (Theorem 1), the evaluation of the induced entailments (Section 5.2 and several parts in Section 6), the incorporation of strict premises in addition to the defeasible ones, and the coverage of all the Dung-style completeness-based semantics (including stage, eager, and ideal semantics, which are not considered in [8]).

Conclusion
Postulate-based studies are a common approach to evaluate and compare different formalisms sharing a similar purpose. In some cases (like the AGM postulates for belief revision [1]) this is a cornerstone of a formal discipline that serves as a standard setting and a trigger for a variety of related works and formalisms. In this work we have provided a comprehensive postulate-based study of logical (sequentbased) argumentation frameworks, based on propositional languages and Tarskian logics that satisfy some very basic assumptions. This study covers all the central postulates for argumentation frameworks in this context (as well as some new ones), and refers to all the completeness-based Dung-style semantics, as well as the main (support-based) attack rules between arguments. It therefore lays the foundations for logical argumentation frameworks allowing for the making of logically justified inferences in the presence of possibly conflicting defeasible and of strict assumptions.
The results of this paper are summarized in Tables 2-7. Our findings also allow us to provide full characterization results concerning the extensions of sequent-based frameworks (Theorems 1 and 3) and the induced entailment relations (Theorems 4 and 5).
Future work involves, among other, the study of more expressive formalisms, like those that are based on first-order logics, or formalisms that incorporate priorities among the arguments (see, e.g., [9] for a description of how this can be done in the context of sequent-based argumentation).