A Generalized Proof-Theoretic Approach to Logical Argumentation Based on Hypersequents

In this paper we introduce hypersequent-based frameworks for the modelling of defeasible reasoning by means of logic-based argumentation and the induced entailment relations. These structures are an extension of sequent-based argumentation frameworks, in which arguments and the attack relations among them are expressed not only by Gentzen-style sequents, but by more general expressions, called hypersequents. This generalization allows us to overcome some of the known weaknesses of logical argumentation frameworks and to prove several desirable properties of the entailments that are induced by the extended (hypersequent-based) frameworks. It also allows us to incorporate as the deductive base of our formalism some well-known logics (like the intermediate logic LC, the modal logic S5, and the relevance logic RM), which lack cut-free sequent calculi, and so are not adequate for standard sequent-based argumentation. We show that hypersequent-based argumentation yields robust defeasible variants of these logics, with many desirable properties.


Introduction
Argumentation theory has been described as "a core study within artificial intelligence" [27]. Among others, it is one of the standard methods for modeling defeasible reasoning. Logical argumentation (sometimes called deductive or structural argumentation) is a branch of argumentation theory in which arguments have a specific structure. This includes rule-based argumentation, such as the ASPIC + framework [71], assumption-based argumentation (ABA) systems [34], defeasible logic programming (DeLP) systems [52], and methods that are based on Tarskian logics, like Besnard and Hunter's approach [31], in which classical logic is the deductive base (the so-called core logic). The latter method was generalized in [9] to sequent-based argumentation, where Gentzen's sequents [53], extensively used in proof theory, are incorporated for representing arguments, and attacks are formulated by special inference rules called sequent elimination rules. The result is a generic and modular approach to logical argumentation, in which any logic with a corresponding sound and complete sequent calculus can be used as the underlying core logic. A dynamic proof theory as a computational tool for sequent-based argumentation was introduced in [10,11]. This allows for reasoning with these argumentation frameworks in a fully automatic way.
In this paper we further extend sequent-based argumentation to hypersequents [13,66,69]. These are a powerful generalization of Gentzen's sequents which may be regarded as disjunctions of sequents. This generalization turned out to be applicable for a large variety of non-classical logics (see, e.g., [45,62,65]), allows a high degree of parallelism in constructing proofs, and has some applications in the proof theory of fuzzy logics (see, e.g., [65]). In our context, there are several further advantages of generalizing sequent-based argumentation to hypersequents.
• It allows us to consider other logics as the deductive base of the argumentation system. For some well-known logics, like the modal logic S5, the relevance logic RM, and Gödel-Dummett logic LC, an ordinary cutfree sequent calculus is not available, but they do have cut-free hypersequent calculi. Cut-free calculi have multiple proof-theoretic benefits, e.g., they allow for resolution, guarantee the strong normalization property, and imply the subformula property. The latter, meaning that for constructing/proving an argument only its subformulas have to be taken into account, is essential for reducing the proof space when looking for counter arguments, in which case the cut rule should be avoided.
• The incorporation of hypersequents enables us to split sequents into different components, and so different rationality postulates [1,40] can be satisfied, some of which are not available otherwise. For instance, the long-standing problem of deductive argumentation frameworks, whose extensions may be inconsistent (see [2,43]) may be resolved by switching to hypersequent-based argumentation frameworks (see Note 6 and Section 7).
The above-mentioned advantages of hypersequential argumentation frameworks are demonstrated in what follows both for particular and for general cases. First, we demonstrate the usefulness of logical argumentation with hypersequents on frameworks whose core logic is either classical logic (CL) or one of the logics mentioned above (namely, S5, RM, and LC). Then, we consider general entailment relations that are obtained by the hypersequential argumentation-based approach, and show how the following ingredients affect their properties: (1) the set of assumptions (premises) at hand; (2) the core logic and its (hyper)sequent calculus, according to which arguments are introduced; (3) the interplay among arguments, namely: how an argument challenges another argument; and (4) considerations that are related to the semantics of the argumentation framework (in particular, what set of arguments should be taken into account when inferences are made).
This paper revises and largely extends the papers [35] and [36], where S5 and RM (respectively) were studied as the core logics. In addition to providing full proofs and further explanations to the results in these papers, and incorporating also the logic LC, we take here a more abstract approach (i.e., define a general setting to which all the specific core logics fit) and consider some rationality postulates from [1,40], and [41], expressed in terms of the induced entailment relations. In particular, we prove that hypersequentbased formalisms for a number of logics, including CL and LC, avoid the problem of logical argumentation raised in [43], and further discussed in [2]. We also investigate the relation of some entailments that are induced by specific frameworks to reasoning with maximally consistent subsets [74], resulting in a generalization of the results in [6]. A byproduct of our approach is therefore a defeasible variant of a large variety of logics and entailment relations with many desirable properties.
The rest of the paper is organized as follows. The next two sections contain some preliminary material: in Section 2 we recall some basic notions of abstract and sequent-based argumentation, and in Section 3 we review the notion of hypersequents. Then, in Section 4 we extend sequent-based argumentation frameworks to hypersequent-based ones and in Section 5 we discuss the logics LC, S5 and RM as possible core logics of such frameworks. In Section 6 we consider some general properties of hypersequent-based calculi that are needed for the results in the next sections. Then, in Section 7 we study some interesting properties of hypersequential frameworks and the entailment relations induced by them. Relations to reasoning with maximal consistent subsets are discussed in Section 8. Finally, in Section 9 we make some concluding remarks. The appendices contain some auxiliary material.

Preliminaries
We start by introducing the notation and some basic logical notions that we will use in the remainder of the paper. Then we review abstract argumentation frameworks (Section 2.1), and their structured representation in terms of sequents (Section 2.2).
Throughout the paper we consider propositional languages, denoted by L. Sets of formulas are denoted by S, T , finite sets of formulas are denoted by Γ, Δ, Π, Θ, formulas are denoted by φ, ψ, δ, γ, and atomic formulas are denoted by p, q, r, all of which can be primed or indexed. In what follows, we shall assume that L contains at least a unary operator (¬) and two binary operators (∧ and ∨).
Given a language L, L-entailments are relations between sets of formulas in L and formulas in L, intuitively indicating that the latter follow from the former. Common kinds of entailments are considered next. Definition 1. A (Tarskian) consequence relation for a language L is an Lentailment satisfying, for every S, S in L, the following three conditions: • reflexivity: if φ ∈ S then S φ; • transitivity: if S φ and S , φ ψ, then S, S ψ; • monotonicity: if S φ and S ⊆ S, then S φ.
Some further properties that the consequence relation is sometimes required to fulfill are the following: • compactness: if S φ then there is a finite Γ ⊆ S for which Γ φ; • non-trivialilty: there is a set of formulas S = ∅ and a formula φ for which S φ; • structurality (closure under substitutions): for every substitution θ and every S and φ, if S φ then {θ(ψ) | ψ ∈ S} θ(φ).

Definition 2.
A logic for a language L is a pair L= L, , where is a non-trivial and structural L-entailment relation.
Given a logic L= L, , we say that a formula φ ∈ L is an L-theorem if ∅ φ (in short: φ), and that it is an L-consequence of S if S φ. Note 1. The requirements for a logic in Definition 2 are very weak. It is usual to assume, in addition, that the entailment relation of a logic is a Tarskian consequence relation (in the sense of Definition 1), which is indeed the case for all the specific logics (i.e., CL, S5, LC and RM) that are considered in this paper (see Section 5). To keep the presentation as general as possible, we have decided to require these common additional properties only when they are really necessary for the results. For instance, in the general metatheoretic part of this paper (Sections 6-8) we shall suppose that the logics under consideration are also compact and monotonic.
The following notions will be useful in what follows.
Definition 3. Let L= L, be a logic and let S be a set of L-formulas.
• The finitary -closure of S is the set CN L (S) = {φ | Γ φ for some finite Γ ⊆ S}. 2 selecting sets of arguments (called extensions) from a given argumentation framework. 5 Definition 5. Let AF = Args, A be an argumentation framework, let S ⊆ Args be a set of arguments, and let a ∈ Args. It is said that: • S attacks a if there is an a ∈ S such that (a , a) ∈ A; • S defends a if S attacks every attacker of a; • S is conflict-free if there are no a 1 , a 2 ∈ S such that (a 1 , a 2 ) ∈ A; • S is admissible if it is conflict-free and it defends all of its elements.
An admissible set that contains all the arguments that it defends is a complete extension of AF . Below are definitions of some particular complete extensions of AF: • the grounded extension of AF is the minimal (with respect to ⊆) complete extension of AF; • a preferred extension of AF is a maximal (with respect to ⊆) complete extension of AF; • a stable extension of AF is a conflict-free set of argument in Arg L (S) that attacks every argument not in it. 6 In what follows we shall refer to either complete (cmp), grounded (grd), preferred (prf) or stable (stb) semantics as completeness-based semantics. We denote by Ext sem (AF) the set of all the extensions of AF under the semantics sem ∈ {cmp, grd, prf, stb}. 7 Example 1. Figure 1 shows an argumentation framework in which, for instance, a defends a 1 from the attack of a ⊥ . The set S = {a 1 , a } is thus a complete (and even the grounded) extension of this argumentation framework. We note, however, that S is not a stable nor a preferred extension of this framework, since it does not attack the arguments not in it, and since, 5 The argumentation semantics serve thus a different purpose than semantics that give meaning to the formal language underlying logics (such as those logics used for the purpose of generating arguments in Section 2.2). 6 As shown in [48], the grounded extension is unique for a given framework and every stable extension is also a preferred extension. Moreover, while grounded and preferred extensions exist for every argumentation framework, stable extensions may not be available in some cases. 7 Other extensions and their properties are discussed, e.g., in [21][22][23].
a 4 a 8 a 9 Figure 1. An abstract argumentation framework (Example 1). The nodes with a gray background are a stable/preferred extension of the framework e.g., S = {a 1 , a 2 , a 3 , a 4 , a 5 , a 6 , a } (nodes with gray background) is a complete set of the framework that properly contains S. The set S is both a stable and a preferred extension of this framework.

Sequent-Based Argumentation
In the frameworks as they were introduced by Dung, the arguments and attacks are abstract objects: there is no structure in the arguments nor is there a specific nature of the attacks. In logical argumentation a formal language provides the structure for arguments and an entailment relation determines the arguments' validity and the nature of the attacks. Some specific approaches to logical argumentation are introduced, e.g., in [9,31,34,68,70,75]. Surveys on the subject have appeared in [30] and [72]. As noted previously, our setting is based on sequent-based argumentation [4,9], where arguments are represented by the well-known proof theoretical notion of a sequent [53] (see, e.g., [9] and the discussion below for some justification of this choice). Definition 6. Let L= L, be a logic and let S be a set of formulas in L.
• An L-sequent (sequent for short) is an expression of the form Γ ⇒ Δ, where Γ and Δ are finite sets of formulas in L and ⇒ is a symbol that does not appear in L.
• An L-argument (argument for short) is an L-sequent of the form Γ ⇒ ψ, 8 where Γ ψ. We say that Γ is the support set of Γ ⇒ ψ and that ψ Note 2. One of the advantages of sequent-based argumentation is that any logic with a corresponding sequent calculus can be used as the core logic. 9 The use of sequent calculi as the basis for an argumentation system opens up the possibility to incorporate other methods from proof theory as well. Moreover, unlike some other logical approaches to argumentation (e.g., [2]), in sequent-based argumentation, the support set of the argument does not have to be consistent, nor ⊆-minimal. 10 Another advantage of the sequent-based approach is that it allows us to define a variety of attacks between sequents, which are expressed in the form of sequent-based inference rules. More specifically, in our case attacks are represented by sequent elimination rules. Such a 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) 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 for the exact meaning of this). The elimination of a sequent a = Γ ⇒ Δ is denoted by a or by Γ ⇒ Δ.

Definition 7.
A sequent elimination rule (or attack rule) is a rule R of the form (where n ≥ 2): Let L= L, be a logic with corresponding sequent calculus C, θ an Lsubstitution, and S a set of L-formulas. An elimination rule R of the form 9 Note that this implies, in particular, that for a given S, all the elements in Arg L (S) are C-provable. 10 See [9] for a detailed overview and further advantages of this approach.
above is Arg L (S)-applicable 11 (with respect to θ), if θ(Γ 1 ) ⇒ θ(Δ 1 ) and θ(Γ n ) ⇒ θ(Δ n ) are in Arg L (S), and for each 1 Example 2. We refer to [9,79] for a definition of many sequent elimination rules. Below are some of them (assuming that Γ 2 = ∅ 12 ): Defeat: Rebuttal: Direct undercut: Consistency undercut: The rules above indicate different cases in which the attacker challenges the attacked argument. For instance, an argument (sequent) defeats another argument, if the conclusion of the former implies, according to the underlying logic, the negation of the support set of the latter. Likewise, according to Consistency Undercut, arguments whose support set is inconsistent in the base logic, are unconditionally attacked by arguments that depict the inconsistency of the attacked support set.
Example 3. Suppose that {p, ¬p} ⊆ S. When classical logic (CL) is the core logic, the sequents p ⇒ p and ¬p ⇒ ¬p attack each other according to Defeat and Undercut (see Example 2). The tautological sequent ⇒ ψ ∨¬ψ is not defeated or undercut by any argument in Arg CL (S), since it has an empty support set. The sequent p, ¬p ⇒ q is Consistency Undercut by ⇒ ¬(p∧¬p), which expresses the inconsistency of its support.
A sequent-based argumentation framework is now defined as follows: 11 Or just applicable, for short. 12 Many of these rules suppose to have available an implication ⊃. Where the ⊃ connective is missing from the language, one may define compact versions of the elimination rules: for instance, we may replace ⇒ ψ1 ⊃ ¬ Γ2 in Defeat by ψ1 ⇒ ¬ Γ2. Clearly, for core logics for which the deduction theorem holds, these two notions of attack will coincide. Definition 8. A sequent-based argumentation framework for a set of formulas S based on a logic L= L, and a set AR of sequent elimination rules, is a pair AF L,AR (S) = Arg L (S), A , where A ⊆ Arg L (S) × Arg L (S) and (a 1 , a 2 ) ∈ A iff there is an R ∈ AR such that a 1 R-attacks a 2 . The subscripts AR and/or L will be omitted when clear from the context or arbitrary.
Some examples of sequent-based argumentation frameworks are considered next.
Example 4. Let AF CL (S) = Arg CL (S), A be a sequent-based argumentation framework for S = {p, q, ¬p ∨ ¬q, r} and let A be based on a nonempty set AR ⊆ {Defeat, Undercut, ConUcut}. Then the following sequents are in Arg CL (S): Note that Figure 1 above may serve also as a graphical representation of (part of) the sequent-based argumentation framework AF CL  We are now ready to define the entailment relations that are induced from a given sequent-based argumentation framework and its semantics.

Hypersequents and Their Calculi
Ordinary sequent calculi do not capture all interesting logics. For some logics, which have a clear and simple semantics, no standard cut-free sequent calculus is known. Notable examples are the Gödel-Dummett intermediate logic LC, the relevance logic RM and the modal logic S5. As indicated in the introduction, in our context a cut-free calculus is very important, e.g., for reducing the proof space in a quest for appropriate arguments and counterarguments. Indeed, cut-elimination frequently implies the subformula property, thus for producing a counterargument for a particular argument a, one has to consider only the (sub)formulas that are mentioned in a.
A large range of extensions of sequent calculi have been introduced for providing decent proof systems for different non-classical logics. Here we consider a natural extension of sequent calculi, called hypersequent calculi. Hypersequents were independently introduced by Mints [66], Pottinger [69] and Avron [13]. Nowadays Avron's notation is mostly used (see, e.g., [15]). Intuitively, a hypersequent is a finite set (or sequence) of sequents, which is valid if and only if at least one of its component sequents is valid. This allows us to define new inference (and elimination) rules for "multi-processing" different sequents. These types of rules increase the expressive power of hypersequents compared to ordinary sequent calculi, and as a result the corresponding argumentation systems have some desirable properties that are not available for ordinary sequent-based frameworks (we refer to Section 7 for more details about this).
In this section we formally define what a hypersequent is and show how to translate ordinary sequent rules to hypersequent versions. Argumentation frameworks that are based on hypersequents are defined in the next section and some useful test cases are considered in Section 5. General properties of hypersequential frameworks and their relations to reasoning with maximal consistency are discussed in Sections 7 and 8.
Note that every ordinary sequent is a hypersequent as well. Also, the conversion of a sequent calculus to a hypersequent calculus is usually a standard matter (see Example 8 below). Provability in a hypersequent calculus is defined like in standard sequent calculi.
Example 8. To see how sequent rules can be translated into hypersequent versions, consider for instance the right implication rule of Gentzen's calculus LK for classical logic (on the left below). The corresponding hypersequent rule is similar, now with added components (on the right below): As noted in [15], many sequent rules can be translated like this. Often there are two versions (an additive form and a multiplicative form), which are equivalent if contraction, exchange and weakening are part of the system.
See Figure 2 for the hypersequent version of LK, which we will refer to as GLK. Note that in addition to the adjustments to hypersequents of the logical rules, as described above, this calculus also contains adjustments to hypersequents of standard structural rules, like internal contraction (IC) and internal weakening (IW), some structural rules that are specific to hypersequents, such as external contraction (EC) and external weakening (EW), and the splitting rule (Sp), which will be discussed in greater details in what follows.
In order to define hypersequent-based argumentation frameworks, it is not enough to simply take hypersequent inference rules to create arguments.
A new definition of arguments is required and sequent elimination rules should be turned into hypersequent elimination rules. This is what we will do in the next section.

Hypersequent-Based Argumentation
Just as sequent-based argumentation frameworks were based on sequents as arguments (and on attacks as corresponding sequent-based rules), we now define hypersequent-based argumentation frameworks based on hypersequents as arguments (and on attacks as corresponding hypersequent-based rules).
Definition 11. Given a logic L= L, , a set S of L-formulas, and a hypersequent calculus H for L, we define: In what follows, since the underlying hypersequent calculus H will be clear from the context or arbitrary, we shall omit it from the notations. We shall therefore continue to denote by Arg L (S) the set of arguments that are based on S. Note 3. Unlike sequent-based arguments that are determined solely by the underlying logic (recall Definition 6), hypersequent-based arguments, as defined above, depend also on the underlying hypersequent calculus. This is so, since different calculi for the same logic might result in different arguments (which, furthermore, may be attacked by different arguments). 16 To see this, consider for instance the case where classical logic (CL) is taken as the base logic. Since sequents are a particular case of hypersequents, one may consider the standard sequent-based calculus LK for classical logic [53] as the underlying hypersequent calculus. Another option would be to use the hypersequent calculus GLK (Example 8 and Figure 2). Note that while both calculi are sound and complete for CL, they produce different hypersequents (for instance, in LK only hypersequents with one component are derivable). As shown e.g. in Example 10 below, this has far reaching consequences on the structure of the argumentation frameworks that are obtained in each case and on their properties. In Section 7 we shall give a detailed analysis showing how the content of the calculus at hand affects the properties of the induced argumentation framework (for instance, Theorem 1 provides some justification for preferring GLK over LK for argumentation-based reasoning; see also Example 20). Note 4. The adequacy of hypersequent calculi with respect to a logic L = L, is usually established relative to a translation function τ which associates hypersequents with formulas in L and for which holds: H is derivable iff τ (H) is a theorem of the given logic. Premise-abiding adequacy is a stronger requirement in that it requires the support and conclusion of H to directly correspond to the -entailment it represents. As will be shown later (see Example 19), the hypersequent calculus GRM for RM is not premiseabiding adequate, although it is adequate relative to a translation τ .
As before, arguments are constructed by the inference rules of the hypersequent calculus under consideration (see Section 3). For the elimination rules, we keep the notations and their structure as in the sequent-based case: H denotes the elimination of the hypersequent H, the first hypersequent in the conditions of an elimination rule is the attacking argument, the last hypersequent in the conditions is the attacked argument, and the rest of the conditions are to be satisfied for the attack to take place (cf. Definition 7). Some elimination rules for hypersequents are given below. Applications of elimination rules and attacks between hypersequents are defined as in Definition 7, except that sequents are replaced by hypersequents and the sequent calculus C is replaced by a hypersequent calculus H. Note that when both the attacking and the attacked arguments are ordinary sequents, that is, when both of them have only one component, these rules are the same as their ordinary sequent-based counterparts. In the general case, these rules reflect the nature of hypersequents, and in particular the disjunctive reading of their components.
Example 9. Below are hypersequent counterparts of the rules in Example 2. Let G, H be two arguments, where Supp(H) = {Δ 1 , . . . , Δ m } and j ∈ {1, . . . , m}. We define: This approach may seem overly cautious since, in our example, the premise of the second component is sufficient to establish the conclusion p ∨ q. In view of this, one may consider an alternative approach to defeats and undercuts according to which all supports of the attacked argument have to be falsified by the conclusion of the attacking argument. Note, however, that such a definition would not allow ¬p ⇒ ¬p to attack the GLK-derivable hypersequent p ⇒ p ∧ q | q ⇒ p ∧ q, which seems counter-intuitive and therefore problematic.
• The definitions in Example 9 are conservative in the sense that when applied to ordinary sequents they yield the same results as the corresponding elimination rules in Example 2.
Hypersequent-based argumentation frameworks will be defined in a similar way to sequent-based argumentation frameworks (cf. Definition 8).
As before, we shall always omit the subscript H from the above notations, 17 and the subscripts L and/or AR when they are clear from the context or arbitrary.  18 With the possibility of splitting components, we get in addition to the arguments a ⊥ , a , a 1 − a 9 in the sequent-based setting, also the following (hypersequential) arguments: Figure 3 for the extension of the graph in Figure 1 (the dashed graph) with the additional hypersequent arguments and attacks (the solid parts of the graph). To avoid clutter we have omitted all attacks on a ⊥ except for the one from a (i.e., the attacks from a 7 , a 8 , a 9 , H 10 , H 11 and H 12 on a ⊥ are omitted. It is interesting to note that in the sequent-based setting considered in Example 4 it can be proven that E = {a , a 1 , a 2 , a 3 , a 4 , a 5 , a 6 Figure 3. Part of the hypersequent-based argumentation graph for S = {p, q, ¬p ∨ ¬q, r}, with core logic CL and ConUcutH and DefeatH as attack rules, from Example 10. To avoid clutter we omit all attacks on a ⊥ except for the one from a . The dashed graph is the same graph as the one in Figure 1 (in the context of Example 4, referring to the same S), the solid nodes and arrows become available when generalizing to the hypersequent setting in AF CL,{Def} (S) (see also Figure 1). 19 However, Concs(E) is inconsistent. This problem may be avoided by using a hypersequent-based framework as in the current example. Indeed, in the present setting, the following three sets of arguments are part of different complete extensions: E 1 = {a , a 1 , a 2 , a 3 , a 5 , a 6 , a 7 , H 12 }, E 2 = {a , a 1 , a 3 , a 4 , a 6 , a 8 , H 10 } and E 3 = {a , a 1 , a 2 , a 4 , a 5 , a 9 , H 11 } (see Figure 3). Now, E = {a , a 1 , a 2 , a 3 , a 4 , a 5 , a 6 } is no longer admissible, since, for instance, a 2 is attacked by H 10 . In order to defend a 2 , E must be extended with a hypersequent like a 7 , a 9 , or H 11 , and so the new set of arguments is not conflict-free anymore. We note, furthermore, that in this hypersequent-based framework each extension contains the argument a 1 , therefore not only that inconsistent extensions are avoided, but also free arguments (i.e., those that are not involved in any contradictory set of premises) are preserved by the extensions. Note 6. The fact that extensions of structured argumentation frameworks may not be consistent is a well-known problem, discussed e.g. in [2,43]. As argued in the last example, a switch to a hypersequent-based argumentation frameworks may resolve the problem. Intuitively, this is so due to the possibility of introducing new arguments by splitting hypersequents into different components. Indeed, in Section 7.3 we show that the consistency of extensions of hypersequential frameworks like the ones of Example 10 is guaranteed.
Given a hypersequent-based argumentation framework AF L,AR (S), Dungstyle semantics are defined in an equivalent way to those in Definition 5. Accordingly, the entailment relations induced by hypersequential frameworks are defined as in the sequent-based case (cf. Definition 9): Definition 14. Given a hypersequent-based argumentation framework AF L,AR (S), we define 20 : The subscripts L and sem are omitted when they are clear from the context.

Some Notable Test-Cases
In this section we exemplify reasoning with specific hypersequential frameworks. We shall consider three frameworks that are based on well-known logics, for which an ordinary cut-free sequent calculus is not known. For each logic we recall a corresponding (cut-free) hypersequent-based calculus and illustrate the use of hypersequent-based attack rules from Definition 9 by means of some examples.

LC-Based Hypersequential Frameworks
We start by considering Gödel-Dummett logic LC (also known as Gödel Logic or G) as the base logic of the framework. This logic is sometimes considered to be the most important intermediate logic (see [14]), namely: a logic that includes intuitionistic logic and is included in classical logic (see [44]). Specifically, LC = L, LC is obtained by adding the axiom (φ ⊃ ψ)∨(ψ ⊃ φ) to (propositional) intuitionistic logic. This logic has some connections to relevance logics [49], is used in research on Heyting's Arithmetics [83], and is one of the best known fuzzy logics (see, e.g., [20,57,65]).
As noted above, no finite cut-free sequent calculus is known for LC (the ordinary cut-free sequent calculus of LC that was introduced in [ [15]). The axiom (φ ⊃ ψ) ∨ (ψ ⊃ φ) (which is added to intuitionistic logic to obtain LC) can be derived in GLC as follows: Proposition 1. [15, Theorems 1 and 2] 1. GLC admits cut elimination. 23

Let
Then H is derivable in GLC if and only if the following formula is a theorem of LC: 21 The rules in GLC are the multiplicative versions. The additive versions of the rules are admissible, to see this, apply [IW] to the premises of the given rules, before deriving the conclusion. 22 The prime in the notation of this rule indicates that the rule (unlike, e.g., the rule [⇒ ∨] of GLK in Figure 2) has two variations (see also similar rules in Figure 6 below). 23 That is, cut elimination is admissible in GLC.

Axioms: G | ψ ⇒ ψ
Logical rules: Structural rules: Proof. Follows from the deduction theorem, which is valid in LC.
Hence, by external contraction, Γ ⇒ φ is derivable in GLC as well. By Item 2 of Proposition 1, . Therefore, H = Γ ⇒ φ and so the claim follows.
By the claims above, GLC is premise-abiding adequate for LC, and so hypersequential argumentation frameworks may be built on top of them.
The next examples illustrate such frameworks.
Example 13. Let AF L,AR (S) = Arg L (S), A be an argumentation framework like that of Example 10 (i.e., where S = {p, q, ¬p ∨ ¬q, r} and Defeat H is the attack rule), but now L = LC is the base logic. Note that the arguments and attacks as depicted in Figure 3, can be derived in this framework as well. Moreover, it can be shown that, like for L = CL, it holds that S |∼ ∩ L,H,sem r but S | ∼ ∩ L,H,sem q for sem ∈ {grd, cmp, prf, stb}. Example 14. The differences between hypersequential frameworks that are based on LC and CL are evident already when, e.g., S = {¬¬p} and Undercut H is the sole attack rule. In this case, for every completeness-based semantics sem, we have that S |∼ 24 As noted in the beginning of this section LC is, among others, one of the best known fuzzy logics. Fuzzy argumentation (e.g., by taking a fuzzy knowledge-base or defining attack rules as a fuzzy relation) has also been explored in the literature (see, e.g., [60,77,80,85]). In this paper we take LC as an example to show that the resulting hypersequent-based argumentation 24 Indeed, arguments like ¬¬p ⇒ p (unlike p ⇒ ¬¬p) are not derivable in GLC. framework has some desirable properties. This will be further explained in Sections 7 and 8, where we discuss common properties of entailment relations that are induced by hypersequential frameworks. Some related issues, like how to interpret the different strengths of formulas and how to incorporate this in the arguments and/or the attacks, are left for future work.

S5-Based Hypersequential Frameworks
The second family of hypersequential argumentation frameworks that we consider here is based on the modal logic S5. This logic also lacks a cut-free sequent calculus, but has hypersequent calculi, one of them is defined below.
The obvious advantage of incorporating modal languages and logics is that they allow to qualify statements (such as 'it is necessary that ψ') by means of modal operators. In particular, this allows to express alethic arguments (about necessity and possibility), epistemic ones (about knowledge and belief ) [46,59] and deontic phrases (about obligation and permission) [51,84].
Most of the important systems in propositional modal logic (like K, K4, T, and S4) have ordinary, cut-free Gentzen-type formulations. 25 The sequential system for S4, for example, is obtained from that of classical logic by adding to it the following two rules for : 26 In the usual formulation of S5, the rule [⇒ ] of S4 is strengthened to the following rule: It is easy to see, however, that p ⇒ ¬ ¬p is derivable in this system using a cut on ¬p, but it has no cut-free proof. 27 As shown in [15,19] and [69], the problem of providing a cut-free formulation for S5 can be solved with 25 The logic K is obtained by adding the axiom (p ⊃ q) ⊃ ( p ⊃ q) and the rule φ / φ to the Hilbert axiomatization of classical propositional logic. For K4 we further add p ⊃ p to K, for T we add p ⊃ p to K, and for S4 we add both. The logic S5 further strengthens these systems with the axiom ¬ p ⊃ ¬ p. See e.g., [61, §2.5]. 26 For simplicity we deal only with (representing in the alethic interpretation necessity), taking ♦ (intuitively representing possibility) as a defined connective. 27 Only analytic cut (on subformulas of the proved sequent) suffice for the proof.

A hypersequent
Proof. By the axiom p ⊃ p, the necessitation rule φ / φ, and Modus Ponens (see Footnote 25). For Item (ii) we also need the deduction theorem.
Once again, by the claims above, GS5 is premise-abiding adequate for S5, and so hypersequential argumentation frameworks may be built on top of them. The next example illustrates such a framework.
Example 16. Recall the argumentation framework AF L,AR (S) from Example 10, in which Defeat H and Undercut H are the attack rules, CL is the core logic and S = {p, q, ¬p ∨ ¬q}. In the case that L = S5, the additional arguments H 10 , H 11 and H 12 would not be derivable. Intuitively, this is due to the fact that only boxed formulas can be split into different components. A similar graph as the one in Figure 3 can be obtained when S is replaced by S = { p, q, (¬p ∨ ¬q), r}, since then every formula in the support of an argument is boxed.
The logic S5 is sometimes considered to be the most important modal logic [47, page 11]. It is applied in several fields, such as linguistics, computer science (e.g., model checking and security) and game theory (see, e.g., [46, Part III] and [33,Part 4]). There have been some results on combining modal logics with argumentation theory. For example, in [55,56] several modal logical settings are defined to represent argumentation frameworks. 30 A proof theoretical approach is taken in [42], to represent extensions by means of different (modal) logics. Deontic logic is taken as the core logic of an ordinary sequent-based argumentation system in [79] and several of the well-known problems of deontic logic are discussed. Again, some useful properties of modal hypersequential frameworks will be discussed in a more general context in Sections 7 and 8.

RM-Based Hypersequential Frameworks
The last family of hypersequential frameworks that we consider in this section is based on the relevance logic RM. This logic was introduced by Dunn and McCall and later extensively studied by Dunn, Meyer [49] and Avron [13,17] (see also [3,18,50]). In [50, p.81], RM is regarded as "by far the best understood of the Anderson-Belnap style systems". 31 The basic idea behind this logic (and relevance logics in general) is that the set of premises should be 'relevant' to its conclusion. This way some problematic phenomena of classical logic, such as the paradoxes of material implication, are avoided. In addition, it was shown that RM is semi-relevant (i.e., it satisfies the basic relevance criterion in Definition 23), paraconsistent, decidable and has the Scroggs' property [3, §29.4]. Furthermore, RM has a clear semantics in terms of Sugihara matrices [3, §29.3] and sound and complete Hilbertand Gentzen-type proof systems (see, e.g., [13,17] and [18,Chapter 15]). While an ordinary cut-free sequent calculus for RM is not known, it does have sound and complete hypersequent calculi that admits cut elimination. Such a calculus, called GRM [13], is presented in Figure 6. 32,33 30 These works actually aim at a rather different goal than ours, namely, to codify reasoning about classical Dung-style argumentation in a specific modal logic. 31 The logic RM is obtained by adding the mingle axiom (φ ⊃ (φ ⊃ φ)) to the Hilbert axiomatization of the relevance logic R (see [3]). The consequence relation RM is then defined in terms of the Hilbert axiomatization, or semantically in terms of Sugihara matrices (see Appendix A and [3, §29.3] for more details). 32 Recall that apostrophes in rules notations indicates that the rules have two variations. 33 A full justification of the advantages of taking RM as the core logic is beyond the scope of this paper. We refer, e.g., to [17] and Part V of [18].
Unlike the previous two case studies, premise-abiding soundness for the underlying logic is not assured for GRM (see Example 19 below). However, GRM is premise-abiding complete and weakly sound for RM. We first consider a positive property of RM and GRM compared to CL and GLK.
Example 18. Let AF L,{Def H } (S) be a hypersequent-based argumentation framework for S = {p, q, ¬p ∨ ¬q, r}, like in Examples 4 and 10. When classical logic is the core logic, the argument a ⊥ = p, q, ¬p ∨ ¬q ⇒ ¬r can be derived. Hence the axiom r ⇒ r is attacked in Thus, H ∈ Arg RM (S). However, ¬p, p ∨ q q, thus GRM is not premiseabiding sound (but only premise-abiding complete) for RM.
To the best of our knowledge, relevance logics have never been considered as being core logics of logical argumentation system, though relevance in argumentation frameworks has been discussed in the literature. For instance, in [54] such issues are considered and paraconsistent logics are taken to overcome trivialization, a weaker version of crash-resistance from [41]. Recently, in [37], properties of some well-known structured argumentation systems (including sequent-based argumentation) that warrant several relevance desiderata are investigated. 34 In [9] similar problems are discussed and resolved by introducing relevant attack rules.

Properties of Hypersequent Calculi
Our next goal is to examine some general properties of hypersequent-based argumentation frameworks and the entailment relations that are induced by them. We will turn to this in Sections 7 and 8 below. For this, we first need to consider some properties that are related to the hypersequent calculi on which the argumentation frameworks are based. This is what we do in this section.
We begin with some notations that will be important in what follows. Since these notations will be applied to single-as well as to multipleconclusioned (hyper)sequent calculi, we shall use the following conventions: • Π denotes a set of formulas which is empty when the underlying calculus is single-conclusioned, • Δ denotes a set of formulas which is a singleton when the underlying calculus is single-conclusioned.

Definition 16. A hypersequent calculus H is called:
• cautiously reflexive, iff it admits 35  • two-sided splitting, iff it admits of [Sp] from Figure 6; 36 35 We say that a (hyper)sequent calculus H admits a rule if there are other rules in H with which the sequent in the consequent of the rule is derivable from the sequent in the premise of the rule. 36 For single-conclusion calculi the coresponding rule would state that if • support splitting, iff it admits of the following rule: 37 • left-conjunctive, iff it admits [∧⇒ ] from Figure 6; • right conjunctive, iff it admits [⇒∧] from Figure 6; • conjunction eliminating iff it admits at least one of the following rules: • right-disjunctive, iff it admits [⇒∨ ] from Figure 6; • left-negative, iff it admits [¬⇒] from Figure 6; • right-negative, iff it admits [⇒¬] from Figure 6; • deductive, iff it admits the following rules: In the presence of external weakening [EW], trivialization absorption implies non-triviality, that is, that the empty sequent " ⇒ " is not derivable, since otherwise from the empty sequent one would be able to derive any sequent, in contradiction to the non-triviality of the underlying logic (see Definition 2).
Given a hypersequential calculus H we say that it is: Normal iff it is cautiously reflexive, trivialization absorptive, externally weakening, contractive, cut admitting, left-conjunctive, right-conjunctive, conjunction eliminating, right-disjunctive, left-negative, right-negative, and deductive.
Weakening normal iff it is normal and internally weakening.

Support splitting (weakening) normal: iff it is (weakening) normal and support splitting.
Two-sided splitting (weakening) normal: iff it is (weakening) normal and two-sided splitting. 37 For single-conclusion calculi the coresponding rule is [SI] as in Figure 4. A graphic representation of the different types of calculi is given in Figure 7. The next lemma considers some specific cases of these types. In particular, it shows that all the calculi discussed in the previous section are normal. The following proof shows that GRM also admits [Sp∧⇒] and it is therefore conjunction eliminating.
Thus, GRM is normal. Since it contains [Sp], GRM is also two-sided splitting normal.
The cases of GLK and GLC are similar and left to the reader.
In the rest of this section we show properties of (normal) hypersequent calculi that will be needed in what follows. This part of the paper may be skipped on a first reading.

Transitivity: if
Proof. Suppose that H is a normal hypersequent calculus. 38 Here and in what follows, by 'derivable' we mean 'derivable in H'.

Lemma 6.
Let H be a support splitting normal hypersequent calculus, Θ a finite set of formulas, and H = Γ 1 ⇒ | · · · | Γ n ⇒ or H = Γ 1 ⇒ | · · · | Γ n ⇒ is derivable in H. Then, in either case, Proof. This follows by multiple applications of support splitting and conjunction elimination (in case of H ).
The following lemma shows that for both variants of a strongly normal calculus, components of hypersequents can be combined. This will be useful in the proofs of the rationality postulates in the next section. • if H is two-sided splitting normal, the following hypersequents are derivable in H: • if H is weakening normal, then the following hypersequents are derivable in H: Proof. For Γ 1 , . . . , Γ n , Θ 1 , . . . , Θ m , Δ 1 , . . . , Δ k = ∅ and n, m, k ≥ 0, H has the following form (assuming that empty sequents have been removed by trivialization absorption): In the following we assume that n, k, m ≥ 1. The other cases are similar.
be a logic and let H be a normal hypersequent calculus for L that is premise-abiding adequate for L. We have: Since H is premise-abiding sound, φ 1 . . . , φ n ψ.
Recall that our requirements for a logic L according to Definition 2 were rather minimal: we only required structurality and non-triviality. A natural question to ask is whether a logic with an adequate normal hypersequential calculus is Tarskian (Definition 1). Clearly, since the calculus only deals with finite sets of formulas, we cannot answer the question for the full consequence relation , but we can answer it positively for its finitary restriction fin , which we define next. Since arguments have finite support sets, if L fin is a logic, every valid entailment of it can be represented by an argument. By the following lemma we can use the properties of a Tarskian consequence relation for normal, premise-abiding adequate calculi.
be a logic with a normal and premise-abiding adequate calculus H, and let L fin = L, fin be the finite reduction of L. Then: • H is also premise-abiding adequate for L fin .
• fin is a compact, monotonic and reflexive consequence relation.
• if H is strongly normal, fin is a Tarskian consequence relation (Definition 1).
Proof. Clearly fin is compact. H is premise-abiding adequate for L fin since it is premise-abiding adequate for L and fin is the same as on ℘ fin (L) × L (Definition 17). We now show that fin is monotonic, reflexive and transitive.
• Monotonicity Suppose that T fin ψ. If T is infinite, by the definition of fin there is a finite Γ ⊆ T for which Γ ψ and T fin ψ for any T ⊇ T . Suppose now that T is finite and Θ is a finite set of formulas. Thus, T ψ. By the premise-abiding completeness of H and by Lemma 5, • Reflexivity Let T be an arbitrary set of formulas and φ ∈ T . Since H is cautiously reflexive, φ ⇒ φ is derivable in H. Thus, by premise-abiding soundness φ φ and thus φ fin φ. By monotonicity (Item 1), T fin φ.
to the core logic (Section 7.1), paraconsistency, and non-monotonicity (Section 7.2). Then, in Section 7.3, we show that in many cases hypersequentbased argumentation overcomes a shortcoming of some other frameworks (including sequent-based ones), namely that under some completeness-based semantics extensions may not always be consistent (see also [2,43], Example 10 and Note 6). In Section 7.4 we consider two properties that concern a non-trivializing handling of inconsistent data: crash-resistance and noninterference [41].
In the rest of this section we suppose that AF L,AR (S) is a hypersequentbased argumentation framework, induced by a set of formulas S, a logic L= L, with corresponding normal calculus H (Definition 16), and the attack rules We will consider any semantics sem in {cmp, grd, prf, stb}.

Note 8.
A justification of the choice of the above-mentioned setting is in order here. Concerning the underlying logic, we believe that those with normal calculi cover the majority of the underlying formalisms that one would like to consider. The argumentation semantics in our setting cover the most common Dung-style extensions in the literature (although there exist other options; see e.g. the surveys in [21][22][23]. As for the attack rules, Def H and Ucut H are hypersequential versions of two of the most investigated attack rules, namely Defeat and Undercut (respectively). In turn, the latter two generalize several other common rules (for instance, Direct Undercut is a special case of Undercut). As we indicate in what follows, Defeat/Undercut are known for being problematic when it comes to some of the rationality postulates (Definition 21), in particular the consistency postulate. In our setting these problems are resolved.
Concerning ConUcut H , as we show below, this rule turns out to be very useful in generalizing known results and obtaining new ones (see, e.g., Theorem 3 on non-interference). As for the other rules mentioned in this paper, it can be easily shown that Rebuttal causes even more problems when it comes to rationality postulates. 40 40 As an illustration of problems that Rebuttal may cause, consider a sequent-based argumentation framework based on S = {p ∧ s, ¬p ∧ t}, classical logic (with LK), and Rebuttal as the attack rule. Absent (ConUcut)-attacks, arguments with the inconsistent support S will contaminate the framework. But even in the presence of (ConUcut) we have problems with closure: while p ∧ s ⇒ p and ¬p ∧ t ⇒ ¬p rebut each other and so never occur in the same extension, p ∧ s ⇒ s and ¬p ∧ t ⇒ t will occur in the same extension. Note, however, that any argument with conclusion s ∧ t will have an inconsistent support and be ConUcut-attacked by ⇒ ¬((p ∧ s) ∧ (¬p ∧ t)).

Relations to the Core Logic
For showing the relations between the consequence relation of the core logic L and the entailment relation induced by AF L,AR (S), we first recall the notion of a conflict-dependent attack relation [1] (adjusted here to hypersequents), which will be useful also in Section 8. Proof. Let G, H ∈ Arg L (S) such that G attacks H. Note that, by Definition 3 and since H is weakly adequate for L= L, , a set of L-formulas T is -inconsistent, iff ¬ Γ for some finite Γ ⊆ T , iff ⇒ ¬ Γ is derivable in H for some finite Γ ⊆ T . We now consider each of the elimination rules at our disposal: • Def H . In this case, the fact that G defeats H means that ⇒ Conc(G) ⊃ ¬ Θ, for Θ ∈ Supp(H). Since H is deductive, Conc(G) ⇒ ¬ Θ is derivable. By applying Lemma 5 to G we have that, Proof. Suppose that S is a -consistent set of formulas. By Lemma 11, Arg L (S) is conflict-free. Thus, Ext sem (AF L,AR (S)) = {Arg L (S)} for every sem ∈ {cmp, grd, prf, stb}. Suppose first that H ∈ Arg L (S). Since H is premise-abiding sound for L, Supp(H) Conc(H). Assume for a contradiction that S Conc(H). Thus, by the monotonicity of fin (Lemma 9) and the compactness of , Supp(H) Conc(H) which is a contradiction. This shows that the condition in Proposition 5, that H should be premise-abiding adequate for the underlying core logic, is indeed necessary.

Paraconsistency and Non-monotonicity
We turn now to two basic properties of |∼ ∪ L,sem and |∼ ∩ L,sem -paraconsistency and non-monotonicity.
Definition 19. Let |∼ be an entailment relation and H a hypersequential calculus.
• We say that |∼ is paraconsistent, if it is not trivialized in the presence of inconsistency: for all atoms p = q it holds that p, ¬p | ∼ q.
• We say that |∼ is non-monotonic, if there are S 1 , S 2 and φ such that S 1 |∼φ but S 1 ∪ S 2 | ∼ φ. Proof. To show the proposition we need to incorporate some results that are shown later in the paper, so we postpone the proof to Appendix B.

Rationality Postulates
In this section we consider the rationality postulates from [1,40]  • consistency: for all E ∈ Ext sem (AF L (S)), Concs(E) is -consistent.
• exhaustiveness: for all E ∈ Ext sem (AF L (S)), for all H ∈ Arg L (S) such that
Next, we examine what rationality postulates from Definition 21 hold in hypersequential argumentation frameworks, and under which conditions. For instance, recall from Example 10 and Note 6 that the sequent-based argumentation framework for CL as core logic with corresponding calculus LK and Defeat as attack rule does not satisfy the consistency postulate. We will show below that the consistency of the extensions in the hypersequentbased setting in that example is not a coincidence.
In (most of) the lemmas below we suppose that H is a strongly normal (that is, H is either support splitting weakening normal or two-sided splitting normal) hypersequent calculus for the core logic L= L, and that AF L,AR (S) = Arg L (S), A is an argumentation framework for a set S of L- 43 For the postulates we use the notations in Definition 3. 44 Recall from Definition 3 that CN(T ) is defined as the finitary -closure of T . E denotes a sem-extension of AF L,AR (S) for some sem ∈ {cmp, grd, prf, stb}. Furthermore, in what follows when saying that a hypersequent G attacks a hypersequent H in Γ, we mean that the set Γ contains the formulas that are the 'reason' for the attack. For instance, a statement that G Ucut H -attacks H in Γ means that there is some Γ ∈ Supp(H) such that Γ ⊆ Γ , and the sequent ⇒ Conc(G) ↔ ¬ Γ (that is, the condition for the attack) is provable in the underlying calculus.

Lemma 15. (Consistency) If H is strongly normal, premise-abiding complete and weakly sound, then Concs(E) is -consistent.
By applying cut with G and G , the empty sequent is derivable, which contradicts the non-triviality of H (recall Note 7). So, Supp(H) = ∅. By [¬⇒],  Example 20. Classical logic CL with the calculus GLK from Figure 2 and LC with the calculus GLC from Figure 4 fulfill the requirements of the above theorem. Moreover, since we do not require the logic to be premise-abiding sound, only weakly sound, the theorem holds for RM with the calculus GRM from Figure 6 as well. On the other hand, CL with its standard sequentcalculus LK does not fulfill the theorem, since LK is not support splitting and thus not a strongly normal hypersequent calculus. This demonstrates the importance of choosing an appropriate calculus when formulating a hypersequent-based argumentation framework.
Note that the logic S5 is not covered by Theorem 1, since it does not have a support splitting normal hypersequent calculus. 45 Despite the fact that the hypersequent calculus GS5 for S5 is not (support) splitting, it does admit the weaker rule [MS]. As we show in what follows, this implies that some weaker versions of the postulates still hold for hypersequent-based argumentation frameworks induced by S5 and similar logics. This is demonstrated in the next example.
See Figure 8 for a graphical representation of the above arguments and the attacks between them. As in Figured 3 we omit the attacks from a 5 , a 6 , a 7 , H 8 , H 9 and H 10 to a ⊥ to avoid clutter. For this set of premises, no inconsistent extensions exist. Moreover, in every complete extension, a 1 is one of the arguments.
Instead of the full consistency and closure postulate, we will consider modular versions here. These will formally justify the results from the example above.

Notation 2.
Let AF L (S) = Arg L (S), A be an argumentation framework for the logic L= L, and set S of L-formulas. Let E ∈ Ext sem (AF L (S)). We denote: Definition 22. Let AF L (S) = Arg L (S), A be a hypersequent-based argumentation framework for the logic L= L, . We say that AF L (S) satisfies: • modal closure: for each E ∈ Ext sem (AF L (S)), Concs(E ) = CN L (Concs(E )); • modal consistency: for each E ∈ Ext sem (AF L (S)), Concs(E ) is consistent.
Instead of a general logic L= L, , we will consider a logic L = L , with a modal language and its corresponding hypersequent calculus H . We say that H is modal normal if it is weakening normal and the rules [MS], [ ⇒] and [⇒ ] are admissible in it.

Lemma 22. (Modal Consistency)
If H is modal normal and premiseabiding complete, then Concs(E ) is consistent.
is derivable in H , which means that H is ConUcut H -attacked. This is impossible, since then H cannot be defended and be in E at the same time.
We therefore obtain the following theorem: Theorem 2. Let L = L , be a modal logic with a corresponding modal normal, premise-abiding complete and weakly sound calculus H , and let Then AF AR,L (S) satisfies modal closure, closure under sub-arguments, modal consistency and free precedence under any completeness-based semantics from Definition 5.

Crash-Resistance and Non-interference
In [41], Caminada, Carnielli and Dunne consider two postulates that are concerned with the 'collapsing' (or trivialization) of formalisms in the presence of inconsistent information. In this section we examine these properties for entailment relations that are induced by hypersequent-based frameworks. For that, we first give some definition and notations.
• We denote by Atoms(S) the set of atoms that occur in the formulas in S and by Atoms(L) the set of all the atoms of the language.
In what follows we will call logics that satisfy the basic relevance criterion semi-relevant.
Example 22. To see the difference between RM and CL in view of the relevance criterion consider the following example: let S 1 = {p 1 , p 2 , p 1 ⊃ p 2 } and S 2 = {q 1 , q 2 , ¬q 1 ∨ ¬q 2 }. Note that S 1 and S 2 are syntactically disjoint. It can be shown that there is no Γ ⊆ S 1 ∪ S 2 such that Γ ⇒ ¬p 2 is derivable in GRM. On the other hand, in GLK for Γ = S 2 , Γ ⇒ ¬p 2 is derivable. However Γ is inconsistent and thus Γ ⇒ ¬p 2 will be ConUcut H attacked. If we would take one formula out of S 2 , say q 1 , then S 2 is consistent and no such Γ exists anymore.
Definition 24. A logic L= L, is said to be uniform [63,81] if for every two sets of L-formulas S 1 , S 2 and a formula φ, if S 1 , S 2 φ and S 2 is a -consistent set of formulas that is syntactically disjoint from S 1 ∪ {φ}, then S 1 φ. Note 11. Clearly, a logic that satisfies the basic relevance criterion is uniform, but the converse does not hold (as can be shown by considering CL).  11). The reason is that arguments for r such as a 1 = r ⇒ r are attacked by arguments with inconsistent supports, such as p ⇒ | q ⇒ | ¬p ∨ ¬q ⇒ | ⇒ ¬r. As a consequence, the only grounded arguments in AF CL,{Def H } (S) will be those with empty supports. In view of this, the additional premises {p, q, ¬p ∨ ¬q} contaminate our set S relative to |∼ ∩ CL,grd . (Note that the additional premises are syntactically disjoint to S .) The two postulates from [41], in our notations, are then the following: Definition 26. Let L be a language and |∼ ⊆ ℘(L) × L an entailment relation. We say that |∼ satisfies: • non-interference, if for every syntactically disjoint sets S 1 , S 2 of L-formulas, and any L-formula φ such that Atoms(φ) ⊆ Atoms(S 1 ), S 1 |∼ φ if and only if S 1 ∪ S 2 |∼ φ; • crash-resistance, if there is no set S of L-formulas that is contaminating with respect to |∼.
In what follows we assume a hypersequent-based argumentation framework AF L,AR (S) = Arg L (S), A for a uniform logic L= L, with corresponding support splitting normal and premise-abiding adequate calculus H, a set of L-formulas S and attack rules AR for which AR∩{Def H , Ucut H } = ∅. If L does not satisfy the basic relevance criterion we assume ConUcut H to be part of AR.
Example 24. Logics with calculi that satisfy the requirements above are for instance CL with GLK and LC with GLC. At the end of Section 8 we motivate the introduction of the logic RM * = L, * RM , associated with RM, and in Appendix A, we show that this logic has a premise-abiding adequate calculus. For this logic with the corresponding calculus GRM, the results in this section hold as well. To show this theorem in the above-mentioned and other cases, we first prove some lemmas. For the first lemma, we need the following definition.
Definition 27. Given a hypersequent H = Γ 1 ⇒ Δ 1 | · · · | Γ n ⇒ Δ n we say that a hypersequent G = φ 1 ⇒ Θ 1 | · · · | φ m ⇒ Θ m is in splitting normal form of H iff it fulfills the following requirements:  Proof. We first show the case where sem = cmp and i = 1 (the case i = 2 is analogous). Let E ∈ Ext sem (AF L,AR (S 1 )), E 2 ∈ Ext cmp (AF L,AR (S 2 )) and let E be the set of all arguments that are defended by E ∪ E 2 . We show that E is complete in AF L,AR (S Since E by definition also includes all arguments it defends, E ∈ Ext cmp (AF L,AR (S )). We now show that indeed E = E ∩ Arg L (S 1 ). Let H ∈ E ∩ Arg L (S 1 ). Suppose G ∈ Arg L (S 1 ) attacks H. Thus, there is an H ∈ E ∪ E 2 that attacks G. Let H be based on H as in Lemma 25. Thus, also H ∈ E ∪ E 2 . If H ∈ E 2 , Supp(H ) = ∅ and hence H ∈ E since it has no attackers. So, in any case H ∈ E and thus E defends H and by the completeness of E, H ∈ E, therefore E ∩ Arg L (S 1 ) ⊆ E. Now, suppose that H ∈ E ⊆ Arg L (S 1 ), since E ∈ Ext cmp (AF L,AR (S 1 )), H is defended by E. It follows immediately that H ∈ E ∩ Arg L (S 1 ), and so E ⊆ E ∩ Arg L (S 1 ). Altogether, then, E = E ∩ Arg L (S 1 ).
The case where sem = prf (and i = 1) is similar to the previous case, where sem = cmp. As before, we get an extension E † ∈ Ext cmp (AF L,AR (S )) for which E † ∩ Arg L (S 1 ) = E. Thus, there is an E ⊇ E † for which E ∈ Ext prf (AF L,AR (S )). Since E is a ⊂-maximal complete extension, also E ∩ Arg L (S 1 ) = E. Lemma 27. Let sem ∈ {cmp, prf} and E ∈ Ext sem (AF L,AR (S )). Then: is the set of arguments in Arg L (S ) defended by E 1 ∪ E 2 .
Proof. We first show (i) for the case sem = cmp and i = 1 (the case for i = 2 is analogous). Let E 1 = E ∩ Arg L (S 1 ). Clearly, E 1 is conflictfree since E is conflict-free. For completeness, suppose that E 1 defends H ∈ Arg L (S 1 ) in AF L,AR (S 1 ), we shall show that H ∈ E 1 . Suppose that some G ∈ Arg L (S ) attacks H in AF L,AR (S ). Let G ∈ Arg L (S 1 ) be the argument from Lemma 25 that attacks H in AF L,AR (S 1 ). Thus, there is an H ∈ E 1 that attacks G in AF L,AR (S 1 ). The same argument also attacks G in AF L,AR (S ).
Hence, E defends H, and so H ∈ E. Since H ∈ Arg L (S 1 ), this implies that indeed H ∈ E 1 . For (ii), suppose that some H ∈ E is attacked by some G ∈ Arg L (S ) for which Supp(G) = {γ 1 , . . . , γ m }. Let G be in splitting normal form of G as in Lemma 23. Clearly G also attacks H. Thus, some H ∈ E attacks G in some γ j ∈ S 1 ∪ S 2 . Without loss of generality, suppose that γ j ∈ S 1 . Hence, Atoms(Conc(H )) ⊆ Atoms(S 1 ). By Lemma 25, there is an H ∈ E 1 that attacks G and hence also G. Thus H is defended by E 1 and so by E 1 ∪ E 2 . Hence, E ⊆ Defended(E 1 ∪ E 2 , AF L,AR (S )) and since E 1 ∪ E 2 ⊆ E (by Item (i)) and E ⊇ Defended(E, AF L,AR (S )) (since E is complete), E = Defended(E 1 ∪ E 2 , AF L,AR (S )).
Let now sem = prf and i = 1. Consider Item (i) (the proof of Item (ii) carries over). We have shown that E 1 = E ∩Arg L (S 1 ) ∈ Ext cmp (AF L,AR (S 1 )). Assume for a contradiction that there is an E ∈ Ext cmp (AF L,AR (S 1 )) for which E 1 ⊂ E . Note that since E 1 = E ∩ Arg L (S 1 ), E \ E = ∅. As shown above, E ∩ Arg L (S 2 ) ∈ Ext cmp (AF L,AR (S 2 )). Let E be the set of arguments in Arg L (S ) defended by E ∪ E 2 . By items (i) and (ii) above we know that E 1 ∪ E 2 = E = Defended(E 1 ∪ E 2 , AF L,AR (S )) and E = Defended(E ∪ E 2 , AF L,AR (S )) ⊇ E ∪E 2 and since E \E = ∅ and Defended(E 1 ∪E 2 , AF L,AR (S )) ⊆ Defended(E ∪ E 2 , AF L,AR (S )) ⊇ E , E ⊂ E . This is a contradiction to the ⊆-maximality of E.
We now turn to the proof of Theorem 3.
Proof. Let sem ∈ {cmp, prf}. Note that, by Definition 14, S |∼ ∩ cmp φ iff there is some H with Conc(H) such that H ∈ Ext cmp (AF L,AR (S)) = Ext grd (AF L,AR (S)). Hence, the case of sem = grd is covered by the discussion of |∼ ∩ cmp . As before, we assume that S = S 1 ∪ S 2 , where S 1 and S 2 are syntactically disjoint. We also assume that φ is an L-formula such that Atoms(φ) ⊆ Atoms(S 1 ).

Suppose that S | ∼
Thus, for all [some] E ∈ Ext sem (AF L,AR (S )) there is no H ∈ E with conclusion φ. By Lemma 26 [Lemma 27], for all [some (namely E ∩ Arg L (S 1 ))] E 1 ∈ Ext sem (AF L,AR (S 1 )) there is no H ∈ E with conclusion φ. Thus, Thus, for some E ∈ Ext sem (AF L,AR (S 1 )) there is no H ∈ E with conclusion φ. By Lemma 26 there is an E ∈ Ext sem (AF L,AR (S )) for which E = E ∩ Arg L (S 1 ). Assume for a contradiction that there is an H ∈ E for which Conc(H) = φ. By Lemma 25, there is an H ∈ E∩Arg L (S 1 ), with conclusion φ, which is a contradiction to the assumption that there is no H ∈ E with conclusion φ. Thus, S |∼ ∩ sem φ. Suppose that S |∼ ∪ sem φ. Thus, there is an E ∈ Ext sem (AF L,AR (S )) for which there is an H ∈ E with conclusion φ. By Lemma 27, E ∩ Arg L (S 1 ) ∈ Ext sem (AF L,AR (S 1 )). By Lemma 25, there is a H ∈ E ∩ Arg L (S 1 ) with conclusion φ. Thus, Proof. We show the cases for ∩ and ∪ simultaneously, and so we let π ∈ {∩, ∪}. Assume for a contradiction that S is a contaminating set. Then, by Definition 25(i), there is a p ∈ Atoms(L) \ Atoms(S

Reasoning with Maximally Consistent Subsets
A well-known method for handling inconsistent sets of formulas is by taking the maximally consistent subsets of such a set [74]. In this section we study the relations between this approach and the semantics of hypersequential argumentation frameworks. Proof. Let φ ∈ S.
∈ Free L (S) as well.
Entailment relations for reasoning with maximally consistent subsets of premises may be defined as follows: The close relations between structured argumentation and reasoning with maximally consistent subsets have been identified in a number of works, including [2,8,43,58,82], see [5] for a survey. In particular, it has been shown that sequent-based argumentation is a useful platform for reasoning with maximally consistency [6,8]. Here we will extend these results to the hypersequent-based setting.
Theorem 5. Let AF L,AR (S) be a hypersequent-based argumentation framework for a logic L= L, with a fixed corresponding normal hypersequent calculus H that is premise-abiding adequate for L, a set S of L-formulas, and a set of attack rules Then, for every L-formula ψ, it holds that: In what follows we omit the subscript L from the notations of the entailment relations.
The next lemma is needed for the proof of Theorem 5. In what follows we shall assume that AF L,AR (S) is a hypersequent-based argumentation framework that satisfies the conditions in the theorem. Now, assume that there is some H = Γ 1 ⇒ φ 1 | . . . | Γ n ⇒ φ n ∈ Arg L (S) \ E. Therefore, there is a formula ψ ∈ Supp(H) \ T . Let ψ ∈ Γ i for some 1 ≤ i ≤ n. By the maximal consistency of T , and by Definition 3, there are ψ 1 , . . . , ψ m ∈ T such that ¬(ψ 1 ∧ . . . ∧ ψ m ∧ ψ).
We now turn to the proof of Theorem 5: Proof. Let AF L,AR (S) be a hypersequent-based argumentation framework for the logic L= L, with a corresponding normal hypersequent calculus H that is premise-abiding adequate for L. Let S be a set of L-formulas and let Then there is an argument H ∈ E and an E ∈ Ext sem (AF L,AR (S)), such that Conc(H) = ψ. However, since S | ∼ ∪ mcs ψ, it follows that there is no T ∈ MCS L (S) such that ψ ∈ CN L (T ). Since H is premise-abiding sound, for each T ∈ MCS L (S) there is no G ∈ Arg L (T ) with Conc(G) = ψ. Therefore, Supp(H) is -inconsistent. Thus, there is a finite Θ ⊆ Supp(H) for which ¬ Θ. Since H is premise-abiding complete, there is an  ∈ CN RM (S), and so S |∼ ∩ mcs q. However, as mentioned in the same example, H = p ∨ q, ¬p ⇒| p ∨ q ⇒ q ∈ Arg RM (S). Moreover, H is not attacked, since there is no argument G ∈ Arg RM (S) such that ⇒ Conc(G) ⊃ ¬(p ∨ q) or ⇒ Conc(G) ⊃ ¬¬p is derivable. Hence, H is in the grounded extension of AF RM (S) for Defeat H and/or Undercut H as the attack rule(s), and so S |∼ π sem q for every π ∈ {∩, ∪} and sem ∈ {grd, cmp, prf, stb}.
Below are two possible directions to obtain results similar to that of Theorem 5 in the context of RM:

Adjust the notion of an argument.
One way of doing so would be as follows: Arg RM (S) contains all hypersequents derivable in GRM of the form

Summary, Related Work and Future Research
In this paper we have presented a generalization of sequent-based argumentation [9], in which hypersequents represent the arguments of a framework. Like sequent-based argumentation, this approach avoids certain limitations of some other approaches to logic-based argumentation (e.g., those in [31]), where the support set of an argument has to be consistent and ⊆-minimal. The use of hypersequents allows us to incorporate base logics that lack cutfree calculi, like the ones considered in Section 5. In such cases, the search for (hypersequent-based) arguments and counter-arguments is more effective than in the sequent-based counterparts. Moreover, hypersequent-based argumentation allows a great flexibility in the specification of the attack rules and in some cases it also allows us to construct argumentation frameworks with desirable properties that are not available otherwise (see, e.g., Example 10 and Note 6). Indeed, it was shown that frameworks for logics like CL, LC and RM satisfy the logic-based rationality postulates from [1,40] and thus that a problem raised in [43] (and further discussed in [2]), in which complete extensions may not be consistent, is avoided. For logics with a modal language, like S5, some modifications to two of the rationality postulates were necessary in order to prove them. Additionally, for yet another set of assumptions on the calculus of the core logic, non-interference and crash-resistance from [41] were shown. Hypersequent calculi are just one of a variety of sequent calculi introduced to formulate cut-free calculi for logics like S5. Other calculi are display calculi [26], nested sequents [39] and labeled sequents [67]. Although hypersequent calculi are not among the most expressive ones (e.g., by definition any hypersequent is a nested sequent and it has been shown that 48 Note that * RM CL . For example, CL p ⊃ (q ⊃ p), but GRM p ⊃ (q ⊃ p). Also, RM * is neither paraconsistent nor does it satisfy the basic relevance criterion. For instance, p, ¬p * RM q since p ⇒ | ¬p ⇒ |⇒ q is derivable in GRM. To see this, take the axiom p ⇒ p, apply [¬⇒] to obtain p, ¬p ⇒, [Sp] to obtain p ⇒ | ¬p ⇒, and finally [EW] to get p ⇒ | ¬p ⇒ | ⇒ q. they can be embedded into display calculi [73]) due to its intuitive interpretation as a disjunction of ordinary sequents, and since the system is still expressive enough to capture interesting logics such as the three discussed here, we believe that hypersequent-based argumentation is a useful generalization of ordinary sequent-based argumentation. Moreover, hypersequents have been shown useful for the proof theory of fuzzy logics [65], and because of their disjunctive nature, they have also been linked to parallel processing [12]. These relations suggest that there may be useful applications of hypersequent-based argumentation frameworks in these areas and for similar purposes.
In the literature there are several approaches to proving non-interference and crash-resistance. For example, in [86] all the inconsistent arguments are filtered out of the argumentation framework. As a result non-interference, and thus crash-resistance, is shown for complete semantics. Other semantics are not considered because it is necessary that at least one extension would exist, and this is not always the case for stable semantics in their framework. Instead of proving full crash-resistance, in [54] a weaker version is proven, called non-triviality. By doing so, any completeness-based semantics can be used for the framework. Recently, in [37] a general framework is defined, in which several well-known structured argumentation frameworks can be represented. It is shown that for this general framework and under a few further assumptions, both crash-resistance and non-interference are obtained for many completeness-based semantics. Here we were able to prove full non-interference and crash-resistance for grounded, complete and preferred semantics, for logics with a corresponding calculus that fulfills several requirements. Among those logics are CL with GLK and LC with GLC (as long as Consistency Undercut is part of the attack rules) and RM * with GRM (because it satisfies the basic relevance criterion).
Reasoning with maximally consistent subsets (MCS) has been studied since its introduction in [74] (see, e.g., [28,29,38]) and applied in different areas of artificial intelligence. Connections between Dung-style argumentation and reasoning with maximally consistent subsets have been investigated e.g., in [2,43,82], though [43,82] only discuss classical logic as the core logic of the system and in [2,82] the support set of an argument has to be consistent and ⊆-minimal. In [6,8] it is shown that ordinary sequent-based argumentation is useful to represent reasoning with maximally consistent subsets. In this paper we have generalized to hypersequent-based frameworks some of the results from [8] that relate reasoning with MCS and the entailment relations |∼ ∩ mcs and |∼ ∪ mcs . The proof theoretical discussion here is more general than that in [8] also in the sense that less is assumed about the base logic.
As mentioned at the beginning of this section, sequent-based and hypersequent-based argumentation have several advantages over other approaches to structured argumentation. In addition, ordinary sequent-based argumentation is equipped with a dynamic proof theory [11], which provides a proof-theoretic approach to formal argumentation. Dynamic proof theories allow for the automatic derivation of arguments and attacks and it turns out that Dung-style semantics are related to notions of derivability. These dynamic derivations benefit from the availability of cut-free calculi, as they rely on the proof-theoretic properties of the calculus. When moving to first-order level, where classical logic is no longer decidable, dynamic proof theory can still be applied and help obtaining approximations of, e.g., maximally consistent subsets. In future work we plan to extend the dynamic proof theory for sequent-based argumentation to the hypersequent setting.
Additional future research directions include investigations of further argumentation semantics and hypersequential attack rules, the integration of priorities among arguments (extending [7] to the hypersequent setting), and we plan to examine the use of assumptions, such as default assumptions [64] and assumptions taken in adaptive logics [24,78], for further extending the expressive power of hypersequent-based argumentation. Concerning application considerations, it would be interesting to see how argumentation theory benefits from frameworks with core logics like S5, for which a huge amount of research on, e.g., dynamic epistemic logic and agent-based settings is available, and whether LC, which among others is known to be a central fuzzy logic [57], can be successfully incorporated as a base logic for fuzzy argumentation.
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A Uniformity of RM *
The results for non-interference (Theorem 3) and crash-resistance (Theorem 4) suppose that the given hypersequent-calculus is premise-abiding adequate for a uniform logic. In the following we show that although GRM is not premise-abiding sound for RM it is premise-abiding adequate for RM * = L, * RM (see Item 2 at the end of Section 8), which is a uniform logic. 49 Definition 30. We denote RM * = L, * RM , where L is the standard propositional language over {¬, ∨, ∧, ⊃}, and * RM is defined by Γ * RM ψ iff there is a hypersequent H that is derivable in GRM, for which Supp(H) ⊆ Γ and Conc(H) = ψ.
It is not difficult to verify that * RM is a Tarskian consequence relation, thus RM * is a logic. To show that RM * is uniform we first recall the semantics of RM (see, e.g., [17,18]).

Definition 31.
A Sugihara chain is a triple: V, ≤, − where: • V contains at least two elements, • ≤ is a linear order on V, and • |a| = max(−a, a), and • a + b if and only if either |a| < |b| or |a| = |b| and a < b. 49 For the material in this appendix we assume familiarity with propositional matrices and their basic theory (see Chapter 3 of [18]). Then This is in contradiction with ( †). We have shown that Items 1-4 cannot hold together. Therefore, Items 1,2,4 imply that Item 3 does not hold. This shows that RM * is uniform.

School of Computer Science
The Academic College of Tel-Aviv Tel-Aviv Israel oarieli@mta.ac.il