1 Introduction
Argumentation is a form of reasoning that makes explicit the reasons for the conclusions that are drawn, and how conflicts between reasons are resolved. Recent years have witnessed intensive study of both logicbased and human orientated models of argumentation and their use in formalising agent reasoning, decision making, and interagent dialogue ArgInAI ; RahSim . Much of this work builds on Dung’s seminal theory of abstract argumentation dung95acceptability . A Dung argumentation framework () dung95acceptability is essentially a directed graph relating arguments by binary, directed forms of conflict, called attacks. The sceptically or credulously justified arguments are those in the intersection, respectively union, of sets—called extensions—of ‘acceptable’ arguments evaluated under various semantics (see baroni11introduction for an overview). Extensions are evaluated based on two core principles. Firstly, a set of arguments should not contain internal attacks, that is, it should be conflictfree. Secondly, it should defend itself in the sense that any argument in the set is either unattacked, or if attacked by some argument , there is then an argument in the set that defends by attacking (in which case is said to be defended by, or acceptable with respect to, the set of arguments). Arguments and attacks may be seen as primitive or assumed to be defined by an instantiating set of sentences in natural language or in a formal logical language. The former case thus provides for characterisations of more humanorientated uses of argument in reasoning, while in the latter case, the claims of the justified arguments identify the nonmonotonic inferences from the instantiating set of logical formulae, thus providing for dialectical characterisations of nonmonotonic logics dung95acceptability .
Dung’s theory has been extended in a number of directions. In particular, ‘exogenously’ given information about the relative strength of arguments has been used to determine which attacks succeed as defeats, so that the acceptable arguments are evaluated with respect to the arguments related by defeats rather than attacks. In this way one can effectively arbitrate amongst credulously justified conflicting arguments. Examples include AmCay ; ModgilAIJ ; Modgil2013361 that make use of a preference relation over arguments (where in the case of ModgilAIJ , preferences are expressed by arguments that attack attacks amongst arguments) and BC03
that makes use of an ordering over the values promoted by arguments. Other notable developments include approaches that associate probabilities with arguments
Hunter2017 ; Li2012 , and weights on attacks so that extensions do not necessarily comply with the conflictfreeness requirement on Dung extensions; arguments in an extension may attack each other provided that the summative weight of attacks does not exceed a given ‘inconsistency budget’ dunne11weighted .^{2}^{2}2See also Janssen which account for the relative strength of attacks amongst arguments.Context: graduality in argumentation
It has long been recognized—since at least BesnardHunter01 and CLS04 —that one of the drawbacks of Dung’s theory of abstract argumentation is the limited level of granularity the theory offers in differentiating the strength or status of arguments (essentially three, but cf. wu10labelling ). Consider the following informal example:
Example 1.
Suppose an argument concluding the presumed innocence of a suspect. is attacked by an argument which consists of two subarguments and that respectively conclude that a suspect had opportunity and motive, where defeasibly extends these subarguments to conclude that the suspect is guilty. Suppose an argument (in support of an alibi) that attacks on the assumption that the suspect does not have an alibi. Suppose then an additional argument that attacks an assumption in . The level to which is defended, and so said to be justified, is intuitively increased in the case that we have counterarguments and that argue against the suspect having motive and opportunity, as compared with when one only has the counterargument .
The above example highlights one amongst a number of intuitions that are formalised by the above mentioned BesnardHunter01 and CLS04 , and more recently AmgoudNaim ; Amgoud:2016 ; conf/jelia/MattT08 , in which the numbers of attackers and defenders are used to give a more fine grained assignment of status to (and hence ranking of) arguments. While these approaches are defined wth respect to s, the status of arguments is not determined using the standard ‘Dungian’ concepts of sets of arguments (extensions) and arguments defended by these sets; rather, measures of the strengths of arguments based on the numbers of attackers and/or defenders are propagated through the graph^{3}^{3}3The exception being conf/jelia/MattT08 in which the strength of arguments is evaluated by reference to two person games in which a proponent defends an argument against counterattacks by an opponent.. These approaches, which we refer to here as ‘propagation based approaches’ (see bonzon16comparative for a recent comparative overview), have also been developed to account for exogenously given information about the strength of arguments, which provide the initial measures that are then adjusted based on how they are propagated through the graph. Notable examples of the latter include gabbayrodrigues:13 ; Gabbay2015 and conf/ijcai/LeiteM11 .
Paper Contribution
The central aim of this paper is to show that a more fine grained assignment of status to arguments that does not rely on exogenous information, and thus only on the numbers of attackers and defenders of arguments, can be formalised as a natural generalisation of the Dungian notions of conflict free sets of arguments and the defense of arguments by sets of arguments. This aim is achieved in the following way. First, our starting point is the classical definition of an admissible set of arguments as one that is conflict free and that defends all its contained arguments dung95acceptability . We show that the logical structure of this definition naturally generalises so as to yield more fine grained graded notions of conflict freeness and defense that account for the number of attackers and defenders, thereby obtaining a graded variant of the concept of admissibility. Second, this graded form of admissibility serves as a basis for defining graded variants of all the classic semantics of abstract argumentation studied in dung95acceptability : complete, grounded, stable and preferred. Dung’s definitions of the standard semantics can all be retrieved as special cases of our graded variants, showing that the form of graduality this paper studies is rooted in a principled way in the classical theory of abstract argumentation. These graded semantics are, intuitively, ways of interpreting the standard Dung semantics in ‘stricter’ or ‘looser’ ways. For instance, the grounded semantics can be interpreted more ‘ strictly’ by requiring that all attackers be counterattacked by at least two arguments, instead of just one as in the classic case. So each Dung semantics now comes equipped with a family of strengthenings and weakenings, which we call graded semantics. Third, these strengthenings and weakenings define a natural (partial) ordering dictated by setinclusion: a stricter semantics will define sets of arguments which are subsets of the sets defined by a weaker one. This natural ordering then induces an ordering on the arguments themselves, thereby defining a ranking over the arguments in the framework. Such a ranking enables arbitration amongst arguments without recourse to exogenous preference information. As this ranking is induced from a generalization of Dung standard semantics, our approach is at the outset methodologically distinct from propagationbased approaches, where argument rankings are defined directly by reference to the argument graph. Fourth, we study the application of graded semantics and their induced rankings to two key instantiations of Dung s: classical logic instantiations that, under the standard Dung semantics, yield a dialectical characterisation of nonmonotonic inference in Preferred Subtheories bre89 , and instantiations that accommodate more human orientated uses of argumentation through the use of schemes and critical questions Wal96 . In so doing, we seek to substantiate some of the intuitions captured by our generalisation of Dung semantics, and show how the graded semantics capture a simple form of counting based accrual of arguments, which has traditionally been regarded as being incompatible with Dung’s theory PrakAccrual .
Outline of the paper
The paper is structured in three parts. Part 1 concerns the development of the abstract theory of graded argumentation. It starts with Section 2, where we review Dung’s theory, giving prominence to its fixpointtheoretic underpinnings, which have remained relatively underinvestigated in the literature, and that we thus consider to be of some independent interest. Section 3 then generalises Dung’s notions of conflict freeness and defense, yielding grading variants of these notions in terms of the number of arguments attacking and defending any given whose acceptability with respect to a given set of arguments is at issue. This yields a ranking among types of conflict freeness and defense. Section 4 then generalises Dung’s standard semantics, so that extensions are graded with respect to the attacks and counterattacks on their contained arguments. These semantics—which we call graded—are shown to generalise Dung’s theory and are studied providing constructive existence results. Part 2 first shows, in Section 5, how the new graded semantics yield a natural way of ranking arguments according to how strongly they are justified under different graded semantics, thereby enabling endogenous arbitration among credulously justified arguments. Then, Section 6 illustrates application of these type of rankings to ASPIC+ instantiations of s Modgil2013361 that formalise stereotypical patterns of argumentation encoded in schemes and critical questions Wal96 , thus accounting for more humanorientated uses of argument, and ASPIC+ instantiations of s that provide dialectical characterisations of nonmonotonic inference in Preferred Subtheories bre89 . Both types of instantiation are then shown to capture a simple form of counting based accrual, whereby multiple arguments in support of the same conclusion mutually strengthen each other. In Part 3, Section 7 develops a thorough comparison of our approach to the existing approaches to graduality and rankings in argumentation, leveraging the systematization recently introduced in bonzon16comparative . This allows us to place more precisely graded argumentation in the growing landscape of rankingbased semantics. We conclude in Section 8 outlining some avenues for future research in graded argumentation.
2 Preliminaries: Abstract Argumentation
This preliminary section reviews key concepts and results from Dung’s abstract argumentation theory dung95acceptability . The presentation we provide gives prominence to the fixpoint theory underpinning Dung’s theoretical framework. After dung95acceptability the fixpoint theory of abstract argumentation has remained relatively underinvestigated in the literature. It is, however, the most natural angle from which to pursue the objectives of this paper. We are unaware of any comprehensive exposition to date of the fixpoint theory of abstract argumentation, and so hope this preliminary section is of independent interest.
Since this paper will provide a generalization of Dung’s original theory, all results presented in this section can actually be obtained as direct corollaries of the results we will establish later in Section 4. This, we argue, should be a desirable feature for any theory of graduality in argumentation which bases itself on Dung’s original proposal. For completeness of the exposition, direct proofs of the results dealt with in this section can be found in Section Appendix. Proofs of Section 2.
2.1 Basic Definitions
Definition 1 (Frameworks).
An argumentation framework (AF) is a tuple where , and is a binary attack relation on . Notation denotes that attacks , and denotes that s.t. . Similarly, denotes that s.t. . For a given we write to denote (the direct attackers of ), and to denote (the direct defenders of ). Also, as is customary, denotes the transitive closure of a given relation, so that stands for ‘there exists a path of attacks from to ’. Finally, an AF such that for each , is finite, is called finitary, whereas an AF with a finite number of arguments is called finite, else infinite.
We will sometimes refer to s as attack graphs.^{4}^{4}4Although note that all definitions in this paper equally apply to ‘defeat’ graphs which assume a binary defeat relation on arguments, obtained through use of preferences in deciding which attacks succeed as defeats. Figure 1 depicts three s. An argument is said to be acceptable w.r.t. , if any argument attacking is attacked by some argument in , in which case is said to defend . An ’s characteristic (also called ‘defense’) function, applied to some , returns the arguments defended by dung95acceptability (henceforth, ‘’ denotes powerset):
Definition 2 (Defense Function).
The defense function for is defined as follows. For any :
Where no confusion arises we may drop the subscript in .
An argument is not attacked by a set if no argument in attacks . One can define a function which, applied to some in an , returns the arguments that are not attacked by . This function was introduced by Pollock in pollock87defeasible for his theory of defeasible reasoning, and we refer to it here as the ‘neutrality function’.
Definition 3 (Neutrality Function).
The neutrality function for is defined as follows. For any :
Again, where no confusion arises we may drop the subscript in .
One final bit of terminology. In what follows we will often use the notion of function iteration for and which we define in the standard inductive way, for : ; .
Example 2 (Defense and neutrality in Figure 1).
The functions applied to the symmetric graph of Figure 1 (left) yield the following equations:
Notice that the output of on corresponds to the whole set of arguments, as no arguments can be attacked by . Notice also that while and are included in and , is not.
One can define the extensions of an under Dung’s semantics, in terms of the fixpoints ( and ) or postfixpoints ( and ) of the defense and neutrality functions, as recapitulated in Table 1. The justified arguments are then defined under various semantics:
Definition 4 (Justification under Semantics).
Let = .Then for semantics grounded, stable, preferred^{5}^{5}5Typically, the justified arguments are not defined w.r.t. the complete semantics, which subsume each of grounded, stable and preferred., is credulously, respectively sceptically, justified under , if is in at least one, respectively all, extensions of .
is conflictfree in  iff  
is selfdefended in  iff  
is admissible in  iff  and 
is a complete extension in  iff  and 
is a stable extension of  iff  
is the grounded set in  iff  is the smallest complete ext. of () 
is a preferred extension of  iff  is a largest complete extension of 
Finally, we recapitulate some wellknown properties, first established in dung95acceptability , of the defense and neutrality functions that will be referred to later:
Fact 0.
Let be an and . The following holds:
That is, function is monotonic, function is antitonic, and the composition of with itself, which we will also denote , is function . For example, in Figure 1 (right), we have that = , and .
Fact 0 (continuity^{6}^{6}6Cf. (dung95acceptability, , Lemma 28).).
If is finitary, then is (upward)continuous for any , i.e., for any upward directed set of finite subsets of :^{7}^{7}7We recall that an upward directed set is a set of sets such that any two elements and in have an upper bound in , that is, there also exists a superset in . A downward directed set is defined dually in the obvious way.
(1) 
Similarly, is (downward)continuous for any , i.e., for any downward directed set :
(2) 
The above fact establishes important properties of the behaviour of the defense function with respect to sequences of sets of arguments and their limits. As we will also later see in the graded generalization of Dung’s theory, these properties are key in the construction of extensions through the iteration of the defense function.
2.2 Rudiments of fixpoint theory of Dung’s extensions
In light of Facts 1 and 2 we can rely on general results from order theory to establish the existence of the least fixpoint (
) of the characteristic function, that is, the existence of the grounded extension. The monotonicity of
guarantees the existence of the least fixpoint of as the intersection of all prefixpoints of :(KnasterTarski Theorem)  (3) 
Given a set of arguments , the fold iteration of is denoted for and its (countably) infinite iteration is denoted . For a given , an infinite iteration generates an infinite sequence, or stream, . A stream is said to stabilize if and only if there exists such that . Such set is then called the limit of the stream. The monotonicity and continuity of the characteristic function, in finitary frameworks, guarantee together that such least fixpoint can be computed ‘from below’ through a stream :
(4) 
The reader is referred to davey90introduction for a detailed presentation of these results. We will come back later in some more detail to equation (4), which is the stepping stone of some of the results the paper presents.
2.3 The Fixpoint Theory of Acceptability and Conflictfreeness
We show how any admissible set of arguments can be saturated to a complete extension through a process of fixpoint approximation. This establishes a general result concerning the computation of complete extensions in (finitary) attack graphs which, to the best of our knowledge, has never been reported in the literature. It is, however, a generalization of wellknown existing results such as (dung95acceptability, , Lemma 46, Theorem 47) (cf. also lifschitz96foundations ).
2.3.1 Construction of Fixpoints from Admissible Sets
Fix a framework and take a set such that and (i.e., an admissible set). By iterating , consider the stream of sets and Since is admissible, is monotonic and antitonic (Fact 1), the first stream (see the lower stream in Figure 2) is nondecreasing and the second stream (see the upper stream in Figure 2) is nonincreasing, with respect to set inclusion. In finite attack graphs, these streams must therefore stabilize reaching a limit at state . In infinite but finitary attack graphs, we will see that the limit can be reached at . We will see (Lemma 2) that the limits of these streams correspond to the smallest fixpoint of containing the admissible set and, respectively, the largest fixpoint of which is contained in . We denote the first one by and the second one by . Intuitively, the two sets denote the smallest superset of which is equal to the set of arguments it defends^{8}^{8}8Theorem 1 will show this set is also conflictfree and it is therefore the smallest complete extension containing . and, respectively, the largest set which is not attacked by and which is equal to the set of arguments it defends.^{9}^{9}9Note that such a set is not necessarily conflictfree. E.g., consider , , that is and . Then and . But clearly it is not the case that , that is, is not conflictfree. The construction is illustrated in Figure 2 below.
Example 3.
Consider the cycle of length three in Figure 1 (center), and take the admissible set . By applying the above construction we obtain immediately as limit of the lower stream, and as limit of the upper stream. is the smallest selfdefended set containing , is the largest selfdefended set contained in .
We prove now the correctness of the above construction by showing that the limits of the above streams correspond indeed to the desired fixpoints. First of all the following important lemma shows how conflictfreeness is preserved by the above process of iteration of the defense function:
Lemma 1.
Let be a finitary attack graph and be admissible. Then for any s.t. ,
That is, each in the stream of iteration of the defense function from an admissible set is a conflictfree set. The lemma can be seen as a reformulation of (dung95acceptability, , Lemma 10), known as Dung’s fundamental lemma.^{10}^{10}10It also generalises (dung95acceptability, , Lemma 46) to the case of admissible, instead of .
We can show that the above streams obtained through the process of iteration of the defense function construct the desired fixpoints:
Lemma 2.
Let be a finitary attack graph and be admissible:
(5)  
(6) 
Notice that since (Fact 1), a stream generated by the indefinite iteration of the defense function can actually be viewed as a stream generated by the indefinite iteration of the neutrality function. So equations (5) and (6) of Theorem 2 can be rewritten as follows:
(7)  
(8) 
In this light, Lemmas 1 and 2 capture several of the key features of the stream generated by the indefinite iteration of the neutrality function on an admissible set
. First, the stream can be split into two parts, the part consisting of even and, respectively, odd iterations of
. Second, the stream of even iterations converges to a limit which is the smallest complete set including , and the stream of odd iteration converges to a limit which is the largest selfdefended set contained in (that is, not attacked by ).^{11}^{11}11Again the finitariness assumption in the theorem could be lifted by making use of transfinite induction. Cf. Remark 1 below. Notice that such a set is just free of conflict with respect to , but it is not necessarily conflictfree, and hence it is not necessarily a complete extension. Third, the two streams can actually be viewed as streams of the defense function applied to and, respectively, to . Fourth, the two parts grow towards each other as the stream of even iterations is increasing, while the one of odd iterations is decreasing. See Figure 2 for an illustration.Remark 1.
The proof of Lemma 2 relies in an essential manner on the finitariness assumption on the underlying framework. The assumption simplifies the proof but, it should be stressed, could be lifted. For infinite graphs which are not finitary, the lemma could be proved by resorting to transfinite induction:
By the monotonicity of it can then be shown that there exists an ordinal of cardinality at most such that: . A proof of this statement in the general setting of complete partial orders can be found in (venema08lectures, , Ch. 3). Transfinite induction relies on the Axiom of Choice (or equivalent formulations such as Zorn’s Lemma or the WellOrdering Principle), which is known to be required for the existence results of the standard Dung semantics (cf. baumann15infinite ).
2.3.2 Construction of Complete Extensions
With the above results in place, one can then show how complete extensions can be constructed through a process of fixpoint approximation.
Theorem 1.
Let be a finitary and be admissible. Then the limit is the smallest complete extension of that includes .
The theorem establishes that, in finitary frameworks, any complete set can be computed via a process of iteration at of the defense function, starting with some admissible set. It is a novel simple generalization of the earlier result in dung95acceptability for the case of = . This process starts by including the arguments that have no attackers or that belong to an initial admissible set, then including those arguments that are defended by the first set of arguments included, and so on.^{12}^{12}12Cf. Remark 1. At an intuitive level, the theorem states that the indefinite iteration of the defense from an admissible can be considered as a formalization of the process whereby an agent constructs a rational argumentative position—a complete extension—starting from .
2.3.3 Construction of Other Dung Extensions
Theorem 1 also yields a constructive proof of existence (in finitary graphs) for the grounded extension. By setting (the trivially admissible set), the theorem returns the known result for the construction of the grounded extension (dung95acceptability, , Th. 47). In this section we show how the theorem relates to the other classical Dung extensions.
Specific conditions can be identified which guarantee that the indefinite iteration of the defense function constructs preferred and stable extensions from a given admissible set . It is fairly easy to see that if the chosen admissible set of is ‘big enough’ in the precise sense that it contains enough arguments to be able, from some argument in , to reach any argument in the graph via the attack relation, i.e., if , then the stream of iterations of from converges to a complete extension containing (by Theorem 1), but this extension is now maximal as all arguments in can be reached from .
The condition under which Theorem 1 constructs stable extensions is particularly interesting. If the streams of even and odd iterations of the neutrality function (recall Figure 2) converge to the same limit, then the process of fixpoint approximation defines a stable extension:
Fact 0.
Let be a finitary and be admissible. If , then both sets coincide with the unique stable extension of that includes .
Proof.
By Theorem 1, is the smallest complete extension containing . However, as by assumption, the set is therefore also a fixpoint of the neutrality function, and therefore a stable extension. ∎
This observation is, to the best of our knowledge, novel and provides a characterization of the existence of a stable extension that includes a given admissible set.
Example 4 (Construction of complete extensions in Figure 1).
Consider the rightmost . Starting with the admissible set , the nondecreasing stream
converges after one step to the smallest complete extension containing . The nonincreasing stream converges to , i.e., to the same set which is also the largest fixpoint of included in . As is also conflictfree, it is a preferred extension. Notice also that if we were to start with , the resulting streams would be and . The first one constructs the grounded extension of the , and the second the largest fixpoint of in the , that is, .
3 Graded Acceptability
We now turn to the main contribution of this paper: a graded generalisation of Dung’s acceptability semantics. We first introduce the intuitions behind our generalisation, and then define and study graded variants of Dung’s defense and neutrality functions, which capture the proposed intuitions. These functions will then be used later (Section 4) to define and study a family of graded variants of Dung’s semantics, and the rankings they enable (Section 5).
3.1 Introducing Graded Acceptability: Intuitions
The central tenet of argumentation theory is that any individual argument cannot, in and of itself, constitute definitive grounds for believing that a claim is true. Rather, the status of an epistemic claim as true justified belief is not established by an individual argument, but through the dialectical consideration of counterarguments and defenders of these counterarguments SMReasoner . Similarly in practical reasoning, the status of (a claim representing) a decision option supported by an individual practical argument is not considered to be the option that simply maximises a given objective, but rather the best option contingent on having dispreferred alternative options and refuted challenges made to the assumptions made in support of the argument. Pragmatically however, we operate under the assumption that the claim is true/the decision option is the best available, to the extent that as of yet we know of no good reason to suppose otherwise. Argumentatively, a claim can be considered established only in as much as it is the claim of a justified argument included in a network of interrelated arguments and counterarguments. Dung’s abstract argumentation theory captures these principles by assuming sets of arguments, rather than individual arguments, as the units of analysis, and studying formal criteria (semantics) for sets of arguments to be acceptable. Apart from trivial cases (unattacked arguments), arguments are acceptable only as members of a set of acceptable arguments. The graded theory of acceptability that we aim at, captures a notion of graduality while at the same time retaining the notion of a set of arguments as the central unit of analysis.
3.1.1 Graded neutrality
According to the standard definition of neutrality (Definition 3) a set is neutral with respect to if is not attacked by any argument in . A less demanding criterion of neutrality of with respect to would require that there exists at most one attacker of in , a yet less demanding one (or at least not ‘as demanding as’) would require that there exist at most two attackers of in , and so on. Intuitively, these weakened neutrality criteria capture the idea that one (two, three, …) attackers are not enough to rule out the coacceptability of and its attacking arguments.^{13}^{13}13Using terminology from logic, this may be viewed as an argumentative form of paraconsistency. In more ‘human orientated’ argumentation formalisms (e.g., dunne11weighted and developments thereof), this may be viewed as accounting for an attacking argument not establishing definitive grounds for its claim (as discussed at the beginning of Section 3.1), and hence not definitively ruling out the claim of the attacked argument. One then obtains a natural way to generalise the neutrality function (Definition 6 below) by making explicit a numerical level of neutrality of a set of arguments with respect to a given argument, as depicted in Figure 3 (left).^{14}^{14}14Weighted Argument Systems dunne11weighted propose a somewhat similar idea, whereby an inconsistency budget sets a threshold on the number of attacks that can be tolerated within a given set. However, notice that our notion of a threshold set, yielded by graded neutrality, is local in the sense that it pertains to the incoming attacks on each individual argument. We will later compare this and other related approaches in more detail (Section 7). So we say that is neutral with respect to whenever there are at most attackers of in .
3.1.2 Graded defense
According to the standard notion of defense, an argument is defended by a set of arguments whenever every attacker of is attacked by some argument in . The quantification pattern (‘for all’, ‘some’) involved in this definition offers again a natural handle to generalise the notion of defense. If all but at most one attackers of are attacked by at least one argument in , the quality of such defense (and hence the extent to which is acceptable w.r.t. ) can reasonably be considered ‘lower’ than in the case in which all arguments are counterattacked by at least one argument in . But the former quality of defense is still ‘higher’ than (or at least not ‘as low as’) the case in which all but at most two attackers are counterattacked by at least one argument in , and so on. Similarly, if all attackers of are counterattacked by at least two arguments in , then the quality of this defense can reasonably be considered ‘higher’ than in standard acceptability, but ‘lower’ than (or at least not ‘as high as’) the case in which all attackers of are counterattacked by at least three arguments in . Combining these intuitions—depicted in Figure 3 (right)—one obtains a way to generalise the defense function (Definition 5 below) by making explicit, through numeric grades ( and ) of the above type, how well a set defends an argument . So we say that defends whenever there are at most attackers of , which are not counterattacked by at least arguments in .
This notion of graded defense is related in a natural way to the above notion of graded neutrality: the set of arguments that are defended by , is the set of arguments which is not attacked by at least arguments, that are not in turn attacked by at least arguments in (i.e., arguments that are neutral with respect to the set of arguments that are neutral with respect to ). In other words, the notion of tolerance towards attack (graded neutrality) can be iterated to obtain a notion of graded defense. Fact 4 will establish this claim formally. We illustrate the above intuitions with a few examples.
Example 5.
In Figures 4i) – 4iv), the encircled set defends () under Dung’s Definition 2. However we can differentiate these cases based on the number of attackers and defenders of . For instance, more strongly defends than defends , as is defended by two arguments whereas is defended by one argument (i.e., the standard of defense that allows at most 0 attackers to not be defended by 2 arguments is met by ’s defense of but not by ’s defense of ). We will later, in Example 12, reference the defense of by and of by to illustrate that neither can be said to be a more strong defense than the other. While neither or defend , respectively , under Dung’s Definition 2, observe that ’s defense of is stronger than ’s defense of . The former meets a standard of defense that allows at most one attacker () not to be defended by at least one defender (which goes hand in hand with accommodating the coacceptability of with at most one undefended attacker; i.e., is neutral with respect to ). This standard is not met by ’s defense of , since is attacked by two undefended attacks (from and ). In the latter case, a weaker standard of defense is met, which again goes hand in hand with accommodating the coacceptability of with its two undefended attackers. These notions then naturally generalise so that one can discriminate standards of defense based only on the number of attackers. Allowing at most one attacker not to be defended by two arguments, is a standard of defense met by s defense of , but not ’s defense of (the former defense thus being stronger than the latter).
3.2 Graded Defense and Neutrality Functions
Let us move now to the formal definitions of graded defense and neutrality. Take an argument and a set of arguments . Let be the number of ’s attackers () and, for each () let be the (non zero) number of attackers of (i.e., defenders of ) in . Finally, let be the minimum among the s, i.e., . We can now count the number () of attackers of , which are counterattacked by at least arguments in . Integers and therefore encode information about how strongly is defended by , in the sense that they express a maximum number (i.e., ) of attackers of which are not counterattacked by a minimum given number (i.e., ) of arguments in . We can now generalise Definition 2 as follows:
Definition 5 (Graded defense).
Let be an and let and be two positive integers (). The graded defense function for is defined as follows. For any :
where , for integers (‘there exist at least arguments ’) are the standard firstorder logic counting quantifiers.^{15}^{15}15Cf. dalen80logic . The definition can be reformulated without counting quantifiers as follows:
So, is the set of arguments (in the given framework) which have at most attackers that are not counterattacked by at least arguments in .
Example 6.
In Figure 4, and since in both cases the following holds: at most arguments attacking , respectively , are not attacked by at least one argument in , respectively . However if we increment by we have that: but since it is the case that at least one argument attacking is not attacked by at least two arguments in . Intuitively, this standard of defense allows for up to attackers to not be counterattacked by two defenders, a standard met by ’s defense of , but not by ’s defense of . Continuing with Figure 4, and , since in both cases the standard of defense that allows for no more than 2 unattacked arguments is met. However, and since this standard of defense accommodates up to a maximum of 1 unattacked attackers, and in the latter case there is more than one unattacked attacker of . Finally, and since the standard of defense requires that all attackers of () are attacked by at least two arguments. However, and , since the standard of defense allowing at most one attacker not to be defended by two counterattackers is met by but not by . Notice that in this last case the two arguments and are discriminated based on the number of their attackers.
By the same logic, Definition 3 can be generalised as follows:
Definition 6 (Graded neutrality function).
Let be an and let be any positive integer. The graded neutrality function for is defined as follows. For any :
So, given a set of arguments , denotes the set of arguments which have at most attackers in .^{16}^{16}16Equivalently, graded neutrality can be defined as follows, without the use of counting quantifiers:
Example 7.
In Figure 4, . Notice that . Also, .
3.3 Properties of Graded Defense and Neutrality
The following two facts show that the graded defense and neutrality functions are generalisations of the standard functions defined in Definitions 2 and 3, and that such generalisations remain wellbehaved in the sense that they retain many of the key features of their standard variants.
Fact 0.
For any , , and positive integers:
(9)  
(10)  
(11)  
(12)  
(13) 
Proof.
Equation (9) follows from the fact that Definition 2 can be retrieved from Definition 5 by setting . Similarly (10) follows from the fact that Definition 3 can be retrieved from Definition 6 by setting . Equation (13) follows from Definitions 5 and 6 by the following series of equations:
Formulae (11) and (12) are direct consequences of Definitions 5 and 6. ∎
Equation (9) reformulates as the set of arguments for which it is not the case that there are one or more attackers, which are not counterattacked by one or more arguments in ; that is, no attacker is not attacked by some argument in . So does Equation (10) for . The remaining formulae generalise Fact 1 to the graded setting. In particular, graded defense is monotonic (12), graded neutrality is antitonic (11), and equation (13) shows that, as in the standard case, the defense function is the twofold iteration of the neutrality function (as in the standard case we may use the notation to denote this composition).
Importantly, the continuity of the defense function is also preserved in the graded setting:
Fact 0 (continuity of graded defense).
If is finitary, then function is (upward) continuous for any , and positive integers. I.e., for any upward directed set of finite subsets of :
(14) 
Similarly, is (downward)continuous for any , and positive integers. I.e., for any downward directed set of finite subsets of :
(15) 
Sketch of proof.
Finally, we establish some properties showing how the values for the defence and neutrality functions are affected by varying the parameters and .
Fact 0.
For any , , and , and positive integers:
(16)  
(17)  
(18) 
Proof.
Recall the definition of the neutrality function (Definition 6). To establish (16) it suffices to notice that the property expresses the contrapositive of the following statement: if there exist at least attackers in then there exist at least attackers in . Property (17) then follows directly by (16) above and (13) (Fact 4), through the following series of relations:
A similar argument applies to establish (18), which follows by (16) above, (13), and the antitonicity of (Fact 4):
This completes the proof. ∎
Intuitively, (16) states that the set of arguments attacked by at most arguments in is included in the set of arguments attacked by at most arguments in . This establishes an ordering, in terms of logical strength, among the values of different neutrality functions: the lower is the stricter is the value of applied to a same set of arguments . Properties (17) and (18) then follow by combining this simple fact with the fact that is the composition of with (13).
3.4 Comparing Graded Defense and Neutrality Functions
Fact 6 provides ground for a natural way in which different graded defense and neutrality functions can be ordered as their parameters and vary. The choice of these parameters determines the logical strength of different ‘types’ or ‘standards’ of conflictfreeness, which is based on neutrality, and acceptability, which is based on defense.
In light of Fact 6, comparing different neutrality functions is straightforward. Any relaxation on the requirement that no argument in a set be attacked by other arguments in that set leads to weaker forms of conflictfreeness. For any , for and positive integers whenever . So neutrality functions can simply be ordered linearly like natural numbers, with lower numbers denoting ‘stronger’ forms of neutrality and hence conflictfreeness.
The ordering of defense functions is more interesting, as these functions are parameterized by two integers:
Definition 7.
(to be read “is at least as strong as”) iff for any , , with positive integers.
Relation orders the set of all graded defense functions in a wellbehaved manner:
Fact 0.
Let be an , and let be defined as above. Then:

iff and ;

Relation is a partial order, i.e., reflexive, antisymmetric and transitive.
Proof.
(i) is a direct consequence of Fact 6. (ii) follows directly from how relation is defined and the properties of set inclusion. ∎
The relation is depicted in its generality in Figure 5. Expressions may be read as follows: ‘being defended is weakly preferable over being defended’ or ‘the defense function is at least as strong as the defense function’. Intuitively, the partial order uses logical strength as a way to order graded defense functions. This equates with the intuition that if an argument meets a demanding standard of defense it also meets a less demanding one.
Example 8.
3.4.1 On the Partiality of
As the relation is a partial order, some defense functions may be incomparable (see Figure 5) and this, we claim, is intuitive. By way of example, consider Dung’s defense . This standard of defense is strengthened by (higher parameter) and weakened by (higher parameter). But under the definition of , defines a standard of defense which is incomparable with respect to : it demands more defenders per attacker, but tolerates more attackers that are not counterattacked to the desired level. In general, incomparability arises every time the parameters of the functions do not meet the condition and of Fact 7.
It should be clear, however, that the partial order over graded defense functions could be further refined to a total order by resolving incomparability. This can be done in two ways: by either giving priority to parameter or to parameter . For example, if a set of arguments is defended and another one is defended, where and (i.e., they are incomparable w.r.t. ) then the first one can be stipulated to be more strongly defended because it is less tolerant with respect to the failure of defense. Therefore, for and , belonging to is ‘better’ than belonging to . One could then redefine as follows: iff either , or and . This yields a lexicographic order over graded defense functions giving priority to the parameter over the parameter. We do not investigate such refinements further in this paper.
4 Graded Semantics for Abstract Argumentation
By means of the graded defense and neutrality functions, Dung’s notions of acceptability and conflictfreeness can be generalised to graded variants in a natural way. A set of arguments is said to be conflictfree at grade (or, conflictfree) whenever none of its arguments is attacked by at least arguments in . A set of arguments is said to be acceptable at grade (or, acceptable) whenever all of its arguments are such that at most of their attackers are not counterattacked by at least arguments in . A graded notion of admissibility follows (conflictfreeness plus acceptability) and we thereby obtain graded variants of all the main admissibilitybased semantics, which are simply Dung’s standard semantics based on graded admissibility instead of standard admissibility. The first part of this section formally defines graded semantics. The rest of the section then develops a core theory of graded semantics. In the tradition of abstract argumentation, our results focus on the central questions of the existence and construction of graded extensions, and provides positive results under certain constraints on the parameters , and .
4.1 Graded Generalisation of Dung’s Semantics
is conflictfree in  iff  
is selfdefended  iff  
is admissible in  iff  and 
is an complete extension of  iff  and 
is an stable extension of  iff  
is the grounded extension of  iff  is the smallest complete ext. of 
is an preferred extension of  iff  is a largest complete ext. of 
We are now in the position to generalise Definition 4 as follows:
Definition 8 (Graded Extensions).
Let be an , , and , and be positive integers. Graded extensions are defined as in Table 2. We may write , and to denote, respectively, the set of admissible, preferred. and stable extensions of , and to denote the grounded extension of . Finally, for an extension type , we say that is credulously justified w.r.t. S if ; and sceptically justified w.r.t. S if Henceforth we assume the sceptical definition when referring to an argument simply as being justified.
The definition deserves some comment. Note first of all that when , we recover the standard definition of conflictfreeness, admissibility and extensions (Definition 4), which we henceforth refer to as ‘Dung conflictfreeness’ and ‘Dung admissibility’ and ‘Dung extensions’. The key notion is graded admissibility, which is obtained by parameterizing the conflictfreeness requirement by — i.e., —, and parameterizing the selfdefense requirement by and — i.e., . The remaining graded semantics are defined by extending graded admissability in exactly the same way in which Dung admissibility is extended to define the standard Dung semantics. So, a graded complete extension, with parameters and , is a fixpoint of , which is also conflictfree, the graded grounded extension is the smallest complete extension, and the graded preferred extensions are the largest complete extensions. Finally, a graded stable extension, with parameters and , is a fixpoint of and (and therefore of ), which is also conflictfree. Constructive existence results for these semantics are provided in the next section.
Each graded extension type should then be interpreted as a class of weakenings and strengthenings of its standard Dung counterpart. For example: Dung complete extensions are strengthened by complete extensions, with , which require a higher number of defenders for each attacked argument (that is, the requirements for acceptability are strengthened); and are weakened by complete extensions, with , which tolerate a higher level of internal conflict (that is, weakening the conflictfreeness requirement), or by complete extensions, with , which tolerate a higher level of undefended arguments (that is, weakening the acceptability requirement). So for each Dung extension type, we now have an ordered family of extensions incorporating a form of graduality.
4.2 Fixpoint Construction for Graded Exensions
We proceed as in the standard case (cf. Section 2). The basic idea is as follows: given a graded admissible set, we show that, and under what assumptions on the parameters , and , this can be expanded into a graded complete set through a process of fixpoint approximation.
Fix a framework and take a set such that
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