An abstract framework for argumentation with structured arguments

1.Introduction

In 1995, Phan Minh Dung introduced an abstract formalism for argumentation-based inference (Dung 1995), which assumes as input nothing else but a set (of arguments) ordered by a binary relation (of attack). Although he thus fully abstracted from the structure of arguments and the nature of the attack relation, he was still able to develop an extremely interesting theory. His article was a breakthrough in three ways: it provided a general and intuitive semantics for the consequence notions of argumentation logics (and for non-monotonic logics in general); it made a precise comparison possible between different systems (by translating them into his abstract format) and it made a general study of formal properties of systems possible, which are inherited by instantiations of his framework. In consequence, Dung's work has given an enormous boost to research in computational argumentation. Yet it has also been criticised for not specifying the structure of arguments and the nature of the attack relation, which makes it less suitable for modelling specific argumentation problems. I believe that such criticism fails to appreciate the nature of Dung's formalism. It is best seen not as a formalism for directly representing argumentation-based inference problems but as a tool for analysing particular argumentation systems and for developing a meta-theory of such systems. As such it has been very successful: differences between particular systems can be characterised in terms of some simple notions, and formal results established for the framework are inherited by its instantiations. This was already illustrated by Dung (1995) with reconstructions of Pollock's (1987) system, various logic-programming semantics and Reiter's (1980) default logic in his formalism.

Nevertheless, it is true that when actual argumentation-based inference has to be modelled, Dung's framework is by itself usually too abstract and instead an instantiated version of his approach should be used. However, here too abstraction is still possible and worthwhile. The aim of this paper is to instantiate Dung's abstract approach with a general account of the structure of arguments and the nature of the defeat relation.11 The framework defines arguments as inference trees formed by applying two kinds of inference rules, strict and defeasible rules. This naturally leads to three ways of attacking an argument: attacking a premise, a conclusion and an inference. To resolve such attacks, preferences may be used, which leads to three corresponding kinds of defeat: undermining, rebutting and undercutting defeats. To characterise them, some minimal assumptions on the logical object language must be made, namely that certain well-formed formulas are a contrary or contradictory of certain other well-formed formulas. Apart from this, the framework is still abstract: it applies to any set of inference rules, as long as it is divided into strict and defeasible ones, and to any logical language with a contrary relation defined over it.

The choice for tree-structured arguments based on two types of inference rules arguably is very natural both in light of logic and argumentation theory and when looking at argumentation as it occurs in human thinking and dialogue. The notion of arguments as trees of inferences is very common in standard logic and in argumentation theory and is the basis of many software tools for argument visualisation. Moreover, in actual argumentation, humans often express their arguments as claims supported with one or more premises, which can in turn be supported with further premises, and so on. Finally, as will be further explained in Section 4, the setup with general defeasible inference rules is very suited for modelling reasoning with argumentation schemes (Walton, Reed, and Macagno 2008).

The account offered in this paper is not completely new. In fact, a rhetorical aim of the paper is to counter the idea that the computational study of argumentation started with Dung's abstract approach and that only then researchers made it more concrete with accounts of the structure of arguments and the nature of defeat. As a matter of fact, much work on these two issues was already done or going on at the time when Dung wrote his paper, and some of this work is still state-of-the-art. For instance, both Pollock (1987, 1994) and Vreeswijk (1993, 1997) did important work on the structure of arguments, while Pollock (1974, 1987) introduced an important distinction between two kinds of defeat, namely rebutting defeat (attack on a conclusion) and undercutting defeat (attack on an inference rule). One aim of the present paper is to profit from, integrate and build on this and other important work as much as possible. As such, this paper is a further development of the integration attempt that was undertaken in the European ASPIC project (Amgoud et al. 2006). In this project, Vreeswijk's formalisation of the structure of arguments was combined with Pollock's definitions of rebutting and undercutting defeat in a way that also used insights from other work. The result was a characterisation of a set of tree-structured arguments ordered with a binary defeat relation, so that an instantiation of Dung's abstract approach was achieved and any of Dung's semantics could be used to compute the acceptability status of the structured arguments.

The ASPIC framework was developed by Leila Amgoud, Martin Caminada, Claudette Cayrol, Marie-Christine Lagasquie-Schieux, myself and Gerard Vreeswijk and was first reported in a European project deliverable (Amgoud et al. 2006). The added expressiveness compared with Dung's abstract formalism gave rise to further work by Caminada and Amgoud (2007) on rationality postulates for systems instantiating the ASPIC framework. The aim of this work was to propose the idea of rationality postulates and to criticise some specific rule-based argumentation systems for failing to satisfy them. For this aim, only a simplified version of the ASPIC framework was needed, without preferences and without the notion of a knowledge base. Moreover, the examples discussed by Caminada and Amgoud (2007) were all with domain-specific inference rules instead of with general inference patterns, which in effect somewhat obscured the potential of the framework to be a general account of structured argumentation.

In contrast, the present paper aims to present the ASPIC framework as a general abstract model of argumentation with structured arguments.22 To achieve this aim, the ASPIC framework will be extended and generalised in four respects.

2.Dung's abstract argumentation frameworks

First without explanation, the basic concepts and insights of Dung's abstract argumentation approach are listed. For a state-of-the-art introduction, see Baroni and Giacomin (2009).

3.Argumentation systems with structured arguments

In this section, the arguments of Dung's argumentation frameworks are given structure and its defeat relation is defined in terms of the structure of arguments plus external preference information. Apart from this, the resulting formalism is still as abstract as possible, allowing for different logical languages, different sets of inference rules for building arguments and different preference orderings. The framework uses Vreeswijk's (1993, 1997) definition of the structure of arguments and then adds Pollock's (1987, 1994) distinction between rebutting and undercutting attack, as well as a variant of the notion of premise attack proposed by Vreeswijk (1993, chap. 8). These notions are then generalised to languages with arbitrary relations of contrariness and contradiction between well-formed formulas. Then the three notions of attack are combined into a notion of defeat in a way inspired by Vreeswijk (1993, chap. 8) and Prakken and Sartor (1997). It is this combination that makes it possible to regard the system as an instantiation of Dung's abstract framework.

The resulting framework unifies two ways to capture the defeasibility of reasoning. Some, e.g. Amgoud and Cayrol (2002), Besnard and Hunter (2008), Bondarenko et al. (1997), Verheij (2003a), locate the defeasibility of arguments in the uncertainty of their premises, so that arguments can only be attacked on their premises. Others, e.g. Pollock (1994), Vreeswijk (1997), instead locate the defeasibility of arguments in the riskiness of their inference rules: in these logics, inference rules are of two kinds, being either deductive or defeasible, and arguments can only be attacked on their applications of defeasible inference rules. Typically, in this approach inconsistency of the knowledge base makes the system collapse. Vreeswijk (1993, chap. 8) called these two approaches plausible and defeasible reasoning: he described plausible reasoning as sound (i.e. deductive) reasoning on an uncertain basis and defeasible reasoning as unsound (but still rational) reasoning on a solid basis. In Chapter 8, Vreeswijk attempted to combine both forms of reasoning in a single formalism, but since then most formal accounts of argumentation have modelled either only plausible or only defeasible reasoning.

3.2.Arguments

Next the arguments that can be constructed from a knowledge base in an argumentation system are defined. Arguments can be constructed step-by-step by chaining inference rules into trees. Arguments thus contain subarguments, which are the structures that support intermediate conclusions (plus the argument itself and its premises as limiting cases). In what follows, for a given argument, the function Prem returns all the formulas of K (called premises) used to build the argument, Conc returns its conclusion, Sub returns all its subarguments, DefRules returns all the defeasible rules of the argument and, finally, TopRule returns the last inference rule used in the argument.

Definition 3.6

Definition 3.6argument

An argument A on the basis of a knowledge base (K,⩽′) in an argumentation system (L,0¯,R,⩽) is

• (1) ϕ if φ∈K with

  Prem(A)={φ},

  Conc(A)=φ,

  Sub(A)={φ},

  DefRules(A)=∅,

  TopRule(A)=undefined.

• (2) A1,…,An→ψifA1,…,An are arguments such that there exists a strict rule

  Conc(A1),…,Conc(An)→ψ in Rs,

  Prem(A)=Prem(A1)∪⋯∪Prem(An),

  Conc(A)=ψ,

  Sub(A)=Sub(A1)∪⋯∪Sub(An)∪{A},

  DefRules(A)=DefRules(A1)∪⋯∪DefRules(An),

  TopRule(A)=Conc(A1),…,Conc(An)→ψ.

• (3) A1,…,An⇒ψ if A1,…,An are arguments such that there exists a defeasible rule Conc(A1),…,Conc(An)⇒ψ in Rd,

  Prem(A)=Prem(A1)∪⋯∪Prem(An),

  Conc(A)=ψ,

  Sub(A)=Sub(A1)∪⋯∪Sub(An)∪{A},

  DefRules(A)=DefRules(A1)∪⋯∪DefRules(An)∪{Conc(A1),…,Conc(An)⇒ψ},

  TopRule(A)=Conc(A1),…,Conc(An)⇒ψ.

Example 3.7

Consider a knowledge base in an argumentation system with

• Rs={p,q→s;u,v→w}

• Rd={p⇒t;s,r,t⇒v}

• Kn={q}

• Kp={p,u}

• Ka={r}

An argument for w is displayed in a traditional proof-tree format in Figure 1, where a single line stands for a strict inference and a double line for a defeasible inference. The type of a premise is indicated with a superscript. Formally, the argument and its subarguments are written as follows:

• A1:pA5:A1⇒t

• A2:qA6:A1,A2→s

• A3:rA7:A5,A3,A6⇒v

• A4:uA8:A7,A4→w

We have that

• Prem(A8)={p,q,r,u}

• Conc(A8)=w

• Sub(A8)={A1,A2,A3,A4,A5,A6,A7,A8}

• DefRules(A8)={p⇒t;s,r,t⇒v}

• TopRule(A8)=v,u→w

Figure 1.

An argument.

Definition 3.8

Definition 3.8argument properties

An argument A is

• • strict if DefRules(A)=∅;

• • defeasible if DefRules(A)≠∅;

• • firm if Prem(A)⊆Kn;

• • plausible if Prem(A)⊈Kn.

We write S⊢φ if there exists a strict argument for ϕ with all premises taken from S, and S|∼φ if there exists a defeasible argument for ϕ with all premises taken from S.

Example 3.9

In Example 3.7 the argument A2 is strict and firm, while A1,A3,A4 and A6 are strict and plausible and A5, A7 and A8 are defeasible and plausible. Furthermore, we have that K⊢p, K⊢q, K⊢r, K⊢u, K⊢s and K|∼t, K|∼v, K|∼w.

(From hereon, the theory will be left implicit if there is no danger for confusion.)

Now that the notion of an argument has been defined, orderings on arguments can be considered. Below ⪯ is a partial preorder such that A⪯B means that B is at least as ‘good’ as A. As usual A≺B means A⪯B and B⋠A.

In Section 6, two ways will be discussed to define ⪯ as a function from the orderings ⩽ on Rd and ⩽′ on K. However, the present framework allows for any partial preorder on arguments that satisfies two basic assumptions (taken from Vreeswijk (1993)).

Definition 3.10

Definition 3.10admissible argument orderings

Let A be a set of arguments. Then a partial preorder ⪯ on A is an argument ordering iff

• (1) if A is firm and strict and B is defeasible or plausible, then B≺A;

• (2) if A=A1,…,An→ψ, then for all 1⩽i⩽n,A⪯Ai and for some 1⩽i⩽n, Ai⪯A.

(Vreeswijk also assumes that an argument cannot be stronger than its weakest subargument but in Section 6 the so-called ‘last-link’ principle will be discussed, which violates this assumption.) The first condition says that strict-and-firm arguments are stronger than all other arguments, while the second condition says that a strict inference cannot make an argument weaker or stronger.

Definition 3.11

Definition 3.11argumentation theories

An argumentation theory is a triple AT=(AS,{KB,⪯), where AS is an argumentation system, KB a knowledge base in AS and ⪯ an argument ordering on the set of all arguments that can be constructed from KB in AS (below called the set of arguments on the basis of AT).

4.Using the framework: domain-specific vs. general inference rules

The framework defined in the previous section can be used in two ways, depending on whether the inference rules are domain-specific or not. The inference rules of argumentation systems are not part of the logical language L but are metalevel constructs. The usual practice in standard logic is that inference rules express general patterns of reasoning, such as modus ponens, universal instantiation and so on. Yet Caminada and Amgoud (2007) use the inference rules to represent domain knowledge, in line with a long tradition in non-monotonic logic of using domain-specific inference rules (e.g. Reiter 1980; Loui 1987; Nute 1994; Garcia and Simari 2004). The difference between both approaches is illustrated with the following example. Consider the information that all Frisians are Dutch, that the Dutch are usually tall and that Wiebe is Frisian. With domain-specific inference rules, this can in a propositional language be represented as follows:

The argument that Wiebe is tall then has the form as displayed on the left in Figure 2.

Figure 2.

Domain-specific vs. general inference rules.

With general inference rules, the two rules must instead be represented in the object language L. The first one can be represented with the material implication but for the second one a connective for defeasible conditionals must be added to L and a defeasible modus-ponens inference rule must be added for this connective. For example:

Then the argument that Wiebe is tall has the form as displayed on the right in Figure 2.

Although the present system can be used both ways, both Vreeswijk and Pollock intended their inference rules to express general patterns of reasoning, which is much more in line with the role of inference rules in standard logic. Indeed, an important part of John Pollock's work was the study of general patterns of (epistemic) defeasible reasoning, which he called prima facie reasons. He formalised prima facie reasons for reasoning patterns involving perception, memory, induction, temporal persistence and the statistical syllogism, as well as undercutters for these reasons. The ASPIC framework allows for such general use of inference rules, by expressing the rules through schemes (in the logical sense, with metavariables ranging over L). When used thus, the framework becomes a general framework for argumentation with structured arguments. It thus is also suitable for modelling reasoning with argument schemes, which currently is an important topic in the computational study of argument (cf. Walton et al. 2008). Argument schemes are stereotypical non-deductive patterns of reasoning, consisting of a set of premises and a conclusion that is presumed to follow from them. Uses of argument schemes are evaluated in terms of critical questions specific to the scheme. An example of an epistemic argument scheme is the scheme from expert opinion (Walton et al. 2008, p. 310):

This scheme has six critical questions:

A natural way to formalise reasoning with argument schemes is to regard them as defeasible inference rules and to regard critical questions as pointers to counterarguments (this approach was earlier defended by Bex, Prakken, Reed, and Walton (2003) and Verheij (2003b). More precisely, the three kinds of attack on arguments correspond to three kinds of critical questions of argument schemes. Some critical questions challenge an argument's premise and therefore point to undermining attacks, others point to undercutting attacks, while again other questions point to rebutting attacks. In the scheme from expert opinion questions (2) and (3) point to underminers (of, respectively, the first and second premise), questions (4), (1) and (6) point to undercutters (the exceptions that the expert is biased or incredible for other reasons and that he makes scientifically unfounded statements) while question (5) points to rebutting applications of the expert opinion scheme. Thus, we also see that Pollock's prima facie reasons are examples of epistemic argument schemes and that his undercutters are negative answers to one kind of critical question.

Now one benefit of having undermining attack in addition to rebutting and undercutting attack can be discussed in more detail: if the inference rules are supposed to be domain-independent, then representing facts with non-conditional inference rules (as done by Caminada and Amgoud (2007)) does not make sense.

5.Transposition and contraposition

Before it can be studied to what extent the present framework satisfies the rationality postulates of Caminada and Amgoud (2007), first some technicalities concerning strict inference rules must be discussed. To start with, Caminada and Amgoud define the notions of a transposition of a strict rule and closure of sets of strict rules under transposition.

If P=ClRs(P), then P is said to be closed.

It is also relevant whether strict inference satisfies contraposition.

In general, it neither holds that closure under contraposition implies closure under transposition, as shown by the following counterexample.

So Rs satisfies contraposition.

However, contraposition does imply transposition in the following special case.

Note that the proposition does not hold if the condition ‘ Rs consists of all valid propositional inferences’ is changed to ‘ ⊢ corresponds to propositional logic’. A counterexample is any argumentation theory with a sound and complete axiomatisation of propositional logic with modus ponens as the only inference rule.

6.Rationality postulates

Dung's semantics can be seen as rationality constraints on evaluating arguments in abstract argumentation frameworks. The refinement of his abstract approach with structured arguments naturally leads to the question whether this additional structure gives rise to additional rationality constraints. Caminada and Amgoud (2007) gave a positive answer to this question by proposing a number of ‘rationality postulates’ for what they called ‘rule-based argumentation’. Four of their postulates formulate constraints on any extension of an argumentation framework corresponding to an argumentation theory:55

The postulates of closure under subarguments and strict-rule application still hold unconditionally for the present framework. (Here that a given semantics is subsumed by complete semantics means that any of its extensions also is a complete extension).

As for the two consistency postulates, Caminada and Amgoud's results do not generalise unconditionally. Consider the following example.

However, this line of reasoning does not hold without a further assumption on the argument ordering. Consider a more complex variant of Example 6.3.

However, assuming contraposition or transposition, direct consistency can still be proved if it can also be assumed that there is a way to extend A with all but one of B's maximal defeasible subarguments that is not weaker than the remaining one. In our example, this means that either A extended with B′ is not weaker than B′′ or A extended with B′′ is not weaker than B′. Intuitively, this assumption seems acceptable given that A is stronger than both B′ and B′′. It is therefore to be expected that it will be satisfied by many reasonable argument orderings. Since similar situations can arise with undermining attack, the notion of a maximal fallible subargument is needed.

The set of maximal fallible subarguments of an argument A will be denoted by M(A).

Condition (2) in effect says that assumptions can only be contraries of other assumptions. An example of an argumentation theory that is not well formed is

Rs={p→q},Rd={r⇒s,t⇒u},Kp={p,r},Ka={v}

and such that s is a contrary of q and v is a contrary of u. Then condition (1) of Definition 6.8 is violated since we have arguments A: p→q and B: r⇒s. Moreover, condition (2) is violated since v∈Ka and t⇒u∈Rd.

Now it can be proved that under certain conditions an argumentation theory satisfies the postulate of direct consistency.

Caminada and Amgoud (2007) also prove that their system satisfies the postulate of indirect consistency. This follows from their Proposition 7, which says that if an argumentation theory satisfies closure and direct consistency, it also satisfies indirect consistency. Since in the present case, the conditions of the proof of direct consistency had to be strengthened, the same holds for indirect consistency.

Concluding this section, two intuitively plausible argument orderings will be shown to be reasonable, namely, the weakest-link and last-link orderings from Amgoud et al. (2006). The versions below are slightly revised to make the principles arguably more intuitive. Both orderings define a strict partial order ≺s on sets in terms of a partial preorder ⩽e on their elements, as follows: S1≺sS2 iff there exists an e1∈S1 such that for all e2∈S2 it holds that e1<ee2.

The last-link principle prefers an argument A over another argument B if the last defeasible rules used in B are less preferred than the last defeasible rules in A or, in case both arguments are strict, if the premises of B are less preferred than the premises of A. The concept of ‘last defeasible rules’ is defined as follows and is essentially the same as Prakken and Sartor's (1997) notion of a ‘relevant set’.

An example with more than one last defeasible rule is with K={p;q} and Rd={p⇒r;q⇒s}. Then for argument A for r∧s, we have LastDefRules(A)={p⇒r;q⇒s}.

The above definition is now used to compare pairs of arguments as follows.

(Amgoud et al. 2006 do not include the second condition so if both arguments are strict the ordering on the knowledge base is ignored.) This definition in effect compares sets on their weakest elements.

The weakest-link principle considers not the last but all uncertain elements in an argument. It prefers an argument A over an argument B if A is preferred to B on both their premises and their defeasible rules.

(Amgoud et al. (2006) do not have condition (2), so that with two strict arguments neither of them can be preferred.)

7.Self-defeat

As discussed by Pollock (1994) and Caminada and Amgoud (2007), self-defeating arguments can cause problems if argumentation systems are not carefully defined, particularly if they include standard propositional logic. In the present framework, two types of self-defeating arguments are possible: serial self-defeat occurs when an argument defeats one if its earlier steps, while parallel self-defeat occurs when the contradictory conclusions of two or more arguments are taken as the premises for ⊥. Pollock (1994) gives an example of serial self-defeat of the following form.

8.The relation with assumption-based argumentation

After having presented his fully abstract approach to argumentation, Dung joined Kowalski, Toni and others in their development of a more concrete version of his approach (e.g. Bondarenko et al. 1997; Dung et al. 2006, 2007). In this approach, arguments essentially are sets of formulas called ‘assumptions’, from which conclusions can be drawn with strict inference rules. Arguments can be attacked with arguments that conclude to the ‘contrary’ of one of their assumptions. In fact, the extensions defined by the various semantics of Bondarenko et al. (1997) are not sets of arguments but sets of assumptions. However, Dung et al. (2007) showed that an equivalent fully argument-based formulation can be given.

In this section, it will be shown that assumption-based argumentation is a special case of the present framework with only strict inference rules, only assumption-type premises and no preferences. The proof will be given for the argument-based version of Dung et al. (2007) and carries over to Bondarenko et al. (1997) by the equivalence result of Dung et al. (2007).

First the main definitions of ABA are recalled (in the formulation of Dung et al. (2007)).

The third condition amounts to a restriction to so-called flat ABFs. This restriction is not entirely innocent, since in debates it may occur that someone first assumes a premise and, after it is defeated, constructs an argument for it, in an attempt to rebut the defeater. To make Dung et al.'s analysis apply to all stages of such a debate, assumptions should be deleted from 𝒜 as soon as they are supported with an argument.

Since the notion of an argument is central to the present concerns, the informal explanation of Dung et al. (2007, p. 646) will be quoted in (almost) full.

Now an assumption-based argument is defined as follows:

As for notation, the existence of an argument for a conclusion α supported by a set of assumptions A is denoted by A⊢α, or by A⊢ABFα if it has to be distinguished from the existence of a strict argument according to Definition 3.6 with the same premises and conclusion; the latter will below be denoted by A⊢ATα.

Finally, Dung et al.'s notion of argument attack is defined as follows.

The argumentation theory corresponding to an assumption-based framework is now defined as follows.

Note that ATABF is well formed and all ATABF arguments are strict and plausible.

The main task now is to prove that there is an ABF-argument for α from P if and only if there is an ATABF-argument for α with premises P. In fact, this can only be proved for the special case of argumentation theories that do not allow for arguments with an infinite number of subarguments. Technically, the present framework allows for such arguments even if they are non-circular. For example, an AT with Rs={pi+1→pi|i⩾1} allows for an argument for p1 with an infinite number of subarguments (and an empty set of premises). So far no proof has depended on finiteness of arguments. In an ABF, however, arguments are by definition finite even if the set of inference rules allows for infinite ones, as in the just-given example.

From this it follows that

Now the main correspondence result can be proved.

9.Other related research

As was said above, the present framework is inspired by the work of Pollock (1987, 1994) and Vreeswijk (1993, 1997). Essentially, it takes from both the idea that defeasible reasoning proceeds by chaining two kinds of inference rules into inference trees. The present mathematical formulation of this idea is directly adopted from Vreeswijk (1993, 1997). The present notions of undercutting and rebutting defeat are taken from Pollock's work and then generalised to arbitrary preference relations on arguments (Pollock only has a notion of probabilistic strength), and to logical languages with arbitrary contrary mappings. They are then combined with a notion of undermining defeat.

In fact, the system of Pollock (1994) is not formalised in terms of arguments but in terms of the so-called ‘inference graphs’, in which nodes are connected either by inference links (applications of inference rules) or by defeat links. The nodes are ‘lines of argument’, which are propositions plus an encoding of the argument lines from which they are derived. So if a proposition is derived in more than one way, it occurs in more than one line of argument. Such duplications cannot be avoided, since defeat relations depend on the strength of a proposition, which in turn depends on the way in which it is derived. Nodes are evaluated in terms of the recursive structure of the graph. Jakobovits and Vermeir (1999) proved that Pollock's system can be given an equivalent formulation as an instance of Dung's abstract argumentation frameworks with preferred semantics.

With Vreeswijk's framework, the relation with Dung-style semantics is still an open issue, since it models conflict not as a relation between two individual arguments but as a property of sets of arguments: a set of arguments is said to be in conflict if there exists a strict argument from their conclusions for ⊥. Vreeswijk then defines a notion of warrant for arguments which resembles stable semantics.

Gordon et al. (2007) proposed the Carneades framework ‘of argument and burden of proof’. Carneades' main structure is that of an argument graph, which, despite its name, is similar to Pollock's inference graphs. Statement nodes are linked to each other via argument nodes, which record the inferences from one or more nodes to another. This notion of an argument does not have the recursive structure of Definition 3.6 but instead stands for a single inference step. As explained in Section 3.1, the premises of an argument can be of three types: presumptions (similar to the present issues), assumptions (similar to the present ordinary premises) and exceptions (similar to contradictories of the present assumptions). Carneades has no distinction between strict and defeasible inference rules and, unlike Pollock, does not express conflicts as a special type of link between statement nodes. Instead, inferences (i.e. arguments) can be either pro or con a statement. Because of this, statements occur only once in the graph. Also, attack relations are thus expressed either as arguments pro and con the same statement or as an argument pro an exception-type premise of another argument. Carneades thus allows for rebutting and undermining but not for undercutting; instead, undercutters are simulated by arguments pro exceptions. Carneades' inference graphs are assumed to contain no cycles, which excludes the representation of mutual attack relations through exceptions.

In Carneades, the evaluation of statements in an argument graph is, as with Pollock's inference graphs, defined in terms of the recursive structure of the graph. Statements are acceptable if they satisfy their ‘proof standard’. The general framework abstracts from their nature but Gordon et al. (2007) give several examples of proof standards. The proof standards are at the heart of Carneades' acceptability notion, just like the notions of defence and admissibility are at the heart of Dung-style semantics. None of the examples given by Gordon et al. (2007) have a known relation with any existing Dung-style semantics or the present framework, which thus is an issue for future research. Here it is also relevant that Carneades incorporates dialogical elements since it matters whether a statement is ‘stated’, ‘questioned’, ‘accepted’ or ‘rejected’. These statuses of a statement are assumed to be provided by a dialogical context in which Carneades is embedded.

Verheij (2003a) presents a ‘sentence-based’ (as opposed to ‘argument-based’) logic for defeasible reasoning, called DefLog. Verheij assumes a logical language with just two connectives, a unary connective × which informally stands for ‘it is defeated that’ and a binary connective ↝ for expressing defeasible conditionals. He then assumes a single inference scheme for this language, namely, modus ponens for ↝. A set of sentences T is said to support a sentence ϕ if ‘ ϕ is in T or follows from T by repeated application of ↝ -modus ponens’ (Verheij 2003a, p. 327). It seems reasonable to formalise this as the backward deductions of assumption-based argumentation or the strict arguments of the present framework. Moreover, T is said to attack ϕ if T supports ×φ. Verheij then considers partitions (J,D) of sets of sentences Δ which he calls dialectical interpretations and which are such that J (the ‘justified’ sentences) is conflict-free and attacks every sentence in D (the ‘defeated’ sentences).

As already suggested by Verheij, there is a close formal relation between DefLog and assumption-based argumentation. First, dialectical interpretations are easily proved to be equivalent to stable labellings, which are known to be equivalent to stable semantics (first proved by Verheij (1996); see also Jakobovits and Vermeir (1999), and Caminada (2006)). Furthermore, DefLog theories can be mapped onto assumption-based frameworks by letting an ABF contrary mapping be ×φ=φ‾ for any ϕ, by regarding any set of dialectically interpreted sentences as the assumptions A of an ABF and by having ϕ, ϕ ↝ ψ → ψ, for any ϕ and ψ in DefLog's language, as the set R of inference rules of the ABF. The result is an assumption-based framework in the sense of Definition 8.2 with stable semantics. The correspondence results of Dung et al. (2007) with Bondarenko et al. (1997) then also apply to the special case of a DefLog-style ABF so that by the above Theorem 8.9 DefLog is a special case of the present framework with only strict arguments and only undermining defeat.

Several argumentation systems model deductive argumentation. Here arguments are proofs according to some deductive logic with consistent premises taken from a possibly inconsistent knowledge base expressed in the language of the logic (usually taken to be standard propositional or first-order logic). In Amgoud and Cayrol (2002), which is based on propositional logic, the structure of arguments is left undefined, except that the premises imply the conclusion according to propositional logic. Several notions of defeat are then considered. One of them corresponds to the present undermining defeat, where arguments are compared in terms of a partial preorder on the belief base from which their premises are taken. Argument acceptability is defined according to grounded semantics.

This variant of Amgoud and Cayrol (2002) can be reconstructed as a special case of the present framework as follows. First, ℒ is any propositional language closed under classical negation, where φ=ψ‾ if φ=¬ψ or ψ=¬φ. Then Rs consists of all valid propositional inferences while Rd is empty. The knowledge base equals Kp. Finally, as with Deflog, it seems reasonable to formalise arguments as the strict arguments of the present framework, although the extra constraint must be added that such arguments have classically consistent premises. This consistency constraint makes that not all results of this paper hold without further qualification. It is easy to verify that Propositions 5.8, 6.1 and 6.2 still hold with this constraint (for Proposition 5.8 note that in this case S⊢φ by definition implies that the strict argument that exists for ϕ has consistent premises). However, the proofs of Theorems 6.9 and 6.10 do not apply to this case, for similar reasons as explained above in Section 7 with Example 7.4. It remains to be investigated whether these theorems can be proved for this case under alternative conditions.

Besnard and Hunter's (2008) version of deductive argumentation is similar to that of Amgoud and Cayrol (2002), except for a generalised notion of undermining: an argument is undermined by any argument of which the conclusion negates the conjunction if its premises. It remains to be seen whether this version of undermining can be reduced to the present version.

Two other logics for defeasible reasoning with both (domain-specific) strict and defeasible inference rules are Defeasible Logic (DL), first proposed by Nute (1994), and Defeasible Logic Programming (DeLP; e.g. Garcia and Simari 2004). In both systems, the logical language is restricted in logic-programming style. DL is not explicitly argument-based but defines the notion of a proof tree, which interleaves support and attack. Governatori, Maher, Antoniou, and Billington (2004) investigated the relation with Dung-style semantics. One variant of DL is proved to instantiate grounded semantics. In DeLP, the only way to attack an argument is on a (sub-)conclusion. DeLP's notion of argument acceptability has no known relation to any of the current argumentation semantics.

Prakken and Sartor (1997) presented an argument-based version of extended logic programming, designed as an instance of Dung's abstract argumentation frameworks with grounded semantics. Their system comes close to being a special case of the present framework. It has (domain-specific) strict and defeasible inference rules and allows for rebutting and undercutting defeat. Furthermore, its notion of an argument comes close to a ‘deduction’ version of Definition 3.6, i.e. it represents a particular order in which an argument can be constructed. A difference is that in Prakken and Sartor (1997) two parallel subarguments do not need to be completed with an inference from their conclusion, so that, for example (in the present notation), p,p⇒q,r,r⇒s is an argument with conclusions q and s. In Prakken and Sartor (1997), this was convenient for modelling reasoning about defeasible priorities in the system. A more substantial difference is that while the present framework considers rebutting and undercutting attack on equal footing, Prakken and Sartor (1997) give priority to undercutting attack, so that if A undercuts B while B rebuts A, A strictly defeats B. It seems that the present results do not crucially rely on this difference, but this should be further investigated.

A final difference with the present framework is that in Prakken and Sartor (1997) the role of strict rules in defeat is different. As in the present framework, only defeasible inferences can be attacked, but an argument A with conclusion ϕ rebuts an argument B with conclusion φ′ if there exists sets of strict rules Sa and Sb and a formula ψ such that (with present notation) Sa∪{φ}⊢ψ and Sb∪{φ′}⊢ψ‾. The difference can be best explained with Examples 3.18 and 6.20. The motivation behind the definition of Prakken and Sartor (1997) was that intuitively the ‘real’ conflict is between the two defaults on whether someone is a bachelor or married. This is captured by their definition of rebutting attack, since A2 can be extended with A3 to contradict B2's conclusion and vice versa. Hence the rule priorities are applied to A2 and B2. By contrast, in the present framework these arguments do not rebut each other since their top rules are strict. Instead, we saw that their conflict is decided indirectly, by comparing A3 with B2 and B3 with A2. The present treatment of such examples can be defended by saying that conflicts are recognised only when they are made explicit in an argument's conclusion, which seems to better respect the general nature of argumentation as providing explicit grounds for conclusions. It remains to be investigated whether this difference affects the present results on the rationality postulates (note that, although Prakken and Sartor (1997) do not assume that the strict rules are closed under transposition, this assumption can be easily added).

In one respect, Prakken and Sartor (1997) go beyond the present framework, namely, in making the preference relation on the set of defeasible inference rules defeasible and derivable within the framework. In this respect, the system is a forerunner of Modgil's (2009) extended AFs.

10.Conclusion

The main rhetorical aim of this paper has been to present the ASPIC framework as a general abstract framework for rule-based argumentation. In previous publications on the ASPIC framework its unifying potential was underexposed because of a focus on domain-specific inference rules instead of on general inference patterns. Here it has been argued that ASPIC, although it can be used as a specific logic at the same level of abstraction as systems such as DeLP, DL and Prakken and Sartor (1997), can also be used as an abstract framework for reasoning with general inference rules, including argument schemes. Moreover, it has been shown that by including undermining attack and generalising negation to arbitrary contrary mappings, the ASPIC framework unifies rule- and assumption-based approaches to argumentation. The latter claim has been backed by a formal proof that assumption-based argumentation (Bondarenko et al. 1997; Dung et al. 2007) is a special case of the framework and by semi-formal explanations that the same holds for Verheij's (2003) DefLog and (to a large extent) Amgoud and Cayrol's (2002) version of deductive argumentation.

In addition, the following technical contributions have been made:

Finally, as touched upon at the end of Section 9, an important extension of the present framework is making the preference relations that are used for resolving conflicts defeasible and derivable within the framework. This could be done along the lines of Prakken and Sartor (1997), after which it should be investigated whether Modgil's (2009) reconstruction of Prakken and Sartor (1997) as an instance of his extended argumentation frameworks can be adapted to the extended ASPIC framework.