Relating Carneades with abstract argumentation via the ASPIC⁺ framework for structured argumentation

2.1.Abstract argumentation frameworks

In 1995, Dung introduced his seminal theory of argumentation, with which he, due to its abstract nature, was able to model logic programming and several of the contemporary approaches to non-monotonic reasoning. By showing that these forms of reasoning can be represented as a form of argumentation, the relationship between these approaches was clarified.

Dung's abstract argumentation frameworks consist of a set of arguments ordered by a binary relation of defeat.22

Finally, we use Dung's definition of a well-founded argumentation framework. A well-founded AF is an AF without cycles or an infinite defeating chain of arguments. We will later prove the translation of Carneades to be well-founded.

The differences between the semantics collapse in an argumentation framework in which there are no cycles.

2.2.Structured argumentation frameworks

The abstract argumentation frameworks by Dung (1995) keep the structure and nature of arguments and the attack relation unspecified. This allows for general reasoning about the acceptability status of arguments, but provides no guidance for the modelling of actual argumentation problems. Other research has therefore taken a structured approach to argumentation (Amgoud 2005, Gordon et al. 2007).

Structured argumentation frameworks (ASPIC+) as defined by Prakken (2010) are a further development of the ASPIC framework as defined by Amgoud et al. (2006). Prakken's frameworks instantiate33 the abstract argumentation model of Dung, defining the internal structure of arguments, defining multiple types of attack and adding preferences to the attack relation, and resulting in a defeat relation.

The basic building block of a structured argumentation framework is the concept of an argumentation system, extending the standard notion of a proof system. In argumentation systems, the logical language is left unspecified except for the existence of a contrariness relation (generalisation of logical negation to asymmetric conflict). Inference rules are divided into strict and defeasible rules, respectively, of the form φ1,…,φn→φ and φ1,…,φn⇒φ. Strict rules are interpreted as “if the antecedents φ1,…,φn hold, then without exception the consequent ϕ holds”, where for defeasible rules this presumably holds. Finally, the relative strength of defeasible rules can be determined by means of a partial preorder.

Since this definition leaves the nature of the logical language and the inference rules largely unspecified, it is possible to reformulate specific argumentation systems as instances of ASPIC+. For example, Prakken (2010) has shown that assumption-based argumentation (Bondarenko et al. 1997, Dung, Mancarella, and Toni 2007), a structured argumentation approach using assumptions from which conclusions are drawn using strict inference rules, is a special case of ASPIC+, and Modgil and Prakken (2011) have proved the same for variants of classical argumentation (cf. Besnard and Hunter 2008).

With the argumentation system defined, we can now look at the construction of arguments by means of a knowledge base in an argumentation system. The set of rules contains both a strict and defeasible kind and the knowledge base can be inconsistent. In addition to the possible inconsistency, the knowledge base also contains four different types of facts, inspired by a similar distinction of Gordon et al. (2007). Similar to the axioms in deductive logic, there are (unattackable) premises called necessary axioms (𝒦_n), (attackable) ordinary premises (𝒦_p), assumptions (𝒦_a) – which are a weak type of premise always defeated by an attack – and issues (𝒦_i) – which are premises that are not acceptable unless backed by further argument.

With the knowledge base and inference rules defined as above, the construction of arguments can be defined by adopting Vreeswijk's (1993, 1997), definition of an argument. The smallest argument is simply a fact from the knowledge base. More complex arguments can be constructed by chaining inference rules on previous arguments, resulting in an argument in tree form (containing sub-arguments).

Figure 1.

An argument for t using an issue premise.

Figure 2.

Another argument for t.

Now we can define the notion of an argumentation theory.

With the internal structure of arguments defined, it is now possible to distinguish between types of attack.

Undermining attack is an attack on the premises on an argument and is the only attack possible in the context of strict rules. An undercutting attack is an attack on the (defeasible) inference step and is a way to provide “exceptions to the rule”. Finally, a rebutting attack is done by constructing a contrary or contradictory conclusion for the attacked argument's (sub)conclusion.

The previous notions can be combined in an overall definition of defeat.

To deal with issue premises, an argument is acceptable only if it contains no issue premises; therefore, changing Definition 2.2 to the following.

An argument A∈Args is acceptable with respect to a set S of arguments, or alternatively S defends A, iff A contains no issue premises and for all arguments B∈S: if defeats(B,A) holds then there is a C∈S for which defeats(C,B) holds.44

With arguments and the defeat relation fully defined, it is possible to link the argumentation theories of the structured approach to Dung's abstract argumentation frameworks, thereby formally making the correspondence between the structured and abstract approach.

Finally, the acceptability of conclusions (of a mathematical language ℒ) is defined in the corresponding argumentation framework.

Figure 3.

Corresponding argumentation framework.

4.1.Translation of stage-specific Carneades

We will start with relating the premises and exceptions of the arguments in Carneades to a knowledge base in an argumentation system. The assumptions of the audience in a CAES are propositional literals which are unattackable and furthermore, as can be seen in Definition 3.9, part of the logical theory. Combining these characteristics, assumptions in Carneades are closely related to the concept of axioms in a knowledge base and thus will be modelled as necessary axioms, 𝒦_n, in our knowledge base. Next, the use of conclusions as a premise in a later argument is similar to the chaining of sub-arguments to construct more complex arguments and can, therefore, be handled by the argument generation part of ASPIC+. Finally, a premise with no backing, also called an issue premise in Gordon et al. (2007), maps exactly to the issue premises in our knowledge base.

Combining these insights, we can now define the knowledge base corresponding to a CAES.

There is no need to differentiate in the strength of premises, making our preference relation on premises just the reflexive closure on non-axiom premises.

As shown in our visualisation of Carneades’ arguments in Example 3.10, the link between the premises, the exceptions and the conclusion is a two-part inference. The first part – applicability of the argument – is solely determined by the acceptability of the premises and exceptions. The second step – acceptability of the conclusion – requires the argument to be applicable and furthermore to satisfy the demands of the proof standard that is assigned to the conclusion.

So for every argument a=⟨P,E,c⟩ in a CAES, a defeasible rule going from the premises to the applicability of the argument is added, P⇒appaarga, saying that if P then a is applicable.99 The other inference is represented by a defeasible rule arga⇒accac, saying that if a is applicable, its conclusion is acceptable. As before, app_a and acc_a are rule names, which will need to be added to the language, ℒ, of the CAES (rule names are assumed to be disjoint with ℒ).

Exceptions in Carneades’ arguments express exceptions to inferring the conclusion. If we have an argument containing an exception that is acceptable or assumed by the audience, then that argument is made inapplicable, so the argument cannot make the conclusion acceptable. Given an acceptable argument containing exception p¯, it is not implied that p can be assumed to be true; so two arguments with conflicting exceptions can both be acceptable. This use of exceptions, similar to the concept of justifications in default logic (Reiter 1980), implies that negations of exceptions cannot be modelled as an assumption, but instead need to be modelled as an undercutter to the inference rule. So in our translation of argument a, for each exception e∈E, an undercutter e⇒¬appa is added to ℛ_d.

Although it might seem natural to include the negation relation of Carneades into the contrariness relation of the corresponding argumentation system, this does not actually work. With scintilla of evidence as a proof standard that can determine acceptable literals of a CAES, both p and p¯ are allowed to be acceptable, for example, arguments={⟨∅,∅,p⟩,⟨∅,∅,¬p⟩}. It is furthermore possible to construct an acceptable argument for ¬ c while c∈assumptions. 1010 To retain the properties of the original Carneades system, the negation relation of Carneades will, therefore, not be imported into the contrariness relation. Instead the contrariness relation in our argumentation system is used to let applicability conclusions for one argument defeat the acceptability of conflicting arguments, depending on the proof standards of their conclusions. This is essentially where the proof standards are encoded.

To illustrate the translation of one proof standard, notice that in a CAES, an argument a with standard clear-and-convincing evidence is unacceptable if either weight(a)⩽α, or there is a contradictory applicable argument b for which weight(a)⩽weight(b)+β holds. This is then translated by extending the set of contraries for the acceptability, acc_a, with an arg_b for every contradictory argument b for which weight(a)⩽weight(b)+β holds. If the weight of a is less than α, there is also an inference →¬acca added to the strict rules, R_s, that together with the contrary, ¬ acc_a, will undercut the acceptability.

Having built up the corresponding argumentation system, we can now relate an argumentation theory and consequently an argumentation framework to a CAES.

To demonstrate our translation, we will show in detail how the CAES in Example 3.10 can be translated into its corresponding argumentation system, generating the corresponding argumentation framework.

Figure 5.

Structured arguments corresponding to Example 3.10.

4.2.Translation properties

Now that we have defined the argumentation framework corresponding to a CAES, we can look at some interesting properties of the translation.

4.2.4.Ambiguity-blocking and ambiguity-propagating

The stage-specific part of Carneades can be called “ambiguity-blocking” in contrast to “ambiguity-propagating” (see Gordon, Prakken, and Walton 2007, Section 7.1). Here a non-monotonic logic is ambiguity-blocking if, when a conflict between two lines of reasoning with contradictory conclusions cannot be resolved, both lines of reasoning are cut-off and neither of the conclusions can be used for further reasoning. In such logics, it may happen that other lines of reasoning remain undefeated even though one of the cut-off lines of reasoning interferes with it and is not weaker.

Consider the following example, containing an ambiguity between q and ¬ q that does not interfere with the inference of ¬ s even though ¬ q is used as an argument for s.

Example 4.12

Consider the CAES C=⟨arguments,audience,standard⟩ and audience =⟨assumptions,weight⟩ with (Figure 6):

arguments={a1,a2,a3,a4},a1=⟨{p},∅,q⟩,a2=⟨{r},∅,¬q⟩,⟩,a3=⟨{¬q},∅,s⟩,a4=⟨{t},∅,¬s⟩,assumptions={p,r,t},weight(a1)=weight(a2)=weight(a3)=weight(a4)=0.5,standard(q)=standard(¬q)=standard(s)=standard(¬s)=preponderance.

With the proof standard of q, ¬ q, s and ¬ s being preponderance, we can see that q, ¬ q and s will not be acceptable, but ¬ s will be acceptable. Now consider a naive, direct translation of the arguments into defeasible inference rules in ASPIC+, that is, Kn={p,r,t} and Rd={p⇒q,r⇒¬q,¬q⇒s,t⇒¬s}. This translation would instead make no corresponding arguments acceptable. The translation according to Definition 4.4 solves this by using an explicit argument node, yielding undefeated undercutters for the acceptability of q and ¬ q, thereby yielding an undefeated undercutter for the argument for s constructed by using the argument for q, so that ¬ s is acceptable in the corresponding AF.

Figure 6.

Ambiguity-blocking in Carneades.

The main difficulty in finding the translation of Carneades to ASPIC+ was dealing with the ambiguity-blocking nature of Carneades, while ASPIC+ is ambiguity-propagating. We have largely solved this problem by introducing additional argument nodes, allowing for an explicit representation of applicability and acceptability. We note that to our knowledge, we are the first to have achieved a translation of an ambiguity-blocking non-monotonic logic to a standard Dung semantics.

4.3.Generalisation of the translation

Important future work mentioned by Gordon and Walton (2009b) is to generalise Carneades to cycle-containing structures. Although it was claimed by Brewka and Gordon (2010a) that Carneades would need a cyclic representation in other frameworks, such as Dung's argumentation frameworks, our translation of Carneades translates to cycle-free, or well-founded argumentation frameworks. This same well-foundedness allows for an easy extension of Carneades's argument set to a possibly cycle-containing structure.

Since our translation of a CAES to an argumentation framework does not depend on possible cycles in the set of arguments, we can use the same translation for cycle-containing CAES and deal with the resulting cycles by using the standard Dung semantics.

We will demonstrate our generalisation of Carneades by translating Example 2 of Brewka and Gordon (2010a) to an argumentation framework, showing intermediate steps.

Figure 7.

Greece versus Italy argument trees.

Figure 8.

Greece versus Italy argumentation framework.

4.4.Related work

Concurrent to the work done in this article and the paper by van Gijzel and Prakken (2011), there have been translations of Carneades to other argumentation approaches. First of all, there is the translation of Carneades to abstract dialectical frameworks by Brewka and Gordon (2010a). In this translation, premises and exceptions, respectively, have a support and attack relation with the argument node, much in the same way that sub-arguments and undercuts are used in our translation. Carneades’ proof standards are encoded as acceptance conditions from the argument node, supporting the conclusion and attacking the contradictory conclusion.

Although the translation of Brewka and Gordon clarified the relation between Carneades and abstract argumentation by relating it to ADFs, one of the main concerns about this translation was that it needed the full power of abstract dialectical frameworks, thus obscuring the direct relation with Dung's argumentation frameworks. This connection has now been made explicit by the paper of Brewka et al. (2011), developing a translation of ADFs to AFs, using Boolean networks (Dunne 1988). The paper concerns itself mostly with the computational complexity of the translation, to keep a polynomial complexity in both size and time. However, this translation introduces additional technical nodes in the final argumentation framework that have no intuitive meaning. So even though the translation gives a formal connection between the two argumentation models, the intuitive relation is mostly lost.

Very recently, Carneades has been translated to defeasible logic (Nute 1994) by Governatori (2011). Defeasible logic is a computational approach to non-monotonic reasoning with an argumentation-like flavour. Defeasible logic has the possibility to handle both ambiguity-blocking and ambiguity-propagating behaviour, allowing for a rather direct representation of Carneades’ proof standards. The translation by Governatori maps proof standards to a single inference mechanism, giving a natural representation of the proof standards.

While Governatori thus establishes an intuitive relation between Carneades and defeasible logic, he only partly relates Carneades to abstract argumentation, since only the ambiguity-propagating part of defeasible logic has an established direct formal relation with Dung's argumentation frameworks. Its ambiguity-blocking variant has instead been translated to a Dung-like semantics using a different notion of acceptability (Governatori, Maher, Antoniou, and Billington 2000).

5.1.Future work

Through this article and by the work of Prakken (2010), several approaches to structured argumentation have been developed and subsequently related through a single framework called ASPIC+. Although the theoretical relations between these approaches have thus been clarified, we have seen that the actual step of generating arguments has not been given a concrete, efficient implementation. A useful path to take for future research would therefore be to develop a class of efficient argument generation algorithms for ASPIC+. The possibility to efficiently generate arguments for a large class of argumentation approaches would give a better integration of these approaches.

The translation of Carneades to ASPIC+ gives us access to the full power of an ASPIC+ argumentation theory, thereby gaining the possibility to use strict and defeasible rules or use different types of knowledge. This additional power could also be used to show how the concept of argument generators Gordon (2010) and the existing argument schemes of Carneades Gordon and Walton (2009a) relate to structured argumentation by translating them into schemes for defeasible inference rules (following the suggestion of Prakken 2010 that most argument schemes can be seen as such). This would make the relation between ASPIC+ and the Carneades argumentation system as a whole more complete.

Our results raise the question whether it would now be better to use ASPIC+ directly, instead of Carneades, to model argumentation when variable proof standards and the other features of Carneades are required. The answer depends on whether Carneades is sufficient as a model of reasoning with variable proof standards. Prakken and Sartor (2011) claim that Carneades’ ambiguity-blocking nature prevents an adequate modelling of the distinction between the burdens of production and persuasion. If they are right, then there is reason to change Carneades in the direction of ASPIC+.