This chapter discussed the Carneades argumentation model and an imple- mentation of it in Haskell. This implementation effort should be seen as a case study and a step towards a generic DSL for argumentation theory, pro- viding a unifying framework in which various argumentation models can be implemented and their relationships studied. We have seen that the original mathematical definitions can be captured at a similar level of abstraction by Haskell code, thereby allowing for greater understanding of the imple-
mentation. At the same time we obtained a domain specific language for the Carneades argumentation framework, allowing argumentation theorists to realise arguments essentially only using a vocabulary with which they are already familiar.
The experience from this work has been largely positive. Comments from Tom Gordon [81], one of the authors and implementers of the Carneades ar- gumentation model, suggests that the implementation is intuitive and would even work as an executable specification, which is an innovative approach in argumentation theory as a field. Furthermore, the literate programming paper on which this chapter is based [72], has been used as an educational tool to teach Bachelor students the Carneades model, implementation of ar- gumentation models and argumentation theory in general49.
The implementation can also be envisioned as being used as a testing framework for computational argumentation theorists and as an intermediate language between implementations, providing a much more formal alterna- tive to the existing Argument Interchange Format [156].
One avenue of future work is the generalisation of the DSL to other related argumentation models. It is relatively common in argumentation theory to define an entirely new model to realise a small extension. However, this hurts the meta-theory as lots of results will have to be re-established from scratch. By reducing such an extension to an existing implementation/DSL such as the previously discussed, for instance by providing an implementation of an existing formal translation such as [77, 145], we effectively formalise a translation between both models, while gaining an implementation of this generalisation at the same time.
This could be taken even further by transferring the functional definitions of an argumentation model into an interactive theorem prover, such as Agda. First of all, the formalisation of the model itself would be more precise. While the Haskell model might seem exact, note that properties such as the acyclicity of arguments, or that premises and exceptions must not overlap, are not inherently part of this model. Second, this would enable formal, machine-checked, reasoning about the model, such as establishing desirable properties like consistency of the set of derivable conclusions. We will see exactly this in Chapter 7.
Then, if multiple argumentation models were to be realised in a theorem prover, relations between those models, such as translations, could be for- malised. As mentioned in the introduction, there has recently been much work on formalisation of translations between conceptually very different ar- 49School of Informatics, University of Edinburgh, AILP 2012–2013, 2013–2014, 2014– 2015: http://www.inf.ed.ac.uk/teaching/courses/ailp/
gumentation models [19, 77, 145, 88]. But such a translation can be very dif- ficult to verify if done by hand. Using a theorem prover, the complex proofs could be machine-checked, guaranteeing that the translations preserve key properties of the models. An argumentation theorist might also make use of this connection by inputting an argumentation case into one model and, through the formal translation, retrieve a specification in another argumen- tation model, allowing the use of established properties (such as rationality postulates [29]) of the latter model.
Finally, we are interested in the possibility of mechanised argumentation as such; e.g., as a component of autonomous agents. We thus intend to look into realising various argumentation models efficiently by considering suitable ways to implement the underlying graph structure and exploiting sharing to avoid unnecessarily duplicated work. Ultimately we hope this would allow us to establish results regarding the asymptotic time and space complexity inherent in various argumentation models, while providing a framework for empirical evaluations and testing problems sets at the same time. Especially the latter is an area that has only recently received attention [24, 19], due to the lack of implementations and automated conversion of problem sets.
Chapter 6
Relating Carneades with
abstract argumentation via the
ASPIC
+
framework for
structured argumentation
Carneades is a recently proposed formalism for structured argumentation with varying proof standards, inspired by legal reasoning but more gener- ally applicable. Its distinctive feature is that each statement can be given its own proof standard, which is claimed to allow a more natural account of reasoning under burden of proof than existing formalisms for structured argumentation, in which proof standards are defined globally. In this chap- ter Carneades and the ASPIC+ framework for structured argumentation are formally related by translating the former into the latter. Since ASPIC+ is
defined to generate Dung-style abstract argumentation frameworks, this in effect translates Carneades graphs into abstract argumentation frameworks. For this translation, a formal correspondence is proven and it certain ratio- nality postulates are shown to hold. It is furthermore proven that Carneades always induces a unique Dung extension, which is the same in all of Dung’s semantics, allowing us to generalise Carneades to cycle-containing structures.
6.1
Introduction
Argumentation involves the construction of arguments in favour of and against statements, selecting the acceptable arguments, and in the end determining which statements hold. How arguments support their conclusion depends on the knowledge they use and the inference rules they apply, so any full
theory of argument evaluation should take the structure and content of ar- guments into account. One way to do so is to define a defeat relation be- tween arguments that takes into account the structure and content of ar- guments and (if available) information on their relative strength. This ap- proach thus results in an abstract argumentation framework in the sense of Dung [48] (see Section 3.2), so that the full theory of abstract argu- mentation can be applied. Two frameworks for structured argumentation that are designed following this approach are assumption-based argumenta- tion [17, 49] and ASPIC+[145] (see Section 3.3). In fact, Prakken [145] shows that assumption-based argumentation can be translated into ASPIC+ as a special case.
However, there have also been advances in structured argumentation that diverge from this approach. A recent application in legal reasoning is the Carneades argumentation system, both a logical model [86, 83] and a soft- ware toolbox for structured argument evaluation, construction and visuali- sation [80] (see Section 3.4). Carneades innovates models of structured argu- mentation by allowing varying proof standards to be assigned to individual propositions. It is claimed that this allows for a more natural account of reasoning under burden of proof than existing formalisms for structured ar- gumentation, in which proof standards are defined globally [7, 13]. This makes the Carneades formalism potentially very attractive, as signified by the large number of citations due to its proof standards.
Recently, Brewka and Gordon [22] translated Carneades into Brewka and Woltran’s [23] abstract dialectical frameworks. Moreover, Brewka and Gor- don [19] have proved a formal correspondence between abstract dialectical frameworks and Dung’s abstract argumentation frameworks. By combining these results, a formal relation between Carneades and Dung’s semantics can be obtained. However, this relation is rather indirect50. In this chapter we therefore take a different approach, by translating Carneades into the ASPIC+framework. Since ASPIC+ is defined to generate abstract argumen-
tation frameworks, which are the input of Dung’s approach, a translation of Carneades into ASPIC+ provides a more direct way to translate Carneades’
graphs into Dung’s frameworks. It will furthermore be proved that Carneades can be modelled cycle-free, thus always inducing a unique Dung extension, which is the same in all of Dung’s semantics. This allows us to generalise Carneades’ argument evaluation structures to cycle-containing structures, addressing an important issue left for future research by Gordon and Wal- 50The translation from Brewka and Gordon [22] is comparable to the translation in this chapter. However, the polynomially sized translation from ADFs to AFs for stable models [19] is achieved through an intermediary representation (boolean networks) and creates various administrative nodes in the AF whose meaning is entire technical.
ton [86]. An additional advantage of translating Carneades to ASPIC+ is
that the results of Prakken [145] on the rationality postulates of Caminada and Amgoud [29] can be shown to hold for the translation.
This chapter is structured as follows. In Section 6.2 a formal relation between Carneades and Dung’s frameworks by developing a translation and proving formal results51. In Section 6.3 related work is treated. Finally, Section 6.4 concludes and discusses future work.