We motivate and introduce a new method of abduction, Matrix Abduction, and apply it to modelling the use of non-deductive inferences in the Talmud such as Analogy and the rule of Argumentum A Fortiori. Given a matrix $${\mathbb {A}}$$ with entries in {0, 1}, we allow for one or more blank squares in the matrix, say a i,j =?. The method allows us to decide whether to declare a i,j = 0 or a i,j = 1 or a i,j =? (...) undecided. This algorithmic method is then applied to modelling several legal and practical reasoning situations including the Talmudic rule of Kal-Vachomer. We add an Appendix showing that this new rule of Matrix Abduction, arising from the Talmud, can also be applied to the analysis of paradoxes in voting and judgement aggregation. In fact we have here a general method for executing non-deductive inferences. (shrink)

In this paper, we prove the correspondence between complete extensions in abstract argumentation and 3-valued stable models in logic programming. This result is in line with earlier work of [6] that identified the correspondence between the grounded extension in abstract argumentation and the well-founded model in logic programming, as well as between the stable extensions in abstract argumentation and the stable models in logic programming.

This paper provides equational semantics for Dung's argumentation networks. The network nodes get numerical values in [0,1], and are supposed to satisfy certain equations. The solutions to these equations correspond to the ?extensions? of the network. This approach is very general and includes the Caminada labelling as a special case, as well as many other so-called network extensions, support systems, higher level attacks, Boolean networks, dependence on time, and much more. The equational approach has its conceptual roots in the nineteenth (...) century following the algebraic equational approach to logic by George Boole, Louis Couturat, and Ernst Schroeder. (shrink)

This paper is intended first for the formal argumentation community (see https://comma.csc.liv.ac.uk/). This community develops logics and systems modelling argumentation and dialogues. The community is in search of major applications areas for their models. One such application area e.g. is Law. The message of this paper is that there is another major application area for formal argumentation. There is an international community of sex offender therapist that is well established and well funded, and their therapy methods use (methods that can (...) be modelled by) formal argumentation and logic. This community presents a natural application area for formal argumentation. We thus describe in this paper how the sex offender therapists work, to give the formal argumentation researcher a view of this application area. What is especially important about this application area is that in order to model it and learn from it, the formal argumentation community have to evolve their formal methods and adapt to this new application. Part of this enhancement is to modify and import certain methods from other areas of Logic e.g. from Non-Monotonic logic. The members of the formal argumentation community are not familiar, on average, with other areas of logic, and so we also describe in this paper, what we need from neighbouring logics. This makes this paper of interest also to sex offender therapist as well. They may be already familiar with their own practices, but the additional logics described will be of interest to them. (shrink)

In 2005 the author introduced networks which allow attacks on attacks of any level. So if a → b reads a attacks 6, then this attack can itself be attacked by another node c. This attack itself can attack another node d. This situation can be iterated to any level with attacks and nodes attacking other attacks and other nodes. In this paper we provide semantics to such networks. We offer three different approaches to obtaining semantics. 1. The translation approach (...) This uses the methodology of ' Logic by translation'. We translate faithfully the new networks into ordinary Dung networks with more nodes and extract the semantics from the translation. 2. The labelling approach This method regards the arrows as additional entities to be attacked and to mount attacks and applies a variation of the usual machinery of Camindada like labelling to the network. The new concept we need to employ here is that of 'joint attacks'. 3. The logic programming approach We translate the higher level network into a logic program and obtain semantics for it through known semantics for logic programs. We then compare our methods with those of S. Modgil and P. M. Dung et al. (shrink)

This paper is part of a research program centered around argumentation networks and offering several research directions for argumentation networks, with a view of using such networks for integrating logics and network reasoning. In Section 1 we introduce our program manifesto. In Section 2 we motivate and show how to substitute one argumentation network as a node in another argumentation network. Substitution is a purely logical operation and doing it for networks, besides developing their theory further, also helps us see (...) how to bring logic and networks closer together. Section 3 develops the formal properties of the new kind of network and Section 4 offers general discussion and comparison with the literature. (shrink)

Inheritance diagrams are directed acyclic graphs with two types of connections between nodes: x → y and x ↛ y . Given a diagram D, one can ask the formal question of “is there a valid path between node x and node y?” Depending on the existence of a valid path we can answer the question “x is a y” or “x is not a y”. The answer to the above question is determined through a complex inductive algorithm on paths (...) between arbitrary pairs of points in the graph. This paper aims to simplify and interpret such diagrams and their algorithms. We approach the area on two fronts. Suggest reactive arrows to simplify the algorithms for the winning paths. We give a conceptual analysis of inheritance diagrams, and compare our analysis to the “small” and “big sets” of preferential and related reasoning. In our analysis, we consider nodes as information sources and truth values, direct links as information, and valid paths as information channels and comparisons of truth values. This results in an upward chaining, split validity, off-path preclusion inheritance formalism of a particularly simple type. We show that the small and big sets of preferential reasoning have to be relativized if we want them to conform to inheritance theory, resulting in a more cautious approach, perhaps closer to actual human reasoning. We will also interpret inheritance diagrams as theories of prototypical reasoning, based on two distances: set difference, and information difference. We will see that some of the major distinctions between inheritance formalisms are consequences of deeper and more general problems of treating conflicting information. It is easily seen that inheritance diagrams can also be analysed in terms of reactive diagrams - as can all argumentation systems. AMS Classification: 68T27, 68T30. (shrink)

In this paper we show that some versions of Dung’s abstract argumentation frames are equivalent to classical propositional logic. In fact, Dung’s attack relation is none other than the generalised Peirce–Quine dagger connective of classical logic which can generate the other connectives ${\neg, \wedge, \vee, \to}$ of classical logic. After establishing the above correspondence we offer variations of the Dung argumentation frames in parallel to variations of classical logic, such as resource logics, predicate logic, etc., etc., and create resource argumentation (...) frames, predicate argumentation frames, etc., etc. We also offer the notion of logic proof as a geometrical walk along the nodes of a Dung network and thus we are able to offer a geometrical abstraction of the notion of inference based argumentation. Thus our paper is also a contribution to the question: “What is a logical system” in as much as it integrates logic with abstract argumentation networks. (shrink)

Formal models of argumentation have been investigated in several areas, from multi-agent systems and artificial intelligence (AI) to decision making, philosophy and law. In artificial intelligence, logic-based models have been the standard for the representation of argumentative reasoning. More recently, the standard logic-based models have been shown equivalent to standard connectionist models. This has created a new line of research where (i) neural networks can be used as a parallel computational model for argumentation and (ii) neural networks can be used (...) to combine argumentation, quantitative reasoning and statistical learning. At the same time, non-standard logic models of argumentation started to emerge. In this paper, we propose a connectionist cognitive model of argumentation that accounts for both standard and non-standard forms of argumentation. The model is shown to be an adequate framework for dealing with standard and non-standard argumentation, including joint-attacks, argument support, ordered attacks, disjunctive attacks, meta-level attacks, self-defeating attacks, argument accrual and uncertainty. We show that the neural cognitive approach offers an adequate way of modelling all of these different aspects of argumentation. We have applied the framework to the modelling of a public prosecution charging decision as part of a real legal decision making case study containing many of the above aspects of argumentation. The results show that the model can be a useful tool in the analysis of legal decision making, including the analysis of what-if questions and the analysis of alternative conclusions. The approach opens up two new perspectives in the short-term: the use of neural networks for computing prevailing arguments efficiently through the propagation in parallel of neuronal activations, and the use of the same networks to evolve the structure of the argumentation network through learning (e.g. to learn the strength of arguments from data). (shrink)