Valued preference-based instantiation of argumentation frameworks with varied strength defeats

A Dung-style argumentation framework aims at representing conflicts among elements called arguments. The basic ingredients of this framework is a set of arguments and a Boolean abstract (i.e., its origin is not known) binary defeat relation. Preference-based argumentation frameworks are instantiations of Dung's framework in which the defeat relation is derived from an attack relation and a preference relation over the arguments. Recently, Dung's framework has been extended in order to consider the strength of the defeat relation, i.e., to quantify the degree to which an argument defeats another argument. In this paper, we instantiate this extended framework by a preference-based argumentation framework with a valued preference relation. As particular cases, the latter can be derived from a weight function over the arguments or a Boolean preference relation. We show under some reasonable conditions that there are “less situations” in which a defense between arguments holds with a valued preference relation compared to a Boolean preference relation. Finally, we provide some conditions that the valued preference relation shall satisfy when it is derived from a weight function.     

Theory of Semi-Instantiation in Abstract Argumentation

We study instantiated abstract argumentation frames of the form (S, R, I), where (S, R) is an abstract argumentation frame and where the arguments x of S are instantiated by I(x) as well formed formulas of a well known logic, for example as Boolean formulas or as predicate logic formulas or as modal logic formulas. We use the method of conceptual analysis to derive the properties of our proposed system. We seek to define the notion of complete extensions for such systems and provide algorithms for finding such extensions. We further develop a theory of instantiation in the abstract, using the framework of Boolean attack formations and of conjunctive and disjunctive attacks. We discuss applications and compare critically with the existing related literature.     

A probabilistic approach to modelling uncertain logical arguments

Argumentation can be modelled at an abstract level using a directed graph where each node denotes an argument and each arc denotes an attack by one argument on another. Since arguments are often uncertain, it can be useful to quantify the uncertainty associated with each argument. Recently, there have been proposals to extend abstract argumentation to take this uncertainty into account. This assigns a probability value for each argument that represents the degree to which the argument is believed to hold, and this is then used to generate a probability distribution over the full subgraphs of the argument graph, which in turn can be used to determine the probability that a set of arguments is admissible or an extension. In order to more fully understand uncertainty in argumentation, in this paper, we extend this idea by considering logic-based argumentation with uncertain arguments. This is based on a probability distribution over models of the language, which can then be used to give a probability distribution over arguments that are constructed using classical logic. We show how this formalization of uncertainty of logical arguments relates to uncertainty of abstract arguments, and we consider a number of interesting classes of probability assignments.     

Bipolarity in argumentation graphs: Towards a better understanding

Different abstract argumentation frameworks have been used for various applications within multi-agents systems. Among them, bipolar frameworks make use of both attack and support relations between arguments. However, there is no single interpretation of the support, and the handling of bipolarity cannot avoid a deeper analysis of the notion of support.In this paper we consider three recent proposals for specializing the support relation in abstract argumentation: the deductive support, the necessary support and the evidential support. These proposals have been developed independently within different frameworks. We restate these proposals in a common setting, which enables us to undertake a comparative study of the modellings obtained for the three variants of the support. We highlight relationships and differences between these variants, namely a kind of duality between the deductive and the necessary interpretations of the support.     

Looking-ahead in backtracking algorithms for abstract argumentation

We refine implemented backtracking algorithms for a number of problems related to Dung's argumentation frameworks. Under admissible, preferred, complete, stable, semi stable, and ideal semantics we add enhancements, what are so-called global looking-ahead pruning strategies, to the-state-of-the-art implementations of two problems. First, we tackle the extension enumeration problem: constructing some/all set(s) of acceptable arguments of a given argumentation framework. Second, we address the acceptance decision problem: deciding whether an argument is in some/all set(s) of accepted arguments of a given argumentation framework. The experiments that we report show that the speedup gain of the new enhancements is quite significant.     

Dung’s Argumentation is Essentially Equivalent to Classical Propositional Logic with the Peirce–Quine Dagger

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 ?, ù, ú, ?{\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:     

Comparison of path-complete Lyapunov functions via template-dependent lifts

This paper investigates, in the context of discrete-time switched systems, the problem of comparison for path-complete stability certificates. We introduce and study abstract operations on path-complete graphs, called lifts, which allow us to recover previous results in a general framework. Moreover, this approach highlights the existing relations between the analytical properties of the chosen set of candidate Lyapunov functions (the template) and the admissibility of certain lifts. This provides a new methodology for the characterization of the ordering relation of path-complete Lyapunov functions criteria, when a particular template is chosen. We apply our results to specific templates, notably the sets of primal and dual copositive norms, providing new stability certificates for positive switched systems. These tools are finally illustrated with the aim of numerical examples.     

Generalized positive sets and abstract monotonicity

The theory of q-positive sets on SSD spaces has been introduced by Simons (J Convex Anal, 14:297–317, 2007; From Hahn–Banach to monotonicity, Springer, Berlin, 2008). Monotone sets can be considered as special case of q-positive sets. In this paper, we develop a theory of q-positive sets in the framework of abstract monotonicity. We use generalized Fenchel’s duality theorem and give some criteria for maximality of abstract q-positive sets. Finally, we investigate the relation between abstract q-positive sets and abstract convex functions.     

Eliminating counterexamples: A Grade 8 student’s learning trajectory for contrapositive proving

This study uses a teaching experiment and retrospective analysis to develop a learning trajectory for improving a Grade 8 student’s ability to construct, critique, and validate contrapositive arguments. The study is predicated on the hypothesis that adolescents perform poorly on contrapositive reasoning tasks because they lack sufficient ways of justifying contrapositive argumentation as a viable mode of argumentation. By studying a student’s actions and comments as she develops, critiques, and validates not-the-conclusion-implies-the-conditions-are-impossible arguments for conditional claims, a promising learning trajectory for contrapositive argumentation is developed. The student’s learning trajectory demonstrates how a conception of contrapositive proving as eliminating counterexamples can be useful in developing, critiquing, and validating contrapositive arguments.     

AFRA: Argumentation framework with recursive attacks

The issue of representing attacks to attacks in argumentation is receiving an increasing attention as a useful conceptual modelling tool in several contexts. In this paper we present AFRA, a formalism encompassing unlimited recursive attacks within argumentation frameworks. AFRA satisfies the basic requirements of definition simplicity and rigorous compatibility with Dung’s theory of argumentation. This paper provides a complete development of the AFRA formalism complemented by illustrative examples and a detailed comparison with other recursive attack formalizations.     

Modelling Inference in Argumentation through Labelled Deduction: Formalization and Logical Properties

Artificial Intelligence (AI) has long dealt with the issue of finding a suitable formalization for commonsense reasoning. Defeasible argumentation has proven to be a successful approach in many respects, proving to be a confluence point for many alternative logical frameworks. Different formalisms have been developed, most of them sharing the common notions of argument and warrant. In defeasible argumentation, an argument is a tentative (defeasible) proof for reaching a conclusion. An argument is warranted when it ultimately prevails over other conflicting arguments. In this context, defeasible consequence relationships for modelling argument and warrant as well as their logical properties have gained particular attention. This article analyzes two non-monotonic inference operators C_arg and C_war intended for modelling argument construction and dialectical analysis (warrant), respectively. As a basis for such analysis we will use the LDS_ar framework, a unifying approach to computational models of argument using Labelled Deductive Systems (LDS). In the context of this logical framework, we show how labels can be used to represent arguments as well as argument trees, facilitating the definition and study of non-monotonic inference operators, whose associated logical properties are studied and contrasted. We contend that this analysis provides useful comparison criteria that can be extended and applied to other argumentation frameworks. Mathematics Subject Classification (2000): Primary 03B22; Secondary 03B42.     

Postulates for logic-based argumentation systems

Logic-based argumentation systems are developed for reasoning with inconsistent information. Starting from a knowledge base encoded in a logical language, they define arguments and attacks between them using the consequence operator associated with the language. Finally, a semantics is used for evaluating the arguments.In this paper, we focus on systems that are based on deductive logics and that use Dung's semantics. We investigate rationality postulates that such systems should satisfy. We define five intuitive postulates: consistency and closure under the consequence operator of the underlying logic of the set of conclusions of arguments of each extension, closure under sub-arguments and exhaustiveness of the extensions, and a free precedence postulate ensuring that the free formulas of the knowledge base (i.e., the ones that are not involved in inconsistency) are conclusions of arguments in every extension. We study the links between the postulates and explore conditions under which they are guaranteed or violated.     

Algorithms for generating arguments and counterarguments in propositional logic

A common assumption for logic-based argumentation is that an argument is a pair 〈Φ,α〉 where Φ is minimal subset of the knowledgebase such that Φ is consistent and Φ entails the claim α. Different logics provide different definitions for consistency and entailment and hence give us different options for formalising arguments and counterarguments. The expressivity of classical propositional logic allows for complicated knowledge to be represented but its computational cost is an issue. In previous work we have proposed addressing this problem using connection graphs and resolution in order to generate arguments for claims that are literals. Here we propose a development of this work to generate arguments for claims that are disjunctive clauses of more than one disjunct, and also to generate counteraguments in the form of canonical undercuts (i.e. arguments that with a claim that is the negation of the conjunction of the support of the argument being undercut).     

Classifying efficient alternatives in SMAA using cross confidence factors

Stochastic multicriteria acceptability analysis (SMAA) is a family of methods for aiding multicriteria group decision making. These methods are based on exploring the weight space in order to describe the preferences that make each alternative the most preferred one. The main results of the analysis are rank acceptability indices, central weight vectors and confidence factors for different alternatives. The rank acceptability indices describe the variety of different preferences resulting in a certain rank for an alternative; the central weight vectors represent the typical preferences favouring each alternative; and the confidence factors measure whether the criteria data are sufficiently accurate for making an informed decision.In some cases, when the problem involves a large number of efficient alternatives, the analysis may fail to discriminate between them. This situation is revealed by low confidence factors. In this paper we develop cross confidence factors, which are based on computing confidence factors for alternatives using each other’s central weight vectors. The cross confidence factors can be used for classifying efficient alternatives into sets of similar and competing alternatives. These sets are related to the concept of reference sets in Data Envelopment Analysis (DEA), but generalized for stochastic models. Forming these sets is useful when trying to identify one or more most preferred alternatives, or suitable compromise alternatives. The reference sets can also be used for evaluating whether criteria need to be measured more accurately, and at which alternatives the measurements should be focused. This may cause considerable savings in measurement costs. We demonstrate the use of the cross confidence factors and reference sets using a real-life example.     

Probabilistic qualification of attack in abstract argumentation

An argument graph is a graph where each node denotes an argument, and each arc denotes an attack by one argument on another. It offers a valuable starting point for theoretical analysis of argumentation following the proposals by Dung. However, the definition of an argument graph does not take into account the belief in the attacks. In particular, when constructing an argument graph from informal arguments, where each argument is described in free text, it is often evident that there is uncertainty about whether some of the attacks hold. This might be because there is some expressed doubt that an attack holds or because there is some imprecision in the language used in the arguments. In this paper, we use the set of spanning subgraphs of an argument graph as a sample space. A spanning subgraph contains all the arguments, and a subset of the attacks, of the argument graph. We assign a probability value to each spanning subgraph such that the sum of the assignments is 1. This means we can reflect the uncertainty over which is the actual subgraph using this probability distribution. Using the probability distribution over subgraphs, we can then determine the probability that a set of arguments is admissible or an extension. We can also obtain the probability of an attack relationship in the original argument graph as a marginal distribution (i.e. it is the sum of the probability assigned to each subgraph containing that attack relationship). We investigate some of the features of this proposal, and we consider the utility of our framework for capturing some practical argumentation scenarios.     

Generating possible intentions with constrained argumentation systems

Practical reasoning (PR), which is concerned with the generic question of what to do, is generally seen as a two steps process: (1) deliberation, in which an agent decides what state of affairs it wants to reach - that is, its desires; and (2) means-ends reasoning, in which the agent looks for plans for achieving these desires. The agent’s intentions are a consistent set of desires that are achievable together.This paper proposes the first argumentation system for PR that computes in one step the possible intentions of an agent, avoiding thus the drawbacks of the existing systems. The proposed system is grounded on a recent work on constrained argumentation systems, and satisfies the rationality postulates identified in argumentation literature, namely the consistency and the completeness of the results.     

Optimization of dialectical outcomes in dialogical argumentation

When informal arguments are presented, there may be imprecision in the language used, and so the audience may be uncertain as to the structure of the argument graph as intended by the presenter of the arguments. For a presenter of arguments, it is useful to know the audience's argument graph, but the presenter may be uncertain as to the structure of it. To model the uncertainty as to the structure of the argument graph in situations such as these, we can use probabilistic argument graphs. The set of subgraphs of an argument graph is a sample space. A probability value is assigned to each subgraph such that the sum is 1, thereby reflecting the uncertainty over which is the actual subgraph. We can then determine the probability that a particular set of arguments is included or excluded from an extension according to a particular Dung semantics. We represent and reason with extensions from a graph and from its subgraphs, using a logic of dialectical outcomes that we present. We harness this to define the notion of an argumentation lottery, which can be used by the audience to determine the expected utility of a debate, and can be used by the presenter to decide which arguments to present by choosing those that maximize expected utility. We investigate some of the options for using argumentation lotteries, and provide a computational evaluation.     

Restricted domains,arrow-social welfare functions and noncorruptible and non-manipulable social choice correspondences: The case of private alternatives

An n-person social choice problem is considered in which the alternatives are n dimensional vectors, with the ith component of such a vector being the part of the alternatives affecting individual i alone. Assuming that individuals are selfish (individual i must be indifferent between any two alternatives with the same components), that they may be indifferent among alternatives and that each individual may choose his preferences out of a different set of permissible preferences, we prove that any set of restricted domains of preferences admits an n person non-dictatorial Arrow-type social welfare function if and only if it admits a two-person Arrow-type social welfare function: we characterize all the sets of restricted domains of preferences which admit two-person Arrow-type social welfare functions (and therefore also admit n-person Arrow-type social welfare functions) and then we prove that we also characterized all the sets of restricted domains of preferences which admit nondictatorial, nonmanipulable, noncorruptible and rational social choice correspondences.     

Explanation, Entailment, and Leibnizian Cosmological Arguments

I argue that there are Leibnizian-style cosmological arguments for the existence of God which start from very mild premises which affirm the mere possibility of a principle of sufficient reason. The utilization of such premises gives a great deal of plausibility to such types of argumentation. I spend the majority of the paper defending three major objections to such “mild” premises viz., a reductio argument from Peter van Inwagen and William Rowe, which proffers and defends the idea that a necessary proposition cannot explain a contingent one. I, then, turn to an amelioration of the Rowe/van Inwagen argument which attempts to appeal to an entailment relation between explanans and explanandum that is fettered out in terms of relevance logic. Subsequent to dispelling with that worry, I tackle objections to the utilization of weak principles of sufficient reason that depend essentially upon agglomerative accounts of explanation.