A Logical Account of Formal Argumentation

In the current paper, we re-examine how abstract argumentation can be formulated in terms of labellings, and how the resulting theory can be applied in the field of modal logic. In particular, we are able to express the (complete) extensions of an argumentation framework as models of a set of modal logic formulas that represents the argumentation framework. Using this approach, it becomes possible to define the grounded extension in terms of modal logic entailment.     

论证的形式模型概述及其在哲学上的应用

论证是一个用理由支持观点以及回应其所受攻击的过程。在过去的几十年里,论证已经成为哲学和人工智能研究领域中的一个重要主题。在哲学方面,十九世纪50年代和60年代图尔敏和佩雷尔曼对形式逻辑的批判,促进了非形式逻辑这一学科的产生,它研究推理和论证的非形式模型。在人工智能方面,论证的形式模型也发展成为常识推理和多主体冲突解决的基本模型。本文将讨论后一领域中所研究发展的形式模型如何能够用以澄清一些哲学领域、以及非形式逻辑领域中的理论问题和争议。本文的一个重要观点是,图尔敏和佩雷尔曼时代的形式逻辑只关注数学化的推理,但那些非数学化的推理形式其实同样也能够被形式化。     

Two approaches to the problems of self-attacking arguments and general odd-length cycles of attack

The problems that arise from the presence of self-attacking arguments and odd-length cycles of attack within argumentation frameworks are widely recognized in the literature on defeasible argumentation. This paper introduces two simple semantics to capture different intuitions about what kinds of arguments should become justified in such scenarios. These semantics are modeled upon two extensions of argumentation frameworks, which we call sustainable and tolerant. Each one is constructed on the common ground of the powerful concept of admissibility introduced by Dung in [P.M. Dung, On the acceptability of arguments and its fundamental role in non-monotonic reasoning, logic programming, and n-person games, Artificial Intelligence 77 (1995) 321–357]. The novelty of this approach consists in viewing the admissibility of a subset of arguments as relative to potentially challenging subsets of arguments. Both sustainable and tolerant semantics are more credulous than preferred semantics (i.e. they justify at least the same arguments, and possibly more). Given certain sufficient conditions they coincide among them as well as with other semantics introduced by Dung.     

The Beth Property in Algebraic Logic

The present paper is a study in abstract algebraic logic. We investigate the correspondence between the metalogical Beth property and the algebraic property of surjectivity of epimorphisms. It will be shown that this correspondence holds for the large class of equivalential logics. We apply our characterization theorem to relevance logics and many-valued logics. Dedicated to the memory of Willem Johannes Blok     

Abstract argumentation and explanation applied to scientific debates

Abstract argumentation has been shown to be a powerful tool within many fields such as artificial intelligence, logic and legal reasoning. In this paper we enhance Dung’s well-known abstract argumentation framework with explanatory capabilities. We show that an explanatory argumentation framework (EAF) obtained in this way is a useful tool for the modeling of scientific debates. On the one hand, EAFs allow for the representation of explanatory and justificatory arguments constituting rivaling scientific views. On the other hand, different procedures for selecting arguments, corresponding to different methodological and epistemic requirements of theory evaluation, can be formulated in view of our framework.     

The Craig Interpolation Theorem in abstract model theory

The Craig Interpolation Theorem is intimately connected with the emergence of abstract logic and continues to be the driving force of the field. I will argue in this paper that the interpolation property is an important litmus test in abstract model theory for identifying “natural,” robust extensions of first order logic. My argument is supported by the observation that logics which satisfy the interpolation property usually also satisfy a Lindström type maximality theorem. Admittedly, the range of such logics is small.     

A First Order Nonmonotonic Extension of Constructive Logic

Certain extensions of Nelson's constructive logic N with strong negation have recently become important in arti.cial intelligence and nonmonotonic reasoning, since they yield a logical foundation for answer set programming (ASP). In this paper we look at some extensions of Nelson's .rst-order logic as a basis for de.ning nonmonotonic inference relations that underlie the answer set programming semantics. The extensions we consider are those based on 2-element, here-and-there Kripke frames. In particular, we prove completeness for .rst-order here-and-there logics, and their minimal strong negation extensions, for both constant and varying domains. We choose the constant domain version, which we denote by QN^c₅, as a basis for de.ning a .rst-order nonmonotonic extension called equilibrium logic. We establish several metatheoretic properties of QN^c₅, including Skolem forms and Herbrand theorems and Interpolation, and show that the .rst-oder version of equilibrium logic can be used as a foundation for answer set inference.     

Modal Provability Foundations for Argumentation Networks

Given an argumentation network we associate with it a modal formula representing the ‘logical content’ of the network. We show a one-to-one correspondence between all possible complete Caminada labellings of the network and all possible models of the formula.     

Abstract argument games via modal logic

Inspired by some logical considerations, the paper proposes a novel perspective on the use of two-players zero-sum games in abstract argumentation. The paper first introduces a second-order modal logic, within which all main Dung-style semantics are shown to be formalizable, and then studies the model checking game of this logic. The model checking game is then used to provide a systematic game theoretic proof procedure to test membership with respect to all those semantics formalizable in the logic. The paper discusses this idea in detail and illustrates it by providing a game for the so-called skeptical preferred and skeptical semi-stable semantics.     

Modal Correspondence for Models

This paper considers the correspondence theory from modal logic and obtains correspondence results for models as opposed to frames. The key ideas are to consider infinitary modal logic, to phrase correspondence results in terms of substitution instances of a given modal formula, and to identify bisimilar model-world pairs.     

The neuronal basis of bimanual coordination: recent neurophysiological evidence and functional models

Recent physiological studies of the neuronal processes underlying bimanual movements provide new tests for earlier functional models of bimanual coordination. The recently acquired data address three conceptual areas: the generalized motor program (GMP), intermanual crosstalk and dynamic systems models. To varying degrees, each of these concepts has aspects that can be reconciled with experimental evidence. The idea of a GMP is supported by the demonstration of abstract neuronal motor codes, e.g. bimanual-specific activity in motor cortex. The crosstalk model is consistent with the facts that hand-specific coding also exists and that interactions occur between the motor commands for each arm. Uncrossed efferent projections may underlie crosstalk on an executional level. Dynamic interhemispheric interactions through the corpus callosum may provide a high-level link at the parametric programming level, allowing flexible coupling and de-coupling. Flexible neuronal interactions could also underlie adaptive large-scale systems dynamics that can be formalized within the dynamic systems theory approach.

The correspondence of identified neuronal processes with functions of abstract models encourages the development of realistic computational models that can predict bimanual behavior on the basis of neuronal activity.     

A neural cognitive model of argumentation with application to legal inference and decision making

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).     

Kripke Models,Distributive Lattices,and Medvedev Degrees

We define a variant of the standard Kripke semantics for intuitionistic logic, motivated by the connection between constructive logic and the Medvedev lattice. We show that while the new semantics is still complete, it gives a simple and direct correspondence between Kripke models and algebraic structures such as factors of the Medvedev lattice. Presented by Daniele Mundici     

Meta-Argumentation Modelling I: Methodology and Techniques

In this paper, we introduce the methodology and techniques of meta-argumentation to model argumentation. The methodology of meta-argumentation instantiates Dung’s abstract argumentation theory with an extended argumentation theory, and is thus based on a combination of the methodology of instantiating abstract arguments, and the methodology of extending Dung’s basic argumentation frameworks with other relations among abstract arguments. The technique of meta-argumentation applies Dung’s theory of abstract argumentation to itself, by instantiating Dung’s abstract arguments with meta-arguments using a technique called flattening. We characterize the domain of instantiation using a representation technique based on soundness and completeness. Finally, we distinguish among various instantiations using the technique of specification languages.     

Capturing equilibrium models in modal logic

Here-and-there models and equilibrium models were investigated as a semantical framework for answer-set programming by Pearce, Valverde, Cabalar, Lifschitz, Ferraris and others. The semantics of equilibrium logic is given in an indirect way: the notion of an equilibrium model is defined in terms of quantification over here-and-there models. We here give a direct semantics of equilibrium logic, stated for a modal language embedding the language of equilibrium logic.     

Predicate Logical Extensions of some Subintuitionistic Logics

The paper presents predicate logical extensions of some subintuitionistic logics. Subintuitionistic logics result if conditions of the accessibility relation in Kripke models for intuitionistic logic are dropped. The accessibility relation which interprets implication in models for the propositional base subintuitionistic logic considered here is neither persistent on atoms, nor reflexive, nor transitive. Strongly complete predicate logical extensions are modeled with a second accessibility relation, which is a partial order, for the interpretation of the universal quantifier. Presented by Melvin Fitting     

A formal account of Socratic-style argumentation

In traditional mathematical models of argumentation an argument often consists of a chain of rules or reasons, beginning with premisses and leading to a conclusion that is endorsed by the party that put forward the argument. In informal reasoning, however, one often encounters a specific class of counterarguments that until now has received little attention in argumentation formalisms. The idea is that instead of starting with the premisses, the argument starts with the propositions put forward by the counterparty, of which the absurdity is illustrated by showing their (defeasible) consequences. This way of argumentation (which we call S-arguments) is very akin to Socratic dialogues and critical interviews; it also has applications in modern philosophy. In this paper, various examples of S-arguments are provided, as well as a treatment of the problems that occur when trying to formalize them in existing formalisms. We also provide general guidelines that can serve as a basis for implementing S-arguments into various existing formalisms. In particular, we show how S-arguments can be implemented in Pollock's formalism, how they fit into Dung's abstract argumentation approach and how they are related to the issue of self-defeating arguments.     

Constructing Natural Extensions of Propositional Logics

The proofs of some results of abstract algebraic logic, in particular of the transfer principle of Czelakowski, assume the existence of so-called natural extensions of a logic by a set of new variables. Various constructions of natural extensions, claimed to be equivalent, may be found in the literature. In particular, these include a syntactic construction due to Shoesmith and Smiley and a related construction due to ?o? and Suszko. However, it was recently observed by Cintula and Noguera that both of these constructions fail in the sense that they do not necessarily yield a logic. Here we show that whenever the ?o?–Suszko construction yields a logic, so does the Shoesmith–Smiley construction, but not vice versa. We also describe the smallest and the largest conservative extension of a logic by a set of new variables and show that contrary to some previous claims in the literature, a logic of cardinality \({\kappa}\) may have more than one conservative extension of cardinality \({\kappa}\) by a set of new variables. In this connection we then correct a mistake in the formulation of a theorem of Dellunde and Jansana.     

On Argumentation Logic and Propositional Logic

This paper studies the relationship between Argumentation Logic (AL), a recently defined logic based on the study of argumentation in AI, and classical Propositional Logic (PL). In particular, it shows that AL and PL are logically equivalent in that they have the same entailment relation from any given classically consistent theory. This equivalence follows from a correspondence between the non-acceptability of (arguments for) sentences in AL and Natural Deduction (ND) proofs of the complement of these sentences. The proof of this equivalence uses a restricted form of ND proofs, where hypotheses in the application of the Reductio of Absurdum inference rule are required to be “relevant” to the absurdity derived in the rule. The paper also discusses how the argumentative re-interpretation of PL could help control the application of ex-falso quodlibet in the presence of inconsistencies.     

Program Constructions that are Safe for Bisimulation

It has been known since the seventies that the formulas of modal logic are invariant for bisimulations between possible worlds models — while conversely, all bisimulation-invariant first-order formulas are modally definable. In this paper, we extend this semantic style of analysis from modal formulas to dynamic program operations. We show that the usual regular operations are safe for bisimulation, in the sense that the transition relations of their values respect any given bisimulation for their arguments. Our main result is a complete syntactic characterization of all first-order definable program operations that are safe for bisimulation. This is a semantic functional completeness result for programming, which may be contrasted with the more usual analysis in terms of computational power. The 'Safety Theorem' can be modulated in several ways. We conclude with a list of variants, extensions, and further developments.