ASPIC-END: Structured Argumentation with Explanations and Natural Deduction

Scientific research in the formal sciences comes in multiple degrees of formality: fully formal work; rigorous proofs that practitioners know to be formalizable in principle; and informal work like rough proof sketches and considerations about the advantages and disadvantages of various formal systems. This informal work includes informal and semi-formal debates between formal scientists, e.g. about the acceptability of foundational principles and proposed axiomatizations. In this paper, we propose to use the methodology of structured argumentation theory to produce a formal model of such informal and semi-formal debates in the formal sciences. For this purpose, we propose ASPIC-END, an adaptation of the structured argumentation framework ASPIC+ which can incorporate natural deduction style arguments and explanations. We illustrate the applicability of the framework to debates in the formal sciences by presenting a simple model of some arguments about proposed solutions to the Liar paradox, and by discussing a more extensive—but still preliminary—model of parts of the debate that mathematicians had about the Axiom of Choice in the early twentieth century.

‘Natural deduction’ is natural in virtue of the way that it allows formal proofs to closely mirror the structure of familiar ordinary language arguments.1 For example, the sentences of the ordinary language argument (1) correspond one-to-one with the lines of the proof (2): (1) Whoever committed the murder left by the window. Anyone who left by the window would have had mud on his shoes. Suppose that the butler committed the murder. Then he left by the window. In that case, he has mud on his shoes. So if the butler committed the murder, he has mud on his shoes.

John Buridan was a fourteenth-century philosopher who enjoyed an enormous reputation for about two hundred years, was then totally neglected, and is now being 'rediscovered' through his relevance to contemporary work in philosophical logic. The final chapter of Buridan's Sophismata deals with problems about self-reference, and in particular with the semantic paradoxes. He offers his own distinctive solution to the well-known 'Liar Paradox' and introduces a number of other paradoxes that will be unfamiliar to most logicians. Buridan also moves on from these problems to more general questions about the nature of propositions, the criteria of their truth and falsity and the concepts of validity and knowledge. This edition of that chapter is intended to make Buridan's ideas and arguments accessible to a wider range of readers. The volume should interest many philosophers, linguists and logicians, who are increasingly finding in medieval work striking anticipations of their own concerns.

The conception of the very first decision procedures for first-order sentences brought about the need for giving citizenship to function symbols (e.g. Skolem functions). We argue that a closer look at proof procedures for first-order sentences with equality brings about the need for introducing (function) symbols for rewrites. This is appropriately done via the framework of labelled natural deduction which allows to formulate a proof theory for the “logical connective” of propositional equality. The basic idea is that when analysing an equality sentence into (i) proof conditions (introduction) and (ii) immediate consequences (elimination), it becomes clear that we need to bring in identifiers (i.e. function symbols) for sequences of rewrites, and this is what we claim is the missing entity in P. Martin-Löf’s equality types (both intensional and extensional). What we end up with is a formulation of what appears to be a middle ground solution to the ‘intensional’ versus ‘extensional’ dichotomy which permeates most of the work on characterizing propositional equality in natural deduction style. (Part of this material was presented at the Logical Methods in the Humanities Seminar, Stanford University, and the authors would like to thank Solomon Feferman and Grigori Mints for their comments and suggestions.)

The introduction of safety cases was a step in the right direction in regards to safety assurance. As presently practiced, safety cases aim at making a serious attempt to explicate, and to provide some structure for, the reasoning involved in assuring that a system is safe, generally in terms of so-called structured arguments. However, the fact current notations for expressing these structured arguments have no formal semantics and, at best, are loosely linked to goal structuring ideas and to Toulmin’s notion of an argument pattern, is a crucial issue to be addressed. History clearly demonstrates that languages that have no formal semantics are deficient in relation to the requirements of a serious approach to engineering. In other words, one can only go so far with intuition, and certainly not far enough to justify the safety of complex systems, such as Cyber Physical Systems or autonomous cars. By rehearsing Gentzen’s program for formalizing mathematical reasoning, his famous Calculus of Natural Deduction, we show how we can begin a program of formalizing safety reasoning by developing a working definition of a structured argument in a safety case and a calculus for safety reasoning.

The ease and speed of spreading misinformation and propaganda on the Web motivate the need to develop trustworthy technology for detecting fallacies in natural language arguments. However, state-of-the-art language modeling methods exhibit a lack of robustness on tasks like logical fallacy classification that require complex reasoning. In this paper, we propose a Case-Based Reasoning method that classifies new cases of logical fallacy by language-modeling-driven retrieval and adaptation of historical cases. We design four complementary strategies to enrich input representation for our model, based on external information about goals, explanations, counterarguments, and argument structure. Our experiments in in-domain and out-of-domain settings indicate that Case-Based Reasoning improves the accuracy and generalizability of language models. Our ablation studies suggest that representations of similar cases have a strong impact on the model performance, that models perform well with fewer retrieved cases, and that the size of the case database has a negligible effect on the performance. Finally, we dive deeper into the relationship between the properties of the retrieved cases and the model performance.

The purpose of this study is to suggest and explain an engaging approach to three distinct types of (alleged or genuine) semantic paradoxes, the White-Horse-Not-Horse Paradox, the Ultimate-Unspeakable Paradox, and the Liar Paradox, in a unifying way that is sensitive to distinct features of them. Although the three types of semantic paradoxes address distinct types of objects, and although their seemingly paradoxical features are different (alleged or genuine), their distinct structures and contents can be understood and treated on the same common ground, which is jointly conceived in people’s pretheoretic understandings of truth and of the double-reference feature of people’s basic employment of language (saying something about an object), and from the unifying vantage point of a double-reference approach.

Dialeteias são contradições verdadeiras. Dialeteísmo é a visão de que há dialeteias e dialeteístas são aqueles que defendem tal visão. Uma das principais motivações para o dialeteísmo encontra-se nos paradoxos semânticos, como o paradoxo do Mentiroso. A sentença do mentiroso, na abordagem dialeteísta, instancia uma dialeteia. O problema é: como aceitar contradições verdadeiras? Como dialeteísmo é a visão que algumas, mas não todas, contradições são verdadeiras, o dialeteísmo demanda um tratamento paraconsistente. Porém, não é qualquer sistema paraconsistente que pode ser aplicado ao dialeteísmo. A interpretação do símbolo de negação deve ser um operador formador de contradição (ofc) com o sentido relevante para a aplicação dialeteísta. Especificamente, na versão de dialeteísmo que discutiremos nesse artigo, defendida por Graham Priest, temos que, para que o dialeteísmo faça sentido, a negação da lógica subjacente deve atender a dois requisitos: (i) ser um ofc e (ii) não ser explosiva. A Lógica do Paradoxo (LP) tem sido apontada como a lógica adequada ao dialeteísmo. Veremos que, a fim de lidar com contradições sem trivialidade, dialeteístas “esticam” a verdade de modo que a verdade possa incluir, em alguns casos, a falsidade também. Nesse caso, haveria sentenças verdadeiras e falsas, chamadas de aglutinações de valores de verdade (truth-value gluts). Em LP, aglutinações de valores de verdade são fundamentais para garantir os requisitos (i) e (ii). Todavia, vamos argumentar que tal procedimento de esticar a verdade, permitindo aglutinações, garante a paraconsistência ao custo de distorcer a interpretação da noção de contradição envolvida no dialeteísmo. Especificamente, tal distorção enfraquece a interpretação da negação comprometendo o sentido de contradição relevante para o dialeteísmo. Vamos argumentar que as próprias restrições dialeteístas a uma compreensão adequada do Mentiroso mostram que as condições (i) e (ii) são incompatíveis e que, com isso, o projeto dialeteísta enfrenta consideráveis obstáculos.

This book aims to provide a solution to the semantic paradoxes. It argues for a unified solution to the paradoxes generated by the concepts of reference or denotation, predicate extension, and truth. The solution makes two main claims. The first is that our semantic expressions ‘denotes’, ‘extension’, and ‘true’ are context-sensitive. The second, inspired by a brief, tantalizing remark of Gödel’s, is that these expressions are significant everywhere except for certain singularities, in analogy with division by zero. A formal theory of singularities is presented and applied to a wide variety of versions of the definability paradoxes, Russell’s paradox, and the Liar paradox. The book argues that the singularity theory satisfies the following desiderata: it recognizes that the proper setting of the semantic paradoxes is natural language, not regimented formal languages; it minimizes any revision to our semantic concepts; it respects as far as possible Tarski’s intuition that natural languages are universal; it responds adequately to the threat of revenge paradoxes; and it preserves classical logic and semantics. The book examines the consequences of the singularity theory for deflationary views of our semantic concepts, and concludes that if we accept the singularity theory, we must reject deflationism.

In structured argumentation frameworks such as ASPIC+, rebuts are only allowed in conclusions produced by defeasible rules. This has been criticized as counter-intuitive especially in dialectical contexts. In this paper we show that ASPIC-, a system allowing for unrestricted rebuts, suffers from contamination problems. We remedy this shortcoming by generalizing the attack rule of unrestricted rebut. Our resulting system satisfies the usual rationality postulates for prioritized rule bases.

Logics for ‘generally’ were introduced as extensions of First-Order Logic (FOL) for handling assertions with vague notions (e.g., ‘generally,’ ‘most,’ and ‘several’) expressed by formulas with generalized quantifiers. Deductive systems have been developed for such logics. Here, we characterize the structure of derivations in natural deduction style for filter logic. This characterization extends the familiar one for FOL.

An abstract framework for structured arguments is presented, which instantiates Dung's (‘On the Acceptability of Arguments and its Fundamental Role in Nonmonotonic Reasoning, Logic Programming, and n-Person Games’, Artificial Intelligence, 77, 321–357) abstract argumentation frameworks. Arguments are defined as inference trees formed by applying two kinds of inference rules: strict and defeasible rules. This naturally leads to three ways of attacking an argument: attacking a premise, attacking a conclusion and attacking an inference. To resolve such attacks, preferences may be used, which leads to three corresponding kinds of defeat: undermining, rebutting and undercutting defeats. The nature of the inference rules, the structure of the logical language on which they operate and the origin of the preferences are, apart from some basic assumptions, left unspecified. The resulting framework integrates work of Pollock, Vreeswijk and others on the structure of arguments and the nature of defeat and extends it in several respects. Various rationality postulates are proved to be satisfied by the framework, and several existing approaches are proved to be a special case of the framework, including assumption-based argumentation and DefLog.

Abstract argumentation systems are formalisms for defeasible reasoning where some components remain unspecified, the structure of arguments being the main abstraction. In the dialectical process carried out to identify accepted arguments in the system some controversial situations may appear. These relate to the reintroduction of arguments into the process which cause the onset of circularity. This must be avoided in order to prevent an infinite analysis. Some systems apply the sole restriction of not allowing the introduction of previously considered arguments in an argumentation line. However, repeating an argument is not the only possible cause for the risk mentioned. A more specific restriction needs to be applied considering the existence of subarguments. In this work, we introduce an extended argumentation framework where two kinds of defeat relation are present, and a definition for progressive defeat path.KeywordsAbstract ArgumentationArgumentation FrameworkAbstract FrameworkPreference CriterionDefeasible ReasoningThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This article outlines what a formal theory of truthshouldbe like, at least at first glance. As not all of the stated constraints can be satisfied at the same time, in view of notorious semantic paradoxes such as the Liar paradox, we consider the maximal consistent combinations of these desiderata and compare their relative advantages and disadvantages.

Computational argumentation is primed to strengthen the current hot research field of Explainable Artificial Intelligence (XAI), e.g., by dialectical approaches. In this paper, we extend and discuss a recently proposed approach of so-called strong acceptance on abstract argumentation that aims to support explaining argumentative acceptance. Our goal is to push these results into the realm of structured argumentation. In this setting, a knowledge base induces an abstract argumentation framework (AF) via instantiation. We investigate how and under which conditions it is possible to transfer results regarding strong acceptance between the given knowledge base and the induced AF. To this end we consider generic functions formalizing the interaction of the AF and the knowledge base. This approach helps us to infer rather general results making basic assumptions rather than dealing with the technical details of several structured argumentation formalisms. Along the way, we apply our techniques to the concrete approach of assumption-based argumentation (ABA) which constitutes one of the primal structured argumentation formalisms.