Dynamics, Uncertainty and Reasoning: The Second Chinese Conference on Logic and Argumentation [1st ed.] 978-981-13-7790-7;978-981-13-7791-4

Table of contents :
Front Matter ....Pages i-xii
Structural Analysis of Extension-Based Argumentation Semantics with Joint Acceptability (Yuming Xu, Lidong Xu, Claudette Cayrol)....Pages 1-20
A Dynamic Approach for Combining Abstract Argumentation Semantics (Jérémie Dauphin, Marcos Cramer, Leendert van der Torre)....Pages 21-43
Local Expansion Invariant Operators in Argumentation Semantics (Stefano Bistarelli, Francesco Santini, Carlo Taticchi)....Pages 45-62
Updating Argumentation Frameworks for Enforcing Extensions (Kang Xu, Beishui Liao, Huaxin Huang)....Pages 63-79
Open Reading and Free Choice Permission: A Perspective in Substructural Logics (Huimin Dong, Norbert Gratzl, Olivier Roy)....Pages 81-115
A Road to Ultrafilter Extensions (Jie Fan)....Pages 117-133
Soft Presuppositions as Scalar Implicatures in Signaling Games (Mengyuan Zhao)....Pages 135-151
Abstract Argumentation in Dynamic Logic: Representation, Reasoning and Change (Sylvie Doutre, Andreas Herzig, Laurent Perrussel)....Pages 153-185
Computational Hermeneutics: An Integrated Approach for the Logical Analysis of Natural-Language Arguments (David Fuenmayor, Christoph Benzmüller)....Pages 187-207

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Logic in Asia: Studia Logica Library Series Editors: Fenrong Liu · Hiroakira Ono

Beishui Liao Thomas Ågotnes Yi N. Wang   Editors

Dynamics, Uncertainty and Reasoning The Second Chinese Conference on Logic and Argumentation

Logic in Asia: Studia Logica Library Editors-in-Chief Fenrong Liu, Tsinghua University and University of Amsterdam, Beijing, China Hiroakira Ono, Japan Advanced Institute of Science and Technology (JAIST), Ishikawa, Japan Editorial Board Natasha Alechina, University of Nottingham, Nottingham, UK Toshiyasu Arai, Chiba University, Chiba Shi, Inage-ku, Japan Sergei Artemov, City University of New York, New York, NY, USA Mattias Baaz, Technical university of Vienna, Austria, Vietnam Lev Beklemishev, Institute of Russian Academy of Science, Russia Mihir Chakraborty, Jadavpur University, Kolkata, India Phan Minh Dung, Asian Institute of Technology, Thailand Amitabha Gupta, Indian Institute of Technology Bombay, Mumbai, India Christoph Harbsmeier, University of Oslo, Oslo, Norway Shier Ju, Sun Yat-sen University, Guangzhou, China Makoto Kanazawa, National Institute of Informatics, Tokyo, Japan Fangzhen Lin, Hong Kong University of Science and Technology, Hong Kong Jacek Malinowski, Polish Academy of Sciences, Warsaw, Poland Ram Ramanujam, Institute of Mathematical Sciences, Chennai, India Jeremy Seligman, University of Auckland, Auckland, New Zealand Kaile Su, Peking University and Griffith University, Peking, China Johan van Benthem, University of Amsterdam and Stanford University, The Netherlands Hans van Ditmarsch, Laboratoire Lorrain de Recherche en Informatique et ses Applications, France Dag Westerstahl, Stockholm University, Stockholm, Sweden Yue Yang, Singapore National University, Singapore Syraya Chin-Mu Yang, National Taiwan University, Taipei, China

Logic in Asia: Studia Logica Library This book series promotes the advance of scientific research within the field of logic in Asian countries. It strengthens the collaboration between researchers based in Asia with researchers across the international scientific community and offers a platform for presenting the results of their collaborations. One of the most prominent features of contemporary logic is its interdisciplinary character, combining mathematics, philosophy, modern computer science, and even the cognitive and social sciences. The aim of this book series is to provide a forum for current logic research, reflecting this trend in the field’s development. The series accepts books on any topic concerning logic in the broadest sense, i.e., books on contemporary formal logic, its applications and its relations to other disciplines. It accepts monographs and thematically coherent volumes addressing important developments in logic and presenting significant contributions to logical research. In addition, research works on the history of logical ideas, especially on the traditions in China and India, are welcome contributions. The scope of the book series includes but is not limited to the following: • • • •

Monographs written by researchers in Asian countries. Proceedings of conferences held in Asia, or edited by Asian researchers. Anthologies edited by researchers in Asia. Research works by scholars from other regions of the world, which fit the goal of “Logic in Asia”.

The series discourages the submission of manuscripts that contain reprints of previously published material and/or manuscripts that are less than 165 pages/ 90,000 words in length. Please also visit our webpage: http://tsinghualogic.net/logic-in-asia/background/

Relation with Studia Logica Library This series is part of the Studia Logica Library, and is also connected to the journal Studia Logica. This connection does not imply any dependence on the Editorial Office of Studia Logica in terms of editorial operations, though the series maintains cooperative ties to the journal. This book series is also a sister series to Trends in Logic and Outstanding Contributions to Logic. For inquiries and to submit proposals, authors can contact the editors-in-chief Fenrong Liu at [email protected] or Hiroakira Ono at [email protected].

More information about this series at http://www.springer.com/series/13080

Beishui Liao Thomas Ågotnes Yi N. Wang •

•

Editors

Dynamics, Uncertainty and Reasoning The Second Chinese Conference on Logic and Argumentation

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Editors Beishui Liao Center for the Study of Language and Cognition Zhejiang University, Xixi Campus Hangzhou, China

Thomas Ågotnes University of Bergen Bergen, Norway

Yi N. Wang Zhejiang University Hangzhou, China

ISSN 2364-4613 ISSN 2364-4621 (electronic) Logic in Asia: Studia Logica Library ISBN 978-981-13-7790-7 ISBN 978-981-13-7791-4 (eBook) https://doi.org/10.1007/978-981-13-7791-4 © Springer Nature Singapore Pte Ltd. 2019 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

Logic and argumentation are two highly related but somewhat different research areas. In our daily life, argumentation is ubiquitous, including not only deliberation and decision-making of individual agents, but also civil debate, dialogue, conversation and persuasion among a group of agents. It is about how conclusions or agreements can be reached through logical reasoning and/or dialogues. Unlike classical logics by which a set of inconsistent premises can lead to arbitrary and useless conclusions, (formal) argumentation can properly handle conflicting information, returning a set of conclusions satisfying some rational criteria. So, argumentation has a close relation to traditional non-monotonic logics and logic programming, such as default logic, circumscription, answer set programming, etc., where knowledge used for reasoning is allowed to be inconsistent. It has been shown that a number of non-monotonic logics and approaches of logic programming can be represented in argumentation. Besides inconsistency, another two important properties of the reasoning systems with incomplete information are uncertainty and dynamics. In recent years, different approaches have been proposed for combining argumentation and uncertainty, and for formulating dynamics of argumentation. This volume includes 6 papers selected from 16 submissions of the Second Chinese Conference on Logic and Argumentation (CLAR 2018) in 2018 held in Hangzhou, China, and 3 invited papers contributed by leading researchers in related fields. These papers nicely cover the fields of logic and argumentation, and the connection between them. On the one hand, from the perspective of argumentation, reasoning is realized by constructing, comparing and evaluating arguments. While the construction and comparison of arguments are studied in the direction of structured argumentation, which concerns how arguments and the relations over them are defined to satisfy some rationality postulates, the evaluation of the status of arguments is handled in an abstract argumentation framework (AF), which consists of a set of arguments and an attack relation over them. Among a set of conflicting arguments, a subset of collectively accepted arguments is called an extension, and a function mapping an AF to a set of extensions is called an (extension-based) argumentation semantics. v

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In terms of different evaluating criteria, there are different argumentation semantics, including complete, grounded, preferred, stable, naive, etc. Although argumentation semantics have been extensively studied in the argumentation community, there are still some interesting research problems, especially the ones related to uncertainty and dynamics. In this volume, Yuming Xu, Lidong Xu and Claudette Cayrol provide a structural analysis of extension-based argumentation semantics, based on a notion of joint acceptability. They show that using this new approach, an admissible set, a complete extension, or a preferred extension can be iteratively built from a conflict-free collection of initial sets. Jérémie Dauphin, Marcos Cramer and Leendert van der Torre introduce a dynamic approach for combining argumentation semantics. They show that the merging of preferred semantics and grounded semantics is complete semantics, and that features of naive-based and complete-based semantics can be meaningfully combined. Stefano Bistarelli, Francesco Santini and Carlo Taticchi study the dynamics of argumentation by introducing local-expansion invariant operators in argumentation semantics, such that for conflict-free sets or admissible sets of arguments in an AF, applying such operators to the AF does not produce any change to the sets. Kang Xu, Beishui Liao and Huaxin Huang present an approach to update an AF such that a given set of arguments can be enforced to be an extension of the AF, by defining general rules. This approach is based on a notion of characterizing an AF with respect to a given set of arguments. On the other hand, from the perspectives of logic and the interplay of logic and argumentation, in addition to trying to understand the various logics themselves, it is interesting to investigate how various logics can be represented by argumentation, and how some aspects (e.g. semantics and computation) of argumentation can be formulated by logics. Along these lines, in this volume, Huimin Dong, Norbert Gratzl and Olivier Roy propose a new solution to the well-known Free Choice Permission Paradoxes, by combining ideas from substructural logics and non-monotonic reasoning. Jie Fan introduces a uniform method for constructing ultrafilter extensions from canonical models, based on the similarity between ultrafilters and maximal consistent sets. Mengyuan Zhao develops game-theoretic models as a unified account for both implicatures and soft presuppositions, and explains why soft presuppositions project non-uniformly through different types of quantifiers, and why soft presuppositions are easily defeasible. It is interesting to note that although these three papers do not directly focus on the connections between various logics and argumentation, they share some important aspects with argumentation, including utilizing principles of non-monotonic reasoning, defining maximal consistent sets and handling defeasibility of reasoning. Besides these three papers, the remaining two papers are exactly in the direction of connecting logic and argumentation. Sylvie Doutre, Andreas Herzig and Laurent Perrussel study abstract argumentation in dynamic logic, by encoding AFs and their dynamics in Dynamic Logic of Propositional Assignments (DL-PA). David Fuenmayor and Christoph Benzmüller utilize higher order automated deduction technologies for the logical analysis of natural language arguments.

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We would like to thank all the authors who submitted contributions, the invited speakers (Natasha Alechina, Anthony Hunter, Xudong Luo and Tjitze Rienstra), as well as the program committee of CLAR 2018 (Pietro Baroni, Stefano Bistarelli, Alexander Bochman, Dragan Doder, Shanshan Du, Massimiliano Giacomin, Jiahong Guo, Andreas Herzig, Fengkui Ju, Fenrong Liu, Hu Liu, Minghui Ma, Nir Oren, Olivier Roy, Katsuhiko Sano, Guillermo Simari, Christian Straßer, Matthias Thimm, Hans van Ditmarsch, Leon van der Torre, Serena Villata, Stefan Woltran, Yun Xie, Fan Yang and Junhua Yu). We thank local organizers (Huimin Dong, Fanghong Shi and Teng Ying) for their excellent work, and Zhe Yu for her help in the process of editing this volume. Meanwhile, we are grateful to Fenrong Liu and Hiroakira Ono, the editor in chief of this book series ‘Logic in Asia’ (LIAA), for their supportive recommendation of this volume to LIAA, and to Fiona Wu and Leana Li, for their support in the process of publication of this volume. Finally, we acknowledge that CLAR 2018 is financially supported by the Fundamental Research Funds for the Central Universities of China for the project Big Data, Reasoning and Decision Making. Hangzhou, China Bergen, Norway Hangzhou, China January 2019

Beishui Liao [email protected] Thomas Ågotnes Yi N. Wang

Contents

Structural Analysis of Extension-Based Argumentation Semantics with Joint Acceptability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yuming Xu, Lidong Xu and Claudette Cayrol

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A Dynamic Approach for Combining Abstract Argumentation Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jérémie Dauphin, Marcos Cramer and Leendert van der Torre

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Local Expansion Invariant Operators in Argumentation Semantics . . . . Stefano Bistarelli, Francesco Santini and Carlo Taticchi

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Updating Argumentation Frameworks for Enforcing Extensions . . . . . . Kang Xu, Beishui Liao and Huaxin Huang

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Open Reading and Free Choice Permission: A Perspective in Substructural Logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Huimin Dong, Norbert Gratzl and Olivier Roy

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A Road to Ultrafilter Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 Jie Fan Soft Presuppositions as Scalar Implicatures in Signaling Games . . . . . . 135 Mengyuan Zhao Abstract Argumentation in Dynamic Logic: Representation, Reasoning and Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 Sylvie Doutre, Andreas Herzig and Laurent Perrussel Computational Hermeneutics: An Integrated Approach for the Logical Analysis of Natural-Language Arguments . . . . . . . . . . . 187 David Fuenmayor and Christoph Benzmüller

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Contributors

Christoph Benzmüller Freie Universität Berlin, Berlin, Germany; University of Luxembourg, Esch-sur-Alzette, Luxembourg Stefano Bistarelli Università degli Studi di, Perugia, Italy Claudette Cayrol IRIT, Universite de Toulouse, CNRS, Toulouse, France Marcos Cramer TU Dresden, Dresden, Germany Jérémie Dauphin University of Luxembourg, Luxembourg, Luxembourg Huimin Dong Department of Philosophy, Zhejiang University, Hangzhou, China Sylvie Doutre IRIT, Université Toulouse 1 Capitole, Toulouse, France Jie Fan School of Humanities, University of Chinese Academy of Sciences, Beijing, China David Fuenmayor Freie Universität Berlin, Berlin, Germany Norbert Gratzl MCMP, LMU Munich, Munich, Germany Andreas Herzig IRIT, CNRS, Toulouse, France Huaxin Huang Center for the Study of Language and Cognition, Zhejiang University, Hangzhou, China Beishui Liao Center for the Study of Language and Cognition, Zhejiang University, Hangzhou, China Laurent Perrussel IRIT, Université Toulouse 1 Capitole, Toulouse, France Olivier Roy Institut für Philosophie, Universität Bayreuth, Bayreuth, Germany Francesco Santini Università degli Studi di, Perugia, Italy Carlo Taticchi Gran Sasso Science Institute, L’Aquila, Italy

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Leendert van der Torre University of Luxembourg, Luxembourg, Luxembourg Kang Xu Zhejiang University of Water Resources and Electric Power, Hangzhou, China Lidong Xu School of Mathematics, Shandong University, Jinan, China Yuming Xu School of Mathematics, Shandong University, Jinan, China Mengyuan Zhao School of Marxism, University of Shanghai for Science and Technology, Shanghai, China

Structural Analysis of Extension-Based Argumentation Semantics with Joint Acceptability Yuming Xu, Lidong Xu and Claudette Cayrol

Abstract Dung’s abstract argumentation provides us with a general framework to deal with argumentation. For extension-based semantics, the central issue is how to determine the extensions w.r.t. various semantics. Motivated by the acceptability and reinstatement criterion, we propose the notions of J-acceptability and J-reinstatement. Correspondingly, we introduce the J-complete semantics which reveals the gap between complete semantics and preferred semantics. It is shown that acceptability together with J-acceptability forms the foundation of extension-based semantics. For example, any admissible set can be built starting from a conflict-free collection of initial sets by iteratively applying some functions based on acceptability and Jacceptability. This novel idea enables to give a clear description of the extensions of the standard semantics. Keywords Argumentation · Semantics · Initial · J-acceptable · J-complete

1 Introduction Based on the theory of non-monotonic reasoning and logic programming, Dung (1995) proposed the abstract argumentation framework in which the acceptability of arguments plays a central role. Since then, many enrichments were introduced by defining new semantics, i.e., different ways of accepting arguments (see for instance Baroni and Giacomin 2007) or by adding properties to the framework (see, for instance, Amgoud et al. 2004; Wyner and Bench-Capon 2007; Martinez et al. 2007). A number of papers investigated and compared the properties of different semanY. Xu (B) · L. Xu School of Mathematics, Shandong University, Jinan, China e-mail: [email protected] L. Xu e-mail: [email protected] C. Cayrol (B) IRIT, Universite de Toulouse, CNRS, Toulouse, France e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2019 B. Liao et al. (eds.), Dynamics, Uncertainty and Reasoning, Logic in Asia: Studia Logica Library, https://doi.org/10.1007/978-981-13-7791-4_1

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tics which have been proposed for abstract argumentation frameworks Baroni and Giacomin (2009), Baroni et al. (2005), Caminada (2014), Dunne (2007). In recent years, dynamic argumentation has gained an increased interest, thus broadening the field of application of the original contribution. Much work has been done in extension-based semantics and dynamic argumentation Baroni et al. (2014), Baroni et al. (2011), Baroni and Giacomin (2007), Baroni et al. (2014), Cayrol et al. (2010), Liao et al. (2011). For a comprehensive and up-to-date view of the state of the art, we refer the reader to Baroni et al. (2018). The concept of acceptability plays a fundamental role in the extension-based semantics. That is, each extension under the traditional semantics is an admissible set. The reinstatement principle and the characteristic function are developed from the acceptability. But acceptability is not enough when we extend a known admissible set to a new admissible set which has more arguments. There exists the case when an admissible set is combined with some additional arguments to form a new admissible set, whereas each of the additional arguments is not accepted w.r.t. the given admissible set. This case motivated us to propose the notion of joint acceptability, J-acceptability for short. Based on this novel concept, we define the J-characteristic function and the J-reinstatement criterion which play a similar role as the characteristic function and reinstatement criterion do in the theory of extension-based semantics. Furthermore, we define a new semantics called J-complete semantics which exactly reveals the gap between complete semantics and preferred semantics. The notion of initial set was first introduced by Xu and Cayrol (2018). It is a generalization of the notion of initial argument and can be seen as the most basic germ of the building of various extensions. Based on this idea, our aim is to introduce a new fundamental criterion called J-acceptability and to combine it with the acceptability and initial sets to set up a procedure, by which any admissible set, complete extension, and preferred extension can be built and described. Comparing with the known methods of finding extensions, we make it clear that initial sets should be seen as the starting points and discover that the J-acceptability is an essential supplement of acceptability. And so, our work provides a more workable method than the known approaches to construct all admissible sets. The paper is organized as follows. Section 2 recalls the basic notions of argumentation frameworks. Section 3 introduces the J-acceptability criterion and J-complete semantics, and studies their properties. Section 4 discusses our method for building and describing admissible sets, complete extensions and preferred extensions by the acceptability, J-acceptability, and initial sets. Section 5 is devoted to concluding remarks and perspectives. The proofs can be found in the Appendix.

2 The Basic Notions of Argumentation Frameworks For each argumentation framework, the arguments are produced by agents and the attack relations between them are set up according to some specific rules. We do not consider the origin and structure of arguments and the practical interaction of them.

Structural Analysis of Extension-Based Argumentation Semantics …

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Definition 1 An argumentation framework is a pair AF = (A, R), where A is a finite set of arguments and R ⊆ A × A represents the attack relation. Let AF = (A, R) be an argumentation framework, a, b ∈ A and S ⊆ A. a is attacked by b if (b, a) ∈ R, denoted by b → a; a is called initial if a is not attacked; a is attacked by S if there is some b ∈ S such that (b, a) ∈ R, denoted by S → a; R + (S) denotes the set of arguments attacked by S; a attacks S if there is some b ∈ S such that (a, b) ∈ R, denoted by a → S. An argumentation framework AF = (A, R) can be represented by a directed graph. Nodes stand for the arguments and edges represent the attack relation. An extension is a set of arguments which can stand together. The basic requirement for any extension is conflict-freeness. That is, if an argument a attacks another argument b, then they cannot stand together. Another requirement is known as admissibility and lies at the heart of all traditional extension-based semantics. It is based on the notions of acceptable argument and admissible set. Definition 2 Let AF = (A, R) be an argumentation framework, S, T ⊆ A, a ∈ A. – S is conflict-free if there are no a, b ∈ S such that (a, b) ∈ R. Furthermore, S is conflict-free with T if S ∪ T is a conflict-free set. – By extension, a set B of subsets of A is said to be conflict-free if the union of the elements of B, denoted by ∪B, is a conflict-free subset of A. – a is acceptable w.r.t. S (or defended by S) if each attacker b of a is attacked by S. – S is an admissible set if S is conflict-free and each a ∈ S is defended by S. The collection of all admissible sets of AF is denoted by A S (AF). The reinstatement principle Baroni and Giacomin (2007) can be viewed as the converse of admissibility principle. Definition 3 Given σ a semantics and AF an argumentation framework, Eσ (AF) denotes the set of extensions of AF under the semantics σ . σ satisfies the reinstatement principle if and only if ∀AF such that Eσ (AF) is non-empty; for each σ -extension E of AF, there is no subset of A\E to be acceptable w.r.t. E. Both principles lead to the following semantics. Definition 4 Let AF = (A, R) be an argumentation framework and S ⊆ A. – S is a complete extension if S ∈ A S (AF) and for each a ∈ A defended by S, a ∈ S. The collection of all complete extensions is denoted by C O(AF). – S is the grounded extension if it is the least element (w.r.t. set inclusion) of C O(AF). The grounded extension of AF is unique and denoted by G E(AF). – S is a preferred extension of AF if it is a maximal element (w.r.t. set inclusion) of C O(AF). The collection of all preferred extensions is denoted by PR(AF). The complete and grounded extensions can also be defined using the characteristic function. Let AF = (A, R), the function F : 2 A → 2 A which, given a set S ⊆ A,

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returns the set of the acceptable arguments w.r.t. S, is called the characteristic function1 of AF. Complete extensions are exactly conflict-free fixed points of F and the grounded extension of AF is the least fixed point of F . In the following, we need to restrict to a subset of an argumentation framework. Definition 5 Let AF = (A, R) be an argumentation framework, and S ⊆ A. The restriction of AF to S, denoted by AF | S , is the sub-argumentation framework (S, R ∩ (S × S)). We also recall the I-maximality and directionality principles first introduced in Baroni and Giacomin (2007). The directionality principle is based on the sets of arguments which do not receive any attack from outside. Definition 6 Given σ a semantics and AF an argumentation framework, Eσ (AF) denotes the set of extensions of AF under the semantics σ . – A set E of extensions is I-maximal if and only if ∀E 1 , E 2 ∈ E , if E 1 ⊆ E 2 then E 1 = E 2 . A semantics σ satisfies the I-maximality principle if and only if ∀AF such that Eσ (AF) is non-empty, Eσ (AF) is I-maximal. – A non-empty set S ⊆ A is unattacked in AF if and only if there exists no a ∈ (A\S) such that a → S. – A semantics σ satisfies the directionality principle if and only if ∀AF such that Eσ (AF) is non-empty, ∀S unattacked in AF, Eσ (AF| S ) = {(E ∩ S) : E ∈ Eσ (AF)}. The I-maximality principle is satisfied by the grounded and the preferred semantics but not satisfied by the complete semantics. The directionality principle is satisfied by the grounded, the preferred, and the complete semantics. Now, let us turn to the notion of initial set first introduced in Xu and Cayrol (2018). Definition 7 Let AF = (A, R) be an argumentation framework. A non-empty admissible set I is initial if it has no non-empty proper subset to be admissible. The collection of all initial sets of AF is denoted by I S (AF). For any initial argument i of AF, {i} is obviously an initial set. Two initial sets of AF may be conflicting. So, we usually consider conflict-free subsets of I S (AF). Example 1 Let AF = (A, R) with A = {1, 2, 3, 4, 5, 6, 7, 8} and R = {(1, 2), (2, 3), (3, 4), (4, 1), (2, 5), (2, 6), (6, 7), (6, 8), (7, 6)}. The directed graph is as follows:

1 Strictly

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speaking, it should be denoted by F AF . The subscript will be omitted in the following.

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There are three initial sets: I1 = {1, 3}, I2 = {2, 4}, I3 = {7}. I1 and I2 are con flicting, whereas {I1 , I3 } is a conflict-free subset of I S (AF). Due to Definition 7, any non-empty admissible set contains at least one initial set. Certainly, any two initial sets contained in an admissible set are conflict-free. So, we have the following proposition. Proposition 1 Let AF = (A, R) and E be an admissible set of AF. If B is a collection of initial sets contained in E, then ∪B is an admissible set. If B is the collection of all initial sets contained in E, then E\(∪B) contains no initial set. The grounded extension can be built starting from the set of all initial arguments by iteratively adding acceptable arguments. The initial arguments can be seen as starting points for building the grounded extension by acceptability. For an admissible extension E, the initial sets contained in E play the same role. That is, the initial sets can be regarded as the starting points for constructing an admissible set (certainly including complete and preferred extensions) by adding acceptable arguments and the so-called joint acceptable sets which we will introduce below.

3 The J-Acceptability and Related Topics Given an argumentation framework AF = (A, R) and an admissible set E, there are usually two ways to construct a new admissible set having more arguments. If S is an admissible set which is conflict-free with E, then E ∪ S is an admissible set. If S ⊆ F (E), then E ∪ S is an admissible set. In fact, there is another way to construct new admissible sets. There may be some set S of arguments, which is not admissible and has no argument contained in F (E), such that E ∪ S is admissible. Example 1 (cont’d) For the admissible set I1 = {1, 3}, we have that I3 = {7} is admissible and conflict-free with I1 and thus I1 ∪ I3 = {1, 3, 7} is admissible. Since S1 = {5} ⊆ F (I1 ), I1 ∪ S1 = {1, 3, 5} is also admissible. Let S2 = {6}. Although S2 is not admissible and S2 ∩ F (I1 ) = ∅, we also have that I1 ∪ S2 = {1, 3, 6} is an admissible set. In the above example, S2 is not admissible and its arguments are not acceptable w.r.t. I1 . But the argument 6 can be acceptable w.r.t. I1 ∪ S2 . In words, all the arguments of S2 are acceptable w.r.t. the union of I1 and S2 . This situation leads us to propose the joint acceptability so as to distinguish from the acceptability. Definition 8 Given AF = (A, R), a non-empty admissible set E ⊂ A and a nonempty subset S ⊂ A. If S is not admissible, F (E) ∩ S = ∅ and E ∪ S is admissible, then we say that S is joint acceptable w.r.t. E (J-acceptable w.r.t. E for short). The collection of all J-acceptable sets w.r.t. E is denoted by J A (E).

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Although a J-acceptable set S w.r.t. an admissible E is not admissible, S is an admissible set in the modified argumentation AF obtained by deleting the arguments of E and the arguments attacked by E. Proposition 2 Let E be an admissible set of AF. If S is J-acceptable w.r.t. E, then S is admissible in the framework AF = AF | A\B where B = E ∪ R + (E). Conversely, each admissible set S of AF , which is not admissible in AF and has an empty intersection with F (E), is J-acceptable w.r.t. E in AF. The notion of J-acceptability induces a new principle, called the J-reinstatement principle. Definition 9 A semantics σ satisfies the J-reinstatement principle if ∀AF = (A, R) such that Eσ (AF) is non-empty, for each σ -extension E of AF, there is no subset of A\E to be J-acceptable w.r.t. E. Just as the description of complete extensions by the reinstatement criterion, we define a new class of extensions based on the J-reinstatement criterion. This class includes the preferred extensions. Definition 10 Let E be an admissible set of AF = (A, R). E is called a J-complete extension if there is no S ⊆ A\E to be J-acceptable w.r.t. E. The set of all J-complete extensions of AF will be denoted by EJ C O (AF). Example 1 (cont’d) Consider the admissible set E 1 = {1, 3, 5}. E 1 is complete. Since there is a subset {6} ⊆ (A\E 1 ) which is J-acceptable w.r.t. E 1 , we claim that E 1 is not J-complete. On the other hand, there is an admissible set E 2 = {1, 3, 6} which is J-complete but not complete. This proves that J-complete semantics does not satisfy the reinstatement principle. Note also that the admissible set E 3 = {1, 3, 6, 5} is both J-complete and complete. It is also a preferred extension. The following proposition identifies the role of J-reinstatement criterion in the theory of extension-based semantics. In words, J-complete semantics reveals the gap between complete semantics and preferred semantics. Proposition 3 A complete extension E of AF = (A, R) is preferred if and only it is J-complete and A\E has no non-empty admissible subset to be conflict-free with E. Just as complete semantics, the J-complete semantics does not satisfy the Imaximality criterion. On the other hand, it can be proved that the J-complete semantics satisfies the directionality criterion. Proposition 4 The J-complete semantics satisfies the directionality criterion. The notion of J-acceptability also enables to define the J-characteristic function for an argumentation framework AF. A

Definition 11 Let AF = (A, R). The function F J : 2 A → 22 which, given an admissible set E ⊆ A, returns the collection of the J-acceptable sets w.r.t. E, is called the J-characteristic function of AF. In words, F J (E) = J A (E).

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Note that F J (E) is usually not conflict-free, so E ∪ (∪F J (E))2 may not be an admissible set, which in contrast is always true for the characteristic function F . If we select a conflict-free collection B of J-acceptable sets w.r.t. E, then it is easy to check that E ∪ (∪(B)) is admissible. Based on these two facts, we define a selection J-function R J on each argumentation framework as follows. Definition 12 Let E be an admissible set of AF = (A, R). A selection function R J assigns to E a set R J (E), which is the union of a conflict-free collection of Jacceptable sets w.r.t. E. That is, R J (E) = ∪C where C is a conflict-free collection of elements of F J (E). Proposition 5 Let E be an admissible set of AF = (A, R). Then, E ∪ R J (E) is admissible. Example 2 Let AF = (A, R) with A = {1, 2, 3, 4, 5, 6, 7, 8} and R = {(1, 2), (2, 3), (2, 4), (3, 7), (4, 5), (5, 6), (6, 7), (7, 3), (7, 6), (7, 8)}. The directed graph is as follows:

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Consider the admissible set E 1 = {1, 4}. J1 = {3} and J2 = {6} are two Jacceptable sets w.r.t. E 1 . {J1 , J2 } ⊆ F J (E 1 ). If R J (E 1 ) = ∪{J1 }, then E 1 ∪ R J (E 1 ) = {1, 4, 3} is admissible. If R J (E 1 ) = ∪{J2 }, then E 1 ∪ R J (E 1 ) = {1, 4, 6} is admissible. If R J (E 1 ) = ∪{J1 , J2 }, then E 1 ∪ R J (E 1 ) = {1, 4, 3, 6} is admissible. In particular, if we let R J (E 1 ) = ∅, then E 1 ∪ R J (E 1 ) = {1, 4} = E 1 is obviously admissible. 

4 The Structure of Traditional Semantics The study of extension-based semantics is a central topic in argumentation. Dung first introduced the admissible, complete, preferred, and stable semantics. After that, new semantics have been introduced in order to meet some special requirements or make up the drawback of known traditional semantics for specific applications. Here, we mainly focus on the structural analysis of various traditional semantics based on the initial sets, acceptability, and J-acceptability. For each considered type of extension, we give a procedure for which we prove the correctness by the construction of the extensions, and the completeness by the description of the extensions. 2 Where

∪F J (E) = ∪{S : S ∈ F J (E)}.

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4.1 The Structure of Admissible Semantics Admissibility is a common feature for the traditional semantics. We first classify the admissible extensions into three types based on initial sets, acceptability, and J-acceptability.

4.1.1

I-Type Admissible Sets

For any admissible set E, F (E) is obviously admissible and can be expressed as E ∪ (F (E)\E). Note that each argument of F (E)\E satisfies: being accepted by E and not belonging to E. This leads to define a new acceptance notion. Definition 13 Given AF = (A, R) and S ⊆ A. An argument i is regularly accepted by S if i ∈ / S and i is defended by S. If there exists an argument regularly accepted by the admissible set S, then S is a proper subset of F (S). And any initial argument is regularly accepted by the empty set. Definition 14 Let E be an admissible set and B the collection of all initial sets contained in E. If each i ∈ E\ ∪ B is regularly accepted by some admissible set B ⊆ E, then we say that E is an I-type admissible set. It is easy to check that the union of conflict-free initial sets is a I-type admissible set. Example 1 (cont’d) For the admissible set I1 = {1, 3}, as 5 ∈ F (I1 )\I1 the argument 5 is regularly accepted by I1 . Let us consider the admissible set E 1 = {1, 3, 5} and the collection B = {I1 } of initial sets contained in E 1 . Obviously, the only argument 5 of E 1 \ ∪ B is regularly accepted by the admissible set I1 ⊆ E 1 . So, E 1 is a I-type admissible set. The grounded extension is also a I-type admissible set. More generally, we have the following proposition. Proposition 6 Any admissible set F k (∅) is a I-type admissible set. Next, we analyze the structure of I-type admissible sets from two different points of view. One is to construct a I-type admissible set starting from initial sets, and another one is to describe a given I-type admissible set starting from initial sets contained in it. From any admissible set, the characteristic function F can be iteratively applied to obtain a complete extension. We try to replace F by some specific operator related to the acceptability so as to obtain a larger admissible set. This goal can be reached by defining a selection function as follows.

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Definition 15 Given an admissible set E of AF = (A, R). A selection function R A assigns to E a random selected subset R A (E) of F (E)\E. It is easy to check that E ∪ R A (E) is certainly a I-type admissible set whenever E is a I-type admissible set. Example 2 (cont’d) Obviously, E 2 = {1} is a I-type admissible set and F (E 2 )\E 2 = {4}. Let R A (E 2 ) = {4}, then E 2 ∪ R A (E 2 ) = {1, 4} is a I-type admissible set. Now, let us start from initial sets to construct I-type admissible sets using the selection function R A . Constructing the I-Type Admissible Sets Let B be a conflict-free collection of initial sets, define E 0 = ∪B and E k+1 = E k ∪ R A (E k ) for each natural number k. Then, E 0 is the union of some conflict-free initial sets, E 1 is the set obtained by adding to E 0 some arguments regularly accepted by E 0 , and so on. So, starting from a conflict-free collection of initial sets, we can construct many different I-type admissible sets by iteratively applying the selection function R A . Theorem 1 Let B be a conflict-free collection of initial sets and E k as above, then E k is a I-type admissible set for each natural number k. Example 3 Let AF = (A, R) with A = {1, 2, 3, 4, 5, 6, 7} and R = {(1, 2), (1, 3), (2, 5), (2, 6), (2, 7), (3, 4), (4, 5), (5, 4)}. The directed graph is as follows: 3

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Obviously, E = {1} is the unique initial set of AF and F (E) = {1, 6, 7}. If R A (E) = {6}, then E ∪ R A (E) = {1, 6} is a I-type admissible set. If R A (E) = {7}, then E ∪ R A (E) = {1, 7} is a I-type admissible set. If R A (E) = {6, 7}, then  E ∪ R A (E) = {1, 6, 7} is a I-type admissible set. Describing a Given I-Type Admissible Set In order to describe a given I-type admissible set E, we need to define another operation also based on acceptability but restricted to E. Definition 16 Let S and E be two admissible sets of AF such that S ⊆ E. An operation P on S w.r.t. E assigns to S the subset of E: P(S, E) = E ∩ (F (S)\S).

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Example 2 (cont’d) For the admissible set E 2 = {1, 3}, we have that F (E 2 )\E 2 = {4, 8}. With the admissible set E 3 = {1, 3, 8}, P(E 2 , E 3 ) = {8}. With the admissible set E 4 = {1, 3, 4}, P(E 2 , E 4 ) = {4}. With the admissible set E 5 = {1, 3, 4, 8}, P(E 2 , E 5 ) = {4, 8}. In Definition 16, if S is a I-type admissible set, then S ∪ P(S, E) is certainly a I-type admissible set. And thus, we can describe a given admissible set from the initial sets contained in it using the operation P(·, ·). Let E be a I-type admissible set and B the collection of all initial sets contained in E, define E 0 = ∪B and E k+1 = E k ∪ P(E k , E) for each natural number k. Then, E 0 is the union of all initial sets contained in E, E 1 is the set obtained by adding to E 0 all the arguments which are regularly accepted by E 0 and contained in E, and so on. The following theorem indicates that any I-type admissible set E can be described as starting from the collection B of all initial sets contained in E and iteratively applying the function P(·, E). Theorem 2 Let E be a I-type admissible set and B the collection of all initial sets contained in E, then E k defined above is a I-type admissible set contained in E for each natural number k. Furthermore, there is some natural number m such that E = E m+1 = E m+2 and E m+1 = E 0 ∪ P(E 0 , E) ∪ P(E 1 , E) . . . ∪ P(E m , E). As a consequence of the above results, we can provide a way for building strongly admissible sets, first introduced in Baroni and Giacomin (2007). An argument i ∈ A is strongly defended by a set S ⊆ A iff each attacker j of i is attacked by some k ∈ S\{i} such that k is strongly defended by S\{i}. Then, S is said strongly admissible iff it strongly defends each of its arguments. An equivalent definition has been proposed by Caminada (2014). S ⊆ A is strongly admissible iff S consists of some initial arguments, or every i ∈ S is defended by some subset T ⊆ (S\{i}) which in its turn is again strongly admissible. From the definitions, it is easy to see that every strongly admissible set is a I-type admissible set and thus can be constructed starting from some initial arguments by applying the function P iteratively.

4.1.2

II-Type Admissible Sets

Except for the I-type admissible sets, there is another kind of admissible sets called II-type admissible sets. They are the admissible sets which have no argument to be regularly accepted by non-empty admissible sets contained in them. Definition 17 Let E be an admissible set of AF = (A, R) and B the collection of all initial sets contained in E. If each i ∈ E\(∪B) belongs to a J-acceptable set S ⊆ E w.r.t some admissible set D ⊆ E, then we say that E is a II-type admissible set.

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Note that the union of a conflict-free collection of initial sets is not only a I-type admissible set but also a II-type admissible set. Example 1 (cont’d) Let us consider the admissible set E 2 = {1, 3, 6} and the collection B = {I1 } of initial sets contained in E 2 , where I1 = {1, 3}. It is easy to check that {6} is a J-acceptable set w.r.t. I1 , and thus E 2 is a II-type admissible set. Note that 6 ∈ / F (I1 )\I1 and is not regularly accepted by the admissible sets I1 and E 2 which are the non-empty admissible sets contained in E 2 . In contrast, the admissible set E 3 = {1, 3, 5, 6}, which is a J-complete extension, is not a II-type admissible set, as 5 ∈ F (I1 )\I1 . Generally speaking, the membership of each argument of a II-type admissible set E does not only depend on other arguments of E but also on itself. That is exactly what the notion of J-acceptability states. The following proposition gives a description for a II-type admissible set having no regularly accepted arguments except initial arguments. Proposition 7 Let E be an admissible set of AF and B the collection of all initial sets contained in E. Then E\(∪B) has no regularly accepted arguments if and only if for each admissible set D ⊂ E containing ∪B as a subset, E\D is J-acceptable w.r.t. D. Next, we analyze the structure of II-type admissible sets from two different points of view. One is to construct a II-type admissible set starting from initial sets, and another is to describe a given II-type admissible set starting from the initial sets contained in it. Just as I-type admissible sets, when constructing II-type admissible sets we need to use a selection function related to the J-acceptability. Interestingly, the selection function R J in Definition 12 is exactly the one we need. Furthermore, Proposition 5 is also true if we substitute “admissible” by “II-type admissible.” This result gives us the theoretical support for constructing II-type admissible sets by applying the selection function R J . Constructing the II-Type Admissible Sets Let B be a conflict-free collection of initial sets, define E 0 = ∪B and E k+1 = E k ∪ R J (E k ) for each natural number k. Then, E 0 is the union of some conflict-free initial sets, E 1 is the set obtained by adding to E 0 some conflict-free J-acceptable sets w.r.t. E 0 , and so on. So, starting from a conflict-free collection of initial sets we construct many different II-type admissible sets by iteratively applying the selection function R J . Theorem 3 Let B a conflict-free collection of initial sets and E k as above, then E k is a II-type admissible set for each natural number k. Example 3 (cont’d) For the unique initial set E = {1} of AF, we have F J (E) = {J1 , J2 } with J1 = {4} and J2 = {5}. If R J (E) = J1 , then E ∪ R J (E) = {1, 4} is a IItype admissible set. If R J (E) = J2 , then E ∪ R J (E) = {1, 5} is a II-type admissible set. Since J1 and J2 are in conflict, R J (E) = J1 ∪ J2 .

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Describing a Given II-Type Admissible Set Describing a given I-type admissible set mainly requires iteratively applying the function P(·, ·). This idea can be further extended to the case of J-acceptability so as to describe a given II-type admissible set. Definition 18 Let S and E be two admissible sets of AF such that S ⊆ E. An operation PJ on S w.r.t. E assigns to S the subset of E: PJ (S, E) = ∪{T : T is J-acceptable w.r.t. S and T ⊂ E}. Example 2 (cont’d) For the admissible set E 1 = {1, 4}, we have that {J1 , J2 } ⊆ F J (E 1 ) where J1 = {3} and J2 = {6}. With the admissible set E 6 = {1, 4, 3}, PJ (E 1 , E 6 ) = ∪{J1 } = {3}. With the admissible set E 7 = {1, 4, 6}, PJ (E 1 , E 7 ) = ∪{J2 } = {6}. With the admissible set E 8 = {1, 4, 3, 6}, PJ (E 1 , E 8 ) = ∪{J1 , J2 } = {3, 6}. Note that S ∪ PJ (S, E) is a II-type admissible set whenever S is a II-type admissible set. And so, we can describe any II-type admissible set E starting from the initial sets contained in it by iteratively applying the function PJ (·, ·). Let E be a II-type admissible set and B the collection of all initial sets contained in E, define E 0 = ∪B and E k+1 = E k ∪ PJ (E k , E) for each natural number k. Then, E 0 is the union of some conflict-free initial sets, E 1 is the set obtained by adding to E 0 all the J-acceptable sets w.r.t. E 0 which are contained in E, and so on. So, any II-type admissible set E can be described starting from the collection B of all initial sets contained in E and iteratively applying the function PJ (·, E). Theorem 4 Let E be a II-type admissible set and B the collection of all initial sets contained in E, then E k defined above is a II-type admissible set contained in E for each natural number k. Furthermore, there is some natural number m such that E = E m+1 = E m+2 and E m+1 = E 0 ∪ PJ (E 0 , E) ∪ PJ (E 1 , E) . . . ∪ PJ (E m , E).

4.1.3

The Mixed-Type Admissible Sets

Roughly speaking, the I-type and II-type admissible extensions have onefold structure. So, we usually call them simple admissible extensions. The other admissible extensions may be given the name of mixed-type admissible sets as follows. Definition 19 An admissible set E is mixed-type if it is neither I-type no II-type. Example 1 (cont’d) Let us consider the admissible set E 3 = {1, 3, 5, 6} and the collection B = {I1 } of initial sets contained in E 3 . Obviously, the argument 5 of E 3 \ ∪ B is regularly accepted by the admissible set I1 ⊆ E 3 . So, E 3 is not a II-type admissible set. We also note that the argument 6 is not regularly accepted by any nonempty admissible set S contained in E 3 , where S = {1, 3} or S = {1, 3, 5}. It follows that E 3 is not a I-type admissible set. To sum up, E 3 is a mixed-type admissible set.

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Constructing an Admissible Extension of Mixed-Type Starting from a conflict-free collection B of initial sets, many mixed-type admissible sets can be constructed by iteratively applying the functions R A and R J . Let B be a conflict-free collection of initial sets, define S0 = ∪B, T0 = S0 ∪ R J (S0 ). For each natural number k, let Sk+1 = Tk ∪ R A (Tk ) and Tk+1 = Sk+1 ∪ R J (Sk+1 ) until Sm+1 = Tm or Tm+1 = Sm+1 for some m (when one of these equations holds, the constructing process ends). Note that both Sk and Tk are admissible. Example 3 (cont’d) For the unique initial set E = {1} of AF, we have that S0 = {1}. Note that F J (S0 ) = {J1 , J2 } with J1 = {4} and J2 = {5}. If we let R J (S0 ) = J1 , then T0 = {1, 4} is a II-type admissible set. Furthermore, F (T0 ) = {6, 7}. If we let R A (T0 ) = {6}, then S1 = {1, 4, 6} is a mixed-type admissible set. Again, we have F (S1 ) = {J2 }. If we let R J (S1 ) = J2 , then T1 = {1, 4, 6, 5} is a mixed-type admissible set. Describing a Given Admissible Extension of Mixed-Type We can describe an admissible set of mixed-type starting from the initial sets contained in it by iteratively applying the functions P(·, ·) and PJ (·, ·). Let E be a mixed-type admissible set and B the collection of all initial sets contained in E, define S0 = ∪B and T0 = S0 ∪ PJ (S0 , E). For each natural number k, let Sk+1 = Tk ∪ P(Tk , E) and Tk+1 = Sk+1 ∪ PJ (Sk+1 , E) until E = Sm or Tm for some natural number m. That is, we have E = S0 ∪ PJ (S0 , E) ∪ P(T1 , E) . . . ∪ PJ (Sm−1 , E) ∪ P(Tm−1 , E) or E = S0 ∪ PJ (S0 , E) ∪ P(T0 , E) . . . ∪ P(Tm−1 , E) ∪ PJ (Sm , E).

4.2 The Structure of Complete and Preferred Semantics Since both the complete extensions and the preferred extensions are all admissible sets, they also can be classified into different types just like we have done for admissible sets. Meanwhile, we have similar results for constructing them and describing them from conflict-free initial sets by iteratively applying the related functions R A , R J , P, and PJ . Definition 20 Given a complete (or preferred) extension E of AF = (A, R). E is said to be of I-type (resp. II-type, mixed-type) if it is a I-type (resp. II-type, mixedtype) admissible set. Constructing a I-Type Complete (or Preferred) Extension Let B be a conflict-free collection of initial sets, define E 0 = ∪B and E k+1 = F (E k ) for each natural number k. Then, E 0 is the union of conflict-free initial sets and thus a I-type admissible set. E 1 is the set obtained by adding to E 0 all arguments regularly accepted by E 0 and thus a I-type admissible set, and so on.

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By definition of the characteristic function F , E k ⊆ F (E k ) = E k+1 for each natural number k. If there is no m such that E m = E m+1 , then the cardinality of E k will be strictly increasing. This contradicts with the fact that A is a finite set. Suppose that E m = E m+1 for some natural number m, that is, E m = F (E m ), then E m is a fixed point of the characteristic function F . This indicates that E m is a complete extension. So we have proved that there is some natural number m such that E m = E m+1 which is exactly the I-type complete extension we want. Furthermore, if F J (E m ) = ∅ and there is no initial set in A\E m conflicting with E m , then E m is a I-type preferred extension according to Proposition 3. Example 1 (cont’d) Considering the conflict-free collection B = {I1 } where I1 = {1, 3} is an initial set. Let E 0 = ∪B = {1, 3} and E 1 = F (E 0 ) = {1, 3, 5}, then E 2 = F (E 1 ) = E 1 is a I-type complete extension. Describing a I-Type Complete (or Preferred) Extension Let E be a I-type complete (preferred) extension and B the collection of all initial sets contained in E. As E is admissible, E can be described by iteratively applying the function P, as for any I-type admissible set. Applying Theorem 2, we have some natural number m such that E = E m+1 = E m+2 where E 0 = ∪B and E k+1 = E k ∪ P(E k , E) for each natural number k. By definition, P(E k , E) = E ∩ (F (E k )\E k ). Since E is complete, F (E k ) ⊆ F (E) = E, so P(E k , E) = F (E k )\E k , and thus E k+1 = E k ∪ P(E k , E) = F (E k ). As a consequence, we can also provide a description of E using the characteristic function F . E = E m+1 = E m+2 , where E 0 = ∪B and E k+1 = F (E k ) for each natural number k. Describing a II-Type Complete (or Preferred) Extension Note that we cannot always obtain a II-type complete (resp. preferred) extension starting from a conflict-free collection of initial sets. In fact, there are many argumentation frameworks which have no II-type complete (resp. preferred) extension. Example 1 (cont’d) There are six II-type admissible sets: E 1 = {1, 3}, E 2 = {2, 4}, E 3 = {7}, E 4 = {1, 3, 6}, E 5 = {1, 3, 7}, E 6 = {2, 4, 7}. But not all of them are complete extensions. In fact, F (E 1 ) = {1, 3, 5}, F (E 2 ) = {2, 4, 8}, F (E 3 ) = {7, 8}, F (E 4 ) = {1, 3, 6, 5}, F (E 5 ) = {1, 3, 7, 5, 8}, F (E 6 ) = {2, 4, 7, 8}. Indeed, F (E 4 ), F (E 5 ), and F (E 6 ) are the preferred extensions, but they are not II-type preferred. Next, we only talk about the description of II-type complete (resp. preferred) extensions. Let E be a II-type complete (preferred) extension and B the collection of all initial sets contained in E, define E 0 = ∪B and E k+1 = E k ∪ PJ (E k , E) for each natural number k. Then, E 0 ⊆ E is the union of initial sets contained in E and thus a II-type

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admissible set. E 1 ⊆ E is the set obtained by adding to E 0 all the J-acceptable sets w.r.t. E 0 which are contained in E and thus a II-type admissible set, and so on. It holds that there is some natural number m such that E = E m = E m+1 . The proof is as follows: By the definition of PJ (E k , E), E k ⊆ E k+1 = E k ∪ PJ (E k , E) ⊆ E for each k. If there is no m such that E m = E m+1 , then the cardinality of E k will be strictly increasing. This contradicts with the fact that E is a finite set. Since E is a II-type complete (preferred) extension, it is certainly a II-type admissible extension. By Theorem 4, there is some natural number r such that E = Er +1 = Er +2 . Next, we prove that E m = Er +1 and thus E = E m = E m+1 . By the definition of PJ (E k , E), E m ⊆ Er +1 or Er +1 ⊆ E m . Without loss of generality, we suppose that E m ⊆ Er +1 . If E m = Er +1 , then E m ⊂ Er +1 . Note that E 0 ⊆ E 1 ⊆ . . . ⊆ E m = E m+1 = E m+2 = . . ., so we have that Er +1 ⊂ Er +1 , a contradiction. Just as II-type complete (resp. preferred) extensions, the mixed complete (resp. preferred) extensions may not exist. So we only talk about the description of mixedtype complete (resp. preferred) extensions. Example 4 Let AF = (A, R) with A = {1, 2, 3, 4, 5} and R = {(1, 2), (2, 3), (3, 4), (4, 1), (4, 2), (2, 5)}. The directed graph is as follows. 1

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It is easy to check that I = {1, 3} is the only initial set and E = {1, 3, 5} is the unique preferred extension, but E is not mixed-type preferred. Describing a Complete (or Preferred) Mixed-Type Extension Let E be a mixed-type complete (preferred) extension and B the collection of all initial sets contained in E, define S0 = ∪B and T0 = S0 ∪ PJ (S0 , E). For each natural number k, let Sk+1 = F (Tk , E) = Tk ∪ P(Tk , E) and Tk+1 = Sk+1 ∪ PJ (Sk+1 , E). Then, S0 is the union of initial sets contained in E, and T0 is the union of S0 with all J-acceptable sets w.r.t. S0 which are contained in E. S1 is the union of T0 with the set of arguments which are regularly defended by T0 and contained in E, T1 is the union of S1 with all J-acceptable sets w.r.t. S1 which are contained in E, and so on. It holds that there is some m such that E = Sm or Tm . That is, we have E = S0 ∪ PJ (S0 , E) ∪ P(T1 , E) . . . ∪ PJ (Sm−1 , E) ∪ P(Tm−1 , E) or E = S0 ∪ PJ (S0 , E) ∪ P(T0 , E) . . . ∪ P(Tm−1 , E) ∪ PJ (Sm , E).

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5 Conclusion and Future Works In this paper, a special type of acceptability called J-acceptability is studied. The reinstatement criterion is extended to the J-reinstatement criterion, and the characteristic function is extended to the J-characteristic function. Based on the J-reinstatement criterion, we introduce the J-complete semantics which reveals the gap between complete semantics and preferred semantics. It is claimed that an admissible set is preferred if and only if it is complete, J-complete and A\E has no non-empty admissible subset to be conflict-free with E. With acceptability and J-acceptability, we figure out the structure of admissible sets, complete extensions, and preferred extensions. That is, any admissible (resp. complete, preferred) extension can be obtained starting from a conflict-free collection of initial sets by iteratively applying some operators related to the acceptability and J-acceptability. This work sheds some further light on the nature of extensionbased semantics and suggests that for any argumentation framework, the problem of determining extensions will be solved if we can find out all initial sets. In Xu and Cayrol (2018), the properties of initial sets were studied and the initial sets semantics was introduced. It was shown that complete and preferred extensions can be generated from initial sets by the characteristic function and adding J-acceptable sets. In the current paper, the focus is rather put on the J-complete semantics and the role of J-acceptability for classifying admissible sets in three types. J-acceptability combined with acceptability has been proved useful for giving a clear description for the extensions of the standard semantics. We plan to study the role of J-acceptability in other non-standard semantics, and more particularly for the structure analysis of their extensions. For instance, SCC recursiveness Baroni et al. (2005) is a general schema that generalizes admissibility-based semantics. We plan to combine our J-complete semantics with the SCC recursive schema in order to overcome the limitations of preferred semantics. Another direction for further research concerns the use of J-acceptability in dynamic argumentation frameworks. In literature, several works have proposed efficient ways for handling dynamics, such as Liao et al. (2011) which introduces the division-based method, and Xu and Cayrol (2015) where a matrix approach allows for a decomposition of standard extensions, using unattacked sets of arguments. We are going to investigate the role of initial sets and J-acceptability in the construction of the extensions of an updated argumentation framework. Recently, Liao and van der Torre (2018) defined equivalences between argumentation frameworks, based on several representations of extensions. In short, an extension can be represented by a subset of it from which the extension can be obtained by iteratively applying the characteristic function. The representation of an extension has some relation with initial sets, as initial sets can be viewed as building blocks for defining semantics. In Liao and van der Torre (2018), the purpose is rather to differentiate argumentation frameworks having the same extensions. The representation

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equivalences between argumentation frameworks could be enriched by introducing the concept of J-acceptability. Finally, we will study the computational complexity of J-complete semantics and related topics, following previous work on the computational complexity of the extension-based semantics such as Dunne and Bench-Capon (2002) which considers the complexity of decision problems in preferred and stable semantics, and Dimopoulos et al. (2002) which gives the upper bounds for decision problems in assumptionbased argumentation.

Appendix Proof of Proposition 1 Suppose that B = {Bi : 1 ≤ i ≤ n}. As ∪B ⊆ E, ∪B is conflict-free. Therefore, B is a conflict-free set. Let c ∈ A\ ∪ B and a ∈ ∪B such that (c, a) ∈ R, then there is some 1 ≤ i ≤ n such that a ∈ Bi . Since Bi is initial and so admissible, there is some b ∈ Bi such that (b, c) ∈ R. Certainly, we have b ∈ ∪B. So, we prove that ∪B is admissible. The second point is obvious.  Proof of Proposition 2 Since S is J-acceptable wr t E, E ∪ S is admissible. This implies that S is conflict-free in AF and thus also conflict-free in AF . Suppose that i ∈ S and is attacked by j in AF , then i is attacked by j in AF. Because E ∪ S is admissible, there is some k ∈ (E ∪ S) such that k attacks j. On the / E and thus we have k ∈ S. Therefore, other hand, j ∈ / (E ∪ R + (E)) means that k ∈ S is defended by itself in AF , i.e., S is admissible in AF . Conversely, we only need to prove that E ∪ S is admissible in AF. First, S is admissible and implies that it is conflict-free in AF , and thus S is conflict-free in AF. Also note that S is conflict-free with E, so we have E ∪ S is conflict-free in AF. Second, let i ∈ (E ∪ S) being attacked by j ∈ (A\(E ∪ S)). If i ∈ E, then obviously there is some k ∈ E which attacks j. Otherwise, i ∈ S. When j ∈ F (E), then there is some k ∈ E which attacks j. When j ∈ / F (E), then j ∈ (A\(E ∪ R + (E))) and there is some k ∈ S which attacks j according to the fact that S is admissible in AF . By definitions, we conclude that E ∪ S is admissible in AF, and S is J-acceptable wr t E in AF.  Proof of Proposition 3 It is an immediate consequence of Theorem 5 in Xu and Cayrol (2018).  Proof of Proposition 4 Let U be an unattacked subset of AF = (A, R). • (⇐) Let E be a J-complete extension of AF. If E ∩ U is not a J-complete extension of AF = AF |U , then there is some non-empty subset S ⊆ U which is Jacceptable w.r.t E ∩ U in AF . By definition, S is not admissible, F (E ∩ U ) ∩ S = ∅ and S ∪ (E ∩ U ) is admissible in AF . Next, we prove that S is J-acceptable w.r.t E in AF and thus E is not J-complete in AF. This contradicts with the assumption.

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Since S is not admissible in AF , there is some i ∈ S and j ∈ (U \S) such that ( j, i) ∈ R |U but no argument of S attacks j. This means that S is not admissible in AF. Furthermore, U is an unattacked subset in AF and implies that F (E\(E ∩ U )) ∩ U = ∅ and thus F (E\(E ∩ U )) ∩ S = ∅. Based on the fact F (E ∩ U ) ∩ S = ∅, we have F (E) ∩ S = ∅. Finally, F (E) ∩ S = ∅ indicates that no argument of E attacks the argument of S. And S is J-acceptable w.r.t E ∩ U means that S ∪ (E ∩ U ) is conflict-free in AF . That is, no argument of S attacks the argument of E ∩ U . If there is some i ∈ S attacking an argument j ∈ (E\(E ∩ U )), then there must be some k ∈ (E\(E ∩ U )) attacking j according to the fact that E is admissible in AF. Therefore, E ∪ S is conflict-free. Let i ∈ (E ∪ S) be attacked by an argument j ∈ (A\(E ∪ S)). If i ∈ E, then it is obviously defended by E. Otherwise, i ∈ S and thus j ∈ U because of the facts U is an unattacked subset and S ⊆ U . Since (E ∩ U ) ∪ S is admissible in AF , j must be attacked by some argument of (E ∩ U ) ∪ S. Therefore, E ∪ S is admissible in AF. • (⇒) Conversely, let S be a J-complete extension of AF = AF |U . We have to prove that there is some J-complete extension E of AF such that S = E ∩ U . If S is a J-complete extension of AF, then E = S satisfies the requirement. Otherwise, S is not J-complete in AF and there is some J-complete extension E of AF such that S ⊂ E. Suppose that (E\S) ∩ U = ∅, then S ∪ ((E\S) ∩ U ) = E ∩ U is J-complete in AF by the first part of the proof. This is in contradiction with S being J-complete in AF . Therefore, we have (E\S) ∩ U = ∅ and thus S = E ∩ U.  Proof of Proposition 5 Let C = {S1 , S1 , . . . , Sk }. Since Sr is J-acceptable w.r.t E for each 1 ≤ r ≤ k, E ∪ Sr is admissible. Note that {S1 , S1 , . . . , Sk } is a conflict-free collection, so E ∪ R J (E) = ∪{E ∪ Sr : 1 ≤ r ≤ k} is conflict-free. Furthermore, it  is easy to check that E ∪ R J (E) defends itself and thus is admissible. Proof of Proposition 6 Let B = {{i} : i is an initial argument}, then F (∅) = ∪B is obviously a I-type admissible set. Suppose F t (∅) is a I-type admissible set for each t < r , we next prove that F r (∅) is a I-type admissible set. And thus, the result is true for each natural number k by mathematical induction. Let i ∈ (F r (∅)\F (∅)), then there is some t < r such that i ∈ F t+1 (∅)\F t (∅). So, i is regularly accepted  by the admissible set F t (∅) contained in F r (∅). Proof of Theorem 1 E 0 is obviously a I-type admissible set. Suppose Er is I-type admissible for each r < t, we prove that E t is a I-type admissible set. Let i ∈ (E t \ ∪ / Er and i ∈ R A (Er ). B), then there is some r < t such that i ∈ (Er +1 \Er ). That is, i ∈ So, i is regularly accepted by the admissible set Er . By mathematical induction, we claim that E k is I-type admissible for each natural number k. By definition of E k , it is obvious that E k+1 = E k ∪ R A (E k ) = (E k−1 ∪ R A (E k−1 )) ∪ R A (E k ) = E k−1 ∪  R A (E k−1 ) ∪ R A (E k ) = . . . = E 0 ∪ R A (E 0 ) ∪ R A (E 1 ) ∪ . . . ∪ R A (E k ). Proof of Theorem 2 E 0 is obviously a I-type admissible set. Suppose Er is Itype admissible for each r < t, we prove that E t is a I-type admissible set.

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Let i ∈ (E t \ ∪ B), then there is some r < t such that i ∈ (Er +1 \Er ). That is, i∈ / Er and i ∈ P(Er , E). So, i is regularly accepted by the admissible set Er . By mathematical induction, we claim that E k is I-type admissible contained in E for each natural number k. Obviously, there is some natural number m such that E m  E m+1 = E m+2 and E m+1 = E m ∪ P(E m , E) = (E m−1 ∪ P(E m−1 ), E) ∪ P(E m , E) = E m−1 ∪ P(E m−1 , E) ∪ P(E m , E) = . . . = E 0 ∪ P(E 0 , E) ∪  P(E 1 , E) ∪ . . . ∪ P(E m , E). Proof of Proposition 7 • (⇒) Let D be an admissible set such that ∪B ⊆ D ⊂ E, then E\D is not admissible. Otherwise, there is some initial set I in E\D. This contradicts with ∪B ⊆ D. Since ∪B ⊆ D, each i ∈ (E\D) is not regularly accepted by D admissible. And thus, we have F (D) ∩ (E\D) = ∅. Therefore, E = D ∪ (E\D) is admissible implies that E\D is J-acceptable w.r.t D. • (⇐) Suppose some i ∈ (E\ ∪ B) is regularly accepted by an admissible set T ⊆ E, then i is regularly accepted by the admissible set D = T ∪ (∪B) ⊆ E. Note that E\D is J-acceptable w.r.t D, so F (D) ∩ (E\D) = ∅. That is i is not accepted by D and thus by T , a contradiction.  Proof of Theorem 3 E 0 is obviously a II-type admissible set. For k ≥ 1, let i ∈ (E k \(∪B). Then, there is some r < k such that i ∈ (Er \Er −1 ). That is, i ∈ R J (Er ). So, i belongs to some J-acceptable set S ⊆ R J (Er ) w.r.t Er . By definition, we claim that E k is a II-type admissible set. By definition of E k , it is obvious that E k+1 = E k ∪ R J (E k ) = (E k−1 ∪ R J (E k−1 )) ∪ R J (E k ) = E k−1 ∪ R J (E k−1 ) ∪ R J (E k ) = . . . =  E 0 ∪ R J (E 0 ) ∪ R J (E 1 ) ∪ . . . ∪ R J (E k ). Proof of Theorem 4 E 0 is obviously a II-type admissible set contained in E. For k ≥ 1, let i ∈ (E k \(∪B). Then, there is some r < k such that i ∈ (Er \Er −1 ). That is, i ∈ PJ (Er ). So, i belongs to some J-acceptable set S ⊆ PJ (Er ) w.r.t Er . By definition, we claim that E k is a II-type admissible set contained in E. Obviously, there is some natural number m such that E m  E m+1 = E m+2 and E m+1 = E m ∪ PJ (E m , E) = (E m−1 ∪ PJ (E m−1 ), E) ∪ PJ (E m , E) = E m−1 ∪ PJ (E m−1 , E)  ∪ PJ (E m , E) = . . . = E 0 ∪ PJ (E 0 , E) ∪ PJ (E 1 , E) ∪ . . . ∪ PJ (E m , E).

References Amgoud, L., Cayrol, C., Lagasquie-Schiex, M.C.: On the bipolarity in argumentation frameworks. In: Delgrande, J., Schaub, T. (eds.) Proceedings of the 10th NMR Workshop (Non Monotonic Reasoning), Uncertainty Framework subworkshop, pp. 1–9. Whistler, BC, Canada (2004) Baroni, P., Boella, G., Cerutti, F., Giacomin, M., van der Torre, L., Villata, S.: On the input/output behavior of argumentation frameworks. Artif. Intell. 217, 144–197 (2014) Baroni, P., Dunne, P.E., Giacomin, M.: On the resolution-based family of abstract argumentation semantics and its grounded instance. Artif. Intell. 175(3–4), 791–813 (2011) Baroni, P., Gabbay, D., Giacomin, M., van der Torre, L.: Handbook of Formal Argumentation. College Publications (2018) Baroni, P., Giacomin, M.: On principle-based evaluation of extension-based argumentation semantics. Artif. Intell. 171(10–15), 675–700 (2007)

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Baroni, P., Giacomin, M.: Skepticism relations for comparing argumentation semantics. Int. J. Approx. Reason. 50(6), 854–866 (2009) Baroni, P., Giacomin, M., Guida, G.: SCC-recursiveness: a general schema for argumentation semantics. Artif. Intell. 168(1–2), 162–210 (2005) Baroni, P., Giacomin, M., Liao, B.: On topology-related properties of abstract argumentation semantics. A correction and extension to dynamics of argumentation systems: a division-based method. Artif. Intell. 212, 104–115 (2014) Caminada, M.: Strong admissibility revisited. In: Proceedings of International Conference on Computational Models of Argument (COMMA 2014), pp. 197–208. IOS Press (2014) Cayrol, C., de Saint-Cyr, F.D., Lagasquie-Schiex, M.: Change in abstract argumentation frameworks: adding an argument. J. Artif. Intell. Res. 38(1), 49–84 (2010) Dimopoulos, Y., Nebel, B., Toni, F.: On the computational complexity of assumption-based argumentation for default reasoning. Artif. Intell. 141(1), 57–78 (2002) Dung, P.: On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games. Artif. Intell. 77, 321–357 (1995) Dunne, P.: Computational properties of argument systems satisfying graph-theoretic constrains. Artif. Intell. 171, 701–729 (2007) Dunne, P.E., Bench-Capon, T.J.: Coherence in finite argument system. Artif. Intell. 141(1–2), 187– 203 (2002) Liao, B., Li, J., Koons, R.: Dynamics of argumentation systems: a division-based method. Artif. Intell. 175, 1790–1814 (2011) Liao, B., van der Torre, L.: Representation equivalences among argumentation frameworks. In: Proceedings of Computational Models of Argument, COMMA 2018, Warsaw, Poland, 12–14 September 2018, pp. 21–28 (2018). https://doi.org/10.3233/978-1-61499-906-5-21 Martinez, D.C., Garcia, A.J., Simari, G.R.: On defense strength of blocking defeaters in admissible sets. In: Proceedings of International Conference on Knowledge Science, Engineering and Management (KSEM). LNAI, vol. 4798, pp. 140–152 (2007) Wyner, A.Z., Bench-Capon, T.J.: Towards an extensible argumentation system. In: Proceedings of European Conference on Symbolic and Quantitative Approaches to Reasoning with Uncertainty (ECSQARU)–LNAI 4724, pp. 283–294 (2007) Xu, Y., Cayrol, C.: The matrix approach for abstract argumentation framework. In: Black, E., Modgil, S., Oren, N. (eds.) Theory and Applications of Formal Argumentation, Buenos Aires, Argentine, 25–26 July 2015. LNAI, vol. 9524, pp. 243–259. Springer, Cham (2015). http://www. springerlink.com Xu, Y., Cayrol, C.: Initial sets in abstract argumentation frameworks. J. Appl. Non Class. Log. 28(2–3), 260–279 (2018). https://doi.org/10.1080/11663081.2018.1457252

A Dynamic Approach for Combining Abstract Argumentation Semantics Jérémie Dauphin, Marcos Cramer and Leendert van der Torre

Abstract Abstract argumentation semantics provide a direct relation from an argumentation framework to corresponding sets of acceptable arguments, or equivalently to labeling functions. Instead, we study stepwise update relations on argumentation frameworks whose fixpoints represent the labeling functions on the arguments. We make use of this dynamic approach in order to study novel ways of combining abstract argumentation semantics. In particular, we introduce the notion of a merge of two argumentation semantics, which is defined in such a way that the merge of the preferred and the grounded semantics is the complete semantics. Finally, we consider how to define new semantics using the merge operator, in particular, how meaningfully combine features of naive-based and complete-based semantics. Keywords Artificial intelligence · Knowledge representation and reasoning · Formal argumentation · Computational argumentation · Symbolic reasoning · Reasoning dynamics

1 Introduction Following the methodology in non-monotonic logic, logic programming, and belief revision, formal argumentation theory defines a diversity of semantics. This diversity has the advantage that a user can select the semantics best fitting her application, but The work of Jérémie Dauphin and Leendert van der Torre was supported by the H2020 Marie Skłodowska-Curie grant number 690974 for the project MIREL. J. Dauphin (B) · L. van der Torre University of Luxembourg, Luxembourg, Luxembourg e-mail: [email protected] L. van der Torre e-mail: [email protected] M. Cramer TU Dresden, Dresden, Germany e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2019 B. Liao et al. (eds.), Dynamics, Uncertainty and Reasoning, Logic in Asia: Studia Logica Library, https://doi.org/10.1007/978-981-13-7791-4_2

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it leads also to various practical challenges. First of all, how to choose among the considerable number of semantics existing in the argumentation literature for a particular application? The behavior of semantics on examples can already be insightful, and Baroni and Giacomin (2007) address the need for more systematic comparison of semantics based on a set of principles. However, what to do when no currently considered semantics is perfect? May there be a better semantics that has not been discovered yet? How to guide the search for new and hopefully better argumentation semantics? In this paper, we propose a new approach: the combination of abstract argumentation semantics. We focus on the following three research questions: 1. How to combine two abstract semantics to yield the third semantics? 2. In particular, how to obtain the complete semantics by combining the preferred and grounded semantics? 3. Can we meaningfully combine features of naive-based and complete-based semantics? Concerning our first research question, there are various ways in which abstract argumentation semantics can be combined. For example, in multi-sorted argumentation (Rienstra et al. 2011; Arisaka et al. 2018; Giacomin 2017), one part of the framework can be evaluated according to, for example, grounded semantics, whereas another part of the framework is evaluated according to the preferred semantics. Another approach manipulates directly the sets of extensions. For example, the grounded and preferred can be combined by simply returning both the grounded and preferred extensions. Both of these approaches have drawbacks. For multi-sorted argumentation, we need to specify explicitly which semantics must be applied to which part of the framework. For the direct combination method, the approach seems too coarse-grained and the number of ways to combine semantics seems relatively limited. We therefore introduce a dynamic approach in this paper, which is based on the labeling approach to argumentation semantics, in which the three labels in, out, and undec are used. In our dynamic approach, we define stepwise versions of standard semantics based on epistemic labelings, which associate with each argument a nonempty set of labels from {in, out, undec}. Intuitively, the set represents uncertainty about the label. We start with labeling each argument of the framework with the set {in, out, undec}. This represents that we do not know the labeling yet. Then in each step we refine the labels by removing some of the labels. Finally, we end up with a single label for each argument, and thus with a standard labeling. To represent the possibility of multiple extensions, the steps are not deterministic. The steps are represented by an abstract update relation, which mathematically is simply a binary relation among epistemic labelings. Note that there are many distinct update relations representing the same standard semantics, and it is this additional expressive power that we will use in our first approach to combining abstract argumentation semantics. Concerning our second research question, it is well known that the grounded semantics outputs the smallest complete extension, and that the preferred semantics outputs maximal complete extensions (Dung 1995). This suggests that there is potential to recover all complete extensions using a mixture of the grounded and preferred

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semantics. Note that there may be complete extensions that are neither minimal nor maximal, and that it is therefore nontrivial to recover all the complete extensions using the grounded and the preferred semantics, without loosing any complete extensions. Though the derivation of the complete semantics from the grounded and preferred semantics does not serve any practical purpose, it serves to show that our dynamic semantic framework has sufficient expressive power to combine abstract semantics. We therefore pursue this second question to showcase our combination operation. Note that we do not claim it to be possible to retrieve the full set of complete extensions from preferred and grounded extensions alone, as they do not provide sufficient information, even when represented as labelings. Indeed, some argumentation frameworks have the same preferred and the same grounded labelings, yet differ in their complete labelings. We present two such frameworks in Sect. 2. Hence, the approaches we propose still take the structure of the framework into account when combining the different semantics. Concerning the third research question, note that recently naive-based semantics like stage semantics (Verheij 1996) and CF2 semantics (Baroni et al. 2005) have received some attention, for example, in the work of Gaggl and Dvoˇrák (2016), who define a new semantics (stage2) that combines features of stage and CF2 semantics, and in the work of Cramer and Guillaume (2018), who performed an empirical study that showed that these naive-based semantics are better predictors of human argument acceptance than complete-based semantics like the grounded and preferred semantics. For argumentation frameworks without odd cycles, the stage semantics fully agrees with the preferred semantics. One difference between the preferred semantics and the stage semantics is that the stage semantics generally provides a way to select accepted arguments even when odd cycles are around, whereas the preferred semantics tends to mark as undecided all arguments that are in an odd cycle or attacked by an odd cycle. One difference between the preferred semantics and the complete semantics is that the complete semantics allows one to locally not make choices for some unattacked even cycles while making choices for other unattacked even cycles, whereas in the preferred semantics one has to make choices for all unattacked even cycles. This motivates the following research question: Is there a sensible semantics that allows one to locally make choices for some unattacked odd or even cycles while not making choices for other unattacked odd or even cycles? The layout of this paper is as follows. After providing some preliminaries about argumentation semantics in Sect. 2, we introduce our dynamic approach based on epistemic labelings and update relations in Sect. 3. Section 4 addresses the second research question by showing how grounded and preferred semantics can be combined to obtain the complete semantics using an algorithmic approach to updates. As this approach is dependent on the choice of algorithm on which the update relation is based, we proceed in Sect. 5 to defining the merge of two argumentation semantics, a modification of our first approach that is applicable to any pair of semantics independently of any algorithmic considerations. In Sect. 5.1, we motivate the definition of the merge operator by considering how to use it to get the complete semantics from the grounded and preferred semantics without adding any algorithmic

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information. In Sect. 5.2, we show how the merge operator can be used to give rise to novel argumentation semantics, and, in particular, how it can be used to meaningfully combine features of naive-based and complete-based semantics. We conclude with an overview of further work in Sect. 6.

2 Preliminaries An argumentation framework (AF) is a directed graph A, R, where A is called the set of arguments, and R is called the attack relation. In this work, we do not consider enriched AFs such as bipolar AFs and weighted AFs. Standard argumentation semantics come in two variants. Extension-based semantics associates with each AF a set of extensions (sets of the arguments). Labeling-based semantics attribute to each argument the label in, out or undecided. The two approaches are inter-definable, in the sense that an argument is labeled in when it is in the extension, it is labeled out when it is not in the extension and there is an argument in the extension attacking it, and it is undecided otherwise. Our dynamic approach uses an epistemic labeling, which associates with each argument a non-empty set of labels. Intuitively, the set represents uncertainty about the label. We assume familiarity with 3-labeling semantics of argumentation frameworks as defined in (Baroni et al. 2011). Note that we will make use of the multi-labeling approach, where a set of labels is assigned to each argument. This set represents the possible labels for a given argument. The standard approach corresponds to the case where arguments are given singleton sets as labels. We define L = {in, out, undec} to be the set of possible labels. Definition 1 Let F = A, R be an AF. We say that any function L from A to L is a 3-labeling of F. The 3-labeling approach makes use of the notions of legal labels. Definition 2 Let F = A, R be an AF, a ∈ A an argument and L a 3-labeling of F. We say that a is: • legally in with respect to L iff L(a) = in and for all b ∈ A such that (b, a) ∈ R, L(b) = out; • legally out with respect to L iff L(a) = out and for some b ∈ A such that (b, a) ∈ R, L(b) = in; • legally undecided with respect to L iff L(a) = undec and for all b ∈ A such that (b, a) ∈ R, L(b) = in and for at least one such b, L(b) = undec. If all arguments in A are legally labeled with respect to L, then we say that L is a complete labeling of F. A complete labeling with a minimal set of in-labeled arguments is called a grounded labeling. A complete labeling with a maximal set of in-labeled arguments is called a preferred labeling. A complete labeling without undec-labeled arguments is called a stable labeling. A complete labeling with a minimal set of undec-labeled arguments is called a semi-stable labeling.

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An argumentation semantics is a function that maps an argumentation framework to a set of labelings. The above-defined notions give rise to the complete, preferred and grounded argumentation semantics. We call an argumentation semantics σ complete-based if all σ -labelings are complete labelings. We will also refer to the stage semantics defined in its extension-based form by Verheij (1996). We adapt it to the labeling-based form by assigning the label out to all arguments that are not in the stage extension in Verheij’s definition, even those which are not attacked by in arguments. This labeling-based form of the stage semantics can be defined as follows. Definition 3 Let F = A, R be an AF and L a 3-labeling of F. Define L in to be the set {a ∈ A | L(a) = in}. Define L + in to be the set {a ∈ A | ∃b ∈ L in .(b, a) ∈ R}. We say that L is a stage labeling of F if L in is conflict-free, L in ∪ L + in is maximal with respect to set inclusion and L(a) = out for all a ∈ A \ L in . We also make use of the notions of transitive closure of a relation and restriction of a relation to a subset of its domain. Definition 4 Let r el be a relation. We define the transitive closure of r el to be the smallest set r el ∗ such that r el ⊆ r el ∗ and if (a, b), (b, c) ∈ r el ∗ , then (a, c) ∈ r el ∗ . Definition 5 Let A, B be sets, A ⊆ A and R ⊆ A × B. We define the restriction of R to A to be:  {(a, b) ∈ R | a, b ∈ A } if A = B R ↓ A = otherwise {(a, b) ∈ R | a ∈ A } The definition of restriction handles separately two cases: if the domain and range of the relation are the same, it then applies the restriction to both of them, for example, in the case of the attack relation of an AF. In the case where the domain and range are different sets, it only performs the restriction on the domain set, for example, in the case of a labeling function. In the introduction, we have pointed out that retrieving the set of complete labelings from the preferred and grounded labelings alone is not feasible. We now provide a concrete example of two argumentation frameworks with the same preferred and grounded labelings, but different complete labelings. Example 1 Consider the two AFs F1 and F2 depicted in Fig. 1. Both have {(a, undec), (b, undec), (c, undec), (d, undec)} as their grounded labeling, and {(a, in), (b, out), (c, in), (d, out)} and {(a, out), (b, in), (c, out), (d, undec)} as their preferred labelings. While these are also all the complete labelings for F1 , F2 also has {(a, in), (b, out), (c, undec), (d, undec)} as a complete labeling which is neither preferred nor grounded. Hence, given nothing other than the preferred and grounded labelings of a framework, it is not feasible to always accurately retrieve the set of complete labelings.

26 Fig. 1 Two AFs with the same preferred and grounded labelings but different complete labelings

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3 Update Relations Standard labeling semantics provide a direct relation between an argumentation framework and a set of labeling functions, which attach to each argument exactly one label. We will now define update relations, which formalize the idea that the final labelings can be determined in a stepwise fashion. For this purpose, we introduce epistemic labelings, which associate with each argument a non-empty set of labels from {in, out, undec}. The intuitive idea is that at a certain step in the update process, the set of labels associated with an argument tells us which labels we consider possible for this argument at this step. The steps in an update relation can be interpreted as moves in a dialogue, or as steps in an algorithm, or as learning a framework, or otherwise. Our dynamic semantic framework does not depend on such particular interpretations. Notice that it makes little sense to separate the labeling function from the underlying framework, as the labeling is meaningless without it. We will hence consider pairs of argumentation framework and labeling functions. Definition 6 We define a labeled argumentation framework (LAF) to be a pair (A, R, Lab) where A, R is a finite argumentation framework and Lab a function from A to P(L) \ {∅}, called an epistemic labeling. Additionally, let F be the class of all labeled argumentation frameworks. Observe that a labeling function cannot assign the empty set of labels to an argument, as the set of labels represents the possible final labels for that argument, and thus the empty set would mean that no label can be attached to it, which prevents us from having a final labeling for the framework. We now introduce the notions of initial and final labeled frameworks, which correspond to the starting point and endpoint of a labeling process. In an initial LAF, every label is possible for each argument, while in a final LAF, every argument is assigned a singleton set of labels, representing the fact that a unique label has been selected. Definition 7 Let F = (A, R, Lab) be a LAF. If for all a ∈ A, Lab(a) ∈ {{in}, {out}, {undec}}, we say that F is final. If for all a ∈ A, Lab(a) = L, we say that F is initial. Note that there is a one-to-one correspondence between the epistemic labelings Lab of the final LAFs (A, R, Lab) and the 3-labelings of A, R. This one-to-one

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correspondence can be formally defined by constructing singletons out of a given 3-labeling as follows. Definition 8 Let A, R be an AF and L a 3-labeling of A, R, define the epistemic labeling T (L) by T (L)(a) := {L(a)} for all a ∈ A. In this section with the basic definitions of our approach, we will be careful to make the formal distinction between a 3-labeling L, the corresponding epistemic labeling T (L), and the corresponding final LAF (A, R, T (L)). In order to improve readability, we will not always make this distinction in later section, but instead identify the 3-labeling L with the corresponding epistemic labeling T (L) and the corresponding final LAF (A, R, T (L)). For example, we might speak of an LAF being a complete labeling of a given argumentation framework, even though formally a complete labeling is a 3-labeling. We now define a precision ordering on the LAFs based on the subset relation between the argument multi-labels, such that the final LAFs are the most precise and the initial LAFs are the least precise. Note, however, that only LAFs with the same underlying AF are comparable. Definition 9 Let F = (A, R, Lab) and F = (A , R , Lab ) be two labeled argumentation frameworks. We say that F is at least as precise as F (F ≥ p F ), iff A, R = A , R , and for all a ∈ A, ∅ ⊂ Lab(a) ⊆ Lab (a). We say that F is more precise than F (F > p F ) iff F ≥ p F and F = p F . We will now define the central notion of this paper, namely, update relations, i.e., relations between LAFs which, starting from an initial LAF, monotonically increase precision, until a fixpoint is reached, at which point the LAF should be final and correspond to a desired output. Definition 10 We say that upd ⊆ F × F is an update relation iff: • for all F ∈ F such that upd(F, F ), F ≥ p F; • if upd(F, F), then F is final. Notice that by the definition of ≥ p , if F is final then upd(F, F ) implies F = F . We now define correspondence between update relations and direct semantics that formalizes the idea that an update relation can be viewed as a stepwise procedure that gives rise to a certain direct semantics. For this we first need an auxiliary definition. Definition 11 Let Rel be a relation on F and F an LAF. We say that F is reachable in Rel iff there exists an initial LAF Fi such that (Fi , F) ∈ Rel ∗ . We say that F is a reachable fixpoint in Rel iff F is reachable in Rel and (F, F) ∈ Rel. Definition 12 Let upd be an update relation and sem a semantics. We say that upd gives rise to sem iff for each 3-labeling L of A, R, (A, R, T (L)) is a reachable fixpoint in upd iff L is a sem labeling of A, R. The following theorem, which easily follows from Definition 10, provides a simple way of combining two given update relations to yield a third update relation.

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Lemma 1 If upd1 and upd2 are update relations, then upd1 ∪ upd2 is an update relation. In Sect. 4, we will present an example where combining two update relations with a union operation gives us not only the union of the final labelings reachable by either of them but also additional labelings. This means that the semantics that upd1 ∪ upd2 gives rise to is not necessarily induced by the semantics that upd1 and upd2 separately give rise to. We are now interested in the comparison of updates in terms of precision increase per step, i.e., in the granularity of update relations. The idea is that an update relation is more granular than another if it takes more steps to reach its final LAFs. First of all, notice that such a comparison only makes sense for updates which output the same final LAF, i.e., updates which give rise to the same semantics. Definition 13 Let upd be an update relation. We define the restriction of upd to relevant paths (upd) to be the set of pairs in upd that are in some upd-path from an initial to a final LAF. Definition 14 Let upd1 and upd2 be two update relations. We say that upd1 is at ∗ least as fine-grained as upd2 (upd1 ≥g upd2 ) iff upd1 ⊇ upd2 . We then abstractly define the most fine-grained update relation for a given labeling semantics. Definition 15 Let sem be a labeling semantics. We define mfgsem to be the smallest update relation such that for all update relations upd that give rise to sem, we have mfgsem ≥g upd. Lemma 2 For every standard semantics, there exists a unique mfgsem . Proof Define mfgsem as follows: (F, F ) ∈ mfgsem iff either F = F is a sem labeling, or the following three properties are satisfied: • F > p F; • F

such that F > p F

> p F; • there exists a final F f which is a sem labeling such that F f ≥ p F . By definition, mfgsem includes all possible links in any relevant path from an initial to a final LAF which encompasses a sem labeling. Hence, for any update relation ∗ upd which gives rise to sem, mfgsem ⊇ upd. Also, mfgsem includes by definition only pairs which are on a relevant path, as the first alternative adds the endpoints of these paths and the third item of the second alternative ensures that the pairs are on a relevant path. The first and second items of the second alternative ensure also that a minimal amount of pairs are added, making mfgsem as small as possible. Also, note that m f gsem is well-defined since we only consider finite AFs, and thus ≥ p is finite.  In subsequent sections, we will need the following notion of a sub-framework. Definition 16 Let F = (A, R, Lab) be a LAF and S ⊆ A. We define the subframework of F generated by S to be Sub(F, S) = (S, R ↓ S , Lab ↓ S ).

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4 Case Analysis: An Algorithmic Approach for Combining Preferred and Grounded In this section, we consider update relations which give rise to the preferred and grounded semantics, and which are motivated by algorithms for computing these semantics that have been described by Dauphin and Schulz (2014). The algorithmic update relation for the grounded semantics first identifies the arguments which are only being attacked by arguments which are already labeled {out}, labels them as {in}, and any argument they attack as {out}, and then repeats this process until no arguments can be further labeled, at which point it will label all remaining arguments as {undec}. Definition 17 For any labeled argumentation framework F = (A, R, Lab), we define the set of unattacked arguments to be unattacked(F) = {a ∈ A | Lab(a)  {in} ∧ ∀b ∈ A.((b, a) ∈ R → Lab(b) = {out})}. In an initial AF, the set of unattacked arguments will correspond to the arguments which do not have any attackers in the framework, while for final AFs, this set will be empty since it only considers arguments which are not finally labeled. Definition 18 We define step_grnd ⊆ F × F to be the relation such that ((A, R, Lab), (A, R, Lab )) ∈ step_grnd iff one of the following conditions holds: • unattacked((A, R, Lab)) = ∅, and (A, R, Lab ) is the least precise LAF that is more precise than (A, R, Lab) such that for all a ∈ unattacked((A, R, Lab)), Lab (a) = {in} and for all c ∈ A such that (a, c) ∈ R and out ∈ Lab(c), Lab (c) = {out}. • unattacked((A, R, Lab)) = ∅, there is an a ∈ A such that Lab(a)  {undec}, and (A, R, Lab ) is the least precise LAF that is more precise than (A, R, Lab) such that for all a ∈ A such that Lab(a)  {undec}, Lab (a) = {undec}. • (A, R, Lab) = (A, R, Lab ) is a final LAF. Note that before labeling arguments out, we ensure that it is a possibility, e.g., by having the condition out ∈ Lab(c) in the first item of Definition 18. While this requirement will straightforwardly be fulfilled in any reachable LAF, it is required to ensure that the increase in precision is satisfied even for those LAFs that are not reachable from an initial LAF. The following lemma now easily follows from the above definition. Lemma 3 step_grnd is an update relation. The following theorem states that step_grnd does indeed have the intended property that it gives rise to the grounded labeling. Theorem 1 step_grnd gives rise to the grounded semantics.

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Proof One can easily see that whenever step_grnd changes the label of an argument a to {in}, {out}, or {undec}, argument a is legally labeled {in}, {out}, or {undec}, respectively. Thus the final labeling reachable in step_grnd is a complete labeling. To show that the final labeling reachable in step_grnd is the complete labeling that maximizes undec, suppose that there is some complete labeling Lab of A, R and let A = {a ∈ A | Lab(a) = undec}. It is now enough to show that step_grnd never labels any a ∈ A {in} or {out}. Consider for a proof by contradiction the first step where step_grnd does label some a ∈ A {in}, respectively {out}. Since a is legally labeled undec in Lab, some a ∈ A must attack a, so by Definitions 17 and 18, a

must already be labeled {out} in a previous step, respectively, there must exist an a

which has been labeled {in} in a previous step, which is a contradiction.  Let us now examine step_pref, a similar update relation which computes the preferred labelings. For this, we first define the notion of minimal nontrivial admissible sets of arguments, which resembles the notion of initial-like sets (Xu and Cayrol 2018), but takes also the partial labels into account. Definition 19 Let F = (A, R, Lab) be a labeled argumentation framework. We define min_adm(F) ⊆ P(A) to be the set of all minimal subsets S of A that satisfy the following conditions: • • • •

S = ∅; for all a ∈ S, Lab(a)  {in}; for all a, b ∈ S, (a, b) ∈ / R; for all a ∈ S and b ∈ A such that Lab(b) = {out} and (b, a) ∈ R, there exists a ∈ S such that (a , b) ∈ R.

So the function min_adm(F) returns all minimal non-empty admissible sets of arguments whose label could still be changed to {in}. The update relation step_pref proceeds with a process similar to the one in the step_grnd update, iteratively labeling {in} all arguments with all attackers {out}, and then labeling all arguments attacked by those as {out}. The difference lies in the case where unattacked(F) is empty, where the preferred update relation looks for minimal nontrivial admissible sets, label them {in} and arguments they attack {out}. Definition 20 We define step_pref ⊆ F × F to be the relation such that ((A, R, Lab), (A, R, Lab )) ∈ step_pref iff one of the following conditions holds: • unattacked((A, R, Lab)) = ∅, and (A, R, Lab ) is the least precise LAF that is more precise than (A, R, Lab) such that for all a ∈ unattacked((A, R, Lab)), Lab (a) = {in} and for all c ∈ A such that (a, c) ∈ R and out ∈ Lab(c), Lab (c) = {out}. • unattacked((A, R, Lab)) = ∅, and for some S ∈ min_adm(F), (A, R, Lab ) is the least precise LAF that is more precise than (A, R, Lab) such that for all a ∈ S, Lab (a) = {in} and for all c ∈ A such that (a, c) ∈ R and out ∈ Lab(c), Lab (c) = {out}.

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• unattacked((A, R, Lab)) = min_adm(F) = ∅, and there is an a ∈ A such that Lab(a)  {undec}, and (A, R, Lab ) is the least precise LAF that is more precise than (A, R, Lab) such that for all a ∈ A such that Lab(a)  {undec}, Lab (a) = {undec}. • (A, R, Lab) = (A, R, Lab ) is a final LAF. The following lemma now easily follows from the above definition. Lemma 4 step_pref is an update relation. The following theorem, which can be proved in a similar way as Theorem 1, states that step_pref has the intended property that it gives rise to the preferred labeling. Theorem 2 step_pref gives rise to the preferred semantics. We now find the interesting result that combining these two update relations with a union operation gives us not only the union of the final labelings reachable by either of them but also the complete labelings which are neither grounded nor preferred. Theorem 3 step_grnd ∪ step_pref gives rise to the complete semantics. Proof sketch One can easily see that any final labeling reachable in step_gr nd ∪ step_ pr e f is a complete labeling, as the two update relations preserve the legality of argument labels. So we only prove that each complete labeling Lab is reachable in step_gr nd ∪ step_ pr e f . Let F = (A, R, Lab) be an initial LAF and Labc the complete labeling we want to reach. First, apply either step_grnd or step_pref until we reach F where the set of unattacked arguments is empty. At this point, the set S of in arguments is the grounded extension, and thus these arguments must also be in in Labc , since the grounded extension is the intersection of all complete extensions (this follows from it being the unique smallest complete extension). Let S be the set of arguments which are in in Labc but not {in} in F . S ∪ S forms an admissible set, since it is a complete extension. Hence, there is a minimal, non-empty subset of S , S1 , such that S ∪ S1 is admissible. There is an edge in the relation step_pref which labels the arguments in S1 as in and any argument they attack as out, according to Definition 20 second item. The rest of the arguments in S are labeled in via Definition 20, either with the first item, or again with the second item as above. Once we have reached the LAF where all in arguments from Labc are {in} and any argument they attack {out}, we can make a step with step_grnd following Definition 18, second item, to label all remaining arguments as {undec}. We have then reached the fixpoint F f = (A, R, T (Labc )), as desired.  Example 2 Let us examine the initial LAF F = (A, R, Lab) where A = {a, b, c, d}, R = {(a, b), (b, a), (b, c), (c, d), (d, c)}. Since unattacked(F) = ∅, step_grnd will send F to the fixpoint where all arguments are labeled {undec}. This is depicted in Fig. 2.

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Fig. 2 Example path from the initial LAF F to the corresponding final LAF in step_grnd

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Let us now consider the same LAF F under the step_pref update relation this time. Again, unattacked(F) = ∅, but min_adm(F) = {{a}, {b}, {d}}. The relation hence branches out in three paths. Let us focus the path with {a}. So the relation step_pref sends F to the LAF Fpref1 where a is {in} and b is {out}, as depicted in Fig. 3. unattacked(Fpref1 ) = ∅, but min_adm(Fpref1 ) = {{c}, {d}}, which gives us once again two possible directions in which to branch out. We will examine the one which selects {c}. This then gives us the final fixpoint Fpref2 = (A, R, Labpref2 ), where Labpref2 (a) = Labpref2 (c) = {in} and Labpref1 (b) = Labpref1 (d) = {out}. We now consider the union of both relations. We can first send F to Fpref1 using the same step from step_pref as above. However this time we can apply step_grnd to Fpref1 , and since unattacked(Fpref1 ) = ∅, the remaining arguments c and d are

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assigned the {undec} label, sending Fpref1 to the fixpoint Fcomp , where a is {in}, b is {out} and c, d are {undec}. Notice that Fcomp corresponds to a complete labeling of F which is neither preferred nor grounded. This situation is depicted in Fig. 4.

5 Merging Semantics Through the Most Fine-Grained Update Relation In the previous section, we have shown that we can obtain the complete semantics by taking the union of two algorithmically motivated update relations giving rise to the grounded and the preferred semantics, respectively. The success of this approach was dependent on the details of the algorithmic update relations that we defined, so it cannot be generalized to combine arbitrary semantics. In this section, we want to generalize our methodology to make it applicable to the combination of arbitrary semantics. For this purpose, we will examine a way to combine any two standard semantics via their most fine-grained update and a combination operation we call merging.

5.1 Merging Preferred and Grounded If we were to attempt to combine mfgpref and mfggrnd by simply taking their union, as we have done in the algorithmic approach, it follows from their definition that we would simply obtain as reachable fixpoints the labelings which are either preferred or grounded. The main issue is that mfgpref and mfggrnd are not applicable to LAFs which do not agree with some final LAF of that semantics. For an example of this issue, we consider again the same LAF F as in Example 2. Suppose we want to reach the same complete labeling that we reached in Fig. 4, i.e., the one in which a is {in}, b is {out}, and c and d are {undec}. We could start by doing those six steps of mfggrnd that are compatible with the complete labeling that we want to reach, as depicted in Fig. 5, yielding the intermediate LAF F . Now we would like to apply mfgpref to F in order to delete the undec-labels from a and b. However, mfgpref cannot be applied at all to F , as F is not compatible with any preferred labeling of F. So instead of just taking the union of mfgpref and mfggrnd , we will define a more complicated operation called the merge of two update relations, which we denote by upd1  upd2 . The idea is that once neither mfgpref nor mfggrnd allow us to get closer to a desired complete labeling, we will focus on a particular sub-framework and draw analogies with another framework which also contains that sub-framework. This operation resembles the way input is imposed in multi-sorted argumentation semantics (Baroni et al. 2014). The details of this approach are somewhat complicated, so let us first sketch the approach by seeing how it can be applied to the example that we just looked at.

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Fig. 5 Example path from the initial LAF F to an intermediate LAF F in mfggrnd

The idea is that we focus on the set S = {a, b} of arguments, as we want to remove labels from a and b. In order to work with mfgpref on the sub-framework Sub(F , S) induced by S, we consider an alternative framework F2 that also has Sub(F , S) as a sub-framework, but to which mfgpref can be applied. A suitable choice of F2 is depicted on the left in Fig. 6. Now we apply mfgpref twice to F2 as depicted in Fig. 6, removing the labels from a and b that we wanted to remove. If certain conditions are satisfied, we may import the changes we have made to F2 back to F, as depicted in Fig. 7.

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Fig. 6 Example path on a parallel F2 framework with S = {a, b} and I = {c}, where m f g pr e f is applicable in out undec

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Now what are the conditions that need to be satisfied in order to allow for this import of changes from one framework to another? In order to describe these conditions, we need to split the original framework into three parts, based on sets of arguments: • S, the arguments we will focus on; • I , called the interface, which is a set of arguments which already have a maximally precise label (i.e., a singleton) and which separate the set S from the rest of the framework; • A \ (S ∪ I ), the rest of the framework, on which the two frameworks may differ. The basic idea is that in order to import some change that an update relation mfgsem can make on F2 to the LAF F, we have to choose F2 in such a way that in both F and F2 , the interface I separates S from the rest of the framework. Furthermore, we have to choose F2 in such a way that mfgsem can actually be applied to F2 , which is only possible if the maximally precise labels of the arguments in I are possible labels for these arguments in F2 under the semantics sem. We are now ready to present the formal definition of the merge upd1  upd2 . Definition 21 Let upd1 and upd2 be two update relations. We define the merge of upd1 and upd2 (upd1  upd2 ) as the smallest relation such that 1. upd1  upd2 ⊇ upd1 ∪ upd2 ; 2. For F = (A, R, Lab) and F = (A, R, Lab ), (F, F ) ∈ upd1  upd2 if there exist disjoint sets S, I ⊆ A and two LAFs F2 = (A2 , R2 , Lab2 ) and F2 = (A2 , R2 , Lab 2 ) such that the following conditions are satisfied: a. b. c. d. e. f. g. h.

(F2 , F2 ) ∈ upd1 ∪ upd2 ; Sub(F2 , S ∪ I ) = Sub(F, S ∪ I ); ∀s ∈ S, ∀a ∈ A \ (I ∪ S), (s, a), (a, s) ∈ / R, R2 ; ∀a ∈ I , Lab(a) = Lab2 (a) is a singleton; Lab 2 ↓ S = Lab2 ↓ S ; Lab 2 ↓ A2 \S = Lab2 ↓ A2 \S ; Sub(F, A \ S) is reachable by upd1  upd2 ; Lab ↓ S = Lab 2 ↓ S .

3. if F is final and reachable by upd1  upd2 , then (F, F) ∈ upd1  upd2 . Given the complexity of this definition, let us explain it a bit more: Item 1 expresses the fact that we can still perform any step which is available in either one of the base updates. However, as we have seen previously, this is not enough in order to obtain meaningful combinations of most fine-grained updates, which is why we have item 2. Given a labeled argumentation framework F, additional changes are potentially possible if we can identify two disjoint sets of arguments S and I , where S is the set of arguments we are interested in and where the update will be occurring and I is a fully labeled interface between S and the rest of the framework, meaning that no argument in S attacks nor is attacked by an argument in A \ (S ∪ I ). Once such sets have been identified, we observe other labeled argumentation frameworks F2

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which also contain S ∪ I with the same structure and epistemic labels but can differ in structure and labels in the rest of the framework. If an update with upd1 ∪ upd2 is possible in such a framework, we then allow this change to be imported into F to produce F . In more details, subitem a specifies that there must be a upd1 ∪ upd2 step which relates F2 to F2 . Subitem b ensures that the parallel framework F2 agrees with F on the structure and epistemic labels of S ∪ I . Subitem c guarantees that there are no connections between S and A \ (S ∪ I ) in neither F nor F2 . Subitem d ensures that I is fully labeled, which is required in order to ensure the well behavior of the merge operation. The idea is that once this interface has been fully labeled by one of the two updates, if we can modify A \ (S ∪ I ) in order to make sense of these labels for the second update, then we can also perform steps from this second update inside S, and then by perhaps modifying A \ (S ∪ I ) again we can switch back to using the first update again, and so on. Subitem e ensures that change happens inside S, while subitem f ensures that no change is made outside of S, so that change happens in S and exclusively there. Subitem g provides an additional restriction on the partitioning to ensure that for an argument i in the interface I which has a justification a ∈ S for its label which has not been assigned yet, we do not introduce a new justification in A \ (S ∪ I ) for i’s label and hence allow for a different label to be assigned to a, leaving i with no justification for its label in F . This is clarified in Example 3. Subitem h simply specifies that the change made in the parallel LAF be imported into the original one to produce F , and combined with the subitem e entails that a change within S is necessary between F2 and F2 . Finally, with item 3 we ensure that reachable final frameworks are also fixpoints, which is needed since the second item of the definition does not produce any fixpoints, as it requires some change to happen in the LAF with the first subitem. Example 3 Consider the AF F depicted in Fig. 8. Since {(a, in), (b, out), (c, out), (d, in), (e, out)} is a preferred labeling, it is possible to assign the out label to c via m f gpref , as well as the in label to a and the out label to b. From there, it would be possible to set I = {c} and S = {d, e}, allowing one to import changes from the parallel framework F depicted in Fig. 9. Here, a few steps in m f ggrnd would assign the undec label to d and e. This would, however, produce a labeling where c is out, but has no reason to be labeled so, since d is undec. This kind of scenario is prevented by item (g) of Definition 21 as no sub-LAF consisting of {a, b, c} where c is out is reachable with m f gpref or m f ggrnd . Thus, item (g) forces what we informally refer to as a justification for the interface’s label to be either part of it, or contained in S.  In Definition 21, we have defined the merge between two arbitrary update relations. In this paper, we always apply this merge operation to two maximally finegrained update relations and focus on the semantics that the resulting update relation

Fig. 8 Example AF F to illustrate the need for item (g) in Definition 21

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Fig. 9 AF F parallel to F from Fig. 8. Here c is out due to f , allowing m f ggr nd to assign the undec label to both d and e

gives rise to. In this way, the notion of a merge between two update relations gives rise to the following notion of a merge between two argumentation semantics. Definition 22 Given two argumentation semantics sem1 and sem2 , we define sem1  sem2 to be the semantics that mfgsem1  mfgsem2 gives rise to. We originally motivated the definition of the merge operation with the goal to combine the grounded and preferred semantics to yield the complete semantics. The following theorem establishes that this is indeed the case for the merge operation as we have defined it. Theorem 4 preferred  grounded = complete. Proof By Definition 22, we need to show that mfgpref  mfggrnd gives rise to the complete semantics. So we need to prove that every complete labeling is a reachable fixpoint in mfgpref  mfggrnd and that every labeling that is a reachable fixpoint in mfgpref  mfggrnd is a complete labeling. We start by proving that every complete labeling is a reachable fixpoint in mfgpref  mfggrnd . Let AF = A, R be an argumentation framework, and let L be the complete labeling of AF we want to reach. We want to show that F f = (A, R, Lab f ) is a reachable fixpoint, where Lab f = T (L). Let C = {a ∈ A | L(a) = in}, I = {b ∈ A | L(a) = out} and S = A \ (C ∪ I ). Consider the LAF Fi = (A, R, Labi ), where for all a ∈ C, Labi (a) := {in}, for all b ∈ I , Labi (b) := {out}, and for all c ∈ S, Labi (c) := {in, out, undec}. Since L is a complete labeling, C is admissible, so there exists a preferred labeling where all arguments in C are in. Thus Fi is reachable with mfgpref . We now want to apply item 2 of Definition 21 to Fi multiple times in order to remove all the in and out labels from the arguments in S. For this purpose, we choose a “new” argument z, i.e., an argument z ∈ / A, and consider the LAF F2 = (A2 , R2 , Lab2 ) where A2 = (A \ C) ∪ {z}, R2 = R ↓ A2 ∪ {(z, a) | a ∈ I } and Lab2 = Lab ↓ A2 ∪ {(z, {in})}. Consider the final LAF F2 f = (A2 , R2 , Lab2 f ), where for all a ∈ A2 \ S, Lab2 f (a) = Lab2 (a) and for all a ∈ S, Lab2 f (a) = {undec}. We want to show that F2 f is grounded. For this purpose, we first establish that F2 f is complete, i.e., that all labels in F2 f are legal labels: z is unattacked and is therefore legally labeled in in F2 f . All arguments in I are attacked by z, so they are legally labeled out in F2 f . Furthermore, since C does not defend any arguments it does not contain, every argument in S is attacked by at least one other argument in S. Additionally, the only in argument, z, does not attack any arguments in S. Thus, the arguments in S are legally labeled undec in F2 f . Therefore, F2 f is a complete LAF, and since the only in argument, z, has to be labeled in, it is also grounded.

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Therefore, F2 f is reachable in mfggrnd from F2 . So by multiple applications of mfgpref  mfggrnd , using item 2 of Definition 21, one can reach F f from Fi . Since F f is final, F f is a fixpoint, and thus F f is a reachable fixpoint. So far, we have shown that every complete labeling is a reachable fixpoint in mfgpref  mfggrnd . Now we still need to show that every labeling that is a reachable fixpoint in mfgpref  mfggrnd is a complete labeling. Let F = (A, R, Lab) be a reachable LAF in mfgpref  mfggrnd . We show by induction on |A| that there exists a final complete LAF which is at least as precise as F. Induction hypothesis 1: Assume that for every LAF F = (A , R , Lab ) such that |A | < |A| and F is reachable in mfgpref  mfggrnd , there exists a final complete LAF which is at least as precise as F . We now use the second induction on the steps required to reach F. Base case: F is initial. Since there always exists a complete labeling for any framework, there exists a final complete LAF more precise than F. Inductive step: F is not initial, but is reached in mfgpref  mfggrnd through an LAF F ∗ = F for which the required property holds. In other words, we have the following induction hypothesis for F ∗ . Induction hypothesis 2: Assume that for F ∗ = (A, R, Lab∗ ) such that F ∗ = F and (F ∗ , F) ∈ mfgpref  mfggrnd , there exists a final complete LAF F ∗f = (A, R, Lab∗f ) such that F ∗ ≤ p F ∗f . We distinguish three cases: 1. (F ∗ , F) ∈ mfgpref . Then, by the definition of mfgpref , there exists a final LAF which represents a preferred labeling of A, R and is at least as precise as F. Since preferred labelings are also complete, we are done. 2. (F ∗ , F) ∈ mfggrnd . Similarly to the case above, it follows from the definition of mfggrnd that there exists a complete final LAF which is at least as precise as F. / mfgpref ∪ mfggrnd . Since F ∗ = F, the item 2 of Definition 21 must be 3. (F ∗ , F) ∈ satisfied. In other words, there exist disjoint sets S, I ⊆ A and two LAFs F2 and F2 that satisfy the conditions a to h from item 2 of Definition 21. By condition a, (F2 , F2 ) ∈ mfgpref ∪ mfggrnd , so by the same reasoning as in cases 1 and 2 above, we can conclude that there exists a final LAF F f 2 = (A2 , R2 , Lab f 2 ) which is complete and at least as precise as F2 . Also, by condition g, Fs = Sub(F ∗ , A \ S) is reachable by mfgpref  mfggrnd , and by condition e, S = ∅, i.e., |A \ S| < |A|. So by induction hypothesis 1, there exists a final complete LAF Fs f = (A \ S, R ↓ A\S , Labs f ) such that Fs f ≥ p Fs . We now construct the final LAF F f = (A, R, Lab f ) as follows: For all a ∈ A \ S, Lab f (a) := Labs f (a), and for all a ∈ S, Lab f (a) = Lab f 2 (a). From the construction of F f and conditions f and h of Definition 21, it follows that F f is more precise than F. To complete the proof, we now still need to show that F f is a complete labeling, i.e., that all arguments in A are legally labeled in F f . According to the definition of legal labeling (Definition 2), a labeling being legal depends only on the label of the arguments it is directly attacking or attacked by. According to condition c of Definition 21, the only arguments which are attacking

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or attacked by arguments in S are in S ∪ I . The arguments in S are legally labeled in F2 f , and thus they are also legally labeled in F f , since Sub(F2 f , S ∪ I ) = Sub(F f , S ∪ I ). Similarly, since the arguments in A \ (S ∪ I ) are legally labeled in Fs f , they are also legally labeled in F f . Now take an arbitrary a ∈ I . We distinguish three cases: a. Lab f (a) = {out}: Then, since Fs f is complete, there exists an argument b ∈ A \ S such that (b, a) ∈ R and Lab f (b) = {in}. So a is legally out in F f . b. Lab f (a) = {in}: Then, since Fs f is complete, for all b ∈ A \ S such that (b, a) ∈ R, Lab f (b) = {out}. Also, since F2 f is complete, for all b ∈ S such that (b, a) ∈ R, Lab f (b) = {out}. Hence, a is legally in in F f . c. Lab f (a) = {undec}: Then, since Fs f is complete, for all b ∈ A \ S such that (b, a) ∈ R, Lab f (b) = {in}, and for at least one such b, Lab f (b) = {undec}. Also, since F2 f is complete, for all b ∈ S such that (b, a) ∈ R, Lab f (b) = {in}. Hence, a is legally undec in F f . So all arguments in F f are legally labeled and thus F f is complete. Hence, there exists a final complete LAF which is at least as precise as F. Therefore, for all reachable LAFs, there exists a complete LAF which is at least as precise. Since mfgpref  mfggrnd is an update relation, every reachable fixpoint is final, and thus every reachable fixpoint is complete. 

5.2 Defining New Semantics via Merging The merge operation defined in Definition 22 can be used to combine two arbitrary argumentation semantics to yield another argumentation semantics. So far, we have shown that merging grounded and preferred semantics yields the complete semantics. In this section, we show how applying this merge operation to other pairs of semantics gives rise to completely new argumentation semantics. First, notice that the second part of the proof of Theorem 4 only makes use of the fact that the labelings reached the different stages are complete, but not of any other properties particular to preferred or grounded. Hence, the merge of any two complete-based semantics is a complete-based semantics itself, i.e., all the labelings it returns are also complete. Theorem 5 Let sem1 and sem2 be two complete-based argumentation semantics. Then sem1  sem2 is also a complete-based semantics. For example, by merging stable and grounded, we obtain labelings which are complete. However, in this case, we do not recover all complete labelings as we did when merging grounded and preferred. Let us examine this short example to see why.

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Example 4 Consider the following AF, which we call F: a

c

b

Using mfgstab  mfggrnd , one can reach the labelings {(a, out), (b, in), (c, out)} and {(a, undec), (b, undec), (c, undec)}. However, suppose we wish to reach the complete labeling Lab = {(a, in), (b, out), (c, undec)}. Since there is no stable labeling, we cannot make any steps via mfgstab from the initial LAF. Also, attempting to find a similar framework from which one could import changes will not work at this point where the LAF is initial, because the interface I would have to be empty, which only works for disconnected AFs. Hence, one can only make steps in mfggrnd in order to reduce c’s epistemic labeling to {undec}, A’s to {in, undec} and B’s to {out, undec}. This, however, is as close as one can get to Lab using mfgstab  mfggrnd . Any F2 satisfying the conditions of item 2 of Definition 21 must have I = {c}. In this case, the set S on which we want to make changes would have to be {a, b}. But then I ∪ S includes all arguments, so that F2 would have to be identical to F, so that we cannot use item 2 of Definition 21 to make any change that we cannot already make with item 1 of Definition 21.  Therefore, Lab is unreachable with mfgstab  mfggrnd . An interesting note to make is that all labelings reachable by mfgstab  mfggrnd are complete, according to Theorem 5, and hence this combination provides a novel complete-based semantics which returns more labelings than both the stable semantics and the grounded semantics. Similarly, the merge of the semi-stable and grounded semantics returns a novel complete-based semantics. One can check this by replacing stable by semi-stable in the situation described in Example 4. The desired complete labeling is still unreachable. As motivated in the introduction, we are also interested in the following research question related to combining features of naive-based and complete-based semantics: Is there a sensible semantics that allows one to locally make choices for some unattacked odd or even cycles while not making choices for other unattacked odd or even cycles. Given our methodology for merging semantics, an obvious candidate for such a semantics is stage  grounded, i.e., the semantics resulting from merging the stage semantics with the grounded semantics. By considering its application to an example, we show that this semantics does indeed have this feature. Example 5 Consider the following AF, which we call F : b c a

d

e

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The stage labelings of F are Lab1 = {(a, in), (b, out), (c, out), (d, in), (e, out)}, Lab2 = {(a, in), (b, out), (c, out), (d, out), (e, in)}, Lab3 = {(a, out), (b, in), (c, out), (d, in), (e, out)}, Lab4 = {(a, out), (b, in), (c, out), (d, out), (e, in)}, Lab5 = {(a, out), (b, out), (c, in), (d, out), (e, in)}. Its grounded labeling is Lab6 = {(a, undec), (b, undec), (c, undec), (d, undec), (e, undec)}. Additionally to these six labelings, it has three further stage  groundedlabelings: Lab7 = {(a, in), (b, out), (c, out), (d, undec), (e, undec)}, Lab8 = {(a, out), (b, in), (c, out), (d, undec), (e, undec)}, Lab9 = {(a, undec), (b, undec), (c, undec), (d, out), (e, in)}. Lab7 can be reached using mfgstage  mfggrnd by first applying mfgstage several times to reduce the epistemic labels on a, b, and c to {in}, {out}, and {out}, respectively, and then applying item 2 of Definition 21 with the interface I := {c}, the set S := {d, e} and the following parallel framework F27 . c

f

e

d

We can then apply mfggrnd multiple times to this parallel framework to reduce the epistemic labels of d and e to {undec} and import these changes to the labeling on the main framework F using item 2 of Definition 21. Lab8 can be reached using mfgstage  mfggrnd in a similar way using the same parallel framework. Lab9 can be reached using mfgstage  mfggrnd by first applying mfgstage several times to reduce the epistemic labels on d and e to {out} and {in}, respectively, and then applying item 2 of Definition 21 with the interface I := {d}, the set S := {a, b, c} and the following parallel framework F29 . b c

d

e

a

We can then apply mfggrnd multiple times to this parallel framework to reduce the epistemic labels of a, b, and c to {undec} and import these changes to the labeling on the main framework F using item 2 of Definition 21. The stage semantics forces us to make a choice on the odd cycle {a, b, c}, and unless we choose to accept the argument c that attacks the even cycle, we are also forced to make a choice on the even cycle {d, e}. In the grounded semantics, there

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are no choices and all arguments become undecided. In stage  grounded, we can combine these features of stage and grounded: We can, for example, choose a from the odd cycle, but stay undecided about the arguments in the even cycle—this possible choice is formalized by Lab7 . So stage  grounded allows one to locally make choices for some unattacked odd or even cycles while not making choices for other unattacked odd or even cycles. It thus provides a positive answer to our third research question from the introduction.

6 Conclusion and Future Work In this paper, we introduce a dynamic approach to combine two argumentation semantics to yield the third one. In particular, we provide a formal environment for the analysis of stepwise relations between labeled framework with an increase in the label precision, whose reachable fixpoints correspond to some standard direct semantics. We define and discuss two approaches to combining two given update relations to yield the third update relation, an approach based on algorithmically motivated update relations and an approach based on merging maximally fine-grained update relations. For both approaches, we examine how to obtain update relations for the complete labeling by combining update relations for the preferred and grounded labelings. Furthermore, we have defined novel semantics using the merge approach, including a semantics that meaningfully combines features of naive-based and complete-based semantics. Our paper gives rise to various topics for further research. Concerning the combination of argumentation semantics, many questions remain. Further, new semantics can be defined using our approach, and properties of the newly defined semantics can be studied systematically using the principle-based approach (Baroni and Giacomin 2007; van der Torre and Vesic 2018). Though we introduced our update relations to combine argumentation semantics, we believe that this dynamic semantics framework can be used for other applications as well. Most importantly, one of the main challenges in formal argumentation is the gap between graph based semantics and dialogue theory. Our more dynamic semantics framework may be used to decrease or even close the gap. In particular, in dialogue each statement may increase the knowledge and thus the set of arguments of participants. This is also related to the formalization of learning in the context of formal argumentation. Moreover, an important approach in argumentation semantics is the SCC recursive scheme. This scheme can be represented naturally using update relations. Various algorithms have been proposed for argumentation semantics, and these algorithmic approaches may be expressed naturally using update relations. Work has also been done on dynamic modifications to be made on a framework in order to enforce a certain set of arguments to become an extension, or prevent it from being so (Baumann and Brewka 2010; Coste-Marquis et al. 2014). Parallels could

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be made between their work and the combination operation presented in this paper. Finally, the principle-based analysis of argumentation semantics can be extended to the more fine-grained update relations.

References Arisaka, R., Satoh, K., van der Torre, L.: Anything you say may be used against you in a court of law. In: Artificial Intelligence and the Complexity of Legal Systems (AICOL). Springer, Cham (2018) Baroni, P., Boella, G., Cerutti, F., Giacomin, M., van der Torre, L., Villata, S.: On the input/output behavior of argumentation frameworks. Artif. Intell. 217, 144–197 (2014). https://doi.org/10. 1016/j.artint.2014.08.004 Baroni, P., Giacomin, M.: On principle-based evaluation of extension-based argumentation semantics. Artif. Intell. 171(10–15), 675–700 (2007) Baroni, P., Giacomin, M., Guida, G.: Scc-recursiveness: a general schema for argumentation semantics. Artif. Intell. 168(1–2), 162–210 (2005) Baroni, P., Caminada, M., Giacomin, M.: An introduction to argumentation semantics. Knowl. Eng. Rev. 26(4), 365–410 (2011) Baumann, R., Brewka, G.: Expanding argumentation frameworks: enforcing and monotonicity results. In: COMMA, vol. 10, pp. 75–86 (2010) Coste-Marquis, S., Konieczny, S., Mailly, J.G., Marquis, P.: On the revision of argumentation systems: minimal change of arguments statuses. In: KR, vol. 14, pp. 52–61 (2014) Cramer, M., Guillaume, M.: Two approaches to dialectical argumentation: admissible sets and argumentation stages. In: Proceedings of the International Conference on Computational Models of Argument (COMMA) (2018) Dauphin, J., Schulz, C.: Arg teach—a learning tool for argumentation theory. In: 2014 IEEE 26th International Conference on Tools with Artificial Intelligence (ICTAI), pp. 776–783. IEEE (2014) Dung, P.M.: On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games. Artif. Intell. 77(2), 321–357 (1995) Gaggl, S.A., Dvoˇrák, W.: Stage semantics and the SCC-recursive schema for argumentation semantics. J. Log. Comput. 26(4), 1149–1202 (2016). https://doi.org/10.1093/logcom/exu006 Giacomin, M.: Handling heterogeneous disagreements through abstract argumentation. In: International Conference on Principles and Practice of Multi-Agent Systems, pp. 3–11. Springer (2017) Rienstra, T., Perotti, A., Villata, S., Gabbay, D.M., van der Torre, L.: Multi-sorted argumentation. In: International Workshop on Theory and Applications of Formal Argumentation, pp. 215–231. Springer (2011) van der Torre, L., Vesic, S.: The principle-based approach to abstract argumentation semantics. In: Baroni, P., Gabbay, D., Giacomin, M., van der Torre, L. (eds.) Handbook of Formal Argumentation. College Publications (2018) Verheij, B.: Two approaches to dialectical argumentation: admissible sets and argumentation stages. In: Proceedings of the Biannual International Conference on Formal and Applied Practical Reasoning (FAPR) Workshop, pp. 357–368 (1996) (Universiteit) Xu, Y., Cayrol, C.: Initial sets in abstract argumentation frameworks. J. Appl. Non-Classical Logics 1–20 (2018)

Local Expansion Invariant Operators in Argumentation Semantics Stefano Bistarelli, Francesco Santini and Carlo Taticchi

Abstract We study invariant local expansion operators for conflict-free and admissible sets in Abstract Argumentation Frameworks (AFs). Such operators are directly applied on AFs, and are invariant with respect to a chosen ‘semantics’ (that is, w.r.t. each of the conflict-free/admissible sets of arguments). Accordingly, we derive a definition of robustness for AFs in terms of the number of times such operators can be applied without producing any change in the chosen semantics. Keywords Argumentation · Semantics · Invariant · Robustness

1 Introduction An Abstract Argumentation Framework (Dung 1995) (AF) is represented by a pair A, R consisting of a set of arguments and a binary relationship of attack defined among them. Given a framework, it is possible to examine the question on which set(s) of arguments can be accepted, hence collectively surviving the conflicts defined by R. A very simple example of AF is {a, b}, {(a, b), (b, a)}, where two arguments a and b attack each other. In this case, each of the two positions represented by either {a} or {b} can be intuitively valid. AFs can also provide a basis for handling the evolution of situations in which instances of particular problems undergo changes; variations on the underlying information can be interpreted as modifications in the This work has been supported by: “ComPAArg” (Ricerca di base 2016–2018), “Argumentation 360” (Ricerca di Base 2017–2019) and “RACRA” (Ricerca di base 2018–2020). S. Bistarelli (B) · F. Santini Università degli Studi di, Perugia, Italy e-mail: [email protected] F. Santini e-mail: [email protected] C. Taticchi Gran Sasso Science Institute, L’Aquila, Italy e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2019 B. Liao et al. (eds.), Dynamics, Uncertainty and Reasoning, Logic in Asia: Studia Logica Library, https://doi.org/10.1007/978-981-13-7791-4_3

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corresponding graph. Such modifications can be performed through operations of addition or subtraction of nodes and edges in the AF. As one can expect, introducing these changes might lead to obtain different semantics for the considered AF. We can classify the operations that can be performed on a framework in two types: the ones that change the semantics of the system and the ones that do not. In this paper, we focus on this latter type of operations (which leave the semantics unchanged), and reducing to the case of addition (or subtraction) of an attack. Our aim is to study a set of local expansion (Baumann 2012) operators with respect to which the semantics is not altered. Due to the dynamic nature of certain problems, settling for a solution (in a particular AF) could not be sufficient to guarantee a good outcome in case the problem evolves. In a dynamic setting, it may happen that new arguments change the meaning (and the outcome) of the conversation itself. Think, for example, to negotiation or persuasion dialogues. With invariant operators at dispose, one could test and possibly “enforce” (Baumann and Brewka 2010) the strength of its position, which amounts to add attacks such that the current semantics is stronger with respect to additional modification of the framework. Also, invariant operators could be successfully exploited for computing, in an efficient way, the semantics of an evolving AF. This paper is focused on a notion of “robustness” related to AFs. The main idea is that every argument (and set of arguments) is more or less suitable to undergo changes in a corresponding belief base (Dix et al. 2016). Robustness gives a measure of how many changes an AF can withstand before changing its semantics. The main part the work we carried out concerns the study of particular modifying operators for which the semantics is invariant: through such operators, it is possible to bring changes in the structure of a framework without changing its meaning. In particular, we study such operators for conflict-free and admissible sets (Dung 1995), since they are at the centre of any classical semantics and they have never been studied before in these terms. Differently from other works done in this direction (see the Related Work section), we consider how difficult is to modify the whole set of extensions instead of a single one, for instance, as in Rienstra et al. (2015). The motivations behind our study involve several perspectives of implementation of such operators in order to deal with different problems related to argumentation. For instance, the frameworks in which semantics are invariant with respect to the same operations can be grouped in the same class: in this way, further properties of AFs could be studied. The notion of robustness (Bistarelli et al. 2018) can be exploited to look for stronger clusters of arguments among frameworks within the same class. At the same time, a measure of robustness can be defined starting from the number of invariant operations admitted. This paper extends the work in Bistarelli et al. (2018) with proofs and additional examples and descriptions. The content is structured as follows: in Sect. 2, we summarise the necessary notions of Abstract Argumentation, by presenting extensionbased Dung (1995) and labelling-based Caminada (2006) semantics. In Sect. 3, we then describe the characteristics of the invariant operators we want to design and we

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define such operators for conflict-free and admissible sets. Finally, we conclude the paper by showing that our approach is novel with respect to the related work (Sect. 4), and by providing conclusive thoughts and ideas about future work (Sect. 5).

2 Abstract Argumentation Frameworks Argumentation deals with both the problems of representing knowledge and deriving information from it, using inference and logic to draw conclusions in a fashion really close to the human way of reasoning. For this, argumentation can be applied in a wide area of different disciplines concerning civil debate, dialogue, conversation and persuasion and can be considered the means by which people protect their beliefs. By neglecting the internal structure of each argument (e.g. premises and a claim), the framework becomes ‘abstract’, that is, we are not interested in the meaning of arguments any more, but we just focus on their relations and we look for general properties. Hence, snapshots of such discussions can be caught using Abstract Argumentation Frameworks (Dung 1995), namely, directed graphs that clearly show the exchange of opinions as attacks between arguments/nodes. Then, working with an AF means to identify subsets of nodes, called extensions, which share certain properties, according to a given semantics. Below, we give the fundamental definitions for AFs and extension-based semantics. Definition 1 (Abstract Argumentation Framework Dung 1995) An abstract argumentation framework is a pair G = A, R where A is a set of arguments and R is a binary relation on A, i.e. R ⊆ A × A. We denote the set of all AFs with the set of argument A with FA . For two arguments a, b ∈ A, (a, b) ∈ R represents an attack directed from a against b. We can interchangeably use a → b. Moreover, we say that a set of arguments E ∈ A attacks an argument a if a is attacked by an argument b ∈ E. Definition 2 (Acceptable argument Dung 1995) An argument a ∈ A is acceptable with respect to E ⊆ A if and only if ∀b ∈ A such that (b, a) ∈ R, ∃c ∈ E such that (c, b) ∈ R and we say that E defends a. P. M. Dung defined several semantics of acceptance for computing subsets of arguments, called extensions, that share characteristic properties (Dung 1995). Respectively, cf , adm, com, stb, prf and gde stand for conflict-free, admissible, complete, stable, preferred and grounded extensions. Definition 3 (Extension-based semantics) Let G = A, R ∈ FA be an AF. A set E ⊆ A is conflict-free in G, denoted E ∈ Sc f (G) if and only if there are no a, b ∈ A such that (a, b) ∈ R. For E ∈ Sc f (G) it holds that • E ∈ Sadm (G) if each a ∈ E is defended by E; • E ∈ Scom (G) if E ∈ Sadm (G) and ∀a ∈ A defended by E, a ∈ E;

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• E ∈ Sstb (G) if ∀a ∈ A \ E, S attacks a; • E ∈ S pr f (G) if E ∈ Sadm (G) and E ∈ Sadm (G) such that E ⊂ E ; • E = Sgde (G) if E ∈ Scom (G) and E ∈ Scom (G) such that E ⊂ E. For the sake of consistency, in the following we will interchangeably use the terms semantics and sets (as conflict-free and admissible sets), annotating them simply with S: even if the results developed in this paper concern sets, the same methodology can be applied to semantics as well (see conclusions). Many of these semantics exploit the notion of defence in order to decide whether an argument is part of an extension or not. Such a phenomenon, for which an argument becomes accepted in some extension after being defended by another argument belonging to that extension, is known as “reinstatement”. This mechanism plays a fundamental role when one needs to understand how a semantics changes in case an AF is modified. We exploit the notion of reinstatement labelling introduced in Caminada (2006).

2.1 Reinstatement Labelling Reinstatement is the process through which a non-accepted argument a becomes accepted w.r.t. a given extension when its attackers are defeated by some argument in that extension. Dung provides several reinstatement approaches, as the stable, preferred, complete and grounded semantics Dung (1995). In Caminada (2006), Caminada strengthens Dung’s theory by introducing “reinstatement labelling”, namely, a function that partitions the set of arguments of an AF into three classes: “in”, “out” and “undec”. An argument is labelled in if all its attackers are labelled out, and it is labelled out if it has at least an attacker which is labelled in. Otherwise, it is labelled undec. Caminada also proved that reinstatement labelling coincides with Dung’s notion of complete extension (Caminada 2006). Further restrictions similarly allow for obtaining the stable, grounded and preferred semantic. Moreover, the same author specifies an additional semantics, called semi-stable, and provides a partial ordering between the various semantics. In particular, we know that every stable extension is a semistable extension, every semi-stable extension is a preferred extension, every preferred extension is a complete extension and every grounded extension is a complete extension. In Definition 4, we provide a formal definition of reinstatement labelling. Definition 4 (AF-labelling Caminada 2006) Let G = A, R be a Dung-style argumentation framework. A labelling is a function L : A → {in, out, undec} such that in(L) ≡ {a ∈ A such that L(a) = in}, out (L) ≡ {a ∈ A such that L(a) = out} and undec(L) ≡ {a ∈ A such that L(a) = undec}. We say that L is a reinstatement labelling if and only if it satisfies the following: • ∀a ∈ A such that a ∈ in(L), ∀b ∈ A such that (b, a) ∈ R, b ∈ out (L); • ∀a ∈ A such that a ∈ out (L), ∃b ∈ A such that (b, a) ∈ R, b ∈ in(L).

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Fig. 1 An AF-labelling in which in(L) = {1, 4}, out (L) = {2, 3} and undec(L) = {5}

Table 1 Reinstatement labelling versus semantics

Labelling restrictions

Semantics

No restrictions Empty undec Maximal in Maximal out Maximal undec Minimal in Minimal out Minimal undec

Complete Stable Preferred Preferred Grounded Grounded Grounded Semi-stable

In Fig. 1, we show an example of reinstatement labelling obtained with the web interface of ConArg1 (Bistarelli et al. 2016; Bistarelli and Santini 2011, 2012): in ({1, 4}), out ({2, 3}) and undec ({5}) arguments are coloured in three different colours. Note that there exist more than one possible labelling for an AF. Moreover, there exists a connection between reinstatement labellings and Dung-style semantics. This connection is summarised in Table 1.

3 Invariant Operators for Robustness Argumentation pursues the objective of studying how conclusions can be reached through a process of logical reasoning, starting from a set of assumptions. In the most common form of argumentation, a part (which can be, for instance, an interlocutor) in a debate tries to affirm some kind of information and defends it from the attacks of other parts. The purpose of each part (or actor) is to persuade the rest of participants in the dialogue, defeating their assertions through non-monotonic inference. Indeed, this model can be applied to a wide range of situations, for instance, in order to solve the issue of finding a suitable outcome (e.g. an agreement in negotiation) for the actors. This raises a number of very interesting questions about the capabilities 1 ConArg

website: http://www.dmi.unipg.it/conarg.

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of argumentation applied to such problems. In particular, some of the questions we used as a starting point for this work are • Is it possible to change the outcome of a debate according to a particular semantics or meaning? • If so, how easy could it be to perform such a change? • And which consequences does it bring? In order to answer these questions and investigate the behaviour of AFs in a dynamic environment, we introduce the notion of robustness, i.e. the property of an AF to withstand changes in terms of alterations in its extensions set. The robustness of an AF, w.r.t. a given semantics, is measured by computing the minimum number of changes in the graph needed to change the corresponding extensions set (for instance, addition or subtraction of nodes or edges). Specifically, we focus on changes in terms of edges (corresponding to attacks relations) and look at the consequences they cause in the chosen semantics. We consider robustness as the property of an argumentation graph to withstand changes in terms of classical extension-based semantics. The robustness of a graph, w.r.t. a given semantics, is measured by computing the number of changes; for instance, addition of attacks needed to change a given extension. The robustness of a graph, w.r.t. a given semantics, will be measured by computing the number of changes needed to change the corresponding extensions.

3.1 Operators While AFs represent a powerful means to deal with argument-based semantics in static environments, possible evolutions of the supporting knowledge base are usually not taken into account. In this section, we present different modifying operators able to make changes in the AF by adding an attack between a couple of arguments and for which the semantics is invariant. In this way, one can identify the strongest extensions, where strong means to be more resistant to changes, and possibly to strengthen the weakest ones. A change in an AF can consist in addition (or subtraction) of nodes or edges. In this work, we focus on modifications concerning attacks between arguments (in particular additions). This kind of transformation coincides with the notion of local expansion (Baumann 2012). Introducing this type of changes in an AF may produce or not alterations on sets of extensions. This behaviour depends on two factors: the semantics we choose in computing the extensions and the change in the framework itself. After a modification, either a set of arguments is no more acceptable w.r.t. to a given semantics, or a new extension is generated, so the semantics of the AF will change in turn. On the contrary, if we consider the case in which extensions are preserved, further non-trivial observations can be made for what concerns the semantics of the

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Fig. 2 On the left an AF G, on the right the same AF after have been modified by the operator m adding the attack from argument 1 to argument 2. Argument 2 becomes out from being in

AF. For instance, even if the subsets of arguments remain unchanged, an admissible set can become also complete, if the right modifications are applied. Formally, an operator can be defined as follows. Definition 5 (Local expansion operator) Let G = A, R ∈ FA be an AF. A local expansion operator is a function m : FA → FA such that m(G) = A, m(R), where m(R) ⊇ R. In other words, a local expansion operator is a function m that takes an AF G in input and outputs a modified version of G in which the set of relation m(R) contains all the attacks in R, plus any new attacks. If we consider those operators taking into account also Dung’s semantics, we can study changes in the AFs from the point of view of sets of extensions. An explanatory example is provided in Fig. 2: in the framework G, the extension {1, 2} is both admissible and complete while the extension containing argument 1 is only admissible. After the modification m = {1 → 2} consisting in the addition of attack 1 → 2, the extension {1, 2} in the framework m(G) is no longer admissible. On the other hand, after the change on the relations set, the extension {1} that in G was just admissible also becomes complete in m(G).

3.2 Semantics Equivalence Our purpose is to find local expansion operators that leave the whole semantics unchanged, so instead of considering changes on the semantics induced by modifications on the graph, we look for the set of allowed changes that leave the semantics unmodified. In this way, we define semantics homomorphisms, namely, operators w.r.t. which the semantics is invariant. In order to preserve the whole semantics, it is necessary to ensure that all the sets will not be modified; hence, every set of extensions has to be, in turn, invariant w.r.t. these operators. We say that if two AFs have the same set of extensions w.r.t. a certain semantics, then the two frameworks are equivalent for such semantics. For this reason, we need the following definitions.

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Definition 6 (Semantics inclusion) Let S and S be two sets of extensions. We say that S ⊆ S if and only if ∀E ∈ S ∃E ∈ S such that E ⊆ E . Definition 7 (σ -equivalence) Let G and G be two AFs and σ a semantics. We say that • G ≡σ G if Sσ (G) = Sσ (G ); • G σ G if Sσ (G) ⊆ Sσ (G ). The equivalence we consider is referred to as standard in Oikarinen and Woltran (2011). Adding an attack in an AF can have different consequences. The most intuitive one is that the new attacked argument becomes defeated, and so it is forced to be removed from an extension. If we, instead, consider semantics in which the notion of acceptability is taken into account, defeating an argument could lead to accept another argument. In both the cases in which an argument become acceptable or is removed from an extension, the semantics would change. To distinguish the operators that reduce the set of extensions from those that expand it, we provide Definition 8. Definition 8 (Invariant operators) A local expansion operator m is said nondecreasing w.r.t. a semantics σ and an argumentation framework G = A, R ∈ FA if G σ m(G), and it is said non-increasing if m(G) σ G. If m is both non-decreasing and non-increasing, it is an invariant: G ≡σ m(G). The last case may occur when an attack has no effect on the set of extensions. Our purpose is exactly to find local expansion operators that guarantee this last outcome when adding an attack. In the following, an invariant operator will be referred to as h. It is necessary to understand how extensions react to changes in the AF. Since the main issue to deal with is due to the reinstatement, the idea we develop in order to define an invariant operator h is to use the notion of reinstatement labelling. Once the arguments of the AF are labelled (with in, out or undec), there are nine (32 ) different ways an edge can be added among nodes, according to labels of the source and the target of the attack. It is therefore essential to know in advance the possible labels of an argument in the framework and, in particular, if a certain argument is never labelled in, out or undec.

3.3 Operators for Conflict-Free Sets The conflict-free property is very fragile: introducing a relation between two nonconflicting nodes is sufficient to change the conflict-free sets. These sets can only be reduced: no new conflict-free set can be generated after the addition of an attack in the AF. Thus, every operator m able to perform the addition of an edge in a graph G produces another graph m(G) in which the semantics is “smaller” (in the sense that in some extensions of the new AF an argument disappears) or at most equal to the set deriving from G. R is the set of relations belonging to G, while m(R) is the same set after the addition of an attack introduced by m. We avoid describing the

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trivial case in which m(G) = G, and we only consider the effective transformation of adding an attack (symmetrical conclusions can be drawn in case of subtraction). Proposition 1 Every local expansion operator m is non-increasing w.r.t. the conflictfree sets for any argumentation framework G = A, R ∈ FA . Proof We have to show that G c f m(G) for every local expansion operator m, with G = A, R ∈ FA . This comes directly from the definition of conflict-free extension, since m is such that m(R) ⊇ R. Corollary 1 Any local expansion operator m which is non-decreasing for an argumentation framework G = A, R ∈ FA w.r.t. conflict-free sets is also invariant: G ≡c f m(G). Proof We know from Proposition 1 that every local expansion operator m is nonincreasing w.r.t. the conflict-free sets, that is, G c f m(G). If m is also nondecreasing, we have G c f m(G), and thus it is also invariant (G ≡c f m(G)). We conclude that an operator m preserves the semantics only if it adds attacks between arguments which already were in conflict. We define an invariant operator h for conflict-free sets with the following theorem. Theorem 1 (Invariant for conflict-free sets) Let G = A, R ∈ FA be an AF. We have G ≡c f h(G) if and only if ∀(a, b) ∈ h(R) • (b, a) ∈ R, or • (a, a) ∈ R, or • (b, b) ∈ R. Proof We show that all conflict-free extensions are preserved if the above conditions hold and vice versa. “=⇒”: Let us suppose to have an h such that G ≡c f h(G). If the conditions are not satisfied, then it would exist a relation in h(R) between two arguments belonging to the same extension in Sc f (G) and so G c f h(G). Indeed, if a and b are never in relation in G and do not attack themselves, then they are also in the same conflict-free extension. We therefore reach a contradiction. “⇐=”: Suppose that the conditions hold. If (b, a) ∈ R, then a and b are already in conflict and do not appear together in any conflict-free extension of G. Thus, no extension will be lost in Sc f (h(G)) when adding an attack between those arguments. On the other hand, if (a, a) ∈ R (or (b, b) ∈ R) then the argument a (b respectively) cannot be in and so it is not possible to change the conflict-free extensions set by adding an attack between a and b. If we consider any of the semantics in Dung (1995), we can conclude that adding an attack between two arguments belonging to a certain set always requires those arguments to be removed from that set, changing the semantics in turn. Hence, denying attacks between nodes within the same set (which do not attack each other in G) is a necessary condition in order to leave the semantics unchanged in h(G).

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3.4 Operators for Admissible Sets Contrary to conflict-free sets, for the admissible ones it is not possible to provide a theorem for the inclusion between semantics without taking into account reinstatement. Since arguments can be defended and consequently accepted w.r.t. a certain extension, we need to consider different types of interaction in order to find an operator capable of maintaining the semantics unchanged. Reinstatement labelling provides a powerful means to overcome the issue of comprehending how arguments defend each other inside an extension. Indeed, labellings are a more expressive way than extensions to suggest the acceptance of arguments. We exploit the notion of in, out and undec arguments to define the invariant operator h for the admissible sets. In order to preserve this semantics, we have to guarantee that neither existent extensions will be destroyed nor new one will be created. To achieve this, an operator h has to ensure that extensions in the set remain conflict-free, in arguments are not defeated from outside and out and undec arguments do not become acceptable. We distinguish between modifications that reduce the semantics from modifications that enlarge it and we give the conditions under which an operator does not allow to perform either kind of change. One of the key points for preserving the in arguments in admissible extensions is to consider the sequences of attacks with even length, which correspond to defence paths. Theorem 2 Let G = A, R be an AF. A local expansion operator h is nondecreasing w.r.t. the admissible set if and only if ∀(a, b) ∈ h(R), there does not exist a labelling L of G such that • a, b ∈ in(L) or • a ∈ out (L), b ∈ in(L), (b, a) ∈ / R and either c ∈ out (L) such that (c, b) ∈ R or there exists a disjoint maximal2 sequence of attacks towards b in which no argument c ∈ in(L) is such that (c, a) ∈ R or • a ∈ undec(L), b ∈ in(L). Proof We have to show that if G adm h(G) then the condition holds and vice versa. “=⇒”: An operator h is non-decreasing w.r.t. admissible sets. Suppose that there exists an attack relation (a, b) ∈ h(R) such that in some labelling L of G we have a, b ∈ in(L) or a ∈ undec(L), b ∈ in(L). In both these cases, the admissible extension corresponding to the labelling L is lost in h(G) and thus h(G) adm G, so we have a contradiction. When a ∈ out (L), b ∈ in(L) and (b, a) ∈ / R we have two cases: if c ∈ out (L) such that (c, b) ∈ R, then the extension containing the only argument b would be lost. In the other case we have that there exists at least one sequence of attacks with even length that ends in b. If no argument in this sequence attacks a, then the admissible extension composed of b and all the other in arguments in the sequence is no longer admissible in h(G). Both cases lead to a contradiction. “⇐=”: 2 Disjoint

w.r.t. edges of the graph G, maximal w.r.t. the number of attacks.

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If the condition holds, it is not possible that an extension in Sadm (G) is also in Sadm (h(G)). Consider any labelling L of G. If a is in or undec and b is not in, then the addition of an attack a → b cannot make an admissible extension of G to become unacceptable in Sadm (h(G)). If instead a is out, it means that it is already defeated, so every argument b belonging to some admissible extension of G remains acceptable w.r.t. such extension also in h(G). Theorem 3 Let G = A, R be an AF. A local expansion operator h is nonincreasing w.r.t. the admissible set if and only if ∀(a, b) ∈ h(R), there does not exist a labelling L of G such that • a, b ∈ in(L) and ∃c ∈ out (L) such that (a, c) ∈ / R and (b, c) ∈ R or • a ∈ in(L), b ∈ out (L) and ∃c ∈ in(L) such that (b, c) ∈ R or • a ∈ in(L), b ∈ undec(L) and ∃c ∈ undec(L) such that (c, c) ∈ / R and (b, c) ∈ R or • a ∈ out (L), b ∈ in(L), there is an odd length sequence of attacks from b to a and c = b such that there is an odd length sequence of attacks from c to a but not from a to c. Proof We show evidence that no new admissible extensions are generated for G applying the operator h if the conditions of the theorem are satisfied and vice versa. “=⇒”: Suppose that h(G) adm G. If there exists a labelling L for which a, b ∈ in(L) and ∃c ∈ out (L) such that (a, c) ∈ / R and (b, c) ∈ R, then arguments a and c would become acceptable together, forming a new admissible extension. The same would happen whenever a ∈ in(L), b ∈ out (L) and ∃c ∈ in(L) such that (b, c) ∈ R or in the case a ∈ in(L), b ∈ undec(L) and ∃c ∈ undec(L) such that (c, c) ∈ / R and (b, c) ∈ R. If instead the last condition does not hold, then a would be defended from all the incoming attacks and so it would be accepted in some admissible extension of h(G). In all these cases we reach a contradiction. “⇐=”: We will see that if the conditions hold, it is not possible that a new admissible extension can be generated. For every labelling L of G, a non-increasing operator h is allowed to add an attack between arguments a and b only in the following cases: 1. a, b ∈ in(L) and c ∈ out (L) such that (a, c) ∈ / R and (b, c) ∈ R; 2. a ∈ in(L), b ∈ out (L) and c ∈ in(L) such that (b, c) ∈ R; / R and (b, c) 3. a ∈ in(L), b ∈ undec(L) and c ∈ undec(L) such that (c, c) ∈ ∈ R; 4. a ∈ out (L) and there is no odd length sequence of attacks from b to a; 5. a ∈ out (L) and c = b such that there is an odd length sequence of attacks from c to a but not from a to c. 6. a ∈ undec(L). Item 4 means that b is not responsible for a being out, so the attack a → b is not sufficient to make a acceptable in a new admissible extension. In case 5, even if a defeats b, it will not become admissible without also defeating c. In all the

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remaining cases no argument can be defended by a (neither itself), thus no new admissible extensions can be obtained. Given Theorems 2 and 3, the following holds. Corollary 2 Let G = A, R be an AF. A local expansion operator h is invariant w.r.t. the admissible set, and we write G ≡adm m(G), if and only if ∀(a, b) ∈ h(R), there does not exist a labelling L of G such that • a, b ∈ in(L), or • a ∈ in(L), b ∈ out (L) and ∃c ∈ in(L) such that (b, c) ∈ R, or • a ∈ in(L), b ∈ undec(L) and ∃c ∈ undec(L) such that (c, c) ∈ / R and (b, c) ∈ R, or / R and either c ∈ out (L) such that (c, b) ∈ R or • a ∈ out (L), b ∈ in(L), (b, a) ∈ there exists a disjoint maximal sequence of attacks towards b in which no argument c ∈ in(L) is such that (c, a) ∈ R or • a ∈ out (L), b ∈ in(L), there is an odd length sequence of attacks from b to a and c = b such that there is an odd length sequence of attacks from c to a but not from a to c, or • a ∈ undec(L), b ∈ in(L). Proof The proof of this corollary is straightforward and comes from the proofs of Theorems 2 and 3. In particular, if a labelling L of G satisfying the properties above does not exist, then the local expansion operator h is both non-decreasing (for Theorem 2) and non-increasing (for Theorem 3) w.r.t. the admissible semantics. Then h is invariant w.r.t. the admissible semantics, because the modification on the set of relations does not allow any change in the semantics. Vice versa, if h is invariant w.r.t. the admissible semantics, then G adm h(G) and G adm h(G) must hold. If a labelling exists such that at least one of the given properties is satisfied, then h could be neither non-decreasing nor non-increasing (or both), according to Theorems 2 and 3. Thus, such a labelling cannot exist. We provide an example of how the conditions given in Corollary 2 allow to know in advance if a modification in an AF will change its semantics. Consider the AF in Fig. 3. The addition of the following attacks does not change the admissible semantics. • (e, c), indeed c does not attack any other argument, while a (which defends c from the argument b) also attacks e; • (d, c), same considerations as before; • (e, d), because d does not attack any other argument and it is never labelled in; • (c, d), for same reasons of (e, d). On the other hand, the modifications below change the set of admissible extensions. • (a, c), because both arguments are in in some extension;

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Fig. 3 An example of an AF G for which Sadm (G) = {{}, {a}, {e}, {a, c}, {b, e}}. The depicted labelling corresponds to the extension {}

• (e, b), both e and b are in in some extension; moreover, e defends the arguments c and d, forming a new extension; • (c, e), indeed e alone cannot defend itself from c; • (b, a), because b would defend itself against a, forming in this way a new extension. Remark 1 In order to determine if an operator m is invariant w.r.t. the admissible semantics, it is sufficient to consider only labelling in which in(L) is maximal, that is, the preferred extensions. In fact, in non-maximal extensions, some arguments remain labelled undec even if they have different labels in more inclusive extensions (w.r.t. set inclusion). Thus, looking directly at the most inclusive extension allows for establishing rules able to preserve all the sets. Invariant operators can be used as a metric to measure the robustness of AFs. The idea is that, starting from G, different invariant operators can be applied in sequence, until no more h exists for the last obtained AF: for example, h 4 (h 3 (h 2 (h 1 (G)))) and no h 5 exists (as in Fig. 4). Thus, the more operations are allowed for a framework, the more difficult it will be to change the extensions set for such semantics. We define the expansion-based robustness of a graph for a generic σ as follows. Definition 9 (Local expansion robustness) The local expansion robustness degree of an AF G = A, R w.r.t. a semantics σ is measured as the maximum number k of invariant operators h i that can be applied on G such that G ≡σ h k (h k−1 (. . . (h 1 (G) . . . )).

4 Related Work In the following of this section, we review the most meaningful works related to what presented in this paper. Rienstra et al. (2015) focus on finding conditions under which the evaluation of an AF remains unchanged when an attack is added or removed. The authors consider grounded, complete, preferred, stable and semi-stable semantics and, for

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Fig. 4 From figure (a)–(e) the AFs obtained starting from G and each time adding an attack through invariant operator h: in(L) = {a, c} persists

each of them, a set of properties for which extensions are preserved is given. Those properties are in the form: “given a certain labelling, attacks between two arguments with labels X and Y, respectively, are allowed (or not) for the semantics σ ”. Invariance is intended w.r.t. a single extension and not w.r.t. the whole semantics (as we do). Given conditions work in two directions: existent extensions cannot be cancelled and new extensions cannot be created. In the latter case, invariant properties are defined only for arguments which have the sale labels in all σ labelling. The problem of finding principles stating whether an extension does not change after adding/removing an attack between two arguments is also addressed by Boella et al. (2009a, b). Differently from us, the authors consider only the case in which the semantics of an AF contains exactly one extension, using the grounded semantics as example. A general theory for handling dynamics in AFs is devised in Liao et al. (2011), and extended in Baroni et al. (2014). The proposed approach consists in dividing the modified AF into three different parts in which the arguments are (i) unaffected by the modifications, (ii) affected or (iii) in relation with the affected arguments, respectively. On the one hand, this kind of division, made on a syntactic level, is different from our work on invariant operations, which is based on the acceptance status of the arguments. On the other hand, it could be interesting to understand how invariant operators behave with respect to the subsets identified through the division-based method of Liao et al. (2011). Cayrol et al. (2010) studied the impact on the evaluation of an AF when new arguments and attacks are added. They define a number of properties for the change

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operations according to how the extensions are modified. For instance, a change operation can be “conservative” if the set of extensions is the same after a change. Differently from our work, the conditions under which the addition of argument does not change the semantics are not studied. The work in Cayrol et al. (2008) addresses the problem of revising AFs when a new argument is added. In particular, they focus on the impact of new arguments on the set of initial extensions, introducing various kinds of revision operators that have different effects on the semantics. For instance, decisive revision allows for making a decision by providing a revised extension’s set with a unique non-empty extension. In Baumann and Brewka (2010), the problem of revising argumentation frameworks according to acquisition of new knowledge is taken into account. While attacks among the old arguments remain unchanged, new arguments and attacks among them can be added. In particular, the authors introduce the notion of enforcing, namely, the process of modifying an AF (and possibly changing its semantics) in order to obtain a desired set of extensions. This notion departs from our work, in which we instead look for operations that leave the semantics unchanged. Also, Baumann introduces the concepts of update and deletion Baumann (2014), focusing on modifications that retract arguments and attacks from an AF. New notions of equivalence are characterised through the so-called kernels, namely, functions that delete redundant attacks from a given framework. We instead concentrate on devising operators that permit both to modify AFs without changing their semantics, and to give a measure of how robust is a given AF, w.r.t. changes on the attack relations set. The concept of desire set is also studied by Boella et al. in (2008) with a work on persuasion in multi-agent systems, addressing the problem of choosing arguments to add into a system in order to maximise their acceptability w.r.t. the receiving agent. To this purpose, the notion of “more appealing” argument is introduced: in making the choice of a belief to add, an argument is more appealing than another if it does not interact with previous goals and beliefs of the agent. This has a different aim w.r.t. our work that consists in keeping unaltered the set of extension. The authors of Croitoru and Kötzing (2013) show that every AF can be augmented in a normal form preserving the semantic properties. In such normal form, no argument attacks a conflicting pair of arguments. A σ -augmentation is an alteration of an AF that leaves unchanged the semantics σ . The changes in the AF can involve arguments (the only allowed operation is the addition) and attacks. A different and more restrictive kind of equivalence is introduced in Oikarinen and Woltran (2011): two AFs G and G are considered strongly equivalent to each other when they are equivalent after the conjunction with the third AF H (similarly to the notion of bisimulation in state transition systems). Since our intent is to provide a method for building equivalent AFs through the addition/deletion of attacks on the same framework, the notion of standard equivalence results to be more fitting than the strong equivalence.

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5 Conclusions and Future Work We have defined invariant operators for AFs w.r.t. the semantics: these operators allow for performing changes on AFs while preserving the semantics. In particular, we have defined two operators, one for the conflict-free and one for the admissible sets, which can be applied to AFs for adding attack relations without resulting in changes to the set of extensions. The operators we have introduced exploit the notion of reinstatement labelling, and thus can be applied without even being aware of the extensions admitted for a given semantics. Moreover, we gave the definition for the semantic equivalence between AFs, and we presented a method for computing the expansion-based robustness degree of a framework. Our study has a very wide set of future perspectives. First, we plan to design invariant operators w.r.t. the complete, stable, semi-stable, preferred and grounded semantics (until now studied only w.r.t. single extensions Rienstra et al. (2015)). We would like to find the sets of arguments which are essential to preserve the whole semantics. Every change inside those sets modifies the semantics, while changes outside do not cause any alteration. By removing the non-core part of AFs, it is possible to obtain equivalent frameworks for which the computation of extensions is faster, especially for checking credulous/sceptical acceptance of arguments. As further work, different notions of equivalence, e.g. local equivalence Oikarinen and Woltran (2011), could be taken into account, and additional modifications of AFs could be considered, as the deletion of attack or the addition/removal of arguments. We also plan to devise a more general notion of robustness, involving the new modifications proposed above. By relaxing the conditions underlying invariant operators, and thus allowing the semantics to change, other operators could be obtained, that allow to reach ‘compromises’: if two parts of a debate desire two different outcomes in terms of semantics, a compromise can be reached as a third semantics, that is, the closest one w.r.t. those desired by both the counterparts. Definitions of closeness could be devised as well. Finally, we want to study local expansion operators also for semiring-based weighted AFs Bistarelli et al. (2018, 2009).

References Baroni, P., Giacomin, M., Liao, B.: On topology-related properties of abstract argumentation semantics. A correction and extension to dynamics of argumentation systems: A division-based method. Artif. Intell. 212, 104–115 (2014). https://doi.org/10.1016/j.artint.2014.03.003 Baumann, R.: Normal and strong expansion equivalence for argumentation frameworks. Artif. Intell. 193, 18–44 (2012). https://doi.org/10.1016/j.artint.2012.08.004 Baumann, R.: Context-free and context-sensitive kernels: update and deletion equivalence in abstract argumentation. In: Schaub, T., Friedrich, G., O’Sullivan, B. (eds.) 21st European Conference on Artificial Intelligence, ECAI 2014, 18–22 August 2014, Prague, Czech Republic—Including Prestigious Applications of Intelligent Systems (PAIS 2014). Frontiers in Artificial Intelligence and Applications, vol. 263, pp. 63–68. IOS Press (2014). https://doi.org/10.3233/978-1-61499419-0-63

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Baumann, R., Brewka, G.: Expanding argumentation frameworks: enforcing and monotonicity results. In: Computational Models of Argument: Proceedings of COMMA, FAIA, vol. 216, pp. 75–86. IOS Press (2010) Bistarelli, S., Faloci, F., Santini, F., Taticchi, C.: Studying dynamics in argumentation with rob. In: Modgil, S., Budzynska, K., Lawrence, J. (eds.) Computational Models of Argument— Proceedings of COMMA 2018, Warsaw, Poland, 12–14 September 2018. Frontiers in Artificial Intelligence and Applications, vol. 305, pp. 451–452. IOS Press (2018). https://doi.org/10.3233/ 978-1-61499-906-5-451 Bistarelli, S., Pirolandi, D., Santini, F.: Solving weighted argumentation frameworks with soft constraints. In: Larrosa, J., O’Sullivan, B. (eds.) Recent Advances in Constraints—14th Annual ERCIM International Workshop on Constraint Solving and Constraint Logic Programming, CSCLP 2009, Barcelona, Spain, 15–17 June 2009, Revised Selected Papers. Lecture Notes in Computer Science, vol. 6384, pp. 1–18. Springer, Heidelberg (2009). https://doi.org/10.1007/ 978-3-642-19486-3_1 Bistarelli, S., Rossi, F., Santini, F.: Conarg: a tool for classical and weighted argumentation. In: Baroni, P., Gordon, T.F., Scheffler, T., Stede, M. (eds.) Proceedings of Computational Models of Argument, COMMA 2016, Potsdam, Germany, 12–16 September, 2016. Frontiers in Artificial Intelligence and Applications, vol. 287, pp. 463–464. IOS Press (2016). https://doi.org/10.3233/ 978-1-61499-686-6-463 Bistarelli, S., Rossi, F., Santini, F.: A novel weighted defence and its relaxation in abstract argumentation. Int. J. Approx. Reason. 92, 66–86 (2018). https://doi.org/10.1016/j.ijar.2017.10.006 Bistarelli, S., Santini, F.: Modeling and solving AFs with a constraint-based tool: ConArg. In: Modgil, S., Oren, N., Toni, F. (eds.) First International Workshop on Theorie and Applications of Formal Argumentation, TAFA 2011. Barcelona, Spain, 16–17 July 2011, Revised Selected Papers. Lecture Notes in Computer Science, vol. 7132, pp. 99–116. Springer, Heidelberg (2011).https:// doi.org/10.1007/978-3-642-29184-5_7 Bistarelli, S., Santini, F.: Conarg: Argumentation with constraints. In: Ossowski, S., Toni, F., Vouros, G.A. (eds.) Proceedings of the First International Conference on Agreement Technologies, AT 2012, Dubrovnik, Croatia, 15–16 October 2012, CEUR Workshop Proceedings, vol. 918, pp. 197–198. CEUR-WS.org (2012). http://ceur-ws.org/Vol-918/111110197.pdf Bistarelli, S., Santini, F., Taticchi, C.: On looking for invariant operators in argumentation semantics. In: Proceedings of the Thirty-First International Florida Artificial Intelligence Research Society Conference, FLAIRS., pp. 537–540. AAAI Press (2018) Boella, G., da Costa Pereira, C., Tettamanzi, A., van der Torre, L.W.N.: Making others believe what they want. In: Artificial Intelligence in Theory and Practice II, IFIP 20th World Computer Congress, IFIP, vol. 276, pp. 215–224. Springer, Boston (2008) Boella, G., Kaci, S., van der Torre, L.W.N.: Dynamics in argumentation with single extensions: abstraction principles and the grounded extension. In: 10th European Conference on Symbolic and Quantitative Approaches to Reasoning with Uncertainty, ECSQARU. LNCS, vol. 5590, pp. 107–118. Springer, Heidelberg (2009a) Boella, G., Kaci, S., van der Torre, L.W.N.: Dynamics in argumentation with single extensions: attack refinement and the grounded extension (extended version). In: 6th International Workshop on Argumentation in Multi-Agent Systems, ArgMAS. LNCS, vol. 6057, pp. 150–159. Springer, Heidelberg (2009b) Caminada, M.: On the issue of reinstatement in argumentation. In: 10th European Conference on Logics in Artificial Intelligence, JELIA pp. 111–123 (2006). https://doi.org/10.1007/ 11853886_11 Cayrol, C., de Saint-Cyr, F.D., Lagasquie-Schiex, M.: Revision of an argumentation system. In: Proceedings of the Eleventh International Conference on Principles of Knowledge Representation and Reasoning, KR, pp. 124–134. AAAI Press (2008) Cayrol, C., de Saint-Cyr, F.D., Lagasquie-Schiex, M.: Change in abstract argumentation frameworks: adding an argument. J. Artif. Intell. Res. 38, 49–84 (2010). https://doi.org/10.1613/jair. 2965

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Croitoru, C., Kötzing, T.: A normal form for argumentation frameworks. In: Second International Workshop on Theory and Applications of Formal Argumentation, TAFA. LNCS, vol. 8306, pp. 32–45. Springer, Heidelberg (2013) Dix, J., Hansson, S.O., Kern-Isberner, G., Simari, G.R.: Belief change and argumentation in multiagent scenarios. Ann. Math. Artif. Intell. 78(3–4), 177–179 (2016) Dung, P.M.: On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games. Artif. Intell. 77(2), 321–358 (1995). https://doi.org/10. 1016/0004-3702(94)00041-X Liao, B.S., Jin, L., Koons, R.C.: Dynamics of argumentation systems: a division-based method. Artif. Intell. 175(11), 1790–1814 (2011). https://doi.org/10.1016/j.artint.2011.03.006 Oikarinen, E., Woltran, S.: Characterizing strong equivalence for argumentation frameworks. Artif. Intell. 175(14–15), 1985–2009 (2011). https://doi.org/10.1016/j.artint.2011.06.003 Rienstra, T., Sakama, C., van der Torre, L.W.N.: Persistence and monotony properties of argumentation semantics. In: Third International Workshop on Theory and Applications of Formal Argumentation, TAFA. LNCS, vol. 9524, pp. 211–225. Springer, Cham (2015)

Updating Argumentation Frameworks for Enforcing Extensions Kang Xu, Beishui Liao and Huaxin Huang

Abstract There are mainly two research directions for the dynamic of argumentation: revision and enforcement. The former studies how the semantics of an argumentation framework changes when the framework is updated, while the later studies how to enforce a given status of arguments by updating the argumentation framework. In the second direction, little attention has been paid to the problem of providing general rules for updating an argumentation framework aiming to enforce an extension of arguments. In this paper, we address this problem by a notion of characterizing the updated argumentation framework with respect to a given set of arguments. Keywords Updating · Argumentation frameworks · Enforcement

1 Introduction Formal argumentation is a very active research area in the field of knowledge representation and reasoning. Within this area, Dung’s abstract argumentation (Dung 1995) has been extensively studied in the past two decades, including argumentation semantics, algorithms and computational complexity, dynamics, etc. Among them, the dynamic of argumentation is a relatively undeveloped area. Abstract argumentation can be modelled as a framework (A, R), where A represents a set of arguments and R a binary ‘attack’ relation. The reasoning of abstract argumentation is from an argumentation framework to a set of extensions, where every extension is a set of arguments and means a feasible choice of actions. There is a rich variety of semantics which define extensions, following with a lot of attentions K. Xu (B) Zhejiang University of Water Resources and Electric Power, Hangzhou, China e-mail: [email protected] B. Liao · H. Huang Center for the Study of Language and Cognition, Zhejiang University, Hangzhou, China e-mail: [email protected] H. Huang e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2019 B. Liao et al. (eds.), Dynamics, Uncertainty and Reasoning, Logic in Asia: Studia Logica Library, https://doi.org/10.1007/978-981-13-7791-4_4

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paid to the properties and interrelations between them (Baroni et al. 2011a, b). Among those who are interested in specifying sets of acceptable arguments of argumentation frameworks, some scholars started to fix their eyes on dynamics of argumentation frameworks. There are mainly two directions in this area. The first direction is called revision, i.e. how the semantics of an argumentation framework changes when the framework is updated, while the second direction is called enforcement, i.e. how to enforce a given status of arguments by updating the framework. Updating an argumentation framework (A, R) can only be changing A and R. In the direction of revision, Bisquert et al. proposed a typology to classify different properties describing a change operation (Bisquert et al. 2013). Oikarinen et al. studied strong equivalence between argumentation frameworks (Oikarinen and Woltran 2011). They defined that an argumentation framework is equivalent to another one, if and only if they have the same semantics whenever they expand in the same way. In the direction of enforcement, Bolla et al. proposed abstraction principles for the attack relation and the set of arguments of an argumentation framework, such that when an argument or an attack is removed, the extension remains unchanged under single-assigned semantics (Boella et al. 2009, 2010)1 . Cayrol et al. provided principles of updating an argumentation framework for satisfying various evolutions on semantics, with a constraint that only a single argument or attack is added or removed (Cayrol et al. 2010; Bisquert et al. 2013). Baumann et al. investigated whether enforcing an extension is possible. They specified necessary and sufficient conditions under which an enforcement is possible (Baumann and Brewka 2010), and characterized minimal modification of an argumentation framework for enforcing an extension with a numerical measure (Baumann 2012). The present approaches on enforcement stipulate that the change on an argumentation framework is carried out within one step: adding or deleting one argument or one attack (Boella et al. 2009, 2010), or one direction: expanding an argumentation framework (Baumann and Brewka 2010). None of them provides general rules for updating an argumentation framework without constraining the number of arguments and attacks. In this paper, we will propose a new approach to deal with this problem. More specifically, we will find rules for updating argumentation frameworks to enforce certain extensions and monotonicity. Here is an example. Example 1 Let AF1 be an argumentation framework as illustrated in Fig. 1. Let E = {b} be a set of arguments that needs to be enforced as an extension under grounded semantics. It is obvious that E is not a grounded extension of AF1 . One way to enforce E is deleting a from AF1 to get AF2 . When an agent wants someone else to accept a set of arguments in an argumentation, this set of arguments can be treated as a certain extension that needs to be enforced. Our work will enable us to choose the right way to revise argumentation frameworks in order to get the objective extension (outcome). 1 Single-assigned

semantics define only one extension. The opposite is multi-assigned semantics which define more than one extension.

Updating Argumentation Frameworks for Enforcing Extensions Fig. 1 The first example of enforcing an extension

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The remaining of this paper is structured as follows. Section 2 introduces some basic notions of abstract argumentation that will be used in this paper. Section 3 presents an approach to update an argumentation framework for enforcing an extension. Section 4 concludes.

2 Preliminaries To make this paper self-contained, in this section, we introduce some basic notions of abstract argumentation such as argumentation frameworks and semantics. Definition 1 An argumentation framework G is a tuple (A, R) where A is a set of arguments, and R ⊆ A × A is a binary relation on A. Let B ⊆ A and R B = R ∩ (B × B). We say that G ↓ B = (B, R B ) is the restriction of G to B. For all a, b ∈ A, (a, b) ∈ R means a attacks b, denoted as a Rb. For all B, C ⊆ A, B Ra (a R B) if and only if there exists b ∈ B such that b Ra (a Rb). B RC if and only  means if there exist b ∈ B and c ∈ C such that b Rc. R is the negation of R. a Rb B (B Ra)  means there is a series x1 , x2 , ... xn such that a Rx1 , x1 Rx2 , ... xn Rb. a R  (b Ra).    is the negation of R.  that there exists b ∈ B such that a Rb R Definition 2 Let G = (A, R) be an argumentation framework, and a set B ⊆ A. B b and b R B a. The G ↓ B is a circle of G if and only if for any argument a, b ∈ A, a R set of all circles of G is denoted as C I RG , and we denote that SC I RG = {B | G ↓ B is a circle of G }.  Example 2 Let AF3 be an argumentation framework illustrated in Fig. 2. C I R AF3 = {AF3 ↓{a,b} , AF3 ↓{b} , AF3 ↓{c,d} } and SC I R AF3 = {{a, b}, {b}, {c, d}}. Given an argumentation framework, a fundamental problem is to determine which arguments can be regarded as acceptable together. There are mainly two approaches:

Fig. 2 An example of circles in an argumentation framework

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extension-based approach and labelling-based approach. The idea underlying the extension-based approach is to identify sets of arguments, called extensions, that can be regarded as collectively acceptable according to some criterion. The idea underlying the labelling-based approach is to assign a label to each argument, according to a certain criterion. On the one hand, in the extension-based approach, under different semantics that represent different evaluation criteria, an argumentation framework may have different sets of extensions. The notion of admissible set is fundamental to the definitions of other semantics. It is defined on the basis of conflict-freeness and the defending relation. Definition 3 Let G = (A, R) be an argumentation framework, a, b ∈ A and E ⊆ A. E is conflict-free if and only if for all a, b ∈ E, a Rb. E defends a if and only if for any b Ra, B Rb. E is an admissible set if and only if E is conflict-free and each argument in E is defended by E. Definition 4 The characteristic function, denoted as F , of an argumentation framework G = (A, R) is defined as F : 2 A −→ 2 A with F (B) = {a | B defends a}. Based on the notion of admissible set and characteristic function, some argumentation semantics such as complete, grounded and preferred can be defined as follows. Definition 5 Let G = (A, R) be an argumentation framework, a ∈ A and E ⊆ A. • E is a complete extension if and only if E is admissible and F (E) ⊆ E. • E is a grounded extension if and only if E is the least complete extension with respect to set inclusion. • E is a preferred extension if and only if E is the maximal admissible extension with respect to set inclusion. • E is an ideal extension if and only if E is the greatest admissible extension included in any preferred extension of G . When an extension is requested to attack all arguments outside itself, it is called stable extension. Definition 6 Let G = (A, R) be an argumentation framework, a ∈ A and E ⊆ A. E is a stable extension if and only if E is an admissible extension and E R(A \ E). On the other hand, labelling-based approach is defined in terms of labellings. A labelling assigns a label to each argument to indicate the status of an argument. There are usually three labels: in, out and undec, where the label in indicates that the argument is accepted, the label out indicates that the argument is rejected and the label undec indicates that one abstains from an opinion on whether the argument is accepted or rejected (Baroni et al. 2011b).

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Definition 7 Let G = (A, R) be an argumentation framework. A labelling L is a total function L : A −→ {in, out, undec}. Let in(L) = {a | L(a) = in}, out (L) = {a | L(a) = out} and undec(L) = {a | L(a) = undec}, L is often represented as a triple (in(L), out (L), undec(L)). Let B ⊆ A. L ↓ B = (in(L) ∩ B, out (L) ∩ B, undec(L) ∩ B) is the restriction of L to B. One of the criteria for labelling-based semantics is whether a label assigned to an argument is legal. Definition 8 Let G = (A, R) be an argumentation framework, a ∈ A, and L be a labelling of G . • L(a) = in is legal if and only if for all b ∈ A, b Ra implies L(b) = out. • L(a) = out is legal if and only if there exists b ∈ A such that b Ra and L(b) = in. • L(a) = undec is legal if and only if – there exists b ∈ A such that b Ra and L(b) = undec. – it is not the case that there exists c ∈ A with b Ra and L(b) = in. Based on the notion of legal labellings, various labelling-based semantics can be defined as follows. Definition 9 Let G = (A, R) be an argumentation framework, and L be a labelling of G . L is admissible if and only if • for any argument a ∈ in(L), L(a) = in is legal. • for any argument b ∈ out (L), L(b) = out is legal. Definition 10 Let G = (A, R) be an argumentation framework, and L be a labelling of G . • L is a complete labelling if and only if it is an admissible labelling, and for any a ∈ undec(L), L(a) = undec is legal. • L is a grounded labelling if and only if it is a complete labelling, and in(L) is minimal among all complete labellings with respect to set inclusion. • L is a preferred labelling if and only if it is an admissible labelling, and in(L) is maximal among all complete labellings with respect to set inclusion. • L is an ideal labelling if and only if it is the greatest admissible labelling that is smaller than or equal to each preferred labelling. Here, we say that a labelling L is smaller than or equal to another labelling L  if and only if in(L) ⊆ in(L  ) and out (L) ⊆ out (L  ). • L is a stable labelling if and only if it is a complete labelling, and undec(L) = ∅. In this paper, we use σ to denote a semantics that could be admissible (ad), complete (co), preferred ( pr ), grounded (gr ), ideal (id) or stable (st). The set of all σ -labellings on G is denoted as Lσ (G ), and Lσ (G ) = {L | L is a σ -labelling of G }. It has been proved that any two ad-labellings of an argumentation framework have relations as follows.

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Proposition 1 Dunne and Bench-Capon (2002) Let G = (A, R) be an argumentation framework, and L 1 , L 2 be ad-labellings of G . • in(L 1 ) ⊆ in(L 2 ) if and only if out (L 1 ) ⊆ out (L 2 ). • in(L 1 ) ⊂ in(L 2 ) if and only if out (L 1 ) ⊂ out (L 2 ). Under each σ -semantics, there is a bijective correspondence between the set of labellings and the set of extensions of an argumentation framework (Baroni et al. 2011a). Given an argumentation framework, the label in can be understood as identifying the members of an extension in extension-based semantics. In the following part of this paper, we call in(L) a σ -extension while L is a σ -labelling.

3 Updating Argumentation Frameworks to Enforce Certain Extensions In this section, we study how to enforce an extension of an argumentation framework. More specifically, given an argumentation framework G = (A, R) and a set E ⊆ A, we identify conditions under which an updated argumentation framework G  = (A , R  ) has E as its one extension. Updating G to G  is just a process of adding a set of arguments or attacks, and deleting a set of arguments or attacks. Suppose G  is adding a set of arguments B and a set of attacks U to G , and deleting a set of arguments D and a set of attacks J from G , we stipulate that the adding and deleting sets are minimal, which means B ∩ A = ∅, U ∩ R = ∅, D ⊆ A and J ⊆ R. For the purpose of making arguments in E acceptable, under semantics σ , we define a notion of updated argumentation framework with E as its σ -extension, a notion borrowed from Liao et al. (2017) in the setting of formalizing semantics of probabilistic argumentation. Definition 11 Let G = (A, R) and G  = (A , R  ) be argumentation frameworks, and E ⊆ A. Under a semantics σ , if E is a σ -extension of G  , then we call G  is G ’s σ -updated argumentation framework with respect to E. Example 3 Let AF4 be an argumentation framework illustrated in Fig. 3, and E = {c}. AF41 and AF42 presented in Fig. 4 are AF4 ’s pr -updated argumentation frameworks with respect to E. AF41 is formed by deleting {a, b} and {(a, b), (b, a), (b, c)} from AF4 , and adding an empty set of arguments and an empty set of attacks to AF4 . AF42 is formed by deleting {a, b, d} and {(a, b), (b, a), (b, c), (c, d), (d, c), (d, d)}  from AF4 , and adding {e} and {(c, e), (e, e)} to AF4 . Given an argumentation framework G , updating G to enforce a σ -extension E depends on characterizing G  defined in Definition 11. E is the key point to the construction of G  . First, E is just requested as a σ -extension of G  , and thus whether E is an σ -extension of G is not important. Second, all σ -extensions are conflict-free, and thus E is at least conflict-free in G  .

Updating Argumentation Frameworks for Enforcing Extensions Fig. 3 An argumentation framework

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Fig. 4 Updated argumentation frameworks

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Generally speaking, three factors jointly make a contribution to the construction of G  : E, arguments attack E, and arguments attacked by E. In the remaining part of this paper, the following sets related to E within any argumentation framework G = (A, R) will be frequently used. • E G− = {x ∈ AE | x R E}, denoting the set of arguments in A that attacks E. • E G+ = {x ∈ AE | E Rx}, denoting the set of arguments in A that is attacked by E. • E G− \ E G+ , denoting the set of arguments in A that attack E but is not attacked by E. • IGE = A \ (E ∪ E G− ∪ E G+ ), denoting the set of arguments in A which is unrelated to E (neither attacks nor is attacked by E). The following parts of this section are the characterizations of updated argumentation frameworks under each σ -semantics, respectively.

3.1 Admissable Semantics Given two argumentation frameworks G and G  , if G  is G ’s ad-updated argumentation framework with respect to E, then there exists L ∈ Lad (G  ) such that in(L) = E. According to Definition 9, arguments in in(L) and out (L) are legally labelled. Then, for all y ∈ E G− , L(y) = out, and for all z ∈ E G+ , L(z) = out. According to Definition 8, we get E attacks E G− , i.e. E G− ⊆ E G+ . E G− ⊆ E G+  is the necessary condition for G  being G ’s ad-updated argumentation framework with respect to E. On the other side, E G− ⊆ E G+ with E is conflict-free are the sufficient conditions for G  being G ’s ad-updated argumentation framework with respect to E. This is because they imply that {E, E G+  , IGE  } is an ad-labelling of G  . The following theorem shows how to construct G  based on G and E. Theorem 1 Let G = (A, R) be an argumentation framework, and E ⊆ A. Suppose that G  = (A , R  ) is constructed by minimally adding a set of arguments B and a set of attacks U to G , and deleting a set of arguments D and a set of attacks J from G , G  is G ’s ad-updated argumentation framework if and only if

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• R ∩ (E × E) ⊆ J and E U E. • For all x ∈ E G− \ E G+ , ({x} × E) ∩ R ⊆ J with x U E, or EU x. • For all y ∈ E G+ , if E U y and (E × {y}) ∩ R ⊆ J , then ({y} × E) ∩ R ⊆ J and y U E. • for all z ∈ IGE ∪ B, if zU E, then EU z. Proof (⇒) Suppose G  is G ’s ad-updated argumentation framework, then there  R  E. Then R ∩ (E × exists L ∈ Lad (G  ) such that in(L) = E. From Definition 3, E − E) ⊆ J and E U E. From Definitions 8 and 9, E G  ⊆ out (L). Then E G− ⊆ E G+ , moreover, E G− \ E G+ = ∅. For all x ∈ E G− \ E G+ , if ({x} × E) ∩ R  J or xU E, and E U x, then x ∈ E G− \ E G+ , contradicting E G−  \ E G+ = ∅. For all y ∈ E G+ , suppose E U y and (E × {y}) ∩ R ⊆ J , and furthermore suppose ({y} × E) ∩ R  J or yU E, then y ∈ E G− \ E G+ , contradicting E G− \ E G+ = ∅. For all z ∈ IGE ∪ B, suppose zU E, then z R  E and L(z) = out. Then E R  y, and then EU y. (⇐) Suppose all the items in Theorem 1 are satisfied, it is sufficient to prove that there exists L ∈ Lad (G  ) such that in(L) = E. Let L = (E, E G+  , A \ (E ∪ E G+ )) be a labelling of G  . For any x ∈ E G+ , there exists y ∈ E such that y R  x. Since L(y) = in, then L(x) = out is legal. Since J ⊆ R ∩ (E × E) and E U E, then E is conflict-free in G  . For any x ∈ E G− , x ∈ (E G− \ E G+ ) \ D or x ∈ E G+ \ D or x ∈ (IGE ∪ B) \ D. If x ∈ (E G− \ E G+ ) \ D, then x ∈ E G− \ E G+ . Then ({x} × E) ∩ R ⊆ J  R  E. Both of them imply x ∈ / E G− \ E G+ . with x U E, or EU x. Then E R  x or x + + + Then x ∈ E G  . If x ∈ E G \ D, then x ∈ E G . The third condition implies x ∈ E G+ . If x ∈ (IGE ∪ B) \ D, then x ∈ IGE ∪ B. Then xU E, and then EU x. Then x ∈ E G+ . Thus for any x ∈ E G− , x ∈ E G+ . Thus for any y ∈ E, L(y) = in is legal. From Definitions 8 and 9, L ∈ Lad (G  ). All items in Theorem 1 make sure that E is conflict-free in G  and E G− ⊆ E G+ , which are the sufficient and necessary conditions for G  beging G ’s ad-updated argumentation framework with respect to E. We use B, U , D and J to indicate what R  is, and this shows how to construct G  by deleting or adding arguments or attacks, and furthermore provides the rules for updating G to enforce an ad-extension E. Example 4 Let AF5 be an argumentation framework illustrated in Fig. 5, and E = {a, d}. According to Theorem 1, AF51 illustrated in Fig. 6 is AF5 ’s ad-updated argumentation framework with respect to E.  Fig. 5 An argumentation framework

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3.2 Complete Semantics Given two argumentation frameworks G = (A, R) and G  = (A , R  ), if G  is G ’s coupdated argumentation framework with respect to E, then there exists L ∈ Lco (G  ) such that for all in(L) = E, furthermore, for all s ∈ E G+ , L(x) = out. As a colabelling is also an ad-labelling, G  is first G ’s ad-updated argumentation framework with respect to E. Then we know that E G− ⊆ E G+ . Then IGE = AG  \ (E ∪ E G+ ). From Definition 10, apart from arguments in in(L) and out (L) are legally labelled, arguments in undec(L) should be legally labelled too. In this case, for all x ∈ IGE , L(x) = undec should be legal. From Definition 8, arguments legally labelled undec should not be attacked by arguments labelled in and at least be attacked by one argument labelled undec. Then for all x ∈ IGE , x is attacked by IGE . This is the necessary condition for G  being G ’s co-updated argumentation framework with respect to E. On the other side, if for all x ∈ IGE , x is attacked by IGE , and G  is G ’s ad-updated argumentation framework with respect to E, then {E, E G+  , IGE  } is G  ’s co-labelling. Based on Theorem 1, G  can be described as follows. Theorem 2 Let G = (A, R) be an argumentation framework, and E ⊆ A. Suppose that G  = (A , R  ) is constructed by minimally adding a set of arguments B and a set of attacks U to G , and deleting a set of arguments D and a set of attacks J from G , G  is G ’s co-updated argumentation framework with respect to E if and only if • G  is G ’s ad-updated argumentation framework with respect to E. • For all x ∈ IGE  , IGE  R  x. Proof (⇒) Suppose G  is G ’s co-updated argumentation framework with respect to E, then there exists L ∈ Lco (G  ) such that in(L) = E. Since a co-labelling is also an ad-labelling, then G  is G ’s ad-updated argumentation framework with respect to E. Then E G− ⊆ E G+ . Then IGE = A \ (E ∪ E G+ ), out (L) = E G+ and undec(L) = IGE .  R  x, then either x is unattacked or x is only Suppose there exists x ∈ IGE such that IGE + attacked by E ∪ E G  . If x is unattacked, then L(x) = undec is illegal, a contradiction. If x is only attacked by E ∪ E G+ , since L(x) = undec, then x only can be attacked by E G+ . Then L(x) = undec is illegal, a contradiction. Thus, for all x ∈ IGE , IGE  R  x. (⇐) Suppose G  is G ’s ad-updated argumentation framework with respect to E, and for all x ∈ IGE , IGE R  x, it is sufficient to prove there exists L ∈ Lco (G  ) such that in(L) = E. Let L = (E, E G+  , A \ (E ∪ E G+ )) be a labelling of G  . Since G  is G ’s ad-updated argumentation framework with respect to E, then arguments in E and E G+ are legally labelled. E G− \ E G+ = ∅: if not, then E G− \ E G+ ⊆ A \ (E ∪ E G+  ). Then for all y ∈ E G− \ E G+ , L(y) = undec, contradicting to Definition 8. Since E G− \ E G+ = ∅, then IGE = A \ (E ∪ E G+ ). Then for all x ∈ A \ (E ∪ E G+ ), L(x) = undec is legal. Thus L ∈ Lco (G  ). Comparing Theorems 1, 2 adds a new condition stipulating that IGE  is selfattacked, which makes sure that all arguments unrelated to E are legally labelled undec. Theorem 2 provides sufficient and necessary conditions for G  being G ’s co-updated argumentation framework with respect to E.

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Example 5 Consider AF5 in Example 4. Let E = {a, d}. AF51 in Fig. 6 is an adupdated argumentation framework with respect to E. According to Theorem 2, {e, f } should be self-attacked. AF52 illustrated in Fig. 7 is one of the solutions. AF52 is AF5 ’s co-updated argumentation framework with respect to E.

3.3 Grounded Semantics Given two argumentation frameworks G and G  , if G  is G ’s gr -updated argumentation framework with respect to E, then there is L ∈ Lgr (G  ) such that in(L) = E. As a gr -labelling is also complete, G  first is G ’s co-updated argumentation framework with respect to E. Then we know that E G− ⊆ E G+ . From Definition 10, the grounded extension of G  is the least complete extension in which for all L  ∈ Lco (G  ), in(L) ⊆ in(L  ), then we can conclude that L ↓ E∪E +  is G  ↓ E∪E +  ’s gr -labelling. G G This is a necessary condition for G  being G ’s gr - updated argumentation framework with respect to E. On the other side, if L ↓ E∪E +  is G  ↓ E∪E +  ’s gr -labelling, then there is no smaller G G co-extension of AF  ↓ E∪E +  than E. Since G  is G ’s co-updated argumentation G framework with respect to E, then {E, E G+  , IGE  } is G  ’s gr -labelling. In Modgil and Caminada (2009), Modgil and Caminada provided a labelling algorithm for the grounded semantics. This algorithm starts by assigning in to all arguments in an argumentation framework that are not attacked, and then iteratively assign out to any argument that is attacked by an argument that has been made in, and in to those arguments all of whose attackers are out. The iteration continues until no more new arguments are made in or out. Based on the assignment progress, we find that confronting the circles of an argumentation framework, if any circle is attacked by arguments assigned by in, then the semantics of it will be affected, otherwise all arguments in it will be undec. Thus, to make L ↓ E∪E +  G  ↓ E∪E +  ’s G G gr -labelling, all circles in G  ↓ E∪E +  should be attacked by E. G Based on Theorem 2, G  can be described as follows. Theorem 3 Let G = (A, R) be an argumentation framework, and E ⊆ A. Suppose that G  = (A , R  ) is constructed by minimally adding a set of arguments B and a set of attacks U to G , and deleting a set of arguments D and a set of attacks J from G , G  = (A , R  ) is G ’s gr -updated argumentation framework with respect to E if and only if

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• G  is G ’s co-updated argumentation framework with respect to E. • For all C ∈ SC I RG  ↓ E∪E + , (E \ C)R  C. G 

Proof (⇒) Suppose G  is G ’s gr -updated argumentation framework with respect to E, then there exists L ∈ Lgr (G  ) such that in(L) = E. Since a gr -labelling is a co-labelling, G  is G ’s co-updated argumentation framework with respect to E. Then E G− ⊆ E G+ and out (L) = E G+ . From Definitions 8 and 10, L ↓ E∪E +  is a co-labelling G  R  V , then there of G  ↓ E∪E +  . Suppose there exists V ∈ SC I RG  ↓ E∪E + such that E G

G 

will be another co-labelling of G  ↓ E∪E +  L  such that in(L  ) ⊂ in(L ↓ E∪E +  ), a G G contradiction. Thus, for all C ∈ SC I RG  ↓ E∪E + , E R  C. G  (⇐) Suppose the two conditions are satisfied, it is sufficient to prove that there exists L ∈ Lgr (G  ) such that in(L) = E. Let L = (E, E G+  , A \ (E ∪ E G+ )) be a labelling of G  . Since G  is G ’s co-updated argumentation framework with respect to E, then L is a co-labelling of G  . From Definitions 8 and 10, L ↓ E∪E +  is a coG labelling of G  ↓ E∪E +  . Suppose there exists L  ∈ Lco (G  ) such that in(L  ) ⊂ E, G then from Proposition 1, out (L  ) ⊂ out (L). Then there exists V ∈ SC I RG  ↓ E∪E + , G 

and for all x ∈ V , L  (x) = undec and for all y ∈ VG− , L  (y) = out. Then for all y ∈ VG− , L(y) = out, a contradiction. Thus L is a gr -labelling of G  . Based on Theorems 2, 3 adds a new condition to make sure that L ↓ E∪E +  is G G  ↓ E∪E +  ’s gr -labelling and avoids computing semantics of G  ↓ E∪E +  . FurtherG G more, it makes L = (E, E G+  , IGE  ) G  ’s grounded labelling. Theorem 3 provides sufficient and necessary conditions for G  being G ’s gr -updated argumentation framework with respect to E. Example 6 Consider AF5 in Example 4. Let E = {a, d}. AF52 in Fig. 7 is a co-updated argumentation framework of AF5 with respect to E. According to Theorem 3, AF52 ↓{d,e} should be attacked by an argument in E and outside {d, e}. Thus AF52 is not a gr -updated argumentation framework with respect to E. The adjusted argumentation framework AF53 illustrated in Fig. 8 satisfies Theorem 3, and is AF5 ’s gr -argumentation framework with respect to E.

3.4 Preferred Semantics Given two argumentation frameworks G and G  , if G  is G ’s pr -argumentation framework with respect to E, then there exists L ∈ L pr (G  ) such that in(L) = E. Fig. 8 An updated argumentation framework of AF5

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As a pr -labelling is also admissible, G  first is G ’s ad-argumentation framework with respect to E. Then we know that E G− ⊆ E G+ , and furthermore IGE  = AG  \ (E ∪ E G+ ). From Definition 10, any pr -extension of G  is a maximal admissable extension, which means there exists no L  ∈ Lad (G  ) such that in(L  ) ⊃ in(L). According to Proposition 1, there is no L  ∈ Lad (G  ) such that undec(L  ) ⊂ undec(L). Then there should be no arguments labelled in under admissible semantics in IGE  . This is the necessary condition for G  being G ’s pr -updated argumentation framework with respect to E. On the other side, if there are no arguments labelled in under admissible semantics in IGE  , then with the condition that G  is G ’s ad-argumentation framework, E and E G+ make the maximal acceptable and rejectable sets of arguments under admissible semantics. The following theorem shows how to construct G  based on Theorem 1. Theorem 4 Let G = (A, R) be an argumentation framework, and E ⊆ A. Suppose that G  = (A , R  ) is constructed by minimally adding a set of arguments B and a set of attacks U to G , and deleting a set of arguments D and a set of attacks J from G , G  = (A , R  ) is G ’s pr -updated argumentation framework with respect to E if and only if • G  is G ’s ad-updated argumentation framework with respect to E. • Lad (G  ↓ I E  ) = {(∅, ∅, IGE  )}. G

Proof (⇒) Suppose G  is G ’s pr -updated argumentation framework with respect to E, then there exists L ∈ L pr (G  ) such that in(L) = E. Since a pr -labelling is an ad-labelling, G  is G ’s ad-updated argumentation framework with respect to E, furthermore, IGE = A \ (E G+ ∪ E). Then E G− ⊆ E G+ . Then for all y ∈ (IGE  )− , y ∈ E G+ . Suppose there exists L  ∈ Lad (G  ↓ I E  ) such that in(L  ) = ∅, then from G Definitions 8 and 9, L 1 = L ↓ A \I E  ∪L  is G ’s ad-labelling. Then in(L 1 ) ⊃ in(L), G contradicting that L is a pr -labelling of G  . Thus, Lad (G  ↓ I E  ) = {(∅, ∅, IGE  )}. G (⇐) Suppose the two conditions are satisfied, it is sufficient to prove that there exists L ∈ L pr (G  ) such that in(L) = E. Let L = (E, E G+  , A \ (E ∪ E G+ )) be a labelling of G  . Since G  is G ’s ad-updated argumentation framework with respect to E, then L is an ad-labelling of G  , and E G−  \ E G+ = ∅. Then IGE  = A \ (E ∪ E G+ ). Since Lad (G  ↓ I E  ) = {(∅, ∅, IGE  )}, then there is no more arguments in IGE  can be G labelled in or out. Thus L is a pr -labelling of AF  . Based on Theorems 1, 4 makes sure that A \ (E ∪ E G+ ) has no accepted arguments under admissible semantics. In this case, L = (E, E G+  , IGE  ) is a pr -labelling of G  . Theorem 4 provides sufficient and necessary conditions for G  beging G ’s pr -updated argumentation framework with respect to E. Example 7 Consider AF5 in Example 4. Let E = {a, d}. Figure 6 shows an adupdated argumentation framework of AF5 with respect to E. According to the second item in Theorem 4, c and f should be undec. AF52 illustrated in Fig. 7 also satisfies this condition, and makes itself AF5 ’s pr -updated argumentation framework with respect to E.

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3.5 Stable Semantics Given two argumentation frameworks G = (A, R) and G  = (A , R  ), if G  is G ’s st-argumentation framework with respect to E, then there exists L ∈ Lst (G  ) such that in(L) = E. As an st-labelling is also admissible, G  first is G ’s ad-updated argumentation framework with respect to E. Then from Theorem 1, E G− ⊆ E G+ , and then out (L) = E G+ . Furthermore, undec(L) = IGE . From Definition 10, no arguments in G  are labelled undec under stable semantics, i.e. undec(L) = ∅. Then IGE = ∅ which is a necessary condition for G  being G ’s st-updated argumentation framework with respect to E. On the other side, if IGE  = ∅ and G  is G ’s ad-updated argumentation framework with respect to E, then {E, E G+  , IGE  } is the st-labelling of G  . Based on Theorem 1, G  is constructed as follows. Theorem 5 Let G = (A, R) be an argumentation framework and E ⊆ A. Suppose that G  = (A , R  ) is constructed by minimally adding a set of arguments B and a set of attacks U to G , and deleting a set of arguments D and a set of attacks J from G , G  is G ’s st-updated argumentation framework with respect to E if and only if • G  is G ’s ad-updated argumentation framework with respect to E. • IGE  = ∅. Proof (⇒) Suppose G  is G ’s st-updated argumentation framework with respect to E, then there exists L ∈ Lst (G  ) such that in(L) = E. Since an st-labelling is also an ad-lebelling, then G  is G ’s ad-updated argumentation framework with respect to E. Then E G−  ⊆ E G+ , and out (L) = E G+ . Then undec(L) = IGE . Since undec(L) = ∅, then IGE = ∅. (⇐) Suppose the two conditions are satisfied, it is sufficient to prove that there exists L ∈ Lst (G  ) such that in(L) = E. Let L = (E, E G+  , A \ (E ∪ E G+  )) be a labelling of G  . Since G  is G ’s ad-updated argumentation framework with respect to E, then E G− ⊆ E G+ , and L is an ad-labelling of G  . Then A \ (E ∪ E G+  ) = IGE = ∅. Then L is an st-labelling of G  . Based on Theorems 1, 5 makes sure that A \ (E ∪ E G+ ) = ∅. Thus, it makes sure that L = (E, E G+  , IGE  ) is an st-labelling of G  . Theorem 5 provides sufficient and necessary conditions for G  being G ’s st-updated argumentation framework with respect to E. Example 8 Consider AF5 in Example 4. Let E = {a, d}. Figure 6 shows an adupdated argumentation framework of AF5 with respect to E. According to the second item of Theorem 5, c and f should be deleted from AF51 . AF54 illustrated in Fig. 9 is AF5 ’s st-argumentation framework with respect to E, and L (AF54 ) = {({a, d}, {b, e}, ∅), ({a, e}, {b, d}, ∅)}.

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Fig. 9 An updated argumentation framework of AF5

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3.6 Ideal Semantics Given two argumentation frameworks G = (A, R) and G  = (A , R  ), if G  is G ’s idargumentation framework with respect to E, then there exists L ∈ Lid (G  ) such that in(L) = E. As the id-labelling is also admissible, G  first is G ’s ad-argumentation framework with respect to E. Then we know that E G− ⊆ E G+ , and then IGE = A \ (E ∪ E G+  ). From Definition 10, the ideal extension of G  is the maximal admissible extension which is included in every preferred extension. This means there exists no L  ∈ Lad (G  ) such that in(L) ⊃ in(L  ) and in(L  ) includes in every pr -extension. We separate G  into two parts: G  ↓ E∪E +  and G  ↓ I E  . G  ↓ I E  should make sure G G G that its ideal labelling stipulates no accepted arguments, and G  ↓ E∪E +  should be G identified as a framework with id-extension E under the influence of G  ↓ I E  . This G is described in the following theorem. Theorem 6 Let G = (A, R) be an argumentation framework and E ⊆ A. Suppose that G  = (A , R  ) is constructed by minimally adding a set of arguments B and a set of attacks U to G , and deleting a set of arguments D and a set of attacks J from G , G  is G ’s id-updated argumentation framework with respect to E if and only if • G  is G ’s ad-updated argumentation framework with respect to E. • Lid (G  ↓ I E  ) = {(∅, ∅, IGE  )}. G • Lid (G  ↓(E∪E +  )\(I E  )+ ) = {(E, E G+  \ (IGE )+ , ∅)}. G

G

Proof (⇒) Suppose G  is G ’s id-updated argumentation framework with respect to E, then there exists L ∈ Lid (G  ) such that in(L) = E. Since id-labelling is also an ad-lebelling, then G  is G ’s ad-updated argumentation framework with respect to E. Then E G− ⊆ E G+ . Then out (L) = E G+ , undec(L) = IGE and IGE only has attack relations with E G+ . Since L ∈ Lid (G  ), then L pr (G  ) ↓ I E  = L pr (G  ↓ I E  ). Then there is no id-labelling of G  ↓ I E  with accepted G G G arguments, i.e. Lid (G  ↓ I E  ) = {(∅, ∅, IGE  )}. As undec(L) = IGE  , then for any G x ∈ (IGE )+ , L(x) = out. Then L pr (G  ) ↓(E∪E +  )\(I E  )+ = L pr (G  ↓(E∪E +  )\(I E  )+ ). G G G G Then Lid (G  ↓(E∪E +  )\(I E  )+ ) = {(E, E G+  \ (IGE )+ , ∅)}. G G (⇐) Suppose the two conditions are satisfied, it is sufficient to prove that there exists L ∈ Lid (G  ) such that in(L) = E. Let L = (E, E G+  , A \ (E ∪ E G+ )) be a labelling of G  . Since G  is G ’s ad-updated argumentation framework with / Lid (G  ), then for the respect to E, then L is an ad-labelling of G  . Suppose L ∈    real L ∈ Lid (G ), either in(L) ⊂ in(L ) or in(L) ⊃ in(L  ). If in(L) ⊂ in(L  ), then Lid (G  ↓(E∪E +  )\(I E  )+ ) = {(E, E G+  \ (IGE  )+ , ∅)}, a contradiction. If in(L) ⊃ G

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in(L  ), then Lid (G  ↓ I E  ) = {(∅, ∅, IGE  )}, a contradiction. Thus L is an id-labelling G of G  . The second item of Theorem 6 makes sure that AG  \ (E ∪ E G+ ) has no accepted arguments under ideal semantics. As we all know that if an argumentation framework is attacked by an arguments labelled out, then its semantics is uninfluenced. G  ↓ E∪E +  is influenced by IGE , and all the arguments directly attacked by IGE are G labelled out, thus the third item of Theorem 6 picks up the set of (E ∪ E G+ ) \ (IGE  )+ . Furthermore, G  ↓ E∪E +  is identified that E is its id-extension under the influence G of G  ↓ I E  . Both of the two items make L = (E, E G+  , IGE  ) from an ad-labelling to G an id-labelling of G  . Theorem 6 provides sufficient and necessary conditions for G  beging G ’s id-updated argumentation framework with respect to E (Fig. 10). Example 9 Consider AF5 in Example 4. Let E = {a, d}. According to Theorem 6, AF53 illustrated in Fig. 8 is AF5 ’s id-updated argumentation framework with respect to E, while AF52 illustrated in Fig. 7 is not, because Lid (AF52 ↓{d,e} ) = {∅, ∅, {d, e}} But we can correct it as AF55 in the following figure, where Lid (AF52 ↓{d,e} ) = {(d, e, ∅)}.

4 Conclusion In this paper, we have proposed a new approach to address the problem regarding how to provide general rules for updating an argumentation framework to enforce an extension without constraints on the number of arguments and attacks. The main part of this paper is Sect. 3, in which we first define the σ -updated argumentation framework with respect to E. E is the extension being enforced, and σ can be admissible, complete, grounded, preferred, stable and ideal. After giving the definition, we characterize each σ -updated argumentation framework with respect to E to show how to update an argumentation framework. For admissible semantics, we give a way to add a set of arguments B and a set of attacks U to an argumentation framework, and delete a set of arguments D and a set of arguments J from it. From complete semantics to ideal semantics, what we do is to put more conditions on an ad-updated argumentation framework with respect to E. When characterizing pr and id-updated argumentation frameworks with respect to E, computing the semantics of sub-frameworks is not avoid, while under the other semantics, every

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updated argumentation framework can be described only by its structure. Although needed to compute semantics, from Liao et al. (2011), it is always easier to compute semantics of an argumentation framework with less arguments and attacks. Thus, to some extent, we provide a way to simplify updating an argumentation framework for enforcement of an extension. There is some related work in existing literature. Among them, Baumann et al. (2010) specified impossibility and possibility results for enforcing desired extensions. It is very related to our work. A basic difference is that we focus on identifying conditions under which an updated argumentation framework has the desired extensions. The research in this paper is the first step towards the study of enforcement. It is easy to expand our work to reserving extension if we confine E as an σ -extension of the original argumentation framework G . Furthermore, we can explore it in addressing enforcement of monotonicity (Bisquert et al. 2013). Acknowledgements The authors are grateful to the anonymous reviewers for their helpful comments. The research reported in this paper was partially supported by the Fundamental Research Funds for the Central Universities of China for the project Big Data, Reasoning and Decision Making, and the National Social Science Foundation of China (No.18ZDA290, No.17ZDA026).

References Baroni, P., Caminada, M., Giacomin, M.: An introduction to argumentation semantics. Knowl. Eng. Rev. 26(4), 365–410 (2011a) Baroni, P., Caminada, M., Giacomin, M.: Review: an introduction to argumentation semantics. Knowl. Eng. Rev. 26(4), 365–410 (2011b) Baumann, R.: What does it take to enforce an argument? Minimal change in abstract argumentation. In: The 20th European Conference on Artificial Intelligence (2012) Baumann, R., Brewka, G.: Expanding argumentation frameworks: enforcing and monotonicity results. pp. 75–86. COMMA, IOS Press (2010) Bisquert, P., Cayrol, C., de Saint-Cyr, F.D., Lagasquie-Schiex, M.C.: Characterizing change in abstract argumentation systems. In: Fermé, E., Gabbay, D., Simari, G. (eds.) Trends in Belief Revision and Argumentation Dynamics, vol. 48, pp. 75–102. College Publications (2013) Boella, G., Kaci, S., van der Torre, L.: Dynamics in argumentation with single extensions: abstraction principles and the grounded extension. In: European Conference on Symbolic and Quantitative Approaches to Reasoning and Uncertainty, ECSQARU (LNAI 5590) (2009) Boella, G., Kaci, S., van der Torre, L.: Dynamics in argumentation with single extensions: attack refinement and the grounded extension (extended version). In: Argumentation in Multi-Agent Systems, pp. 150–159 (2010) Cayrol, C., de Saint-Cyr, F.D., Lagasquie-Schiex, M.C.: Change in abstract argumentation frameworks: adding an argument. J. Artif. Intell. Res. 38, 49–84 (2010) Dung, P.M.: On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games. Artif. Intell. 77(2), 321–357 (1995) Dunne, P.E., Bench-Capon, T.: Coherence in finite argument systems. Artif. Intell. 141, 187–203 (2002) Liao, B., Jin, L., Koons, R.C.: Dynamics of argumentation systems: a division-based method. Artif. Intell. 175, 1790–1814 (2011)

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Liao, B., Xu, K., Huang, H.: Formulating semantics of probabilistic argumentation by characterizing subgraphs: theory and empirical results. J. Log. Comput. (2017). https://doi.org/10.1093/logcom/ exx035 Modgil, S., Caminada., M.: Proof theories and algorithms for abstract argumentation frameworks. In: Rahwan, I., Simari, G. (eds.) Argumentation in Artificial Intelligence, pp. 105–129. Springer, Boston (2009) Oikarinen, E., Woltran, S.: Characterizing strong equivalence for argumentationg frameworks. Artif. Intell. 175(14–15), 1985–2009 (2011)

Open Reading and Free Choice Permission: A Perspective in Substructural Logics Huimin Dong, Norbert Gratzl and Olivier Roy

Abstract This paper proposes a new solution to the well-known Free Choice Permission Paradoxes (Barker 2010; Hansson 2013; Xin and Dong 2014), combining ideas from substructural logics and non-monotonic reasoning. Free choice permission is intuitively understood as “if it is permitted to do α or β then it is permitted to do α and it is permitted to do β.” Yet, one of its logically equivalent forms allows the following inference which seems unacceptable: if it is permitted to order a vegetarian lunch then it is permitted to order a vegetarian lunch and not pay for it (Hansson 2013). The challenge for a logic of free choice permission is to exclude such counterintuitive consequences while not giving up too much deductive power. We suggest that the right way to do so is using a family of substructural logics augmented with a principle borrowed from non-monotonic reasoning. This follows up on a proposal made in Anglberger et al. (2014). Keywords Free choice permission · Deontic logic · Substructural logics · Non-monotonic reasoning

1 Motivation This paper proposes a new solution to the Free Choice Permission Paradox (Barker 2010; Hansson 2013; Xin and Dong 2014), combining ideas from substructural logics and non-monotonic reasoning. Free choice permission is intuitively understood as “if it is permitted to do α or β then it is permitted to do α, and it is permitted to do H. Dong (B) Department of Philosophy, Zhejiang University, Hangzhou, China e-mail: [email protected] N. Gratzl MCMP, LMU Munich, Munich, Germany e-mail: [email protected] O. Roy Institut für Philosophie, Universität Bayreuth, Bayreuth, Germany e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2019 B. Liao et al. (eds.), Dynamics, Uncertainty and Reasoning, Logic in Asia: Studia Logica Library, https://doi.org/10.1007/978-981-13-7791-4_5

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β” (Hansson 2013). This is usually formalized as follows: P(α ⊎ β) → Pα ∧ Pβ

(FCP)

where ⊎ stands for some form of disjunction of action types. There are many wellknown problems associated with FCP. In this paper, we focus on three of them. First, in many deontic systems adding FCP allows for a form of conjunctive inference which seems unacceptable: if it is permitted to order a vegetarian lunch, then it is permitted to order lunch and not pay for it. This is the so-called “vegetarian free lunch” example (Hansson 2013). Second, many deontic logics become resourceinsensitive in the presence of FCP. They validate inferences of the form “if the patient with stomach trouble is allowed to eat one cookie then he is allowed to eat more than one,” which are also counterintuitive. Third, in its classical form, FCP entails that classically equivalent formulas can be substituted to the scope of a permission operator. This is also implausible: It is permitted to eat an apple or not iff it is permitted to sell a house or not. The challenge for a logic of free choice permission is to exclude such counterintuitive consequences while not giving up too much deductive power. In other words, we need a suitable nonclassical calculi for free choice permission. We suggest one way of doing this is using a family of substructural logics augmented with a principle borrowed from non-monotonic reasoning. The solution that we put forward in this paper is to build a family of logics including FCP on top of a plausible calculus for action types. This calculus must, in our view, be at the same time relevant, non-monotonic, and resource-sensitive. Two of the authors of this paper have already made that suggestion in Anglberger et al. (2014). The present contribution is a technical follow-up. After briefly rehearsing the arguments in favor of a nonclassical calculus of action types, we develop a family of such substructural logics which we augment with principles from non-monotonic reasoning to preserve enough deductive power. We provide sound and complete axiomatizations of these logics and study their proof theory.

2 Background Concepts This section will circumscribe the understanding of FCP that we will study in this paper by correlating it with many different views of permissions: strong permission, weak permission, explicit permission/implicit permission, the “open reading.”

2.1 Strong Permission Free choice permission is often argued as going hand in hand with the idea of strong permission (von Wright 1963; Asher and Bonevac 2005). The notion of strong permission goes back at least to von Wright who stated that an action is strongly permit-

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ted “if the authority has considered its normative status and decided to permit it (von Wright 1963).”1 Recall that in contrast to strong permission, weak permission is defined as the absence of prohibition, i.e., the dual of obligation in standard deontic logic (McNamara, 2010, 2006). Due to these presence/absence features, von Wright roughly classified strong permission as explicit permission and weak permission as tacit permission (von Wright 1981): I think we are well advised to distinguish between things being permitted in the weak sense of simply not being forbidden and things being permitted in some stronger sense. Exactly in what this stronger sense “consists” may be difficult to tell. That which is in the strong sense permitted is, somehow, expressly permitted, subject to norm and not just void of deontic status altogether.

It has been defended that strong permission satisfies free choice, whereas weak permission does not (von Wright 1968; Dignum et al. 1996). More precisely, it is strongly permitted to do α or β, iff it is strongly permitted to do α and it is strongly permitted to do β; while it is weakly permitted to do α or β, iff it is weakly permitted to do α or it is weakly permitted to do β. In von Wright (1968, p. 31), von Wright claimed that strong permission is closely associated with free choice permission. In this paper, we do not take a stand on whether strong permission as illustrated above is free choice permission or the other way around. We do observe, however, that the permissions that we study here share some essential features with strong permissions: they are explicit, they are not the dual of obligations, and they, of course, satisfy a restricted form of FCP.

2.2 Open Reading The so-called open reading of permission is defined as follows: An action type α is permitted iff each instantiation of α is normatively okay. (OR∗ ) The open reading provides a “sufficiency” reading (van de Putte 2019): an instantiation of a permitted type is sufficient for being okay (van Benthem 1979; Trypuz and Kulicki 2010; Kulicki and Trypuz 2019; Anglberger et al. 2015; Dong and Roy 2015). The open reading thus becomes: if an instantiation of α is an instantiation of β, we conclude that if it is permitted to do β then it is also permitted to do α. We propose OR+ as a formal representation of the “sufficiency” feature of OR∗ : α ⊸ β  Pβ → Pα 1 Subsequent

(OR+ )

literature has not always used “strong permissions” in the same sense as von Wright. Asher and Bonevac (2005) use the term in a way which is closer to what we call the “open reading” in Sect. 1.2. In Makinson and van der Torre (2003), two different senses of explicit permissions, static and dynamic, are distinguished and studied. In this paper, we will use “strong permission” in von Wright’s sense unless otherwise specified.

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Here  is a logical consequence relation between deontic statements and ⊸ a nonclassical conditional on action types. Pβ → Pα, on the other hand, is read as “If type β is permitted then so is type α.” The form OR+ captures the core idea of OR∗ : whether α is a permitted action type depends on whether action type α is sufficient for action type β, and whether type β is permitted. In Sect. 2.3, we will argue that the conditional ⊸ should not be classical. FCP follows from OR+ if ⊸ is a material implication (Anglberger et al. 2014, 2015; Dong and Roy 2015). Given that an instantiation of type α is an instantiation of this disjunctive type α ⊎ β, if type α ⊎ β is permitted then type α is also permitted. The “sufficiency” component of open reading goes back to open interpretation in dynamic deontic logic (Broersen 2004) and disjunctive permissions in conditional logic (Hilpinen 1982; Makinson 1984). The open reading also underlies the analysis of rational permission in games developed in Dong and Roy (2015). One of the driving ideas of this paper is to take a controlled version of OR∗ , in the form of OR+ , as the core of free choice permission. In this controlled version, the conditional ⊸ on actions should not be classical.

2.3 Problematic Inferences with Free Choice Permission and Possible Solutions Using OR+ , the three types of undesired counterintuitive consequences of free choice permission can be illustrated as special cases of two general patterns of free choice permission inference: disjunctive and conjunctive free choice inferences. Disjunctive inferences are the canonical forms of inferences involving FCP. The premise in a disjunctive inference contains “or” inside the scope of permission. Here is one instance of the typical disjunctive inferences: (1a) It is permitted to eat an apple or eat a pear. ∴ (1b) It is permitted to eat an apple, and (1c) it is permitted to eat a pear. In the presence of FCP inferences of this type will be valid if the conditionals α ⊸ α ⊎ β and β ⊸ α ⊎ β are valid. As already alluded to having ⊸ logically behave like a material implication in OR+ leads to several problematic disjunctive free choice inferences. It entails in particular unlimited substitutions of classically equivalent formulas (Xin and Dong 2014). An instance derived from OR+ is the following: α ⧟ β  Pα ↔ Pβ

(P-E)

where α ⧟ β means that α ⊸ β and β ⊸ α, and Pα ↔ Pβ means that Pα → Pβ and Pβ → Pα. This principle P-E is a form of the substitution principle. If ⧟ is strong enough to make classical tautologies equivalent, then “eating an apple or not is permitted iff selling a house or not is permitted.” Yet these permissions seem

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different. The permission to sell a house or not involves a different authority than the permission to eat an apple or not. And thus, eating an apple or not is, in our view, irrelevant to selling a house. This means that inferences involving action types are relevant2 in the logical sense of the term (Weingartner and Schurz 1986; Schurz 1991). Authors in that tradition have argued that the root of (some) classical paradoxes is the rules governing (classical) disjunction introduction. In deontic logic, the paradigmatic example is Ross’ paradox. Generally speaking, an inference rule “From ϕ infer ψ” is prone to introducing irrelevance iff ψ contains a subformula which can be replaced in a validity-preserving way by an arbitrary formula. This also holds for reasoning about actions types. The inference from “smoking” to “smoking or walking and notwalking” seems intuitively not valid. In our view, a plausible calculus of entailment between action type should thus give up the substitution of classically equivalent formulas on the ground of introducing irrelevance. By conjunctive inference, we mean the inference of the following form: (2a) It is permitted to order a vegetarian lunch. ∴ (2b) It is permitted to order a vegetarian lunch and not pay for it. whose abstract form is shown as follows:  Pα → P(α ◦ β)

(CI)

where β is an arbitrary action type and ◦ is nonclassical fusion operator. If α ◦ β ⊸ α is valid, then CI is valid under OR+ . The classical “vegetarian free lunch” example is one instance of CI. Moreover, CI is logically equivalent to FCP under OR+ if ⊸ is classical.3 This is not the case in the logic that we develop below. One argument for this is the observation that statements about actions types typically refer to the normal instances of that type. Such statements are made against a set of conventional assumptions (McCarthy 1977; Makinson 2005). This means that inferences about action types will typically be non-monotonic. A normal instance of “ordering vegetarian” might not be a normal instance of “ordering vegetarian and not pay for it.” There is an extensive literature in formal semantics showing how to handle this phenomenon when it comes to action sentences (Pelletier and Asher 1997; Zimmermann 2000; Simons 2005). The logics we study here follow these insights and treat entailment between action types using restricted forms of monotonicity. CI also encounters problems if ⊸ is resource-insensitive. If eating one cookie is permitted, for a patient with stomach trouble, we should not be able to conclude that eating more than one is also permitted. In this case, the classical “contraction” law α ◦ α ⧟ α indicates an unrestricted resource composition. It is by now fairly 2 We take “irrelevance” in a different sense, but still very close to relevant logic (Restall 2006): only

α ⊸ β ⊎ ∼β is not valid in our logic. Our logics validate α ◦ ∼α ⊸ β, which is rejected in relevant logic. In this paper, ∼ and ¬ are different negations, and ∼ will be seen as the negation for actions. 3 By taking α ⧟ (α ◦ β) ⊎ (α ◦ ∼β) and α ⧟ (α ⊎ β) ◦ (α ⊎ ∼β) as classical validities, the logical equivalence between CI and FCP holds after using the substitution of logical equivalences.

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common to hold, however, that linguistic or informational resources cannot be reused or discarded at will (Asudeh 2019; van Benthem 1995). One particular form of substructural logic, i.e., linear logic (Di Cosmo and Miller 2016) has been developed precisely to deal with this phenomena. They typically reject contraction. As our example shows, this applies to action types as well, and this will be reflected in the logics that we develop below. Our solution to the problems above related to FCP and CI is a semantic one. We insist on OR∗ as the semantic core of permissions, and propose a new interpretation for the nonclassical conditional ⊸ as well as for OR+ , in which the conditional is required to be relevant, non-monotonic, and resource-sensitive. By doing this follow the suggestions made by Barker (2010) and two of the authors of the present paper in Anglberger et al. (2014), and move to a substructural interpretation of the conditional and the conjunction. In doing so, one should, however, be careful not to put the deductive barrier too high. In the words of van Benthem (2004, p. 95): “This is like turning down the volume on your radio so as not to hear the bad news. You will not hear much good news either.” Indeed, as we have seen earlier, there are still plausible cases of FCP inferences. Our strategy to do so is to borrow one fundamental principle from non-monotonic logic. We will see that this provides just enough deductive power to keep control of the FCP.

3 Rational Monotonicity for Action Types To retain some deductive power and with it some plausible free choice permission inferences, we will examine two principles from non-monotonic reasoning. The first option is the following weakly monotonic principle from non-monotonic reasoning: (α ⊸ γ ) ∧ (α ⊸ β)  α ◦ β ⊸ γ

(CaM+ )

where, again, fusion ◦ is a nonclassical counterpart of the classical conjunction ∧. This principle is called cautious monotony. It means that, if action type α entails action type γ , and also action type β, then the composite action type α and β still entails γ . Using CaM+ , the selling example can be accommodated as follows. Suppose that selling the house to Ann entails not selling the house to Bob. Applying CaM+ we conclude that selling the house to Ann and not selling the house to Bob entails selling the house. Accepting, however, CaM+ reintroduces the kind of resource-insensitivity that we want to avoid in this paper. We can infer α ◦ α ⊸ α by applying α ⊸ α on CaM+ . Therefore, this weakly monotonic principle does not help to solve the problems we want to address. Instead, we consider a closely related principle: rational monotony. Rational monotony states if action type α entails action type γ , and it is not the case that α entails any actions except β, then the composition of α and β still entails action type γ . We read ∼β as the action type “any actions except β,” and it is equal

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to saying β ⊸ 0, where 0 is the “fail” action type.4 Formally, rational monotony is presented in the following form: (α ⊸ γ ) ∧ ¬(α ⊸ ∼β)  α ◦ β ⊸ γ

(RaM+ )

Under a classical reading of conjunction and implication, rational monotony implies cautious monotony. This is not the case here.

4 A Substructural Calculus of Action Types and Permissions The stage is now set to develop a family of sound and complete substructural logics for actions and permissions. We argued earlier that, in the context of free choice permission, the calculus of actions should belong to the family of substructural logics. All logics in this family exclude the familiar structural rules for left-hand weakening, mingle, and cautious monotony. They can thus avoid the unrestricted monotonic “vegetarian free lunch” example and the resource-insensitive cases. Because the negation on action types is defined by the nonclassical conditional ⊸ together with the impossible action type 0, the irrelevant α ⊎ ∼α ⧟ β ⊎ ∼β fails too. Furthermore, to restore the deductive power, one of these logics will derive RaM+ .

4.1 Substructural Logics for Actions and Permissions The presentation in this section is close to the standard systems and the standard semantics in Restall (2000a). Notice that the following is a calculus of actions as types and permissions as propositions. The language is then of two sorted: one part covers well-formed types for actions, and the other well-formed formulas for permissions. Definition 1 (Types and Formulas) The set L A of well-formed types of actions and the set Ln of well-formed formulas of norms are defined as follows: L A : α∷ = 0 | a | α ⊎ α | α ◦ α | ∼α Ln : ϕ∷ = α ⊸ α | Pα | ¬ϕ | ϕ → ϕ where a ∈ Act0 the (countable) set of action generators, and α ∈ L A . Let L = L A ∪ Ln . The set L A of well-formed action types are constructed out of the set Act0 of primitive action types and the impossible action 0, by taken together with the disjoint 4 In

dynamic logic, “fail” action cannot change anything (Harel et al. 2000; Dignum et al. 1996), and thus it “causes no harm” (Kamp 1973).

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union ⊎ for Boolean disjunction, the fusion ◦ for the nonclassical conjunction, and ∼ for the action negation. So α ⊎ β is a choice between actions α and β, read as “doing α or doing β.” And α ◦ β is an action composition, read as “doing α together with β.”5 The action negation ∼α is read as “doing any actions except α.” This is not a De Morgen negation. Neither α ⊸ ∼∼α nor its converse ∼∼α ⊸ α hold.6 The set Ln of well-formed formulas includes two kinds of atomic formula. The first kind of atomic formula is constructed by the binary conditional ⊸ over action types, which is understood as nonclassical entailment. The set LB of “action formulas” is defined as taking all action types and closing under entailment. Namely, LB : ϕ:: = α ∈ L A | α ⊸ α Observe that the classical disjunction ∨ and the classical conjunction ∧ are definable as usual: ϕ ∨ ψ := ¬ϕ → ψ, and ϕ ∧ ψ := ¬(ϕ → ¬ψ). So are the false constant ⊥ := Pa ∧ ¬Pa and the true constant := ¬⊥. We use the Greek letters α, β, γ , . . . to represent action types, and ϕ, ψ, χ , . . . to represent formulas. Sometimes, in certain particular situations, we use the capital letters A, B, C, . . . to encompass either action types or well-formed formulas, if no confusion arises. In addition, we use A  B to indicate the two cases: A ∨ B or A ⊎ B. We can then introduce another important syntactic component in substructural logics: structures. In our logics, structures are categorized into two sorts, corresponding to action formulas in LB and deontic formulas in Ln . Definition 2 (Structures and Substructures) The set S of structures is defined as follows: §B : X ∷ = ψ | X ; X where ψ ∈ LB § : X ∷ = ϕ | Y | X, X where ϕ ∈ Ln and Y ∈ §B Given a structure X ∈ §, the set Sub(X ) of all substructure of X is defined by induction as follows: • • • •

X If If If

∈ Sub(X ); X ∈ Ln ∪ LB , then X ∈ Sub(X ); X = Z ; Z  ∈ §B , then Z ∈ Sub(X ) and Z  ∈ Sub(X ); X = Z , Z  ∈ §, then Z ∈ Sub(X ) and Z  ∈ Sub(X ).

Observe that L ⊆ SB ⊆ S. And then the set S of structures contains the set SB of all action structures. The semicolon is a binary punctuation mark over action formulas ψ ∈ LB . Usually, X ; Y is read as “a structure X combines with a structure 5 The composition operator ◦ is not the sequent composition operator in propositional dynamic logic

(PDL) Harel et al. (2000, p. 168). This operator ◦, sometimes, may be understood as a non-standard concurrency operator of actions Harel et al. (2000, p. 268, 276). For example, Listen ◦ W riteN ote. We suggest reading α ◦ β as “doing action α and action β (together).” 6 Later, we will see that neither the double negation introduction  α ⊸ ∼∼α nor the double negation elimination  ∼∼α ⊸ α are valid in the (bi)-frame class.

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Y .” The comma is a binary punctuation mark over well-formed formulas ϕ ∈ Ln . Then X, Y is read as “a structure X and a structure Y .” Here, we use X, Y, Z , H, . . . to represent structures in S. We write X [Y ] as the structure X with a substructure Y which occurs in it at least one time, i.e., Y ∈ Sub(X ), while writing X [Z /Y ] as the structure X replacing at most one occurrence of the substructure Y by Z .7 For instance, α ◦ ∼α is a substructure for the structure ((α ◦ ∼α; α); α ◦ ∼α), and then ((α ◦ ∼α; α); α ◦ ∼α)[0/α ◦ ∼α] = ((α ◦ ∼α; α); 0). Moreover, for simplification, we use 1≤i≤n X i to indicate the structure (X 1 , . . . , X n ). When X i ∈ L A in 1≤i≤n X i is an action type, then it is obvious that n = 1. The models for our logics are the standard models in substructural logics (Restall 2000a). The basic ontological entities in these models are states, which, as we will show in Sect. 4.4, are consistent but possibly incomplete. Here states play two roles. They instantiate action types and give truth value to well-formed formulas. In other words, action types can be executed or non-executed at states, while formulas can be true or false at states. Moreover, two different types of relations should be contained in the substructural models. One is the ternary relation L, which is understood as a semantic counterpart of the entailment relation. Since we are now mainly interested in developing the technical aspects of the nonclassical calculus of action types, we leave open the specific interpretation of the ternary relation. The second main relation is the binary relation O for interpreting (free choice) permission. O x y is read as “a state x is normatively fine w.r.t. a state y.” Definition 3 (Models) A model is a tuple M = W, L , O, V  where • • • •

W is a non-empty set of states. L ⊆ W × W × W is a ternary relation on W . O ⊆ W × W is a binary relation on W . V : Act0 → ℘ (W ) is a valuation function.

The O-relation is a normative standard to evaluate the deontic facts at each state,8 while the L-relation is used to interpret our nonclassical connectives. As usual, a frame for our substructural logics is a model minus the valuation V . A class of frames satisfying the first-order property r is called the r -frame class. Observe that the language L is two sorted, and so are the truth conditions. Now we can interpret action types and formulas in L and structures in S as in substructural logics (Restall 2000a). Definition 4 (Truth Conditions) Let M = W, L , O, V  be a model. A well-formed type α ∈ L A is brought about at state w in model M, written M, w |= α, iff

7 Although after one replacement by substructure Z to Y the possibility of X [Z /Y ] is not unique, it will not affect the sequent calculus introduced later. 8 Notice that O-relation is not a serial relation as in standard deontic logic does. One reason is that we have not considered obligation, and to interpret it using O. Another reason is that seriality is not necessary for the characterization of (free choice) permission. The consistency is ensured by the action operators rather than the deontic one. Please refer to Theorem 9 for more details.

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M, w M, w M, w M, w M, w

|= a iff |= 0 for no |= α ⊎ β iff |= α ◦ β iff |= ∼α iff

w ∈ V (a) w∈W M, w |= α or M, w |= β ∃y, z ∈ W (L yzw, M, y |= α & M, z |= β) ∀y, z ∈ W (M, y |= α ⇒ ¬Lwyz)

Now we define the truth conditions for ϕ ∈ Ln : ϕ is true at state w in model M, written M, w |= ϕ, iff M, w M, w M, w M, w

|= α ⊸ β iff |= Pα iff |= ¬ϕ iff not |= ϕ → ψ iff

∀y, z ∈ W (Lwyz & M, y |= α ⇒ M, z |= β) ∀y ∈ W (M, y |= α ⇒ O yw) M, w |= ϕ M, w |= ϕ ⇒ M, w |= ψ

So we can extend the truth conditions to all structures such that M, w |= X ; Y iff ∃y, z ∈ W (L yzw, M, y |= X & M, z |= Y ) M, w |= X, Y iff M, w |= X & M, w |= Y Action types are interpreted via the concept “being brought about,” as what have been done in the standard literature of action logic (Horty 2000; Perloff and Xu 2001; Segerberg et al. 2016). So V (a) is a set of states that can bring about action generator a ∈ Act0 , in the sense that w ∈ V (a) indicates that at w some instantiation of action generator a is brought about. Then α ⊎ β is brought about at state w iff w is either sees α or β is brought about. The nonclassical connectives for action types are for the fusion ◦ and the action negation ∼, which are interpreted using the ternary relation. So α ◦ β is executed at w, iff there exist y and z such that α is brought about in y, β in z, and L yzw. For instance, ordering a lunch together with paying for it is brought about at w, iff there are two states y and z such that in y a lunch is ordered, in z the meal is paid, and L yzw. The action negation ∼α is instantiated at w iff, if y has α being brought about, then it is not the case that Lwyz. As usual well-formed formulas are interpreted via the concept of truth. The truth condition of ⊸ uses the ternary relation. α ⊸ β is true at w iff if Lwyz and α is brought about at y entails that β is brought about at z. The truth conditions for ¬ and → are defined by V as usual. Now we define a new relation ⊆ over states as follows: s ⊆ u iff M, s |= A implies M, u |= A where A ∈ LB . So s = u iff s ⊆ u and u ⊆ s. The truth conditions for structures are similar as for formulas. Validity is defined as usual. A structure X is valid in a model M, denoted as M |= X , iff M, w |= X for all w ∈ W . M ∈ F indicates that M is a model based on a frame F with a valuation function. A structure X is valid in a frame F, denoted as F |= X , iff M |= X for all M ∈ F. We define X |=F A iff if M, w |= X then M, w |= A where M ∈ F and w ∈ M. We define the turnstile  ⊆ S × L as the basic syntactic consequence of actions and permissions. The consequence X  A is a consequence from X ∈ S to A ∈ L,

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and is read as “if X then A,” where X is the antecedent and A is the succedent. We call A  B the equivalent consequence when A  B and B  A. A rule of consequences is a set of ordered pairs of consequences, encoded in the following way: X 1  A1 , . . . , X n  An XA where n ∈ ω. The upper consequences are called premises, and the lower consequence is called the conclusion. If n = 0, then a rule becomes an axiom. If n = 1, then we call it a proper rule.9 If X [Y ]  A is one of the premises and X [Z ]  A is the conclusion, then X [Z ]  A means X [Z /Y ]  A. If X [Y ]  A and X [Z ]  A both are premises, then they are X [Y/ X  ]  A and X [Z / X  ]  A where X  ∈ Sub(X ). An instance of a rule is a member of this set. An instance of a given premise of a rule is a member of an instance of this rule w.r.t. the given premise. An instance of the conclusion is similar. A sequent calculus, a system, or a logic consists of axioms and rules (Negri et al. 2008; Paoli 2013). See Table 1 on page 12 for the sequent calculus N for action type and permission. See Table 2 on page 12 for the additional rules. We define derivations and their heights in Definitions 5 and 6, which are inspired by the notions in Negri et al. (2008), Paoli (2013). They are the key notions for cut elimination. Definition 5 (Derivations) X  A is a derivation in a system, iff 1. either X  A is an instance of an axiom in this system 2. or there are derivations X 1  A1 , . . . , X n−1  An−1 as instances of premises of a rule in this system such that X  A is the instance of the conclusion, where n > 1. X  A is a theorem of a system iff it is a derivation in this system. It is also said to be a derived consequence. A rule is derivable in a system iff given its premises of one instance, and its conclusion is a theorem of this system. Definition 6 (Height of Derivations) X  A is a derivation with the height n ∈ ω in a system, denoted as X n A, iff 1. either n = 0 and X  A is an instance of an axiom in this system 2. or n = k1 + · · · + kn−1 + 1 and there are derivations X 1 k1 A1 , . . ., X n−1 kn−1 An−1 as premises of an instance of a rule in this system such that X  A is the conclusion. So an instance of an axiom is a derivation with the height 0. In Table 1, the left rules and right rules are used to introduce operators or modality from the upper consequences to its derived consequence. The left rules show how to introduce an operator or a modality on the left side of  and the right rules show how to introduce an operator or a modality in the right side of . For instance, (◦L) and (◦R) are the left introduction and the right introduction for the fusion ◦. If no confusion arises, it is no harm to omit the brackets to identify rules. 9 As

usual, when it is a double line, then the lower one is the consequent and the upper one is the conclusion.

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Table 1 The sequent calculus N for action types and permissions (Id) a  a where a ∈ Act0 (◦L) (⊎L)

X [α; β]  γ X [α ◦ β]  γ

X [α]  γ X [β]  γ X [α ⊎ β]  γ (∼L)

(⊎R1 )

X α Y [β]  γ Y [α ⊸ β; X ]  γ

X α⊸β X, Pβ  Pα (¬L) (→ L)

(Den)

(◦R)

X α Y β X; Y  α ◦ β X α X α⊎β

X α , for all γ ∈ L A Y [∼α; X ]  γ

(⊸L) (OR)

(0) X [0]  γ , for all γ ∈ L A

(⊎R2 )

(∼R)

(⊸R)

X φ Y [ψ]  χ Y [φ → ψ, X ]  χ

X α⊸β Y β ⊸γ X, Y  α ⊸ γ

(¬R)

(0) X  P0

X, φ  ⊥ X  ¬φ

(→ R) (Cut)

X; α  0 X  ∼α

X; α  β X α⊸β

(FC) Pα, Pβ  P(α ⊎ β) X φ Y [¬φ, X ]  ψ

X β X α⊎β

X, φ  ψ X φ→ψ

XA Y [A]  B Y [X ]  B

Table 2 Additional rules (BI) (CaM)

Z [X ; Y ]  A Z [Y ; X ]  A

(B)

H [(X ; Y ); Z ]  A H [X ; (Y ; Z )]  A

X α⊸β Y α⊸γ X, Y  α ◦ β ⊸ γ

(RaM)

(M)

H [X ]  A H [X ; X ]  A

X α⊸γ X, ¬(α ⊸ ∼β)  α ◦ β ⊸ γ

System S is a subsystem of system S’, or system S’ is an extension of system S, denoted as S ⊆ S’, iff T h(S) ⊆ T h(S ), where T h(S) is the set of all theorems of system S. Now we define the other systems according to the additional rules in Table 2. System NE contains all axioms in N and is closed under all rules in N together with BI and B (which are called exchange rules). System NERaM contains all axioms in NE and is closed under all rules in NE together with RaM. System NEM contains all axioms in NE and is closed under all rules in NE together with M. System NEMCaM contains all axioms in NEM and is closed under all rules in NEM together with CaM. We therefore have two families of substructural logics of

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actions and permissions, and their relations are presented in the following branching chains: N ⊆ NE

⊆ NERaM ⊆ NEM ⊆ NEMCaM

All substructural logics in the upper branch exclude left-hand weakening, mingle, and cautious monotony. The lower branch shows how cautious monotony correlates with mingle, and why it is resource-insensitive. In addition, the upper family of substructural logics has the following useful properties. Theorem 1 1. ID is a theorem of N: AA

for all A ∈ LB .

2. (OR+ ) is a theorem of N: α ⊸ β  Pβ → Pα 3. Action composition is commutative and associative in NE: A; B  C B; A  C

(A; B); C  D A; (B; C)  D

4. The following rule is derivable in N: (⊥) X [⊥]  φ for all φ ∈ Ln 5. The following rules are also derivable in N: (+0) ∼α  α ⊸ 0 (−0) α ⊸ 0  ∼α 6. We have the following consequences in N: (0) X  0 ⊸ α for any α ∈ L A (OR) X  Pα → P0 for any α ∈ L A Proof 1. We will show that A  A for all A ∈ LB . • If A = a ∈ Act0 , by (Id), we have a  a. • If A = 0, by (0), we have 0  0. • If A = β ⊎ γ , then the proof is γ γ ββ (⊎R1 ) (⊎R2 ) β β ⊎γ γ β ⊎γ (⊎L) β ⊎γ β ⊎γ

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• If A = β ◦ γ , then the proof is ββ γ γ (◦R) β; γ  β ◦ γ (◦L) β ◦γ β ◦γ • If A = β ⊸ γ , then the proof is ββ γ γ (⊸L) β ⊸ γ;β  γ (⊸R) β ⊸γ β ⊸γ 2. This derivation goes by the rules (OR) and (→ R). 3. The derivations go by BI and B. 4. The derivation goes as follows: By applying ID, α ⊸ α  α ⊸ α (OR) α ⊸ α, Pα  Pα (¬L) X [α ⊸ α, Pα, ¬Pα]  ψ (Definition) X [⊥]  ψ 5. The derivations go as follows: • For the first one, according to (ID), we have αα (∼L) ∼α; α  0 (⊸R) ∼α  α ⊸ 0 • For the second one, according to (ID), we have αα 0  0 (⊸L) α ⊸ 0; α  0 (∼R) α ⊸ 0  ∼α 6. The first consequence is derived by (0) and (⊸R). The second is implied from (0) and (OR). Theorem 2 1.  α ⊸ α is invalid. 2. α ⊸ β, β ⊸ γ  α ⊸ γ is a theorem of N. 3. α ⊸ γ , ¬(α ⊸ ∼β)  α ◦ β ⊸ γ is a theorem of NERaM. Proof We only prove the first case here. The other two are clear using Den and RaM. Construct a model M = W, L , O, V  s.t. W = {x, y, z}, L x yz, O = ∅, and V (a) = {y}. Obviously M, x ⊧̸ a ⊸ a.

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4.2 Standard Translation and Frame Correspondence We now study the standard translation and frame correspondence for L A . We follow the framework suggested in Kurtonina (1995, p. 29), and consider OR’s frame condition, as well as for rules Den, BI, B, M, CaM, and RaM’s. First of all, we recall the standard first-order translation in the natural correspondence language for L A . Definition 7 (Standard Translation) := STx (a) := STx (0) STx (α ⊎ β) := STx (α ◦ β) := STx (α ⊸ β) := := STx (Pα) := STx (¬ϕ) STx (ϕ → ψ) := STx (X ; Y ) := STx (X, Y ) :=

A(x) A(x) ∧ ¬A(x) STx (α) ∨ STx (β) ∃yz(L yzx ∧ STx (α)[x := y] ∧ STx (β)[x := z]) ∀yz(L x yz ∧ STx (α)[x := y] → STx (β)[x := z]) ∀y(STx (α)[x := y] → O yx) ¬STx (ϕ) STx (ϕ) → STx (ψ) ∃yz(L yzx ∧ STx (X )[x := y] ∧ STx (Y )[x := z]) STx (X ) ∧ STx (Y )

It is straightforward to check that this standard translation is adequate (Blackburn et al. 2002). Now we turn to frame correspondence. We will present in turn the correspondents of OR, Den, BI, B, M, CaM, and RaM. As usual, given a rule R is valid in F iff when the premises of R are true in a given model of F then so is the conclusion. Now that a frame F satisfies a first-order property r is denoted as F |= r . Definition 8 (Correspondences) A frame condition (r) corresponds to a rule R if and only if for any frame F, F is an (r)-frame iff R is valid in F. We now consider the frame correspondents of our key rules. Definition 9 (Open Reading) F is an open-reading frame iff F satisfies the following condition: ∀wz[∀y(Lwzy → O yw) → Ozw] (or) The open-reading frame condition can be taken as a closure of the normative relation O along the ternary relation L. Theorem 3 (or) corresponds to OR: α ⊸ β |=F Pβ → Pα iff F |= ∀wz[∀y(Lwzy → O yw) → Ozw] Proof The proof is a standard application of the method of minimal valuations developed, for instance, in Sahlqvist correspondence theory, c.f. (Blackburn et al. 2002). We illustrate it for this case. For the Left-to-Right direction, we adopt the contrapositive method in the constructive argument in Kurtonina (1995, p. 43). Suppose

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α ⊸ β |=F Pβ → Pα. That is, given a frame F, for any valuation V w.r.t. F and any state w ∈ F, if w ∈ / V (Pβ → Pα) then w ∈ / V (α ⊸ β). Let A, B be predicates denoting the sets of possible states where α, β hold, respectively. This is reflected in the corresponding second-order formula ∀AB∀w[∀y(By → O yw) ∧ ∃z(Az ∧ ¬Ozw) → ∃su(Lwsu ∧ As ∧ ¬Bu)] This formula is equal to ∀AB∀wz∃ysu((By → O yw) ∧ Az ∧ ¬Ozw → Lwsu ∧ As ∧ ¬Bu) Then we define a minimal valuation V ∗ w.r.t. F as follows: • V ∗ (α) = {v | v = z} • V ∗ (β) = {v | Ovw} This corresponds to the syntactic substitution • A∗ v := v = z • B ∗ v := Ovw The required frame condition is obtained by instantiation: ∀wz∃ysu((B ∗ y → O yw) ∧ A∗ z ∧ ¬Ozw → Lwsu ∧ A∗ s ∧ ¬B ∗ u) After the instantiation, the first-order condition becomes ∀wz∃su(¬Ozw → Lwsu ∧ s = z ∧ ¬Ouw) This can be simplified as ∀wz∃u((Lwzu → Ouw) → Ozw) As universal second-order formulas imply all their instantiations, this shows that the original OR rule implies this frame property. But also conversely, the Right-to-Left direction, we can prove that OR is true in each frame satisfies condition (or). Let M, w |= α ⊸ β. We need to show that M, w |= Pβ → Pα. Suppose M, w |= Pβ. Let M, z |= α. Assume that it is not the case that Ozw. Applying this to (or), it follows that ∃y s.t. Lwzy and ¬O yw. From Lwzy, M, z |= α and M, w |= α ⊸ β, we have M, y |= β. But we already know that ¬O yw. It implies that M, w ⊧̸ Pβ, which contradicts our assumption M, w |= Pβ. Definition 10 (L-Density) F is an L-dense frame iff F satisfies the following condition: ∀x yz[L x yz → ∃u(L x yu ∧ L xuz)]

(den)

Open Reading and Free Choice Permission: A Perspective in Substructural Logics Fig. 1 Two main compositions of normative expectation starting at x

s

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u

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y z

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Theorem 4 The (den) condition corresponds to Den: α ⊸ β, β ⊸ γ |=F α ⊸ γ iff F |= ∀x yz[L x yz → ∃u(L x yu ∧ L xuz)] Proof The argument uses the same techniques as the previous correspondence argument. Let F = W, L , O be a frame. Then • F is a (bi)-frame iff it satisfies that ∀x yz(L x yz → L yx z). • F is a (b)-frame iff it satisfies that ∀x yzw[L x(yz)w → L(x y)zw]. • F is a (mig)-frame iff it satisfies that ∀x yz(L x yz → x = z ∨ y = z). The following shorthands will now be useful. These are illustrated in Fig. 1. • L x(su)z := ∃y(L x yz ∧ Lsuy). • L x y(su) := ∃z(L x yz ∧ Lzsu). Definition 11 (Cautious Monotonicity) F is a cautiously monotonic frame iff F satisfies the following condition: ∀x zsu[L x(su)z ∧ ∀z  (L xsz  → z  ∩ u = ∅) → L xsz]

(cam)

Theorem 5 The (cam) condition corresponds to CaM: α ⊸ β, α ⊸ γ |=F α ◦ β ⊸ γ iff F |= ∀x zsu[L x(su)z ∧ ∀z  (L xsz  → z  ∩ u = ∅) → L xsz] Proof The argument proceeds again like in Theorem 3. The details are available on request. Definition 12 (Rational Monotonicity) Let a tuple F = W, L , O be a frame. F is a rationally monotonic frame iff F satisfies the following condition: ∀xsuzu  [L x(su)z ∧ L(xs)uu  → L xsz]

(ram)

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Theorem 6 The (ram) condition corresponds to RaM: α ⊸ γ , ¬(α ⊸ ∼β) |=F α ◦ β ⊸ γ iff F |= ∀xsuzu  [L x(su)z ∧ L(xs)uu  → L xsz] Proof We apply the same method as above. The details are again available on request. The two last results can be used to show that (ram) and (cam) are logically independent in this framework. Notice that the idea of incompleteness (shown in Theorems 8 and 9) makes it possible to have a model to invalidate this property: M, x |= α ⊸ β ⇔ M, x ⊧̸ α ⊸ ∼β. Take the counterexample in Theorem 8 (or, refer to Fig. 2). The state x1 makes pay ⊸ ∼ pay and pay ⊸ pay both true. Of course, if the model is complete, then this will not happen.

4.3 Soundness and Completeness Now we move on to the proof of soundness and completeness of the logics N, NE, NERaM, NEM, and NEMCaM with respect to the class of frames just defined. We define the so-called proper theories as canonical states. They are based on structures rather than formulas (as L ⊆ S). Most of the results in this section follow from applying standard techniques from modal and substructural logics. We nevertheless include the details of some of the key arguments, for illustration. Definition 13 (Proper Theories and Consequents) • A set w of structures S is a theory for a logic iff X ∈ w and X  B in this logic imply B ∈ w, where X ∈ S and B ∈ L. • A theory w is prime iff α ⊎ β ∈ w implies α ∈ w or β ∈ w. • A theory w is fine iff [X ∈ w and X  ∈ w iff (X, X  ) ∈ w]. • A theory w is trivial iff either 0 or ⊥ is contained in w; otherwise, it is nontrivial. • A theory w is normatively empty iff P0 ∈ / w; otherwise, it is normatively nonempty. • A proper theory is a normatively non-empty, nontrivial, fine, and prime theory. • A ∈ L is a consequent of a theory w, denoted as w  A, iff ∃X 1 , . . . , X n ∈ w s.t. X 1 , . . . , X n  A where X i ∈ S and 1 ≤ i ≤ n. Observe that Definition 13 is a revision of those in Restall (2000a, p. 89–90). Now we construct the canonical model as follows. To illustrate the relation between structures and formulas, we first translate the ;-structures into the ◦-formulas, and the ,-structures into the ∧-formulas. More precisely, S : S → LB is a translation from structures to formulas defined as follows:

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⎧ if X = ϕ ∈ LB ⎨ϕ S(X ) = S(Y ) ◦ S(Z ) if X = (Y ; Z ) ∈ SB ⎩ S(Y ) ∧ S(Z ) If X = (Y, Z ) ∈ / SB We can observe that X  S(X ) using the rules in above. We then borrow the important notion the so-called -pairs for consequences and the relevant results from Restall (2000b, p. 93), in order to replace the standard notion of consistency and maximality in modal logic (Blackburn et al. 2002). This step gives a general version of Lindenbaum Lemma, as shown in Lemma 4 later. Definition 14 (-Pairs) An ordered pair x, y of sets of formulas is called a -pair if and only if ¬∃X 1 , . . . , X n ∈ x∃B1 , . . . , Bm ∈ y s.t.  X i   B j . 1≤i≤n

1≤ j≤m

The turnstile  is the consequence relation defined earlier. Definition 15 (Pair Extension) A -pair u, v extends x, y, denoted as x, y ⊆ u, v, if and only if u ⊇ x and v ⊇ y. Lemma 1 (Theories and -Pairs) If x is a theory s.t. A ∈ / T , then x, {A} is a -pair. Definition 16 (Full -Pair) A -pair x, y is a full -pair if and only if x ∪ y = S. Lemma 2 (Proper Theories from Full -Pairs) If x, y is a full -pair, then x is a proper theory. Proof The proving strategy here is very similar to that in Restall (2000b). We first need to show that x is a theory. Assume that X ∈ x and X  B. If B is not in x, then B ∈ y because of that this -pair is full. But then this contradicts the assumption that x, y is a -pair. Second, we will show that x is prime. Let α ⊎ β ∈ x. If α, β are not in x, then, again, α, β ∈ y since the pair x, y is full. But we know that α ⊎ β  α ⊎ β, against our assumption that x, y is a pai. We then show that x is fine. First let substructures X, X  both be in x. Suppose the structure (X, X  ) is not in x, then it must be in y. We already know that X, X   S(X ) ∧ S(X  ). This contradicts the assumption that x, y is a -pair. The argument for the converse direction is entirely similar. x is not trivial. Otherwise, either 0 or ⊥ is contained in x. But the axiom (0) tells us that 0  γ for any γ ∈ L A ∩ y, and the rule (¬L) tells us that ⊥  ψ for any ψ ∈ LB ∩ y. Of course either implies that x, y is not a -pair, which is not possible. x is not normatively empty. Suppose that P0 ∈ / x, and so is in y. Axiom (0) gives us directly that X  P0 for all S(X ) ∈ x, against our assumption.

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Lemma 3 (Step Lemma) If x, y is a -pair, then so is either x ∪ {X }, y or x, y ∪ {X } for X ∈ S. Proof Our proof is very close to that in Restall (2000a), except the differences in notation. Similarly, the rule (Cut) and the distribution of boolean operators on structures are used. We only need to translate the structures with their correspondent formulas when it is necessary, to keep the same format of consequence. Notice that our Step Lemma does not require “ is pair extension acceptable” as (Restall 2000a) does, because the rule and property mentioned in this proof are both contained in our system (already in the most basic system N). Lemma 4 (Pair Extension Lemma) Any -pair x, y can be extended into some full -pair v, u. Proof We enumerate X 1 , . . . , X n , . . . ∈ S. We define the series of -pairs xn , yn  as follows: 1. x0 , y0  = x, y  2. xn+1 , yn+1  =

xn ∪ {X n }, yn  if xn ∪ {X n }, yn  is a  -pair, xn , yn ∪ {X n } otherwise.

According to the Step Lemma, each xn+1 , yn+1  is a -pair if its predecessor is. Now we xn , yn   can see that for any xi , yi  where i ∈ ω it is a -pair. Then their limits  i∈ω xi , i∈ω yi  are also a -pair. Otherwise, it contradicts the previous claim. And this limit covers the whole structure S by the way it is constructed. Corollary 1 (Proper Theories Excluding Formulas) If w  A, then there is a proper / w . theory w ⊇ w s.t. A ∈ Proof Let w  A. By Definition 13, for any structures X 1 , . . . , X n ∈ w that X 1 , . . . , X n  A. This means that w, {A} is a -pair. From Lemma 4, w, {A} can be extended into a full -pair w , u   where w ⊇ w and u  ⊇ {A}. So w is the desired proper theory according to Lemma 2. It is obvious to see that A ∈ / w . Definition 17 (Canonical Model) Given a logic S, the canonical model MC w.r.t. S is a structure W C , L C , O C , V C  defined as follows: • • • •

W C be the set of all proper theories on S for S. L C wsu iff α ⊸ β ∈ w and α ∈ s imply β ∈ u. O C uw iff ∃α ∈ L A s.t. Pα ∈ w and α ∈ u. V C : AC T0 → ℘ (W C ) s.t. w ∈ V C (a) iff a ∈ w.

If no confusion has arisen, the notation of S is omitted for the related canonical model. Lemma 5 Let MC = W C , L C , O C , V C  be the canonical model, and define L C; suw as: X ∈ s and Y ∈ u imply S(X ) ◦ S(Y ) ∈ w, wher e X, Y ∈ SB . Then L C; is included in L C .

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Proof Given L C; x yz. We need to show L C x yz. Suppose α ⊸ β ∈ x and α ∈ y. Because L C; x yz, we then have α ⊸ β; α ∈ z. By (⊸L) we know that α ⊸ β; α  β.   It implies β ∈ z. Now we can conclude L C x yz. This lemma is similar to the lemma in Restall (2000a, p. 254) proved using the rule (⊸ L). Lemma 6 (Witness Lemma) Let MC = W, L , O, V  be the canonical model, we have 1. If X ; Y ∈ w, then exist s, u ∈ W s.t. Lsuw, X ∈ s and Y ∈ u. 2. If α ⊸ β ∈ / w, then exist s, u ∈ W s.t. Lwsu, α ∈ s and β ∈ / u. 3. If Pα ∈ / w, then exists u ∈ W s.t. α ∈ u and u ∈ / O C [w]. / w}. It is a -pair. Otherwise, Proof 1. Let s0 , s0  = {X }, {X  | S(X  ) ◦ S(Y ) ∈ ∃X 1 , . . . , X n ∈ s0 s.t. X  1≤i≤n S(X i ). Applying the (◦R) rule on this result, we have X ; Y  [1≤i≤n S(X i )] ◦ S(Y ). But we know that [1≤i≤n S(X i )] ◦ S(Y ) ∈ w implies at least one X i satisfies that S(X i ) ◦ S(Y ) ∈ w. So w is prime. By construction / w. However, it gives us X ; Y ∈ / w, of s0 , s0  we know that [1≤i≤n S(X i )] ◦ S(Y ) ∈ which contradicts our assumption. So s0 , s0  is a -pair, and it can be extended into a full -pair s, s  , where s is a proper theory. / w}. If this We then construct u 0 , u 0  = {Y }, {Y  | ∃X  ∈ s s.t. S(X  ) ◦ S(Y  ) ∈ is not a pair, then ∃Y1 , . . . , Yn ∈ u 0 s.t. Y  1≤i≤n S(Yi ). Similarly, we have X ; Y  S(X ) ◦ [1≤i≤n S(Yi )]. But from the construction of u 0 , u 0  and that w is prime, / w. Again we have X ; Y ∈ / w, which is not it implies that S(X ) ◦ [1≤i≤n S(Yi )] ∈ possible. So u 0 , u 0  is a -pair. And it can be extended into a full -pair u, u   with a proper theory u. Notice that Lsuw holds, because of Lemma 5. Suppose X  ∈ s and Y  ∈ u. / w. Because X  ∈ s, according to the construction of u, Assume that S(X  ) ◦ S(Y  ) ∈  / u, which contradicts our assumption. we know that Y ∈ 2. Let u 0 , {β} = {γ | α ⊸ γ ∈ w}, {β}. Then u 0 , {β} is a -pair. Otherwise, ∃γ ∈ u 0 s.t. γ  β.10 From α ⊸ γ ∈ w, γ  β, the (Cut) rule, and the definition of consequents of a theory, we get α ⊸ β ∈ w. But this contradicts our assumption. So u 0 , {β} is a -pair, and it can be extended into a full pair u, u  , where u is a proper theory. Notice that β ∈ / u. / u s.t. γ ⊸ γ  ∈ w}. So s0 , s0  is a -pair. Now let s0 , s0  = {α}, {γ | ∃γ  ∈  / Otherwise, ∃γ1 , . . . , γn ∈ s0 s.t. α  ⨄1≤i≤n γi . Notice that for each γi there is a γi ∈ u s.t. γi ⊸ γi ∈ w. Using (⊎L), (⊎R1 ) and (⊎R2 ), we have ⨄1≤i≤n γi ⊸ ⨄1≤i≤n γi ∈ w. This implies α ⊸ ⨄1≤i≤n γi ∈ w. But by the construction of u, we know that ⨄1≤i≤n γi ∈ u, which, according to u’s primeness, contradicts that each γi is not in u where 1 ≤ i ≤ n. Thus s0 , s0  is a -pair, and it can be extended into a full pair s, s  , in which s is a proper theory that contains α. / u, then by the conNow we check Lwsu. Given γ ⊸ γ  ∈ w and γ ∈ s. If γ  ∈ struction of s we have γ ∈ / s, which contradicts the previous assumption. So γ  ∈ u, which is desired. 10 Recall

that  is a single-consequence relation.

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3. Let u 0 , u 0  = {α}, {γ | Pγ ∈ w}. We say that u 0 , u 0  is a -pair. Otherwise, ∃γ1 , . . . , γn ∈ u 0 s.t. α  ⨄1≤i≤n γi . This implies P ⨄1≤i≤n γi  Pα (@) using (OR). Since Pγi ∈ w for each i, using (FC) we have P ⨄1≤i≤n γi ∈ w. Combining this with the previous result (@), it gives Pα ∈ w, which contradicts the assumption. Thus u 0 , u 0  is a -pair, and it can be extended into a full pair u, u  , in which u is a proper theory that contains α. Moreover, according to the pair construction, we can see that ¬Ouw. Therefore, this u is what we desire. Lemma 7 (Canonical Properties) The canonical model satisfies the conditions (or) and (den). The canonicity of (cam) and (ram) also holds in the canonical models w.r.t. the corresponding logics. Proof 1. We first will show that: ∀wu[¬O C uw → ∃y(L C wuy ∧ ¬O C yw)] / w (#). Let y0 , y0  = {γ | ∃α ∈ u s.t. α ⊸ Suppose ¬O C uw. So ∀α ∈ u.Pα ∈   γ ∈ w}, {γ | Pγ ∈ w}. If y0 , y0  is not a pair, then ∃γ1 , . . . , γn ∈ y0 ∃γ1 , . . . , γm ∈ y0 s.t. 1≤i≤n γi  ⨄1≤ j≤m γ j (@). As the previous observation shown, we know that n = 1. So γ1  ⨄1≤ j≤m γ j (@). Notice that Pγ j ∈ w for each j. Using (FC), we have P ⨄1≤ j≤m γ j ∈ w. Applying (→ R) and (OR) onto this result together with (@), we have Pγ1 ∈ w. Because α1 ⊸ γ1 ∈ w, it follows that Pα1 ∈ w according to (OR). But this contradicts the assumption (#). Now we can conclude that y0 , y0  is a pair. As usual, we can extend this pair into a full pair y, y  , where y satisfies the desired properties Lwuy and ¬O yw by the construction. 2. Second, we have to show ∀x yz[L C x yz → ∃u(L C x yu ∧ L C xuz)] / z s.t. Suppose L C x yz. Let u 0 , u 0  = {γ | ∃α ∈ y s.t. α ⊸ γ ∈ x}, {γ  | ∃β ∈ γ  ⊸ β ∈ x}. If u 0 , u 0  is not a pair, then ∃γ1 , . . . , γn ∈ u 0 ∃γ1 , . . . , γm ∈ u 0 s.t. 1≤i≤n γi  1≤ j≤m γ j . Similarly, it must be n = 1. So γ1  1≤ j≤m γ j (!). Let X  α  β be an abbreviation for two consequences X  α and X  β. / z s.t. γ j ⊸ β j ∈ x for each j. Given α1 ∈ y with α1 ⊸ γ1 ∈ x (@). Let β j ∈ Using (⊎R1 ), (⊎R2 ), (⊸L), and (⊸R), we have γ j ⊸ ⨄1≤ j≤m β j ∈ x. Notice that ⨄1≤ j≤m β j ∈ / z because that z is prime. Let β = ⨄1≤ j≤m β j . After applying (⊎L) on this result, we have γ j ⊸ β  (⨄1≤ j≤m γ j ) ⊸ β for each 1 ≤ j ≤ m. Now we have (⨄1≤ j≤m γ j ) ⊸ β ∈ x (#). Applying (!), (@), (#) using (Cut) and (Den), we get α1 ⊸ β ∈ x. Since α1 ∈ y, we have β ∈ z according to L x yz. But this contradicts the previous result that β∈ / z. Now we can conclude that u 0 , u 0  is a pair. As usual, we can extend this pair into a full pair u, u  , where u satisfies the desired properties L x yu and L xuz.

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3. We then need to show ∀x zsu[L C x(su)z ∧ ∀z  (L C xsz  → z  ∩ u = ∅) → L C xsz] Suppose L C x(su)z and ∀z  (L C xsz  → z  ∩ u = ∅). From L C x(su)z it follows: ∃y s.t. L C x yz and L C suy. Given α ⊸ β ∈ x and α ∈ s. We will show β ∈ z. / u}. If z 0 , z 0  is not Construct z 0 , z 0  = {β | ∃α ∈ s s.t α ⊸ β ∈ x}, {β | β ∈  / u s.t. β  ⨄1≤i≤n βi (@), where ∃α ∈ s s.t. a pair, then ∃β ∈ z 0 ∃β1 , . . . , βn ∈ α ⊸ β ∈ x (!). From (@) and (!) it follows α ⊸ ⨄1≤i≤n βi ∈ x, which gives that ⨄1≤i≤n βi ∈ z 0 from the construction of z 0 . But this implies that ⨄1≤i≤n βi ∈ u. / u. Now a However, as we already known from the primeness of u, ⨄1≤i≤n βi ∈ contradiction follows. So z 0 , z 0  is a pair. Similarly to the previous argument, we know that z 0 , z 0  can be extended into a full pair z  , z   with the proper theory z  ⊇ z 0 . Notice that this z  satisfies the desired property L C xsz  → z  ∩ u = ∅. Now given a γ ∈ L A s.t. α ⊸ γ ∈ x, this γ must be contained in z  . And so is u from the construction. According to CaM, it follows that α ◦ γ ⊸ β ∈ x. We already know that L C x(su)z. This follows that β ∈ z, which is desired. 4. We also need to show ∀xsuzu  [L C x(su)z ∧ L C (xs)uu  → L C xsz] Suppose L C x(su)z and L C (xs)uu  . Given α ⊸ γ ∈ x and α ∈ s. We need to show γ ∈ z. From L C (xs)uu  , it follows ∃t s.t. L C xst and L C tuu  . Suppose β ∈ / t. In this case, α ⊸ ∼β ∈ / x. In addition u. From L C tuu  , it implies that ∼β ∈ with α ⊸ γ ∈ x and RaM, we have α ◦ β ⊸ γ . By the construction of L C x(su)z, it implies γ ∈ z. Lemma 8 (Truth Lemma) Given the canonical model MC = W, L , O, V . Then for any X ∈ S, MC , w X iff X ∈ w Proof The truth lemma can be proved by applying our Witness Lemma in the standard way (Blackburn et al. 2002). One difference of our proof from the standard one is that the canonical states are a set of structures instead of formulas. We prove it using induction on the complexity of X . 1. In the case of X ∈ LB , the way of proof is standard as in Blackburn et al. (2002). We only show the most interesting cases α ⊸ β and ∼α here. • Suppose MC , w |= α ⊸ β. Assume that α ⊸ β ∈ / w, then according to Witness Lemma (Lemma 6) we know that ∃x, y s.t. Lwx y, α ∈ x but β ∈ / y. By inductive hypothesis, we then have MC , x |= α and MC , y ⊧̸ β. It implies that M, w ⊧̸ α ⊸ β, which contradicts our assumption. On the other hand, suppose α ⊸ β ∈ w, and let Lwx y and M, x |= α. By the inductive hypothesis, we have α ∈ x. In addition to the canonical definition of L, we then have β ∈ y. This implies M, y |= β, which is desired.

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• Suppose MC , w |= ∼α. So for any y, z ∈ W that MC , y |= α implies ¬Lwyz. So α ∈ y implies ¬Lwyz for any y, z ∈ W by inductive hypothesis. Assume that ∼α ∈ / w. Then it implies α ⊸ 0 ∈ / w, in accordance with that w is a theory and the derivable rule (−0). As what Witness Lemma tells us, there exists / x  . But this contradicts our previous result x, x  s.t. Lwx x  , α ∈ x and 0 ∈ from the assumption. On the other hand, let ∼α ∈ w. So we have α ⊸ 0 ∈ w according to the derivable rule (+0) and that w is a theory. Suppose MC , x |= α. According to the inductive hypothesis, we have α ∈ x. But we know that there is no y ∈ W C s.t. 0 ∈ y. According to the canonical definition of L, we know that ¬Lwx y for any y. This is what we want to have. 2. In the case of X = Y ; Z ∈ SB , • Suppose MC , w |= Y ; Z , then ∃s, u ∈ W C s.t. Lsuw, MC , s |= Y , and MC , u |= Z . By applying the inductive hypothesis, it follows that Y ∈ s and Z ∈ u. According to the definition of canonical relation, it implies Y ; Z ∈ w. • Suppose Y ; Z ∈ w. According to Lemma 6, it follows that ∃s, u ∈ W C s.t. Lsuw, Y ∈ s, and Z ∈ u. As what the inductive hypothesis says, it means MC , s |= Y , and MC , u |= Z . So we have MC , w |= Y ; Z . 3. In the case of X ∈ Ln , the way of the proof is standard as in Blackburn et al. (2002) too. We only show the case of X = Pα here. • Suppose MC , w |= Pα. From Witness Lemma, we can have Pα ∈ w. • Suppose Pα ∈ w. Let MC , u |= α. By the inductive hypothesis, we have α ∈ u. By the canonical definition of O C , we have O C uw. This implies that MC , w |= Pα. 4. In the case of X = Y, Z , • Suppose MC , w |= Y, Z , then MC , w |= Y and MC , w |= Z . According to the inductive hypothesis, so Y ∈ w and Z ∈ w. As w is fine, it implies that Y, Z ∈ w. • Suppose Y, Z ∈ w. As w is fine, it follows that Y ∈ w and Z ∈ w. By inductive hypothesis again, we then have M, w |= Y and M, w |= Z . It implies M, w |= Y, Z . Lemma 9 (Substitution) If M, w |= Z implies either M, w |= Z 1 , or . . ., or M, w |= Z n , then M, w |= X [Z /Y ] implies either M, w |= X [Z 1 /Y ], or . . ., or M, w |= X [Z n /Y ]. Proof We prove this lemma by induction on the number of punctuation marks over X . Let n be the number of punctuation marks over X , and that M, w |= Z implies M, w |= Z i for at least a i. • When n = 0, then either X ∈ Ln or X ∈ LB . Then X [Y ] indicates that Y ∈ Sub(X ), and so Y = X . In this case M, w |= X [Z /Y ] is equal to M, w |= Z . Similarly, M, w |= X [Z i /Y ] is M, w |= Z i where 1 ≤ i ≤ n. And thus the desired property holds.

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• Assume that for any X that has n punctuation marks satisfies this property. • Suppose X has n + 1 punctuation marks. 1. If X = X 1 ; X 2 , then M, w |= (X 1 ; X 2 )[Z /Y ] where both X 1 and X 2 have less than n + 1 punctuation marks. Suppose it is M, w |= X 1 [Z /Y ]; X 2 , then there are x, y s.t. L x yw, M, x |= X 1 [Z /Y ] and M, y |= X 2 . Because X 1 has less than n + 1 punctuation marks, by inductive hypothesis we then have M, x |= X 1 [Z i /Y ] where 1 ≤ i ≤ n. From this we can have M, w |= X 1 [Z i /Y ]; X 2 , which is M, w |= (X 1 ; X 2 )[Z i /Y ], where 1 ≤ i ≤ n. 2. If X = X 1 , X 2 , we can have similar proof of the desired property. Theorem 7 (Soundness and Completeness) The logics N, NE, NERaM, NEM, and NEMCaM are sound and complete w.r.t. their corresponding frame classes. Proof The soundness and completeness can be proved in a standard way, using the Witness Lemma, the Truth Lemma, the Proper Theories Excluding Formulas Corollary, and the Canonical Properties Lemma. We first prove the soundness of N. • For (Id) it is clear. • For (0), we have induction on the number of punctuation marks over X . Let n be the number of punctuation marks over X . – When n = 0, then either X ∈ Ln or X ∈ LB . Since X [0] indicates that 0 ∈ Sub(X ), it then has X = 0. In this case, we have 0 |= γ for any γ ∈ L A trivially. – Assume that for any X that has n punctuation marks satisfies X [0] |= γ for all γ ∈ LA. – Let X has n + 1 punctuation marks, and w s.t. M, w |= X [0]. Then there are two subcases: 1. If X = Y ; Z , then let Y has k ≤ n punctuation marks and Z has n − k ≤ n. Because (Y ; Z )[0] indicates that 0 ∈ Sub(Y ; Z ), it implies either 0 ∈ Sub(Y ) or 0 ∈ Sub(Z ). Suppose 0 ∈ Sub(Y ). Then Y [0] |= 0 by inductive hypothesis. According to this result and Definition 4, we know that there is no w s.t. M, w |= Y [0]; Z , and so neither M, w |= (Y ; Z )[0]. Obvious that (Y ; Z )[0] |= γ for any γ ∈ L A trivially. 2. If X = Y, Z , the proof is similar to the previous one. • For (⊎L), we have induction on the number of punctuation marks over X . Let n be the number of punctuation marks over X . Assume that X [α] |= γ and X [β] |= γ . First of all we know that Y is a substructure of X (e.g., Y ∈ Sub(X )), and X [α], X [β] and X [α ⊎ β] indicate that α, β and α ⊎ β replace Y in X . Assume that X [α] |= γ and X [β] |= γ . – When X has zero punctuation mark, then it is either X ∈ Ln or X ∈ LB . It then implies that X [α/Y ] = α and X [β/Y ] = β. Suppose M, w |= α ⊎ β, we have either M, w |= α or M, w |= β. As we have α |= γ and β |= γ , both cases imply M, w |= γ , which we desire. – Assume that for any X that has n punctuation marks satisfies (⊎L).

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– Suppose X has n + 1 punctuation marks. 1. If X = X 1 ; X 2 where X 1 and X 2 both have less than n + 1 punctuation marks. We then let M, w |= (X 1 ; X 2 )[α ⊎ β]. Suppose (X 1 ; X 2 )[α ⊎ β] = X 1 [α ⊎ β]; X 2 . So there are x, y s.t. L x yw, M, x |= X 1 [α ⊎ β], and M, y |= X 2 . As what we know, M, s |= α ⊎ β implies either M, s |= α or M, s |= β by Definition 4. By Lemma 9, we then have M, x |= X 1 [α] or M, x |= X 1 [β]. This implies that either M, w |= X 1 [α ⊎ β]; X 2 or M, w |= X 1 [α ⊎ β]; X 2 by inductive hyphothesis. As that X [α] |= γ and X [β] |= γ both hold, they imply M, w |= γ . We therefore get the desired result. 2. If X = X 1 , X 2 , we can have similar proof of the desired conclusion. • For (∼L), we have induction on the number of punctuation marks over X . Let n be the number of punctuation marks over Y . Suppose X |= α. – When Y has zero punctuation mark, then either Y ∈ Ln or Y ∈ LB . We then have Y [∼α; X ] = ∼α; X . Let M, w |= ∼α; X . This implies that there are x, y s.t. L x yw, M, x |= ∼α and M, y |= X . So we have M, y |= α from X |= α. On the other hand, we also have M, z 1 |= α implies ¬L x z 1 z 2 for any z 1 , z 2 by Definition 4 and M, x |= ∼α. Together with M, y |= α, we then have ¬L x yz 2 for any z 2 . It implies ¬L x yw, which contradicts our previous result. We therefore can have M, w ⊧̸ ∼α; X . We can conclude that desired result trivially. – Assume that (∼L) holds when Y has n punctuation marks. – When Y has n + 1 punctuation marks, then 1. If Y = Y1 ; Y2 where Y1 and Y2 both have less than n + 1 punctuation marks. Suppose Y [∼α; X ] = Y1 [∼α; X ]; Y2 and let M, w |= Y1 [∼α; X ]; Y2 . So there exist x, y s.t. L x yw, M, x |= Y1 [∼α; X ] and M, y |= Y2 . From M, x |= Y1 [∼α; X ], this gives us M, x |= 0 according to the inductive hypothesis, but which cannot happen. So it leads to M, w ⊧̸ Y1 [∼α; X ]; Y2 . It suggests the desired result. 2. If Y = Y1 , Y2 , we have a similar proof. • For (⊸L), we have induction on the number of punctuation marks over X . Let n be the number of punctuation marks over Y . Suppose X |= α and Y [β] |= γ . – When Y has zero punctuation mark, then either Y ∈ Ln or Y ∈ LB . In this case we have Y [α ⊸ β; X ] is equal to α ⊸ β; X . Let M, w |= α ⊸ β; X . So there exist x, y s.t. L x yw, M, x |= α ⊸ β and M, y |= X . So M, y |= α. From L x yw and M, x |= α ⊸ β, we then have M, w |= β. Then by the assumption this implies M, w |= γ , which is desired. – Assume that (⊸L) holds when Y has n punctuation marks. – When Y has n + 1 punctuation marks, then 1. If Y = Y1 ; Y2 where Y1 and Y2 both have less than n + 1 punctuation marks. We suppose that (Y1 ; Y2 )[α ⊸ β; X ] is equal to Y1 [α ⊸ β; X ]; Y2 , and let M, w |= Y1 [α ⊸ β; X ]; Y2 . So there are x, y s.t. L x yw, M, x |= Y1 [α ⊸ β; X ] and M, y |= Y2 . As what we can learn from Definition 4 and X |= α, it gives that M, s |= α ⊸ β; X implies M, s |= β. By Lemma 9 and M, x |=

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Y1 [α ⊸ β; X ], we have M, x |= Y1 [β]. It implies that M, w |= Y [β]. This leads to the desired result M, w |= γ . 2. If Y = Y1 , Y2 , we have a similar proof. • The other cases like the rule (FC) are standard, except for O. But this case is ensured by Theorem 3. The soundnesses of NE, NERaM, NEM, and NEMCaM are proven in a similar way. The completeness is proved in the standard method (Blackburn et al. 2002).

4.4 Applications to Free Choice Inferences The substructural logics that we have just studied provide a good balance between cautiousness and deductive power. They avoid the unwelcome FCP inferences, while still allowing plausible ones. Here we call the class of all frames satisfying (or), (den), (bi), (b), and (ram) the open-reading frame class. Recall from the previous section that the system NERaM is sound and complete w.r.t. the open-reading frame class. Theorem 8 The following consequences are not valid in the open-reading frame class: given α, β, γ ∈ L A , 1. 2. 3. 4. 5. 6.

α⊸β α◦γ ⊸β α⊸α α◦α⊸α α ⊸ β, γ ⊸ β  α ◦ γ ⊸ β

 α ⊎ ∼α

 α ⊎ ∼α ⧟ β ⊎ ∼β

 α ⊸ β ⊎ ∼β

Proof Here, we only provide a counter model for α ⊸ β  α ◦ γ ⊸ β by illustrating the case “vegetarian free lunch.” The model M = W, L , O, V  is constructed as follows: • W = {x1 , x2 , x3 , x4 , x5 } • L = {(x1 , x1 , x1 ), (x1 , x3 , x1 ), (x3 , x1 , x1 ), (x1 , x1 , x4 ), (x2 , x2 , x2 ), (x2 , x3 , x2 ), (x3 , x2 , x2 ), (x2 , x2 , x5 )}. • O = {(x2 , x2 ), (x3 , x2 ), (x5 , x2 )} • V (vega) = {x2 , x3 }, V (or der ) = {x1 , x2 , x3 }, V ( pay) = {x2 , x5 } It is not difficult to check that M is a model in the open-reading frame class.11 This model M can be used to illustrate the “free lunch” example. First of all, the model M validates the action formula vega ⊸ or der , which admits that normally ordering a vegetarian lunch is ordering a lunch. Further, this model also validates this statement: vega ◦ pay ⊸ or der (@), which means that having a vegetarian lunch and paying 11 This

model M does not satisfy (cam), but it does satisfy (ram). Here is the case to invalidate (cam). We have L x1 (x3 x1 )x4 . And for all z  (L x1 x3 z  → z  ⊇ x1 ) because this z  must be x1 if L x1 x3 z  exists. However, we do not have L x1 x3 x4 in this model.

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Fig. 2 The model invalidates the “vegetarian free lunch” case at x1

for it is normally entails an instance of ordering a meal for lunch. But there is a state in the model falsifies this undesired statement vega ◦ ∼ pay ⊸ or der . Figure 2, as a part of model M, illustrates this situation. Moreover, as Fig. 2 describes, a state can be incomplete: neither pay nor ∼ pay is true at state x3 (because L x3 x2 x2 where / V ( pay)). Notice that this model M satisfies (ram). In fact, x2 ∈ V ( pay) and x3 ∈ we have ¬(vega ⊸ ∼ pay) is true at x2 . It offers us a reason to conclude the desired claim (@). The above invalidities are part of our solution to the three counterintuitive FCP inferences discussed earlier. First, since the monotonic α ⊸ β  α ◦ γ ⊸ β is not valid in the open-reading frame class, α ⊸ β  Pβ → P(α ◦ γ ) is not valid either. So the logics N, NE, and NERaM can exclude the unrestricted monotonic cases like “if it is permitted to order a vegetarian lunch, then it is permitted to order a vegetarian lunch and not pay for it.” Second, because the resource-insensitive α ⊸ α  α ◦ α ⊸ α is invalid in the open-reading frame class, α ⊸ α  Pα → P(α ◦ α) is also not valid. We then can exclude resource-insensitive cases like “if it is permitted to eat one cookie then it is permitted to eat more than one” in the logics N, NE, and NERaM too. Third, the irrelevant cases are also excluded in the sequent calculus N, because  α ⊎ ∼α ⧟ β ⊎ ∼β and  α ⊸ β ⊎ ∼β are not valid in the openreading frame class. Thus we cannot conclude the following two consequences  P(α ⊎ ∼α) → P(β ⊎ ∼β) and  P(β ⊎ ∼β) → Pα, which contain irrelevant cases like “it is permitted to eat an apple or not iff it is permitted to sell a house or not” and “if it is permitted to eat an apple or not then it is permitted to sell a house.” The family of substructural logics N, NE, and NERaM studies in the previous section avoid the aforementioned unrestricted monotonic, resource-insensitive, and irrelevant FCP inferences. The family of substructural logics N, NE, and NERaM also invalidates the following: α ⊸ β, γ ⊸ β  Pβ → Pα ◦ γ because it also invalidates α ⊸ β, γ ⊸ β  α ◦ γ ⊸ β. This avoids further unwelcome consequences. Suppose that both “selling the house to Ann is entails selling the house” and “selling the house to Bob entails selling the house.” We should not

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be able to conclude from that “if it is permitted to sell the house then it is permitted to sell the house to Ann and sell the house to Bob.” This indeed does not follow in N, NE, and NERaM. On the other hand, the logic NERaM in this family can derive some restricted monotonic cases in the following form: α ⊸ β, ¬α ⊸ ∼γ  Pβ → P(α ◦ γ ) For instance, the good variant of the vegetarian lunch example can be derived in logic NERaM. From a common situation “it is not the case that ordering a lunch entails any other action except paying for the lunch,” this logic can derive “if it is permitted to order a lunch then it is permitted to order the lunch and pay for it.” Theorem 9 (Incomplete and Consistent) 1. In the open-reading frame class the consequence  α ⊎ ∼α is not valid, so the states are not complete. In this sense, this property ensures that the FCP inferences are relevant. 2. In the (bi)-frame class α ◦ ∼α  0 is valid.12 So the states are consistent. The family of N, NE, and NERaM not only excludes the undesired properties discussed in Sect. 2.3 and saves the deductive power of the FCP inferences. It also satisfies some interesting properties: states are incomplete but consistent.13

4.5 Proof Theory Cut elimination states that, if X  A is a derivation by the application of the cut rule, then it can be a derivation without using the cut rule. Cut elimination is important because it implies the subformula property, which in turn allow for a purely syntactic proof of the consistency of the system. The definition of cut height that we use is inspired by Negri et al. (2008, p. 35), and the definition of principal formulas is inspired by Negri et al. (2008, p. 29). Notice that the exchange rules are heightpreserving, and this ensures the following technique work. Definition 18 (Cut Height) The cut height of a derivation with an application of the rule of cut in a proof is the sum of heights of the two premises of cut plus 1. Definition 19 (Principal Formulas) A formula is principal in a derivation iff it is introduced in a conclusion derived from a rule in the system. The cut elimination can be formalized in the following way, using the cut height. 12 The action negation ∼ still satisfies the ex contradictione quodlibet

rule (ECQ)  α ◦ ∼α ⊸ β in the (bi)-frame class, which is rejected in relevant logics. 13 Given arbitrary frame F in the open reading frame class, all states in F are consistent and not complete, which can be understood in this way: α ◦ ∼α ⊸ β is valid in F , but not α ⊸ β ⊎ ∼β.

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Definition 20 (Cut Elimination) A system satisfies the cut elimination iff for arbitrary conclusion Y [X ]  B concluded from the cut rule with premises X n A and Y [A] m B in a cut height n + m + 1, the system can derive Y [X ]  B in a cut height at most n + m, where n, m ∈ ω. Theorem 10 The systems N and NE satisfy cut elimination. Proof Given a derivation X  A which is derived from the cut rule. The proof of cut elimination is proved in the standard way by inducing on the cut height of X  A. Here, we only present the interesting cases below. • X  A is a derivation by applying cut with at least one premise such that its height of derivation is 0. Consider two subcases: either (1) the left premise X 0 A is an instance of either (Id) or (0); Or (2) the right premise Y [A] 0 B is an instance of either (Id) or (0). For the first subcase, we only present the derivation by (0) as an example for the first subcase: Y [γ ] n β X [0] 0 γ (Cut) Y [X [0]] n+1 β It is clear that Y [X [0]] 0 β according to (0). For the second subcase, we have Y n 0 X [0] 0 γ X [Y ] n+1 γ

(Cut)

Two possibilities of this proof: 1. If 0 in Y n 0 is principal. The nontrivial case is that Y n 0 is derived by (∼L). Then Y n 0 is in the form Z [∼α; Z  ] n 0, and the above proof is in the following form: Z  n−k−1 α Z [0] k 0 (∼L) Z [∼α; Z  ] n 0 X [0] 0 γ  X [Z [∼α; Z ]] n+1 γ

(Cut)

where 0 ≤ k ≤ n − 1. This proof can be transferred into X [0] 0 γ Z [0] k 0 (Cut) Z n−k−1 α X [Z [0]] k+1 γ (∼L) X [Z [∼α; Z  ]] n+1 γ 

whose cut height is at most n. 2. If 0 in Y n 0 is not principal. For example, Y n 0 is derived by (◦L), and the proof concerned is presented as follows:

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(Cut)

Obviously, this proof can be transferred into X [0] 0 γ Y [α; β] n−1 0 (Cut) X [Y [α; β]] n γ (◦L) X [Y [α ◦ β]] n+1 γ whose cut height is at most n. • Cut with premises which are derived with height is at least 1. So its premises cannot be derived from either (Id) nor (0). In this case, there are three subcases: Cut formula A is not principal in the left premise; Cut formula A is principal in the left premise only; and Cut formula A is principal in both premises. Given n, m ≥ 1: 1. For the first subcase, we only show the application of (⊸L) as an example. Suppose that the concluded derivation is in the cut height n + m + 1 where n ≥ k ≥ 1: X [β] k−1 α Z n−k γ (⊸L) X [γ ⊸ β; Z ] n α Y [α] m δ (Cut) Y [X [γ ⊸ β; Z ]] n+m+1 δ This proof can be transformed into the following proof such that the concluded derivation is in a cut height m + k, which is of course at most n + m: Y [α] m δ X [β] k−1 α (Cut) Z n−k γ Y [X [β]] m+k δ (⊸L) Y [X [γ ⊸ β; Z ]] n+m+1 δ 2. For the second subcase, we show two applications as examples. – Application of (⊸R). Suppose that the concluded derivation is in a cut height n + m + 1: Y [α]; β m−1 γ (⊸R) X n α Y [α] m β ⊸ γ (Cut) Y [X ] n+m+1 β ⊸ γ This proof can be transformed into the following proof such that the concluded derivation in a cut height is n + m, which is at most n + m: Y [α]; β m−1 γ X n α (Cut) Y [X ]; β n+m γ (⊸R) Y [X ] n+m+1 β ⊸ γ

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3. For the third subcase, we show two applications: – Application of (⊸R) and (⊸L): Suppose that the concluded derivation is in a cut height n + m + 1 where m ≥ k ≥ 1: Y [β] k−1 γ Z m−k α X ; α n−1 β (⊸L) (⊸R) X n α ⊸ β Y [α ⊸ β; Z ] m γ (Cut) Y [X ; Z ] n+m+1 γ It can be transformed into the proof which concluded the original derivation in cut heights n + m − k and n + m, which are both at most n + m: X ; α n−1 β Z m−k α (Cut) X ; Z n+m−k β Y [β] k−1 γ (Cut) Y [X ; Z ] n+m γ – Application of OR: Suppose that the concluded derivation is in a cut height n + m + 1: X n−1 β ⊸ α Y m−1 γ ⊸ β (OR) (OR) X, Pα n Pβ Y, Pβ m Pγ (Cut) Y, X, Pα n+m+1 Pγ It can be transformed into the proof which concluded the original derivation in the cut height at most n + m: X n−1 β ⊸ α Y m−1 γ ⊸ β (Den) (Y, X ) n+m−1 γ ⊸ α (OR) Y, X, Pα n+m Pγ – Application of Den: Suppose that the concluded derivation is in a cut height n + m + 1: X ; α n−k−1 β Z  [Y ; β] k Pγ Z ; δ m−1 γ (Den) (OR)  Z , Pγ m Pδ Z [(X, Y ); α] n Pγ (Cut) Z , Z  [(X, Y ); α] n+m+1 Pδ It can be transformed into the proof concluding the original derivation in the cut height at most n + m: Z  [Y ; β] k Pγ Z , Pγ m Pδ (Cut) Z , Z  [Y ; β] k+m+1 Pδ X ; α n−k−1 β (Den) Z , Z  [(X, Y ); α] n+m+1 Pδ

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5 Conclusion and Discussion of Related Work This paper suggests a family of substructural logics for excluding the unrestricted monotonic, resource-insensitive, and irrelevant consequences of action types in free choice permission. One logic among this family can derive the consistent monotonic free choice permission by accepting our version of rational monotony RaM. The driving idea of this family of substructural logics comes from the open reading of free choice permission. Following this semantic core, these logics offer various sound and complete calculus on action types and permissions, such that the calculus is not only monotonic, resource-sensitive, and relevant, but also consistent and incomplete. The logic we develop in this paper is close to but different from the one developed by Barker (2010). First, our substructural logics are weaker than Barker’s linear logic (Barker 2010), though similar in that Barker’s and our proposal both exclude contraction and left-hand weakening.14 Statements like “if it is permitted to do α and β then it is permitted to do β and α” are derivable using the multiplicative “and” in Barker’s linear logic. These statements are not derivable in our substructural logic N.15 Also our logic NERaM gains deductive power in comparison to Barker’s using the reformed rational monotony. Our logic NERaM is thus able to derive more intuitive free choice permission statements like “if it is permitted to order a lunch then it is permitted to order lunch and pay for it,” which are not derivable in Barker’s logic. In this paper, we have not considered conditional permissions (Makinson and van der Torre 2003; Governatori et al. 2013). Instead, we focus on the analysis of prima facie permissions through the study of the consequences on action types. Even though our logics are inspired by non-monotonic reasoning, both in motivation and its proof system (by RaM), they are substantially weaker because of the resource-sensitive and relevant aspects of them. Obligations are missing in this paper. This is the natural step for future work. For instance, we can define obligations as the dual of permissions, viewing permissions as weak permissions. This logic would be still weaker than standard deontic logic since negation is not classical in our logics. Another option is to see permissions as strong permissions.16 In this case, it is not clear whether obligations should be defined as the dual of permissions. And so the relationship between obligations and permissions might become weaker.

14 Removing OR and Den from the logic N, this system is a basic substructural logic for full Lambek calculus FL (Galatos et al. 2007). It is weaker than Barker’s linear logic because our fusion is neither associative nor commutative in N. 15 Barker’s system does not contain the rules (OR) and (Den), which are required in our system N. Even so, N cannot be seen as an extension of Barker’s logic, as the case, we argued here. So our system N has a substantive difference from Barker’s linear logic. 16 As we discussed earlier, strong permissions are not the dual of obligations.

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Acknowledgements We would like to thank Albert J. J. Anglberger, Sabine Frittella, Fei Liang, Johannes Korbmacher, Piotr Kulicki, Clayton Peterson, and Robert Trypuz for their comments and suggestions on the early version, and thanks to Johan van Benthem, O. Foisch, Xiaowu Li, and the anonymous reviewers for their insightful comments and suggestions on the latest version. All authors are supported by the PIOTR research project [No. RO 4548/4-1]. Huimin Dong is supported by the China Postdoctoral Science Foundation funded project [No. 2018M632494], the MOE Project of Key Research Institute of Humanities and Social Sciences in Universities [No. 17JJD720008], the National Social Science Fund of China [No. 18ZDA290], and by the Fundamental Research Funds for the Central Universities of China.

References Anglberger, A.J., Dong, H., Roy, O.: Open reading without free choice. In: Cariani, F., Grossi, D., Meheus, J., Parent, X. (eds.) Deontic Logic and Normative Systems. Lecture Notes in Computer Science, vol. 8554, pp. 19–32. Springer International Publishing (2014). https://doi.org/10.1007/ 978-3-319-08615-6_3 Anglberger, A.J., Gratzl, N., Roy, O.: Obligation, free choice, and the logic of weakest permissions. Rev. Symb. Log. 8, 807–827 (2015). https://doi.org/10.1017/S1755020315000209 Asher, N., Bonevac, D.: Free choice permission is strong permission. Synthese 145(3), 303–323 (2005) Asudeh, A.: Linear logic, linguistic resource sensitivity and resumption (2006) Barker, C.: Free choice permission as resource-sensitive reasoning. Semant. Pragmat. 3(10), 1–38 (2010). https://doi.org/10.3765/sp.3.10 Belnap, N.D., Perloff, M., Xu, M.: Facing the Future: Agents and Choices in our Indeterminist World. Oxford University Press on Demand (2001) Blackburn, P., De Rijke, M., Venema, Y.: Modal Logic, vol. 53. Cambridge University Press (2002) Broersen, J.: Action negation and alternative reductions for dynamic deontic logics. J. Appl. Log. 2(1), 153–168 (2004) Di Cosmo, R., Miller, D.: Linear logic. In: Zalta, E.N. (ed.) The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University, Winter 2016 edn (2016) Dignum, F., Meyer, J.J.C., Wieringa, R.J.: Free choice and contextually permitted actions. Stud. Logica 57(1), 193–220 (1996) Dong, H., Roy, O.: Three deontic logics for rational agency in games. Stud. Log. 8(4), 7–31 (2015) Galatos, N., Jipsen, P., Kowalski, T., Ono, H.: Residuated Lattices: An Algebraic Glimpse at Substructural Logics, vol. 151. Elsevier (2007) Governatori, G., Olivieri, F., Rotolo, A., Scannapieco, S.: Computing strong and weak permissions in defeasible logic. J. Philos. Log. 42(6), 799–829 (2013) Hansson, S.O.: The varieties of permissions. In: Gabbay, D., Horty, J., Parent, X., van der Meyden, R., van der Torre, L. (eds.) Handbook of Deontic Logic and Normative Systems, vol. 1. College Publication (2013) Harel, D., Kozen, D., Tiuryn, J.: Dynamic Logic. MIT Press (2000) Hilpinen, R.: Disjunctive permissions and conditionals with disjunctive antecedents. Acta Philos. Fenn. 35, 175–194 (1982) Horty, J.F.: Agency and Deontic Logic. Oxford University Press (2000) Kamp, H.: Free choice permission. In: Proceedings of the Aristotelian Society, vol. 74, pp. 57–74 (1973) (JSTOR) Kulicki, P., Trypuz, R.: On deontic action logics based on boolean algebra. J. Log. Comput. 25(5) (2015) Kurtonina, N.: Frames and labels. A modal analysis of categorial deduction. Ph.D. thesis, PhD Thesis and University of Amsterdam (1995)

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Makinson, D.: Stenius’ approach to disjunctive permission. Theoria 50(2–3), 138–147 (1984) Makinson, D.: Bridges from Classical to Nonmonotonic Logic. King’s College (2005) Makinson, D., van der Torre, L.: Permission from an input/output perspective. J. Philos. Log. 32(4), 391–416 (2003) McCarthy, J.: Epistemological problems of artificial intelligence. In: Proceedings of the 5th International Joint Conference on Artificial Intelligence, vol. 2, pp. 1038–1044. Morgan Kaufmann Publishers Inc. (1977) McNamara, P.: Deontic logic. In: Handbook of the History of Logic, vol. 7, pp. 197–289 (2006) McNamara, P.: Deontic logic. In: Zalta, E.N. (ed.) The Stanford Encyclopedia of Philosophy, Fall 2010 edn (2010) Negri, S., Von Plato, J., Ranta, A.: Structural Proof Theory. Cambridge University Press (2008) Paoli, F.: Substructural Logics: A Primer, vol. 13. Springer Science & Business Media (2013) Pelletier, F.J., Asher, N.: Generics and defaults. In: Handbook of Logic and Language, pp. 1125– 1177. Elsevier (1997) Restall, G.: An Introduction to Substructural Logics. Routledge (2000a) Restall, G.: An Introduction to Substructural Logics. Psychology Press (2000b) Restall, G.: Logic: An Introduction. McGill-Queen’s University Press (2006) Schurz, G.: Relevant Deduction. In: Erkenntnis Orientated: A Centennial Volume for Rudolf Carnap and Hans Reichenbach, pp. 391–437. Springer (1991) Segerberg, K., Meyer, J.J., Kracht, M.: The logic of action. In: Zalta, E.N. (ed.) The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University, Winter, 2016 edn (2016) Simons, M.: Dividing things up: the semantics of or and the modal/or interaction. Nat. Lang. Semant. 13(3), 271–316 (2005) Trypuz, R., Kulicki, P.: Towards metalogical systematisation of deontic action logics based on boolean algebra. In: Deontic Logic in Computer Science, pp. 132–147. Springer (2010) van Benthem, J.: Minimal deontic logics. Bull. Sect. Log. 8(1), 36–42 (1979) van Benthem, J.: Language in Action: Categories, Lambdas and Dynamic Logic. MIT Press (1995) van Benthem, J.: What one may come to know. Analysis 64(282), 95–105 (2004) van de Putte, F.: “That will do”: logics of deontic necessity and sufficiency. Erkenntnis (2016) (in print) von Wright, G.H.: Norm and Action-A Logical Enquiry. Routledge (1963) von Wright, G.H.: An Essay in Deontic Logic and the General Theory of Action. North-Holland Publishing Company (1968) Weingartner, P., Schurz, G.: Paradoxes solved by simple relevance criteria. Logique et Analyse 113, 3–40 (1986) von Wright, G.H.: On the logic of norms and actions. In: New Studies in Deontic Logic: Norms, Actions, and the Foundations of Ethics, pp. 3–35. Springer Netherlands, Dordrecht (1981). https:// doi.org/10.1007/978-94-009-8484-4_1 Xin, S., Dong, H.: The deontic dilemma of action negation, and its solution. In: The Eleventh Conference on Logic and the Foundations of Game and Decision Theory (LOFT 2014). http:// hdl.handle.net/10993/19577 (2014) Zimmermann, T.E.: Free choice disjunction and epistemic possibility. Nat. Lang. Semant. 8(4), 255–290 (2000)

A Road to Ultrafilter Extensions Jie Fan

Abstract We propose a uniform method of constructing ultrafilter extensions from canonical models, which is based on the similarity between ultrafilters and maximal consistent sets. This method can help us understand why the known ultrafilter extensions of models for normal modal logics and for classical modal logics are so defined. We then apply this method to obtain ultrafilter extensions of models for Kripke contingency logics and for neighborhood contingency logics. Keywords Ultrafilter extensions · Canonical models · Contingency logic · Modal logic · Kripke semantics · Neighborhood semantics

1 Introduction The notion of ultrafilter extensions goes back to Stone’s representation theorem (1936) and the Jónsson–Tarski Theorem (1951), and it is introduced in van Benthem (1979) (see Chagrov and Zakharyaschev 1997, p. 372). As a classical result in model theory, ultrafilter extensions have played an important role in various nonclassical logics, such as modal logic, temporal logic, dynamic logic, and so on. For instance, by using a notion of ultrafilter extension, van Benthem (1979) semantically characterizes a kind of complete modal logics, called “canonical modal logics” introduced in Fine (1975). Besides, it is a crucial concept in various important results, such as the abovementioned Jónsson–Tarski theorem, a second bisimilarity-somewhere-else result, Goldblatt–Thomason Theorem, van Benthem Characterization Theorem, and many others (e.g., Goldblatt and Thomason 1975; van Benthem 1984, 1988; Venema 1999; Blackburn et al. 2001; Kupke et al. 2005; Fan et al. 2014; Bakhtiari et al. 2017).

J. Fan (B) School of Humanities, University of Chinese Academy of Sciences, Beijing, China e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2019 B. Liao et al. (eds.), Dynamics, Uncertainty and Reasoning, Logic in Asia: Studia Logica Library, https://doi.org/10.1007/978-981-13-7791-4_6

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Usually, a suitable notion of ultrafilter extensions should have the following nice properties (cf. e.g. Blackburn et al. 2001): • The ultrafilter extension and its original model are logically equivalent. • Ultrafilter extensions are saturated in some proper sense, i.e., in the sense that the class of saturated models has the Hennessy–Milner property: logical equivalence implies bisimilarity. The two properties follow that • Logical equivalence can be characterized as bisimilarity-somewhere-else, that is, between ultrafilter extensions. Compared to other model operations like disjoint unions, generated submodels, and bounded morphisms, constructing ultrafilter extensions is seen as a far less natural job. Despite the fact that there are various notions of ultrafilter extensions of models, for instance, Goranko (2007), Saveliev (2011) for first-order logic, Goldblatt and Thomason (1975), Blackburn et al. (2001) for normal modal logics, Hansen et al. (2009) for classical modal logics, Jacobs (2001), Kupke et al. (2005) for coalgebras, to our knowledge, however, there has been no uniform method of constructing ultrafilter extensions in the literature, and it is sometimes hard to find a suitable notion of ultrafilter extensions (e.g. Blackburn et al. 2001 for normal modal logics). This paper proposes a uniform method of constructing ultrafilter extensions of models.1 The method is via a two-step transformation from the notion of canonical models. That is, given a canonical model, we can transform it into a desired ultrafilter extension in two steps. This is due to the similarity between ultrafilters and maximal consistent sets. For instance, an ultrafilter contains the whole domain, is closed under intersection and supersets, does not contain the empty set, and contains exactly one of any set and its complement; and a maximal consistent set contains the tautologies, is closed under conjunction and logical implication, does not contain the contradictions, and contains exactly one of any formula and its negation.2 It is these similar properties that make the transformation from a canonical model to the corresponding ultrafilter extension workable. As we shall see, via this method, we can construct the desired ultrafilter extensions quite easily, in an automatic way.3 1 Note

that ultrafilter extensions of frames follow directly by leaving out the valuations. researchers may say that such a similarity is well known for a long time, and in fact, maximal consistent sets of formulas are exactly ultrafilters in Lindenbaum–Tarski algebras of formulas modulo equivalence of formulas. That may be true; however, to our knowledge, there has been no method yet in the literature to construct ultrafilter extensions from canonical models based on this similarity. 3 It should be noted that Goranko (2007) introduces a construction of ultrafilter extensions for arbitrary structures from universal-algebraic perspective, which is quite different from ours. His construction is more like a definition based on ultrafilter extensions of the domain, relations, and functions of structures, rather than a (transformation) method. Besides, as the author himself remarked (see Goranko 2007, pp. 10–11), the “approaches” to ultrafilter extensions of functions either lead to complications in the algebraic theory of ultrafilter extensions, or is not completely satisfactory and have at least two natural rivals for the title “ultrafilter extensions” of functions. 2 Some

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The structure of the paper is outlined as follows. After the basics of ultrafilters and maximal consistent sets followed by an important theorem, we summarize the method roughly (Sect. 2). We then illustrate the method of constructing the known notions of ultrafilter extensions of models for normal modal logics (Subsect. 3.1) and for classical modal logics (Subsect. 3.2). Then, we apply the method to contingency logic, and obtain the ultrafilter extensions of Kripke models (Subsect. 4.1) and of neighborhood models (Subsect. 4.2). We conclude with some discussions in Sect. 5.

2 Preliminaries Throughout the paper, we use P to denote a fixed nonempty set of propositional variables. Definition 1 (Ultrafilters) Let S be a nonempty set. A set U ⊆ P(S) is an ultrafilter over S, if U (i) (ii) (iii) (iv) (v)

Contains the whole set: S ∈ U . Closed under intersection: if X, Y ∈ U , then X ∩ Y ∈ U . Closed under supersets: if X ∈ U and X ⊆ Z ⊆ S, then Z ∈ U . Does not contain empty set: ∅ ∈ / U. / U , where X denotes the complement of the set for all X ⊆ S, X ∈ U iff X ∈ X with respect to S.

If we drop the condition (v) from the above definition, then we obtain the notion of proper filters. There is an important class of ultrafilters, called “principal ultrafilters”. Given a nonempty set S and an element w ∈ S, the principal ultrafilter πw generated by w is the filter generated by the singleton set {w}; in symbol, πw = {X ⊆ S | w ∈ X }. Theorem 1 (Ultrafilter Theorem) Any proper filter over S can be extended to an ultrafilter over S. As a corollary, any subset of P(S) with the finite intersection property can be extended to be an ultrafilter over S. Definition 2 (Maximal consistent sets) Let Σ be a set of formulas. Σ is said to be consistent, if Σ  ⊥; it is said to be maximal, if for all formulas ϕ, we have ϕ ∈ Σ or ¬ϕ ∈ Σ; it is said to be maximal consistent, if it is maximal and also consistent.4 Here is a list of some properties of maximal consistent sets (not exclusively). We choose these properties but not others, which is because we would like to make clear the similarity between ultrafilters and maximal consistent sets.5 4 Strictly speaking, notions of consistency and maximality, respectively, refer to a proof system and

a language. But we leave out the references for simplicity. the aforementioned similarity, one natural question would be if there is any (essential) difference between ultrafilters and maximal consistent sets (The author would like to thank Christoph Benzmüller for posing this question on the conference of CLAR 2018). The author conjectures that the answer is negative.

5 Given

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Fact 1 (Properties of maximal consistent sets) Let Σ be a maximal consistent set. Then (i) (ii) (iii) (iv) (v)

 ∈ Σ. if ϕ, ψ ∈ Σ, then ϕ ∧ ψ ∈ Σ. if ϕ ∈ Σ and ϕ → ψ, then ψ ∈ Σ. ⊥∈ / Σ. for all formulas ϕ, ϕ ∈ Σ iff ¬ϕ ∈ / Σ.



Lemma 1 (Lindenbaum’s Lemma) Every consistent set can be extended to a maximal consistent set. We can summarize the similarity between ultrafilters and maximal consistent sets as follows: an ultrafilter U

a maximal consistent set Σ

S∈U X, Y ∈ U ⇒ X ∩ Y ∈ U X ∈U &X ⊆ Z ⇒ Z ∈U ∅∈ /U X ∈ U ⇐⇒ X ∈ /U Ultrafilter Theorem

∈Σ ϕ, ψ ∈ Σ ⇒ ϕ ∧ ψ ∈ Σ ϕ∈Σ& ϕ→ψ⇒ψ∈Σ ⊥∈ /Σ ϕ ∈ Σ ⇐⇒ ¬ϕ ∈ /Σ Lindenbaum’s Lemma

Let  be an arbitrary unary modal operator, and L () be the extension of the language of classical propositional logic L enriched with the primitive modality .6 Theorem 2 Let Σ be a maximal consistent set and V a valuation on the domain of a model. If for all ϕ ∈ L (), we have (∗)

for each p ∈ P ∪ {ϕ}, p ∈ Σ iff Σ ∈ V ( p),

then for all ϕ ∈ L (), we have ϕ ∈ Σ iff V (ϕ) ∈ πΣ . Proof Suppose (∗) holds. We proceed with induction on ϕ ∈ L (). • ϕ = p ∈ P ∪ {ψ}. By (∗), p ∈ Σ iff Σ ∈ V ( p). This is equivalent to V ( p) ∈ πΣ according to the definition of πΣ . • ϕ = ¬ψ. We have the following equivalences:

6 For

the sake of simplicity, we here only consider logics with a sole unary modality, but it is worth remarking that our method also applies for multimodal logics and polymodal logics.

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¬ϕ ∈ Σ Fact 1(v)

⇐⇒ ϕ ∈ /Σ IH

⇐⇒ V (ϕ) ∈ / πΣ Def. 1(v)

⇐⇒ V (ϕ) ∈ πΣ ⇐⇒ V (¬ϕ) ∈ πΣ

• ϕ = ψ ∧ χ. We have the following equivalences: ψ∧χ∈Σ Fact 1(ii)(iii)

⇐⇒

ψ ∈ Σ and χ ∈ Σ

IH

V (ψ) ∈ πΣ and V (χ) ∈ πΣ

⇐⇒ Def. 1(ii)(iii)

⇐⇒ ⇐⇒

V (ψ) ∩ V (χ) ∈ πΣ V (ψ ∧ χ) ∈ πΣ

The relationship between maximal consistent sets and ultrafilters provides us with a road to the construction of ultrafilter extensions from that of canonical models. To give the reader a warm-up, we summarize the method roughly as follows. Let M = S, R, V  be an arbitrary model for a language L. We construct its ultrafilter extension via the canonical model construction in the following way. Given is the canonical model M  = S  , R  , V   for a logic  of L, where • S  consists of all maximally consistent sets (MCSs), • R  is a relation (or a neighborhood function, for instance) on S  , the definition of which depends on the logic, and • V  is a valuation of atomic propositions in S  . At first step, from the above canonical model, we replace MCSs with the corresponding principal ultrafilters and formulas with their extensions/truth sets (due to Theorem 2), respectively; at second step, we generalize each principal ultrafilter and the extension to an arbitrary ultrafilter and an arbitrary set, respectively, and replace V  in the valuation function with V .

3 Examples: Ultrafilter Extensions in Modal Logic 3.1 Normal Modal Logics Familiarity with the language L (), Kripke semantics of modal logic, and also normal modal logics is assumed (cf. e.g. Blackburn et al. 2001). Given a Kripke model S, R, V , define m  (X ) = {s ∈ S | for all t, if Rst, then t ∈ X }. Then, m  (V (ϕ)) = V (ϕ) for all V and ϕ. Recall (cf. e.g. Blackburn et al. 2001) that given a normal modal logic , the canonical model M  = S  , R  , V   is defined as follows:

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• S  = {Σ | Σ is a maximal consistent set}. • R  ΣΓ iff for all ϕ, if ϕ ∈ Σ, then ϕ ∈ Γ . • For each p ∈ P and Σ ∈ S  , Σ ∈ V  ( p) iff p ∈ Σ. The following is a standard result in normal modal logics (cf. e.g. Blackburn et al. 2001). Proposition 1 Let Σ ∈ S  . Then, for all ϕ ∈ L (), for each p ∈ P ∪ {ϕ}, we have p ∈ Σ iff Σ ∈ V  ( p). Applying Theorem 2, we immediately have the following result. Proposition 2 Let Σ ∈ S  . Then, for all ϕ ∈ L (), we have ϕ ∈ Σ iff V  (ϕ) ∈ πΣ . Recall that maximal consistent sets are thought of as a state in the canonical model construction, and from every state one may generate a principal ultrafilter, we thus can obtain the definition of ultrafilter extension of normal modal logics from that of the above canonical model. At first step, from the canonical model above, we replace maximal consistent sets with the corresponding principal ultrafilters and formulas with their extensions, by using Proposition 2 and the fact that V  (ϕ) = m  (V  (ϕ)) (because for all valuations V , V (ϕ) = m  (V (ϕ))). We obtain a model ue(M ) = U f (S), R ue , V ue , where • U f (S) = {πΣ | πΣ is an ultrafilter over S}. • R ue πΣ πΓ iff for all V  (ϕ), if m  (V  (ϕ)) ∈ πΣ , then V  (ϕ) ∈ πΓ . • For each p ∈ P and πΣ ∈ U f (S), πΣ ∈ V ue ( p) iff V  ( p) ∈ πΣ . At second step, we polish the above model, by generalizing each principal ultrafilter and V  (ϕ) to an arbitrary ultrafilter and an arbitrary set, respectively, and replace V  in the valuation function with V . By doing so, we obtain the notion of ultrafilter extension in normal modal logics (e.g. Blackburn et al. 2001, Definition 2.57). • U f (S) = {s | s is an ultrafilter over S}. • R ue st iff for all X , if m  (X ) ∈ s, then X ∈ t. • For each p ∈ P and s ∈ U f (S), s ∈ V ue ( p) iff V ( p) ∈ s. It is then shown that the notion of ultrafilter extension is the required one in normal modal logics. That is, the constructed ultrafilter extension and the original model are L ()-equivalent; moreover, L ()-equivalence can be characterized as -bisimilarity-somewhere-else—namely, between ultrafilter extensions. For the proof details, refer to, e.g. Blackburn et al. (2001, Proposition 2.59, Theorem 2.62).

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3.2 Classical Modal Logics When modal logic is interpreted on neighborhood semantics, we obtain a class of classical modal logics.7 In this part, we apply our method to construct the ultrafilter extension of a neighborhood model for modal logic. Given a neighborhood model M = S, N , V , the necessity operator is interpreted in the following: M , s  ϕ ⇐⇒ V (ϕ) ∈ N (s), where V (ϕ) = {w ∈ S | M , w  ϕ}. Define m N (X ) = {s ∈ S | X ∈ N (s)}. It is clear that m N (V (ϕ)) = V (ϕ) for all V and ϕ. Recall that in the completeness of classical modal logic E (cf. e.g. Chellas 1980), the canonical model M E = S E , N E , V E  is defined as follows: • S E = {Σ | Σ is a maximal consistent set}. • N E (Σ) = {|ϕ| | ϕ ∈ Σ}, where |ϕ| = {Σ ∈ S E | ϕ ∈ Σ}. • For each p ∈ P and Σ ∈ S E , Σ ∈ V E ( p) iff p ∈ Σ. The following is a standard result in classical modal logic. Proposition 3 Let Σ ∈ S E . Then, for all ϕ ∈ L (), for each p ∈ P ∪ {ϕ}, we have p ∈ Σ iff Σ ∈ V E ( p). Applying Theorem 2, we immediately have the following result. Proposition 4 Let Σ ∈ S E . Then, for all ϕ ∈ L (), we have ϕ ∈ Σ iff V E (ϕ) ∈ πΣ . Now, applying the method in Sect. 2, we obtain the notion of ultrafilter extensions in classical modal logics Hansen et al. (2009, Definition 4.20). Definition 3 (Ultrafilter extensions) Let M = S, N , V  be a neighborhood model. The triple ue(M ) = U f (S), N ue , V ue  is the ultrafilter extension of M , if • U f (S) = {s | s is an ultrafilter over S}. • N ue (s) = { Xˆ | m N (X ) ∈ s}, where Xˆ = {s ∈ U f (S) | X ∈ s}. • For each p ∈ P and s ∈ U f (S), s ∈ V ue ( p) iff V ( p) ∈ s. It is then shown that the above notion of ultrafilter extension is the required one in classical modal logics. That is, the constructed ultrafilter extension and the original model are L ()-equivalent; moreover, L ()-equivalence can be characterized as behavioral-equivalence-somewhere-else—namely, between ultrafilter extensions. For the proof details, refer to, e.g., Hansen et al. (2009, Lemma 4.24, Theorem. 4.27). 7 Here by “classical modal logics” it means the systems which contain the axiom ♦ϕ

ϕ↔ψ closed under the inference rule , c.f. e.g. Chellas (1980, Sect. 8). ϕ ↔ ψ

↔ ¬¬ϕ are

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4 Applications: Ultrafilter Extensions in Contingency Logic 4.1 Kripke Contingency Logics In this section, we construct the ultrafilter extension out of a Kripke model in contingency logic. Since in the literature of contingency logic, there are mainly two ways of defining the canonical model, this leads to two notions of ultrafilter extensions. The language of contingency logic is denoted L (), where  is read “it is contingent that”, and the non-contingency operator  is abbreviated as ¬. Semantically, given a Kripke model M = S, R, V  and a state s ∈ S: M , s  ϕ ⇐⇒ there are t, u ∈ S such that Rst, Rsu and M , t  ϕ and M , u  ϕ.

And consequently, M , s  ϕ ⇐⇒ for all t, u ∈ S, if Rst, Rsu, then (M , t  ϕ ⇐⇒ M , u  ϕ).

A Kripke model M is said to be L ()-saturated, if for any s in M , for any Γ ⊆ L (), if Γ is finitely satisfiable in the set of successors of s, then Γ is satisfiable in the set of successors of s.8 Similar to the case of normal modal logics, we here need some requisite notation. Definition 4 Given any nonempty set S and any X ⊆ S, m  (X ) = {s ∈ S | Rst, Rsu for some t ∈ X and some u ∈ / X} m  (X ) = {s ∈ S | for all t, u, if Rst and Rsu, then t ∈ X iff u ∈ X }. It is straightforward to verify that m  (V (ϕ)) = V (ϕ) and m  (V (ϕ)) = V ( ϕ). The definition of m  is very natural, in that just as “Rst, Rsu for some t ∈ X and some u ∈ / X ” corresponds to the Kripke semantics of , s ∈ m  (X ) corresponds to the Kripke semantics of . The naturalness of m  is similar. In fact, the definitions of m  and m  can be followed from the semantics of ϕ and  ϕ, by generalizing V (ϕ) to an arbitrary set X . Also, m  (X ) = m  (X ), m  (X ) = m  (X ) and m  (X ) = m  (X ). Moreover, Proposition 5 For all X, X  , Y ⊆ S, we have (a) m  (X ) ∩ m  (Y ) ⊆ m  (X ∩ Y ). (b) m  (X ∩ Y ) ∩ m  (X  ∩ Y ) ⊆ m  (Y ). Proof Item (a) is direct from Definition 4. For item (b), let s ∈ m  (X ∩ Y ) and s ∈ m  (X  ∩ Y ). Then, there are t, u ∈ S such that s Rt and s Ru and t ∈ X ∩ Y and u∈ / X ∩ Y , and there are t  , u  ∈ S such that s Rt  and s Ru  and t  ∈ X  ∩ Y and  u ∈ / X  ∩ Y . Now, focus on t and t  : t ∈ Y and t  ∈ / Y . Therefore, s ∈ m  (Y ). set of formulas Γ is said to be satisfiable in a set of states X , if there is a state x ∈ X such that for every formula ϕ ∈ Γ , ϕ is true at x. 8A

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∃∀-Version

One version is inspired by the canonical model proposed in Fan et al. (2014, Definition 5.6), which is in turn inspired by an “almost-definability” schema χ → (ϕ ↔ ϕ∧  (χ → ϕ)) (Fan et al. 2014, Definition 2.4), see also (Fan et al. 2015, Proposition 3.5, Definition 4.5) for the multimodal case. Recall that the canonical model M c = S c , R c , V c  for minimal contingency logic CL in (Fan et al., 2014, Definition 5.6) (see also Fan et al. 2015, Definition 4.5 for the multimodal case) is defined as follows: • S c = {Σ | Σ is a maximal consistent set}. • R c ΣΓ iff – there exists χ such that χ ∈ Σ, and – for all ϕ, if  ϕ∧  (χ → ϕ) ∈ Σ, then ϕ ∈ Γ . • For each p ∈ P and Σ ∈ S c , Σ ∈ V c ( p) iff p ∈ Σ. The following result is shown in Fan et al. (2014, Lemma 5.7) (also see Fan et al. 2015, Lemma 4.6 for the multimodal case). Proposition 6 Let Σ ∈ S c . Then, for all ϕ ∈ L (), for each p ∈ P ∪ {ϕ}, we have p ∈ Σ ⇐⇒ Σ ∈ V c ( p). Applying Theorem 2, we get the following result. Proposition 7 Let Σ ∈ S c . Then, for all ϕ ∈ L (), we have ϕ ∈ Σ ⇐⇒ V c (ϕ) ∈ πΣ . Now, with the method in Sect. 2 in hand, we can obtain the ultrafilter extension in Kripke contingency logics. We call it “∃∀-version” since the accessibility relation R ue has the ∃∀-form. Definition 5 (Ultrafilter Extension) Let M = S, R, V  be a Kripke model. The triple ue(M ) = U f (S), R ue , V ue  is the ultrafilter extension of M , if • U f (S) = {s | s is an ultrafilter over S}. • R ue st iff there exists X such that m  (X ) ∈ s, and for all Y , if m  (Y ) ∩ m  (X ∪ Y ) ∈ s, then Y ∈ t. • V ue ( p) = {s ∈ U f (S) | V ( p) ∈ s}. In what follows, we verify that the notion of ultrafilter extension constructed above is a desired one in Kripke contingency logics. That is, ultrafilter extension and the original model are L ()-equivalent; moreover, L ()-equivalence can be characterized as -bisimilarity-somewhere-else—namely, between ultrafilter extensions.

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Proposition 8 For all formulas ϕ ∈ L () and all ultrafilters s over S, we have V (ϕ) ∈ s iff ue(M ), s  ϕ. As a corollary, for all w ∈ S, we have (M , w) ≡L () (ue(M ), πw ). Proof By induction on ϕ ∈ L (). The only nontrivial case is ϕ. First suppose ue(M ), s  ϕ. Then, there are t, u ∈ U f (S) such that R ue st and ue R su and ue(M ), t  ϕ and ue(M ), u  ϕ. By induction hypothesis, V (ϕ) ∈ t and V (ϕ) ∈ / u. From R ue st it follows that there exists X such that m  (X ) ∈ s and, for all Y , if m  (Y ) ∩ m  (X ∪ Y ) ∈ s, then Y ∈ t. Since V (ϕ) ∈ t and t is an ultrafil/ t. Then, m  (V (ϕ)) ∩ m  (X ∪ V (ϕ)) ∈ / s. Similarly, from R ue su we can ter, V (ϕ) ∈  / s. We claim that obtain that there exists X such that m  (V (ϕ)) ∩ m  (X  ∪ V (ϕ)) ∈ m  (V (ϕ)) ∈ s: if not, then m  (V (ϕ)) ∈ s. Since m  (V (ϕ)) = m  (V (ϕ)), by the / s and fact that ultrafilters are closed under intersection, we have m  (X ∪ V (ϕ)) ∈ / s. This means that m  (X ∩ V (ϕ)) ∈ s and m  (X  ∩ V (ϕ)) ∈ s. m  (X  ∪ V (ϕ)) ∈ Again, by the fact that ultrafilters are closed under intersection, we have m  (X ∩ V (ϕ)) ∩ m  (X  ∩ V (ϕ)) ∈ s. Then, using Proposition 5(b) and the fact that ultrafilters are closed under supersets, we infer that m  (V (ϕ)) ∈ s: an contradiction. Therefore, m  (V (ϕ)) ∈ s, viz. V (ϕ) ∈ s. Now, assume V (ϕ) ∈ s, to show ue(M ), s  ϕ. By induction hypothesis, this means that we need to find two states t, u in U f (S) such that R ue st and R ue su and V (ϕ) ∈ t and V (ϕ) ∈ / u. By assumption and m  (V (ϕ)) = V (ϕ), we have m  (V (ϕ)) ∈ s. Define  = {Y | m  (Y ) ∩ m  (V (ϕ) ∪ Y ) ∈ s}. We first show that  is closed under intersection. Let Y, Z ∈ . Then, m  (Y ) ∩ m  (V (ϕ) ∪ Y ) ∈ s and m  (Z ) ∩ m  (V (ϕ) ∪ Z ) ∈ s. By the fact that ultrafilters are closed under supersets, we have m  (Y ), m  (V (ϕ) ∪ Y ) ∈ s and m  (Z ), m  (V (ϕ) ∪ Z ) ∈ s. Then, by the fact that ultrafilters are closed under intersection, m  (Y ) ∩ m  (Z ) ∈ s and m  (V (ϕ) ∪ Y ) ∩ m  (V (ϕ) ∪ Z ) ∈ s. Again, by Proposition 5(a) and the fact that ultrafilters are closed under supersets, m  (Y ∩ Z ) ∈ s and m  (V (ϕ) ∪ (Y ∩ Z )) ∈ s. Therefore, Y ∩ Z ∈ . We now show that for any Y ∈ , Y ∩ V (ϕ) = ∅ and Y ∩ V (ϕ) = ∅. Let Y ∈ , then by definition of , m  (Y ) ∩ m  (V (ϕ) ∪ Y ) ∈ s. By assumption and the fact that ultrafilters are closed under intersection, m  (Y ) ∩ m  (V (ϕ) ∪ Y ) ∩ V (ϕ) ∈ s. Since s does not contain the empty set, we obtain m  (Y ) ∩ m  (V (ϕ) ∪ Y ) ∩ V (ϕ) = ∅. Then, there is an element in m  (Y ) ∩ m  (V (ϕ) ∪ Y ) ∩ V (ϕ), say x. From x ∈ V (ϕ), it follows that there exist y, z ∈ R(x) with y ∈ V (ϕ) and z ∈ / V (ϕ), and then z ∈ V (ϕ) ∪ Y . Then, by x ∈ m  (V (ϕ) ∪ Y ), we infer y ∈ V (ϕ) ∪ Y , and thus y ∈ Y . Therefore, Y ∩ V (ϕ) = ∅. Since x ∈ m  (Y ) and x Ry, x Rz, we infer z ∈ Y , and hence z ∈ Y ∩ V (ϕ), therefore Y ∩ V (ϕ) = ∅. This indicates that  ∪ {V (ϕ)} and  ∪ {V (ϕ)} both have the finite intersection property. Using the Ultrafilter Theorem, there are t, u ∈ U f (S) such that  ∪ {V (ϕ)} ⊆ t and  ∪ {V (ϕ)} ⊆ u. Then, by definition of R ue , we conclude that / u). R ue st and R ue su and V (ϕ) ∈ t and V (ϕ) ∈ u (thus V (ϕ) ∈ Therefore, for all w ∈ S, for all ϕ ∈ L (), we have M , w  ϕ iff w ∈ V (ϕ) iff  V (ϕ) ∈ πw iff ue(M ), πw  ϕ.

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Proposition 9 Let M be a Kripke model. Then, ue(M ) is L ()-saturated. Proof Given any set Γ ⊆ L (), and any ultrafilter s over S, we show that if Γ is finitely satisfiable in the set of successors of s, then Γ is satisfiable in the set of successors of s. W.l.o.g. we may assume that R ue (s) = ∅, since otherwise the statement holds vacuously. By definition of R ue , m  (X ) ∈ s for some X . Suppose that Γ is finitely satisfiable in the set of successors of s, to show that Γ is satisfiable in the set of successors of s. For this, we define the following set Θ = {V (ϕ) | ϕ ∈ Γ  } ∪ {Y | m  (Y ) ∩ m  (X ∪ Y ) ∈ s}, where Γ  is the set of finite conjunctions of formulas in Γ . In the following, we will show that Θ has the finite intersection property. The proof of Proposition 8 has shown that {Y | m  (Y ) ∩ m  (X ∪ Y ) ∈ s} is closed under intersection. Moreover, {V (ϕ) | ϕ ∈ Γ  } is closed under intersection, as shown below. Let A, B ∈ {V (ϕ) | ϕ ∈ Γ  }. Then, A = V (ϕ) and B = V (ψ) for some ϕ, ψ ∈ Γ  . One may easily verify that A ∩ B = V (ϕ ∧ ψ) and ϕ ∧ ψ ∈ Γ  , and thus A ∩ B ∈ {V (ϕ) | ϕ ∈ Γ  }. Next, we show that for any ϕ ∈ Γ  and Y ⊆ S for which m  (Y ) ∩ m  (X ∪ Y ) ∈ s, we have V (ϕ) ∩ Y = ∅. By ϕ ∈ Γ  and supposition, there is an ultrafilter t over S such that R ue st and ue(M ), t  ϕ. By Proposition 8, V (ϕ) ∈ t; by R ue st and the definition of R ue , Y ∈ t. Since t is closed under intersection, we obtain V (ϕ) ∩ Y ∈ t. Since t does not contain the empty set, V (ϕ) ∩ Y = ∅. We have thus shown that Θ has the finite intersection property. Then, by the Ultrafilter Theorem, Θ can be extended to an ultrafilter u. From m  (X ) ∈ s and {Y | m  (Y ) ∩ m  (X ∪ Y ) ∈ s} ⊆ u, it follows that R ue su; moreover, from {V (ϕ) | ϕ ∈ Γ  } ⊆ u, it follows that ue(M ), u  Γ : since for any ϕ ∈ Γ , ϕ ∈ Γ  , then V (ϕ) ∈ u, applying Proposition 8, we derive ue(M ), u  ϕ.  Since the class of L ()-saturated models has the Hennessy–Milner property (Fan et al. 2014), Proposition 3.9, we obtain the main result of this section: L ()equivalence can be thought of as L ()-bisimilarity between ultrafilter extensions. For L ()-bisimilarity, which is denoted by ↔ , we refer to Fan et al. (2014, Definition 3.3). Theorem 3 Let M and M  be models and w ∈ M , w  ∈ M  . Then, (M , w) ≡L () (M  , w  ) ⇐⇒ (ue(M ), πw ) ↔ (ue(M  ), πw ).

4.1.2

∀-version

Another version is inspired by the canonical model given in Kuhn (1995, p. 232), which is in fact equal to the canonical model originally given in Humberstone (1995), as shown in Fan (2017).

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The canonical model M c = S c , R c , V c  for minimal contingency logic in Kuhn (1995) is defined as follows: • S c = {Σ | Σ is a maximal consistent set}. • R c ΣΓ iff for all ϕ, if  (ψ ∨ ϕ) ∈ Σ for all ψ, then ϕ ∈ Γ . • For each p ∈ P and Σ ∈ S c , Σ ∈ V c ( p) iff p ∈ Σ. The following result is shown in Kuhn (1995, Lemma 2). Proposition 10 Let Σ ∈ S c . Then, for all ϕ ∈ L (), for each p ∈ P ∪ {ϕ}, we have p ∈ Σ ⇐⇒ Σ ∈ V c ( p). Theorem 2 then gives us the following result. Proposition 11 Let Σ ∈ S c . Then, for all ϕ ∈ L (), we have ϕ ∈ Σ ⇐⇒ V c (ϕ) ∈ πΣ . Once again, with the method in Sect. 2 in hand, we can obtain the ultrafilter extension in Kripke contingency logic. We call it “∀-version”, since the accessibility relation R ue has the ∀-form. Definition 6 (Ultrafilter Extension) Let M = S, R, V  be a Kripke model. The triple ue(M ) = U f (S), R ue , V ue  is the ultrafilter extension of M , if • U f (S) = {s | s is an ultrafilter over S}. • R ue st iff for all Y , if m  (X ∪ Y ) ∈ s for all X , then Y ∈ t. • V ue ( p) = {s ∈ U f (S) | V ( p) ∈ s}. Proposition 12 For all formulas ϕ ∈ L () and all ultrafilters s over S, we have V (ϕ) ∈ s iff ue(M ), s  ϕ. As a corollary, for all w ∈ S, we have (M , w) ≡L () (ue(M ), πw ). Proof By induction on ϕ ∈ L (). We only consider the case ϕ. First, assume that ue(M ), s  ϕ. Then, there exist t, u ∈ U f (S) such that R ue st and R ue su and ue(M ), t  ϕ and ue(M ), u  ϕ. By induction hypothesis, V (ϕ) ∈ t / t) and V (ϕ) ∈ / u. By definition of R ue , m  (X ∪ V (ϕ)) ∈ / s for some X (i.e., V (ϕ) ∈ / s for some Y . Thus, m  (X ∩ V (ϕ)) ∈ s and m  (Y ∩ V (ϕ)) ∈ and m  (Y ∪ V (ϕ)) ∈ s. Since s is closed under intersection, we have m  (X ∩ V (ϕ)) ∩ m  (Y ∩ V (ϕ)) ∈ s. Since s is closed under supersets, by Proposition 5(b), we infer that m  (V (ϕ)) ∈ s, and thus V (ϕ) ∈ s. Conversely, assume that V (ϕ) ∈ s, to show ue(M ), s  ϕ. By induction hypothesis, we need to find two ultrafilters t, u over S such that R ue st and R ue su and V (ϕ) ∈ t and V (ϕ) ∈ / u. Now, define  = {Y | m  (X ∪ Y ) ∈ s for all X }. We first show that  is closed under intersection. Let Y, Z ∈ . Then, m  (X ∪ Y ) ∈ s and m  (X ∪ Z ) ∈ s for all X . Since s is closed under intersection, m  (X ∪ Y ) ∩ m  (X ∪ Z ) ∈ s. Since

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m  (X ∪ Y ) ∩ m  (X ∪ Z ) ⊆ m  (X ∪ (Y ∩ Z )) (by Proposition 5(a)) and s is closed under supersets, m  (X ∪ (Y ∩ Z )) ∈ s. Since X is arbitrary, we have Y ∩ Z ∈ . We now show that for any Y ∈ , Y ∩ V (ϕ) = ∅ and Y ∩ V (ϕ) = ∅. Let Y ∈ . Then, m  (X ∪ Y ) ∈ s for all X . Letting X = ∅, we obtain m  (Y ) ∈ s; letting X = V (ϕ), we get m  (V (ϕ) ∪ Y ) ∈ s. By assumption and the fact that s is closed under intersection, m  (Y ) ∩ m  (V (ϕ) ∪ Y ) ∩ V (ϕ) ∈ s. Since s does not contain the empty set, we have m  (Y ) ∩ m  (V (ϕ) ∪ Y ) ∩ V (ϕ) = ∅. Thus, there is an x such that x ∈ m  (Y ) ∩ m  (V (ϕ) ∪ Y ) ∩ V (ϕ). By x ∈ V (ϕ), there exist / V (ϕ) (thus z ∈ V (ϕ)). y, z ∈ R(x) such that y ∈ V (ϕ) (thus y ∈ / V (ϕ)) and z ∈ Then, z ∈ V (ϕ) ∪ Y . From this and x ∈ m  (V (ϕ) ∪ Y ) it follows that y ∈ V (ϕ) ∪ Y , and hence y ∈ Y . Combing this and the fact that x ∈ m  (Y ), we infer that z ∈ Y . Now, we have shown that y ∈ Y ∩ V (ϕ) and z ∈ Y ∩ V (ϕ), which implies Y ∩ V (ϕ) = ∅ and Y ∩ V (ϕ) = ∅. Then, similar to the corresponding part of the proof of Proposition 8, we can find / u.  two states t, u ∈ U f (S) with R ue st and R ue su and V (ϕ) ∈ t and V (ϕ) ∈ Proposition 13 Let M be a Kripke model. Then, ue(M ) is L ()-saturated. Proof Given any set Γ ⊆ L (), and any ultrafilter s over S, suppose that Γ is finitely satisfiable in the set of successors of s, to show that Γ is satisfiable in the set of successors of s. For this, we define the following set: Θ = {V (ϕ) | ϕ ∈ Γ  } ∪ {Y | m  (X ∪ Y ) ∈ s for all X }, where Γ  is the set of finite conjunctions of formulas in Γ . In the following, we will show that Θ has the finite intersection property. Similar to the proof of Proposition 12, we can show that {Y | m  (X ∪ Y ) ∈ s for all X } is closed under intersection. Moreover, {V (ϕ) | ϕ ∈ Γ  } is closed under intersection, as shown below. Let A, B ∈ {V (ϕ) | ϕ ∈ Γ  }. Then, A = V (ϕ) and B = V (ψ) for some ϕ, ψ ∈ Γ  . One may easily verify that A ∩ B = V (ϕ ∧ ψ) and ϕ ∧ ψ ∈ Γ  , and thus A ∩ B ∈ {V (ϕ) | ϕ ∈ Γ  }. Next, we show that for any ϕ ∈ Γ  and Y ⊆ S for which m  (X ∪ Y ) ∈ s for all X , we have V (ϕ) ∩ Y = ∅. By ϕ ∈ Γ  and supposition, there is an ultrafilter t over S such that R ue st and ue(M ), t  ϕ. By Proposition 12, V (ϕ) ∈ t; by R ue st and the definition of R ue , Y ∈ t. Since t is closed under intersection, we obtain V (ϕ) ∩ Y ∈ t. Since t does not contain the empty set, V (ϕ) ∩ Y = ∅. We have thus shown that Θ has the finite intersection property. Now, by the Ultrafilter Theorem, Θ can be extended to an ultrafilter u. From {Y | m  (X ∪ Y ) ∈ s for all X } ⊆ u, it follows that R ue su; moreover, from {V (ϕ) | ϕ ∈ Γ  } ⊆ u, it follows that ue(M ), u  Γ : since for any ϕ ∈ Γ , ϕ ∈ Γ  , then V (ϕ) ∈ u, applying Proposition 12 we derive ue(M ), u  ϕ.  Since the class of L ()-saturated models has the Hennessy–Milner property (Fan et al. 2014, Proposition 3.9), we obtain the main result of this section: modal equivalence can be thought of as L () equivalence between ultrafilter extensions.

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Theorem 4 Let M and M  be models and w ∈ M , w  ∈ M  . Then (M , w) ≡L () (M  , w  ) ⇐⇒ (ue(M ), πw ) ↔ (ue(M  ), πw ).

4.2 Neighborhood Contingency Logics Given a neighborhood model M = S, N , V , the non-contingency operator is interpreted in the following. M , s  ϕ ⇐⇒ V (ϕ) ∈ N (s) or V (ϕ) ∈ N (s), where V (ϕ) = {w ∈ S | M , w  ϕ}. Define m C (X ) = {s ∈ S | X ∈ N (s) or X ∈ N (s)}. It is obvious that m C (X ) = m C (X ) and m C (V (ϕ)) = V ( ϕ). Recall that in the completeness of classical contingency logic CCL (Fan and Ditmarsch 2015), the canonical model M c = S c , N c , V c  is defined as follows: • S c = {Σ | Σ is a maximal consistent set}. • N c (Σ) = {|ϕ| | ϕ ∈ Σ}, where |ϕ| = {Σ ∈ S c | ϕ ∈ Σ}. • For each p ∈ P and Σ ∈ S c , Σ ∈ V c ( p) iff p ∈ Σ. The following is a standard result in classical contingency logic (Fan and Ditmarsch 2015, Lemma 1). Proposition 14 Let Σ ∈ S c . Then, for all ϕ ∈ L (), for each p ∈ P ∪ {ϕ}, we have p ∈ Σ iff Σ ∈ V c ( p). Then, Theorem 2 gives us the following result. Proposition 15 Let Σ ∈ S c . Then, for all ϕ ∈ L (), we have ϕ ∈ Σ iff V c (ϕ) ∈ πΣ . Now, applying the method in Sect. 2, we obtain the notion of ultrafilter extensions in classical contingency logics. Definition 7 (Ultrafilter extensions) Let M = S, N , V  be a neighborhood model. The triple ue(M ) = U f (S), N ue , V ue  is the ultrafilter extension of M , if • U f (S) = {s | s is an ultrafilter over S}. X | m C (X ) ∈ s}, where  X = {s ∈ U f (S) | X ∈ s}. • N ue (s) = {  • For each p ∈ P and s ∈ U f (S), s ∈ V ue ( p) iff V ( p) ∈ s. For all s ∈ U f (S) and for all X ⊆ S, we can show Xˆ = Xˆ . With this and the fact that m C (X ) = m C (X ), we can show that N ue is closed under complements, in that for all U ⊆ U f (S), if U ∈ N ue (s), then U ∈ N ue (s).

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Proposition 16 For all formulas ϕ ∈ L () and all ultrafilters s over S, we have V (ϕ) ∈ s iff ue(M ), s  ϕ. As a corollary, for all w ∈ S, we have (M , w) ≡L () (ue(M ), πw ). Proof By induction on ϕ ∈ L (). The only nontrivial case is  ϕ. For this, we have the following equivalences: ue(M ), s  ϕ ⇐⇒ V ue (ϕ) ∈ N ue (s) or V ue (ϕ) ∈ N ue (s) IH ˆ ∈ N ue (s) ˆ ∈ N ue (s) or V (ϕ) ⇐⇒ V (ϕ) ˆ ∈ N ue (s) ˆ ∈ N ue (s) or V (ϕ) ⇐⇒ V (ϕ) ⇐⇒ m C (V (ϕ)) ∈ s or m C (V (ϕ)) ∈ s ⇐⇒ m C (V (ϕ)) ∈ s ⇐⇒ V ( ϕ) ∈ s

In the remainder of this section, we show that L ()-equivalence can be characterized as nbh--bisimilarity-somewhere-else.9 For this, we need to introduce a notion of -saturation for L () in the neighborhood setting, called “L -saturation” in Bakhtiari et al. (2017, Definition 11).  Definition 8 (-saturation) Let M = S, N , V  be a neighborhood model. A set X ⊆ S is -compact, if every set of L ()-formulas that is finitely satisfiable in X is itself also satisfiable in X . M is said to be -saturated, if for all s ∈ S and all ≡L () -closed10 neighborhoods X ∈ N (s), both X and X are -compact. It is shown in Bakhtiari et al. (2017, Theorem 1) that the class of -saturated models is a Hennessy–Milner class. That is, on -saturated models M and M  and states s in M and s  in M  , if (M , s) ≡L () (M  , s  ), then (M , s) ∼ (M  , s  ).11 Proposition 17 Let M be a neighborhood model. Then, ue(M ) is -saturated. Proof Given any s ∈ U f (S) and any ≡L () -closed neighborhood Xˆ ∈ N ue (s), to show Xˆ and Xˆ are both -compact. Since N ue is closed under complements, it is sufficient to show that Xˆ is -compact (Fan 2018, p. 43). Suppose that  ⊆ L () is finitely satisfiable in Xˆ . This means that for any finite set {ϕ1 , · · · , ϕm } ⊆ , there exists u ∈ Xˆ such that M ue , u  ϕ1 ∧ · · · ∧ ϕm . By Proposition 16, V (ϕ1 ) ∩ · · · ∩ V (ϕm ) = V (ϕ1 ∧ · · · ∧ ϕm ) ∈ u. By u ∈ Xˆ , X ∈ u. Since u is closed under intersection, V (ϕ1 ) ∩ · · · ∩ V (ϕm ) ∩ X ∈ u. Since u does not contain the empty set, it follows that V (ϕ1 ) ∩ · · · ∩ V (ϕm ) ∩ X = ∅. 9 Here “nbh” abbreviates neighborhood. As for the definition of nbh--bisimilarity, refer to Bakhtiari

et al. (2017, Definition 9); this definition is simplified in Fan (2018). R be a binary relation and X a set. We say X is R-closed, if for any (x, y) ∈ R, we have x ∈ X iff y ∈ X . 11 ∼ denotes nbh--bisimilarity.  10 Let

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Because ϕ1 , · · · , ϕm are arbitrary, the set {V (ϕ) | ϕ ∈ } ∪ {X } has the finite intersection property. By the Ultrafilter Theorem, there is an u  ∈ U f (S) such that {V (ϕ) | ϕ ∈ } ∪ {X } ⊆ u  . From {V (ϕ) | ϕ ∈ } ⊆ u  and Proposition 16, it follows that ue(M ), u   ; moreover, from X ∈ u  , it follows that u  ∈ Xˆ . Therefore,  is satisfiable in Xˆ .  Since the class of -saturated models is a Hennessy–Milner class, we obtain a characterization of L ()-equivalence as nbh--bisimilarity-somewhere-else— namely, between ultrafilter extensions. Theorem 5 Let M and M  be neighborhood models. Then, (M , w) ≡L () (M  , w  ) ⇐⇒ (ue(M ), πw ) ∼ (ue(M  ), πw ).

5 Conclusion In this paper, based on the similarity between ultrafilters and maximal consistent sets, we proposed a uniform method of constructing ultrafilter extensions out of canonical models. We illustrated this method with ultrafilter extensions of models for normal modal logics and for classical modal logics, which can help us understand why the known ultrafilter extensions are so defined. We then applied it to obtain ultrafilter extensions of models for Kripke contingency logics and for neighborhood contingency logics. Our results also hold for multimodal cases. Although we only investigated ultrafilter extensions of any Kripke/neighborhood model for standard modal logic and contingency logic, we believe our method also works for many other logics, special models (monotonic models, regular models, reflexive models, etc.), and many other semantics (algebraic semantics, coalgebraic semantics, etc.). Once we have the canonical model of a logic, we can construct the notion of ultrafilter extension, by using the abovementioned uniform method, in an automatic way. With suitable notions of bisimilarity and saturation, we can show the ultrafilter extension for this logic is as desired, that is to say, it has the nice properties mentioned in the introduction. Last but not least, it is worth remarking that although the notion of ultrafilter extension for standard modal logic may be obtained from an algebraic aspect (e.g., Blackburn et al. 2001, Chap. 5), one cannot similarly obtain notions of ultrafilter extensions for those logics whose algebraic semantics have not been explored yet, such as contingency logic. Acknowledgements This research is funded by the project 17CZX053 of National Social Science Foundation of China. The author would like to thank three anonymous referees of CLAR 2018 for their insightful comments.

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References Bakhtiari, Z., van Ditmarsch, H., Hansen, H.H.: Neighbourhood contingency bisimulation. In: Indian Conference on Logic and its Applications, pp. 48–63. Springer, Berlin, Heidelberg (2017) van Benthem, J.: Canonical modal logics and ultrafilter extensions. J. Symb. Log. 44(1), 1–8 (1979) van Benthem, J.: Correspondence theory. In: Handbook of philosophical logic, pp. 167–247. Springer, Dordrecht (1984) van Benthem, J.: Notes on modal definability. Notre Dame J. Form. Log. 30(1), 20–35 (1988) Blackburn, P., de Rijke, M., Venema, Y.: Modal Logic. Cambridge Tracts in Theoretical Computer Science 53. Cambridge University Press, Cambridge (2001) Chagrov, A., Zakharyaschev, M.: Modal Logic. Oxford University Press (1997) Chellas, B.F.: Modal Logic: An Introduction. Cambridge University Press (1980) Fan, J.: A family of neighborhood contingency logics. Accepted by Notre Dame J. Form. Log. (2018) Fan, J.: Neighborhood contingency logic: a new perspective. Stud. Log. 11(4), 37–55 (2018) Fan, J., van Ditmarsch, H.: Neighborhood contingency logic. In: Banerjee, M., Krishna, S. (eds.) Logic and Its Application. Lecture Notes in Computer Science, vol. 8923, pp. 88–99. Springer (2015) Fan, J., Wang, Y., van Ditmarsch, H.: Almost necessary. Adv. Modal Log. 10, 178–196 (2014) Fan, J., Wang, Y., van Ditmarsch, H.: Contingency and knowing whether. Rev. Symb. Log. 8(1), 75–107 (2015) Fine, K.: Some connections between elementary and modal logic. In: Proceedings of the Third Scandinavian Logic Symposium, Uppsala 1973, pp. 15–31. North-Holland, Amsterdam (1975) Goldblatt, R.I., Thomason, S.K.: Axiomatic classes in propositional modal logic. Algebra and logic. Lecture Notes in Mathematics, vol. 450, pp. 163–173. Springer, Berlin, Heidelberg and New York (1975) Goranko, V.: Filter and ultrafilter extensions of structures: universal-algebraic aspects. Preprint (2007) Hansen, H.H., Kupke, C., Pacuit, E.: Neighbourhood structures: bisimilarity and basic model theory. Logical Methods Comput. Sci. 5(2), 1–38 (2009) Humberstone, L.: The logic of non-contingency. Notre Dame J. Form. Log. 36(2), 214–229 (1995) Jacobs, B.: Many-sorted coalgebraic modal logic: a model-theoretic study. Theor. Inform. Appl. 35, 31–59 (2001) Jónsson, B., Tarski, A.: Boolean algebras with operators. Part I. Am. J. Math. 73(4), 891–939 (1951) Kuhn, S.: Minimal non-contingency logic. Notre Dame J. Form. Log. 36(2), 230–234 (1995) Kupke, C., Kurz, A., Pattinson, D.: Ultrafilter extensions for coalgebras. In: International Conference on Algebra and Coalgebra in Computer Science, pp. 263–277. Springer, Berlin, Heidelberg (2005) Saveliev, D.I.: Ultrafilter extensions of models. In: Indian Conference on Logic and its Applications, LNCS, vol. 6521, pp. 162–177. Springer, Berlin, Heidelberg (2011) Stone, M.H.: The theory of representations for boolean algebra. Trans. Am. Math. Soc. 40, 37–111 (1936) Venema, Y.: Model definability, purely modal. In: JFAK. Essays Dedicated to Johan van Benthem on the Occasion of his 50th Birthday (1999)

Soft Presuppositions as Scalar Implicatures in Signaling Games Mengyuan Zhao

Abstract In this paper, I will discuss why soft presuppositions behave differently from hard presuppositions: the former are easily defeasible and project nonuniformly in quantificational sentences. I assume that soft triggers should be associated with alternatives, and thus share many similarities with scalar implicatures. As a generalization of Parikh’s games of partial information, I develop game-theoretic models, which provide a unified account for both scalar implicatures and soft presuppositions. I argue that iterated best response (IBR) reasoning allows us to analyze the behaviors of soft presuppositions in accordance with rational inferences. The models yield the following predictions of soft presuppositions: projection happens unless it is common knowledge that the speaker is ignorant about it, in which case the presupposition is defeasible; projection depends on the type of quantifiers, which may lead to nonuniform behaviors. Keywords Signaling games · Iterated best reasoning · Presuppositions · Scalar implicatures

1 Introduction Presupposition, which expresses the information taken for granted, is a common pragmatic phenomenon in our daily communication. For example, when one says Jerry won the race, he or she presupposes that Jerry participated in the race. The word won induces the presupposition, and so is called a presupposition trigger. Presupposition triggers induce some conventional meaning of presuppositions, and a selected list of presupposition triggers can be found in Soames (1982). M. Zhao (B) School of Marxism, University of Shanghai for Science and Technology, 516 Jun Gong Road, Shanghai, China e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2019 B. Liao et al. (eds.), Dynamics, Uncertainty and Reasoning, Logic in Asia: Studia Logica Library, https://doi.org/10.1007/978-981-13-7791-4_7

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Though presupposition triggers are traditionally treated uniformly (see Heim 1983; Beaver 2001; van der Sandt 1992 among many others), it has been noticed that certain triggers behave differently since Karttunen (1971) and Stalnaker (1974). As suggested by Levinson (1983), there are two crucial properties of presuppositional behavior: defeasibility and projection. According to two criteria with respect to these properties, it has been proposed recently that presupposition triggers should be divided into “hard” triggers, such as too, again, it-cleft, and “soft” triggers, such as stop, win, discover (see Abusch 2002, 2010; Romoli 2015). A soft presupposition trigger, comparing to a hard one, is associated with weaker and more contextdependent behavior. First, triggers should be divided based on whether their presuppositions are easily defeasible or cancelable. As illustrated in (1) and (2), the soft presupposition of Linda participated in the context triggered by won is easily canceled by adding a context sentence of I don’t know..., while the hard presupposition of someone saw the cat triggered by it-cleft cannot be easily canceled by the I don’t know... sentence in the sense that (2) is odd.1 (1) I don’t know whether Linda participated in the contest or not. But if she won, she will celebrate. (2) I don’t know whether anybody saw the cat.  But if it is Angelo who saw it, he should let me know. Second, triggers should be divided based on whether they project uniformly in quantificational sentences. As originally suggested by Langendoen and Savin (1971), the projection problem is the problem of determining the presuppositions of complex sentences from presuppositions of their parts. Though presuppositions seem to project uniformly through propositional connectives, questions and modal operators, hard and soft presuppositions project differently in quantificational sentences. The examples in (3) and (4), adapted from Charlow (2009), show that hard presuppositions but not soft ones project uniformly through both universal and existential quantifiers. The hard presupposition of (3d) triggered by too projects uniformly through different quantifiers in (3a–c). The soft presupposition of (4c) triggered by stop only projects through universal quantifiers as in (4a–c) but not through the existential quantifier as in (4d–e). (3)

a. b. c. d. (4) a. b. c. d. e.

All of the students smoke Marlboro too. None of the students smoke Marlboro too. Some of the students smoke Marlboro too. → All of the students smoke something other than Marlboro. All of the students stopped smoking. None of the students stopped smoking. → All of the students used to smoke. Some of the students stopped smoking.  All of the students used to smoke.

1 In English grammar, an it-cleft is a sentence construction that consists of a nonreferential it, a form

of the verb be, a noun phrase, and a relative clause.

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The differences between soft and hard presuppositions constitute a challenge to traditional presupposition theories. Some linguists have proposed a solution based on the idea that soft triggers are identical to strong scalar terms like all, and that soft presuppositions can be accounted for by a theory of scalar implicatures. Their accounts take advantage of grammatical devices including lexical alternatives and pragmatic principles.2 The idea of presenting a unified account of presuppositions and scalar implicatures follows a Gricean view of conversational implicatures.3 One main contribution of Grice is to treat conversation as a cooperative and rational activity, governed by a group of maxims. Based on Gricean assumption, many attempts have been made to formalize the program, among which is the game-theoretic approach. Game-theoretic pragmatics seeks to apply mathematical models of interactive decision making involving agents to a general framework for representing and solving problems of natural language use and interpretation (see Jäger 2011; van Rooij and Michael 2015; Benz and Jon 2018 for selective survey articles). The tradition of game-theoretic pragmatics goes back to the influential work of Lewis (1969). He gives the basic model of signaling games, where communication is taken as an attempt of a speaker to influence decisions of a hearer by transmitting certain signals. As a generalization of Lewisian work, Parikh (1991, 2001) develops the first comprehensive framework of signaling games, named as games of partial information. Compared to Lewisian model, which assumes that messages are associated with a meaning arbitrarily, Parikhian model requires that the use and interpretation of a message is constrained by its literal meaning. On the other hand, Lewis focuses on a standard solution concept of Nash equilibria on the games. A Nash equilibrium is a sequence of strategies that no player has an interest to deviate from it given the choices of other players. However, Lewisian solution is incomplete since it faces the problem of equilibrium selection. To single out the Gricean strategy, Parikh introduces Pareto Nash equilibrium, which assumes that rational agents will select strategies among equilibria based on expected payoffs. There are also many who have explored solution concepts other than equilibrium. Pavan (2013) and Rothchild (2013) adopt dominance reasoning to capture scalar implicatures in a Gricean way. They present the situations including the Some/All Case in terms of signaling games and solve them through iterated elimination of dominated strategies. Franke (2009, 2011) develops the iterated best response (IBR) model and applies it to the analysis of scalar implica2 Abusch

(2002, 2010) assumes that soft triggers are associated with a group of lexical alternatives, which are intuitively contrastive items (for instance, win and lose are a pair of alternatives). She also assumes a principle of disjunctive closure for generating soft presuppositions. Chemla (2010) takes a step further and identifies presuppositional triggers with strong scalar items. He looks at presuppositions as weaker alternatives of their triggers (for instance, participate is a weaker alternative of win). Romoli (2015) adopts an account based on alternatives following Chemla (2010) and an exhaustification principle for generating presuppositions. 3 Grice (1975) first, systematically analyzed a typical case of scalar implicatures, i.e., the Some/All Case. He accounts for the phenomenon by making reference to a set of maxims regulating conversation. In addition, Grice (1981) discussed the relationship between presupposition and implicatures. Following Grice, the neo-Gricean attempted to account for presuppositions in terms of a generalized sense of Gricean implicature (see Atlas 1978; Atlas and Stephen 1981).

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ture.4 The core idea of IBR models is that the players make iterative reasoning based on increasingly sophisticated belief terminating at a fixed point. Franke suggests that the solution of iterated reasoning leads to Gricean strategies. This paper develops game-theoretic models as a unified account applied to scalar implicatures and soft presuppositions. I assume: First, the nature of these pragmatic phenomenons can be better captured by making reference to Gricean maxims rather than by other special grammatical rules; second, Gricean reasoning should be explained in terms of rationality of the agents rather than linguistic principles. These assumptions are consistent with a generalized Gricean view of conversation. I propose that game theory allows us to view soft presuppositions as a form of rational inference in various communicative situations. The models have two parts: signaling games as descriptions of the situations and reasoning accounts as solutions to the games. I argue that IBR reasoning is a better solution than equilibrium for three reasons. First, IBR reasoning will avoid the selection problem of equilibrium. Second, complex calculation of equilibria requires high-level cognitive capabilities of the agents, while IBR reasoning allows bounded rationality of agents by employing a step-by-step reasoning procedure.5 Third, compared to equilibrium which requires an outsider perspective on the calculation of the final solution, IBR reasoning represents a more real reasoning heuristics in situations of language use and interpretation. In Sect. 2, I construct the models for different types of situations mainly based on two standards, namely, speaker’s knowledge about the world and the quantificational context. For each type, I build a signaling game first, then I solve the game in application of IBR reasoning. I draw a comparison with other related models of Gricean game in the end of Sect. 2. In Sect. 3, I apply the models to analyze various cases of scalar implicatures and soft presuppositions. I argue that the predictions of the models can capture the defeasibility and projection behavior of soft presupposition. I conclude in Sect. 4.

2 The Models The game-theoretic models in this section follow the idea of Gricean assumption, that is, conversation should be taken as a cooperative and rational activity. Accordingly, the games are pure coordination games in the sense that the payoff functions of the players are identical. The basic model assumes two possible situations where the players are in and the speaker’s expertise about the situation. The basic model 4 Many

other authors have also suggested an iterative reasoning accounts to capture Gricean reasoning. Jäger (2011) makes contribution to IBR model based on his earlier work of evolutionary dynamics (see Jäger 2007). Benz (2006), Benz and van Rooij (2007) construct the Optimal Answer model, which can be taken as a special version of IBR model. 5 Empirical work on behavioral game theory has shown that agents actually do not play on equilibria. Various researchers, like Selten (1998) and Camerer (2003), provide experimental evidence to show that agents follow a step-by-step reasoning rather than a one-step jump to equilibria while making decisions in game-theoretic situations.

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applies to simple cases of scalar implicature and presupposition projection through negation. As for the extended models, I either expand two situations to three and apply the model to the case of presupposition projection through universal quantifiers, or assume speaker’s non-expertise about the situation and apply the model to defeasibility of presupposition.

2.1 The Basic Model In the basic model, I assume an asymmetric information between interlocutors: the speaker, S, has all relevant information of the world, but the hearer, H , has to judge which world she is in by reasoning on the speaker’s message. There are two situations modeled as worlds: w1 and w2 . The two-situation setting is not only simple but also fair enough to analyze the case where the speaker knows all relevant information about the situation. Let us model the speaker’s knowledge states as her types: t1 is the type where S knows that she is in w1 , t2 is the type where S knows that she is in w2 . In the basic model, t1 and t2 are the only possible types of the speaker. Let us introduce Nature as an impersonal player, say N , which chooses its move to either type of S, ti ∈ T , with a certain probability, pi . In the basic model, I assume that S is in an arbitrary situation, say pi = 21 . S may send a message m i ∈ M to help H to get the information about the current situation. I assume that the speaker will only send messages of which the semantic meanings are consistent with the speaker’s types. Accordingly, I introduce a denotation function, [[·]] : M → f (T ), which maps the semantic meaning of a message to the speaker’s types. In situations involving scalar implicatures or soft presuppositions, the messages are associated with a group of alternatives. Strong and weak scalar items are alternatives, for instance, all and some. Similarly, soft triggers and their presuppositions are alternatives, for instance, win and participate. Lexical alternatives will compositionally form sentential alternatives, which constitute a set of messages for the speaker to choose from. I assume that messages can be distinguished based on their strength, which is defined as follows. Definition 1 For sets of speaker types T and T  , and a denotation function [[·]], message m i is weaker than message m j , say m i < S m j , iff (i) T ⊃ T  ;   / m j ); (ii) ∀ti ∈T (ti ∈ [[m i ]]→ ∃tk ∈  (iii) ∀ti ∈T  (ti ∈ m j → ti ∈ [[m i ]]). A speaker type is the speaker’s knowledge state about the world, and thus can describe the speaker’s private information in a game. Definition 1 suggests: If m 1 is semantically consistent with both t1 and t2 , namely, t1 , t2 ∈ [[m 1 ]], and m 2 is only / [[m 2 ]] and t2 ∈ [[m 2 ]], semantically consistent with t2 but not with t1 , namely, t1 ∈ then m 1 is weaker than m 2 . I assume that m 1 is true in both w1 and w2 , and that m 2 is only true in w2 . According to Definition 1, m 1 < S m 2 . For t1 , S may utter m 1 ; for t2 , S may utter either m 1 or m 2 . I also assume that H may interpret m 1 into either t1 or t2 , and interpret m 2 into t2 .

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Fig. 1 The basic model

In accordance with Gricean Cooperative Principle, I assume that S and H are purely cooperative, in the sense that the utility functions for the players are identical. Specifically, both players prefer H ’s interpretation t j of the received message m corresponding to S’s type ti . I define players’ payoff functions as follows. Definition 2 In basic model, let U N (ti , m, t j ) be the payoff of N ∈ {S, H } given ti , m and t j , where i, j = 1, 2.  U N (ti , m, t j ) =

1, if i = j . 0, if i = j

Definition 2 suggests: both interlocutors will gain 1 if the hearer successfully gets the knowledge of the world through the message received, and both will gain 0 otherwise. Figure 1 illustrates the signaling game tree of the basic model. To solve the basic model, I introduce the IBR reasoning framework. This framework includes two parallel reasoning sequences, that is, the S0 -sequence and the H0 -sequence. For the S0 -sequence, S0 is the level-zero speaker who may play irrationally, but level-1 hearer H1 plays rationally based on her belief in S0 , and level-2 speaker plays rationally based on her belief in H1 , and so on; for the H0 -sequence, H0 is the level-zero hearer who plays according to her prior belief in the speaker’s strategies, and level-1 speaker S1 plays rationally based on her belief in H0 , and so on; this generalizes to that a level k + 1 player plays rationally based on her belief in the strategies of level k player. I now define the scaffolding of IBR reasoning by induction. The S0 -sequence begins from a naïve speaker S0 . I assume that S0 arbitrarily plays a semantically consistent strategy defined as follows. Definition 3 A semantically consistent speaker strategy σ is a function from a speaker type, t (∈ T ), to a message, m(∈ M), that is, semantically consistent with t: σ ∈ M T , where t ∈ [[σ (t)]]. From Definition 1, m 1 is weaker than m 2 , i.e., m 1 is semantically consistent with both t1 and t2 , and m 2 is only semantically consistent with t2 . According to Definition 3, S0 may utter m 1 in t1 , and utter m 1 or m 2 in t2 . The strategies of S0 can be perspicuously illustrated as follows:

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 S0 =

141

 t1 → m 1 . t2 → m 1 , m 2

For clarity, the above illustration means that a naïve speaker will send only m 1 in t1 , and she will send either m 1 or m 2 in t2 . The H0 -sequence begins from a naïve hearer H0 . I assume that H0 may select a strategy ρ ∈ T M that guarantees her the best expected payoff based on an semantically consistent interpretation of messages. Definition 4 A semantically consistent interpretation is a posterior belief μ(t|m) = Pr (t|m), which is to update prior beliefs with semantically consistent meaning of the received message. t is semantically consistent meaning of m iff {t} = [[m]]. Definition 4 suggests that H0 will take into account the semantic meaning of a received message, but will not consider the speaker’s strategies. Let μ0 (t|m) be H0 ’s posterior beliefs updated by the semantic meaning of m. The following results are easily calculated by Bayesian update: μ0 (t1 |m 1 ) = μ0 (t2 |m 1 ) = 21 , μ0 (t1 |m 2 ) = 0, μ0 (t2 |m 2 ) = 1. Let H0 = B R(μ0 ) be the set of all rational responses to semantic interpretations of the messages:  H0 =

 m 1 → t 1 , t2 . m 2 → t2

For level k + 1 players, I assume that they give best responses to their beliefs in their opponent strategies of level k. In S0 -sequence, S0 sends some m to H1 . H1 will choose a strategy according to her posterior belief μ1 (t|m), which is dependent on her belief in S0 ’s strategy and her prior belief in t, say Pr (t): μk+1 (t j |m i ) = 

Pr (t j ) × σk (m i |t j ) .   t  ∈T Pr (t ) × σk (m i |t )

(1)

H1 will select the strategy ρ(m) that guarantees her the best expected payoff EU H1 (t, m):  μ1 (ti |m) × U H (ti , m, t), (2) EU H1 (t, m) = ti ∈T

ρ(m) = B R(μ1 ) ∈ arg max EU H1 (t, m). t∈T

(3)

From (1) and (2), the assumption of pi = Pr (ti ) = 21 , and the corresponding payoffs given in Fig. 1, EU H1 (t1 , m 1 ) = 23 , EU H1 (t1 , m 2 ) = 0, EU H1 (t2 , m 1 ) = 13 , and EU H1 (t2 , m 2 ) = 1. Combining this calculation result with (3), H1 ’s best response is to interpret m 1 into t1 , and m 2 into t2 :  H1 =

 m 1 → t1 . m 2 → t2

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And for S2 who will give a best response according to her belief in H1 :  S2 =

 t1 → m 1 . t2 → m 2

In the basic model, we have finite sets of T and M, which guarantees finitely many pure player strategies. Therefore, the IBR reasoning sequences are bounded to repeat themselves. The idealized solution to IBR reasoning is defined as follows. Definition 5 The idealized solutions of IBR reasoning are infinitely repeated strategies S ∗ and H ∗ : S ∗ = {σ ∈ S|∃i∀ j > i : σ ∈ S j }, H ∗ = {ρ ∈ H |∃i∀ j > i : ρ ∈ H j }. From the reasoning steps above, level-1 hearer H1 has selected a one-to-one mapping strategy between a message and a speaker type, and level-2 speaker S2 who plays according to her belief in H1 will continue this one-to-one mapping, and similarly level-3 hearer H3 will play the same strategy as H1 does, and level-4 speaker will play the same as S2 does, and so on. The reasoning analysis of S0 -sequence shows that repeated strategies will be reached after two rounds of iteration.6 Proposition 1 ∗



S =

   t1 → m 1 m 1 → t1 ∗ ,H = . t2 → m 2 m 2 → t2

Proposition 1 suggests: When it is common knowledge that the speaker has all relevant information of the world, there will be a one-to-one mapping between the weaker message (or the stronger message) and the corresponding speaker type. Proposition 1 is based on an assumption that the speaker has all relevant information about the world and both players know they know this, and know they know they know this, and so on. In other words, they have common knowledge of the speaker’s expertise. In the basic model, this assumption is shown as the player’s common belief in the speaker’s types: there are two types of the speaker, t1 is the type where the speaker knows that she is in w1 , and t2 is the type where the speaker knows that she is in w2 .

2.2 Extended Models 2.2.1

Three-Point Situations

In this extended model, I continue to assume that S has all relevant information of the world. A key difference is that I assume three situations modeled as worlds: w1 , 6 It

is also easy to prove that an analysis of H0 -sequence will lead to the same result.

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Fig. 2 The extended model for three-point situations

w2 , and w3 . The three-situation setting opens a possibility that the speaker is ignorant about the situation. Accordingly, S’s types are t1 , t2 , and t3 . I also assume that the nature will choose its move to either speaker type with an equal probability p = 13 . m 1 is true in all three worlds, m 2 is true in w2 and w3 , and m 3 is only true in w3 . According to Definition 1, m 1 < S m 2 < S m 3 . Players’ payoff function is given in Definition 2. Figure 2 illustrates the signaling game tree of the extended model for three-point situations. The following proposition illustrates the idealized solution to extended model for three-point situations within the IBR reasoning framework. Proofs are shown in Appendix 1. Proposition 2

⎧ ⎧ ⎫ ⎫ ⎨ t1 → m 1 ⎬ ⎨ m 1 → t1 ⎬ S ∗ = t2 → m 2 , H ∗ = m 2 → t2 . ⎩ ⎩ ⎭ ⎭ t3 → m 3 m 3 → t3

Proposition 2 suggests: When it is common knowledge that the speaker has all relevant information about the three possible worlds, there will be a one-to-one mapping between the weakest message (or the medium message or the strongest message) and the corresponding speaker type.

2.2.2

Speaker Non-expertise

In this extended model, I will give up the assumption of speaker expertise in the two models above. In other words, I assume that there is a chance for S to be ignorant about the situation of the world. For two possible situations, w1 and w2 , there may be three speaker types: t1 is the type where S knows that she is in w1 , t2 is the type where S knows that she is in w2 , and t3 is the type where S is uncertain about whether she is in w1 or w2 . I assume that S is of t3 with a probability of p, and that there is equal chance, say q, for S to be of t1 and t2 . It is obvious that q = 1−2 p . m 1 is true in both

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Fig. 3 The extended model for speaker non-expertise

t1 and t2 , but m 2 is only true in t2 , which means m 1 < S m 2 . Note that for t3 , only the weaker message m 1 is true. Players’ payoff function follows Definition 2. Figure 3 illustrates the signaling game tree of the extended model for speaker non-expertise. The following proposition illustrates the idealized solution to extended model for speaker non-expertise within the IBR reasoning framework. Proofs are shown in Appendix 2. Proposition 3 ⎧ ⎧ ⎫ m1 ⎪ ⎪ ⎨ t1 → m 1 ⎬ ⎨ m S ∗ = t2 → m 2 , H ∗ =  2 m ⎩ ⎪ ⎭ 1 ⎪ t3 → m 3 ⎩ m2

→ t1 , if p < → t2 , → t3 , if p > → t2 ,

1 3

⎫ ⎪ ⎪ ⎬

1 3

⎪ ⎪ ⎭

.

Proposition 3 suggests: When it is more probable that the speaker is ignorant about the current situation than that she is certain about her type, the speaker will use the weaker message to express her ignorance; otherwise, she will use the weaker message to express her knowledge about one situation.

2.3 Model Comparison 2.3.1

Parikhian Model

The key differences between my models and Parikhian games of partial information (see Parikh 1991, 2001) are as follows. First, I assume that t ∈ T represents the speaker’s type, that is, speaker’s knowledge about the current situations, while Parikh assumes that t illustrates the situations directly. This improvement makes it possible to describe the speaker’s expertise about the situation in the current models. Second, I introduce the definition of message strength to give an order for alternatives associated with scalar items and presuppositions. The introduction of message

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strength is the key contribution to the formalization work of the game models, based on which the model may offer a uniform account for both scalar implicature and soft presupposition. Third, I adopt the idealized solution of IBR reasoning as the solution concept of the games, while Parikh uses Pareto dominant equilibrium as a solution. The solution of Pareto dominant equilibrium is based on the idea that Gricean strategy is selected because of its salience. In comparison, IBR reasoning selects the intuitive strategy based on player’s rationality, which is a more satisfying account. Comparing to Parikhian model, my models consider the speaker’s expertise of the current situation and how to represent scalar meanings of different messages. I also take pragmatic reasoning as an iterated process rather than a static outcome from an outsider perspective.

2.3.2

Franke’s Model

The key difference between my IBR reasoning framework and Franke’s model (see Franke 2009, 2011) is as follows. Franke assumes that the hearer has unbiased prior beliefs in speaker’s types. In my extended model for speaker non-expertise, I introduce a probability p to indicate the hearer’s belief in speaker’s non-expertise. Sentences with the form of I don’t know... in the context will guarantee the hearer’s belief in the speaker’s ignorance about the situation. And this belief update can be formalized in terms of p. Therefore, the introduction of p is key to analyze the rationality of defeasibility of soft presuppositions. Compared to Franke’s model, my models show a way to quantify the role of common belief in speaker’s expertise. Specifically, it is useful to explain the property of defeasibility of both scalar implicatures and soft presuppositions.

2.3.3

Rothschild’s Model

Rothschild (2013) offers a game-theoretic model in accordance with the idea of Gricean maxims, and he applies the model to an analysis of various situations involving scalar implicatures. In comparison, my models are to offer a unified account for both scalar implicatures and soft presuppositions. Specifically, I assume that two crucial properties of soft presuppositions, namely, defeasibility and projection can be explained by rational inferences in terms of game models. In addition, Rothschild adopts the solution concept based on iterated elimination of weakly dominated strategies, which imposes strong requirements on the reasoning capabilities of the agents. In comparison, I use the idealized solution from IBR reasoning, which also allows bounded rationality of the players by employing a step-by-step reasoning procedure.

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3 Applications 3.1 Scalar Implicatures: Some/All Case

Example Helen gave Sam a small bag of cookies the day before, and Sam says to Helen, next morning, “I ate some of the cookies yesterday.”

I now apply the basic model to analyze the situations shown in Example 1. Here are two situations: Sam ate some but not all of the cookies, say w1 , and Sam ate all, say w2 . Let us model Sam’s knowledge states as his types: t1 is the type where Sam knows that he is in w1 , t2 is the type where Sam knows that he is in w2 . It is equally possible that Sam is of type t1 or t2 . If Sam is of t1 , he may utter I ate some of the cookie, say m 1 ; if Sam is of t2 , he may either utter m 1 or I ate all of the cookie, say m 2 . For t2 , m 1 is an alternative of m 2 , and m 1 contains a weaker scalar item than m 2 . Literally, when Helen hears m 1 , she may interpret it into either t1 or t2 . According to Proposition 1, rational interlocutors will link the message of some to the implicature of some but not all, namely, a one-to-one mapping between m 1 and t1 . For the simple case of scalar implicatures, the basic model predicts the Gricean strategy pair. In this case, the basic game-theoretic model based on IBR reasoning allows us to cash out scalar implicatures as rational inferences. In other words, the basic model may provide an account for the rationality of scalar implicatures.

3.2 Soft Presupposition: Projection Through Negation

Example Sam and Helen are discussing the result of a contest, and Sam says to Helen, “Linda did not win the contest.”

The basic model can be applied to analyze the situations in Example 2. For two situations: Linda participated in but did not win the contest, say w1 , and she did not even participate, say w2 . t1 is the type where Sam knows that he is in w1 , t2 is the type where Sam knows that he is in w2 . It is equally possible that Sam is of type t1 or t2 . If Sam is of t1 , he may utter Linda did not win the contest, say m 1 ; if Sam is of t2 , he may either utter m 1 or Linda did not participate in the contest, say m 2 . For t2 , m 1 is an alternative of m 2 , and m 1 contains a weaker item of not win than not participate

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in m 2 . According to Proposition 1, rational interlocutors will link the message of not win to the presupposition of not win but participated, namely, a one-to-one mapping between m 1 and t1 . For the case of soft presupposition trigger of win, the basic game-theoretic model predicts that its presupposition of participate projects through negation. In other words, the basic model may provide an account for the rationality of projection behavior of soft presuppositions through negation.7

3.3 Soft Presupposition: Projection Through Existential Quantifiers Example Sam and Helen are discussing a computer game, and Sam says to Helen, “Some of the students won the game.”

The basic model can be applied to analyze Example 3. For two situations: Some but not all of the students participated in the game, say w1 , and all participated but only some won, say w2 . t1 is the type where Sam knows that he is in w1 , t2 is the type where Sam knows that he is in w2 . It is equally possible that Sam is of type t1 or t2 . If Sam is of t1 , he may utter Some of the students won the game, say m 1 ; if Sam is of t2 , he may either utter m 1 or All participated though only some won, say m 2 . According to Proposition 1, rational interlocutors will not link the message of some win to the presupposition of all participated. Proposition 1 is in accordance with the idea that presupposition of participate triggered by win will not project through existential quantifiers.

3.4 Soft Presupposition: Projection Through Universal Quantifiers Example Sam and Helen are discussing the result of a marathon, and Sam says to Helen, “None of the students won the marathon.”

7 In this paper, I only apply the basic model to account for the rationality of presupposition projection

through the simplest case of proposition connectives, i.e., negation. A discussion about the rationality of uniform projection through other proposition connectives, such as conditionals, conjunctions, disjunctions, and through questions and modal operators is also interesting but beyond the scope of this paper.

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The extended model for three-point situations can be applied to analyze Example 4. For three situations, all students participated in the marathon, but none of then won, say w1 ; some but not all of the students participated, but none of them won, say w2 ; none of the students participated, say w3 . t1 is the type where Sam knows that he is in w1 , t2 is the type where Sam knows that he is in w2 , and t3 is the type where Sam knows that he is in w3 . It is equally possible that Sam is of type t1 , t2 or t3 . If Sam is of t1 , he may utter None of the students won the marathon, say m 1 ; if Sam is of t2 , he may either utter m 1 or Some but not all of the students participated in the marathon, but none of them won, say m 2 ; if Sam is of t3 , he may utter m 1 or m 2 or None of the students participated in the marathon. For t2 , m 1 is an alternative of m 2 . For t3 , m 1 and m 2 are an alternatives of m 3 . According to Proposition 2, rational interlocutors will link the message of none of the students won to the presupposition of all the students participated, namely, a one-to-one mapping between m 1 and t1 . Proposition 2 is in accordance with the idea that presupposition of participate triggered by win will project through the negation of universal quantifiers. For the case of soft presupposition trigger of win, the extended game-theoretic model predicts that its presupposition of participate projects through the negation of universal quantifier, none, but does not project through the existential quantifier, some (see Sect. 3.2). In other words, the extended model may provide an account for the rationality of nonuniform projection behavior of soft presuppositions through different types of quantifiers.

3.5 Soft Presupposition: Defeasibility

Example Sam and Helen are discussing the result of a contest, and Sam says to Helen, “Linda did not win the contest. Maybe she did not even participated.”

The extended model for speaker non-expertise can be applied to analyze Example 5. For two situations, Linda participated in but did not win the contest, say w1 , and she did not even participate, say w2 . t1 is the type where Sam knows that he is in w1 , t2 is the type where Sam knows that he is in w2 , t3 is the type where Sam is ignorant about whether he is in w1 or w2 . The message Maybe she did not even participated gives Helen a sufficient reason to believe that Sam is of type t3 . In other words, the probability of t3 is as equal to 1, say p = 1. According to Proposition 3, rational interlocutors will link the message of not win to Sam’s ignorance about Linda’s participation. Proposition 3 is in accordance with the idea that presupposition of participate triggered by win can be canceled by further information indicating speaker’s non-expertise.

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For the case of soft presupposition trigger of win, the extended game-theoretic model predicts that when the speaker is ignorant about the current situation, the soft presupposition of participate can be canceled. In other words, the extended model may provide an account for the rationality of easy defeasibility of soft presuppositions.

4 Summary In this paper, I develop game-theoretic models as a unified account for both scalar implicatures and soft presuppositions. I explain why soft presuppositions project nonuniformly through different types of quantifiers, and why soft presuppositions are easily defeasible. The solutions based on IBR reasoning serve to single out intuitive strategies, which suggests that behavioral properties of soft presuppositions should be viewed in terms of rational inferences. Acknowledgements I would like to thank three anonymous referees for very helpful comments. This work is supported by the Shanghai Philosophy and Social Sciences Fund under Grant No. 2017EZX008, the National Social Science Fund under Grant No. 18CZX014, the National Natural Science Foundation of China under Grant No. 61703277, and the Shanghai Sailing Program under Grant No. 17YF1427000.

Appendix 1: Proof of Proposition 2 First consider S0 -sequence. From Definition 2, ⎧ ⎫ ⎨ t1 → m 1 ⎬ S0 = t2 → m 1 , m 2 . ⎩ ⎭ t3 → m 1 , m 2 , m 3 6 3 , EU H1 (t2 , m 1 ) = 11 , EU H1 (t3 , m 1 ) = Given (1) and (2), EU H1 (t1 , m 1 ) = 11 3 2 EU H1 (t2 , m 2 ) = 5 , EU H1 (t3 , m 2 ) = 5 , EU H1 (t3 , m 3 ) = 1. Given (3),

⎧ ⎫ ⎨ m 1 → t1 ⎬ H1 = m 2 → t2 . ⎩ ⎭ m 3 → t3 Since S2 will act according to her belief in H1 , ⎧ ⎫ ⎨ t1 → m 1 ⎬ S2 = t2 → m 2 . ⎩ ⎭ t3 → m 3

2 , 11

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Evidently, the S0 -sequence begins repetition after two round of iteration. In other words, S ∗ = S2 and H ∗ = H1 . The H0 -sequence leads to the same fixed point in a similar way. 

Appendix 2: Proof of Proposition 3 First, consider the S0 -sequence. From Definition 2, ⎧ ⎫ ⎨ t1 → m 1 ⎬ S0 = t2 → m 1 , m 2 . ⎩ ⎭ t3 → m 1 Given (1) and (2), EU H1 (t1 , m 1 ) = 4p , 3+ p

2(1− p) , 3+ p

EU H1 (t2 , m 1 ) =

1− p , 3+ p

EU H1 (t3 , m 1 ) =

EU H1 (t2 , m 2 ) = 1. Given (3), ⎧ m1 ⎪ ⎪ ⎨ m H1 =  2 m ⎪ 1 ⎪ ⎩ m2

→ t1 , if p < → t2 , → t3 , if p > → t2 ,

1 3

⎫ ⎪ ⎪ ⎬

1 3

⎪ ⎪ ⎭

.

Since S2 will act according to her belief in H1 , ⎧ ⎫ ⎨ t1 → m 1 ⎬ S ∗ = t2 → m 2 . ⎩ ⎭ t3 → m 3 Evidently, the S0 -sequence begins repetition after two round of iteration. Then, S ∗ = S2 and H ∗ = H1 . And the H0 -sequence leads to the same fixed point in a similar way. 

References Abusch, D.: Lexical alternatives as a source of pragmatic presupposition. In: Jackson, B., (ed.) Semantics and linguistic theory (SALT) 12, pp. 1–19 (2002) Abusch, D.: Presupposition triggering from alternatives. J. Semant. 27, 1–44 (2010) Atlas, J.D.: On presupposing. Mind 87, 396–411 (1978) Atlas, J.D., Levinson, S.C.: It-clefts, informativeness, and logical form. In: Cole, P. (ed.) Radical Pragmatics, pp. 1–51. Academic Press, New York (1981) Beaver, D.I.: Presupposition and Assertion in Dynamic Semantics. CSLI Publications, Stanford University (2001) Benz, A.: Utility and relevance of answers. In: Benz, A., Jäger, G., van Rooij, R. (eds.) Game Theory and Pragmatics, pp. 195–214. Palgrave Macmillan, Basingstoke, UK (2006)

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Benz, A., van Rooij, R.: Optimal assertions and what they implicate. Topoi 26, 63–78 (2007) Benz, A., Stevens, J.: Game-theoretic approaches to pragmatics. Annu. Rev. Linguist. 4, 173–191 (2018) Camerer, C.: Behavioral Game Theory: Experiments in Strategic Interaction. Princeton University Press (2003) Charlow, S.: “Strong” predicative presuppositional objects. In: Proceedings of ESSLLI 2009, Bordeaux (2009) Chemla, E.: Similarity: towards a unified account of scalar implicatures, free choice permission and presupposition projection. Unpublished manuscript (2010) Franke, M.: Signal to act: game theory in pragmatics. Ph.D. thesis, Institute for Logic, Language and Computation, University of Amsterdam (2009) Franke, M.: Quantity implicatures, exhaustive interpretation, and rational conversation. Semant. Pragmat. 4, 1–82 (2011) Grice, P.: Logic and conversation. In: Cole, P., Morgan, J. (eds.) Syntax and Semantics, pp. 41–58. Academic Press, New York (1975) Grice, P.: Presupposition and conversational implicature. In: Cole, P. (ed.) Radical Pragmatics, pp. 183–198. Academic Press, New York (1981) Heim, I.: On the projection problem for presuppositions. In: Flickinger, D.P. (ed.) Proceedings of WCCFL 2, pp. 114–125. Stanford University, Stanford, CSLI Publications, California (1983) Jäger, G.: Game dynamics connects semantics and pragmatics. In: Pietarinen, A.-V. (ed.) Game Theory and Linguistic Meaning, pp. 89–102. Elsevier, Amsterdam (2007) Jäger, G.: Game-theoretical pragmatics. In: van Benthem, J., ter Meulen, A. (eds.) Handbook of Logic and Language, pp. 467–491. Elsevier, Amsterdam (2011) Karttunen, L.: Some observations on factivity. Pap. Linguist. 4, 55–69 (1971) Langendoen, T., Savin, H.: The projection problem for presuppositions. In: Fillmore, C., Langendoen, T. (eds.) Studies in Linguistic Semantics, pp. 373–388. Holt, Reinhardt and Winston, New York (1971) Levinson, SC.: Pragmatics. Cambridge University Press, Cambridge (1983) Lewis, D.: Convention. Harvard University Press, Cambridge (1969) Parikh, P.: Communication and strategic inference. Linguist. Philos. 14, 473–531 (1991) Parikh, P.: The Use of Language. CSLI Publications, Stanford, CA (2001) Pavan, S.: Quantity implicatures and iterated admissibility. Linguist. Philos. 36, 261–290 (2013) Romoli, J.: The presuppositions of soft triggers are obligatory scalar implicatures. J. Semant. 32, 173–219 (2015) Rothschild, D.: Game theory and scalar implicatures. Philos. Perspect. 27, 438–478 (2013) Selten, R.: Features of experimentally observed bounded rationality. Eur. Econ. Rev. 42, 413–436 (1998) Soames, S.: How presuppositions are inherited: a solution to the projection problem. Linguist. Inq. 13, 483–545 (1982) Stalnaker, R.: Pragmatic presuppositions. In: Munitz, M., Unger, P (eds.) Semantics and Philosophy, pp. 197–213. New York University Press (1974) van der Sandt, R.: Presupposition projection as anaphora resolution. J. Semant. 9, 333–377 (1992) van Rooij, R., Franke, M.: Optimality-theoretic and game-theoretic approaches to implicature. In: The Stanford Encyclopedia of Philosophy. (2015). http://plato.stanford.edu/entries/ implicatureoptimality-games/

Abstract Argumentation in Dynamic Logic: Representation, Reasoning and Change Sylvie Doutre, Andreas Herzig and Laurent Perrussel

Abstract We provide a logical analysis of Dung’s abstract argumentation frameworks and their dynamics. We express attack relation and argument status by means of propositional variables and define acceptability criteria by formulas of propositional logic, which enables us to formulate the standard reasoning problems in logic. While the approaches in the literature express these problems as Boolean or quantified Boolean formulas, we here take advantage of a variant of Propositional Dynamic Logic PDL: Dynamic Logic of Propositional Assignments DL-PA, whose atomic programs are assignments of propositional variables to truth values. One of the benefits is that algorithms computing extensions of argumentation frameworks can be viewed as particular DL-PA programs. This allows us to formally prove the correctness of these algorithms. Another benefit is that in the same logic, we can also design and study programs which modify argumentation frameworks. Indeed, the basic operations on these propositional variables, viz. change of the truth values of the attack variables and the argument status variables, are nothing but atomic programs of DL-PA. We mainly focus on how the acceptance of one or more arguments can be enforced and show how this can be achieved by changing the truth values of the propositional variables describing the attack relation in a minimal way. Keywords Argumentation framework · Update · Revision · Dynamic logic

S. Doutre (B) · L. Perrussel IRIT, Université Toulouse 1 Capitole, Toulouse, France e-mail: [email protected] L. Perrussel e-mail: [email protected] A. Herzig IRIT, CNRS, Toulouse, France e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2019 B. Liao et al. (eds.), Dynamics, Uncertainty and Reasoning, Logic in Asia: Studia Logica Library, https://doi.org/10.1007/978-981-13-7791-4_8

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1 Introduction Argumentation is a reasoning model based on the construction and on the evaluation of arguments. The seminal approach by Dung (1995) defines an argumentation framework (henceforth abbreviated AF) as a set of arguments along with an attack relation between them. The arguments are abstract: their structure and origin are left unspecified. Dung and his followers defined various semantics for the evaluation of the acceptability of arguments. We here focus on semantics in terms of extensions: subsets of the set of arguments that are collectively acceptable. Several such semantics were proposed and discussed in the literature. We refer to Baroni and Giacomin (2007, 2009) for a comprehensive overview of the principles underlying extension-based AF semantics as well as other AF semantics. Extension-based semantics were analysed in several logical frameworks, including propositional logic Besnard and Doutre (2004), Answer-Set Programming Gaggl et al. (2015), Alternating-time Temporal Logic Belardinelli et al. (2019), and quantified Boolean formulas QBF (Arieli and Caminada 2013; Diller et al. 2015). Their encodings of AFs in logic typically make use of two kinds of propositional variables: attack variables such as ra,b express that argument a attacks b and acceptance variables such as ina express that a is accepted. The various semantics can then be characterised in that language by Boolean formulas built from attack and acceptance variables, constraining the valuations to correspond to the extensions under the semantics: each model of the formula stands for an extension of the AF. In the present paper, we advocate another logical framework: Dynamic Logic of Propositional Assignments, abbreviated DL-PA Balbiani et al. (2013). DL-PA is a simple instantiation of Propositional Dynamic Logic PDL (Harel 1984; Harel et al. 2000) whose atomic programs are assignments of propositional variables to either true or false. Complex programs are built as usual from atomic programs by the standard PDL program operators of sequential composition, nondeterministic composition, converse and test. It is shown in Balbiani et al. (2013) that every DL-PA formula can be reduced to an equivalent propositional formula. The aim of our paper is to show that DL-PA provides an interesting tool to reason about AFs and their modification. There are two immediate benefits of the DL-PA framework. The first that a given semantics σ can be characterised by means of a DL-PA formula. While this is a priori not surprising—given that DL-PA and propositional logic have the same expressive power—, it is of interest because DL-PA formulas are sometimes more compact. This applies in particular to the preferred semantics: it is one level higher in the polynomial hierarchy than the other semantics and can therefore not be captured by a polynomial propositional logic formula; we provide a polynomial DL-PA formula. The second benefit is that, being a dynamic logic, DL-PA can account for the dynamics that is at work in the construction of extensions. To start with, we can define a general family of nondeterministic DL-PA programs building all extensions of a given AF. These programs are parametrised by a given semantics σ and follow a generate-and-test schema: they generate all logically possible extensions and then test

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the (propositional or DL-PA) formula characterising the semantics. Such programs exist for every semantics that can be characterised by a DL-PA formula. When executed in a valuation describing an AF they build all the extensions of that AF. The above generic algorithm can be improved by recasting specific more efficient algorithms into DL-PA programs, enabling formal verification of correctness. The third and main benefit of DL-PA is that it provides a suitable framework to account for the dynamics of AFs. Indeed, authors from several places recently investigated that issue, including Baumann and Brewka (2010), Baumann (2012), Baumann and Brewka (2013), Baumann and Brewka (2019) from Bisquert et al. (2011), Bisquert et al. (2013a, b), Bisquert (2013), Dupin de Saint-Cyr et al. (2016) from Toulouse, Coste-Marquis et al. (2013, 2014, 2019) Mailly (2015), Coste-Marquis (2014), Delobelle et al. (2016) from Lens, Booth et al. (2013) from Luxembourg, Fan and Toni (2015) from London, Diller et al. (2019) from Vienna, and Niskanen et al. (2016), Wallner et al. (2016) from Helsinki. They are all based on a representation of AFs in propositional logic. They start by distinguishing several kinds of AF modifications, such as the addition or the removal of attacks, or the enforcement of the acceptability of a given argument a, as well as several kinds of success criteria, e.g. such that a is part of at least one extension, or of all extensions. All these papers build on previous work in belief change, either referring to AGM theory Alchourrón et al. (1985), such as Booth et al. (2013); Coste-Marquis et al. (2013) or to KM theory Katsuno and Mendelzon (1992), such as Bisquert et al. (2013a). They express the modification as a logical formula describing some goal, i.e. a property that the AF should satisfy: the task is to revise/update the AF so that this formula is true. An overview of the contributions to the dynamics of AFs can be found in Doutre and Mailly (2018). The above papers do not provide a single framework encompassing at the same time an AF, the logical definition of the enforcement constraint and the change operations: there is usually one language for representing AFs and another language for representing constraints, plus some definitions in the metalanguage connecting them. As we are going to show, DL-PA provides a general, unified logical framework for both the representation and the update of AFs. We start by showing that the construction of extensions under a given semantics can be performed by a DL-PA program that is parametrised by the formula describing the semantics. Then, we consider modifications of the attack relation and/or of the extensions. Modifications of the extensions are enforced by changing the attack relation only (addition or removal of attacks between the existing arguments). This can be achieved by changing the truth values of the attack variables. More precisely, to every input formula ϕ describing the desired modification, we associate a DL-PA program πϕ implementing the update by ϕ. We can then check whether a formula ψ is true in all (resp. in some) extensions of (the AF resulting from) the update of the AF by the goal ϕ. The paper is organised as follows. In the next section, we introduce DL-PA. In Sect. 3, we recall the definitions of an AF and of various semantics as well as their encoding in propositional logic and in DL-PA. In Sect. 4, we show how extensions can be built by means of DL-PA programs. In Sect. 5, we discuss several possible

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kinds of modifications of an AF. We then focus on Sect. 6 on the most challenging kind of operation: modifications enforcing a goal either in all extensions or in some extension. We apply it to the modification of the attack relation and of the extensions (Sect. 7). After that, we discuss several ways to extend our framework in order to capture other kinds of modifications (Sect. 8). The last section concludes. The present paper extends and generalises previous work that was presented at KR’2014 Doutre et al. (2014) and complemented in Doutre et al. (2017). We have in particular elaborated how the semantics can be compactly described by means of DL-PA formulas and how it can be computed by means of DL-PA programs.

2 Dynamic Logic of Propositional Assignments We now introduce our logical framework: Dynamic Logic of Propositional Assignments DL-PA Herzig et al. (2011); Balbiani et al. (2013, 2014); Herzig (2014). DL-PA is an instance of Propositional Dynamic Logic PDL Harel (1984); Harel et al. (2000). It has the same program operators: sequential and nondeterministic composition, converse, iteration and test. However, the atomic programs of DL-PA are not abstract as in PDL, but concrete assignments of truth values to propositional variables: the assignment of true to p is written p← and the assignment of false to p is written p←⊥. The formulas of DL-PA express what holds after the execution of a program. We here base our work on the star-free version of DL-PA of Herzig et al. (2011) that we extend by the operator of converse execution of programs as done in Herzig (2014). We keep on calling that logic DL-PA. In the rest of the section, we define syntax and semantics of DL-PA, state its relevant properties and introduce some useful programs.

2.1 Language The language of star-free DL-PA is defined by the following grammar: ϕ ::= p |  | ⊥ | ¬ϕ | ϕ ∨ ϕ | π ϕ π ::= p← | p←⊥ | π ; π | π  π | π − | ϕ?

(1)

where p ranges over the set of propositional variables P. The key formula π ϕ reads ‘ϕ holds after some execution of π ’. The atomic programs p← and p←⊥, respectively, make p true and make p false. The operators of sequential composition (‘;’), nondeterministic composition (‘’) and test (‘(.)?’) are familiar from PDL.

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The operator ‘(.)− ’ is the operator of converse execution. So, π − ϕ can be read ‘ϕ was true before some execution of π ’. An expression is a formula or a program. We use  to denote expressions. Let P ⊆ P be a set propositional variables. The set of DL-PA formulas that can be built from P is noted Lang(P). Given an expression , the set of variables from P occurring in  is noted P(). So, P(ϕ) is the set of propositional variables from P occurring in the formula ϕ. For example, P( p←  q←⊥) = { p, q} = P( p←⊥q). The length of an expression , noted ||, is the number of symbols used to write down , without ‘’, ‘’ and parentheses. For example, |¬(q ∨ ¬r )| = 1+(1+1+2) = 5, |q←(q ∨ r )| = 3+3 = 6 and | p←⊥; p?| = 3+1+2 = 6. Conjunction (∧), implication (→) and equivalence (↔) are considered with their usual meaning. We also make use of the exclusive disjunction ϕ ⊕ ψ which abbreviates (ϕ∧¬ψ) ∨ (¬ϕ∧ψ). The formula [π ]ϕ abbreviates ¬π ¬ϕ. So, [π ]ϕ has to be read ‘ϕ holds after every execution of π ’.

2.2 Semantics of DL-PA Models of DL-PA formulas are nothing but models of classical propositional logic, i.e. subsets of the set of propositional variables P, alias valuations. We use v, v ,…for valuations. We sometimes write v( p) = 1 when p ∈ v and v( p) = 0 when p ∈ / v. DL-PA formulas ϕ are interpreted as sets of valuations ||ϕ|| ⊆ 2P . When v ∈ ||ϕ||, we say that v is a model of ϕ or a ϕ-model. DL-PA programs π are interpreted as relations between valuations ||π || ⊆ 2P×P . The definitions of ||ϕ|| and ||π || are by mutual recursion just as in PDL, where the atomic programs p← and p←⊥ are interpreted as update operations on valuations. Table 1 gives the interpretation of the DL-PA connectives. For instance, suppose P is the singleton { p}. Consider the atomic program π = p← and the two valuations v1 = ∅ and v2 = { p}. The execution of π relates v1 to v2 , and v2 to itself. So π is interpreted as ||π || = {(v1 , v2 ), (v2 , v2 )}. Two formulas ϕ1 and ϕ2 are formula equivalent if ||ϕ1 || = ||ϕ2 ||. Two programs π1 and π2 are program equivalent if ||π1 || = ||π2 ||. In that case, we write π1 ≡ π2 . For example, the program equivalence π ; skip ≡ π holds. When we say that two expressions are equivalent we mean program equivalence if we are talking about programs, and formula equivalence otherwise. A formula ϕ is valid if it is equivalent to , i.e. if ||ϕ|| = 2P . It is satisfiable if it is not a formula equivalent to ⊥, i.e. if ||ϕ|| = ∅. For example, the formulas  p← and  p←ϕ ↔ ¬ p←¬ϕ are both valid.

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Table 1 Interpretation of the DL-PA connectives || p←|| = {(v1 , v2 ) : v2 = v1 ∪ { p}} || p←⊥|| = {(v1 , v2 ) : v2 = v1 \ { p}} ||π ; π  || = ||π || ◦ ||π  || ||π  π  || = ||π || ∪ ||π  || ||π − || = ||π ||−1 ||ϕ?|| = {(v, v) : v ∈ ||ϕ||} y || p|| = {v : p ∈ v}x x x x x x |||| = 2P ||⊥|| = ∅ ||¬ϕ|| = 2P \ ||ϕ|| ||ϕ ∨ ψ|| = ||ϕ|| ∪ ||ψ||   ||π ϕ|| = v : there is v such that (v, v ) ∈ ||π || and v ∈ ||ϕ||

2.3 Properties of DL-PA As in any reasonable logic, replacement of equivalents is valid in DL-PA: equivalence is preserved under replacement of one or more sub-expressions by an equivalent expression Balbiani et al. (2013).1 The fact that DL-PA formulas are interpreted through classical propositional logic valuations indicates that modal operators can be eliminated. Indeed, every DL-PA formula is equivalent to a Boolean formula. Theorem 1 (Balbiani et al. (2013); Herzig (2014)) For every DL-PA formula there is an equivalent boolean formula. The proof uses valid equivalences that together make up a complete set of reduction axioms. For example,  p←⊥q is equivalent to  if p and q are identical, and is equivalent to q otherwise. For a more complex example, consider two different propositional variables r and p and the formula  p←⊥( p ∨ r ). It is successively equivalent to  p←⊥ p ∨  p←⊥r , to ⊥ ∨ r , and to r . Note that the Boolean formula resulting from the reduction might be exponentially longer than the original formula. The satisfiability problem is to decide whether ||ϕ|| = ∅ for a given formula ϕ; the model checking problem is to decide whether v ∈ ||ϕ|| for a given formula ϕ and valuation v. Theorem 2 (Balbiani et al. (2014)) The satisfiability problem and the model checking problem for DL-PA are both PSPACE complete.

1 Note

that replacements cannot be applied to variables in assignments such as to p in p←: the result would not be well formed.

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Table 2 DL-PA programs modifying variables in P = { p1 , . . . , pn }  mkTrueOne(P) = ( p?; p←⊥) = ( p1 ?; p1 ←)  · · ·  ( pn ?; pn ←) p∈P  (¬ p?; p←) = (¬ p1 ?; p1 ←⊥)  · · ·  (¬ pn ?; pn ←⊥) mkFalseOne(P) = p∈P  ( p←¬ p) = p1 ←¬ p1  · · ·  pn ←¬ pn flipOne(P) = p∈P

; ( p←  skip) = ( p1 ←  skip) ; · · · ; ( pn ←  skip) mkFalseSome(P) = ; ( p←⊥  skip) = ( p1 ←⊥  skip) ; · · · ; ( pn ←⊥  skip) p∈P flipSome(P) = ; ( p←  p←⊥) = ( p1 ←  p1 ←⊥) ; · · · ; ( pn ←  pn ←⊥) mkTrueSome(P) =

p∈P

p∈P

2.4 Some Useful DL-PA Programs We now define some DL-PA programs that will be useful in the rest of the paper. The first programs are familiar from PDL: if ϕ then π1 else π2 abbreviates (ϕ?; π1 )  (¬ϕ?; π2 ) and while ϕ do π abbreviates (ϕ?; π )∗ ; ¬ϕ?. Moreover, we define π n and π ≤n inductively by  πn =  π

≤n

=

skip if n = 0 π ; π n−1 if n ≥ 1, skip if n = 0 ≤n−1 ( skip  π ); π if n ≥ 1,

where skip stands for ? (‘nothing happens’). Next, we define assignments of literals to variables: p←q = (q?; p←)  (¬q?; p←⊥), p←¬q = (q?; p←⊥)  (¬q?; p←). The former assigns to p the truth value of q, while the latter assigns to p the truth value of ¬q. So p← p does nothing and p←¬ p flips the truth value of p. Note that both abbreviations have constant length, viz. 14. It will be useful to associate sequences of atomic assignments to sets of propositional variables P = { p1 , . . . , pn } by identifying ; p∈P p← with p1 ←; · · · ; pn ← and likewise ; p∈P p←⊥ with p1 ←⊥; · · · ; pn ←⊥. Note that the interpretation does not depend on the order of the variables in P. Note also that this no longer holds if we generalise from atomic assignment programs to complex programs. Finally, Table 2 collects programs that are going to be instrumental in the rest of the paper. They modify the truth values of one or more propositional variables in a given finite subset P = { p1 , . . . , pn } of P: We understand that all these programs

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equal skip when n is zero. The program mkTrueOne(P) makes exactly one variable true that was not true yet; symmetrically for mkFalseOne(P). flipOne(P) nondeterministically chooses a variable from P and nondeterministically sets it to either true or false. It is equivalent to mkTrueOne(P)  mkFalseOne(P). The program mkTrueSome(P) makes zero or more variables of P true; symmetrically for mkFalseSome(P). flipSome(P) nondeterministically sets the value of each variable pi to either to true or false. Observe that the length of each program is linear in the cardinality of P. Observe also that the order of the variables in P does not matter. This can be seen from the following lemma characterising the behaviour of the programs. It uses the symmetric difference between two valuations, which is the number of variables whose ˙ 2 is the set of all those p such that v1 ( p) = v2 ( p). This truth values differs: v1 −v ˙ 2 = (v1 \ v2 ) ∪ (v2 \ v1 ). For example, the symmetric is formally defined as v1 −v difference between the valuations { p, q} and {q, r, s} is { p, r, s}. Lemma 1 Let P ⊆ P be some set of propositional variables. Then, the following hold: ||mkTrueOne(P)|| = {(v, v ) : v = v ∪ { pk } for some pk ∈ P} ||mkFalseOne(P)|| = {(v, v ) : v = v \ { pk } for some pk ∈ P} ˙  = { pk } for some pk ∈ P} ||flipOne(P)|| = {(v, v ) : v−v ||mkTrueSome(P)|| = {(v, v ) : v = v ∪ P for some P ⊆ P} ||mkFalseSome(P)|| = {(v, v ) : v = v \ P for some P ⊆ P} ˙  ⊆ P} ||flipSome(P)|| = {(v, v ) : v−v Proof We only prove the cases of flipSome(P) and flipOne(P). They are stated without proof in Herzig (2014, Proposition 5). For flipSome(P), if the set P = { p1 , . . . , pn } is empty then flipSome(P) = skip and we are done. If it is a singleton, then we have ||flipSome({ p1 })|| = || p1 ←  p1 ←⊥|| = || p1 ←|| ∪ || p1 ←⊥||     = (v, v ) : v = v ∪ { p} ∪ (v, v ) : v = v \ { p}   ˙  ⊆ { p} . = (v, v ) : v−v Then, the result for the general case of an arbitrary set of variables P should be clear (even if the proof is a bit lengthy to spell out). For flipOne(P), if P is empty, then flipOne(P) = skip. For a single flip, we have

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|| pk ←¬ pk || = ||( pk ?; pk ←⊥)  (¬ pk ?; pk ←)|| = || pk ?; pk ←⊥|| ∪ ||¬ pk ?; pk ←||     = (v, v ) : pk ∈ v and pk ∈ / v ∪ (v, v ) : pk ∈ / v and pk ∈ v   ˙  = { pk } . = (v, v ) : v−v ˙  = { pk } for some pk ∈ P}. Therefore, ||flipOne(P)|| equals {(v, v ) : v−v



Consider the set of propositional variables P = { p, q, r }. The set of valuations for that language is 2P . Consider the formula ϕ = p ∧ ¬q. Then, flipSome(P(ϕ)) = flipSome({ p, q}) = ( p←  p←⊥) ; (q←  q←⊥). Its interpretation  varies the truth values of p and q in all possible ways: ||flipSome(P(ϕ))|| = (v, v ) :   ˙ ⊆ { p, q} . v−v The interpretation of the test ϕ? is {({ p}, { p}), ({ p, r }, { p, r })}. The sequential composition of flipSome(P(ϕ)) and ϕ? therefore is   ˙  ⊆ { p, q} . ||flipSome(P(ϕ)); ϕ?|| = (v, v ) : v ∈ ||ϕ|| and v−v So the pair ({ p}, {q}) is a possible execution of the program flipSome(P(ϕ)); ϕ?. More generally, programs of the form flipSome(P(ϕ)); ϕ? accesses all relevant ϕ-models, where ‘relevant’ means that the truth values of variables not occurring in ϕ are keep constant. It follows that the satisfiability of a Boolean formula ϕ can be expressed in DL-PA. The result below is stated without proof in Herzig (2014). Lemma 2 Let ϕ be a DL-PA formula. ϕ is satisfiable iff flipSome(P(ϕ)); ϕ? is valid. Proof Suppose flipSome(P(ϕ)); ϕ? is DL-PA valid. Let v be some valuation. By the interpretation of the dynamic operator there exists a valuation v such that (v, v ) ∈ ||flipSome(P(ϕ)); ϕ?||. This means that there exists a valuation v such that (v, v ) ∈ ||flipSome(P(ϕ))|| and (v , v ) ∈ ||ϕ?||. The latter means that v ∈ ||ϕ||: we have found a valuation where ϕ is true. The other way round, suppose ϕ is propositionally satisfiable, i.e. there is some model vϕ of ϕ. Let v be an arbitrary valuation. Let vϕ be the valuation which interprets the variables of ϕ in the same way as v A and interprets the other variables in the same way as v vϕ = (vϕ ∩ P(ϕ)) ∪ (v ∩ (P \ P(ϕ))). This is clearly also an ϕ-model. (Indeed, for every p that does not occur in ϕ, ˙ ϕ ⊆ P(ϕ) we have (v, vϕ ) ∈ we have v ∪ { p} ∈ ||ϕ|| iff v \ { p} ∈ ||ϕ||.) As v−v ||flipSome(P(ϕ))|| by Item 1 of Lemma 1. And as vϕ is an ϕ-model, we have (vϕ , vϕ ) ∈ ||ϕ?||. So (v, vϕ ) ∈ ||flipSome(P(ϕ)); ϕ?||, from which it follows that v ∈ ||flipSome(P(ϕ)); ϕ?||. As v was arbitrary, flipSome(P(ϕ)); ϕ? is DL-PA valid. 

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3 Describing AFs and Their Semantics by Formulas In the present section, we recall the main definitions of abstract AFs. We do so in terms of the language of DL-PA. We show in particular that minimisation-based and maximisation-based semantics can be handled in an elegant way by taking advantage of the programs of DL-PA.

3.1 AFs and Their Representation in Propositional Logic We suppose given a finite set of arguments A = {a1 , . . . , an }. We associate to each couple of arguments (a, b) ∈ A2 an attack variable ra,b . Furthermore, we associate to every argument a ∈ A an acceptance variable ina expressing that a is accepted. So the respective sets of such variables are ATTA = {ra,b : (a, b) ∈ A × A}, INA = {ina : a ∈ A}. In the rest of the paper, we suppose that the set of propositional variables is P = ATTA ∪ INA . So our language is Lang(ATTA ∪ INA ); the set Lang(ATTA ) are the formulas that can be built from attack variables only, and Lang(INA ) are those that can be built from acceptance variables only. An abstract AF as defined by Dung in Dung (1995) is a pair G = (A, R), where R ⊆ A × A is the attack relation. So AFs are nothing but finite directed graphs with arguments as vertices and attacks as edges. The theory of G = (A, R) is the Boolean formula ⎛ θ (G) = ⎝

⎞



⎛

ra,b ⎠ ∧ ⎝

(a,b)∈R

⎞



¬ra,b ⎠ .

(a,b)∈(A×A)\R

We clearly have that ra,b ∈ ||θ (G)|| if and only if (a, b) ∈ R. Just as θ (G) represents the attack relation in propositional logic, an extension E ⊆ A can be represented by the formula  θ (E, A) =



a∈E



⎛

ina ∧ ⎝



a∈A\E

⎞ ¬ina ⎠ .

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Table 3 Argumentation semantics θ(σ, A) in propositional logic, for σ being the stable, admissible, complete, preferred or grounded semantics  

θ(stable, A) = (rb,a → ¬inb ) ina ↔ a∈A  b∈A  

  θ(admissible, A) = (inc ∧ rc,b ) rb,a → ¬inb ∧ ina → a∈A  b∈A  c∈A  



  θ(complete, A) = ina → rb,a → (rb,a → ¬inb ) ∧ ina ↔ (inc ∧ rc,b ) a∈A b∈A b∈A  c∈A θ(preferred, A) = θ(admissible, A) ∧ mkTrueOne(A); mkTrueSome(A) ¬θ(admissible, A)   θ(grounded, A) = θ(complete, A) ∧ mkFalseOne(A); mkFalseSome(A) ¬θ(complete, A)

3.2 Semantics of an AF Many semantics were defined for acceptability. Some of them are extension based, some others are labelling based. We here only consider the first: they prevail in the literature, and moreover labelling-based semantics have equivalent extension-based formulations Baroni and Giacomin (2009). As proposed in Besnard and Doutre (2004) and extensively discussed in Gabbay (2011), many definitions of extensions can be captured in propositional logic. In this paper, we consider the following. A stable extension of (A, R) is a set of arguments E ⊆ A such that it does not exist two arguments a and b in E such that (a, b) ∈ R (that is, E is conflict free), and for each argument b ∈ / E, there exists a ∈ E such that (a, b) ∈ R (any argument outside the extension is attacked by the extension). An admissible set of (A, R) is a conflict-free set E ⊆ A that defends all its elements: for all a ∈ E, if there exists b such that (b, a) ∈ R, then there is some argument c ∈ E such that (c, b) ∈ R.2 A complete extension of (A, R) is an admissible set E ⊆ A that contains all the arguments it defends, that is, if an argument a is such that, for all b such that (b, a) ∈ R, there exists some c ∈ E such that (c, b) ∈ R, then a ∈ E. A preferred extension of (A, R) is an admissible set E ⊆ A that is maximal w.r.t. inclusion. A grounded extension of (A, R) is a complete extension E ⊆ A that is minimal w.r.t. inclusion. Their definitions in propositional logic are collected in Table 3. Given a definition of a semantics σ (stable, admissible or complete) and a set of arguments A, θ (σ, A) denotes the associated logical characterisation of σ . Computing the extensions of an AF G under semantics σ amounts to finding the models of the conjunction G ∧ θ (σ, A). 2 Admissibility

is usually considered to be a building brick of other, stronger semantics, but we consider it here as a semantics on its own, the construction of admissible sets following the same process as the construction of extensions.

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The following proposition connects extensions and valuations for all the above semantics. Proposition 1 Let G = (A, R) be an AF and let E ⊆ A. Let σ be either the stable, admissible,complete, preferred  or grounded semantics. E is a σ -extension of G if and only if θ (G) ∧ θ (E, A) → θ (σ, A) is DL-PA valid. Proof The result for stable, admissible and complete semantics follow from Propositions 5, 6 and 8 in Besnard and Doutre (2004). The results for the preferred and the grounded semantics follow from the following program equivalences:   ||mkTrueOne(P); mkTrueSome(P)|| = (v, v ∪ P  ) : v ⊆ 2P and ∅ ⊂ P  ⊆ P   ||mkFalseOne(P); mkFalseSome(P)|| = (v ∪ P  , v) : v ⊆ 2P and ∅ ⊂ P  ⊆ P ,



where P is some set of propositional variables.

Corollary 1 Let G = (A, R) be an AF. Let σ be either the stable, admissible, complete, preferred or grounded semantics. 1. E ⊆ A is a σ -extension of G if and only if there is a v ∈ ||θ (G) ∧ θ (σ, A)|| such that E = {a : ina ∈ v}. 2. Let γ ∈ Lang(ATTA ∪ INA ) be a formula describing some property of A. All σ -extensions of G have property γ if and only if θ (G) ∧ θ (σ, A) → γ is DL-PA valid. Proof The first item summarises the preceding proposition. (Observe that when v ∈ ||G||, then E = {a : ina ∈ v} implies that v = v E .) As to the second item, by Proposition 1, γ is true in all extensions of G if and only if γ is true in every model of θ (G) ∧ θ (σ, A), i.e. iff θ (G) ∧ θ (σ, A) → γ is DL-PA valid.  An AF may have none, one or several extensions, depending on the number of models of θ (G) ∧ θ (E, A). Example Consider the set of arguments A1 = {a, b} and the two AFs G1 = (A1 , R1 ) and G2 = (A1 , R2 ), with R1 = {(a, b)} and R2 = {(a, b), (b, a)}. They are depicted in Fig. 1.

a

Fig. 1 G1 (left) and G2 (right)

b

a

b

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The associated propositional theories are as follows: θ (G1 ) = ra,b ∧ ¬rb,a ∧ ¬ra,a ∧ ¬rb,b θ (G2 ) = ra,b ∧ rb,a ∧ ¬ra,a ∧ ¬rb,b . The AF G1 has a single stable extension {a}, two admissible extensions ∅ and {a}, and one complete extension {a}, that is, θ (G1 ) ∧ θ (stable, A1 ) → ina ∧ ¬inb θ (G1 ) ∧ θ (admissible, A1 ) → ¬inb θ (G1 ) ∧ θ (complete, A1 ) → ina ∧ ¬inb .

The justification state of an argument Baroni and Giacomin (2009) depends on the extensions it belongs to: it is credulously justified in AF G under semantics σ if it belongs to at least one of the σ -extensions of G. It is skeptically justified in G under σ if it belongs to every σ -extension of G.

4 Building Extensions by Programs In the last section, we have seen how to define by formulas what an extension is. In practice, such extensions are constructed by algorithms. The DL-PA programs provide a means to recast these algorithms in a logical framework. Remember that such programs modify valuations: our aim is to define programs which when executed at a valuation describing an AF modify the truth values of the acceptance variables, leading to valuations where the formula characterising the semantics holds. Remember that ATTA (ϕ) and INA (ϕ), respectively, are the set of attack variables and the set of acceptance variables occurring in the formula ϕ. For example, ATTA (¬ra,b ∨ ina ) = {ra,b } and INA (¬ra,b ∨ ina ) = {ina }.

4.1 A Generic DL-PA Program The generic DL-PA program mkExtσA below is indexed by a set of arguments A and a semantics σ and builds for every AF G over A the valuations representing the extensions of G w.r.t. σ : mkExt(σ, A) = flipSome(INA ); θ (σ, A)? where n is the cardinality of A, σ is some semantics, and θ (σ, A) is the DL-PA formula characterising σ .

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Proposition 2 Let A be a set of arguments. Let σ be either the stable, complete, admissible, preferred or grounded semantics. Then   ||mkExt(σ, A)|| = (v1 , v2 ) : v2 ∈ θ (σ, A) and v1 ∩ ATTA = v2 ∩ ATTA . Proof When (v1 , v2 ) ∈ ||mkExt(σ, A)|| then there is a valuation that is accessible from v1 via the relation ||flipSome(INA )|| and from which v2 can be accessed via ||θ (σ, A)?||. As the test program θ (σ, A)? does not modify any truth value that valuation must be v2 itself, which moreover must be a model of θ (σ, A). By Item 1 of Lemma 1, v2 differs from v1 only by the truth values of the INA variables. It therefore has the same truth values for the ATTA variables, i.e. v1 ∩ ATTA = v2 ∩ ATTA .  It follows that the DL-PA program mkExt(σ, A) constructs all the extensions of a given AF w.r.t. σ . Indeed, when the set of attack variables of v1 describes G, then the set of v2 such that (v1 , v2 ) ∈ ||mkExt(σ, A)|| characterises all the extensions of G w.r.t. σ . Corollary 2 Let G = (A, R) be an AF and let ϕ be a formula. The formula θ (G) ∧ θ (σ, A) → ϕ is valid if and only if θ (G) → [mkExt(σ, A)]ϕ is valid. Moreover, the equivalence θ (G) ∧ θ (σ, A) ↔ (mkExt(σ, A))− θ (G) is valid. According to Corollary 2, given a description of the attack relation, mkExt(σ, A) constructs all the valuations where the formula describing the semantics is true, and so in a way such that only acceptance variables are changed, while the attack variables do not change. Note that the equivalence takes care of cases where G has no extension. Suppose the formula ϕ is a goal, i.e. a formula that should be satisfied by all or some of the extensions of some AF. By means of the program mkExt(σ, A), one can check what has been called σ -consistency in Coste-Marquis et al. (2013): whether there exists an AF G such that some or every extension w.r.t. σ satisfies ϕ. Adopting the standard terms for characterising a justification state, we distinguish the notions of credulous and skeptical consistency. Definition 1 Let σ be the stable, admissible, complete, preferred or grounded semantics. Let ϕ be an Lang(ATTA ∪ INA ) formula. Then • ϕ is σ -credulously consistent for A iff mkExt(σ, A)ϕ is satisfiable; • ϕ is σ -skeptically consistent for A iff [mkExt(σ, A)]ϕ is satisfiable.

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Table 4 A more efficient DL-PA program  mkExt1 (σ, A) =

  

; ina ←⊥ ; skip  b∈{a } inb → (¬ra1 ,b ∧¬rb,a1 ) ?; ina1 ← ; · · · ; 1     ina ∈INA  ;in ∈INA ina ←⊥ skip  b∈{a ,...,an } inb → (¬ran ,b ∧¬rb,an ) ?; inan ← ; θ(σ, A)? a

1

Example Consider the set of arguments A2 = {a, b} and the goal ϕ = ¬ina ∧ ¬inb ∧ ¬ra,a . Then, mkExt(σ, A)ϕ is unsatisfiable, in particular, it is false in every valuation for θ (G2 ) of Fig. 1. Therefore, the formula ϕ is stable-credulously inconsistent for A2 . The length of mkExt(σ, A) is linear in the length of the logical description of the semantics θ (σ, A) and in the cardinality of the set of arguments A. The program is however heavily nondeterministic. There are other, more efficient algorithms that can also be implemented in DL-PA. We turn to that in the rest of the section.

4.2 More Efficient Algorithms The above programs mkExt(σ, A) implement a generate-and-test schema where all logically possible valuations are generated. An improvement is the program in Table 4 before accepting an argument it checks whether this would lead to a conflict with previously accepted arguments. (Remember that the set of arguments A is {a1 , . . . , an }.) Theorem 3 The program equivalence mkExt1 (σ, A) ≡ mkExt(σ, A) holds. Proof The proof takes the form of a sequence of DL-PA programs that are all DL-PA equivalent: 1. mkExt 1 (σ, A)      

2. ;ina ∈INA ina ←⊥ ; skip  b∈{a1 } inb → (¬ra1 ,b ∧¬rb,a1 ) ?; ina1 ← ; · · · ;       ;ina ∈A ina ←⊥ ; skip  b∈{a1 ,...,an } inb → (¬ran ,b ∧¬rb,an ) ?; inan ← ; θ(σ, A)?       

3. ;ina ∈A ina ←⊥ ; skip  ina1 ←; b∈{a1 } (ina1 ∧inb ) → (¬ra1 ,b ∧¬rb,a1 ) ? ; · · · ; 

 4.



 5.

6.

     

; skip  inan ←; b∈{a ,...,an } (inan ∧inb ) → (¬ran ,b ∧¬rb,an ) ? ; θ(σ, A)? 1

;ina ∈A ina ←⊥

;ina ∈A ina ←⊥

     ; skip  ina1 ← ; b∈{a1 } (ina1 ∧inb ) → (¬ra1 ,b ∧¬rb,a1 ) ?; · · · ;

     ; skip  inan ← ; b∈{a ,...,an } (inan ∧inb ) → (¬ran ,b ∧¬rb,an ) ?; θ(σ, A)?   1  

;ina ∈A ina ←⊥

;ina ∈A ina ←⊥ ; skip  ina1 ← ; · · · ; skip  inan ← ;    ;ina ∈INA ina ←⊥ ; b∈{a1 } (ina1 ∧inb ) → (¬ra1 ,b ∧¬rb,a1 ) ?; · · · ;     ;ina ∈INA ina ←⊥ ; b∈{a1 ,...,an } (inan ∧inb ) → (¬ran ,b ∧¬rb,an ) ?; θ(σ, A)?       ;ina ∈A ina ←⊥ ; skip  ina1 ← ; · · · ; skip  inan ← ; θ(σ, A)? 

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7. flipSome(INA ); θ(σ, A)?

The above theorem states a program equivalence of the logic that is parametrised by the cardinality n of the set of arguments A. While the above proof was by hand, it could as well be done (for a given n) by means of some automated theorem prover for DL-PA. While theoretical results establishing the complexity of model checking and satisfiability exist, there is for the time being no implemented theorem prover for DL-PA. Actually many algorithms were proposed in the literature to build extensions, such as those of Charwat et al. (2015) or of Nofal et al. (2014) (the latter for the preferred semantics). We claim that all of them can be implemented in DL-PA. While we do not work this out here and leave it to future work, we nevertheless sketch how labellingbased algorithms can be captured in DL-PA. We use new propositional variables deca , standing for ‘the status of a is settled (decided)’. The following abbreviations will be useful.   decb ∧ inb ∧ rb,a AttackedByAcc(a) = b∈A

DefendedByAcc(a) =

   rb,a → decc ∧ inc ∧ rc,b

b∈A

c∈A

Then, we define the following program:

;

mkExt2 (σ, A) =

deca ←⊥ ;

deca



if  if



 ¬rb,a1 then ina1 ←; deca1 ← else skip ; · · · ;

b∈A

 ¬rb,an then inan ←; decan ← else skip ;



b∈A



while

a

while 

¬deca do

   AttackedByAcc(a) ∨ DefendedByAcc(a) do a

 if AttackedByAcc(a1 ) then ina1 ←⊥; deca1 ← else skip ;   if DefendedByAcc(a1 ) then ina1 ←; deca1 ← else skip ; · · · ;   if AttackedByAcc(an ) then inan ←⊥; decan ← else skip ;   if DefendedByAcc(an ) then inan ←; decan ← else skip ;    ¬deca ?; (ina ←  ina ←⊥); deca ← ; if deca then skip else a

a∈A

θ(σ, A)?

We do not prove the correctness of the program here.

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To sum it up, we have seen up to now how extensions of a given AF can be constructed via DL-PA programs. In the rest of the paper, we will use DL-PA programs to modify an AF and its extensions in order to fulfil some goal.

5 What is Argumentation Framework Modification? We now introduce and discuss the problem of AF modification. We first distinguish three different kinds of modification and then discuss whether they correspond to a revision or to an update operation.

5.1 What Changes? Cayrol et al. Cayrol et al. (2010) proposed to distinguish several kinds of modifications of a given AF G = (A, R), according to the goal that is pursued: 1. Modifications of the set of arguments A (by adding or removing an argument). 2. Modifications of the attack relation R (by adding or removing an edge between two arguments). 3. Modifications of the extensions of G in order to satisfy some property. Modifications of the semantics that is applied to the AF, to produce the set of extensions, can also be considered, as outlined in Doutre and Mailly (2018). Such modifications are left for future work.

5.1.1

Changes on the Set of Arguments

The first kind of modification presented here, viz. adding an argument to the set of arguments A or removing it, requires additional linguistic resources that have to be added to our language for it to be implemented. First of all, we have to add a further ingredient to AFs: let us consider triples G = (A, A En , R), where A is the finite background set of all possible arguments, while A En ⊆ A is the set of all arguments that are currently under consideration (arguments that are enabled). We add to the logical language a set of enablement variables ENA = {Ena : a ∈ A}, where Ena means that a is enabled (or ‘considered’). Lang(ATTA ∪ ENA ) being the set of all formulas that are built from variables ATTA ∪ ENA , the theory of G = (A, A En , R) is the Boolean formula

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θ (G) =

 

          ra,b ∧ ¬ra,b ∧ Ena ∧ ¬Ena

(a,b)∈R

(a,b)∈R /

a∈A En

a ∈A / En

Note that in the theory, ra,b is true even if a or b are not enabled. The semantic definitions have to be adapted, too, and should only quantify over arguments in A En and not over those in A. Also, attacks must be considered only if they link enabled arguments. To this end, we define the following formula: En = ra,b ∧ Ena ∧ Enb . ATTa,b

Now, we can easily transform the semantic formulas: we check if the argument is enabled, otherwise it will not be included in the extension, and we replace attacks En to ensure they are indeed present. We illustrate this variables ra,b by formulas ATTa,b transformation to capture the stable semantics. Let Lang(ATTA ∪ ENA ∪ INA ) be the language of formulas built from P = ATTA ∪ ENA ∪ INA . The following formula captures the stable semantics3 : θ(stable, A) =

       En ) ∧ ¬En → ¬in Ena → ina ↔ ¬ . (inb ∧ ATTb,a a a a∈A

b∈A

In this setting, the mere addition or deletion of an argument a can be achieved straightforwardly, viz. by changing the status of a from ‘disabled’ to ‘enabled’ by means of the assignments Ena ← and Ena ←⊥. When π is a DL-PA program describing one or a sequence of modifications of that kind then the modification of G by π is described in DL-PA by the formula π − θ (G) which says that the program π was possibly executed and θ (G) was true before. Since attacks are already in the theory even if arguments are not considered, we do not need to include them in our update: all the attacks from (resp. to) a to (resp. from) other enabled arguments are considered. Attack can however be removed or added; this is the kind of modification that we present in the next section. A full description and an illustration of the setting presented here for the modification of the set of arguments can be found in Doutre et al. (2017).

3 An

equivalent way to express these formulas would be to use the set of enabled arguments. For example, to describe the stable extensions   we would

write: θ(stable, A, AEn ) = a∈A En ina ↔ ¬ b∈A En (inb ∧ rb,a ) ∧ a ∈A / En ¬ina . This highlights the fact that when all arguments are enabled, i.e., when A En = A, we indeed retrieve the formulas presented earlier in the paper.

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Changes on the Set of Attacks

The second kind of modification, addition and removal of an attack edge between two arguments a, b ∈ A is rather simple: we just add or subtract (a, b) from R. In our logical representation, this operation corresponds to making propositional variables true or false, which can be immediately captured by DL-PA programs assigning the attack variables to true and false, ra,b ← and ra,b ←⊥. Like for the addition and the removal of arguments, when π is a DL-PA program describing one or a sequence of modifications of attacks, then the modification of G by π is described in DL-PA by the formula π − θ (G) which says that the program π was possibly executed and θ (G) was true before. To illustrate this let us take up our above AF G2 = (A2 , R2 ) with A2 = {a, b} and R2 = {(a, b), (b, a)}. The removal of the attack rb,a results in (A2 , R2 ) with R2 = {(a, b)}. In DL-PA, the modification of G2 by rb,a ←⊥ is described by the formula rb,a ←⊥− θ (G2 ).

5.1.3

Changes on the Extensions

The third kind of modification, adding arguments to or removing them from extensions, is more involved because it has to be achieved indirectly, by changing the underlying attack relation of G (or by changing the set of arguments, but as we said, we disregard this option for the time being). There are moreover two different options here: when an AF has several extensions one may wish to change the justification status of a, so that it be skeptically justified (that is, added in all the extensions), or credulously justified (added in at least one extension), or credulously justified but not skeptically justified (added in at least one extension, but removed from at least one), or even not credulously justified (removed from all extensions). In the rest of the section, we explore the issue of extension modification a bit further. Consider again the above AF G2 = (A2 , R2 ) and the stable semantics. G2 has two stable extensions E a = {a} and E b = {b}. We have seen that the associated Boolean formula θ (G2 ) ∧ θ (stable, A2 ) has two Lang(ATTA ∪ INA ) models: va = {ra,b , rb,a , ina } vb = {ra,b , rb,a , inb }. Suppose we wish to modify G2 such that a is in none of its stable extensions. What we would like to do is to adapt the attack relation of G2 in a way that is minimal and that guarantees that a does not occur in any of its extensions. We view minimality as minimality of the number of modifications of the attack relation; in contrast, the current extensions may be modified in a non-minimal way.

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The papers Coste-Marquis et al. (2013, 2014) take a different perspective: the minimization of the changes to the attack relation is considered to be secondary, whereas the changes on the current extensions should be minimal. More generally, we are interested in enforcing some constraint α on an AF G = (A, R), where we understand that α is a formula in the language Lang(ATTA ∪ INA ), and that enforcing α consists in minimally modifying the attack relation R of G to R in a way such that α holds in all of the extensions of G = (A, R ). Another perspective is however possible, where enforcing consists in minimally modifying R such that α holds not in all but only in some extension of the resulting G = (A, R ). So ‘enforcement’ can be understood in two different ways, leading to two different definitions. In both cases, there is a key difference with standard revision and update operations: in classical belief change, output is limited to one belief set, while here, several G may be produced by the enforcement operations. To sum it up, we identify two kinds of modification operations Cred,σ and Skep,σ . Both map an AF and a Boolean formula of Lang(ATTA ∪ INA ) to a set of AFs. The set G Cred,σ α is the set of credulous enforcements of α and G Skep,σ α is the set of skeptical enforcements of α.4

5.2 Which Postulates? In line with the classical definitions of belief change operations, we now define some postulates that ‘reasonable’ operations  should satisfy. They are mainly inspired by the work of Coste-Marquis et al. Coste-Marquis et al. (2013). Definition 2 Let σ be any semantics. Let  be an operation mapping an AF over A and an Lang(ATTA ∪ INA ) formula to a set of AFs over A. The operation σ is a credulous enforcement operation iff it satisfies the following three postulates: E1.C θ (G ) ∧ θ (σ, A) ∧ α is satisfiable for every G ∈ G σ α.5 E2.C If θ (G) ∧ θ (σ, A) ∧ α is satisfiable then G σ α = {G}. E3 If |= α1 ↔ α2 then for every G1 ∈ G σ α1 there exists G2 ∈ G σ α2 such that G1 = G2 . The operation  is a skeptical enforcement operation iff it satisfies postulate E3 plus the following: E1.S E2.S

|= θ (G ) ∧ θ (σ, A) → α for every G ∈ G σ α. If |= θ (G) ∧ θ (σ, A) → α then G σ α = {G}.

4 Actually it is possible to refine these operations further. We might, e.g. define G Cred,σ,∃

α where α is credulously enforced in some G ∈ G Cred,σ,∃ α on the one hand, and G Cred,σ,∀ α where α is credulously enforced in every G ∈ G Cred,σ,∃ α on the other.   5 This is the same as: there exists a σ -extension E of G such that |= θ(G) ∧ θ(E) → α.

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The postulates E1.C and E1.S say that success in required for credulous and skeptical enforcement. E2 represents a minimal change principle: it states that if α already holds then G is unchanged. Postulate E3 is the postulate of syntax independence: enforcement should be based on the logical content of a goal and not on its syntax. Additional postulates may be formulated; see Bisquert et al. (2013a); CosteMarquis et al. (2013) for more details. A key difference is that we do not consider postulates based on the expansion operation. The main reason is that this operation is actually useless: first, if an attack has to be changed then expansion cannot be used because, as we have seen above, θ (G) is complete for Lang(ATTA ). Now, consider that an argument has to be enforced in a credulous way. We face two cases: (i) either the argument is already credulously acceptable and there is no reason to change the AF or (ii) the argument is not credulously acceptable. Then, as all possible extensions are considered, some attacks must be changed so that new extensions can be constructed (and α will then hold). The same reasoning can be made for skeptical acceptance.

5.3 Which Belief Change Operation? In the literature, two different families of belief change operations were studied: AGM revision operations Alchourrón et al. (1985) and KM update operations Katsuno and Mendelzon (1992). In a nutshell, the former models the incorporation of a new piece of information about a static world, while the latter models a dynamic world. Neither AGM theory Alchourrón et al. (1985) nor KM theory Katsuno and Mendelzon (1992) provide a single, concrete belief change operation: they rather constrain the set of ‘reasonable’ belief change operations by means of a set of postulates. Another way of saying this is: both AGM and KM rely on underlying orderings in order to construct . In the case of AFs, it is not immediately clear where such information comes from. (One may think of preference relations between arguments, it is however not clear whether this matches intuitions.) Several concrete belief change operations satisfying the AGM or KM postulates have been defined in the literature. Among the most prominent are Winslett’s ‘possible models approach’ (PMA) Winslett (1988, 1990), Forbus’s update operation Forbus (1989), Winslett’s standard semantics (WSS) Winslett (1995), and Dalal’s revision operation Dalal (1988); see Herzig and Rifi (1999); Lang (2007) for an overview. According to Katsuno and Mendelzon’s distinction between update and revision operations Katsuno and Mendelzon (1992), the first three are update operations, usually written , while Dalal’s is a revision operation, usually written ∗. Actually, the models of Forbus’s update and Dalal’s revision coincide in the case of complete belief bases, and the same is the case for Winslett’s update and Satoh’s revision. Complete bases contain either p or ¬ p, for every propositional variable p. This observation applies to the modification of AFs: the theory θ (G) is complete for Lang(ATTA ), and this is what matters: the variables of Lang(INA ) only encode

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particular extensions. Therefore, the question whether AF modification is an update or a revision does not really play a role here. We dedicate the next sections to two problems: modifying the framework and modifying an extension.

6 Enforcing a Constraint on All/Some Extensions The aim of this section is to provide a formal definition of two extension change operations. Both are based on Forbus’s update operation Forbus (1989), where minimal change involves counting how many variables change their truth value. Our operations satisfy the enforcement postulates that we have defined in Sect. 5.2. We start by observing that the update of AFs has some specificities: first, we are going to modify only the attack variables while leaving the accept variables unchanged; second, the target formula is not going to be a Boolean formula, but a formula saying that α will be the case after building extensions. When we construct the extensions we are going to minimise the modifications of ATTA (α), while those of the set INA (α) will not be minimised: given the truth values of the attack variables, the truth values of the accept variables are determined by the semantics (or rather, the possible combinations of accept variables, because there may be several extensions).

6.1 The Hamming Distance The Hamming distance between two valuations v and v w.r.t. a set of propositional variables P is the cardinality of the set of all those variables from P whose truth value differs ˙  )). H P (v, v ) = card(P ∩ (v−v ˙ 2= For example, consider v1 = {ra,b , rb,a , inb } and v2 = {ra,b , ina }. Then, v1 −v A A A {rb,a , ina , inb } and therefore H ATT (v1 , v2 ) = 1 and H ATT ∪IN (v1 , v2 ) = 3. When P equals P then H P (v, v ) is nothing but the Hamming distance between v and v . Forbus’s update operation is based on minimisation of that distance: first, the Forbus update of a valuation v by α is the set of those α-models whose Hamming distance to v is minimal; second, the Forbus update of a belief base β by α collects the Forbus updates of all models of β by α:   v forbus α = v : v |= α and there is no v such that v |= α and H P (v, v ) < H P (v, v )    α . β forbus α = v forbus P v|=β

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6.2 Argumentation Framework Update Just as the original Forbus update operation, our two update operations are defined in two steps: first, the update of a valuation in terms of the Hamming distance and then the update of an AF as the union of the updates of each of its models. But first we need a definition: given a valuation v, the AF associated with v is defined as G(v) = {(a, b) : ra,b ∈ v}. Then, the skeptical Forbus update by a Lang(ATTA ∪ INA ) formula α under semantics σ is defined as follows:  v σskep α = v : every σ -extension of G(v ) satisfies α and A

A

there is no v such that H ATT (v, v ) < H ATT (v, v )

 and every σ -extension of G(v ) satisfies α .



G σskep α =

v σskep α

v∈||θ(G)||

So v σskep α is the set of valuations whose σ -extensions all satisfy α and whose Hamming distance to v is minimal w.r.t. the variables in ATTA .6 This can be viewed as a circumscription policy with fixed and varying propositional variables Lifschitz (1994). Symmetrically, we define the credulous Forbus update as  v σcred α = v : some σ -extension of G(v ) satisfies α and A

A

there is no v such that H ATT (v, v ) < H ATT (v, v )

G σcred α =



 and some σ -extension of G(v ) satisfies α .

v σcred α

v∈||θ(G)||

Note that these operations resemble the Forbus update operation but cannot be reduced to them. The reason is that the input would have to be a counterfactual statement of the form ‘if the current valuation is transformed into a σ -extension by modifying the acceptance variables then α results’. Both operations coincide for updates by attack variables and their negation. Proposition 3 Let α be an attack literal, i.e. a propositional variable of the form ra,b or its negation. Then, G σskep α = G σcred α. The identity fails to hold for arbitrary formulas. expression ‘the set of valuations whose σ -extensions all satisfy α’ is a bit imprecise   here; more precisely, it is the set of valuations v such that |= θ((A, v ∩ (A × A))) ∧ θ(σ, A) → α.

6 The

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7 Expressing Extension Modification in DL-PA The aim of this section is to define modification programs in DL-PA implementing the two operations of modification of AFs that we have defined. The exposition follows Herzig (2014).

7.1 Modifications of the Attack Relation Only To warm up, observe that addition and removal of an attack edge between two arguments a, b ∈ A can be directly implemented by the atomic DL-PA programs: the addition of (a, b) to R is obtained by executing ra,b ←, and the removal of (a, b) from R is obtained by executing ¬ra,b ←⊥. Indeed, we have G  ra,b = ra,b ←− θ (G) G  ¬ra,b = ra,b ←⊥− θ (G), where  is any of the above belief change operations σskep and σcred . Once θ (G) has been updated by some ra,b or ¬ra,b , the extensions of the resulting framework can be obtained by conjoining the result with θ (σ, A). This can be generalised to input formulas α that are conjunctions of literals in the language Lang(ATTA ). More generally, when α is a formula in the language Lang(ATTA ), then it can be shown that both the skeptical and the credulous update of G by α are nothing but the classical Forbus update. Proposition 4 Let α be a formula in the language Lang(ATTA ). Then G σskep α = θ (G) forbus α.

7.2 The Hamming Distance Predicate in DL-PA Let us define the DL-PA formula H(α, P, ≥m), where α is a Lang(ATTA ∪ INA ) formula, P is a set of propositional variables and m ≥ 0 is an integer:  H(α, P, ≥m) =

  ≤m−1  ¬ flipOne(P) α

if m = 0 if m ≥ 1

We call H(α, P, ≥m) the Hamming distance predicate w.r.t. the set of variables P: it is true at a valuation v exactly when the α-models v that are closest to v in the sense of the Hamming distance differ in at least m variables from v, where the computation

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of the distance only considers variables from P while the other variables in P \ P cannot be modified. Proposition 5 Let v a valuation, α a Boolean formula, P some set of propositional variables and m ≥ 0. Then 1. v ∈ ||H(α, P, ≥m)|| iff the α-models that are closest to v w.r.t. P have Hamming ˙  ⊆ P. distance at least m, i.e. iff H(v, v ) ≥ mfor every v ∈ ||A|| such that v−v  m   2. (v, v ) ∈ ||H(α, P, ≥m)?; flipOne(P) ; α?|| iff v is one of the α-models that is closest to v w.r.t. P, i.e. iff v ∈ ||α|| and H(v, v ) = m for every v ∈ ||A|| ˙  ⊆ P. such that v−v Proof For Item 1, things are clear for m = 0, and we only consider the case m > 1. From the left to the right, suppose v is a model of H(α, P, ≥m), i.e. of the  ≤m−1 α. Then, there is no α-model v such that (v, v ) ∈ formula ¬ flipOne(P)  k ||flipOne(P)|| , for some k < m. So by Lemma 1, for every valuation v ∈ ||A|| ˙  ⊆ P, we must have H(v, v ) ≥ m. such that v−v ˙ ⊆ From the right to the left, suppose for every valuation v ∈ ||A|| such that v−v   P we have H(v, v ) ≥ m. Then, by Lemma 1, there cannot be a v such that (v, v ) ∈ ||flipOne(P)||k for some k < m and v ∈ ||α||. Therefore, v must be a model of the  ≤m−1 α. formula ¬ flipOne(P) Item 2 then follows from Item 1 and Lemma 1. It follows from the first item of Proposition 5 that when P equals P(α) then v ∈ ||H(α, P(α), ≥m)|| iff the α-models that are closest to v have Hamming distance at least m, i.e. iff H(v, v ) ≥ m for every v ∈ ||A||. This is used in Forbus’s udpate operation. The following DL-PA program performs Forbus’s update operation: forbus(α) =





 m  H(α, P(α), ≥m)?; flipOne(P(α)) ; α?

m≤card(P(α))

The program nondeterministically chooses an integer m, checks if the Hamming distance to α-models is at least m and flips m of the variables of α. Finally, the test α? only succeeds for α-models. Proposition 6 The formula γ is true after the Forbus update of β by α if and only if β → [forbus(α)]γ is DL-PA valid.

7.3 Argumentation Framework Update The update of AFs has some specificities: first, we only modify the attack variables while leaving the accept variables unchanged; second, the target formula is not a Boolean formula but a formula saying that α will be the case after building extensions.

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The programs credEnfσ (α) and skepEnfσ (α) minimally modify a valuation w.r.t. some semantics σ such that the Boolean formula α ∈ Lang(ATTA ∪ INA ) becomes true in some/all σ -extensions. credEnfσ (α) =



    m  H mkExt(σ, A)α, ATTA , ≥m ?; flipOne(ATTA ) ;

m≤card(ATTA )

mkExt(σ, A)α?      m  H [mkExt(σ, A)]α, ATTA , ≥m ?; flipOne(ATTA ) ; skepEnfσ (α) = m≤card(ATTA )

[mkExt(σ, A)]α?

Both programs (1) nondeterministically choose a value m for the Hamming distance to mkExt(σ, A)α, i.e. to valuations  of attack variables having extensions satisfying α, (2) check that m satisfies H mkExt(σ, A)α, ATTA , ≥m , (3) flip m attack variables, and then (4) check that either some extension satisfies α (credulous case), or all extensions satisfy α (skeptical case). The length of these two programs is polynomial in the cardinality of A.7 The next theorem provides an embedding of both skeptical and credulous AF update into DL-PA. It is the main result of our paper. Theorem 4 Let G be an AF. Let α ∈ Lang(ATTA ∪ INA ) be a Boolean formula. Then    G σcred α =  (credEnfσ (α))− θ (G)    G σ α =  (skepEnfσ (α))− θ (G) skep

For α, γ ∈ Lang(ATTA ∪ INA ), to check whether γ is true in all extensions of G modified by α can then be done by checking whether the DL-PA formula θ (G) → [skepEnfσ (α)]γ is valid. We in particular have the following result, which says that every possible execution of the two enforcement programs will succeed. Proposition 7 Let G be an AF. Let α ∈ Lang(ATTA ∪ INA ) be a Boolean formula. Then    |= credEnfσ (α) mkExt(σ, A) α   |= skepEnfσ (α) mkExt(σ, A) α The second key property is that an AF remains unchanged if the goal already holds. Proposition 8 For every goal α that is credulously justified, the credulous update program does not change anything. m

cardinality of the set ATTA is quadratic in that of A, and the length of (flipOne(ATTA ) quadratic in that of A.

7 The

is

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|= (θ (G) ∧ mkExt(σ, A)α ∧ γ ) → [credEnfσ (α)]γ . For every goal α that is skeptically justified, the skeptical update program does not change anything: |= (θ (G) ∧ [mkExt(σ, A)]α ∧ γ ) → [skepEnfσ (α)]γ . The modified AFs can be extracted from the formulas credEnfσ (α)− θ (G) and skepEnfσ (α)− θ (G) representing it in DL-PA (or rather, their reduction) by forgetting the accept variables, as proposed in Coste-Marquis et al. (2013). This operation can be implemented in our framework by the program flipSome(INA ). Definition 3 Let σ be either the stable, admissible or complete semantics. Let enf be either the skepEnfσ or the credEnfσ program. Let σ,enf be an operation mapping an AF and an Lang(ATTA ∪ INA ) formula to a set of AFs. The update of G by α under σ and enf is   G σ,enf α = (A, Rv ) : v ∈ ||(enf(α))− θ (G)|| , where Rv is the attack relation extracted from v, defined as Rv = {(a, b) : ra,b ∈ v}. The two preceding propositions guarantee that our enforcement operations satisfy the postulates. σ

Theorem 5 Operation σ,credEnf satisfies E1.C and E2.C. Operation σ,skepEnf σ σ satisfies E1.S and E2.S. Both operations σ,credEnf and σ,skepEnf satisfy E3.

σ

This result is actually not a surprise, given that our tool for enforcement is a variant of Forbus’s update operation. Example Let us take up the AF G2 of Fig. 1. Remember that G2 = (A1 , R2 ), with A1 = {a, b} and R2 = {(a, b), (b, a)} and that θ (G2 ) = ra,b ∧ rb,a . Let us consider the stable semantics and suppose we want skeptical enforcement of a, i.e. we want to enforce that a is always acceptable.  We disregard self-attacks for the sake of simplicity. The nondeterministic part m≤card(ATTA ) (. . .) of the program skepEnf(ina ) changes one variable from θ (G2 ), either ra,b or rb,a . This corresponds to two candidate extensions: one where a only attacks b and one where b only attacks a. Only the former case gives valuations where a is always acceptable. Hence, G stable,skepEnf

stable

  ina = (A1 , {(a, b)}) .

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8 Going Further We have illustrated how DL-PA offers a fruitful framework for representing AFs and reasoning about them. We now sketch several ways of extending our account.

8.1 Checking (Odd/Even) Cycles One may also wish to identify and modify global properties of argumentation frameworks, such as the existence of cycles, or the existence of odd or even cycles. In propositional logic, the existence of a cycle in a given AF can be characterised by means of the propositional formula 



ExistsCycleA =

(ra1 ,a2 ∧ ... ∧ ran−1 ,an ∧ ran ,a1 ).

n≤card(A) a1 ,...,an ∈A

The length of that formula is however exponential in the number of arguments. Fortunately, the existence of cycles can be characterised in DL-PA by a more succinct formula: it tests if the execution of a program iterating one transitive closure step can lead to a self-attack. Such a closure step is performed by the following program: 

step =

(ra,b ∧ rb,c ?; ra,c ←).

a,b,c∈A

Using that  we characterise the existence of a self-attack in an AF by the formula Loop = a∈A ra,a , the formula ExistsCycleA = step∗ Loop then characterises the existence of a cycle. The length of that formula is polynomial in the number of arguments. Actually, the Kleene star can be replaced by a bounded iteration up to card(A) − 1. This will be useful to characterise even and odd cycles. Observe that when there is an odd cycle in an AF then a self-attack can be produced by an even number of closure steps; symmetrically, an even cycle can be achieved by an odd number of closure steps. Based on that observation, we can check the existence of odd and even cycles by means of the following DL-PA formulas: existsEvenCycle =

!

 " card(A) #

0≤n≤

existsOddCycle = Loop ∨

2

!

 $ ¬step2n Loop?; step2n+1 Loop  " card(A) #

1≤n≤

2

$  ¬step2n−1 Loop?; step2n Loop.

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The formula characterising even cycles checks whether there is an integer n ≥ 0 such that 2n + 1 transitive closure steps may lead to a self-attack. The test ¬step2n Loop? makes sure that this is the smallest such number. The formula characterising odd cycles similarly checks whether there is an n ≥ 0 such that 2n transitive closure steps may lead to a self-attack. The length of these two formulas is still polynomial in the number of arguments.

8.2 A Prioritised Version We may adapt our modification operation in order to accommodate a prioritised version that was proposed in Coste-Marquis (2014). While up to now we only minimised modifications of ATTA , this paper proposes to replace this policy by a ‘first minimise INA , then ATTA ’ policy: first, we minimally change INA variables to make flipSome(INA )γ true, ending up in those minimally ATTA -distant states from which an extension satisfying the goal can be constructed, then we minimally change the ATTA variables in order to make the goal γ true. We capture this in DL-PA by two Forbus updates in sequence:    (A, ATTA ) forbus flipSome(INA )γ forbus INA ATTA γ . We observe that this may lead to multiple extensions and that it might be more appropriate to rather apply Dalal revision.

9 Conclusion The main result of this paper is the encoding of AFs and their dynamics in DL-PA, extending and generalising Doutre et al. (2014, 2017). More precisely, our contribution is threefold. First, as long as argument acceptability can be expressed in propositional logic, finding acceptable arguments and enforcing acceptability can be done in DL-PA. The DL-PA framework moreover enables formal verification of the correctness of an algorithm (which however for the time being has to await implemented DL-PA solvers). Other logical frameworks allow capturing and computing argument acceptability (see Charwat et al. (2015) for an overview), but few of them allow capturing and computing acceptability change as well. Second, as DL-PA formulas can be rewritten as propositional logic formulas, the result of the modification of an AF is described by a propositional formula from the models of which one may retrieve the modified AFs. Our proposal is hence more operational than those of other approaches because we use a formal logic encompassing the representation of change operations.

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Moreover, we consider not only credulous acceptability changes, as most of the current approaches do Baumann (2012), but skeptical acceptability changes as well. Third, our framework takes advantage of the complexity results for DL-PA: both model checking and satisfiability checking are in PSPACE. A closer look at the formulas expressing the modifications shows that the alternation of quantifications is bounded, which typically leads to complexity bounds at the second level of the polynomial hierarchy. The proposed DL-PA encodings may be related to QBF encodings for argumentation Arieli and Caminada (2013). As QBF has the same complexity as star-free DL-PA, all we do in DL-PA must be polynomially encodable into QBF. However, the availability of assignment programs makes a difference: update programs such as forbus(α) and extension construction programs such as mkExt(σ, A) capture things in a more general, flexible and natural way than a QBF encoding. To conclude, the richness of our framework makes it expandable to other kinds of changes, other update semantics and other argumentation semantics beyond those that are detailed in the present paper. We plan to investigate this research avenue in future work. Acknowledgements This work benefited from the support of the AMANDE project (ANR-13BS02-0004) of the French National Research Agency (ANR). We would like to thank Gabriele Kern-Isberner for triggering our work when DL-PA was presented at the Dagstuhl seminar ‘Belief Change and Argumentation in Multi-Agent Scenarios’ in June 2013. The paper benefited from several comments by Christoph Beierle after its presentation at KR 2014. After KR 2014, the ideas in the paper were presented in a workshop on argumentation in Lens in September 2015 and at the Cardiff Argumentation Forum in July 2016.

References Alchourrón, C., Gärdenfors, P., Makinson, D.: On the logic of theory change: partial meet contraction and revision functions. J. Symb. Log. 50, 510–530 (1985) Arieli, O., Caminada, M.W.: A QBF-based formalization of abstract argumentation semantics. J. Appl. Log. 11(2), 229–252 (2013) Balbiani, P., Herzig, A., Schwarzentruber, F., Troquard, N.: DL-PA and DCL-PC: model checking and satisfiability problem are indeed in PSPACE. CoRR abs/1411.7825 (2014). arXiv.org/abs/1411.7825 Balbiani, P., Herzig, A., Troquard, N.: Dynamic logic of propositional assignments: a well-behaved variant of PDL. In: Logic in Computer Science (LICS). IEEE (2013) Baroni, P., Giacomin, M.: On principle-based evaluation of extension-based argumentation semantics. Artif. Intell. 171(10–15), 675–700 (2007). https://doi.org/10.1016/j.artint.2007.04.004 Baroni, P., Giacomin, M.: Semantics of abstract argument systems. In: Simari, G., Rahwan, I. (eds.) Argumentation in Artificial Intelligence, pp. 25–44. Springer, New York (2009). https://doi.org/ 10.1007/978-0-387-98197-0_2 Baumann, R.: What does it take to enforce an argument? Minimal change in abstract argumentation. In: Raedt, L.D., Bessière, C., Dubois, D., Doherty, P., Frasconi, P., Heintz, F., Lucas, P.J.F. (eds.) Frontiers in Artificial Intelligence and Applications, ECAI 2012, vol. 242, pp. 127–132. IOS Press (2012)

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Baumann, R., Brewka, G.: Expanding argumentation frameworks: enforcing and monotonicity results. COMMA 216, 75–86 (2010) Baumann, R., Brewka, G.: Spectra in abstract argumentation: an analysis of minimal change. In: Cabalar, P., Son, T.C. (eds.) Proceedings of 12th International Conference on Logic Programming and Nonmonotonic Reasoning, LPNMR 2013, Corunna, Spain, 15–19 September 2013. Lecture Notes in Computer Science, vol. 8148, pp. 174–186. Springer, Heidelberg (2013). https://doi. org/10.1007/978-3-642-40564-8_18 Baumann, R., Brewka, G.: AGM meets abstract argumentation: Expansion and revision for dung frameworks. In: Yang and Wooldridge [15], pp. 2734–2740. http://ijcai.org/Abstract/15/387 Belardinelli, F., Grossi, D., Maudet, N.: Formal analysis of dialogues on infinite argumentation frameworks. In: Yang and Wooldridge [15], pp. 861–867. http://ijcai.org/Abstract/15/126 Besnard, P., Doutre, S.: Checking the acceptability of a set of arguments. In: 10th International Workshop on Non-Monotonic Reasoning (NMR 2004), pp. 59–64 (2004). http://www.pims. math.ca/science/2004/NMR/papers/paper18.pdf Bisquert, P.: Étude du changement en argumentation. De la théorie à la pratique. Ph.D. thesis, University of Toulouse, Toulouse, France (2013). http://www.irit.fr/publis/ADRIA/ ThesePierreBisquert.pdf Bisquert, P., Cayrol, C., de Saint-Cyr, F.D., Lagasquie-Schiex, M.: Change in argumentation systems: exploring the interest of removing an argument. In: Benferhat, S., Grant, J. (eds.) Proceedings of 5th International Conference on Scalable Uncertainty Management, SUM 2011, Dayton, OH, USA, 10–13 October 2011. Lecture Notes in Computer Science, vol. 6929, pp. 275–288. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-23963-2_22 Bisquert, P., Cayrol, C., Bannay, F., Lagasquie-Schiex, M.C.: Enforcement in Argumentation is a kind of Update. In: Liu, W., Subrahmanian, V., Wijsen, J. (eds.) International Conference on Scalable Uncertainty Management (SUM), Washington DC, USA, No. 8078 in LNAI, pp. 30–43. Springer, Heidelberg (2013a) Bisquert, P., Cayrol, C., de Saint-Cyr, F.D., Lagasquie-Schiex, M.: Goal-driven changes in argumentation: a theoretical framework and a tool. In: 2013 IEEE 25th International Conference on Tools with Artificial Intelligence, Herndon, VA, USA, 4–6 November 2013, pp. 610–617. IEEE Computer Society (2013b). https://doi.org/10.1109/ICTAI.2013.96 Booth, R., Kaci, S., Rienstra, T., van der Torre, L.: A logical theory about dynamics in abstract argumentation. In: Liu, W., Subrahmanian, V.S., Wijsen, J. (eds.) SUM. Lecture Notes in Computer Science, vol. 8078, pp. 148–161. Springer, Heidelberg (2013) Cayrol, C., Bannay, F., Lagasquie-Schiex, M.C.: Change in abstract argumentation frameworks: adding an argument. J. Artif. Intell. Res. 38, 49–84 (2010). http://www.jair.org/papers/paper2965. html Charwat, G., Dvorák, W., Gaggl, S.A., Wallner, J.P., Woltran, S.: Methods for solving reasoning problems in abstract argumentation—a survey. Artif. Intell. 220, 28–63 (2015). https://doi.org/ 10.1016/j.artint.2014.11.008 Coste-Marquis, S., Konieczny, S., Mailly, J., Marquis, P.: On the revision of argumentation systems: minimal change of arguments statuses. In: Baral, C., De Giacomo, G., Eiter, T. (eds.) International Conference on Principles of Knowledge Representation and Reasoning (KR). AAAI Press (2014) Coste-Marquis, S., Konieczny, S., Mailly, J., Marquis, P.: Extension enforcement in abstract argumentation as an optimization problem. In: Yang and Wooldridge [52], pp. 2876–2882. http://ijcai. org/Abstract/15/407 Coste-Marquis, S., Konieczny, S., Mailly, J.G., Marquis, P.: On the revision of argumentation systems: minimal change of arguments status. In: TAFA’13 (2013) Coste-Marquis, S., Konieczny, S., Mailly, J.G., Marquis, P.: A translation-based approach for revision of argumentation frameworks. In: JELIA, pp. 397–411. Springer, Cham (2014) Dalal, M.: Investigations into a theory of knowledge base revision: preliminary report. In: Proceedings of 7th Conference on Artificial Intelligence (AAAI’88), pp. 475–479 (1988) Delobelle, J., Haret, A., Konieczny, S., Mailly, J., Rossit, J., Woltran, S.: Merging of abstract argumentation frameworks. In: Baral, C., Delgrande, J.P., Wolter, F. (eds.) Proceedings of the

184

S. Doutre et al.

Fifteenth International Conference of Principles of Knowledge Representation and Reasoning, KR 2016, Cape Town, South Africa, 25–29 April 2016, pp. 33–42. AAAI Press (2016). http:// www.aaai.org/ocs/index.php/KR/KR16/paper/view/12872 Diller, M., Haret, A., Linsbichler, T., Rümmele, S., Woltran, S.: An extension-based approach to belief revision in abstract argumentation. In: Yang and Wooldridge [52], pp. 2926–2932. http:// ijcai.org/Abstract/15/414 Diller, M., Wallner, J.P., Woltran, S.: Reasoning in abstract dialectical frameworks using quantified Boolean formulas. Argum. Comput. 6(2), 149–177 (2015). https://doi.org/10.1080/19462166. 2015.1036922 Doutre, S., Herzig, A., Perrussel, L.: A dynamic logic framework for abstract argumentation. In: C. Baral, G. De Giacomo, T. Eiter (eds.) International Conference on Principles of Knowledge Representation and Reasoning (KR), pp. 62–71. AAAI Press (2014) Doutre, S., Maffre, F., McBurney, P.: A dynamic logic framework for abstract argumentation: adding and removing arguments. In: Benferhat, S., Tabia, K., Ali, M. (eds.) 30th International Conference on Industrial Engineering and Other Applications of Applied Intelligent Systems, Advances in Artificial Intelligence: From Theory to Practice, IEA/AIE 2017. Lecture Notes in Computer Science, vol. 10351, pp. 295–305. Springer, Cham (2017). https://doi.org/10.1007/ 978-3-319-60045-1_32 Doutre, S., Mailly, J.G.: Constraints and changes: a survey of abstract argumentation dynamics. Argum. Comput. 9, 223–248 (2018) Dung, P.M.: On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games. Artif. Intell. 77(2), 321–357 (1995) Fan, X., Toni, F.: On explanations for non-acceptable arguments. In: Black, E., Modgil, S., Oren, N. (eds.) Third International Workshop on Theory and Applications of Formal Argumentation, TAFA 2015, Buenos Aires, Argentina, 25–26 July 2015, Revised Selected Papers. Lecture Notes in Computer Science, vol. 9524, pp. 112–127. Springer, Cham (2015). https://doi.org/10.1007/ 978-3-319-28460-6_7 Forbus, K.D.: Introducing actions into qualitative simulation. In: Sridharan, N.S. (ed.) Proceedings of 11th International Joint Conference on Artificial Intelligence (IJCAI’89), pp. 1273–1278. Morgan Kaufmann Publishers (1989) Gabbay, D.M.: Dung’s argumentation is essentially equivalent to classical propositional logic with the peirce-quine dagger. Logica Universalis 5(2), 255–318 (2011) Gaggl, S.A., Manthey, N., Ronca, A., Wallner, J.P., Woltran, S.: Improved answer-set programming encodings for abstract argumentation. TPLP 15(4-5), 434–448 (2015). https://doi.org/10.1017/ S1471068415000149 Harel, D.: Dynamic logic. In: Gabbay, D.M., Günthner, F. (eds.) Handbook of Philosophical Logic, vol. II, pp. 497–604. D. Reidel, Dordrecht (1984) Harel, D., Kozen, D., Tiuryn, J.: Dynamic Logic. MIT Press (2000) Herzig, A.: Belief change operations: a short history of nearly everything, told in dynamic logic of propositional assignments. In: Baral, C., De Giacomo, G. (eds.) Proceedings of KR 2014. AAAI Press (2014) Herzig, A., Lorini, E., Moisan, F., Troquard, N.: A dynamic logic of normative systems. In: Walsh, T. (ed.) International Joint Conference on Artificial Intelligence (IJCAI), pp. 228–233. IJCAI/AAAI, Barcelona (2011). Erratum at http://www.irit.fr/~Andreas.Herzig/P/Ijcai11.html Herzig, A., Rifi, O.: Propositional belief base update and minimal change. Artif. Intell. J. 115(1), 107–138 (1999). https://www.irit.fr/~Andreas.Herzig/P/aij99.html Katsuno, H., Mendelzon, A.O.: On the difference between updating a knowledge base and revising it. In: Gärdenfors, P. (ed.) Belief revision, pp. 183–203. Cambridge University Press (1992). (Preliminary version in Allen, J.A., Fikes, R., Sandewall, E. (eds.) Principles of Knowledge Representation and Reasoning, pp. 387–394. Morgan Kaufmann Publishers (1991) Lang, J.: Belief update revisited. In: Proceedings of the 10th International Joint Conference on Artificial Intelligence (IJCAI’07), pp. 2517–2522 (2007)

Abstract Argumentation in Dynamic Logic: Representation, Reasoning and Change

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Lifschitz, V.: Circumscription. In: Gabbay, D.M., Gabbay, D.M., Hogger, C., Robinson, J.A. (eds.) Handbook of Logic in Artificial Intelligence and Logic Programming, vol. 3—Nonmonotonic Reasoning and Uncertain Reasoning, pp. 298–352. Oxford University Press (1994) Mailly, J.G.: Dynamics of argumentation frameworks. University of Artois, Lens, France, ThÃÍe de doctorat (2015) Niskanen, A., Wallner, J.P., Järvisalo, M.: Optimal status enforcement in abstract argumentation. In: Kambhampati, S. (ed.) Proceedings of the Twenty-Fifth International Joint Conference on Artificial Intelligence, IJCAI 2016, New York, NY, USA, 9–15 July 2016, pp. 1216–1222. IJCAI/AAAI Press (2016). http://www.ijcai.org/Abstract/16/176 Nofal, S., Atkinson, K., Dunne, P.E.: Algorithms for decision problems in argument systems under preferred semantics. Artif. Intell. 207, 23–51 (2014) Dupin de Saint-Cyr, F., Bisquert, P., Cayrol, C., Lagasquie-Schiex, M.: Argumentation update in YALLA (yet another logic language for argumentation). Int. J. Approx. Reason. 75, 57–92 (2016). https://doi.org/10.1016/j.ijar.2016.04.003 Wallner, J.P., Niskanen, A., Järvisalo, M.: Complexity results and algorithms for extension enforcement in abstract argumentation. In: Schuurmans, D., Wellman, M.P. (eds.) Proceedings of the Thirtieth AAAI Conference on Artificial Intelligence, 12–17 February 2016, Phoenix, Arizona, USA, pp. 1088–1094. AAAI Press (2016). http://www.aaai.org/ocs/index.php/AAAI/AAAI16/ paper/view/12228 Winslett, M.A.: Reasoning about action using a possible models approach. In: Proceedings of 7th Conferene on Artificial Intelligence (AAAI’88), pp. 89–93. St. Paul (1988) Winslett, M.A.: Updating Logical Databases. Cambridge Tracts in Theoretical Computer Science. Cambridge University Press (1990) Winslett, M.A.: Updating logical databases. In: Gabbay, D.M., Hogger, C.J., Robinson, J.A. (eds.) Handbook of Logic in Artificial Intelligence and Logic Programming, vol. 4, pp. 133–174. Oxford University Press (1995) Yang, Q., Wooldridge, M. (eds.) Proceedings of the Twenty-Fourth International Joint Conference on Artificial Intelligence, IJCAI 2015, Buenos Aires, Argentina, 25–31 July 2015. AAAI Press (2015). http://ijcai.org/proceedings/2015

Computational Hermeneutics: An Integrated Approach for the Logical Analysis of Natural-Language Arguments David Fuenmayor and Christoph Benzmüller

Abstract We utilize higher order automated deduction technologies for the logical analysis of natural-language arguments. Our approach, termed computational hermeneutics, is grounded on recent progress in the area of automated theorem proving for classical and nonclassical higher order logics, and it integrates techniques from argumentation theory. It has been inspired by ideas in the philosophy of language, especially semantic holism and Donald Davidson’s radical interpretation; a systematic approach to interpretation that does justice to the inherent circularity of understanding: the whole is understood compositionally on the basis of its parts, while each part is understood only in the context of the whole (hermeneutic circle). Computational hermeneutics is a holistic, iterative approach where we evaluate the adequacy of some candidate formalization of a sentence by computing the logical validity of (i) the whole argument it appears in and (ii) the dialectic role the argument plays in some piece of discourse. Keywords Computational hermeneutics · Rational argumentation · Universal logical reasoning · Higher order logic · Semantical embeddings · Proof assistants

1 Motivation While there have been major advances in the field of automated theorem proving (ATP) during the past years, its main field of application has mostly remained bounded to mathematics and hardware/software verification. We argue that the use of ATP Christoph Benzmüller: Funded by VolkswagenStiftung under grant CRAP: Consistent Rational Argumentation in Politics. D. Fuenmayor (B) · C. Benzmüller Freie Universität Berlin, Berlin, Germany e-mail: [email protected] C. Benzmüller University of Luxembourg, Esch-sur-Alzette, Luxembourg e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2019 B. Liao et al. (eds.), Dynamics, Uncertainty and Reasoning, Logic in Asia: Studia Logica Library, https://doi.org/10.1007/978-981-13-7791-4_9

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in argumentation (particularly in philosophy) can also be very fruitful,1 not only because of the obvious quantitative advantages of automated reasoning tools (e.g., reducing by several orders of magnitude the time needed to test argument’s validity), but also because it enables a novel approach to the logical analysis (aka formalization) of arguments, which we call computational hermeneutics. As a result of reflecting upon previous work on the application of ATP for the computer-supported evaluation of arguments in metaphysics (Benzmüller and Woltzenlogel Paleo 2014, 2016a; Fuenmayor and Benzmüller 2017a; Bentert et al. 2016; Benzmüller et al. 2017), we have become interested in developing a methodology for formalizing natural-language arguments with regard to their assessment using automated tools. Unsurprisingly, the problem of finding an/the adequate formalization of some piece of natural-language discourse turns out to be far from trivial. In particular, concerning expressive higher order logical representations, this problem has already been tackled in the past without much practical success.2 In spite of the efforts made in this area, carrying out logical analysis of natural-language (particularly of arguments) continues to be considered as a kind of artistic skill that cannot be standardized or taught methodically, aside from providing students with a handful of paradigmatic examples supplemented with commentaries. Our research aims at improving this situation. By putting ourselves in the shoes of an interpreter aiming at “translating” some natural-language argument into a formal representation, we have had recourse to the philosophical theories of radical translation (Quine 2013) and radical interpretation (Davidson 2001, 1994) (the latter being a further development of the former), in which the so-called radical translator (Quine) or interpreter (Davidson), without any previous knowledge of the speaker’s language, is able to find a translation (in her own language) of the speaker’s utterances. The interpreter does this by observing the speaker’s use of language in context and also by engaging (when possible) in some basic dialectical exchange with him/her (e.g., by making utterances while pointing to objects or asking yes/no questions). In our proposed approach, this exchange takes place between a human (seeking to translate “unfamiliar” natural-language discourse into a “familiar” logical formalism) and interactive proof assistants. The questions we ask concern the logical validity, invalidity, and consistency of formulas and proofs (our translation candidates). We also draw upon recent work aimed at providing adequacy criteria for logical formalization of natural-language discourse (e.g., Brun 2003; Baumgartner and Lampert 2008; Peregrin and Svoboda 2013), with a special emphasis on the work of Peregrin and Svoboda (2017), who, apart from providing syntactic and pragmatic (inferential) adequacy criteria, also tackle the problem of providing a systematic methodology for logical analysis. In this respect, they propose the method of reflec1 See

e.g., the results reported in Benzmüller and Woltzenlogel Paleo (2014), Bentert et al. (2016), Benzmüller et al. (2017), Benzmüller and Woltzenlogel Paleo (2016a), Fuenmayor and Benzmüller (2018). 2 See e.g., the research derived from Montague’s universal grammar program (Montague 1974) and some of its followers like Discourse Representation Theory (e.g., Kamp et al. 2011) and Dynamic Predicate Logic (e.g., Groenendijk and Stokhof 1991).

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tive equilibrium,3 which is similar in spirit to the idealized scientific method and additionally, has the virtue of approaching this problem in a holistic way: the adequacy of candidate formalizations for some argument’s sentences is assessed by computing the argument’s validity as a whole (which depends itself on the way we have so far formalized all of its constituent sentences).4 As we see it, this circle is a virtuous one: it does justice to holistic accounts of meaning drawing on the inferential role of sentences. As Davidson has put it: [...] much of the interest in logical form comes from an interest in logical geography: to give the logical form of a sentence is to give its logical location in the totality of sentences, to describe it in a way that explicitly determines what sentences it entails and what sentences it is entailed by. The location must be given relative to a specific deductive theory; so logical form itself is relative to a theory. (Davidson, 2001, p. 140)

2 Radical Interpretation and the Principle of Charity What is the use of radical interpretation in argumentation? The answer is trivially stated by Davidson himself, by arguing that “all understanding of the speech of another involves radical interpretation” Davidson (1994, p. 125). Furthermore, the impoverished evidential position we are faced with when interpreting some arguments (particularly philosophical ones) corresponds very closely to the starting situation Davidson contemplates in his thought experiments on radical interpretation, where he shows how an interpreter could come to understand someone’s words and actions without relying on any prior understanding of them. Davidson’s program builds on the idea of taking the concept of truth as basic and extracting from it an account of interpretation satisfying two general requirements: (i) it must reveal the compositional structure of language and (ii) it can be assessed using evidence available to the interpreter Davidson (1994, 2001). The first requirement (i) is addressed by noting that a theory of truth in Tarski’s style (modified to apply to natural language) can be used as a theory of interpretation. This implies that, for every sentence s of some object language L, a sentence of the form: “s” is true in L iff p (aka T-schema) can be derived, where p acts as a translation of s into a sufficiently expressive language used for interpretation (note that in the T-schema the sentence p is being used, while s is only being mentioned). 3 The notion of reflective equilibrium has been initially proposed by Goodman (1983) as an account

for the justification of the principles of (inductive) logic and has been popularized years later in political philosophy and ethics by Rawls (2009) for the justification of moral principles. In Rawls’ account, reflective equilibrium refers to a state of balance or coherence between a set of general principles and particular judgments (where the latter follow from the former). We arrive at such a state through a deliberative give-and-take process of mutual adjustment between principles and judgments. More recent methodical accounts of reflective equilibrium have been proposed as a justification condition for scientific theories (Elgin 1999) and objectual understanding (Baumberger and Brun 2016). 4 In much, the same spirit of Davidson’s theory of meaning Davidson (2001) and Quine’s holism of theory (dis-)confirmation (Quine 1976) in philosophy.

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Thus, by virtue of the recursive nature of Tarski’s definition of truth (Tarski 1956), the compositional structure of the object-language sentences become revealed. From the point of view of computational hermeneutics, the sentence s is to be interpreted in the context of a given argument (or a network of mutually attacking/supporting arguments). The language L thereby corresponds to the idiolect of the speaker (natural language), and the target language is constituted by formulas of our chosen logic of formalization (some expressive logic XY ) plus the turnstyle symbol XY signifying that an inference (argument or argument step) is valid in logic XY. As an illustration, consider the following instance of the T-schema: “Fishes are necessarily vertebrates” is true [in English, in the context of argument A] iff A1 , A2 , ..., An MLS4 “∀x. Fish(x) → Vertebrate(x)” where A1 , A2 , ..., An correspond to the formalization of the premises of argument A and the turnstyle MLS4 corresponds to the standard logical consequence relation in the chosen logic of formalization, e.g., a modal logic S4 (MLS4).5 This toy example aims at illustrating how the interpretation of a sentence relates to its logic of formalization and to the inferential role it plays in a single argument. Moreover, the same approach can be extended to argument networks. In such cases, instead of using the notion of logical consequence (represented above as the parameterized logical turnstyle XY ), we can work with the notion of argument support. It is indeed possible to parameterize the notions of support and attack common in argumentation theory with the logic used for argument’s formalization (see example in Sect. 5). The second general requirement (ii) of Davidson’s account of radical interpretation states that the interpreter has access to objective evidence in order to judge the appropriateness of her interpretations, i.e., access to the events and objects in the “external world” that cause sentences to be true (or, in our case, arguments to be valid). In our approach, formal logic serves as a common ground for understanding. Computing the logical validity of a formalized argument constitutes the kind of objective (or, more appropriately, intersubjective) evidence needed to secure the adequacy of our interpretations, under the charitable assumption that the speaker follows (or at least accepts) similar logical rules as we do. In computational hermeneutics, the computer acts as an (arguably unbiased) arbiter deciding on the truth of a sentence in the context of an argument. A central concept in Davidson’s account of radical interpretation is the principle of charity, which he holds as a condition for the possibility of engaging in any kind of interpretive endeavor. The principle of charity has been summarized by Davidson by stating that “we make maximum sense of the words and thoughts of others when we interpret in a way that optimizes agreement” (Davidson 2001, p. 197). Hence, the principle builds on the possibility of intersubjective agreement about external facts among speaker and interpreter. The principle of charity can be invoked to make sense of a speaker’s ambiguous utterances and in our case, to presume (and foster) the 5 As

described below, using the technique of semantical embeddings (Benzmüller and Paulson 2013) (cf. also Benzmüller 2019 and the references therein) allows us to work with several different nonclassical logics (modal, temporal, deontic, intuitionistic, etc.) while reusing existing higher order reasoning infrastructure.

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validity of an argument. Consequently, in computational hermeneutics, we assume from the outset that the argument’s conclusions indeed follow from its premises and disregard formalizations that do not do justice to this postulate.

3 Holistic Approach: Why Feasible Now? Following a holistic approach for logical analysis was, until very recently, not feasible in practice; since it involves an iterative process of trial and error, where the adequacy of some candidate formalization for a sentence becomes tested by computing the logical validity of the whole argument. In order to explore the vast combinatoric of possible formalizations for even the simplest argument, we have to test its validity at least several hundreds of times (also to account for logical pluralism). It is here where the recent improvements and ongoing consolidation of modern automated theoremproving technology (for propositional logic, first-order logic and in particular also higher order logic) become handy. To get an idea of this, let us imagine the following scenario: A philosopher working on a formal argument wants to test a variation on one of its premises or definitions and find out if the argument still holds. Since our philosopher is working with pen and paper, she will have to follow some kind of proof procedure (e.g., tableaus or natural-deduction calculus), which depending on her calculation skills, may take some minutes to be carried out. It seems clear that she cannot allow herself many of such experiments on such conditions. Now, compare the above scenario to another one in which our working philosopher can carry out such an experiment in just a few seconds and with no effort, by employing an automated theorem prover. In a best-case scenario, the proof assistant would automatically generate a proof (or the sketch of a countermodel), so she just needs to interpret the results and use them to inform her new conjectures. In any case, she would at least know if her speculations had the intended consequences, or not. After some minutes of work, she will have tried plenty of different variations of the argument while getting real-time feedback regarding their suitability.6 We aim at showing how this radical quantitative increase in productivity does indeed entail a qualitative change in the way we approach formal argumentation, since it allows us to take things to a whole new level (note that, we are talking here of many hundreds of such trial and error “experiments” that would take months or even years if using pen and paper only). Most importantly, this qualitative leap opens the door for the possibility of fully automating the process of argument formalization, as it allows us to compute inferential (holistic) adequacy criteria of formalization in 6 The

situation is obviously idealized, since as is well known, most of theorem-proving problems are computationally complex and even undecidable, so in many cases, a solution will take several minutes or just never be found. Nevertheless, as work in the emerging field of computational metaphysics (Fitelson and Zalta 2007; Rushby 2013; Benzmüller and Woltzenlogel Paleo 2014, 2016a; Benzmüller et al. 2017; Fuenmayor and Benzmüller 2017a) suggests, the lucky situation depicted above is not rare and will further improve in the future.

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real-time. Consider as an example Peregrin and Svoboda’s (2013, 2017) proposed criteria: (i) The principle of reliability: “φ counts as an adequate formalization of the sentence S in the logical system L only if the following holds: If an argument form in which φ occurs as a premise or as the conclusion is valid in L, then all its perspicuous natural-language instances in which S appears as a natural-language instance of φ are intuitively correct arguments.” (ii) The principle of ambitiousness: “φ is the more adequate formalization of the sentence S in the logical system L the more natural-language arguments in which S occurs as a premise or as the conclusion, which falls into the intended scope of L and which are intuitively perspicuous and correct, are instances of valid argument forms of S in which φ appears as the formalization of S.” (Peregrin and Svoboda 2017, pp. 70–71). The evaluation of such inferential criteria clearly involves automatically computing the logical validity or consistency of formalized arguments (proofs). This is the kind of work automated theorem provers are built for. Moreover, our focus on theorem provers for higher order logics is motivated by the notion of logical pluralism. Computational hermeneutics targets the utilization of different kinds of classical and nonclassical logics through the technique of semantical embeddings Benzmüller and Paulson (2013) (cf. also Benzmüller (2019) and the references therein), which allows us to take advantage of the expressive power of classical higher order logic (HOL) as a metalanguage in order to embed the syntax and semantics of another logic as an object language. Using (shallow) semantical embeddings we can, for instance, embed a modal logic by defining the modal  and ♦ operators as meta-logical predicates in HOL and using quantification over sets of objects of a definite type w, representing the type of possible worlds or situations. This gives us two important benefits: (i) we can reuse existing automated theorem-proving technology for HOL and apply it for automated reasoning in nonclassical logics (e.g., free, modal, temporal or deontic logics); and (ii) the logic of formalization becomes another degree of freedom and thus can be fine-tuned dynamically by adding/removing axioms in our metalanguage: HOL. A framework for automated reasoning in different logics by applying the technique of semantical embeddings has been successfully implemented using automated theorem-proving technology (see e.g., Benzmüller 2019; Fuenmayor and Benzmüller 2017a; Gleißner et al. 2017). The following two sections illustrate some exemplary applications of the computational hermeneutics approach. They have been implemented using the Isabelle/HOL (Nipkow 2002) proof assistant for classical higher order logic, aka Church’s type theory (Andrews 2018).

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4 Logical Analysis of Individual Structured Arguments The first application of computational hermeneutics for the logical analysis of arguments has been presented in Fuenmayor and Benzmüller (2018) (with its corresponding Isabelle/HOL sources available in Fuenmayor and Benzmüller 2017b). In that work, a modal variant of the ontological argument for the existence of God, introduced in natural language by the philosopher E. J. Lowe Lowe (2013), has been iteratively analyzed using our computational hermeneutics approach and as a result, a “most” adequate formalization has been found. In a series of iterations (seven in total) Lowe’s argument has been formally reconstructed using slightly different sets of premises and logics, and the partial results have been compiled and presented each time as a new variant of the original argument. We aimed at illustrating how Lowe’s argument, as well as our understanding of it, gradually evolves as we experiment with different combinations of (formalized) definitions, premises, and logics for formalization. We quote from Fuenmayor and Benzmüller (2018) the following paragraph which best summarizes the methodological approach taking us from a natural-language argument to its corresponding adequate logical formalization (and refer the interested reader to Fuenmayor and Benzmüller (2017b, 2018) for further details): We start with formalizations of some simple statements (taking them as tentative) and use them as stepping stones on the way to the formalization of other argument’s sentences, repeating the procedure until arriving at a state of reflective equilibrium: A state where our beliefs and commitments have the highest degree of coherence and acceptability. In computational hermeneutics, we work iteratively on an argument by temporarily fixing truth values and inferential relations among its sentences, and then, after choosing a logic for formalization, working back and forth on the formalization of its premises and conclusions by making gradual adjustments while getting automatic feedback about the suitability of our speculations. In this fashion, by engaging in a dialectic questions-and-answers (“trial-anderror”) interaction with the computer, we work our way toward a proper understanding of an argument by circular movements between its parts and the whole (hermeneutic circle).

5 Logical Analysis of Arguments in their Extended Argumentative Context As mentioned in the previous section, an instance of the ontological argument by E. J. Lowe has previously been employed to showcase the application of our computational hermeneutics approach to the problem of finding an adequate formalization for a natural-language argument (but without considering the surrounding network of arguments where it is embedded) Fuenmayor and Benzmüller (2018). In contrast, the example we discuss in this section7 additionally motivates and illustrates the fruitful combination of our previous work with methods as typically used in abstract 7 The assessment presented here draws on previous work in Fuenmayor and Benzmüller (2017a) and

particularly the more recent, invited paper (Benzmüller and Fuenmayor 2018), which present an

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argumentation frameworks. By doing so, we can now extend our holistic approach to logical analysis to include the dialectic role an argument plays in some larger area of discourse represented as a network of arguments. We see this as a novel contribution, which aligns our work with other prominent structured approaches to argumentation in artificial intelligence (Besnard and Hunter 2001, 2009; Dung et al. 2009). Below we will show how our approach can extend methods from abstract argumentation (Dung 1995; van Eemeran and Grootendorst 2004) by adding a layer for deep semantical analysis to it and vice versa, how our approach to deep semantical argument analysis becomes enriched by augmenting it with methods as developed in argumentation theory. This way, argument analysis becomes supported at both the abstract level and a concrete semantical level (i.e., with fully formalized naturallanguage content). We believe that such a combined, two-level approach can provide a fruitful technological backbone for our computational hermeneutics program. A particular advantage being the logical plurality we achieve at both layers. At the abstract level, for instance, the support or attack relations8 can be replaced with little technical effort with e.g., their intuitionistic logic or relevance logic counterparts (exploiting the technique of shallow semantical embeddings).9 At a concrete level, we will demonstrate how the employed logics can be varied and that it is, in fact, essential to do so, in order to achieve proper assessment results. More precisely, in the example below, we will switch between the higher order modal logics K, KB, S4, and S5.

5.1 Gödel’s Ontological Argument as a Showcase Gödel’s and Scott’s variants of the ontological argument are direct descendants of Leibniz’s, which in turn derives from Descartes. These arguments have a two-part structure: (i) prove that God’s existence is possible (see t3 in Figs. 1 and 2) and (ii) prove that God’s existence is necessary, if possible (t5). The main conclusion (God’s necessary existence, t6) then follows from (i) and (ii), either by modus ponens (in non-modal contexts) or by invoking some axioms of modal logic (notably, but not necessarily as we will see, the so-called modal logic system S5). Gödel’s ontological argument, in its different variants, is amongst the most discussed formal proofs in modern literature, and so most of its premises and inferential steps have been subject to criticism in some way or another (see e.g., Oppy 1996, 2007 and Sobel 2004). We can therefore conceive of this argument as a network of (abstracted) nodes, some updated analysis of Gödel’s and Scott’s modal variants (Gödel 2004; Scott 2004) of the ontological argument and illustrate how our method is able to formalize, assess, and explain those in full detail. 8 See lines 4–5 in Fig. 4, where their definitions are provided for classical logic. 9 The full flexibility of our framework is not illustrated to its maximum in this paper due to space restrictions. For example, for intuitionistic logic we would simply integrate the respective embedding presented in earlier work Benzmüller and Paulson (2010) to model intuitionistic support/attack relations.

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Fig. 1 The definitions (d1, d2, d3), assumptions (a1, a2, t2, a4, a5), and theorems/argumentation steps (t1, t3, t4, t5, t6) of Gödel’s, respectively, Scott’s, modal version of the ontological argument. Both the natural-language statements and the corresponding modal logic formalizations (in Isabelle/HOL) are presented

of them standing for some argument supporting the respective premise and others standing for attacking arguments (cf. bipolar argumentation frameworks Cayrol and Lagasquie-Schiex 2005, 2009).

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Fig. 2 Abstract argumentation network for Gödel’s ontological argument. The displayed arrows indicate support relations. Arrow annotations (e.g., d1, d2 in the support arrow from a4 to t4) indicate which definitions need to be unfolded for the respective support relations to apply

The abstracted nodes of the natural-language argument are introduced in Fig. 1 together with their associated formalizations in higher order modal logic. The corresponding abstract argumentation network is displayed in Fig. 2. The network presented in Fig. 2 only comprises support relations, which is sufficient for the purpose of this paper. Meaningful attack relations could of course be added. For example, it is well known that Gödel’s and Scott’s versions of the ontological argument support the modal collapse (Sobel 2004), which in turn can be interpreted as an attack to free will (see the recent discussion of this aspect in Benzmüller and Fuenmayor 2018). Extending our work below to cover the more elaborate analysis of the ontological argument as presented in Benzmüller and Fuenmayor (2018) will be addressed in future work.

5.2 Embedding a Higher Order Modal Logic in Isabelle/HOL As previously mentioned, higher order modal logic has been employed as a logic for formalization of the natural-language content of the argument nodes. To turn Isabelle/HOL into a flexible modal logic reasoner, we have adopted the shallow semantical embedding approach (Benzmüller 2019; Benzmüller and Paulson 2013). The respective embedding of higher order modal logic in Isabelle/HOL is the content of theory file IHOML.thy, which is displayed in Fig. 3. The base logic of Isabelle/HOL is classical higher order logic (HOL aka Church’s type theory Andrews 2018). HOL is a logic of functions formulated on top of the simply typed λ-calculus, which also provides a foundation for functional programming. The semantics of HOL is well understood (Benzmüller et al. 2004). Relevant for our purposes is that HOL supports the encoding of sets via their characteristic functions represented as λ-terms. In this sense, HOL comes with a in-built notion of (typed) sets that is exploited in our work for the explicit encoding of the truth sets that are associated with the formulas of higher order modal logic. Since Isabelle/HOL-specific extensions of HOL (except

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Fig. 3 Shallow semantical embedding of higher order modal logic in Isabelle/HOL

for prefix polymorphism) are not exploited in our work, the technical framework we depict here can easily be transferred to other HOL theorem-proving environments. Our semantical embedding in Isabelle/HOL encodes in lines 6–24 in Fig. 3 the standard translation of propositional modal logic to first-order logic in form of a few (non-recursive) equations. Formula ϕ, for example, is modeled as an abbreviation (syntactic sugar) for the truth set λwi .∀vi .wr v −→ ϕv, where r denotes the accessibility relation associated with the modal  operator. All presented equations exploit the idea that truth sets in Kripke-style semantics can be directly encoded as predicates (i.e., sets) in HOL. Possible worlds are, thus, explicitly represented in our framework as terms of type i and modal formulas ϕ are identified with their corresponding truth sets ϕi→o of predicate type i → o. Note how validity and invalidity is encoded in

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lines 21 and 24. A modal logic formula ϕ is valid, denoted ϕ, if its truth set is the universal set, i.e., if ϕi→o is true in all words wi . Similarly, ϕ is invalid, denoted ϕinv , if ϕi→o is false in all worlds wi . In lines 26–35, further equations are added to obtain actualist quantification (here only for individuals) and (polymorphic) possibilist quantification for objects of arbitrary type (order). This is where the shallow semantical embedding approach significantly augments the standard translation for propositional modal logics. For example, where ∀xα .φx is shorthand (binder notation in HOL) for (λxα .φx) (the denotation of  test whether its argument denotes the universal set of type α), then ∀xα .P x is now represented as ( (λxα .λwi .P xw)), where  stands for the lambda term (λ.λwi .∀xα .xw) and the  gets resolved as described above. In lines 37–42, some useful relations on accessibility relations are stated, which are used to provide semantical definitions for modal logics KB, S4 and S5. If none of the latter abbreviations is postulated, the content of Fig. 3 introduces higher order modal logic K. Further details of the presented embedding, including proofs of faithfulness, have been presented elsewhere (see e.g., Benzmüller and Paulson 2013, further references are given in Benzmüller 2019).

5.3 The Ontological Argument as an Abstract Argument Network In lines 4–5 of the Isabelle/HOL file ArgumentDefinitions.thy, displayed in Fig. 4, we import two central notions from argumentation theory: the binary relations “Supports” and “Attacks” (Besnard and Hunter 2009; Cayrol and Lagasquie-Schiex 2009). The former states that a valid modal formula A and another valid modal formula B together imply the validity of modal formula ψ. The concretely employed implication and conjunction relations are those from classical logic (the meta-logic HOL). However, as mentioned before, our framework is rich and expressive enough to replace classical consequence here by various other notions of logical consequence. Alternatively, we could parameterize the three contained validity statements for A, B and ψ for different modal logics. In fact, the different notions of logics to be employed in these definitions could be modeled as proper parameters (arguments) of the support and attack relations. This way we would obtain a very expressive and powerful reasoning framework. In order to keep things simple we will not further pursue this here, but leave it for further work. Gödel’s argument is then specified in lines 13–14 of Fig. 4 as a network of abstract nodes (recall Fig. 2). The modal validity of the assumptions “a1”, “a2”, “t2”, “a4”, and “a5” is assumed and the various support relations are stated as depicted graphically in Fig. 3. The nodes itself have been introduced as uninterpreted constant symbols in line 8 of Fig. 4 (and in line 10, we introduce further uninterpreted constant symbols “kb”, “s4”, and “s5” for characterizing the assumed modal logic conditions). The “inner semantics” of these abstract nodes, and also the definitions of the concepts Godlike (G), Essence (E) and Necessary Existence (NE), are subse-

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Fig. 4 Encoding of Gödel’s ontological argument as an abstract argument network, with a specification of the inner semantics of the argument nodes

quently specified in lines 17–33. Since theory file IHOML.thy is imported, we have access to the higher order modal logic notions introduced there. At this point, we still leave it open whether the logic K, KB, S4, or S5 is considered, since we will experiment with the different settings later on (neither Gödel nor Scott explicitly stated in their works which modal logic they actually assumed). We also do not fix the notion of essence here, but introduce the two alternative definitions proposed by Gödel and Scott (see lines 26–27). In line 28, a respective uninterpreted constant symbol for essence, E, is introduced, and then used as a dummy in the formalization of the subsequent argument nodes. In the experiments ahead, we can now switch between Gödel’s and Scott’s notions of essence, by equating this dummy constant symbol with the different concrete definitions considered.

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In lines 36–37, the abstract argument nodes are identified with their formalizations as just introduced. Thus, when postulating the here defined Boolean flag “InstantiateArgumentNodes”, the argument from lines 13–14 is no longer just abstract, but in a sense instantiated through activation of the inner semantics of the argument nodes. In line 39, the abstract logic conditions are analogously instantiated with their concrete realizations by Boolean flag “InstantiateLogics”. The “Instantiate” in line 41 then simply combines them into a single flag.

5.4 Analysis of Gödel’s Variant of the Ontological Argument In Fig. 5, we analyze Gödel’s variant of the ontological argument using the notions and ideas as introduced in the imported theory files IHOML.thy and ArgumentDefinitions.thy from Figs. 3 and 4 respectively. To do so, the “essence” dummy constant symbol is mapped to the definition as proposed by Gödel’s (see line 4). Then, in lines 7–9, we ask the model finder Nitpick Blanchette and Nipkow (2010), integrated with Isabelle/HOL, to compute a model for the abstract Gödel argument. For the call in line 7, a model consisting of one world (accessible from itself—not shown in the window) with one single individual is presented in the lower window of Fig. 5. The duplicated calls in line 8 and 9 are of course redundant, since the Boolean logic flags “kb”, “s4” and “s5” (and also the argument nodes) are still uninterpreted. Hence, at the abstract level, the Gödel argument has a model (we here even present the minimal model) and is thus satisfiable. For illustration purposes we show in line 12 that the abstract argument can easily become unsatisfiable, for example, by adding an attack relation as displayed. Automated theorem-proving technology integrated with Isabelle/HOL can quickly reveal such inconsistencies at the abstract argument level. A much more interesting and relevant aspect is illustrated in lines 25–24 of Fig. 5: the satisfiability of the abstract Gödel argument provides no guarantee at all for its satisfiability at instantiated level, i.e., when the semantics of the argument nodes is added. In line 15, the model finder Nitpick indeed fails to compute a satisfying model for the instantiated argument (it terminates with a timeout). However, if we now study the semantically instantiated (formalized) argument, which is done by activating the link between the abstract nodes and their formalizations, then we find out that the Gödel argument is actually inconsistent (for all logic conditions). And in lines 21–24, the inconsistency of the instantiated abstract argument is then proven automatically by respective automation tools in Isabelle/HOL (here, the prover Metis is employed). The clue to the inconsistency is the “Empty Essence Lemma (EEL)”, which is proven in line 18. The inconsistency of Gödel’s ontological argument was unknown to philosophers until recently, when it was detected by the automated higher theorem prover LEO-II (Benzmüller et al. 2015); for more on this see Benzmüller and Woltzenlogel Paleo (2016a, b).

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Fig. 5 Analysis of Gödel’s variant of the ontological argument

5.5 Analysis of Scott’s Variant of the Ontological Argument We analyze Scott’s version of the argument in Fig. 6. In line 4 of this file, the notion of essence according to Scott is activated. In lines 7–9, the model finder Nitpick again confirms the consistency of the argument at the abstract level, which was expected, since the concretely employed notion of essence does not have an influence at this level. It does so, however, at instantiated level, and this can be seen in lines 13–15 in Fig. 6. In contrast to the instantiated Gödel argument from Fig. 5, where the model finder Nitpick failed to confirm satisfiability, it now succeeds. And in fact, it does so for all logic conditions. The reported model (for logic S5, which in fact works for all logic conditions) is displayed in the lower window of Fig. 6. This model is minimal,

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Fig. 6 Analysis of Scott’s variant of the ontological argument—Part I

it consists of one world (accessible to itself) and one object, and further details on the interpretation of essence and the notion of positive properties are displayed. However, as we illustrate next, the satisfiability of Scott’s argument for logics KB, S4, and S5 does of course not imply the validity of the argument for these logic conditions. In lines 19–29 in Fig. 7, we first prove the validity of the Scott argument for logic KB. This is done by automatically proving that all (instantiated) support relations are validated. Note how concisely the exactly required dependencies are displayed in the justifications of the proof steps. In lines 32–42 of Fig. 7, we analogously assess the validity of the Scott argument for logic S4. However, the attempt to copy and paste the previous proof does fail: in line 38, we obtain a countermodel, and this countermodel comes with a nonsymmetric accessibility relation between worlds. Note that in line 25, in the proof from before, the logic condition KB (assms(1)) was

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Fig. 7 Analysis of Scott’s variant of the ontological argument—Part II

indeed employed in the justification. We here see that the symmetry condition of logic KB indeed plays a role for the validity of Scott’s argument. In a logic such as S4, where symmetric accessibility relations between worlds are not enforced, the argument fails. This is confirmed again in lines 44–45, where the countermodel is computed directly for the stated validity conjecture (excluding the possibility that there might be an alternative proof in S4 to the one attempted in lines 32–42). For logic S5, the Scott argument is valid again, which is not a surprise given that logic S5 entails logic KB. This is confirmed in lines 48–58 in Fig. 7.

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5.6 Section Summary The analysis of nontrivial natural-language arguments at the abstract argumentation level is useful, but of limited explanatory power. Achieving such explanatory power requires the extension of techniques from abstract argumentation frameworks with means for deep semantical analysis as provided in our computational hermeneutics approach. This has been illustrated in this section with the help of Gödel’s and Scott’s versions of the ontological argument for the existence of God. Highly relevant aspects, such as inconsistency of Gödel’s argument, invalidity of Scott’s argument for S4 and validity for KB and S5 could only be shown by the integration of the abstract argumentation layer with our machinery.

6 Ongoing and Future Work In previous work Fuenmayor and Benzmüller (2018), we have illustrated how the computational hermeneutics approach can be carried out in a semi-automatic fashion for the logical analysis of an isolated argument: We work iteratively on an argument by (i) tentatively choosing a logic for formalization; (ii) fixing truth values and inferential relations among its sentences; and (iii) working back and forth on the formalization of its axioms and theorems, by making gradual adjustments while getting real-time feedback about the suitability of our changes (e.g., validating the argument, avoiding inconsistency or question-begging, etc.). These steps are to be repeated until arriving at a state of reflective equilibrium: A state where our arguments and claims have the highest degree of coherence and acceptability according to syntactic and particularly, inferential criteria of adequacy (see Peregrin and Svoboda’s criteria presented above Peregrin and Svoboda 2013, 2017). In this paper, we have sketched another exemplary application of computational hermeneutics to argumentation theory. We exploited the fact that our approach is well suited for the utilization of different kinds of classical and nonclassical logics through the technique of shallow semantical embeddings (Benzmüller 2019; Benzmüller and Paulson 2013), which allows us to take advantage of the expressive power of classical higher order logic (as a metalanguage) in order to embed the syntax and semantics of another logic (object language). This way, it has become possible to parameterize the relations of argument support and attack by adding the logic of formalization as a variable. This parameter can then be varied by adding or removing premises (at a meta-level), which correspond to the embedding of the logic in question. In future work, we aim at integrating our approach with others in argumentation theory, which also take into account the logical structure of arguments (see e.g., Besnard and Hunter 2001, 2009; Dung et al. 2009 and also Arieli and Straßer 2015 for a proof theoretically enriched approach). Computational hermeneutics features the sort of (holistic) mutual adjustment between theory and observation, which is characteristic of scientific inquiry; we are

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currently exploring the way to fully automate this process. The idea is to tackle the problem of formalization as a combinatorial optimization problem, by using (among others) inferential criteria of adequacy to define the fitness/utility function of an appropriate optimization algorithm. Davidson’s principle of charity would provide our main selection criteria: an adequate formalization must (i) validate the argument (among other qualitative requirements) and (ii) do justice to its intended dialectic role in some discourse (i.e., it attacks/supports other arguments as intended). It is worth noting that, for the kind of nontrivial arguments, we are interested in (e.g., from ethics, metaphysics, and politics), such a selection criteria would aggressively prune our search tree. Furthermore, the evaluation of our fitness function is, with today’s technologies, not only completely automatizable, but also seems to be highly parallelizable. The most challenging task remains how to systematically come up with the candidate formalization hypotheses (an instance of abductive reasoning). Here, we see great potential in the combination of automated theorem proving with other areas in artificial intelligence such as machine learning and in particular, argumentation frameworks, by exploiting a layered, structured approach as illustrated above. Acknowledgements We thank the anonymous reviewers for their valuable comments which helped to improve this paper.

References Andrews, P.: Church’s type theory. In: Zalta, E.N. (ed.) The Stanford Encyclopedia of Philosophy, summer, 2018th edn. Stanford University, Metaphysics Research Lab (2018) Arieli, O., Straßer, C.: Sequent-based logical argumentation. Argum. Comput. 6(1), 73–99 (2015) Baumberger, C., Brun, G.: Dimensions of objectual understanding. In: Explaining Understanding. New Perspectives from Epistemology and Philosophy of Science, pp. 165–189 (2016) Baumgartner, M., Lampert, T.: Adequate formalization. Synthese 164(1), 93–115 (2008) Bentert, M., Benzmüller, C., Streit, D., Woltzenlogel Paleo, B.: Analysis of an ontological proof proposed by Leibniz. In: Tandy, C. (ed.) Death and Anti-Death, Volume 14: Four Decades after Michael Polanyi, Three Centuries after G.W. Leibniz. Ria University Press (2016). https:// philpapers.org/rec/TANDAA-10 Benzmüller, C.: Universal (meta-)logical reasoning: recent successes. Sci. Comput. Program. 172, 48–62 (2019). https://doi.org/10.1016/j.scico.2018.10.008, https://doi.org/10.13140/RG.2. 2.11039.61609/2 Benzmüller, C., Brown, C., Kohlhase, M.: Higher-order semantics and extensionality. J. Symb. Logic 69(4), 1027–1088 (2004). https://doi.org/10.2178/jsl/1102022211, http://christoph-benzmueller. de/papers/J6.pdf Benzmüller, C., Fuenmayor, D.: Can computers help to sharpen our understanding of ontological arguments? In: Gosh, S., Uppalari, R., Rao, K.V., Agarwal, V., Sharma, S. (eds.) Mathematics and Reality, Proceedings of the 11th All India Students’ Conference on Science & Spiritual Quest, 6–7 October 2018, IIT Bhubaneswar, Bhubaneswar, India. The Bhaktivedanta Institute, Kolkata. www.binstitute.org (2018). https://doi.org/10.13140/RG.2.2.31921.84323, http:// christoph-benzmueller.de/papers/C74.pdf

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Benzmüller, C., Paulson, L.: Multimodal and intuitionistic logics in simple type theory. Logic J. IGPL 18(6), 881–892 (2010). https://doi.org/10.1093/jigpal/jzp080, http://christophbenzmueller.de/papers/J21.pdf Benzmüller, C., Paulson, L.: Quantified multimodal logics in simple type theory. Logica Universalis (Special Issue on Multimodal Logics) 7(1), 7–20 (2013). https://doi.org/10.1007/s11787-0120052-y, http://christoph-benzmueller.de/papers/J23.pdf Benzmüller, C., Sultana, N., Paulson, L.C., Theiß, F.: The higher-order prover LEO-II. J. Autom. Reason. 55(4), 389–404 (2015). https://doi.org/10.1007/s10817-015-9348-y, http://christophbenzmueller.de/papers/J30.pdf Benzmüller, C., Weber, L., Woltzenlogel-Paleo, B.: Computer-assisted analysis of the AndersonHájek controversy. Logica Universalis 11(1), 139–151 (2017). https://doi.org/10.1007/s11787017-0160-9, http://christoph-benzmueller.de/papers/J32.pdf Benzmüller, C., Woltzenlogel Paleo, B.: Automating Gödel’s ontological proof of God’s existence with higher-order automated theorem provers. In: Schaub, T., Friedrich, G., O’Sullivan, B. (eds.) Frontiers in Artificial Intelligence and Applications, ECAI 2014, vol. 263, pp. 93–98. IOS Press (2014). https://doi.org/10.3233/978-1-61499-419-0-93, http://christoph-benzmueller.de/ papers/C40.pdf Benzmüller, C., Woltzenlogel Paleo, B.: The inconsistency in Gödel’s ontological argument: a success story for AI in metaphysics. In: IJCAI 2016 (2016a). http://christoph-benzmueller.de/ papers/C55.pdf Benzmüller, C., Woltzenlogel Paleo, B.: An object-logic explanation for the inconsistency in Gödel’s ontological theory (extended abstract). In: Helmert, M., Wotawa, F. (eds.) Proceedings of Advances in Artificial Intelligence, KI 2016. LNCS, vol. 9725, pp. 43–50. Springer, Heidelberg (2016b). http://christoph-benzmueller.de/papers/C60.pdf Besnard, P., Hunter, A.: A logic-based theory of deductive arguments. Artif. Intell. 128(1–2), 203– 235 (2001) Besnard, P., Hunter, A.: Argumentation based on classical logic. In: Argumentation in Artificial Intelligence, pp. 133–152. Springer, Boston (2009) Blanchette, J., Nipkow, T.: Nitpick: a counterexample generator for higher-order logic based on a relational model finder. In: Proceedings of ITP 2010. LNCS, vol. 6172, pp. 131–146. Springer, Heidelberg (2010) Brun, G.: Die richtige Formel: Philosophische Probleme der logischen Formalisierung, vol. 2. Walter de Gruyter (2003) Cayrol, C., Lagasquie-Schiex, M.C.: On the acceptability of arguments in bipolar argumentation frameworks. In: European Conference on Symbolic and Quantitative Approaches to Reasoning and Uncertainty, pp. 378–389. Springer, Heidelberg (2005) Cayrol, C., Lagasquie-Schiex, M.C.: Bipolar abstract argumentation systems. In: Rahwan, I., Simari, G.R. (eds.) Argumentation in Artificial Intelligence, pp. 65–84. Springer, Boston (2009) Davidson, D.: Radical interpretation interpreted. Philos. Perspect. 8, 121–128 (1994) Davidson, D.: Essays on Actions and Events: Philosophical Essays, vol. 1. Oxford University Press on Demand, Oxford (2001) Davidson, D.: Inquiries into Truth and Interpretation: Philosophical Essays, vol. 2. Oxford University Press, Oxford (2001) Davidson, D.: Radical interpretation. In: Inquiries into Truth and Interpretation. Oxford University Press, Oxford (2001) Dung, P.M.: On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games. Artif. intell. 77(2), 321–357 (1995) Dung, P.M., Kowalski, R.A., Toni, F.: Assumption-based argumentation. In: Argumentation in Artificial Intelligence, pp. 199–218. Springer, Boston (2009) van Eemeran, F.H., Grootendorst, R.: A Systematic Theory of Argumentation. Cambridge University Press, Cambridge (2004) Elgin, C.: Considered Judgment. Princeton University Press, New Jersey (1999)

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Fitelson, B., Zalta, E.N.: Steps toward a computational metaphysics. J. Philos. Logic 36(2), 227–247 (2007) Fuenmayor, D., Benzmüller, C.: Automating emendations of the ontological argument in intensional higher-order modal logic. In: Kern-Isberner, G., Fürnkranz, J., Thimm, M. (eds.) Advances in Artificial Intelligence, KI 2017, vol. 10505, pp. 114–127. Springer, Cham (2017a) Fuenmayor, D., Benzmüller, C.: Computer-assisted reconstruction and assessment of E. J. Lowe’s modal ontological argument. Archive of Formal Proofs (2017b). http://isa-afp.org/entries/Lowe_ Ontological_Argument.html, Formal proof development Fuenmayor, D., Benzmüller, C.: A case study on computational hermeneutics: E. J. Lowe’s modal ontological argument. IfCoLoG J. Logics Appl. (Special issue on Formal Approaches to the Ontological Argument) (2018). http://christoph-benzmueller.de/papers/J38.pdf Gleißner, T., Steen, A., Benzmüller, C.: Theorem provers for every normal modal logic. In: Eiter, T., Sands, D. (eds.) 21st International Conference on Logic for Programming, Artificial Intelligence and Reasoning, LPAR-21 EPiC Series in Computing, vol. 46, pp. 14–30. EasyChair, Maun, Botswana (2017). https://doi.org/10.29007/jsb9, https://easychair.org/publications/paper/ 340346 Gödel, K.: Appx. A: Notes in Kurt Gödel’s Hand, pp. 144–145. In: [50] (2004). http://books.google. de/books?id=ZQh8QJOQdOQC Goodman, N.: Fact, Fiction, and Forecast. Harvard University Press, Cambridge (1983) Groenendijk, J., Stokhof, M.: Dynamic predicate logic. Linguist. Philos. 14(1), 39–100 (1991) Kamp, H., Van Genabith, J., Reyle, U.: Discourse representation theory. In: Handbook of Philosophical Logic, pp. 125–394. Springer, Dordrecht (2011) Lowe, E.J.: A modal version of the ontological argument. In: Moreland, J.P., Sweis, K.A., Meister, C.V. (eds.) Debating Christian Theism, Chap. 4, pp. 61–71. Oxford University Press (2013) Montague, R.: Formal Philosophy: Selected Papers of Richard Montague. Ed. and with an Introd. by Richmond H. Thomason. Yale University Press (1974) Nipkow, T., Paulson, L.C., Wenzel, M.: Isabelle/HOL—A Proof Assistant for Higher-Order Logic. No. 2283 in LNCS. Springer, Heidelberg (2002) Oppy, G.: Gödelian ontological arguments. Analysis 56(4), 226–230 (1996) Oppy, G.: Ontological Arguments and Belief in God. Cambridge University Press, Cambridge (2007) Peregrin, J., Svoboda, V.: Criteria for logical formalization. Synthese 190(14), 2897–2924 (2013) Peregrin, J., Svoboda, V.: Reflective Equilibrium and the Principles of Logical Analysis: Understanding the Laws of Logic. Routledge Studies in Contemporary Philosophy. Taylor and Francis (2017) Quine, W.V.O.: Two dogmas of empiricism. In: Can Theories be Refuted?, pp. 41–64. Springer, Dordrecht (1976) Quine, W.V.O.: Word and Object. MIT Press, New York (2013) Rawls, J.: A Theory of Justice. Harvard University Press, Cambridge (2009) Rushby, J.: The ontological argument in PVS. In: Proceedings of CAV Workshop “Fun With Formal Methods”. St. Petersburg, Russia (2013) Scott, D.: Appx.B: Notes in Dana Scott’s Hand, pp. 145–146. In: [50] (2004). http://books.google. de/books?id=ZQh8QJOQdOQC Sobel, J.: Logic and Theism: Arguments for and Against Beliefs in God. Cambridge University Press, New York (2004). http://books.google.de/books?id=ZQh8QJOQdOQC Tarski, A.: The concept of truth in formalized languages. Logic Semant. Metamathematics 2, 152– 278 (1956)