Metainferential Logics (Trends in Logic, 61) 3031443802, 9783031443800

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Trends in Logic 61

Federico Pailos Bruno Da Ré

Metainferential Logics

Trends in Logic Volume 61

TRENDS IN LOGIC Studia Logica Library VOLUME 61 Series Editor Heinrich Wansing, Department of Philosophy, Ruhr University Bochum, Bochum, Germany Editorial Board Arnon Avron, Department of Computer Science, University of Tel Aviv, Tel Aviv, Israel Katalin Bimbó, Department of Philosophy, University of Alberta, Edmonton, AB, Canada Giovanna Corsi, Department of Philosophy, University of Bologna, Bologna, Italy Janusz Czelakowski, Institute of Mathematics and Informatics, University of Opole, Opole, Poland Roberto Giuntini, Department of Philosophy, University of Cagliari, Cagliari, Italy Rajeev Goré, Australian National University, Canberra, ACT, Australia Andreas Herzig, IRIT, University of Toulouse, Toulouse, France Wesley Holliday, UC Berkeley, Lafayette, CA, USA Andrzej Indrzejczak, Department of Logic, University of Lódz, Lódz, Poland Daniele Mundici, Mathematics and Computer Science, University of Florence, Firenze, Italy Sergei Odintsov, Sobolev Institute of Mathematics, Novosibirsk, Russia Ewa Orlowska, Institute of Telecommunications, Warsaw, Poland Peter Schroeder-Heister, Wilhelm-Schickard-Institut, Universität Tübingen, Tübingen, Baden-Württemberg, Germany Yde Venema, ILLC, Universiteit van Amsterdam, Amsterdam, Noord-Holland, The Netherlands Andreas Weiermann, Vakgroep Zuivere Wiskunde en Computeralgebra, University of Ghent, Ghent, Belgium Frank Wolter, Department of Computing, University of Liverpool, Liverpool, UK Ming Xu, Department of Philosophy, Wuhan University, Wuhan, China Jacek Malinowski, Institute of Philosophy and Sociology, Polish Academy of Sciences, Warszawa, Poland Assistant Editor Daniel Skurt, Ruhr-University Bochum, Bochum, Germany Founding Editor Ryszard Wojcicki, Institute of Philosophy and Sociology, Polish Academy of Sciences, Warsaw, Poland The book series Trends in Logic covers essentially the same areas as the journal Studia Logica, that is, contemporary formal logic and its applications and relations to other disciplines. The series aims at publishing monographs and thematically coherent volumes dealing with important developments in logic and presenting significant contributions to logical research. Volumes of Trends in Logic may range from highly focused studies to presentations that make a subject accessible to a broader scientific community or offer new perspectives for research. The series is open to contributions devoted to topics ranging from algebraic logic, model theory, proof theory, philosophical logic, non-classical logic, and logic in computer science to mathematical linguistics and formal epistemology. This thematic spectrum is also reflected in the editorial board of Trends in Logic. Volumes may be devoted to specific logical systems, particular methods and techniques, fundamental concepts, challenging open problems, different approaches to logical consequence, combinations of logics, classes of algebras or other structures, or interconnections between various logic-related domains. This book series is indexed in SCOPUS. Authors interested in proposing a completed book or a manuscript in progress or in conception can contact either [email protected] or one of the Editors of the Series.

Federico Pailos · Bruno Da Ré

Metainferential Logics

Federico Pailos IIF-SADAF-CONICET University of Buenos Aires Buenos Aires, Argentina

Bruno Da Ré IIF-SADAF-CONICET University of Buenos Aires Buenos Aires, Argentina

ISSN 1572-6126 ISSN 2212-7313 (electronic) Trends in Logic ISBN 978-3-031-44380-0 ISBN 978-3-031-44381-7 (eBook) https://doi.org/10.1007/978-3-031-44381-7 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed 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 Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Paper in this product is recyclable.

Preface

This book offers a gentle introduction to the investigation of the technical features of the metainferential logics developed over the last years, with their most relevant results and applications. Also, the book provides some new paths to define and investigate metainferential logics. The main goal of this book is to present an approach to the study of the semantics and the proof theory of this new and exciting variety of families of logics, which, among other things, show new ways of being fully classical or fully empty, while also being able to deal with semantic paradoxes. What distinguishes a metainferential logic is that it is characterized by the set of metainferences that it validates. There are many ways to define what a metainference is, and also many ways to determine when a metainference is valid. We will present the different forms of understanding both of these things, while also justifying our preference for metainferences as syntactic objects—i.e., as a kind of inference that relates sets of (traditional) inferences instead of sets of formulas—and for the local way to define metainferential validity—i.e., as preservation of satisfaction (in a valuation or a model). Although metainferential logics can be characterized both semantically and proof-theoretically, all the metainferential logics we will introduce here are first presented from a semantic perspective, and as kinds of mixed logics. All of the metainferential logics which we will discuss in this book are specified through satisfaction standards of logics of previous (meta)inferential levels. Metainferential logics of the first level are built with satisfaction standards of inferential logics. The standards that we will consider here are either those of Strong Kleene or Weak Kleene logics, or even both. Throughout the different chapters, we will show some of the main applications of these logics to some of the most important problems in philosophical logics, such as semantic paradoxes, the monism-versus-pluralism debate, the implications for logical nihilism, the different ways of characterizing dual logics and rules, and the

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different ways in which logics might, or should, be characterized. These different applications of metainferential logics show the fruitfulness of these logics. There are also many open problems and applications for metainferential logics. We hope that this book will help the reader discover them, solve them, and further expand the pages written here. Buenos Aires, Argentina

Federico Pailos Bruno Da Ré

Acknowledgements

We are indebted to many friends, family, and colleagues. Above all, we are grateful to our families, Tobías, Maite, Irina, Greta, and Maru. We want to express our gratitude to all the members of the Buenos Aires Logic Group. In particular, Eduardo Barrio, who has always supported us, discussed the material with us and served as a mentor for this project. He is also a co-author of some of the material this book is based on. We would also like to thank Damián Szmuc, who has also discussed the ideas in this book in detail and is also a co-author of some of the articles that we have reworked for this book. We also thank Paula Teijeiro and Mariela Rubin, also co-authors of two of the articles whose ideas are included in this book. Finally, we would like to thank other members of the group for their feedback, in particular Lucas Rosenblatt, Diego Tajer, Camillo Fiore, Joaquín Toranzo Calderón, and Eliana Franceschini. Also, we are indebted to many logicians and philosophers for giving us the chance to discuss some of the ideas in this book. Especially, we would like to thank Dave Ripley, Luca Tranchini, Pablo Cobreros, Paul Égré, Chris Scambler, Brian Porter, Thomas Ferguson, Andreas Fjellstad, Graham Priest, Melvin Fitting, Elia Zardini, Bas Kortenbach and Bodgan Dicher for their comments. Beyond the input of these individuals, the contents of the present work have been sharpened and refined by the many pages of comments and criticism we have received from anonymous referees, and we want to express our gratitude for their help. We are also grateful to Springer (https://link.springer.com/), Cambridge University Press (https://www.cambridge.org/), and Taylor and Francis (https://taylorandfra ncis.com/) for their permission to reprint the portions of this work that had previously been published in print. We would also like to thank the Humboldt Foundation. While writing this book, Federico Pailos held a Humboldt Research Fellowship for Experienced Researchers (March 2020 to July 2021). This work was supported by PLEXUS, (Grant Agreement no 101086295) a Marie Sklodowska-Curie action funded by the EU under the Horizon Europe Research and Innovation Programme. Views and opinions expressed are however those of the authors only and do not necessarily reflect those of the

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European Union. Neither the European Union nor the granting authority can be held responsible for them. We especially thank two public institutions in our country: CONICET and the University of Buenos Aires. First of all, CONICET has funded this and other related projects, as well as our doctorates and postdoctorates.Today we are proud to be CONICET researchers. On the other hand, we teach at the University of Buenos Aires, which is the institution that has prepared us as logicians and philosophers. We thank our colleagues, professors and students at the University of Buenos Aires. Many of them have contributed explicitly or implicitly to this project. Furthermore, in addition to CONICET and the University of Buenos Aires, we thank the members and authorities of SADAF (Sociedad Argentina de Análisis Filosófico). At SADAF we have met periodically for the last eight years and it has been the place where most of the content of this book has been created. Finally, we would like to express our gratitude to Peter Schroeder-Heister, who has not only discussed with us part of the ideas included here but has also supported us in many different ways. Peter was the host of the Humboldt Research Fellowship for Experienced Researchers for one of us while most of the contents of this book were written.

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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About Metainferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Two Ways of Understanding Metainferences: As Properties and as Inferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Metainferences: A Technical Definition . . . . . . . . . . . . . . . 2.2 Three Ways of Understanding Metainferential Validity . . . . . . . . . 2.2.1 Comparing the Three Notions . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9 9 12 14 19 26

3

Strong Kleene Metainferential Logics . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Four Basic Inferential Logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Strong Kleene Metainferential Logics (of Level 1) . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29 30 33 46

4

Hierarchies of Strong Kleene Metainferential Logics . . . . . . . . . . . . . 4.1 The ST-Hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 The TS-Hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47 47 55 60

5

Weak Kleene Metainferential Logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Four Basic Inferential Logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Weak Kleene Mixed Metainferential Logics (of Level 1) . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63 63 65 70

6

Combining Weak and Strong Kleene Metainferential Logics . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Contents

Hierarchies of Global and Absolutely Global Metainferential Logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Global Hierarchies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Absolutely Global Hierarchies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79 80 84 85

8

Metainferential Sequent Calculi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 8.1 Sequent Calculi for Local Metainferential Validities . . . . . . . . . . . 88 8.2 Sequent Calculi for Global Metainferential Validities (and Invalidities) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

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Metainferential Theories of Truth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

10 Philosophical Reflections: Applications and Discussions . . . . . . . . . . . 10.1 Metainferential Logics and the Monism/Pluralism Debate. New (Metainferential) Collapses . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Duality Between Metainferential Logics . . . . . . . . . . . . . . . . . . . . . 10.3 What Is a (Paraconsistent or Paracomplete) Logic? . . . . . . . . . . . . 10.4 Further Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

113 114 118 120 123 125 126

11 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

Chapter 1

Introduction

Much of the interest that substructural logics held over during the last years revolves around how they deal with metainferences. Most notably, the non-transitive—and, therefore, substructural—logic ST has sometimes been described as a theory that validates every classical inference while making invalid some classical metainferences. But, what exactly is a metainference? There are two basic ways to construe metainferences. Traditionally, logicians have thought about them as closure properties on a set of (valid) inferences. Specifically, as properties under which a set of inferences is closed. For example, this is how Dave Ripley talks about metainferences in [1], and that is why he can justifiably claim that ST—in a language without constants for every truth-value—is just classical logic (CL) in disguise.1 Cut, in particular, turns out to be a valid ST metainference, in the sense that the set of valid inferences is closed under it.2 The other way of thinking about metainferences is syntactic. In this sense, metainferences are just a special type of inferences. The only thing that distinguishes them from ordinary inferences is that they do not relate two collections3 of sentences. Instead, metainferences (of level 1) are defined as relating two collections of inferences. From an abstract point of view, this move can be generalized, at least, to any finite level, and thus, there are metainferences of any finite level. For any level n ≥ 1, the premises are metainferences of level n − 1—or inferences (if n = 1)—, and the conclusions are metainferences of level n − 1. Under this approach to metainferences, they can either be satisfied or dissatisfied by valuations or models—i.e., a valuation or a model can either satisfy, or be a counterexample of, 1

The absence of truth-constants for every truth-value in the language is not innocent. If we added them to ST, with the usual three-valued semantics, it would become non-transitive. For a critical stance towards ST in this respect, see Dicher and Paoli [2]. 2 As Ripley in [3] shows, there are many ways of thinking about transitivity and of characterizing Cut. In this context, our assertion holds for any form of it. 3 Notice that we are assuming multiple conclusions. Also, we talk about collections in general since they might be sets, multisets, or sequences. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. Pailos and B. Da Ré, Metainferential Logics, Trends in Logic 61, https://doi.org/10.1007/978-3-031-44381-7_1

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1 Introduction

the metainference, just as it happens with inferences. Thus, models and valuations turn out to be a pretty decent way of studying the behavior of metainferences. In this book, following the latter point of view, we will consider metainferences to be just special cases of inferences. In this sense, one crucial point to elucidate will be: what does it mean for a metainference to be valid? In the literature, we can find two main answers to this question: the local and the global interpretation of validity. According to the local view, a metainference is valid if and only if every valuation that is a counterexample of some conclusion of the metainference is also a counterexample of some premise. On the other hand, according to the global interpretation, a metainference is valid if and only if, if every valuation satisfies all of the premises, then every valuation satisfies some of the conclusions. Throughout this book, we will present them in detail along with some variations of them. Another very important topic is the nature of metainferences, as they can be either schemas or tokens. Any decision we make on this point has several implications for the logical features of the logics. For instance, Dicher and Paoli in [2] have pointed out that a local interpretation is a better fit for distinguishing between valid and invalid instances of a schema. And, even if it seems intuitive to think about validity as mainly being about schemas, and not instances, schemas are not the most important thing when one moves from logic to, say, theories of truth. In addition, it is not at all clear that logic should be mainly focused on schemas. Also, there are some technical results (see, in particular, Teijeiro [4]) which show that using schemas instead of tokens (plus some linguistic richness) forces the collapse between the local and the global notions of validity at the first metainferential level. Chapter 2 of this book is devoted to understanding metainferences, and we will elaborate on these topics there. This is the first Chapter of the mostly technical part of the book, which goes up to Chap. 8. The last two chapters explore possible applications and some important philosophical consequences of investigating these mixed metainferential logics. Going back to metainferences, a question arises at this point of the debate. If they are so important that it is possible to distinguish some logics from each other solely by their different behavior regarding metainferences, shouldn’t metainferences, and not inferences, be the main focus of logics? That might be going too far. Nevertheless, in Chap. 3 we will present a family of logics whose main focus is exactly that—i.e., metainferences.4 They are called metainferential logics. In a nutshell, a logic is metainferential if and only if it is defined through a standard of validity for metainferences. And it is a fact that most metainferential logics we will deal with in this book are also mixed logics. Chemlá et al., in [5], originally introduced the notion of inferential mixed consequence relation (or mixed logic) as follows: an inference  ⇒  is valid in a mixed logic if and only if, for every valuation v, if v(γ ) meets some standard S1 for all the premises 4

In a sense, this is what is done in Gentzen-style sequent calculi, for there the basic rules are all, in a sense, metainferential. Nevertheless, the focus of those calculi is usually on inferences themselves (represented by sequents), and not on metainferences (represented by the basic rules, but also by every valid—or invalid—rule, no matter how metainferential validity is represented in those calculi—as admissibility, derivability or whatnot. (We thank an anonymous reviewer for this comment.).

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γ ∈ , then v(δ) meets some standard S2 for some conclusion δ ∈ .5 A standard, for [5], is a set of truth-values. And for any standard Si , valuation v and formula ϕ, v(ϕ) satisfies or meets the standard Si if and only if v(ϕ) ∈ Si . Another way of understanding the standards for premises and conclusions is as elements specifying which values each formula in a sound argument or inference can adopt. If S1 = S2 , then the mixed consequence relation is pure. If S1 = S2 , then the mixed consequence relation is impure. What happens if Si ’s are not sets of truth-values, but satisfaction standards for inferentially mixed consequence relations? Then what we get is a metainferential mixed consequence relation. We will present some of them, starting with a family of metainferential logics based on four Strong Kleene logics, two of them structural— K3 and LP—and the other two substructural—TS and ST, previously described by Pailos in [6]. One of the most interesting ones is TS/ST. While ST recovers every classically valid inference, many classically valid metainferences (of level 1) are invalid in ST. TS/ST recaptures every single one of them but fails to locally validate some classically valid metametainferences (metainferences of level 2). But a new metainferential logic based on TS/ST and ST/TS—specified through the same inferential logics as TS/ST, but switching standards between premises and conclusions—recovers them. Nevertheless, it fails to locally validate some classically valid metainferences of higher levels. TS/ST and ST/TS are two out of sixteen different mixed metainferential consequence relations that are specified using K3, LP, ST or TS with a local notion of validity. Nevertheless, there are more interesting mixed metainferential logics of this kind. For example, K3/LP can be understood as a logic where Meta-Identity fails. We will describe and explore these logics in detail in Chap. TS/ST is the first step in a hierarchy of logics—each one belonging to a different finite level—such that each step recaptures more classical validities than the former logics in the hierarchy, although none of them is entirely classical. Eventually, it is possible to define a logic based on the logics in the hierarchy that recaptures every classically valid metainference of any level. Another important hierarchy of metainferential logics is the one that starts with ST/TS. Each of the logics in this new hierarchy validates fewer and fewer metainferences. A logic equivalent to the intersection of them is a truly empty logic. We will present both of these hierarchies of metainferential logics in detail in Chap. 4. The logics introduced so far, based on the Strong Kleene schema, represent just a small fraction of every possible metainferential logic. In Chap. 5 we will expand the landscape with hierarchies of three-valued metainferential logics based on the Weak Kleene schema. We will also evaluate the differences and similarities that these logics have when comparing them with their Strong Kleene relatives that we have already introduced. Moreover, in Chap. 6 we will also introduce two different ways of combining Strong Kleene and Weak Kleene logics in a single metainferential logic. They will define two new families of metainferential logics: the Strong-Weak and the 5

A formal definition of mixed logics will be provided in Chap. 2.

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Weak-Strong logics. We will also describe possible ways of interpreting them that make those logics philosophically relevant. So far, we have talked about metainferential logics—and hierarchies of these logics—defined using a local approach to metainferential validity. Nevertheless, as we already mentioned, this is not the only way of understanding metainferential validity. If we adopt a global point of view, then logics defined through the same standards will validate different sets of metainferences. Chapter 7 is devoted to the study of these different approaches, and to the various ways of recovering classical logic with metainferential logics using just substructural (inferential) standards as the basic building blocks. Up to this point, we will have introduced different semantic ways to characterize mixed metainferential logics. But we would not have mentioned anything related to sound and complete proof systems for them. Chapter 8 explains how several sequent calculi for these logics can be constructed. They will recover either the local or the global metainferential validities of these logics—which might collapse, given that certain conditions are fulfilled. We will present different kinds of sequent calculi. First, we will introduce calculi that prove every local metainferential validity of the ST hierarchy. Rea Golan, in [7], presents a kind of traditional, two-sided sequent calculi that achieves this goal. We will also present and explain the labeled sequent calculi for a truth-theory based on the hierarchy of metainferential logics based on ST, originally developed in Cobreros et al. [8]. We will compare them, and see why, from the point of view of an ST-metainferential logician, the latter is a better option than the former. Finally, we will introduce traditional, two-sided sound and complete calculi for every global metainferential validity (defined using single conclusions) from every metainferential logic of any level, based on any of the logics described in the book. The last two chapters of this book shape the most philosophical part of this book, and starts assessing some possible applications. Metainferential logics have a wide range of applications. Probably the most important of them is the way they are used to build theories of truth that deal in a very satisfactory way with semantic paradoxes. If Hjortland [9] is right, and a solution to semantic paradoxes is better if it recovers more classical logic, applying what he calls the minimal mutilation principle, then any of the theories based on the logics belonging to the hierarchy of metainferential logics of the ST hierarchy—but especially STω +, the truth-theory based on the logic that recaptures every classical metainferential validity—fares much better than the truth-theory based on ST itself—which, arguably, is the best non-classical candidate, or at least one of the best candidates non-classical philosophers can bring to the table. Chapter 9 is devoted to explaining these solutions, and why they seem better than the rest. Chapter 10 (the last one in this book, besides the conclusion) revisits the philosophical motivations, applications, and theoretical consequences of the development of metainferential logics: how they force a new characterization of a logic, how classical logic should be understood, how paraconsistency (and paracompleteness) should be defined, what exactly we mean by an empty logic, and whether, and why,

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these logics should count as genuine logics. Finally, we will explore how metainferential logics affect other important problems in the area of philosophical logic, such as the debate between pluralism/monism/nihilism, whether a logic should be descriptive or normative (and in what sense), and what fixes the meaning of the logical constants—specifically, whether metainferences are relevant of not, and which metainferences are important. This chapter provides, then, a more full-fledged answer to the following question: why should we care about metainferential logics? Why are they interesting, and worth our time? But it might not be fair to ask readers to wait until that chapter to know some reasons that justify the interest in metainferential logics. Thus, we will provide some answers right now, which will hopefully meet this demand for now. A first, quick answer is that, on the one hand, some metainferential logics seem to work “better” than any inferential logic as (the base logic for) a solution to some important philosophical problems. TS/ST provides one clear example. Non-classical theories of truth pursue two conflicting desiderata. On the one hand, they seek a paradox-free transparent truth predicate. On the other hand, they want to retain as much classical logic as possible. This conflict has been recently examined in [9]. There, Hjortland claimed that “nonclassical theories try to recapture classical principles in special cases. This is a form of damage control” ([9], p. 1). Hjortland calls this desideratum “the maxim of minimal mutilation.” Thus, although it might be argued that ST seems to do much better than the other inferential non-classical solutions to paradoxes—precisely because it resolves paradoxes while “mutilating” less classical logic than the other non-classical theories—, TS/ST seems to work even better than every other solution based on any inferential logic, and specifically when compared to ST. TS/ST retains every classically valid inference, just as ST does, but, in addition, it recovers every classically valid metainference—while ST loses Cut (and many other classically valid metainferences).6

6

This does not mean that we subscribe to this view. ST loses Cut—and many other classically valid metainferences—, while other non-classical structural solutions to paradoxes, such as LP or K3, retain the usual structural metainferences, despite losing some classically valid inferences— and, therefore, some non-structural metainferences. Nevertheless, neither LP nor K3—nor any subclassical inferential logic, for that matter—can recapture every classically valid metainference, as they lose every metainference with an empty set of premises, and a classically valid, but nonclassically invalid, inference as its only conclusion. Measuring exactly which non-classical solution “mutilates” less classical logic should involve weighing what is more distinctive of classical logic, for example, the set of its valid inferences or its valid metainferences (or if the two things are equally important). (However, note that, while no non-classical inferential logic can recover every classically valid metainference, ST does at least recover every classically valid inference.) As Ripley [1] and Cobreros et al. [10] think ST is just classical logic, they are implicitly defending the idea that a logic is defined in terms of its valid inferences. If this is true, then it is true that ST is more classical than LP or K3. Certainly, this conception of what a logic is can be challenged—and has been challenged, for example, in Barrio et al. [11]. A logic might also be partially defined in terms of its valid metainferences. Someone who supports TS/ST, then, might defend that TS/ST is just classical logic. Nevertheless, this claim can also be challenged in turn by claiming that a logic is also defined in terms of its valid metametainferences, and so on. [11], once again, defend that every metainferential level should be taken into account.

6

1 Introduction

Another example of how metainferential logics work “better” than inferential logics is the case of ST/TS. TS is a logic that has no valid inferences. Nevertheless, TS is informative about metainferences—e.g., TS validates some, but not all, metainferences. But it is a fair question to ask if TS is “as empty” as a logic can be. In particular, could there be a logic without valid metainferences? ST/TS is a logic of that kind. This does not mean that ST/TS is ‘as empty’ as a logic can be—in fact it is not, as the TS hierarchy explored in Chap. 9 shows. But it is certainly “emptier” than TS. The emergence of these logics raised the question of what a logic is, and how it can be characterized. Are all logics informative? Can there be a non-informative logic? What are the extensional properties necessary (and maybe sufficient) for characterizing a logic? Are validities enough, or should we also consider invalidities, antivalidities, contingencies, etc.? Which ones of those other properties—like the ones described by Cobreros et al. in [12]—are relevant, besides validities? Empty and trivial logics shed light on these issues. Another reason for paying attention to metainferential logics is that they have already been applied to some important philosophical problems in the area. These logics have been presented as a new way to characterize a logic (see for example Barrio et al. [11]), as a way to analyze the debate between global and local validity (in Barrio et al. [13]), as a key for a new version of the collapse argument against logical pluralism (as shown by Barrio et al. [14]), as a central feature of new solutions to semantic paradoxes (as in Pailos [15]), and as a useful way of distinguishing between various substructural solutions to semantic paradoxes (as shown in Pailos [15]). Moreover, as Barrio et al. [16] have shown, there is some connection between Priest’s Logic of Paradox (LP) and Cobreros et al.’s Strict-Tolerant approach ST [17, 18]. In particular, it is possible to design a translation between the valid metainferences of ST and the set of LP’s inferential validities. This result was generalized by Barrio et al. [11], showing that a similar structural feature is exhibited between any pair of logics STn /STn+1 belonging to the ST hierarchy. Finally, in Chap. 11 we will conclude with some final remarks and mention some recent developments that were left out of the content of this book. This book is as friendly an introduction to the study of metainferential logics as possible. Throughout the book, we will go through the latest developments in this field and contribute new ones. Continuing the tradition, we will focus on the semantic presentations of these logics, especially on those metainferential logics based on three-valued schemas. However, we hope that the tools introduced in this book can be applied to any logic based on any kind of schema. Also, we aim to provide an overview of the importance and application of metainferential logics to many different philosophical problems. We hope that we have achieved these goals, at least partially.

References

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References 1. Ripley, D. (2012). Conservatively extending classical logic with transparent truth. Review of Symbolic Logic, 5(2), 354–378. 2. Dicher, B., & Paoli, F. (2019). ST, LP and tolerant metainferences. Graham Priest on dialetheism and paraconsistency (pp. 383–407). Springer. 3. Ripley, D. (2018). On the ‘transitivity’ of consequence relations. Journal of Logic and Computation, 28(2), 433–450. 4. Teijeiro, P. (2021). Strength and stability. Análisis Filosófico, 41(2), 337–349. 5. Chemla, E., Égré, P., & Spector, B. (2017). Characterizing logical consequence in many-valued logic. Journal of Logic and Computation, 27(7), 2193–2226. 6. Pailos, F. (2019). A family of metainferential logics. Journal of Applied Non-Classical Logics, 29(1), 97–120. 7. Golan, R. (2022). Metainferences from a proof-theoretic perspective, and a hierarchy of validity predicates. Journal of Philosophical Logic, 51, 1295–1325. 8. Cobreros, P., La Rosa, E., & Tranchini, L. (2021). Higher-level inferences in the strong-Kleene setting: A proof-theoretic approach. Journal of Philosophical Logic 1–36. 9. Thomassen Hjortland, O. (2021). Theories of truth and the maxim of minimal mutilation. Synthese, 199(Suppl 3), 787–818. 10. Cobreros, P., et al. (2014). Reaching transparent truth. Mind, 122(488), 841–866. 11. Barrio, E., Pailos, F., & Szmuc, D. (2020). A hierarchy of classical and paraconsistent logics. Journal of Philosophical Logic, 49(1), 93–120. https://doi.org/10.1007/s10992-019-09513-z 12. Cobreros, P., Tranchini, L., & La Rosa, E. (2020). (I Can’t Get No) Antisatisfaction. Synthese, 1–15. 13. Barrio, E., Pailos, F., & Szmuc, D. (2019). (Meta)inferential levels of entailment beyond the Tarskian paradigm. Synthese. https://doi.org/10.1007/s11229-019-02411-6 14. Barrio, E., Pailos, F., & Szmuc, D. (2018). Substructural logics, pluralism and collapse. Synthese, 1–17. 15. Pailos, F. (2020). A fully classical truth theory characterized by substructural means. The Review of Symbolic Logic, 13(2), 249–268. 16. Barrio, E., Rosenblatt, L., & Tajer, D. (2015). The logics of strict-tolerant logic. Journal of Philosophical Logic, 44(5), 551–571. 17. Cobreros, P., et al. (2012). Tolerant, classical, strict. Journal of Philosophical Logic, 41(2), 347–385. 18. Ripley, D. (2013). Paradoxes and failures of cut. Australasian Journal of Philosophy, 91(1), 139–164. 19. Golan, R. (2023). On the Metainferential Solution to the Semantic Paradoxes’. Journal of Philosophical Logic, 52(3), 797–820.

Chapter 2

About Metainferences

2.1 Two Ways of Understanding Metainferences: As Properties and as Inferences In this book, as the title suggests, we undertake the task of giving a general account of several aspects of metainferences and metainferential logics. Thus, we need to start by providing a precise definition of what a metainference is. As is commonly assumed, we will take an inference to be an ordered pair of collections of formulas of some formal language.1 Moreover, logics are usually identified with a certain set of inferences, which are the valid ones. Therefore, inferential consequence relations are defined over collections of formulas. In the traditional picture, metainferences appear as properties that emerge from the set of valid inferences. This is what we call the notion of metainferences as properties. For instance, Ripley [1, 2] and Cobreros et al. [3] state that metainferences are the properties under which inferences are (or are not) closed under. Thus, for example, let’s consider the following metainference: φ,  ⇒   ⇒ φ,  ⇒ which is the (additive version of the) metainference schema called Cut. According to the view of metainferences as properties, Cut is always related to some set of inferences, i.e., for every set of inferences, if the premises of Cut are in this set, then the conclusion should also be in this set. So, under this perspective, a metainference would be just a closure property of a set of inferences.

1

As we make clear below, this should be better understood as the content of a possible act of inferring. We will also explain why we chose this term instead of other available options—such as argument or entailment.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. Pailos and B. Da Ré, Metainferential Logics, Trends in Logic 61, https://doi.org/10.1007/978-3-031-44381-7_2

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2 About Metainferences

According to the first view, the focus is on inferences, and metainferences are derived from them. However, the second way of understanding metainferences is as syntactic objects in their own right, emancipated, in some sense, from inferences— though closely related to them in another sense. This is the position we call metainferences as inferences, and it was proposed and defended by Dicher and Paoli [4], Barrio, Pailos, and Szmuc [5], and Pailos [6], among others. To understand what metainferences are, first we need to establish what inferences are, or at least to have a clear idea of what they might be. To start with, the Collins English Dictionary provides the following definition: An inference is a conclusion that you draw about something by using information that you already have about it.2

So it seems that an inference—at least in its ordinary use—is a kind of act, or something that resembles an act.3 A more standard view is presented by Boghossian [7]. There, he introduces and defends what he calls “the Taking Condition”: “Inferring necessarily involves the thinker taking his premises to support his conclusion and drawing his conclusion because of that fact” [7, Sect. 3]. Moreover, he argues that every account of inference must explain why this stance is taken as true. The Taking Condition has been rejected by Wright [8], while Broome [9] denies that an account of inference must necessarily pronounce upon it. Finally, Hlobil [10], though accepting both the Taking Condition and Boghossian’s demand that any theory of inference should accommodate it, rejects Boghossian’s interpretation of it as the agent following a particular rule when making the inference. According to Hlobil, the “taking” involved in the Taking Condition is neither an act (different from the act of inferring itself) nor a belief, but a part of the act of reason of inferring. He claims that to infer is to attach a particular inferential force to the content of the act, namely, what he calls “the argument” (in a similar vein as a judgment involves attaching a judgement force to the content of the judgement, a proposition or a surrogate of it). An argument is the structure of contents that is made up of the premises and the conclusion of an inference. To a first approximation, we can think of arguments as ordered pairs of a set of contents and a content. ([10, p.94])

Though we think something in the ballpark of the Boghossian–Hlobil’s way of understanding an inference is correct, we will neither dive deep into the debate they hold with Boghossian’s critics, nor into the details of any particular theory of what an inference is. Moreover, we have no intention to be exhaustive, but just to review some positions we find worth mentioning about what an inference is. As long as the content of an inference is an argument, or something closely related to it, what we 2

In https://www.collinsdictionary.com/dictionary/english/inference. Not everyone agrees, though. In particular, Neta [11], claims that “every inference is simply a judgment with a certain kind of content” [11, p. 404]. This position, however, has several problems. To name just one, inferences seem not to be truth-apt, but judgments clearly are. For an extensive discussion of Neta’s view on inferences, see [10].

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2.1 Two Ways of Understanding Metainferences: As Properties and as Inferences

11

will say about inferences (and in particular, about their content) in the rest of the book seems correct to us. By the content of the inference, we will not be referring to a collection of premises and a single conclusion, as Hlobil does. Instead, we will adopt a usual generalization of this notion, and consider the possible content of an inference (or an abstract representation of that content) as an ordered pair of collections of formulas of some formal language, i.e., we are admitting multiple conclusions. Finally, we do not intend to represent inferences with (the formal representation of) arguments. If that were our goal, something in the argument should aim at playing the role of what Hlobil calls “the inferential force”. And even though that might be done, we will not explore that path, as we intend to focus on the content of the inference itself—mainly through its formal representation. For the sake of convenience, we will use the term “inference” to refer to the content of an inference, and not—unless otherwise specified—to the act of inferring.4 With this we follow an established use in the literature, as this is the terminology used, for instance, by Dicher and Paoli [4], Barrio et al. [5], [12], Scambler [13], Ripley [14] and others.5,6 When talking about “metainferences”, then, we will also be referring not to a possible act of reason, but to its content. From this viewpoint, metainferences are just like inferences but, instead of relating sets of formulas, they relate sets of inferences or sets of metainferences. But what are, exactly, these types of acts of reason similar to inferences, but with a different content? There is no only way of understanding them, but, in any case, how they are interpreted depends on the particular conception about metainferential validity that is adopted (and there are many ways of understanding it). When making explicit the different approaches to metainferential validity that exist (or at least the main ones), we will provide philosophically relevant ways of understanding them. But if logics are sets of inferences, and if metainferences are just like inferences, then there is a place for a logic of metainferences, or a metainferential logic. Throughout this book, we will take this stance and study metainferences, in the more general way, as inferences, and metainferential logics as determining sets of valid metainferences. However, from a technical point of view, it is common to find in the literature two different frameworks which implement this idea. The first, which can be called S E T − M E T framework, defines a metainference as a relation between a set of metainferences and a particular metainference. This framework is adopted e.g. in 4

Moreover, we will use this term to refer both to the content of an inference and to its formal representation, hoping that the context helps disambiguate the reference. 5 This is not the only option, though. As we already mentioned, Hlobil uses the term “argument” for what we call “inference”, while Elia Zardini [15], prefers to refer to the contents of inferences as “entailments”. 6 An anonymous reviewer pointed out to us that it is misguided to think of inferences and metainferences in this syntactic way. She claims that they are, at best, represented through syntactic objects. We hope that it is now clear that we are not stating that inferences and metainferences are not mental acts or performances, but that the content of those acts might be this type of syntactic entities, or represented by these syntactic entities.

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2 About Metainferences

Barrio, Pailos and Szmuc, and Pailos [5, 6, 12]. Secondly, in the S E T − S E T framework, a metainference is defined as a relation between sets of metainferences. This is the framework used e.g. in Scambler and Da Ré, Pailos, Szmuc, Teijeiro [13, 16]. Interestingly, the preference for one of these two frameworks is only justified by practical considerations, and no one (including the authors of this book) has advanced any philosophical justification for preferring one or the other. However, in view of the fact that the S E T − S E T framework is more general and treats metainferences as relations between collections (in the same vein as inferences are defined), throughout this book we will assume the S E T − S E T framework. Nevertheless, since in the literature many of the results have been provided for the S E T − M E T framework (especially regarding sequent calculi—see Chap. 8), we will also present the technical definitions considering this framework, and we will explicitly state when we assume it.

2.1.1 Metainferences: A Technical Definition Let’s start this technical section by defining what an inference is. Given a language L, we can define an inference as follows: Definition 2.1.1 An inference on L is an ordered pair , , where ,  are two collections7 of formulas (written  ⇒0 ). I N F(L) is the set of all inferences on L. For the sake of simplicity, sometimes I N F(L) in Definition 2.1.1 will be denoted by M E T A0 (L). So although we need to formally distinguish between inferences and metainferences, in a loose way we can think of inferences as metainferences of level 0. Also, when no confusion arises, we drop the subscript and simply write  ⇒  for denoting inferences. Given an inference  ⇒ , we call  its premises and  its conclusions. The former are read conjunctively, while the latter should be interpreted disjunctively. Once we have this formal definition of inference, we can define what a metainference is: Definition 2.1.2 A metainference of level 1 on L is an ordered pair , , where  and  are sets of inferences (written  ⇒1 ). M E T A1 (L) is the set of all metainferences of level 1 on L. Here, we have defined a metainference as an inference between two sets of inferences. However, this is just one particular class of metainferences—say, metainferences of level 1. But we can inductively define metainferences of any finite level n, as follows: 7

The most usual way of considering these collections is as sets. However, we will not specify what these collections are in this definition, since sometimes we will need to work with more fine-grained collections, as multisets. When circumstances require it, we will be explicit about this topic.

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13

Definition 2.1.3 A metainference of level n + 1 on L is an ordered pair , , where  ⊆ M E T An (L) and  ⊆ M E T An (L) (written  ⇒n+1 ). M E T An+1 (L) is the set of all metainferences of level n + 1 on L. As in the case of inferences, given a metainference , , we call  its premises and  its conclusions. Once again, the former are read conjunctively, while the latter should be interpreted disjunctively. So far, we have been talking about inferences and metainferences as tokens. However, before moving forward it is important to make a precise distinction that will be useful throughout the whole book: the difference between tokens and schemas. Recall that an inference token is a pair ,  ⊆ F O R(L) × F O R(L). An inference schema is defined as the set which contains all and only the inference tokens that can be obtained from one of its members—its “basic instance”—by uniformly substituting some propositional variable p in it by some formula ϕ. Also, we will say that an inference schema with contexts is the union of a simple inference schema ρ with a subset of { ∪ ,  ∪  | ,  ∈ ρ}. Metainference schemas can be defined in the same straightforward fashion. Now, it is easy to adapt the definitions given for the S E T − S E T framework to the S E T − M E T framework. More formally, given a language L, we can define an inference as follows: Definition 2.1.4 (Dicher and Paoli [4]) A metainference of level 1 on L (in the S E T − M E T framework) is an ordered pair , φ, where  is a set of inferences, and φ is a particular inference (written  ⇒1 φ). M E T A1 (L) is the set of all metainferences of level 1 on L. Definition 2.1.5 (Dicher and Paoli [4]) A metainference of level n + 1 on L (in the S E T − M E T framework) is an ordered pair , φ, where  ⊆ M E T An (L) and φ ∈ M E T An (L) (written  ⇒n+1 φ). M E T An+1 (L) is the set of all metainferences of level n+1 on L. Once we have stated that metainferences are inferences, we would like to characterize metainferential logics. In order to do so, we need to understand what it means for a metainference to be valid. As Teijeiro [17] has shown in the S E T − M E T framework for the case of metainferences of level 1, the distinction between tokens and schemas has a crucial role when the notion of validity comes into play (and straightforwardly the same happens in the S E T − S E T framework). In the next section, we will explore the three notions of validity that have been considered in the literature.

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2 About Metainferences

2.2 Three Ways of Understanding Metainferential Validity In this book, we will work with particular mixed logics and hierarchies of mixed metainferential logics.8 However, the idea of this section is to be neutral and general about the definitions of validity, regardless of the logics involved. Even though an obvious shortcoming of pursuing this path is the arduousness of being too abstract, it has some major benefit: that of being applicable to any mixed logic whatsoever. Those who find the material on this section too general can wait until the next chapter, when we will introduce logics we are mostly interested in, and all these definitions will be applied. There are three notions of validity for metainferences we will take into account here: Global, Local and Absolute Global. As far as we know, these were introduced and discussed (with a slightly different terminology) by Garson [18, 19] and discussed among others by Humberstone [20], only the first two were much discussed recently, e.g., Dicher and Paoli [4], Barrio et al. [5], Da Ré, Pailos, Szmuc, Teijeiro [16]. However, also recently, Da Ré, Szmuc and Teijeiro [21] have presented some results regarding the last one. So, we will start with the Local and the Global notions and then we will introduce the notion of Absolute Global validity. Before moving on to the metainferential concepts, we need to start by making explicit some technicalities regarding mixed inferential logics. In [22], Chemlá, Egré and Spector introduce the notion of mixed consequence relation,9 in the following way: Definition 2.2.1 An inference  ⇒  is valid in a mixed logic S1 /S2 if and only if, for every valuation v, if v satisfies γ according to the standard S1 for every γ ∈ , then v satisfies δ according to the standard S2 for some δ ∈ . Although there are many ways of understanding the notion of standard, here we will assume that it is just a set of truth-values. For any standard Si , valuation v and formula ϕ, v satisfies ϕ according to the standard Si if and only if v(ϕ) ∈ Si . Standards for premises and conclusions can be interpreted as specifying which values each formula in a sound argument or inference can take. Notice that we use boldface for logics and italics for standards. Also note that, by varying the set of valuations, a pair of standards can determine different logics. Finally, when no confusion arises we use standards and logics interchangeably.

8

There may be metainferential logics that cannot be considered mixed logics. See, for example, Schroeder-Heister’s [23], and Priest and Wansing’s [24]. In this book, however, we will only focus on mixed logics. 9 Q-logics and p-logics constitute two important subsets of these kinds of logics that have been previously introduced in articles by Malinowski [25] and Frankowski [26]. Nevertheless, the generalization and the label are originally from [22].

2.2 Three Ways of Understanding Metainferential Validity

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Further specifications are called for. If S1 = S2 , then the mixed consequence relation is called pure, and the set S1 is called a set of designated values. But if S1 = S2 , then the mixed consequence relation is impure.10 From a pair of standards for formulas S1 and S2 , we can now build a new standard S = S1 /S2 , which is not a set of values, but a pair of sets of values. This new standard can be used to determine when a valuation satisfies an inference (according to it) as follows: Definition 2.2.2 A valuation v ∈ V satisfies an inference  ⇒  according to a standard S (v  S  ⇒ ) if and only if it is not a counterexample of it. Since a set of valuations and an inferential standard S are sufficient to determine one inferential logic L (given a language), when no confusion arises we replace the standard for the corresponding logic, and denote it as follows: v L  ⇒ . Of course, as we will see in future chapters, we will define different counterexample relations that will vary depending on the consequence relation and the set of valuations.11 These mixed inferential logics are based on satisfaction standards, which specify when a particular inference is satisfied in a valuation. In what follows, it will be important to keep them apart from the things they validate—which is another way to (extensionally) define a logic.12 If we change the reference from inferences to metainferences, we obtain some definitions of mixed (either pure or impure) metainferential logics—i.e., logics whose consequence relation is characterized by a satisfaction standard that specifies when a metainference is valid. Let’s start by defining global and local mixed metainferential logics: Definition 2.2.3 A metainference of level 1  ⇒1  is locally valid in (according to) a mixed metainferential logic S1 /S2 of level 1 (defined over a set of valuations V ), if and only if, for every valuation v ∈ V , if v satisfies each inference γ according to the standard S1 , for every γ ∈ , then v satisfies some δ according to the standard S2 , for some δ ∈ . 10

More subtleties have been introduced. For example, if S1 and S2 do not have any elements in common, then the mixed consequence relation is disjoint. Disjoint logics based on Strong Kleene semantics have been explored by Pailos [27]. A systematic study of every pure Strong Kleene logic is given in [28]. 11 Following Scambler [13] we could provide a more formal definition of satisfaction of an inference as follows. Let Inf(L) be the set of all inferences. A validity notion for inferences is a function V : val × MInf n → {1, 0} where val ⊆ V al. We say that val is the validity space of V, where V tells you which valuations in val satisfy which inferences. However, for the sake of simplicity, we will avoid this terminology throughout the book. 12 A similar distinction has been introduced in the context of metainferential logics by Dave Ripley in both [14, 29]. Though he prefers to talk about counterexample relations instead of satisfaction standards, the results he presents are easily adaptable to our setting. Moreover, if a valuation either satisfies or is a counterexample of any sentence, inference or metainference according to a given standard, but not both, then the two notions are equivalent.

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2 About Metainferences

As before, from a pair of standards for inferences S1 and S2 , we can now define a new standard S = S1 /S2 which can be used to determine when a valuation satisfies a metainference of level 1 (according to it) as follows: Definition 2.2.4 A valuation v ∈ V satisfies a metainference of level 1  ⇒1  according to a standard S (v  S  ⇒1 ) if and only if it is not a counterexample of it. As for the inferential case, since a set of valuations and a metainferential standard S are sufficient for determining one metainferential logic of level 1 L (given a language)—i.e., a set of locally/globally valid metainferences, when no confusion arises we replace the standard for the corresponding logic, and denote it as follows: v L  ⇒1 . Now we can define satisfaction of a metainference of level n > 1 by a valuation.13 Definition 2.2.5 A valuation v ∈ V satisfies a metainference of level n  ⇒n  according to a standard S (v  S  ⇒n ) if and only if it is not a counterexample of it. Notice that according to a satisfaction standard for metainferences of level n S = Si /S j (where Si and S j are metainferential standards of level n−1) a given valuation v satisfies a metainference  ⇒n  if and only if, if v satisfies γ according to Si for every γ ∈ , then v satisfies δ according to S j for some δ ∈ . Definition 2.2.6 A metainference of level n > 1  ⇒n  is locally valid in (according to) a mixed metainferential logic S1 /S2 of level n (defined over a set of valuations V ), if and only if, for every valuation v ∈ V , if v satisfies γ according to S1 , for every γ ∈ , then v satisfies δ according to S2 , for some δ ∈ . Definition 2.2.7 A metainference of level n ≥ 1  ⇒n  is globally valid in a mixed metainferential logic Si /Sj of level n (defined over a set of valuations V ), if and only if, if for every valuation v ∈ V , v satisfies γ according to S1 for every γ ∈ , then for every valuation v ∈ V , v must satisfy δ according to S2 , for some δ ∈ . As before, since a set of valuations and a metainferential standard S are sufficient for determining one metainferential logic of level n L (given a language)—i.e., a set of locally/globally valid metainferences—, when no confusion arises we replace the standard for the corresponding logic, and denote it as follows: v L  ⇒n .14 13

As we mentioned for the notion of satisfaction for inferences, following Scambler [13] here we could also provide a formal definition. However, for our purposes, it is enough with this simpler definition. 14 The terms global/local are taken from modal logic, as mentioned by Garson [19, p. 18]. Also, it is connected to supervaluationism. Recall that in supervaluationism there are valuations and models as sets of valuations. Supertruth is defined as true in every valuation of the model. Thus, true preservation in a valuation is called local validity, while supertrue preservation is called global validity (see Williamson [30, p. 148]). Although all these topics can be more deeply related, we leave these considerations for future research. We would like to thank an anonymous reviewer for pointing this connection to us.

2.2 Three Ways of Understanding Metainferential Validity

17

It is worth emphasizing that, in these definitions, the standards are not sets of truth-values anymore, but criteria that specify when a valuation satisfies a given (meta)inference and when it is a counterexample of it. Notice, in particular, that every mixed (meta)inferential logic provides such standards. Also, since S1 and S2 are satisfaction relations, they also determine logics, but of level n − 1, say L1 and L2 . So, when presenting a metainferential logic L of level n, we sometimes denote it L1 /L2 meaning that L1 provides the standard for the premises and L2 for the conclusions. In the case that n = 1, L1 and L2 are mixed inferential logics. Also, every logic of level n determines all the higher levels and also all the lower ones in the following way. Given a logic L of level n we assume that the logic of level n + 1 determined by L is L/L, and so on. On the other hand, for every logic of level n, L = L1 /L2 we assume that the logic of level n − 1 determined by L is L2 .15 Generalizing again the definitions for inferential logics, when a mixed metainferential logic L = L1 /L2 is such that L1 = L2 we say that the metainferential logic is pure. Otherwise, it is impure. For example, classical logic CL is considered to be a pure metainferential logic, since by convention it is assumed to provide the same standard for premises and conclusions, i.e., that of CL.16 We will now provide some explanations of the global and local notions of validity. When working in a S E T − M E T framework, the global notion states that a metainference is valid whenever it preserves validity from its premises to its conclusion (validity of the premises in L1 implies validity of the conclusion in L2 ). In the more general S E T − S E T setting it is a bit different, and there is not a simple way of expressing it. Notice that the property specified in the definition that we have provided cannot be paraphrased in terms of preservation of validity from premises to conclusions. Therefore, there is another natural generalization of global validity in the context of the S E T − S E T framework, which consists in the following one: Definition 2.2.8 A metainference of level 1 is globally2 valid in a mixed metainferential logic L = L1 /L2 of level 1, if and only if some premise is invalid in L1 or some conclusion is valid in L2 . A metainference of level n > 1 is globally2 valid in a mixed metainferential logic L = L1 /L2 of level n, if and only if some premise is globally2 invalid in L1 or some conclusion is globally2 valid in L2 . Notice that global and global2 validity are not equivalent. Consider for instance the following metainference token of level 1:

15

Using Dave Ripley’s terminology in [29], when talking about a logic L of level n we demand that it is both upward closed from level n on, and also downward closed at every level. Moreover, every logic we will be talking about determines a satisfaction criterion for every metainferential level, thus being a full logic, once again in Ripley’s terms. We will come back to these issues—and to Ripley’s article—in Chap. 10. 16 Notice that, as we mentioned in the previous paragraph, the metainferential logic of level 1 determined by CL is CL/CL, that of level 2 is CLCL/CLCL, and so on, and for simplicity we just name it CL assuming all of the higher levels.

18

2 About Metainferences q⇒q ⇒p p⇒

This is such that the premise is obviously valid in classical logic (and of course this means that every Boolean valuation satisfies the premise), but none of the conclusions is classically valid. So the metainference is globally valid in the pure metainferential logic CL, since every Boolean valuation satisfies some of the conclusions, but it is not globally2 valid. This distinction only makes sense once we allow multiple conclusions: in the SETMET framework global validity is understood as preservation of (global) validity, leaving no room for further innovations. In other words, global validity in the SETMET framework is just a degenerate case of global2 validity. In this book, we will usually take global validity as the way to go, mainly due to the collapse results that we will present in the next section (which cannot be applied to the second notion). However, we will revisit global2 validity in Chap. 7. Going back to global validity, as a consequence of its definition, any metainference of level 1 with invalid premises (not meeting the standards of the premises in every valuation) is globally valid. Thus, some theorists have opted to work with a more fine-grained notion, the local one. The idea is that metainferences should preserve satisfaction by each valuation. Then, one should think about a locally valid metainference as a metainference such that any counterexample of all of the conclusions is a counterexample of some of the premises. The notion of local validity is in some sense finer than the notion of global validity as we claimed before. We will illustrate this claim with an example. Let’s consider the following metainference token: ⇒p ⇒q Now take again the well-known classical logic CL. We know that ⇒ p is not valid in CL. This metainference will be globally valid in CL (or CL/CL if you prefer), because not every valuation will satisfy the premise. However, there is a Boolean bivaluation that satisfies ⇒ p and dissatisfies ⇒ q according to CL (i.e., v( p) = 1 and v(q) = 0). So the metainference is locally invalid in CL. More generally, under some plausible assumptions, any metainference which is locally valid is globally valid, but there might be some globally valid metainferences which are locally invalid. As we will see, under certain conditions, at the first metainferential level, the two concepts collapse. Before presenting the collapse results, let’s briefly consider the third notion: the Absolute Global. Definition 2.2.9 Let V be a set of sets of valuations. A metainference of level n ≥ 1  ⇒n  is absolutely globally valid in a mixed metainferential logic L of level n with respect to V if and only if it is globally valid according to L in each set of valuations V ∈ V. Looking at the definition, it is not hard to realize that absolute global validity seems to be an even finer conception of metainferential validity than the local and

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19

the global ones.17 One could wonder what the exact relation between these three notions is. In the next section, we will expand on this.18

2.2.1 Comparing the Three Notions Firstly, it is worth noting that, when working with tokens, the local notion is finer than the global one. In formal terms: Theorem 2.2.10 Let  ⇒n  be any metainference token of level n. If  ⇒n  is locally valid according to L (any mixed metainferential logic defined over a set of valuations V ), then it is globally valid. Proof Suppose  ⇒n  is globally invalid. Then, every valuation satisfies the premises  but there is a valuation v ∈ V which dissatisfies all the conclusions. Thus, the valuation v is a witness of the local invalidity of the metainference.  And also, it is easy to extend the result for metainference schemas: Theorem 2.2.11 Let  ⇒n  any metainference schema of level n. If  ⇒n  is locally valid according to L (any mixed metainferential logic defined over a set of valuations V ), then it is globally valid. Proof Similar to the proof of Theorem 2.2.10.



However, in [17], Teijeiro showed some conditions under which both notions collapse in a S E T − M E T setting and for metainferences of level 1. In a nutshell, she showed that, to ensure collapse, it is sufficient that every truth-constant should be 17

Although we will not go into the details in this book, it is worth mentioning that there are some relations between the notions of validity and very well-known proof-theoretical notions, i.e., the global notion relates to admissibility in a sequent calculus and absolute global validity coincides with derivability. There are some assumptions for these statements to be true; see, e.g., Da Ré et al. [21] for a comprehensive study of the relations between these notions. In that article, Da Ré et al. claim that the notion of absolute global validity lacks an intuitive reading. Nevertheless, they present in a footnote the following way to interpret this notion: An anonymous reviewer suggested the following interpretation of absolute global validity. A given set of metainference schemas S determines a set of valuations, which can be seen as a set of “good” or acceptable scenarios, and adding any axioms constitutes a way of restricting those valuations (alternatively, those acceptable scenarios). Whence, the reviewer suggests that assessing the derivability of a given metainference would amount to quantifying over all those valuations determined (alternatively, those scenarios deemed “good”) by the joint force of S and its premise inferences, and assessing whether in all these cases the conclusion inference of such a metainference is valid (respectively, considering whether these are “good” scenarios too). ([21, p. 1547 fn.19])

18

The following remarks are inspired by Teijeiro [17] and Da Ré et al. [21]. However, in those articles, the authors only focus on the S E T − M E T setting, and in metainferences of level 1.

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2 About Metainferences

able to be expressed in the language and that schemas should be considered instead of tokens. We generalize it for the S E T − S E T framework19 : Theorem 2.2.12 Let V ⊆ {v | v : F O R(L) → {a1 , . . . , an }} be a set of valuations and L a propositional language that contains a truth-value constant ai for each truthvalue ai . Then a metainference schema of level 1  ⇒1  is locally valid according to L (i.e., any mixed metainferential logic defined over the set V ) if and only if it is globally valid. Proof The proof needs both directions: (locally valid ⊆ globally valid) Assume there is a metainference schema  ⇒1 —such that it is not globally valid. Then, there is an instance of the schema let’s call it   ⇒1  such that it is not globally valid, i.e., every valuation satisfies all the premises but not every valuation satisfies all conclusions. Thus, any of the latter valuations shows that the metainference token is not locally valid. (globally valid ⊆ locally valid) Assume there is a metainference schema  ⇒1  which is not locally valid. Then, there must be an instance of the schema— let’s call it   ⇒1  —and a valuation v such that the valuation v satisfies all the premises but it is a counterexample of all the conclusions. Since in the language we have truth-constants, let’s build a new metainference from   ⇒1  and v, by replacing in   ⇒1  each propositional letter p by ai , where v( p) = ai . Notice first that the metainference so built is an instance of the metainference schema  ⇒1  and, since it is built by using truth-constants, it is globally invalid. This implies that the very metainference schema is globally invalid.  Also, this can be generalized for any level: Theorem 2.2.13 Let V ⊆ {v | v : F O R(L) → {a1 , . . . , an }} be a set of valuations and L a propositional language that contains a truth-value constant ai for each truth-value ai . Then a metainference schema of level n  ⇒n  is globally valid according to L (i.e., any mixed metainferential logic defined over the set V ), if and only if it is locally valid. The proof of this theorem is easily adaptable from the proof of Theorem 2.2.12. Regarding global2 validity, Bas Kortenbach [31] has shown that the following schematic metainference of level 2: [⇒ φ] ⇒1 [⇒ ψ] ∅ ⇒1 φ ⇒ ψ

is globally2 invalid (i.e. globally invalid in the S E T − M E T framework), but it is locally valid in classical logic. Thus, the desired collapse result between both notions cannot be obtained.20 19

As a reviewer correctly points out, this is due to the fact that this global notion of metainferential validity is defined in a kind of local way on the right side. 20 The collapse can be recovered if, instead of considering this notion of global validity, one defines a different concept mixing local and global validity. Interesting as it is, we leave this whole matter for another discussion (see [31] for further details).

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21

So far, we have compared the local and the global notion. Now we would like to make a comparison with the absolute global notion. However, for this purpose we need to make some comments. Given that the absolute global notion is defined for a set of sets of valuations, and the local and the global notions are defined for a set of valuations, it is not easy to establish how to compare them. One natural way is to consider the power set of the set of valuations used in the global and the local definition, but this is not the only possible choice. In any case, once we fix some set of valuations V to define the local and the global concepts, the choices from which absolute global validity can be defined are obtained by taking the power set of the power set of V . We get then a lattice of options, where the maximum is the power set ℘ (V ), and the minimum is the empty set. Any of the elements of the lattice can be used to provide a definition of absolute global validity. However, notice that, for some elements of the lattice, the union of all its members will not be equal to V . Thus, defining local and global validity over it will not be equivalent to defining it over V . However, the union of the maximum is of course V , and therefore if we define absolute global validity over ℘ (V ), we can prove the following result21 : Fact 2.2.14 Let V ⊆ {v | v : F O R(L) → {a1 , . . . , an }} be a set of valuations and L a propositional language that contains a truth-value constant ai for each truthvalue ai . Then, a metainference schema  ⇒1  is globally valid in L (i.e., any logic defined over the set V ) if and only if it is absolutely globally valid in L (defined over ℘ (V )). Proof We need to prove both directions: (absolutely globally valid ⊆ globally valid) Assume  ⇒1  is globally invalid in L (defined over V ). Since V ∈ ℘ (V ),  ⇒1  is also absolutely globally invalid in L (defined over ℘ (V )). (globally valid ⊆ absolutely globally valid) Assume  ⇒1  is absolutely globally invalid in L (defined over ℘ (V )). Therefore, there is an instance of it   ⇒1  and a set of valuations Vi ⊆ V such that each v ∈ Vi satisfies all of the premises of   but dissatisfies all the conclusions in  . So, take one of these valuations, say vi and define a new metainference token   ⇒1  , uniformly substituting in   ⇒1  each propositional letter p occurring in it by ak , with vi ( p) = ak . Now, it is easy to see that   ⇒1  is also an instance of  ⇒1 , such that in every valuation of V , all of the premises are satisfied, but the conclusions are not.  Therefore,  ⇒1  is also globally invalid in L (defined over V ). Now, we can obtain a similar result regarding local validity: Fact 2.2.15 Let V be a set of valuations. The set of metainferences which are absolutely globally valid in L (defined over ℘ (V )) is identical to the set of metainferences which are locally valid in L (defined over V ). 21

We adapt this and some of the results in this section from [21].

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2 About Metainferences

Proof Assume a metainference is locally invalid in L (defined over V ). Thus, there is a valuation v ∈ V satisfying the premises but not the conclusions. Take the set {v}. This set belongs to ℘ (V ), and the metainference does not preserve validity in {v}. Thus, it is not absolutely globally valid in L (defined over ℘ (V )). Assume a metainference is absolutely globally invalid in L (defined over ℘ (V )). Thus, there is a Vi ⊆ V such that the metainference is globally invalid in L defined over Vi . Hence, there is a v ∈ Vi such that it is a counterexample of the conclusions but not of the  premises. But v ∈ V and thus the metainference is not locally valid either.22 So, in general, when we define absolute global validity over the power set of a given set of valuations, absolute global validity and local validity will coincide. For the purposes of this book, this will be the case in general, so we can dispense with this complex notion and use the local one. However, it is important to mention that this is not the case in general.23,24 So far we have compared the three notions from a technical point of view. However, in the literature, most metainferential logics are based on a local way of understanding metainferential validity. But, is there anything philosophically prominent about this notion regarding the global one?25 There seems to be some consensus in answering this question in the affirmative. The main advantage is the following: the local notion of metainferential validity provides a uniform account of validity that includes inferences and metainferences. Assuming that metainferences of any level are also inferences, but just that they do not relate sets of sentences, but sets of (meta)inferences (of lower levels, but of course these kinds of details will change if we switch from a non-cumulative to a cumulative way of understanding them), they should all be understood in a uniform way regarding their most important properties. And the way their (in)validity is defined is certainly one of them. Regarding traditional inferences, there seems to be no other reasonable candidate than the local way of interpreting them: as preservation of satisfaction in every valuation. For instance, in propositional classical logic, one usually understands 22

As a reviewer noticed, the proof is similar to the one in [32] that establishes that supervaluationism has the same single-conclusion valid inferences as classical logic for single conclusions. (Also, a similar proof can be found in [33].) 23 See [21] for more details. 24 Very recently, Kortenbach [31], has argued that these are not the only notions of metainferential validity that exist and are worth exploring. In particular, he distinguishes two different ways in which a metainference of level n can be globally valid. As we mentioned, the notion of global validity in the S E T − M E T framework is usually taken to represent the idea of preservation of validity from premises to conclusions. But, which notion of validity should we adopt? How should we evaluate the validity of premises and conclusions? Kortenbach considers two main options: (a) either we evaluate validity locally, or (b) we evaluate it globally. We are mostly working in a S E T − S E T framework. We leave for future work to find correlates of these notions in this context. (We would like to thank Bas Kortenbach for discussing some of these ideas with us.) 25 For the comments that follow, we leave aside the notion of absolute global validity, since it does not seem to have any philosophical interest, other than that of being an interesting technical artifact. However, we leave it for future work to investigate whether some intuitive reading of this notion could be developed.

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23

validity as preservation of truth in every valuation. This is, of course, a local way of understanding validity. On the other hand, a global way to interpret inferential validity demands, of a valid inference with one conclusion, preservation of validity from premises to conclusion. This has undesirable consequences; for instance, the following inference p ⇒ q would be valid in any reasonable logic, including, of course, classical logic (because p is not a tautology). So, for inferences, the usual way of understanding validity is in a local way. Therefore, if we extend the usual inferential notion to the metainferential level, validity is defined as preservation of satisfaction (in every valuation) between metainferences. Thus, in order to provide a uniform account of validity including inferences and metainferences, the most reasonable approach is to extend the inferential notion to the metainferential level, and the result of doing that is to embrace a local notion of metainferential validity. We started this chapter by describing two ways of considering metainferences: as closure properties of sets of inferences and as inferences between inferences. The global way of understanding metainferential validity is closely related to the first conception (see, e.g., [19–21]). On the other hand, once we consider metainferences as inferences and we intend to give a uniform definition of validity for them, the global reading yields undesirable consequences, as mentioned before. Thus, the local reading (or the absolutely global one) is the way to uniformly understand validity for metainferences. There are still some ways to argue for global validity. For instance, Teijeiro [17], rejects this view for many reasons. We will evaluate what we think are the two main ones. It is worth mentioning that Teijeiro’s arguments are formulated for the S E T − M E T framework, and so there is just one global notion. Firstly, she claims that what is usually considered a huge disadvantage of the global notion of metainferential validity, i.e., that metainferences like the following ⇒p ⇒q turn out valid, in (almost) any logic, is not a flaw. On the one hand, because it is not the case that validating many counterintuitive metainferences makes a logic metainferentially trivial, and purportedly this is the only reason for rejecting logics like these ones. On the other hand, this kind of behavior is not at all a flaw, as classical logic itself validates many counterintuitively valid inferences—e.g., inferences like p ⇒ q → p are classical validities. Secondly, Teijeiro claims that the notion of satisfaction of a (meta)inference by a valuation is technical, and does not correspond to any intuitive notion that plays an important role in our research, scientific or everyday epistemic practices, unlike the notion of truth. Thus, as the local notion depends on it, but the global one does not manifest such dependence, then the local notion of metainferential validity seems to be less appealing than the global way of understanding it. Regarding the first criticism, though, adopting a global notion of metainferential validity does not lead to triviality indeed, but it does lead to a set of valid inferences describable as trivial relative to some particular set of sentences. There are many

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2 About Metainferences

ways of being trivial, and validating every (meta)inference is just an extreme form of triviality. This is pretty much the same criticism as the one Urbas [34], introduces. In his article, Urbas criticizes different paraconsistent theories for presenting diverse forms of triviality. For example, Urbas mentions that Minimal Logic validates every instance of explosion with a negated formula as a conclusion, and other logics validate every inference with a conditional as a conclusion, and so on. According to Urbas, though these different forms of triviality are not as bad as validating every inference, they should nevertheless be avoided, if possible, as they seem not to be an adequate way of reflecting on the set of sentences they are trivial about. Similarly, as we have already mentioned, we can describe the set of globally valid metainferences of a logic as being trivial relative to invalid premises. And if Urbas’ forms of triviality should be avoided, we cannot imagine why this metainferential form of triviality should not be avoided if possible, too. As we mentioned, Teijeiro also noticed that this particular form of triviality is a feature of many logics, including classical logic. Nevertheless, this can be turned upside down and described, as in fact it has been, as a flaw of classical logic. For example, inferences and sentences known as paradoxes of material implication—which have things like p ⇒ q → p as one of them—have motivated the development of relevant logics, while what many consider the unsoundness of Explosion or Excluded Middle has also favored the development of paraconsistent and paracomplete logics. Of course, the soundness of, for example, the paradoxes of material implication can be defended. But it seems at least strange to claim that recovering them is, prima facie, a desideratum of a logic. Pretty much the same can be said regarding metainferences like ⇒ p ⇒1 ⇒ q, which is globally valid in classical logic. Moreover, there is some initial plausibility behind the paradoxes of material implication, Explosion and Excluded Middle, which we, at least, have not found here. Regarding the second point, we think that Teijeiro’s argument arises from a pretty extended confusion, which is the following. It is a mistake to interpret inferential validity, understood as truth preservation, as making valid every inference that preserves truth. Instead, it validates inferences that preserve truth in a valuation. If the former were the case, then inferences like “Tübingen is the German capital city. Therefore, the grass is not green” should be regarded as (natural language) inferential validities, as they preserve truth from the premises to the conclusion (just because the only premise is not true). But they are not, as they should, because the relevant notion to evaluate inferential validity is not truth, but truth in a valuation—or maybe truth in a possible situation. And there are valuations that treat the premise of that inference as true and the conclusion as false. But true in a valuation is not the same notion as being true simpliciter, which is an informal and important notion. True in a valuation is as technical as being satisfied in a valuation. Thus, local metainferential validity cannot be dismissed just because it is based on it—at least if we are not willing to discard also the traditional (local) notion of inferential validity. Moreover, though there seems to be no obvious way of understanding local metainferential validity—in contrast to (the single-conclusion version of) global validity, which can be interpreted as preservation of validity—, we think there is an interest-

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25

ing informal way of understanding local metainferential validity. A metainference  ⇒n  (for 1 ≤ j > ω) is locally valid according to a standard of metainferential validity of level n, P/C, if and only if, for every valuation v, if v is not a counterexample of γ (for every γ ∈ ) according to P, then v is not a counterexample of δ (for no δ ∈ ) according to C. We think that a fruitful reading is the following: if the situation represented by v does not discard any premise (because the situation is a counterexample to the (meta)inferential validity of them)—according to the standard relevant for evaluating premises—, then the situation represented by v does not discard every conclusion (because the situation is a counterexample to the (meta)inferential validity of them)—according to the standard relevant for evaluating conclusions. This negative flavor of what is preserved from premises to conclusions is not new, and actually resembles one way of characterizing validity LP, i.e., as preservation of non-falsity. In this case, what is being preserved is the valuation not discarding the metainference’s (local) validity. Certainly, the “preservation” metaphor should be softened, as the standard required to evaluate whether a given inference or metainferece is or is not discarded by the data (i.e., by a given valuation) might vary from premises to conclusions. But this is just a distinctive feature of mixed logics. One last thing regarding Teijeiro’s second criticism is that when we move to the S E T − S E T framework and allow multiple meta-conclusions, things are not so straightforward. As we have mentioned in this Chapter, the most coherent way of reading global metavalidity is the following: a metainference is globally valid if and only if either some valuation doesn’t satisfy some premise or, for every valuation, some conclusion is satisfied. This means that, in this framework, global validity also needs a technical appeal to valuations or interpretations since it is not equivalent to preservation of validity anymore. So someone could still argue that the correct notion is global2 validity: a metainference is globally valid if either some premise is (globally2 ) invalid or some conclusion is (globally2 ) valid. However, this new notion has some counterintuitive consequences regarding its relation with local validity: as we mentioned, the collapse results presented do not apply, and even worse, for some logics there are locally valid metainferences of level 1 that are globally2 invalid. There is one final interesting criticism that has (recently) been raised against the global notion of metainferential validity. It has been presented by Golan [35]. He proves that, while the local notion meets the substitution requirement, the global notion fails to meet it. That means that, if a metainference is locally valid, so is any metainference obtained from it by uniformly substituting sentences for other sentences. But not every globally valid metainference can be uniformly replaced while guaranteeing that the metainference obtained from this procedure is also globally valid. He considers the following metainference: p⇒q ⇒ A ∧ ¬A where p and q are propositional letters, while A can be any sentence that does neither include p nor q as subformulas. Now consider any logic with a satisfaction standard (*)

26

2 About Metainferences

for premises such that any instance of Identity—i.e., φ ⇒ φ—is satisfied by any valuation according to that satisfaction standard. The logic must also make ⇒ A ∧ ¬A antivalid, i.e., it is dissatisfied by every valuation.26 If the logic is such that p ⇒ q is not valid (or, actually, satisfied by every valuation according to the satisfaction standard for premises of a metainference), the metainference (*) is globally valid. Now consider the following metainference, obtained by uniformly substituting q by p in (*): p⇒p ⇒ A ∧ ¬A (**) is globally invalid in this logic. Thus, global validity does not meet the substitution requirement. How serious is this? It depends on whether we think uniform substitution must be met by valid metainferences or not. It is common to ask for a consequence relation for inferences to be closed under uniform substitution. Thus, it seems natural to demand it also of metainferences, in particular if we take metainferences to be just a specific kind of inferences. But if this is so, then, on the one hand, the global notion of metainferential validity seems not to qualify for the job, as there are consequence relations (for metainferences) determined by it that are not closed under uniform substitution. On the other hand, consequence relations (for metainferences) determined by the local notion of metainferential validity are closed under uniform substitution. Thus, the local notion seems to be a considerably better candidate for a notion of metainferential validity. So far so good for the different notions of metainferential validity. In this book, we will mostly focus on the local notion, which is the one commonly used for defining metainferential logics. The next chapter will introduce the best-known family of them. (**)

References 1. Ripley, D. (2013). Paradoxes and failures of cut. Australasian Journal of Philosophy, 91(1), 139–164. 2. Ripley, D. (2013). Revising up. Philosophers’ Imprint, 13(5), 1–13. 3. Cobreros, P., et al. (2014). Reaching transparent truth. Mind, 122(488), 841–866. 4. Dicher, B., & Paoli, F. (2019). ST, LP and tolerant metainferences Graham Priest on dialetheism and paraconsistency (pp. 383–407). Springer. 5. Barrio, E., Pailos, F., & Szmuc, D. (2020). A hierarchy of classical and paraconsistent logics. Journal of Philosophical Logic, 49(1), 93–120. https://doi.org/10.1007/s10992-019-09513-z 6. Pailos, F. (2019). A family of metainferential logics. Journal of Applied Non-Classical Logics, 29(1), 97–120. 7. Boghossian, P. (2014). The nature of inference. Philosophical Studies, 169(1), 1–18. https:// doi.org/10.1007/s11098-012-9903-x Golan chooses, instead of this inference, the inference A ⇒ ¬A, which is not antivalid unless A is valid. But ⇒ A ∧ ¬A is antivalid in many logics, including classical logic. 26

References

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8. Wright, C. (2014). Comment on Paul Boghossian, the nature of inference. Philosophical Studies, 169(1), 27–37. http://www.jstor.org/stable/42920595 9. Broom, J. (2014). Comment on Boghossian. Philosophical Studies, 169(1), 19–25. https://doi. org/10.1007/s11098-012-9894-7 10. Hlobil, U. (2016). What is inference? Or the force of reasoning. Doctoral Dissertation (Unpublished), University of Pittsburgh. http://dscholarship.pitt.edu/28130/ 11. Neta, R. (2013). What is an inference? Philosophical Issues, 23(1), 388–407. https://doi.org/ 10.1111/phis.12020 12. Pailos, F. (2020). A fully classical truth theory characterized by substructural means. The Review of Symbolic Logic, 13(2), 249–268. 13. Scambler, C. (2020). Classical logic and the strict tolerant hierarchy. Journal of Philosophical Logic, 49(2), 351–370. 14. Ripley, D. (2022). One step is enough. Journal of Philosophical Logic, 51(6). 15. Zardini, E. (2022). The final cut. Journal of Philosophical Logic, 51(6), 1583–1611. 16. Da Ré, B., et al. (2020). Metainferential duality. Journal of Applied Non-classical Logics, 30(4), 312–334. 17. Teijeiro, P. (2021). Strength and stability. Anáalisis Filosófico, 41(2), 337–349. 18. Garson, J. W. (1990). Categorical semantics (pp. 155–175). Springer. 19. Garson, J. W. (2013). What logics mean: From proof theory to model-theoretic semantics. Cambridge University Press. 20. Humberstone, L. (1996). Valuational semantics of rule derivability. Journal of Philosophical Logic, 25, 451–461. 21. Da Ré, B., Szmuc, D., & Teijeiro, P. (2022). Derivability and metainferential validity. Journal of Philosophical Logic, 51(6). 22. Chemla, E., Égré, P., & Spector, B. (2017). Characterizing logical consequence in many-valued logic. Journal of Logic and Computation, 27(7), 2193–2226. 23. Schroeder-Heister, P. (2014). The calculus of higher-level rules, propositional quantification, and the foundational approach to proof-theoretic harmony. Studia Logica, 102(6), 1185–1216. https://doi.org/10.1007/s11225-014-9562-3 24. Priest, G., & Wansing, H. (2015). External curries. Journal of Philosophical logic, 44(4), 453–471. 25. Malinowski, G. (1990). Q-Consequence operation. Reports on Mathematical Logic, 24(1), 49–59. 26. Frankowski, S. (2004). Formalization of a plausible inference. Bulletin of the Section of Logic, 33(1), 41–52. 27. Pailos, F. (2021). Disjoint logics. Logic and Logical Philosophy, 30(1). https://doi.org/10. 12775/LLP.2020.014 28. Pailos, F. (2022). Pure strong-kleene logics. Journal of Logic and Computation, exac087. https://doi.org/10.1093/logcom/exac087 29. Ripley, D. A toolkit for metainferential logics. Manuscript. 30. Williamson, T. (2002). Vagueness. Routledge. 31. Kortenbach, B. (2021). The classicality of epistemic multilateral logic. Ph.D. thesis. Institute for Logic, Language and Computation, Amsterdam, Netherlands. 32. Fine, K. (1975). Vagueness, truth and logic. Synthese, 30(3–4), 265–300. 33. Keefe, R. (2000). Theories of vagueness. Cambridge University Press. 34. Urbas, I. (1990). Paraconsistency. Studies in Soviet Thought, 39(3/4), 343–354. 35. Golan, R. (2021). There is no tenable notion of global metainferential validity. Analysis, 81(3), 411–420.

Chapter 3

Strong Kleene Metainferential Logics

In this chapter, we will introduce the most studied family of pure and impure mixed metainferential logics of level 1: the Strong Kleene metainferential logics. We will only focus on some of them: the ones that can be characterized through the inferential logics K3, LP, ST, and TS, which will be introduced below. Let’s start with the Strong Kleene truth tables. For a matter of simplicity—and unless we explicitly do otherwise—, we will work most of the time with a propositional language L with the usual truth-functional connectives defined by the threevalued Strong Kleene schema: ¬ 1 0 1/2 1/2 1 0

∧ 1 1/2 0

1 1 1/2 0

1/2

0 0 1/2 0 0 0

1/2

∨ 1 1/2 0

1 1/2 1 1 1 1/2 1 1/2

0 1 1/2 0

The functions → and ↔ can be defined as usual. Notice that the functions are Boolean normal: restricted to classical inputs, the result of the operations are also classical. In this sense, the trivaluations that assign a classical value to every propositional letter behave as boolean bivaluations.1 This will play an important role in understanding some of the results in this book. Also, another important feature of these tables is that the operations are monotonic.2 Assuming that truth values are ordered according to the informational order, i.e. 21 < 0 and 21 < 1 we have the following:

1

Though strictly speaking, boolean valuations are not Strong Kleene valuations, they are obviously equivalent to these Strong Kleene valuations. 2 Although we won’t expand on this point, it is important to mention it since it helps to deeply understand the connection between ST and CL. See e.g. [1] for more details. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. Pailos and B. Da Ré, Metainferential Logics, Trends in Logic 61, https://doi.org/10.1007/978-3-031-44381-7_3

29

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3 Strong Kleene Metainferential Logics

• if x ≤ y then ¬x ≤ ¬y • for ∗ ∈ {∨, ∧}, if x, y ≤comp x  , y  then (x ∗ y) ≤ (x  ∗ y  ). where ≤comp is the componentwise order obtained from ≤.3 Finally, it is worth mentioning that sometimes we will work with a constant ⊥ denoting the value 0 and for the value 1. This addition does not change any of the results provided for the simpler language. However, in this book, we assume that the language does not contain a constant for the intermediate value 1/2. In fact, most of the results introduced in the next Chapters do not hold for the language thus enriched. The valuations satisfying the Strong Kleene truth-tables will be called Strong Kleene valuations (or SK-valuations). We will now introduce four inferential logics based on the Strong Kleene valuations and then we will present a family of metainferential logics based on the defined inferential logics.

3.1 Four Basic Inferential Logics We will now present four inferential logics: the Tarskians—or structural—LP and K3, and the non-Tarskian—or substructural—TS and ST. These logics are the basic building blocks for characterizing the metainferential logics we will be introducing later. Let’s start by defining the propositional versions of these logics, beginning with LP. Given a propositional language L and assuming the matrices associated to the 3-element Strong Kleene algebra, validity for LP is defined as preservation of the designated values 1 and 1/2. The valuation functions are homomorphisms (with respect to the numerical order) from FOR(L), the set of formulas of the language, to the set of truth-values of the semantic structure—in this case, the set {1, 21 , 0}. Valuations are extended from propositional variables to complex formulae with the aid of the truth-functions for the connectives given by the 3-element Kleene algebra. In the case of LP, we can define what an LP-valid inference is in the following straightforward manner. (Below, LP is a substitution-invariant consequence relation.) Definition 3.1.1 A valuation v satisfies an inference  ⇒  in LP (written v LP  ⇒ ) if and only if, if v(γ) ∈ {1, 21 }, for every γ ∈ , then v(δ) ∈ {1, 21 }, for some δ ∈ . An inference  ⇒  is LP-valid (written LP  ⇒ ) if and only if v LP  ⇒ , for all valuations v. In a similar vein, K3 can be then understood as a propositional language L with the matrices associated to the 3-element Kleene algebra that we have already presented, and a consequence relation characterize as defining sets of inferences that preserves the only designated value, 1. (Once again, K3 is a substitution-invariant consequence relation.) 3

Meaning that x, y ≤lex x  , y  if and only if x ≤ x  and y ≤ y  .

3.1 Four Basic Inferential Logics

31

Definition 3.1.2 A valuation v satisfies an inference  ⇒  in K3 (written v K3  ⇒ ) if and only if, if v(γ) ∈ {1} for every γ ∈ , then v(δ) ∈ {1}, for some δ ∈ . An inference  ⇒  is K3-valid (written K3  ⇒ ) if and only if v K3  ⇒ , for all valuations v. LP and K3 are pure consequence relations since the standards for premises and conclusions are the same. However, ST and TS are impure logics and substructural because at least one structural feature of a Tarskian consequence relation is given up by them. ST abandons Cut,4 while TS drops Reflexivity. These two logics play a key role in what we will call the ST-hierarchy, a hierarchy of metainferential logics that recovers every classical metainferential validity while allowing the introduction of a transparent truth predicate. We will first characterize ST. The logic ST5 is defined as follows: Definition 3.1.3 A valuation v satisfies an inference  ⇒  in ST (written v ST  ⇒ ) if and only if, if v(γ) ∈ {1} for every γ ∈ , then v(δ) ∈ {1, 21 }, for some δ ∈ . An inference  ⇒  is ST-valid (written ST  ⇒ ) if and only if v ST  ⇒ , for all valuations v. It is immediate to check that in this logic reflexivity, i.e. the inference φ ⇒ φ is valid. ST can also be technically presented by appealing to p-matrices (see e.g. Frankowski [2]). Another way to put this requires talking about strict and tolerant satisfaction or truth. A valuation v satisfies tolerantly a formula ϕ if and only if v(ϕ) ∈ {1, 21 }, and satisfies it strictly if and only if v(ϕ) ∈ {1}. There are different philosophical ways to understand the difference between the two notions. Strict satisfaction might be equated to being true (in a valuation), while tolerant satisfaction can be related to not being false (in a given valuation). Or, in a more dialetheist fashion, strict satisfaction can be understood as being just true (in a valuation), while tolerant satisfaction should be interpreted as being true, but also false (in the given valuation). An attitudinal way to interpret both notions relate strict satisfaction with acceptance, and tolerant satisfaction with non-rejecting. Finally, an informational reading might relate strict satisfaction with being-told-true and tolerant satisfaction with not-being-told-false. A valuation v satisfies an inference  ⇒  if and only if, if v strictly satisfies every γ ∈ , then v tolerantly satisfies at least one δ ∈ . (This is where ST gets its name from, i.e. because it has a Strict-Tolerant consequence relation.) Finally, an inference from  to  is valid if and only if for every valuation v, if v satisfies strictly every γ ∈ , then v satisfies tolerantly some δ ∈ . Nevertheless, it is worth mentioning that this is not the only way ST’s supporters explain their position. They prefer to talk about strict and tolerant assertion rather than talking about strict and tolerant satisfaction, or strict and tolerant truth. Fjellstad [3], explains that the reason 4

In the sense that it is locally invalid in ST/ST. For an extensive presentation of ST, see Cobreros, Ripley, Egré and van Rooij [4], Ripley [5, 6] and Cobreros, Ripley, Egré and van Rooij [7]. Though originally applied to vagueness and semantic paradoxes by these authors, Hlobil [8] offers an interpretation of this logic in terms of truth-makers.

5

32

3 Strong Kleene Metainferential Logics

why they use the idea of strict and tolerant assertion instead of any of the last notions is to avoid revenge paradoxes related to the idea of being ‘strictly true’ and ‘strictly false’ in the context of truth-theories based on ST. The logic TS6 is defined as follows: Definition 3.1.4 A valuation v satisfies an inference  ⇒  in TS (written v TS  ⇒ ) if and only if, if v(γ) ∈ {1, 21 } for every γ ∈ , then v(δ) ∈ {1}, for some δ ∈ . An inference  ⇒  is TS-valid (written TS  ⇒ ) if and only if v TS  ⇒ , for all valuations v. Notice that from this definition the logic TS doesn’t validate reflexivity, nor any other inference (since the SK valuation which assigns the intermediate value to every formula is a counterexample to every inference). TS can also be technically presented by appealing to q-matrices (see e.g. Malinowski [9]). The consequence relation of TS can be defined as a Tolerant-Strict one: v satisfies an inference  ⇒  if and only if, if v tolerantly satisfies every γ ∈ , then v strictly satisfies at least one δ ∈ , and it is valid if every valuation satisfies it. Next, we mention some main facts regarding these logics7 : Fact 3.1.5 (Cobreros, Ripley, Egré and van Rooij [4]) TS is a non-reflexive, and thus a substructural, logic. Fact 3.1.6 (Cobreros, Ripley, Egré and van Rooij [4]) ST is a non-transitive, and thus a substructural, logic. Fact 3.1.7 (Cobreros, Ripley, Egré and van Rooij [4]) TS has no valid inferences. Fact 3.1.8 (Girard [10], Ripley [5]) ST and classical propositional logic CL have the same set of valid inferences.8 We should better pause to clarify how CL behaves. One way of defining CL’s consequence relation is by using the {0, 1}-reduct of the Strong Kleene schema: an inference  ⇒  is valid in CL if and only if, for every valuation v in this reduct, either v(γ) = 0 (for some γ ∈ ), or v(δ) = 1 (for some δ ∈ ). Similarly, a valuation v is a counterexample to  ⇒  in CL if and only if for every γ ∈ , v(γ) = 1, and for every δ ∈ , v(δ) = 0. Sometimes, we will talk about Boolean 6

TS is discussed by e.g., Cobreros, Ripley, Egré and van Rooij [4], and also by Chemlá, Egré and Spector [11]. Moreover, it was also discussed by Malinowski [12] as a tool to model empirical inference, and more recently was stressed by Rohan French [13], in connection with the paradoxes of self-reference. 7 Shramko and Wansing offer in [14] a way to read these two kinds of logics. While some logics, like ST, take as valid derivations of conclusions whose degree of strength (i.e., the conviction in its truth) is smaller than the one of the premises, other logics, like TS, are designed to qualify as valid derivations of true sentences from non-refuted premises (understood as hypotheses). 8 Barrio, Rosenblatt and Tajer [15], Dicher and Paoli [16] and Pynko [17] have shown that— through some suitable translation—the set of valid inferences in LP coincides with the set of valid metainferences in ST. Moreover, French [13], have conjectured that—again, through some suitable translation—the set of valid inferences in K3 coincides with the set of valid metainferences in TS.

3.2 Strong Kleene Metainferential Logics (of Level 1)

33

bivaluations to denote this set of valuations and distinguish it from the set of SK valuations. Also, it’s worth noticing that the previous fact holds provided we cannot define the intermediate value in the language of the connectives of Strong Kleene. Of course, if we added such a constant, CL and ST would differ. In the following section, we will show in detail the connection between CL and ST and TS.

3.2 Strong Kleene Metainferential Logics (of Level 1) With the tools we have introduced, we are now ready to present sixteen mixed metainferential logics of level 1. They are mixed metainferential logics L1 /L2 , where L1 and L2 are possibly different inferential logics (or, better, the satisfaction standards that determine its consequence relations).9 L1 stands for the standard that the premises of a sound argument should meet, while L2 represents the canon for the conclusion. Here, a valuation satisfies an inference (or metainference) according to a standard Li if and only if it is not a counterexample to that inference (or metainference) according to Li . From these sixteen logics, twelve are impure metainferential consequence relations built over LP, K3, ST and TS. These are TS/ST, ST/TS, LP/K3, K3/LP, ST/LP, ST/K3, TS/LP, TS/K3, LP/ST, LP/TS, K3/ST and K3/TS, as Fig. 3.1 shows.10 TS/ST and ST/TS are just the very first step of two infinite-sized sets of notions— that form two hierarchies with very interesting features–, resulting from extending the former two to every finite level. Later we will talk extensively about some of these hierarchies of metainferential logics. Moreover, all of these sixteen logics are the very first step of the whole universe of metainferential logics of any finite level defined over ST, LP, TS, and K3. But first, we will present in some detail some of these Strong Kleene metainferential logics. We will just focus on the ones that we Fig. 3.1 Sixteen metainferential consequence relations of level 1

9

L1 /L2 ST TS LP K3

ST ST/ST TS/ST LP/ST K3/ST

TS ST/TS TS/TS LP/TS K3/TS

LP ST/LP TS/LP LP/LP K3/LP

K3 ST/K3 TS/K3 LP/K3 K3/K3

As we have already mentioned, a similar move can be made using different notions of being a counterexample, as Ripley has recently done in [18]. Nevertheless, we will stick to the way they have been initially presented in Pailos [19]. 10 This means that we will be explicit about the four metainferential mixed but pure logics about which Pailos [19] says nothing about—i.e., ST/ST, TS/TS, LP/LP and K3 /K3 . His silence might be explained by the (presumed) fact that he is identifying them with the inferential and well-known logics ST, LP, TS, and K3, respectively. But this identification can and has been, resisted. See, in particular, what Ripley says in [18]. There, Ripley defends the view that inferential logics are silent about higher-order behavior. (And, more generally, that an n-logic is silent about what happens in higher-than-n metainferential levels).

34

3 Strong Kleene Metainferential Logics

think are the most interesting, for one reason or another,11 and afterward, we will present a comprehensive comparison table between all the impure metainferential logics of level 1. In particular, we will discuss whether these logics locally validate the following well-known structural properties: Cut

 ⇒ ϕ,  , ϕ ⇒,  Reflexivity ϕ⇒ϕ ,  ⇒ ,  LW

⇒ ⇒ RW ϕ,  ⇒   ⇒ ϕ, 

where LW and RW stand for the left and right rules of Weakening. It’s worth noticing that in usual presentations of sequent calculus there are also two more metainferences (Contraction and Exchange): LC

ϕ, ϕ,  ⇒   ⇒ ϕ, ϕ,  RC ϕ,  ⇒   ⇒ ϕ, 

LE

 ⇒ ψ, ϕ,  , ψ, ϕ ⇒  RE ϕ, ψ ⇒   ⇒ ϕ, ψ, 

However, since the collections of formulas are sets, in this context these rules cannot even be formulated.12 In other words, Exchange and Contraction are just MetaIdentity: φ ⇒1 φ. However, in what follows we will consider these two rules as if they were distinguishable from Meta-Identity, and the positive consequence of this decision is that if the logics were defined using lists or multisets, all what we will say could be applied directly. 1. TS/ST We will now present TS/ST, a logic that not only validates every classically valid inference—as ST does—but also validates every classically valid metainference of level 1. One way to interpret this result is the following: TS/ST’s metainferential satisfaction standard behavior resembles ST’s inferential satisfaction standard. As Cobreros, Tranchini and La Rosa [20] describes it, ST’s standard for premises is more stringent—or demanding—than its standard for conclusions—, which is, in comparison, less demanding. Recall that an inference  ⇒  is valid in ST if and only if, if for every premise γ ∈ , v(γ) = 1—i.e., if v strictly satisfies every premise—, then, for some conclusion δ ∈ , v(δ) ∈ {1, 21 }—i.e., v tolerantly satisfies one of the conclusions. Conversely, if v strictly satisfies every premise, but does not tolerantly satisfies any conclusion, then v is a counterexample to the

11

A detailed review of each of these impure logics can be found in Pailos [19]. In order to make sense to these rules we would need to take the collections of formulas as lists or sequences of formulas, in the case of Exchange, and as multisets in the case of Contraction.

12

3.2 Strong Kleene Metainferential Logics (of Level 1)

35

validity of that inference in ST. Similarly, TS/ST’s standard for premises is more stringent—or demanding—than its standard for conclusions. TS/ST’s satisfaction standard works similarly, but on the metainferential level. Thus, TS/ST’s standard for the premises is more demanding—or stringent—than its standard for the conclusion. Definition 3.2.1 A metainference  ⇒1  is valid in TS/ST if and only if, for every valuation v, if every γ ∈  is satisfied by v according to TS, then v satisfies some δ ∈  according to ST, where ,  ⊆ M E T A0 . Before showing in what sense TS/ST recovers classical logic not only at the inferential level, but also for metainferences,13 it is worth noticing two facts. The first one relates ST and the validity standard of classical propositional logic, CL. The second one relates TS’s and CL’s. First, every bivaluation v  —i.e., every Boolean valuation—which is a counterexample in CL to an inference  ⇒ , can be transformed into a Strong Kleene trivaluation v that is a counterexample to that inference in ST, and vice versa. On the one hand, a valuation v  is a counterexample to  ⇒  in CL if and only if, for every γ ∈ , v  (γ) = 1, and for every δ ∈ , v  (δ) = 0. But since we are defining CL using the {0, 1}-reduct of the Strong Kleene schema, it easy to see that this bivaluation can be transformed into a trivaluation which is a counterexample in ST (i.e., a trivaluation which coincides with the bivaluation in the truth value they give to any formula). On the other hand, a trivaluation v is a counterexample in ST to an inference  ⇒  only if there is a bivaluation v  such that it is a counterexample to that inference in CL. We will give below a detailed proof of both of these facts. Secondly, every (Boolean) bivaluation v  which satisfies an inference  ⇒  in CL can be transformed into a (Strong Kleene) trivaluation v that satisfies that inference in TS, and vice versa. Recall that v  satisfies  ⇒  in CL if and only if either v(γ) = 0, for some γ ∈ , or, for some δ ∈ , v(δ) = 1. But v  can be trivially transformed into a (Strong Kleene) trivaluation v which satisfies  ⇒  in TS. Finally, a (Strong Kleene) trivaluation v satisfies an inference  ⇒  in TS only if there is a (Boolean) bivaluation v  such that it confirms that inference in TS. Once again, we will show below detailed proofs of these facts.14 We are now ready to introduce the main result for TS/ST. It establishes that a metainference is valid in CL if and only if it is valid in TS/ST, i.e.  ⇒1  is valid in CL if and only if  ⇒1  is valid in TS/ST, where ,  ⊆ M E T A0 . To prove this result, we need first to present the following lemmata. 13

First established in [19], and whose consequences are explored, e.g., in Barrio, Pailos, and Szmuc [3], Pailos [21] or Scambler [22]. 14 Nevertheless, it is not true that a valuation v is a counterexample in CL to an inference  ⇒  if and only if v is a counterexample to that inference in ST. Consider for example the SK-valuation v ∗∗ such that v ∗∗ ( p) = 1 and v(q) = 21 . The valuation v ∗∗ is a counterexample in ST to the inference ψ ⇒, but it surely is not a counterexample in CL to that inference, because v ∗∗ is not a bivaluation. A similar point can be made with respect to the relation between TS and CL. That’s why it is important to keep in mind that we are transforming bivaluations into trivaluations, and vice versa.

36

3 Strong Kleene Metainferential Logics

Lemma 3.2.2 For all , , ,  ⊆ M E T A0 (L), and all SK-valuations v, there is a Boolean bivaluation v ∗ such that: if then

v ST  ⇒  ∗

v CL  ⇒ 

and

v TS  ⇒ ,

and

v ∗ CL  ⇒ 

Proof Assume that there is a SK-valuation v such that v ST  ⇒  and v TS  ⇒ . This implies, on the one hand, that v is such that v(γ) = 1, for all γ ∈  and v(δ) = 0, for all δ ∈  and, on the other hand, that v(θ) = 0, for some θ ∈ ,  or v(π) = 1, for some π ∈ . Consider, now, a SK valuation v that is just like v  except that for each propositional variable p such that v( p) = 21 , we have that v ( p) is 1 or 0—it doesn’t matter which. By the Monotonicity of the operations of the   Strong Kleene schema,15 we know that that v is such that v (γ) = 1, for all γ ∈    and v (δ) = 0, for all δ ∈  and, on the other hand, that v (θ) = 0, for some θ ∈ ,   or v (π) = 1, for some π ∈ . But, by construction, v is extensionally equivalent to a Boolean bivaluation v ∗ (the non-classical value plays no role in the trivaluation),  and then v ∗ CL  ⇒  and v ∗ CL  ⇒ . Lemma 3.2.3 For all , , ,  ⊆ M E T A0 (L), and all Boolean valuations v, there is a SK-valuation v ∗ such that: if then

v CL  ⇒  ∗

v ST  ⇒ 

and

v CL  ⇒ ,

and

v ∗ TS  ⇒ 

Proof Assume that there is a Boolean valuation v such that v CL  ⇒  and v CL  ⇒ . This implies, on the one hand, that v is such that v(γ) = 1, for all γ ∈  and v(δ) = 0, for all δ ∈  and, on the other hand, that v(θ) = 0, for some θ ∈ , or v(π) = 1, for some π ∈ . But, since Boolean valuations are a subset of SK valuations, v is also a SK valuation of the kind we are looking for, i.e. a valuation  v such that v ST  ⇒  and v TS  ⇒ . Lemma 3.2.4 If there is a SK-valuation v such that v TS/ST  ⇒1 , then there is a Boolean bivaluation v ∗ , such that v ∗ CL  ⇒1 , with ,  ⊆ M E T A0 (L). Proof Assume that there is a SK-valuation v such that v TS/ST  ⇒1 . From the fact that v TS/ST  ⇒1  we can infer that v TS γ, for all γ ∈  and that v ST δ, for all δ ∈ . Furthermore, from this and Lemma 3.2.2 it follows that there is a Boolean bivaluation v ∗ such that v ∗ CL γ, for all γ ∈  and v ∗ CL δ, for all δ ∈ . But, by definition, this implies that there is a Boolean bivaluation v ∗ such  that v ∗ CL  ⇒1 .

15

Recall that, as we explained at the beginning of the chapter, this notion is defined over the partial (information) order ≤ of this schema, which is defined by stipulating that 21 ≤ 0 and 21 ≤ 1.

3.2 Strong Kleene Metainferential Logics (of Level 1)

37

Lemma 3.2.5 For every v Boolean bivaluation, and every ,  ⊆ M E T A0 (L), if v such that v CL  ⇒1 , then there is a SK-valuation v ∗ , such that v ∗ TS/ST  ⇒1 . Proof Assume that there is a Boolean bivaluation v such that v CL  ⇒1 . Thus, we know that v is such that v CL γ, for every γ ∈ , and yet v CL δ, for every δ ∈ . Given this, by Lemma 3.2.3 we know that there is a SK valuation v ∗ such that for every γ ∈ , v ∗ TS γ and v ∗ ST δ, for every δ ∈ . Thus, by definition,  we know that this entails that v ∗ TS/ST  ⇒1 . Now we can rephrase and prove The Collapse Result. Theorem 3.2.6 (The Collapse Result) For all ,  ⊆ M E T A0 (L) TS/ST  ⇒1  if and only if

CL  ⇒1 

Proof From left to right, let us suppose that CL  ⇒1 , from which we infer that there is a Boolean bivaluation v such that v CL γ, for all γ ∈ , and yet v CL δ, for every δ ∈ . But, since Boolean bivaluations are a subset of SK valuations (recall it is the {0, 1}-reduct), we know by the definition of satisfaction in TS and ST that v can be trivially seen as a SK trivaluation v  such that v  TS γ, for all γ ∈ , and yet v  ST δ, for some δ ∈ . From this, by the definition of validity of a metainference of level 1 in TS/ST, we know that v  TS/ST  ⇒1 , whence TS/ST  ⇒1 . From right to left, let us suppose that TS/ST  ⇒1 , from which we infer that there is a SK valuation v such that v TS γ, for all γ ∈ , and yet v ST δ, for all δ ∈ . By Lemma 3.2.4, we know that there is a Boolean valuation v ∗ such that v ∗ CL γ, for all γ ∈ , and yet v ∗ CL δ for all δ ∈ . Therefore, we know that  v ∗ CL  ⇒1 . Thus, CL  ⇒1 . Naturally, if a valuation is a Boolean counterexample to a metainference, it is also a SK counterexample to it. This explains the left-to-right side of the proof. From the right-to-left side, Lemma 3.2.4 shows that each time we have a SK (non-Boolean) counterexample v to a metainference in TS/ST, then there is also a Boolean counterexample to it. Specifically, that there is a Boolean valuation v  that matches the SK (non-Boolean) valuation v in every classical value, but also that gives a classical value to each formula that receives a non-classical value in v, and such that v  is also a counterexample to that metainference. This theorem guarantees that TS/ST recovers every classical formula, inference and metainference (of level 1). Now, valid sentences are usually regarded as degenerate cases of valid inferences (i.e., inferences with an empty set of premises).16 In a similar vein, valid inferences can be interpreted as degenerate cases of valid metainferences (i.e., metainferences of level 1 with an empty set of premises).17 Thus, every classically valid inference will 16

Though this is not mandatory. If these two things are regarded as different, then logics that validate one but not the other may be admitted. 17 The previous footnote can be adapted here for inferences and metainferences, and, in general, for metainferences of level n and metainferences of level n + 1. See, in particular, Ripley’s position in [18].

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3 Strong Kleene Metainferential Logics

be valid in TS/ST, and vice-versa. So, a degenerate case of a metainference—e.g., an inference—is valid in TS/ST if and only if every valuation satisfies the conclusion according to the standard for conclusions—e.g., according to ST. And, as we have already mentioned, ST recovers every classically valid inference. As every classically metainference is valid in TS/ST, so is every structural metainferential schema, including the most well-known of them: Cut, Contraction, Identity, Weakening, and Exchange. It is worth noticing that it is possible to expand TS/ST’s language with a transparent truth predicate. The resulting theory, TS/ST+ , will be satisfiable. TS/ST+ ’s satisfiability follows from the fact that the truth predicate can be interpreted as a fixed-point of a jump operator over the Strong Kleene schema, that is the ones used by TS/ST’s—and TS/ST+ —models. In fact, every consequence relation that we will be exploring shares this feature with TS/ST, for exactly the same reasons. We will dive deeper into truth-theories based on these logics in Chap. 9. One last interesting feature of TS/ST is that it is a fully Tarskian logic, though the means used to characterize it are non-Tarskian—or substructural—theories: ST and TS. We will see later that it is also possible to go the other way, i.e., from Tarskian inferential logics, to non-Tarskian metainferential logics. Before exploring that path, though, we will present a close relative of TS/ST: ST/TS. 2. ST/TS The logic ST/TS invalidates not only every inference—as TS does—, but also every metainference (of level 1).18 ST/TS shares with TS a relevant feature that explains both logic’s behavior. An inference is valid in TS if and only, for every valuation v, if v satisfies the premises according to a certain standard, then v meets the conclusion according to some other—more demanding or more stringent (using Cobreros, Tranchini and La Rosa terminology in [23])—satisfaction canon. ST/TS works in a similar way but in a metainferential setting. Thus, ST/TS’s standard for premises is less demanding than its satisfaction criterion for conclusions. This is how it works. Definition 3.2.7 A metainference  ⇒1  is valid in ST/TS if and only if, for every valuation v, if every γ ∈  is satisfied by v according to ST, then v confirms some δ ∈  according to TS, with ,  ⊆ M E T A0 (L). From this definition, it follows that ST/TS is a (level 1) metainferentially empty logic. Fact 3.2.8 ST/TS  ⇒1  for every ,  ⊆ M E T A0 (L). Proof Consider a valuation v such that, for every propositional letter p that appears in  ∪ , v( p) = 21 . As the matrices for the logical constants are the Strong Kleene ones, v assigns the value 1/2 to every formula in the metainference. Thus, every premise γ ∈  is satisfied by v according to ST, but v does not satisfy any conclusion δ ∈  according to TS.  18

But it has many valid metainferences of higher levels, as we will soon show.

3.2 Strong Kleene Metainferential Logics (of Level 1)

39

As no metainference is valid in ST/TS, neither are the structural most well-known metainferential schemas: Cut, Contraction, Identity, Weakening, and Exchange. Thus, ST/TS is a, so to speak, completely non-Tarskian logic, as it does not validate any of the structural metainferences (that correspond to the structural rules of a sequent system) that CL validates. We are now going to present a logic that runs in the opposite direction than TS/ST, a fully Tarskian logic that can be characterized through non-Tarskian—or substructural—logics. LP/K3 is a non-Tarskian metainferential logic characterized using two Tarskian logics: LP and K3. 3. LP/K3 Definition 3.2.9 A metainference  ⇒1  is valid in LP/K3 if and only if, for every valuation v, if v satisfies every γ ∈  according to LP, then v confirms some δ ∈  according to K3. As we have mentioned, LP/K3 is non-Tarskian—or substructural—because not every structural metainference is valid in it. In particular, Cut fails in LP/K3. Fact 3.2.10 Cut is invalid in LP/K3. Proof Consider the following instance of Cut: Cut

⇒ q, ¬q q⇒q ⇒ q, ¬q

Any valuation v such that v(q) =

1 2

is a counterexample to it.



At this point, it might be interesting to consider a structural metainference that has not received much attention: Meta-Identity—i.e. the metainference of level 1, φ ⇒1 φ (here φ ranges over inferences). We will also see how it fails in this logic. Moreover, Weakening, Contraction, and Exchange are also invalid in LP/K3. And Meta-Identity at least partially explains their failure. Fact 3.2.11 Meta-Identity, Weakening, Contraction and Exchange are invalid in LP/K3. Proof Weakening

Consider the following instance of Weakening (RW ): Weakening

⇒ p, ¬ p ⇒ p, ¬ p, p ∨ ¬ p

The valuation V ( p) = 21 is a counterexample to it. It’s easy to formulate a similar counterexample to LW . Meta-Identity Consider now the following instance of Meta-Identity: ⇒ p, ¬ p ⇒ p, ¬ p

40

3 Strong Kleene Metainferential Logics

The valuation v such that v( p) = 21 , satisfies the premise according to LP, but does not meet the conclusion in K3. Therefore, it is a counterexample to Meta-Identity (to this instance, and, furthermore, to the schema) in LP/K3. Contraction Consider the following instance of LC: p, p ⇒ q, ¬q p ⇒ q, ¬q The valuation v, with v( p) = 1 and v(q) = 21 is a counterexample to it. Similar for RC. However, as mentioned, this version of Contraction, defined for sets of formulae as premises and conclusions, is just a notational variant of Meta-Identity, in the sense that every instance of one of them is at the same time an instance of the other. But a different presentation of LP/K3 that defines inferences as multisets of formulae will have the resources to distinguish them. Nevertheless, even in that presentation of LP/K3, both Contraction and Meta-Identity will be invalid. A similar point can be made with respect to K3/LP, ST/LP, K3/TS, and LP/TS. Moreover, this version of Meta-Identity is also a notational variant of Exchange. Nevertheless, moving to a presentation of this logic that defines inferences as multisets of formulae will not be necessary to distinguish them. What we need to accomplish that purpose are inferences understood as pairs of lists of formulae. Those types of logics will be able to distinguish between MetaIdentity and Exchange, but even in them Meta-Identity and Exchange will be invalid. Exchange Consider the following instance of LE: p, q ⇒ r, ¬r q, p ⇒ r, ¬r Any valuation v, such that v( p) = v(q) = 1 and v(r ) = to it. And similar for RE.

1 2

is a counterexample 

To sum up, as every instance of Meta-Identity is also an instance of Contraction and Exchange, then the failure of (an instance of) Meta-Identity is enough to guarantee the failure of the rest of the previously mentioned metainferential schemas. But this is just because we are working with inferences as sets of formulae. Working with sequences instead—even working with multisets will be enough in the cases of Contraction— will be enough to distinguish these metainferential schemas. Nevertheless, even in those cases—i.e., even if these logics were defined with sequences or multisets instead of sets of formulae—they will stand or fail together. 4. K3/LP LP/K3 is not the only non-Tarskian metainferential logic characterized using both LP and K3. K3/LP is the other member of this group. Definition 3.2.12 A metainference  ⇒1  is valid in K3/LP if and only if, for every valuation v, if v confirms every γ ∈  according to SK, then v satisfies some δ ∈  according to LP.

3.2 Strong Kleene Metainferential Logics (of Level 1)

41

K3/LP, in a way, is strongly non-Tarskian, in the sense that most structural metainferences are invalid in it. In fact, even if Identity is valid19 (because it is valid in LP, and K3/LP’s standard for inferences is just LP’s criterion for them),20 Cut, Meta-Identity, Weakening, Contraction and Exchange fail in it. Fact 3.2.13 Cut is invalid in K3/LP. Proof Consider an instance of Cut without logical constants such that the sentences in , , φ share no propositional variables with the sentences in , . The valuation v such that v(γ) = v(σ) = v(φ) = 21 , for every γ ∈ , σ ∈   , v(δ) = v(π) = 0, for every δ ∈ , π ∈ , is a counterexample to Cut’s validity in K3/LP.  Fact 3.2.14 Meta-Identity, Weakening, Contraction, and Exchange are invalid in K3/LP-even though we are working with sets of formulas, and not with multisets or sequences.. Proof As we have already pointed out, if Meta-Identity is invalid, the rest of them are also invalid. In order to prove that Meta-Identity is invalid, it is enough to consider the following instance of Meta-Identity: p⇒q p⇒q The valuation v such that v( p) = 21 and v(q) = 0 satisfies the premise according to K3, but does not confirm the conclusion in LP.  We will introduce just two more metainferential logics. A new thing about them is that both combine a Tarskian, or structural standard, with a non-Tarskian, or substructural standard. 5. ST/LP This logic has the non-Tarskian ST as the standard for premises, but LP as the canon for conclusions, and is a substructural logic, as every structural metainferential schema that we have considered, but Identity—which is valid since it is valid in LP—is invalid in it. Definition 3.2.15 A metainference  ⇒1  is valid in ST/LP if and only if, for every valuation v, if v satisfies every γ ∈  according to ST, then v confirms some δ ∈  according to LP. Fact 3.2.16 Cut is invalid in ST/LP. Proof Consider an instance of Cut such that every formula in the metainference is a propositional variable. The valuation v such that v(γ) = v(σ) = 1, for every γ ∈ , σ ∈ , v(δ) = v(π) = 0, for every δ ∈ , π ∈ , and v(φ) = 21 , is a counterexample to it in ST/LP.  19 20

Therefore, it is not completely non-Tarskian. Similarly, Identity is valid in LP/K3 because it is valid in K3.

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3 Strong Kleene Metainferential Logics

Fact 3.2.17 Meta-Identity, Weakening, Contraction, and Exchange are invalid in ST/LP. Proof Once again, for this, it is enough to prove that Meta-Identity is invalid. Thus, consider the following instance of Meta-Identity: p⇒q p⇒q The valuation v such that v( p) = Identity in ST/LP.

1 2

and v(q) = 0 is a counterexample to Meta

6. K3/TS The last Strong Kleene metainferential logic we will introduce has the Tarskian K3 as a standard for premises, and TS as a criterion for conclusions, and it also invalidates every structural metainferential schema. Definition 3.2.18  ⇒1  is valid in K3/TS if and only if, for every valuation v, if v satisfies every γ ∈  according to K3, then v confirms some δ ∈  according to TS. Fact 3.2.19 Cut, Meta-Identity, Identity, Weakening, Contraction, and Exchange are invalid in K3/TS. Proof We will leave the proof that Cut is invalid as an exercise to the reader. Regarding the other cases, consider the following instance of Meta-Identity: p⇒p p⇒p The valuation v such that v( p) = 21 is a counterexample to Meta-Identity in K3/TS, and, therefore, to the other structural metainferences.  So far, we have introduced some systems in order to illustrate how the metainferential logics of level 1 are defined, and also we have presented with some detail some of the structural properties of these logics. Figures 3.2 and 3.3 summarize all the structural properties of all the impure Strong Kleene metainferential logics: Metainferences CUT IDENTITY META-IDENTITY WEAKENING CONTRACTION EXCHANGE

Fig. 3.2 Comparison table

TS/ST Yes Yes Yes Yes Yes Yes

ST/TS No No No No No No

K3/LP No Yes No No No No

LP/K3 No Yes No No No No

LP/ST Yes Yes Yes Yes Yes Yes

LP/TS No No No No No No

3.2 Strong Kleene Metainferential Logics (of Level 1) Metainferences CUT IDENTITY META-IDENTITY WEAKENING CONTRACTION EXCHANGE

K3/ST Yes Yes Yes Yes Yes Yes

K3/TS No No No No No No

ST/LP No Yes No No No No

43 ST/K3 No Yes No No No No

TS/LP Yes Yes Yes Yes Yes Yes

TS/K3 Yes Yes Yes Yes Yes Yes

Fig. 3.3 Comparison table (cont.)

From the tables above, we can draw the following fact: Fact 3.2.20 Let X/Y be an impure metainferential logic: • If Y = ST then the logic validates all the structural properties. • If X = TS then the logic validates all the structural properties. • If X = ST then the logic invalidates all the structural properties except from identity (which is valid if Y = LP or Y = K3). • If Y = TS then the logic invalidates all the structural properties. These remarks, of course, are non-exhaustive regarding the information summarized in the tables above. However, they illustrate the role the substructural inferential logics play in the metainferences that validate the metainferential logics based on them. Moreover, we can partially order these different logics considering the different strength they have. The measure of the strength is the metainferences they validate. The one that proves more things is TS/ST. Every classically valid metainference is valid in it. Regarding the rest of the logics, for example, it is not the case that the four logics of the second group—e.g., TS/LP, TS/K3, LP/ST, and K3/ST— validates exactly the same metainferences. In fact, for example, TS/LP and TS/K3 are incomparable, as K3 and LP are incomparable at the inferential level. Moreover, Cobreros, Tranchini and La Rosa [20] and Ripley [24] have independently shown that a metainferential logic XY is stronger than a metainferential logic X Y —i.e., that X Y ’s validities are properly contained in XY’s validities—if and only if (i) X is stronger than X, and Y is at least as strong as Y , or (ii) Y is stronger than Y , and X is at least as strong as X. Thus, these four logics are incomparable, and, for example, K3/ST is stronger than ST/ST. In the sequel, we will say more about these logics. To make it easy to visualize this order, we will present a Hasse-diagram of the lattice of these metainferential logics. This diagram can be originally found in Ripley [24] (Fig. 3.4). Before moving on, we would like to dig a little bit deeper into a pattern that we have already mentioned, that these mixed metainferential logics follow. As we have previously mentioned, if Meta-Identity is invalid, then Contraction, Exchange and Weakening will also be invalid, since every instance of the first one (in this setting) is an instance of the lasts. And it does not need much to invalidate Meta-Identity. Take two inferential logics, L1 and L2 . If there is an inference  ⇒ , and one valuation v such that v satisfies  ⇒  in L1 but not in L2 , then Meta-Identity will be invalid

44

3 Strong Kleene Metainferential Logics

Fig. 3.4 Hasse diagram of the inclusion ordering of the Strong Kleene Metainferential Logics

TS/ST

LP/ST

K3/ST

TS/LP

LP/LP

LP/K3

ST/ST

TS/TS

LP/TS

TS/K3

K3/LP

K3/K3

K3/TS

ST/LP

ST/K3

ST/TS

in L1 /L2 . Nevertheless, it is not necessary for L1 to be stronger than L2 . In fact, they may even be incomparable, and Meta-Identity might still be invalid. (If they are, then Meta-Identity is also invalid in L2 /L1 .) We have introduced sixteen different mixed metainferential consequence relations. Each one of them is specified using two different inferential Tarskian or nonTarskian (e.g., substructural) standards for inferential logics: the ones for K3, LP, ST or TS. In twelve of these cases, the standard for premises and consequence is different in each case—i.e., they are impure. Five of these logics—e.g., TS/ST, TS/LP, TS/K3, LP/ST, K3/ST—are Tarskian logics—e.g., they validate Cut, Identity, and Weakening. Moreover, they also validate Meta-Identity, Contraction, and Exchange. Nevertheless, they are characterized through substructural inferential logics. In fact, TS/ST is specified entirely with substructural inferential logics. Moreover, TS/ST collapses with CL at the metainferential level. Do the other four logics share this feature? In fact, they do not. It is not difficult to prove that TS/LP and TS/K3 do not validate every classically valid metainference. Neither LP nor K3 have the same set of valid inferences as classical logic—in fact, they are sublogics of CL. Thus, it is enough to consider any metainference with an empty set of premises, and a conclusion valid in CL but invalid in LP—or K3. For example, the following metainference will be valid in CL, but invalid in TS/LP: p, p → q ⇒ q

3.2 Strong Kleene Metainferential Logics (of Level 1)

45

Whereas the following metainference is invalid in TS/K3, but is nevertheless valid in CL: ⇒p→p The cases of LP/ST and K3/ST are a little bit trickier. While it is true that there are valuations that satisfy some metainferences in CL (or in TS/ST), but not in LP/ST and K3/ST, this does not automatically mean that the set of valid metainferences is actually different in these cases. But in fact, they are. As has been shown in Cobreros, Tranchini and La Rosa [20], the set of valid metainferences of LP/ST and K3/ST is strictly included in the set of valid CL’s (and TS/ST’s) metainferences. It is not hard to realize that, though every possible counterexample in CL is also a counterexample in both LP/ST and K3/ST, these last logics in fact have more possible counterexamples. And some of those valuations invalidate some classically valid metainferences. The following one is a metainference valid in CL but invalid in LP/ST. ⇒ p ∧ ¬p q ⇒r Though no valuation will satisfy its premise according to CL, a valuation v such that v( p) = 21 , v(q) = 1 and v(r ) = 0, confirms the premise, but does not satisfy the conclusion in LP/ST. As this metainference is also valid in K3/ST, this shows that K3/ST’s metainferential validities are not included in LP/ST’s metainferential validities. The following metainferential schema is valid in CL, but not in K3/ST. ¬( p ∧ ¬ p) ⇒ q ⇒r Once again, while no valuation will confirm the premise in CL, a valuation v such that v( p) = 21 , v(q) = 1 and v(r ) = 0 confirms the premise, but not the conclusion, in K3/ST. As this metainference is also valid in LP/ST, this shows that LP/ST’s metainferential validities are not included in K3/ST’s metainferential validities. This completes the proof that these two logics are incomparable (regarding their metainferential validities). On the opposite side of the spectrum, ST/TS, LP/TS, and K3/TS are completely substructural logics—e.g., not even one of the structural metainferential schemas that we have talked about is valid in them. A natural question that may arise at this point is if metainferences of level 1 is the limit for metainferential logics, i.e., if there are metainferential logics defined through a satisfaction standard of level 2, or of any finite level. The answer is yes. The next chapter is devoted to introducing every finite metainferential logic defined

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through the four basic inferential logics we have introduced, and to explore two of the most interesting hierarchies of such metainferential logics: the ST-hierarchy and the TS-hierarchy.

References 1. Da Re, B., et al. On three-valued presentations of classical logic. The Review of Symbolic Logic, 1–26. 2. Frankowski, S. (2004). Formalization of a plausible inference. Bulletin of the Section of Logic, 33(1), 41–52. 3. Fjellstad, A. (2016). Naive modus ponens and failure of transitivity. Journal of Philosophical Logic, 45(1), 65–72. 4. Cobreros, P., et al. (2012). Tolerant, classical, strict. Journal of Philosophical Logic, 41(2), 347–385. 5. Ripley, D. (2012). Conservatively extending classical logic with transparent truth. Review of Symbolic Logic, 5(2), 354–378. 6. Ripley, D. (2013). Paradoxes and failures of cut. Australasian Journal of Philosophy, 91(1), 139–164. 7. Cobreros, P., et al. (2014). Reaching transparent truth. Mind, 122(488), 841–866. 8. Hlobil, U. (2022). A truth-maker semantics for ST: Refusing to climb the strict/tolerant hierarchy. Synthese, 200(1), 1–23. https://doi.org/10.1007/s11229-022-03820-w 9. Malinowski, G. (1990). Q-consequence operation. Reports on Mathematical Logic, 24(1), 49– 59. 10. Girard, J.-Y. (1987). Proof theory and logical complexity. Bibliopolis. 11. Chemla, E., & Égré, P., Spector, B. (2017). Characterizing logical consequence in many-valued logic. Journal of Logic and Computation, 27(7), 2193–2226. 12. Malinowski, G. (2014). Kleene logic and inference. Bulletin of the Section of Logic, 43(1/2), 43–52. 13. French, R. (2016). Structural reactivity and the paradoxes of self-reference. Ergo, an Open Access Journal of Philosophy, 3. 14. Shramko, Y., & Wansing, H. (2010). Truth values. In E. Zalta (Ed.), The stanford encyclopedia of philosophy. Stanford University. http://plato.stanford.edu/archives/sum2010/entries/truthvalues/ 15. Barrio, E., Rosenblatt, L., & Tajer, D. (2015). The logics of strict-tolerant logic. Journal of Philosophical Logic, 44(5), 551–571. 16. Dicher, B., & Paoli, F. (2019). ST, LP and tolerant metainferences. In: Graham priest on dialetheism and paraconsistency (pp. 383–407). Springer. 17. Pynko, A. (2010). Gentzen’s cut-free calculus versus the logic of paradox. Bulletin of the Section of Logic, 39(1/2), 35–42. 18. Ripley, D. (2022). One step is enough. Journal of Philosophical Logic, 51(6) (2022). 19. Pailos, F. (2019). A family of metainferential logics. Journal of Applied Non-Classical Logics, 29(1), 97–120. 20. Cobreros, P., La Rosa, E., & Tranchini, L. (2021). Higher-level inferences in the strong-kleene setting: A proof-theoretic approach. Journal of Philosophical Logic, 1–36. 21. Pailos, F. (2020). A fully classical truth theory characterized by substructural means. The Review of Symbolic Logic, 13(2), 249–268. 22. Scambler, C. (2020). Classical logic and the strict tolerant hierarchy. Journal of Philosophical Logic, 49(2), 351–370. 23. Cobreros, P., Tranchini, L., & La Rosa, E. (2020). (I Can’t Get No) Antisatisfaction. Synthese,, 1–15. 24. Ripley, D. A toolkit for metainferential logics. Manuscript.

Chapter 4

Hierarchies of Strong Kleene Metainferential Logics

We have previously introduced all the metainferential logics of level 1 that can be defined through four Strong Kleene logics. Nevertheless, that is not the whole story, as those sixteen metainferential logics of level 1 can be combined in different ways to generate 256 metainferential logics of level 2, which can be combined in different ways to generate the 65536 metainferential logics of level 3, and so on. Thus, we have a full space of finite Strong Kleene metainferential logics—i.e., the whole space of Strong Kleene metainferential logics of different finite levels. In this chapter, though, we will not explore this space in general, but focus on two hierarchies of metainferential logics that are particularly interesting for different, but related, reasons. We will name the first one “the ST-hierarchy”, while we will refer to the second one as “the TS-hierarchy”. Each of them is such that, for every level n, there is one and only one particular logic of that level that belongs to the hierarchy. Moreover, in these two hierarchies, the conclusion-standard for each metainferential logic of level n + 1, is the metainferential logic of level n that is the previous link in the chain. The other important feature of these hierarchies is how they are related to classical logic CL—defined through the traditional two-valued deterministic models as preservation of value 1. And the relation is the following: for every level n, (i) a metainference is valid in CL if and only if it is valid in STn —i.e., the n-link of the ST-hierarchy—, and (ii) a metainference is antivalid—i.e., unsatisfiable—in CL if and only if it is antivalid in TSn —i.e., the n-link in the TS-hierarchy. We will explore in detail both of these facts, and we will start by presenting the ST-hierarchy.

4.1 The ST-Hierarchy The first link of the ST-hierarchy of metainferential logics is TS/ST, a logic that not only validates every classically valid inference—as ST does—, but also validates every classically valid metainference (of level 1). This is possible because TS/ST’s satisfaction standard embraces a feature of the inferential satisfaction © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. Pailos and B. Da Ré, Metainferential Logics, Trends in Logic 61, https://doi.org/10.1007/978-3-031-44381-7_4

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standard of ST, but applies it to the metainferential level. As a reminder, an inference is valid in ST if and only if, for every valuation v, if the premises satisfy certain— demanding—standard, then the conclusion satisfies some less demanding standard. We have adopted a similar norm for TS/ST’s notion of satisfaction, but for the metainferential level. TS/ST’s standard for the premises, as we have shown, is more demanding than its standard for the conclusion (see Definition 3.2.1). So far, we have shown that TS/ST recovers not only every classically valid inference but also every classically valid metainference (of level 1) (see Theorem 3.2.6). Nevertheless, we still have not mentioned the following drawback of the logic: TS/ST is not fully classical. Many classically valid metametainferences—e.g., metainferences of level 2—are invalid in TS/ST. In particular, the metametainference that [1] and [2] call Meta-Cut1 turns out to be invalid in TS/ST. 11 ⇒ 11 , ...,  1j ⇒ 1j

12 ⇒ 21 , ..., k2 ⇒ 2k

11 ⇒ 11 , φ

φ, 12 ⇒ 21

Meta-Cut

11 ⇒ 11 , ...,  1j ⇒ 1j , 12 ⇒ 21 , ..., k2 ⇒ 2k 11 , 12 ⇒ 11 , 21

The reason for this is the following: a metainferential logic not only determines a validity standard for metainferences (and, furthermore, for inferences and sentences): it also fixes a standard for metainferences of higher levels.2 In particular, if L is a propositional metainferential logic, a metametainference is valid in L if and only if every valuation v that satisfies every premise according to L, satisfies the conclusion in L. Moreover, for any metainference  ⇒n  of any level n (where  and each  are metainferences of level n − 1, and ⇒n stands for the metainferential relation of level n between premises and conclusion),  ⇒n  is valid in L if and only if, for every valuation v, if v satisfies every premise i according to L, then v satisfies the conclusion  according to L. It is not hard to realize, now, why Meta-Cut is invalid in TS/ST. Take any instance of it that involves only propositional letters, where the cut formula φ is different from every other sentence in the premises and the conclusion. The valuation v such j that v(φ) = 21 , v(γ) = v(δ) = v(σ) = 1—for every γ ∈ i , every δ ∈ nm and every l s σ ∈ k —, and v(π) = 0—for every π ∈ r —, satisfies every premise according to TS/ST, but does not confirm the conclusion in TS/ST. Nevertheless, it is possible to design a new logic that recovers not only every classically valid inference—as ST and TS/ST do—and every classically valid 1

Perhaps Meta-Cut is not the most adequate name for the mentioned metainference, since in it what it is cut is a formula and not an inference, as we would expect. However, we follow the usual terminology to avoid any confusion. 2 This claim is not universally accepted, though, as we have already mentioned. In particular, Dave Ripley, in [3, 4] argues for the view that a logic of level n does not pronounce about higherlevel metainferences. One of us thinks this position might lead to triviality under some reasonable assumptions (as he defends in Pailos [5]). Nevertheless, we will stick to the more traditional view, and leave the discussion of Ripley’s position aside for the moment.

4.1 The ST-Hierarchy

49

metainference—as TS/ST does—, but also every classically valid metametainference. The next paragraphs present such logic. In order to improve readability, we will rename ST as ST0 , TS/ST as ST1 , and call the new logic ST2 . This suggests that they are the first steps of a hierarchy of logics based on ST as its first mixed logic.3 Definition 4.1.1 A meta-metainference  ⇒2  is valid in ST2 if and only if, for every valuation v, if every γ ∈  is satisfied by v according to ST/TS, then v confirms some δ ∈  according to TS/ST. As we have already anticipated, the main result about ST2 —or STTS/TSST, if one prefers a more informative name—says that a metametainference (or metainference of level 2) is valid in CL if and only if it is valid in ST2 . Theorem 4.1.2 (The Second Collapse Result) For every metametainference  ⇒2  (being ,  sets of metainferences of level 1),  ⇒2  is valid in CL if and only if it is valid in ST2 . Proof This is a corollary of the more general result—i.e., the Third Collapse Result— that we prove below.  ST2 recovers not only every classically valid metametainference, but also every classically valid metainference, inference, and sentence. But metainferential logics of level n not only determine a set of valid metainferences of level n, but a set of valid metainferences of a level higher than n. In the case of ST2 , for any metainference  ⇒3  of level 3  ⇒3  is valid in ST2 if and only if, for every valuation v, if v confirms every premise γ ∈  according to ST2 , then v satisfies at least one δ ∈  in ST2 . And generalizing to any level n, as we mentioned, for any metainference  ⇒n  of level n  ⇒n  is valid in ST2 if and only if, for every valuation v, if v confirms every premise γ ∈  according to ST2 , then v satisfies at least one δ ∈  in ST2 . In general, for every logic L of level n and for any metainference  ⇒n+1 ,  ⇒n+1  is valid in L if and only if, for every valuation v, if v confirms every premise γ ∈  according to L, then v satisfies at least one δ ∈  in L. And, of course, this operation can be repeated to determine any metainference of any level higher than n.4 Thus, though ST2 recovers more classically valid metametainferences than ST1 —or TSST, as is better known—, it is not fully classical either. Many classically valid metametametainferences—or metainferences of level 3—are invalid in ST2 . In particular, what Pailos in [2] named as Meta-Meta-Cut—which is a classically valid metametametainference—turns out not valid. To understand what Meta-Meta-Cut looks like, we should better recall what Meta-Cut is. 3

We can even go one step further and start with the sentential standard T, as Scambler does in Scambler [6]. Nevertheless, we prefer the route taken by Barrio, Pailos, and Szmuc [1] and Pailos [2] and stick to a hierarchy of mixed logics. 4 This is what Ripley, both in Ripley [4] and on Ripley [3], calls lifting the satisfaction standard (or, in his case, the counterexample relation).

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4 Hierarchies of Strong Kleene Metainferential Logics

11 ⇒ 11 , ... 1j ⇒ 1j

12 ⇒ 21 , ...k2 ⇒ 2k

11 ⇒ 11 , φ

φ, 12 ⇒ 21

Meta-Cut

11 ⇒ 11 , ... 1j ⇒ 1j , 12 ⇒ 21 , ...k2 ⇒ 2k 11 , 12 ⇒ 11 , 21

Now, Meta-Meta-Cut is a metainferential schema of level 3 that has, as a conclusion, a metainference of level 2, that has, as a conclusion, Meta-Cut’s conclusion. The premises of this conclusion are a finite set of metainferences. Each premise has the following structure (for 1 ≤ j > ω, 1 ≤ k ≤ 2):

PREMS

k1 ⇒ k1 , ...kj ⇒ kj k ⇒ k

Meta-Meta-Cut has two premises. The conclusion of each premise is, on the one hand, the first premise of Meta-Cut, and, on the other hand, the second Meta-Cut’s premise. Meta-Meta-Cut’s premises have the same set of premises as the premises of Meta-Meta-Cut’s conclusion. This is how the Meta-Meta-Cut-schema looks like: 11

P R E M S k=1 P R E M S k=2 1 1 1 2 ⇒ 1 , ...,  j ⇒  j 1 ⇒ 21 , ..., k2 ⇒ 2k 11 ⇒ 11 , φ

11

⇒

11 , ...,  1j

φ, 12 ⇒ 21

P R E M S k=1,2 ⇒ 1j , 12 ⇒ 21 , ..., k2 ⇒ 2k

11 , 12 ⇒ 11 , 21 The following is a more formal definition of Meta-Cut: Definition 4.1.3 Consider the following set of sequents: • P01 = 01 ⇒0 10 , φ • P02 = 02 , φ ⇒0 20 • C0 = 01 , 02 ⇒0 10 , 20 . Then we define Cut as: P01 , P02 ⇒1 C0 . Let i1 , i2 be any two metainferential schemas of level i. And inductively, define for every i > 0: 1 • Pi1 = i1 ⇒i Pi−1 2 2 2 • Pi = i ⇒i Pi−1 1 2 • Ci = i , i ⇒i Ci−1 .

Meta-Cuti+1 as follows: Pi1 , Pi2 ⇒i+1 Ci .

4.1 The ST-Hierarchy

51

Meta-Meta-Cut (or Meta-Cut2 ) is an invalid ST2 metametametainference. The j valuation v such that v(φ) = 21 , v(γ) = v(δ) = v(σ) = 1—for every γ ∈ i , every l δ ∈ δmn and every σ ∈ k —, v(π) = 0—for every π ∈ rs —, and v(θ) = v(ξ) = v(φ) = v(φ) = 1—for every θ ∈ ed , every ξ ∈ ed , every φ ∈ e , and every ψ ∈ e —satisfies every premise, but does not confirms its conclusion, both things according to ST2 . Now, it can be shown5 that ST2 cannot recover every classically valid metainference of a level higher than 3. Nevertheless, it is possible to design a hierarchy of metainferential logics STn —one for each n (1 ≤ n < ω)—such that for every logic for metainferences of level n, that logic recovers every classically valid metainferences of level n or less. Thus, for any classically valid metainference of some level j, there is a logic characterized by substructural means that captures it. The next paragraphs are devoted to presenting this hierarchy. In order to define the hierarchy of logics STn we need the following definition. Definition 4.1.4 For any logic Lj /Lk , (Lj /Lk )∗ =Lk /Lj . When applied to a metainferential logic, the star operation ∗ gives another logic. This new logic switches the original logic’s standards for the premises and the conclusion of a sound argument (e.g., a metainference). With the help of the star operation ∗, it is now possible to define the desired hierarchy of metainferential logics. Definition 4.1.5 ST0 =ST, ST1 =TS/ST, and for every n such that 2 ≤ n < ω, a metainference of level n  ⇒n  is valid in STn if and only if, for every valuation v, if every γ ∈  is satisfied by v according to (STn−1 )∗ , then v satisfies  according to STn−1 . Thus, we can prove that each logic STn of the hierarchy coincides with classical logic at the level n. This is what we call the Third Collapse Result.6 In order to show this, we need some Lemmas. Lemma 4.1.6 For all n ≥ 1, for all , , ,  ⊆ M E T An−1 (L), and all SK valuations v, there is a Boolean valuation v such that: if then

v STn  ⇒n  

v CL  ⇒n 

and

v (STn )∗  ⇒n ,

and

v CL  ⇒n 





Similarly, for all Boolean valuations v, there is an SK-valuation v such that: if then

5

v CL  ⇒n  

v STn  ⇒n 

v CL  ⇒n ,

and and



v (STn )∗  ⇒n 

Though we will not do it here, as we will rely on a more general proof that will be given below. This hierarchy, and also the TS-hierarchy we will define in a while, are two types of Thue-Morse sequences, as Eoin Moore (in private conversation) has pointed out to us.

6

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4 Hierarchies of Strong Kleene Metainferential Logics

Proof We prove them by induction on the index of the logic. Base case: n = 1. These results have already been proven in Lemmas 3.2.2 and 3.2.3, respectively. Inductive step: n > 1. We will prove now the first conditional. Assume that there is an SK valuation v such that v STn  ⇒n  and v (STn )∗  ⇒n . From the fact that v STn  ⇒n  we may infer that v (STn−1 )∗ γ, for all γ ∈  and that v STn−1 δ, for every δ ∈ . From the fact that v (STn )∗  ⇒n  we may infer that v STn−1 θ, for some θ ∈ , or v (STn−1 )∗ σ, for some σ ∈ . Furthermore, from all these facts and the Inductive Hypothesis, there is a guarantee that there is a Boolean valuation    v such that, on the one hand, v CL γ, for all γ ∈  and v CL δ, for every δ ∈ ,   and, on the other, v CL θ, for some θ ∈ , or v CL δ, for some δ ∈ . But, by  definition, this is the same as saying that there is a Boolean valuation v such that   v CL  ⇒n  and v CL  ⇒n . The proof of the remaining conditional is similar.



We are now in a position to present the Third collapse result: Theorem 4.1.7 For every level n (1 ≤ n < ω), a metainference of level n,  ⇒n  is valid in CL if and only if it is valid in STn . Proof The proof is by induction on the index of the logic. Base case: n = 1. From left to right, let us suppose that CL  ⇒1 , from which we infer that there is a Boolean valuation v such that v CL γ, for all γ ∈ , and yet v CL δ, for every δ ∈ . But, since Boolean valuations are a subset of SK valuations, we know by the definition of satisfaction in TS and ST that v is also an SK valuation such that v TS γ, for all γ ∈ , and yet v ST δ, for every δ ∈ . Thus, by the definition of validity of a metainference of level 1 in TS/ST, we know that v TS/ST  ⇒1 , whence TS/ST  ⇒1 . From right to left, let us suppose that TS/ST  ⇒1 , from which we infer that there is an SK valuation v such that v TS γ, for all γ ∈ , and yet v ST δ, for all δ ∈ . By Lemma 3.2.4 we know that    there is a Boolean valuation v such that v CL γ, for all γ ∈ , and yet v CL δ,  for every δ ∈ . Therefore, we know that v CL  ⇒1 , whence CL  ⇒1 . Inductive step: n > 1. From left to right, let us suppose that CL  ⇒n , from which we infer that there is a Boolean valuation v such that v CL γ, for all γ ∈ , and yet v CL δ, for every δ ∈ . By Lemma 4.1.6 we know that there is an SK    valuation v such that v (STn−1 )∗ γ, for all γ ∈ , and yet v STn−1 δ, for every  δ ∈ . Therefore, we know that v STn  ⇒n , whence STn  ⇒n . From right to left, let us suppose that STn  ⇒n , from which we infer that there is an SK valuation v such that v (STn−1 )∗ γ, for all γ ∈ , and yet v STn−1 δ, for  every δ ∈ . By Lemma 3.2.4 we know that there is a Boolean valuation v such that   v CL γ, for all γ ∈ , and yet v CL δ, for every δ ∈ . Therefore, we know that   v CL  ⇒n , whence CL  ⇒n .

4.1 The ST-Hierarchy

53

Before moving on, we would like to stress a general fact, that explains why each step in the hierarchy of logics STn recovers increasingly more classically valid metainferences. We have already mentioned that what it takes for a valuation to be a counterexample in ST to an inference  ⇒  is exactly what it takes for it to be a counterexample to that inference in CL and that what is required for it to satisfy an inference  ⇒  in TS is the same as what is required for it to satisfy that inference in CL. Similarly, for any level n, what it takes for a valuation to be a counterexample in STn to a metainference of level n is exactly what it takes for it to be a counterexample to that inference in CL. And, finally, for any level n, what is required for a valuation to satisfy a metainference of level n in CL is exactly what is required for it to satisfy that metainference in (STn )∗ . Nevertheless, none of the logics STn is fully classical. In fact, every STn invalidates a version of Meta-Cut of level n. Theorem 4.1.8 For every n (1 ≤ n < ω), STn invalidates Meta-Cutn . Proof The proof is also by induction. We have already presented the structure of both Meta-Cut and Meta-Meta-Cut, and showed that Meta-Cut—which can be renamed as Meta-Cut 1 —is invalid in TS/ST. Furthermore, it is possible to specify the structure of each Meta-Cut of level j (2 ≤ j < ω) constructively. Each Meta-Cut j —i.e., MetaCut of level j—is a two-premise metainference of level j + 1. Each premise is a metainference of level j that has, as its conclusion, one of the premises of an instance of Meta-Cut of level j − 1. The conclusion of Meta-Cut of level j is also a metainference of level j that has, as a conclusion, the conclusion of an instance of Meta-Cut of level j − 1. But the two premises and the conclusion of the particular version of Meta-Cut j that we will be talking about, have the same (and single) premise. For the sake of simplicity, it will be a metainference of level j − 1 such that every propositional letter in it does not appear in the instance of the metainference Meta-Cut j−1 used to design this instance of Meta-Cut j . Moreover, every inference that occurs in it will be an instance of p ⇒ q. Here, p and q are propositional letters that appear only once in the metainference. Thus, the metainference that is the premise of both premises, and also the premise of the conclusion of Meta-Cut j , can be considered a contingent metainference, in the sense that some valuations will satisfy it, and some others will not. Now we are ready to present a counterexample to Meta-Cut j . It will be a valuation v such that, for every p, q in the premise of both premises of Meta-Cut j —e.g. two metainferences of level j—, and also in the premise of the conclusion, v( p) = v(q) = 1. The rest of the formulas—e.g., the conclusion of each premise, and also the conclusion of the conclusion—will be part of Meta-Cut j−1 . By the inductive hypothesis, there is a valuation v∗ that is a counterexample to Meta-Cut j−1 . For every formula ψ in Meta-Cut j−1 —e.g., the conclusion of each premise, and also the conclusion of the conclusion—, let v(ψ) = v ∗ (ψ). In particular, let the meta-cutformula A be such that v(φ) = v ∗ (φ) = 21 . The valuation v satisfies every premise and does not confirm the conclusion of Meta-Cut j in STj . Remember that Meta-Cut j is a metainference of level j + 1. A valuation v does not satisfy an instance of Meta-Cut j

54

4 Hierarchies of Strong Kleene Metainferential Logics

in STj if and only if it satisfies every premise and does not satisfy the conclusion, both according to STj . The conclusion and each premise are metainferences of level j. Thus, v satisfies any of them in STj if and if either if v does not satisfy some premise according to ST∗j−1 , or it confirms the conclusion according to STj−1 . As v(ψ) = v ∗ (ψ), for every ψ in Meta-Cut j−1 , and v∗ is a counterexample to it, then v∗—and thus v—satisfies every Meta-Cut j−1 ’s premise according to ST∗j−1 . Thus, it satisfies it according to STj−1 , also, as every valuation that satisfies a metainference in any ST∗k , also satisfies it in STk , as it is easy to see.7 Therefore, it satisfies the conclusion of every premise of Meta-Cut j in STk , because it satisfies each conclusion. Moreover, v does not satisfy Meta-Cut j ’s conclusion. Remember that for every p, q in the premise of the conclusion, v( p) = v(q) = 1. But the premise of the conclusion is satisfied by v according to ST∗j−1 . As v(ψ) = v ∗ (ψ), for every ψ in Meta-Cut j−1 , and v∗ is a counterexample to it, then v will not satisfy Meta-Cut j−1 ’s either. But Meta-Cut j−1 ’s conclusion is the conclusion of the conclusion of Meta-Cut j . Then, v will satisfy each premise of Meta-Cut j , but will not confirm its conclusion. Therefore, v is a counterexample to it.  The goal, now, is to define a logic that recovers every classically valid metainference. We will call it STω , and it can be interpreted as “the union of every STn ”—at least if we focus on the things it validates. Definition 4.1.9 For every valuation v, v satisfies a metainference of level n— for any level n— ⇒n  in STω , if and only if  ⇒n  is satisfied by v in STn (1 ≤ n < ω). Theorem 4.1.10 (The General Recovery Result) For every level n (1 ≤ n < ω), a metainference of level n  ⇒n  is valid in CL if and only if it is valid in STω . Proof We know, by the Third Collapse Result, that for every level n (1 ≤ n < ω), a metainference of level n  ⇒n  is valid in CL if and only if it is valid in STn . But if it is valid in STn , then, by the definition of STω , it is valid in STω . Therefore,  every classically valid metainference of any level n will be valid in STω . What is, then, the exact relationship between CL and STω ? Dave Ripley have claimed in [7, 8] that ST is just another way to present classical logic. His point is that ST recovers every classically valid inference. Others, like Barrio, Rosenblatt and Tajer [9] and Dicher and Paoli [10], argue against Ripley’s statement. These authors claim that there are some central features of CL that ST cannot recover. Those features are some classically valid metainferences, mainly of level 1. But ST loses classically valid metainferences in every level.8 TSST—e.g., TS/ST—recovers 7

Or, to put it another way, for every metainference of every level k, the set of counterexamples to it in STk is properly included in the set of counterexamples to it in ST∗k . We leave the proof of these facts as an exercise to the reader. 8 Though ST recovers every classically valid metainference from a global point of view, it loses many of them from a local perspective, which is the right way to think about metainferential validity according to Dicher and Paoli [10].

4.2 The TS-Hierarchy

55

every classically valid metainferences, but still does not recover some classically valid metametainferences, like Meta-Cut. ST2 recovers them, but still cannot recover every classically valid metametametainference. Moreover, for every n, though STn recaptures every classically valid metainference of level n, cannot recover every classically valid metainference of higher levels. In this regard, STω works better than every STn . STω recovers every classically valid metainference of every level. If classical logic is defined through the inferences and metainferences of every level that turns out valid in it—as Barrio, Pailos and Szmuc [1] defend—, then STω is a fully classical logic. The only special thing about it is that it is defined through substructural means.

4.2 The TS-Hierarchy Just like we have defined a hierarchy of metainferential logics that, in a way, is based on ST, we can define, using the exact same procedure, a hierarchy of metainferential logics based on TS. This hierarchy, of course, will not recover classical logic—or at least not in the same way as the ST-hierarchy. In the sequel, we will define it and specify a logic of level ω based on the hierarchy.9 Definition 4.2.1 TS0 =TS, TS1 =ST/TS, and for every n such that 2 ≤ n < ω, a metainference of level n  ⇒n  is valid in TSn if and only if, for every valuation v, if every γ ∈  is satisfied by v according to (TSn−1 )∗ , then v satisfies  according to TSn−1 . Thus, for every consequence relation TSn , the following is true: Fact 4.2.2 (Metainferential Emptiness) For every level n (1 ≤ n < ω), each metainference of level n,  ⇒n  is invalid in TSn —i.e., every TSn is empty up to level n. The following fact shows this. Fact 4.2.3 For every level n (1 ≤ n < ω), the valuation v such that, for every propositional letter p, v( p) = 21 : • is a counterexample to every metainference  ⇒n  in TSn • satisfies every metainference  ⇒n  in STn .

9

This hierarchy was used in both Barrio, Pailos and Szmuc [1] and Pailos [2], but has been highlighted and carefully studied by Chris Scambler in [6]. Moreover, Scambler was the first to defend that the TS-hierarchy recaptures as much classical logic as the ST-hierarchy, because it recovers every classical antivalidity.

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4 Hierarchies of Strong Kleene Metainferential Logics

Proof Regarding the base case, we know that that is the case for both TS1 , i.e.,  ST/TS, and for ST1 , i.e., TS/ST. The inductive step is left to the reader. Despite its emptiness at the nth level, every TSn is informative about the metainferences of level n + 1—i.e., each of them validates at least one such metainferences. As an example of a valid metainference of level n + 1, take what we will call (an instance of) Meta-Identity of level n + 1: a metainference φ ⇒n+1 φ, where φ is a metainference of level n. For any valuation v, either v satisfies φ according to TSn , or it doesn’t. If it does, then it satisfies the conclusion of φ ⇒n+1 φ, and thus satisfies φ ⇒n+1 φ itself. And if it doesn’t, then it does not satisfy the premise of φ ⇒n+1 φ, and so it satisfies the metainference itself. TSn is not trivial at the n + 1 level, either. Consider a metainference of level n + 1,  ⇒ , where both  and  are different instances of Meta-Identity of level n. Moreover, at each stage of the metainference, there will be only one inference, an instance of Identity, i.e., a case of φ ⇒ φ. For the sake of simplicity, let φ be a propositional letter, and a different one in  and . Thus, the only inference that appears in  is, for example, p ⇒ p, and the only inference that is part of  is, say, q ⇒ q. Take a valuation v such that v( p) = 1 and v(q) = 21 . That valuation will satisfy  according to TSn , but would not satisfy  according to TSn . What we just show proves that, though every logic TSn is empty regarding metainferences of level n and below, it is nonetheless informative regarding metainferences of level n + 1—i.e., each TSn validate at least one of those metainferences, but also at least one of them is invalid. Thus, though no logic in the hierarchy is empty at every metainferential level, each metainferential level is empty at some point (and downwards) in the hierarchy. Nevertheless, no logic TSn is empty at every level. In the sequel, we will introduce a logic that has this feature. To define it, we will use the TS-hierarchy. This logic is empty at every metainferential level. We will call it TSω —in part, for its obvious inspiration in STω —, and it can be interpreted as “the intersection of every TSn ”—at least if we focus in the things it validates.10 Definition 4.2.4 For every valuation v, v satisfies a metainference of level n— for any level n— ⇒n  in TSω , if and only if  ⇒n  is satisfied by v in TSn (1 ≤ n < ω). Fact 4.2.5 (The General Emptiness Result) For every level n (1 ≤ n < ω), a metainference of level n  ⇒n  is invalid in TSω . This is immediate from Fact 4.2.3. We have shown how closed CL and STω are. Is there a similar relation between CL and TSω ? Here is where antivalidities come into the picture. TSω is what Scambler calls “the twist logic S”, just as STω is named by him as “the twist logic T”. They are called that way because the hierarchy starts not with a satisfaction standard for inferences, but with satisfaction standards for formulas, strict and tolerant, respectively. We prefer to start the hierarchies with the inferential level because the sentential standard does not fix a logic—i.e., it does not determine which inferences are valid, unless we are willing to extend the meaning of inference so formulas are also inferences.

10

4.2 The TS-Hierarchy

57

Scambler in [6] formulates a challenge against the idea that STω is equivalent, or just another way to present classical logic, as introduced and defended in both Pailos and Barrio, Pailos, and Szmuc [1].11 The authors argue that this hierarchy recovers classical logic. In the case of Barrio, Pailos, and Szmuc [1], this assertion is stated as the claim that every classical (meta)inferential validity is valid at some point—i.e., by some metainferential logic—in the hierarchy. Pailos [2] takes another route, an explicitly defends that STω —or CM! , as he calls it—is the one equivalent to classical logic, because STω shares its validities with classical logic. Now, the idea that this hierarchy recovers classical logic, is identical to classical logic, or is equivalent to it, as we have mentioned, has been defied by Scambler [6]. Scambler argues that, as the hierarchy recovers no classical antivalidities, it cannot be identified with (or being equivalent to) CL. He claims that once we bring antivalidities into the picture, none of the logics in the ST-hierarchy, and neither also STω , can be defined as classical. Scambler argues that the ST-hierarchy provides, at best, a positive characterization of classical logic—i.e., one that recovers every classical validity. But a positive characterization is not better than a negative approach to it— i.e., one that recaptures every classical antivalidity, he claims. A similar conclusion can be presented against STω . Scambler also shows that TSω negatively recovers classical logic, because it shares with classical logic its antivalidities. Therefore, TSω is as good a candidate for recapturing classical logic as STω . If TSω does not recover classical logic, then STω does not recover it either. It seems that, from Scambler’s perspective, a logic A characterizes a logic B only if it characterizes it both positively and negatively. And neither Barrio, Pailos, and Szmuc [1] and Pailos [2] have shown that the ST-hierarchy can be equal to CL in both of these aspects. In order to understand why Scambler argues that the ST-hierarchy does not (negatively) characterize classical logic, we must say more about what antivalidities are. In a nutshell, a formula is antivalid if it receives a designated value in no valuation, where which truth-values are designated is fixed by the logic. Similarly, for any level n, a metainference  ⇒n φ (or an inference  ⇒ φ) is antivalid according to a logic if and only if no valuation satisfies it according to that logic. So, for example, p ⇒ ¬ p, even being classically invalid, is not classically antivalid, because there is at least one valuation that does not satisfy the premise. (Meta)Inferential antivalidity is not invalidity, but (meta)inferential unsatisfiability. In general, antivalidities have validities as premises and antivalidities as conclusions. For example, p ∨ ¬ p ⇒ p ∧ ¬ p is a case of a classical antivalidity. One of the main reasons to focus on antivalidities is the following. It is important to pay attention not only to what a logic accepts but also to what it rejects. And antivalidities—and not invalidities—correspond exactly to what a logic rejects. 11

Though, strictly speaking, [1] claim that it is in fact the hierarchy of logics STn that is somewhat equivalent to classical logic. This statement is, we think, misleading, as they are comparing one logic with a hierarchy of logics. If they are claiming that the validities in the hierarchy are the things valid at some point in the hierarchy, and its invalidities, the things that are invalid everywhere in the hierarchy, then they are, in fact, implicitly defending that it is STω the one equivalent to classical logic.

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At first, this might seem a little surprising—i.e. to separate invalidities from what a logic rejects. But a little thought might shed some light on this issue. At first glance, rational agents should reject invalid inferences. Antivalidities are just a particular kind of invalidities. But there is a more illuminating way to think about them. Antivalidities are formulas, inferences, metainferences, etc, that should be rejected no matter what, in any context. And this is not what happens with every invalid inference. Inductive reasoning, for example, is classically invalid. Nevertheless, we should not always reject it. Moreover: there are situations that demand us to embrace it—i.e., sometimes making inferences based on inductive reasoning is the rational choice. Where is the limit to what can be embraced? A quick and straightforward answer is: antivalidities. Antivalidities—and not invalidities—are what a logic truly rejects. Facing an invalid reasoning, what the logic informs us is that there is no guarantee that we are dealing with a good piece of reasoning, one that can always be trusted, no matter what circumstances. When facing invalid inferences, we should pay attention to the details—i.e., to the reasoning, but also to the context—before adopting an attitude of acceptance or rejection towards it. Antivalid reasoning does not pose such a question: we should reject it straightforwardly. From this bilateralist-inspired point of view—i.e., one that stresses the importance not only of acceptance, but also of rejection when defining a validity relation, and their mutual irreducibility—,12 it is important not only to look at what a logic accepts but also to what it rejects. And if a logic rejects its antivalidities, and TSω ’s antivalidities are just CL’s antivalidities, then TSω reject a lot of (meta)inferences. Though Barrio and Pailos [11] faces Scambler’s challenge, the way the authors deal with it is by a completely different type of mixed-logics: multi-standards logics. And, as not every multi-standard logic is a metainferential logic, we will not delve in these theories here.13 Scambler not only presents this challenge but also does a completely new thing: he extends these hierarchies into the transfinite. Though we will not deal with transfinite hierarchies in the rest of the book, we will briefly explain how Scambler does it. The metainferences that we have met so far are, to use Scambler’s terminology, non-cumulative. This term refers to the fact that every premise and conclusion of every metainference we have been dealing with, belong to the same level. But in both Scambler [6] and, even more explicitly, in [12], Scambler introduce non-cumulative metainferences, and, moreover, non-cumulative hierarchies of metainferential logics. This is not the only way to do it, though. Brian Porter [13] and Isabella McAllister [14] get to the transfinite level in a slightly different way, that is, though, similar with respect to each other. McAllister is explicit about calling her way, non-cumulative. 12

Bilateralism explains validity in terms of acceptance and rejection, which is not what we are doing here. The purported inspiration from bilateralism comes only in claiming that when characterizing a logic, acceptance, and rejection are both worth paying attention to. For more about bilateralism, see Prawitz [15], Smiley [16], Rumfitt [17], Restall [18] and Ripley [19]. 13 Nevertheless, the multi-standard logics defined in Barrio and Pailos [1] are also cases of metainferential logics. What they do is to define new logics involving at least two independent standards: a validity and an antivalidity canon. The first one is give by STω , while the second one is determined by TSω .

4.2 The TS-Hierarchy

59

Though this is mainly a terminological debate, we think that it is less inaccurate to describe both Porter’s and McAllister’s as mixed approaches, as they are noncumulative up to ω, and then admit cumulative stages. But Scambler hierarchy is straightforward cumulative, in the sense that in each stage higher than 1, premises and conclusions can belong to any of the previous levels. Thus, his metainferences are “hybrid”, in the sense that, for example, one premise might belong to level 43, another to level 1000, while the conclusion is a metainference of level 1. And when the hierarchy reaches a limit, then it proceeds in basically the same way. Metainferences of level ω, then, involve metainferences of any of the previous levels, while metainferences of level ω + 1 might also have metainferences of level ω either as premises or as conclusions. Scambler defines two important transfinite logics, one based on the hierarchy that starts with ST, and the other based on the transfinite hierarchy that starts with TS. He calls the former one “the transfinite twist logic Tω→ ”, and the latter, “the transfinite twist logic Sω→ ”. He claims that the former recovers every classical metainferential validity, even in the transfinite level that the non-cumulative hierarchy leaves unexplored, while the former does a similar thing with classical antivalidities. Nevertheless, no member of both hierarchies is fully classical, in the sense of recovering both every metainferential validity and every metainferential antivalidity. With this, we end with the description of how the cumulative hierarchies S and T —and the transfinite logics based on those hierarchies—work.14 To be clear, we are not claiming that it is not possible nor convenient to extend the hierarchy of metainferences beyond ω. In fact, we will not follow this path, not because we think there is something wrong about going into the transfinite, but because it complicates things unnecessarily for the technical and philosophical purposes of this book. For one thing, as we have mentioned, there are at least two different ways to go transfinite, that are different and require different kinds of complications. Both of them demand adopting a decision about how to deal with limit ordinals—i.e., limit levels.15 But no matter how this is done, the limit levels behave quite differently from the successor levels. Moreover, both Scambler and McAllister are forced to deal with what has referred to as “hybrid” metainferences: metainferences with at least some premises and conclusions that belong to different levels. And while this might also be taken as a virtue—in fact, [20] argues for hybrid metainferences—, it is also a contentious position (as [20, 20] themselves acknowledge). Then there is also the question of how to address the metainferential standard of a hierarchy of metainferential standards at the level On. Not doing so (as Scambler does) seems arbitrary. But addressing it forces McAllister to define formulas for On as pairs of sets of formulas, instead of pairs of sets of formulas (of lower levels) plus an ordinal (different from On). There is, then, no uniform way to understand formulas from

14 There is a debate on whether this transfinite hierarchy that goes on up to On or not. Isabella McCallister in [14] claims that it does not, and describes another hierarchy that is, in a way, cumulative in the limits—though not at the successor stages. 15 With the possible exception of On in Scambler’s case.

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metainferential levels (i.e. metainferences), not to mention that the maneuver regarding On seems a little bit ad hoc. We eschew this problem by just talking about finite levels. Moreover, there is not just one single way to extend the ST hierarchy beyond ω, even adopting McAllister’s strategy. In fact, she explores at least two different ways to do it, which validate very different metainferences and can plausibly count as extending the ST hierarchy beyond ω. We face no such indeterminacy if we stick to finite metainferential levels. Finally, exploring the metainferential realm beyond the finite level may imply losing the link with (meta)inferential moves as done by finite beings like us humans (and, ultimately, to the contents of those acts, the things Hlobil in [21] called “arguments” and that we have decided to refer to as “inferences”). For all these reasons, we chose to stick to finite metainferential levels (we insist, just for the purposes of this book).

References 1. Barrio, E., Pailos, F., & Szmuc, D. (2020). A hierarchy of classical and paraconsistent logics. Journal of Philosophical Logic, 49(1), 93–120. https://doi.org/10.1007/s10992-019-09513-z 2. Pailos, F. (2020). A fully classical truth theory characterized by substructural means. The Review of Symbolic Logic, 13(2), 249–268. 3. Ripley, D. (2022). One step is enough. Journal of Philosophical Logic, 51(6). 4. Ripley, D. A toolkit for metainferential logics. In Manuscript. 5. Pailos, F. (2022). Why metainferences matter. In Manuscript. 6. Scambler, C. (2020). Classical logic and the strict tolerant hierarchy. Journal of Philosophical Logic, 49(2), 351–370. 7. Ripley, D. (2012). Conservatively extending classical logic with transparent truth. Review of Symbolic Logic, 5(2), 354–378. 8. Ripley, D. Paradoxes and failures of cut. Australasian Journal of Philosophy, 91(1), 139–164. 9. Barrio, E., Rosenblatt, L., & Tajer, D. (2015). The logics of strict-tolerant logic. Journal of Philosophical Logic, 44(5), 551–571. 10. Dicher, B., & Paoli, F. (2019). ST, LP and tolerant metainferences. In Graham Priest on dialetheism and paraconsistency (pp. 383–407). Springer. 11. Barrio, E., & Pailos, F. (2021). Validities, antivalidities and contingencies: a multi-standard approach. Journal of Philosophical Logic. https://doi.org/10.1007/s10992-021-09610-y 12. Scambler, C. (2020). Transfernite meta-inferences. Journal of Philosophical Logic, 49, 1079– 1089. https://doi.org/10.1007/s10992-020-09548-7 13. Porter, B. (2022). Supervaluations and the strict-tolerant hierarchy. Journal of Philosophical Logic, 51(6), 1367–1386. 14. McAllister, I. (2022). Classical logic is not uniquely characterizable. Journal of Philosophical Logic, 51(6), 1345–1365. 15. Prawitz, D. (1965). Natural deduction: A proof-theoretical study. Stockholm: Almquist and Winksell. 16. Smiley, T. (2015). Rejection. Analysis, 56(1), 1–9. 17. Rumfitt, I. (2000). “Yes” and “no”. Mind, 109(436), 781–823. 18. Restall, G. (2013). Assertion, denial, and non-classical theories. In Tanaka et al. 2013 (pp. 81–100).

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19. Ripley, D. (2017). Bilateralism, coherence, warrant. In F. Moltmann & M. Textor (Eds.), Actbased conceptions of propositional content (pp. 307–324). Oxford University Press. 20. Ferguson, T. M., & Ramírez-Cámara, E. (2022). Deep ST. Journal of Philosophical Logic, 1261–1293. Springer. https://doi.org/10.1007/s10992-021-09630-8 21. Hlobil, U. (2016). What is inference? Or the force of reasoning. Doctoral Dissertation (Unpublished): University of Pittsburgh. http://dscholarship.pitt.edu/28130/

Chapter 5

Weak Kleene Metainferential Logics

As we have mentioned in Chap. 3, in [1], Pailos introduces twelve impure metainferential logics defined using two different inferential Tarskian or non-Tarskian (e.g., substructural) logics: K3, LP, ST or TS. The standard for premises and conclusions is different in each case. Five of these logics—i.e., TS/ST, TS/LP, TS/K3, LP/ST, K3/ST—are Tarskian logics–i.e., they validate Cut, Identity and Weakening. Moreover, they also validate Meta-Identity, Contraction, and Exchange. Nevertheless, they are defined using substructural inferential logics. In fact, TS/ST is built entirely with substructural inferential standards. More significantly, TS/ST collapses with CL at the metainferential level, proving to be the only one of these twelve logics with this feature. So, one could wonder what happens when at least one of these standards is not defined through the Strong Kleene schema. The general goal of this chapter is to develop mixed logics defined using the Weak Kleene schema (instead of the Strong Kleene one). So, if L is a metainferential logic L1 /L2 —L1 and L2 being inferential logics (or metainferential logics of a lower level), we will introduce a family of mixed and impure metainferential logics—in the sense that L1 and L2 are different inferential logics—built over the so-called weak logics. In particular, we use the four logics defined over the Weak Kleene Schema: the structural logics K3w and LPw and the substructural logics STw and TSw .

5.1 Four Basic Inferential Logics Let’s start by introducing the Weak Kleene Schema. Definition 5.1.1 The Weak Kleene algebra is the structure    1 ¬ ∧ ∨ K = 1, , 0 , { f K , f K , f K } 2

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. Pailos and B. Da Ré, Metainferential Logics, Trends in Logic 61, https://doi.org/10.1007/978-3-031-44381-7_5

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where the functions f K¬ , f K∧ , f K∨ are as follows f K¬ f K∧ 1 0 1 1/2 1/2 1/2 1 0 0

1 1 1/2 0

1/2

0 0 1/2 1/2 1/2 0 1/2

f K∨ 1 1/2 0

1 1 1/2 1

1/2

0 1 1/2 1/2 1/2 0 1/2

The functions → and ↔ can be defined in the usual way. A Weak Kleene valuation (hereafter, WK-valuation) is a homomorphism from FOR(L) to the Weak Kleene algebra, just like a Strong Kleene valuation (hereafter, SK-valuation) is a homomorphism from FOR(L) to the Strong Kleene algebra. Notice here that, just like in the Strong Kleene case, the functions are Boolean normal: when restricted to classical inputs, they give the result of the classical operations. So trivaluations assigning a classical value to every propositional letter behave as Boolean bivaluations. Also, the operations are monotonic regarding the informational order. Finally, valuations satisfying the operations of the Weak Kleene truth tables are called Weak Kleene valuations (or just W K -valuations). As we did in Chap. 3, our target logics have a validity relation defined for metainferences. Therefore, to understand exactly how they work, we need to introduce first the new standards for inferential logics that will be used to build them. Thus, we will introduce the Weak Kleene logics we will be working with. They can be understood as the Weak Kleene versions of the previous four Strong Kleene logics, because the main difference between them is that the valuations are now restricted by the operations of the Weak Kleene Algebra (instead of being restricted by the Strong Kleene operators). Definition 5.1.2 An inference  ⇒  is valid in STw if and only if there is no valuation v such that for every γ ∈ , v(γ) = 1, and for every δ ∈ , v(δ) = 0. STw is also non-transitive (as Cut is not valid in STw ). Moreover, just as it happens with its Strong Kleene version, every classically valid inference is STw -valid. This interesting fact has been proved by Ferguson and Szmuc in [2]. Definition 5.1.3 An inference  ⇒  is valid in TSw if and only if for every valuation v, there is a γ ∈ , v(γ) = 0, or there is a δ ∈ , v(δ) = 1. TSw is non-reflexive (Identity is TSw -invalid). Moreover, no inference is valid in TSw , as the valuation v that gives value 1/2 to every propositional letter witnesses. We will be working also with the usual three-valued semantics for Weak Kleene propositional versions of LP and K3, usually referred to as Paraconsistent Weak Kleene (LPw ) and Weak Kleene (K3w ).1 Here is a brief introduction to a three-valued version of them. K3w is sometimes denoted as B3 due to Bochvar [3], WK or even K3w . Also, LPw is sometimes found as H3 due to Halldén [4], or PWK. See, for example, Da Ré, Pailos and Szmuc [5], Szmuc [6] or Szmuc [7], among many others. In this chapter, we choose this somewhat nonstandard terminology K3w and LPw in order to increase the readability of the content.

1

5.2 Weak Kleene Mixed Metainferential Logics (of Level 1)

65

Definition 5.1.4 An inference  ⇒  is valid in LPw if and only if there is no valuation v such that for every γ ∈ , v(γ) ∈ {1, 21 }, and for every δ ∈ , v(δ) = 0. LPw is, of course, a paraconsistent logic. Neither Explosion nor Modus Ponens are valid in LPw . Moreover, every classically valid sentence is LPw -valid. Definition 5.1.5 An inference  ⇒  is valid in K3w if and only if there is no valuation v such that for every γ ∈ , v(γ) = 1 and for every δ ∈ , v(δ) ∈ {0, 21 }. Just as it happens with its Strong Kleene relative, no sentence is valid in K3w (without constants for truth-values). In the next section, we will introduce the first family of new metainferential logics that are defined using only Weak Kleene inferential logics.

5.2 Weak Kleene Mixed Metainferential Logics (of Level 1) We will present the sixteen mixed metainferential logics that can be built with STw , TSw , LPw and K3w . Those are the logics listed in Fig. 5.1: In what follows, we will focus only on the twelve impure metainferential logics, i.e. logics X/Y where X = Y. As we did in Chap. 3 we will describe how the Weak Kleene metainferential logics behave regarding the structural properties they validate. We will only mention the two most important metainferential logics, and after that, we will provide an exhaustive table regarding all the metainferential logics of level 1. The first metainferential logic we are interested in is TSw /STw . Like its Strong Kleene counterpart, this logic not only validates every classically valid inference, as ST and STw do, but also every classically valid metainference (including Cut, Meta-Explosion, Meta-Modus-Ponens, etc.), as the following fact shows: Fact 5.2.1 For every metainference  ⇒1 ,  ⇒1  is valid in CL if and only if  ⇒1  is valid in TSw /STw .

L1 /L2 STw TSw LPw K3w

STw STw /STw TSw /STw LPw /STw K3w /STw

Fig. 5.1 Sixteen metainferential logics

TSw STw /TSw TSw /TSw LPw /TSw K3w /TSw

LPw STw /LPw TSw /LPw LPw /LPw K3w /LPw

K3w STw /K3w TSw /K3w LPw /K3w K3w /K3w

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The proof of this result is similar to the proof of Theorem 3.2.6 regarding the Collapse Result between CL and TS/ST—i.e., the Strong Kleene counterpart of TSw /STw . Therefore, every classically valid metainference is valid in TSw /STw . It is, thus, a structural consequence relation defined through substructural means (e.g., logics). Thus, Cut, Identity, Meta-Identity, Weakening, Contraction, and Exchange are valid in TSw /STw . The second important logic is STw /TSw . This logic, as its Strong Kleene counterpart, is, in a way, as metainferentially empty as it can get —thus being even emptier than TS and TSw . Fact 5.2.2 No metainference  ⇒1  is valid in STw /TSw . Proof Consider a valuation v such that, for every propositional letter p that appears in  ⇒1 , v( p) = 21 . As STw /TSw matrices for the logical constants are the ones of K3w , v gives the value 1/2 not only to every propositional letter but also to every formula φ in the metainference. Thus, every premise γ ∈  is satisfied by v according  to STw , but v does not satisfy any conclusion δ ∈  according to TSw . So TSw /STw is a fully Tarskian logic, despite being characterized through nonTarskian—or substructural—theories. The following are two tables that summarize the structural properties validated by each metainferential logic (Fig. 5.2, 5.3). Metainferences CUT IDENTITY META-IDENTITY WEAKENING CONTRACTION EXCHANGE

TSw /STw Yes Yes Yes Yes Yes Yes

STw /TSw No No No No No No

K3w /LPw No Yes No No No No

LPw /K3w No Yes No No No No

LPw /STw Yes Yes Yes Yes Yes Yes

LPw /TSw No No No No No No

K3w /TSw No No No No No No

STw /LPw No Yes No No No No

STw /K3w No Yes No No No No

TSw /LPw Yes Yes Yes Yes Yes Yes

TSw /K3w Yes Yes Yes Yes Yes Yes

Fig. 5.2 Comparison table Metainferences CUT IDENTITY META-IDENTITY WEAKENING CONTRACTION EXCHANGE

K3w /STw Yes Yes Yes Yes Yes Yes

Fig. 5.3 Comparison table (cont.)

5.2 Weak Kleene Mixed Metainferential Logics (of Level 1)

67

We would like to address the obvious: the similarities between this family of logics and its Strong Kleene metainferential relatives. Firstly, notice that these tables are exactly the same as the Strong Kleene ones, in the following sense: Fact 5.2.3 Xw /Yw validates (invalidates) some structural property if and only if X/Y validates it, where X, Y ∈ {LP, K3, ST, TS} and X = Y. Thus, regarding the structural properties we have listed, the two families of logics behave exactly in the same way. However, this does not mean that, for each of these metainferential logics of level 1, the Strong and Weak Kleene versions of them validate exactly the same (meta)inferences. Actually, they do not. For example, the following metainference: ⇒ p∧q ⇒p is valid in TS/LP, while it is not valid in TSw /LPw . However, the relative strength between the logics is also the same as in the Strong Kleene case as the Fig. 5.4 shows: Also, following the comparison with the Strong Kleene results, Dave Ripley has proved, in [8] two main facts about these Weak Kleene metainferential logics: (1) that the inclusion relations between them are the same as the one exhibited by its Strong Kleene metainferential relatives (as we have illustrated before); (2) that TSw /STw recovers classical logic up to level 1, and, moreover, that a hierarchy of metainferential logics based on this logic will also recover classical logic, just like the hierarchy based on TS/ST.

Fig. 5.4 Hasse diagram of the inclusion ordering of the Weak Kleene Metainferential Logics

TSw /STw

LPw /STw

TSw /LPw

LPw /LPw

LPw /K3w

K3w /STw

STw /STw

TSw /TSw

LPw /TSw

TSw /K3w

K3w /LPw

K3w /TSw

STw /LPw

STw /K3w

STw /TSw

K3w /K3w

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Regarding (1), Ripley says the following: The key upshot of Theorem 31 [i.e., a technical theorem whose result Ripley is about to explain] is this: if we have a collection of meta counterexample relations such that we know which of them are subsets of which others, that alone is enough to fully settle all questions about subsethood among the counterexample relations we can form from these by slashing. For example, suppose we have two meta+1 counterexample relations S and T such that T  S. This is enough to guarantee that inclusions among meta+1 counterexample relations are organized as in Fig. 5.1 [i.e., a figure that shows that TS is properly included in both LP and K3, that the three of them are properly included in ST, but neither LP is properly included in K3, nor the other way around]. And this, in turn, is enough to guarantee that inclusions among meta+2 counterexample relations are organized as in Fig. 5.2 [i.e., our Fig. 5.4. In these figures, lines going up indicate inclusion, all pictured counterexample relations (in Ripley’s paper) or satisfaction relations (in our figure) are distinct, and all inclusions are indicated.] [1] explores the structure of slashed counterexample relations2 built on a particular choice of such S, T requiring a particular choice of language and of models. What we have seen here, though, is that the inclusion structure of these slashed counterexample relations is totally independent of those details; it is fully settled just by the fact that T  S. Any other setting with two counterexample relations related like this would be isomorphic in its inclusions at all levels. ([8, p. 10])

Thus, the whole inclusion relations between these metainferential counterexample relations are given by the fact that, on the Weak Kleene schema, T  S—just like the same inclusion relations of the metainferential logics defined using the Strong Kleene schema is also given by the fact that T  S. Moreover, the relations between the Strong Kleene versions of these Weak Kleene logics are the same as the ones exhibited by the Weak Kleene (metainferential) logics. Regarding (2), Ripley says the following. ... by examining the reasoning just deployed, we can see that just the same reasoning would work if we used weak Kleene models instead of strong, over the same language... So these hierarchy results hold as well of weak Kleene models: we have a new slash hierarchy, now built on weak Kleene models, that agrees exactly with classical logic at every level, and such that each full consequence relation determined by the hierarchy is distinct.

Thus, there is a hierarchy of metainferential logics that starts with TSw /STw , and recovers every classically valid metainference of any level. Moreover, it is possible to define a transfinite logic based on this new hierarchy that recovers all of them. Let’s see now more formally how the above-mentioned hierarchy of logics based on TSw /STw can be defined in order to recover classical logic.

2

Because, strictly speaking, the author talks about counterexample relations and not satisfaction relations, but, in the framework in which we are working, they are equivalent approaches.

5.2 Weak Kleene Mixed Metainferential Logics (of Level 1)

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w Definition 5.2.4 STw 0 =STw , ST1 =TSw /STw , and for every n such that 2 ≤ n < ω, a metainference of level n  ⇒n  is valid in STw n if and only if, for every valuation ∗ v, if every γ ∈  is satisfied by v according to (STn−1 w ) , then v satisfies  according w to STn−1 .

Recall that for any logic Lj /Lk , (Lj /Lk )∗ =Lk /Lj . Thus, for every logic STw n , the following is true: Theorem 5.2.5 (The (Weak) Collapse Result) For every level n (1 ≤ n < ω), a metainference of level n,  ⇒n  is valid in CL if and only if it is valid in STw n. Proof The proof of this Theorem can be easily adapted from the proof of Theorems 3.2.6 and 4.1.7, so we leave it to the reader.  Therefore, every logic in the hierarchy has the same valid inferences as CL. By this, we mean that each STw n (for any finite n) coincides with CL up to its valid (meta)inferences of level n. It is worth noting that the fact that the validities obtained in a given logic are retained in the successive systems appearing in the hierarchy, means that the hierarchy has a cumulative nature. Nevertheless, and just as it happens with its Strong Kleene relatives, none of the w logics STw n is fully classical. In fact, every STn invalidates Meta-Cut of level n. Theorem 5.2.6 For every n (1 ≤ n < ω), STw n invalidates Meta-Cutn . Proof The proof is easily adaptable from the proof of Theorem 4.1.8, and we leave it as an exercise to the reader.  As we have done with the Strong Kleene hierarchy, it is possible to define a logic that recovers every classically valid metainference based on the new STw -hierarchy. w We will call it STw ω , and it can also be interpreted as the union of every STn —at least if we focus on the things it validates. Definition 5.2.7 For every valuation v, v satisfies a metainference of level n—for w any level n— ⇒n  in STw ω , if and only if  ⇒n  is satisfied by v in STn (1 ≤ n < ω). And with this definition, we can introduce a new collapse result, the General (Weak) Recovery Result: Theorem 5.2.8 For every level n (1 ≤ n < ω), a metainference of level n  ⇒n  is valid in CL if and only if it is valid in STw ω. Proof We know, by the (Weak) Collapse Result, that for every level n (1 ≤ n < ω), a metainference of level n  ⇒n  is valid in CL if and only if it is valid in STw n. w w , then, by the definition of ST , it is valid in ST . Therefore, But if it is valid in STw ω ω n  every classically valid metainference of any level n will be valid in STw ω.

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Just as what we have done with CL and STω , we can ask about the nature of the relationship between CL and STw ω . And the answer is pretty much the same w works better than every STw as we have given in Chap. 4. STw ω n , as STω recovers every classically valid metainference of every level. And if classical logic is defined through the inferences and metainferences of every level that turns out valid in it, w then STw ω is a fully classical logic. What is the relationship between STω , STω and CL? If the definition of a logic is purely extensional—i.e., if all that matters is the things it validates—, then they are at least equivalent, if not the same. To discriminate between them more is required—e.g., paying attention to the valuations or models used to characterize them. Finally, it is worth mentioning that the truth theories we will describe in Chap. 9 can be easily translated into this Weak Kleene framework. This means that we have another witness to the possibility of adding a transparent truth predicate to classical logic. So far, we have introduced metainferential logics built entirely either on Strong Kleene logics or entirely on Weak Kleene logics. But, what happens if we combine them? Can we even do that? Can we define Strong-Weak or Weak-Strong mixed metainferential logics? In the next Chapter, we will explain how this can be achieved.

References 1. Pailos, F. (2019). A family of metainferential logics. Journal of Applied Non-Classical Logics 29(1), 97–120. 2. Szmuc, D., & Macaulay Ferguson, T. (2021). Meaningless division. Notre Dame Journal of Formal Logic 62(3), 399–424. 3. Anotolevich Bochvar, D., & Bergmann, M. (1981). On a three-valued logical calculus and its application to the analysis of the paradoxes of the classical extended functional calculus History and Philosophy of Logic 2(1–2), 87–112. 4. Hallden, S. (1949). The logic of nonsense. Uppsala Universitets Arsskrift. 5. Da Ré, B., Pailos, F., Szmuc, D. (2020). Theories of truth based on four-valued infectious logics. Logic Journal of the IGPL 28(5), 712–746. https://doi.org/10.1093/jigpal/jzy057. 6. Szmuc, D. (2015). Classical negation and detachable conditionals in Weak Kleene logic, Paraconsistent Weak Kleene logic and generalized logics of nonsense. Manuscript. 7. Paoli, F., Pra Baldi, M., Szmuc, D. (2021). Pure variable inclusion logics. Logic and Logical Philosophy 30(4), 631–652. 8. Ripley, D. (2023). A toolkit for metainferential logics. Manuscript. Manuscript.

Chapter 6

Combining Weak and Strong Kleene Metainferential Logics

Weak Kleene metainferential logics are not the only kind of three-valued metainferential logics that can be built with the four traditional inferential Weak Kleene logics. In fact, it is possible to combine these Weak Kleene logics with their Strong Kleene relatives to define new metainferential logics. We will present thirty-two mixed and impure metainferential logics of level 1 such that the premise standard is a Strong Kleene logic, and its conclusion standard is a Weak Kleene logic, or the other way around. We will call the first type of them Strong-Weak metainferential logics, and the second type, Weak-Strong metainferential logics. For the sake of simplicity, we will once again stick to logics characterized through ST, TS, LP, K3—all of them Strong Kleene logics—, STw , TSw , LPw and K3w —i.e., their Weak Kleene counterparts. Those logics are (Figs. 6.1 and 6.2). The definition of these logics, though, is not as straightforward as in the previous cases, where the logics combined were defined through the same schema, i.e.,

Fig. 6.1 Sixteen Strong-Weak metainferential logics

L1 /L2 ST TS LP K3

STw ST/STw TS/STw LP/STw K3/STw

TSw ST/TSw TS/TSw LP/TSw K3/TSw

LPw ST/LPw TS/LPw LP/LPw K3/LPw

K3w ST/LPw TS/WK LP/K3w SK/K3w

Fig. 6.2 Sixteen Weak-Strong metainferential logics

L1 /L2 STw TSw LPw K3w

ST STw /ST TSw /ST LPw /ST K3w /ST

TS STw /TS TSw /TS LPw /TS K3w /TS

LP STw /LP TSw /LP LPw /LP K3w /LP

K3 STw /SK TSw /K3 LPw /SK K3w /K3

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. Pailos and B. Da Ré, Metainferential Logics, Trends in Logic 61, https://doi.org/10.1007/978-3-031-44381-7_6

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belonging to either the Strong Kleene or the Weak Kleene family. Thus, we need to introduce further technical apparatus.1 Let an evaluation be any function from propositional letters to (any of our three) truth values. Our definition of how these mixed and impure Strong-Weak and WeakStrong logics work, quantifies over evaluations, and not over valuations—understood as functions from sentences to truth-values. Below, Lsi stands for a Strong Kleene j j logic Li , while Lw stands for a Weak Kleene logic Lw . Notice, also, that every evaluation u can be extended by one and only one valuation v1 according to Strong Kleene matrices, and that each evaluation can be extended by one and only one valuation v2 according to Weak Kleene matrices. j

j

Definition 6.1 A metainference  ⇒1  is valid in Lsi /Lw (Lw /Lsi ) if and only if, for every evaluation u, if the valuation v1 that extends u according to Strong Kleene matrices (the valuation v2 that extends u to every formula according to Weak Kleene j matrices) satisfies every γ ∈  according to Lsi (according to Lw ), then the valuation v2 that extends u to every formula according to Weak Kleene matrices (the valuation j v1 that extends u according to Strong Kleene) satisfies some δ ∈  according to Lw i (according to Ls ). All the facts that we have proved about Weak Kleene versions of these logics in the previous chapter can also be proved for these new mixed Strong-Weak or Weak-Strong logics. This is because (1) we will be talking mainly about schemas, and not about instances or cases. Notice that when in the proof of these facts we ask to consider some sentence φ and valuation v such that v(φ) = 1/ 21 /0, we can just consider atomic instances—i.e., propositional letters—of φ (and of every other schematic letter being considered). Thus, as no logical constant is involved in the proof of these facts—i.e., as we might in fact only consider metainferences and inferences with just atomic sentences in them—, then whether we are working with a Strong or a Weak-schema, the results will be the same. That being said, this is true also because (2) we will only consider structural metainferential schemas—i.e., schemas where no logical constant plays any role. (This is why considering instances involving only atomic sentences delivers the same results in both cases.) Finally, it is worth stressing that, given an evaluation u, there is only one Strong Kleene valuation v1 that extends it, and only one Weak Kleene valuation that also extends it. From this, we get new collapses between these logics and CL. In what follows we will prove not only that both TS/ST and TSw /STw are equivalent to classical logic at the first metainferential level, but also that TSw /ST and TS/STw are. Theorem 6.2 (Strong-Weak and Classic correspondence)  ⇒1  is locally valid in CL if and only if  ⇒1  is locally valid in TS/STw , for every metainference  ⇒1 . 1

The way of defining these logics is based on—or is a variant of—the dual-valuation approach introduced by Lloyd Humberstone in [1], which has been used since then in many places, such as Fjellstad [2, 3] and Rosenblatt [4].

6 Combining Weak and Strong Kleene Metainferential Logics

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Proof (LTR) Assume CL  ⇒1 . Then there is a Boolean bivaluation v such that it is a counterexample of some premise or satisfies the conclusion according to CL. Notice that, given that SK and WK schemas are Boolean normal, there is an SK and WK trivaluation v  such that v(ϕ) = v  (ϕ). It is easy to check that v  is a counterexample of some premise according to TS or satisfies some conclusion according to STw . (RTL) Suppose TS/STw  ⇒1 . Then there is an evaluation u, and two valuations v1 , v2 such that v1 extends u according to Strong Kleene, v2 extends u according to Weak Kleene and v1 satisfies all the inferences in  according to TS and v2 does not satisfy any inference in  according to STw . Let u  be a two-valued evaluation defined as follows:  1 if u( p) = 21 u  ( p) = u( p) otherwise and let v be the Boolean bivaluation that extends u  . We need to show that v satisfies all the inferences in  and doesn’t satisfy any conclusion in . The way to do this is by showing the following four facts: • • • •

if v1 (ϕ) = 1 then v(ϕ) = 1 if v1 (ϕ) = 0 then v(ϕ) = 0 if v2 (ϕ) = 1 then v(ϕ) = 1 if v2 (ϕ) = 0 then v(ϕ) = 0

These facts can be shown by an easy induction on the complexity of the formulas, which we leave to the reader. Theorem 6.3 (Weak-Strong and Classic correspondence)  ⇒1  is locally valid in CL if and only if  ⇒1  is locally valid in TSw /ST, for every metainference  ⇒1 . Proof Similar to the previous Strong-Weak case. So this means that there are other metainferential logics, besides TS/ST and TSw /STw , that also recover classical logic at the first metainferential level, and that can also (though we will not prove it here) be safely expanded with a transparent truth predicate: the two hybrid Strong-Weak and Weak-Strong metainferential logics TSw /ST and TS/STw . Of course, there are other Strong and Weak-mixed metainferential logics that do not involve ST or TS, in either their Strong or Weak versions—i.e., at least one of the two building blocks is none of these four logics. Here are some results regarding these logics. We will not include the proofs of the following facts, as they are pretty similar to the ones we have just introduced. Fact 6.4 Let Xw /Y be a Weak-Strong logic. Then, Xw /Y validates (invalidates) some structural property of those considered in the previous chapters if and only if X/Y does, where X, Y ∈ {LP, TS, K3, ST} and X = Y.

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Fact 6.5 Let X/Yw be a Strong-Weak logic. Then, X/Yw validates (invalidates) some structural property of those considered in the previous chapters if and only if X/Y does, where X, Y ∈ {LP, TS, K3, ST} and X = Y. Notice that this does not mean, again, that the Strong-Weak or the Weak-Strong logics validate (or invalidate) the same metainferences as their Strong Kleene or Weak Kleene counterparts. Once there is some logical vocabulary involved in the formulation of the schematic metainferences the behavior between the logics differs (as we have already shown for the Strong Kleene and the Weak Kleene logics in the previous chapter). A big difference between the Strong-Weak and Weak-Strong logics, on the one hand, and the Strong Kleene or the Weak Kleene logics, on the other, is the following. In the hybrid logics, a sentence can receive a different value if it is acting as a premise than the one it receives when acting as a conclusion of a metainference. So an initially relevant question is: what is the value of the sentence, then? Although we are not entirely sure that this question really makes sense,2 a possible answer is that only propositional letters, or maybe propositional letters and negations, receive a definite truth value in an evaluation. The rest of the sentences— i.e., conjunctions, disjunctions, and material (bi)conditionals—receive a different value depending on the role they play—i.e., as part of a premise or as part of a conclusion of a metainference. Nevertheless, another possible answer is the following. The (so to speak, real) value of a formula in a mixed metainferential logic (in an evaluation), is the one the conclusion standard gave to the formula (relative to that particular evaluation). Asserting a sentence will be, under this interpretation, equivalent to asserting the validity of the metainference of level 1 with an empty set of premises and that inference as its only conclusion. We have presented thirty-two different mixed metainferential logics based on Strong and Weak Kleene logics, plus sixteen mixed metainferential Strong-Weak, and sixteen metainferential Weak-Strong Kleene logics. Each one of them has been specified using two different inferential Tarskian or non-Tarskian (e.g., substructural) logics. In particular, we have shown that the logics TSw /ST and TS/STw can recover all the classical metainferences up to level 1. What remains is to extend these metainferential logics to higher metainferential levels. We will present two ST-like hierarchies of metainferential hybrid logics based on combining Weak and Strong versions of ST and TS, though we will just prove the relevant facts about the one based on the Weak/Strong logic TSw /ST. The results for the hierarchy based on TS/STw can be easily adapted from this. ws Definition 6.6 STws 0 =ST (i.e., ST.), ST1 =TSw /ST, and for every n such that 2 ≤ n < ω, a metainference of level n  ⇒n  is valid in STws n if and only if, for every ∗ evaluation v, if every γ ∈  is satisfied by v according to (STn−1 ws ) , then v satisfies ws  according to STn−1 . 2

For example, if the truth-value of a sentence depends on a context of assessment, which in turn can be expressed through a semantic schema, then no sentence has a truth-value on its own, but relative to a context and a schema.

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sw Definition 6.7 STsw 0 =STw , ST1 =TS/STw , and for every n such that 2 ≤ n < ω, a metainference of level n  ⇒n  is valid in STsw n if and only if, for every evaluation ∗ v, if every γ ∈  is satisfied by v according to (STn−1 sw ) , then v satisfies  according sw to STn−1 .

Before continuing, let’s define what it means for an evaluation v to satisfy or to be a counterexample of a particular metainference, especially in the context of StrongWeak and Weak-Strong metainferential logics. What we mean when we say that, for any evaluation v, v STws  ⇒1  (and similarly for any mixed metainferential logic 1 XY1ws ), is that either the Weak Kleene valuation vw that extends v does not satisfy some γ ∈ , or the Strong Kleene valuation vs that extends v satisfies some δ ∈ .  ⇒n  if and only if the Weak Kleene valuation vw that extends v And v STws 1 satisfies every γ ∈ , and the Strong Kleene valuation vs that extends v satisfies no  ⇒n  (and similarly δ ∈ . Finally, for any n > 1, and any evaluation v, v STws n for any mixed metainferential logic XYnws ), either v does not satisfy some γ ∈ , or  ⇒n  if and only v satisfies every γ ∈  and v satisfies some δ ∈ . And v STws 1 v satisfies no δ ∈ . Now we will prove the (Weak/Strong) Collapse Result, i.e., that for every level n (1 ≤ n < ω), a metainference of level n,  ⇒n  is valid in CL if and only if it is valid in STws n . Before doing that, we need the following Lemma. : Lemma 6.8 For all n ≥ 1, for all , , , ⊆ M E T An−1 (L), and all evaluations  v, there is a Boolean valuation v such that: if then

v STws  ⇒n  n 

v CL  ⇒n 

and

v (STws ⇒n , n )∗

and

v CL ⇒n





Similarly, for all Boolean valuations v, there is an evaluation v such that: v CL  ⇒n 

if then



v STws  ⇒n  n

v CL ⇒n ,

and and



v (STws ⇒n

n )∗

Proof We prove them by induction on the index of the logic. Base case: n = 1. These results have already been stated in Theorem 6.3. Inductive step: n > 1. We will now prove the first conditional. Assume that there is  ⇒n  and v (STws ⇒n . From the fact that an evaluation v such that v STws n )∗ n ws  ⇒  we may infer that v  γ for all γ ∈ , and that v STws δ v STws n (ST )∗ n n−1 n−1 ws θ ⇒

we may infer that v  for every δ ∈ . From the fact that v (STws )∗ n ST n n−1 for some θ ∈ , or v (STws σ for some σ ∈ . Furthermore, from all these facts n−1 )∗  and the Inductive Hypothesis, it follows that there is a Boolean valuation v such that,   on the one hand, v CL γ for all γ ∈ , and v CL δ for every δ ∈ , and, on the   other, v CL θ , for some θ ∈ , or v CL δ, for some δ ∈ . But, by definition, this   is the same as saying that there is a Boolean valuation v such that v CL  ⇒n   and v CL ⇒n . The proof of the remaining conditional is similar.

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Just as it happens with the Strong and the Weak ST-hierarchies of metainferential logics, every logic in this new hierarchy has the same valid inferences as CL. By this, we mean that each STws n (for any finite n) coincides with CL up to its valid (meta)inferences of level n. And, once again, the validities obtained in a given logic are retained in the successive systems appearing in the hierarchy, which means that the hierarchy has a cumulative nature. Finally, we are in a position to prove the main result: Theorem 6.9 For all n ≥ 1, for all  ⊆ M E T An−1 (L),  ⊆ M E T An−1 (L) STws  ⇒n  if and only if n

CL  ⇒n 

Proof The proof is by induction on the index of the logic. Base case: n = 1. From left to right, let us suppose that CL  ⇒1 , from which we infer that there is a Boolean valuation v such that v CL γ , for all γ ∈ , and yet v CL δ, for every δ ∈ . But, since Boolean valuations are a subset of both WK and SK valuations, we know by the definition of satisfaction in TSw and ST that v is also a WK valuation such that v TSw γ , for all γ ∈ , and yet v is also an SK valuation such that v ST δ, for every δ ∈ . Thus, by the definition of validity of a metainference of level 1 in TSw /ST, we know that v TSw /ST  ⇒1 , whence TSw /ST  ⇒1 . From right to left, let us suppose that TSw /ST  ⇒1 , from which we infer that there is an evaluation v extended by a WK valuation vw such that vw TSw γ , for all γ ∈ , and yet that v is extended by an SK valuation vs , vs ST δ, for all δ ∈ . By   Theorem 6.3 we know that there is a Boolean valuation v such that v CL γ , for all   γ ∈ , and yet v CL δ, for every δ ∈ . Therefore, we know that v CL  ⇒1 , whence, CL  ⇒1 . Inductive step: n > 1. From left to right, let us suppose that CL  ⇒n , from which we infer that there is a Boolean valuation vb based on an evaluation v such that vb CL γ , for all γ ∈ , and yet vb CL δ, for every δ ∈ . By Lemma 6.8 we γ , for all γ ∈ , and yet v STws δ, for every δ ∈ , whence, know that v (STws n−1 )∗ n−1  ⇒ . STws n n  ⇒n , from which we infer that there From right to left, let us suppose that STws n γ , for all γ ∈ , and v STws δ, for every is an evaluation v such that v (STws )∗ n−1 n−1 δ ∈ . By Lemma 6.8 we know that there is a Boolean valuation vb that extends v such that vb CL γ , for all γ ∈ , and yet vb CL δ, for every δ ∈ . Therefore, we know that vb CL  ⇒n , whence CL  ⇒n . As expected, and just as it happens with their Strong Kleene and Weak Kleene ws relatives, none of the logics STws n is fully classical, as every STn invalidates a version of Meta-Cut of level n. Theorem 6.10 For every n (1 ≤ n < ω), STws n invalidates Meta-Cutn . Proof The proof is also easily adaptable from the proof of Theorem 4.1.8, and we leave it as an exercise to the reader. It is also possible to define a logic that recovers every classically valid metainferws ence based on the new STws w -hierarchy. We will call it, for obvious reasons, STω , ws and it can also be interpreted as the union of every STn .

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Definition 6.11 For every valuation v, v satisfies a metainference of level n—for ws any level n— ⇒n  in STws ω , if and only if  ⇒n  is satisfied by v in STn (1 ≤ n < ω). Theorem 6.12 (The General (Weak-Strong) Recovery Result) For every level n (1 ≤ n < ω), a metainference of level n  ⇒n  is valid in CL if and only if it is valid in STws ω . Proof We know, by the (Weak-Strong) Collapse Result, that for every level n (1 ≤ n < ω), a metainference of level n  ⇒n  is valid in CL if and only if it is valid ws ws ws in STws n . But if it is valid in STn , then, by the definition of STω , it is valid in STω . Therefore, every classically valid metainference of any level n will be valid in STws ω . ws Regarding the closeness between CL and STws ω , it is worth stressing that STω ws ws works better than every STn , as STω recovers every classically valid metainference of every level. If classical logic is defined through the inferences and metainferences of every level that turns out valid in it, then STws ω is a fully classical logic. And if the definition of a logic is purely extensional—i.e. if all that matters are the things w it validates—, then STws ω , STω , STω and CL are at least equivalent, if not the same. For discriminating between them, more is required—e.g., paying attention to the valuations or models used. Finally, we should also mention that the truth theories described in Chapter 5 can be easily translated into this Weak-Strong and Strong-Weak hybrid logics. Thus, these new hybrid metainferential truth theories work also as interesting solutions to semantic paradoxes related to a transparent truth predicate. Before concluding, we would like to reflect a little bit on why these logics are (or can be) interesting. First, if it is admitted that all the Weak and Strong standards that we have talked about in this book (and, specifically, in the last two chapters) can determine mixed inferential or metainferential logics, we need a reason for forbidding their interaction. The technical difficulty in achieving this goal can be a reason for not engaging in the exploration of a logic that results from combining a Weak standard as a premise standard (conclusion standard) and a Strong standard as a conclusion standard (premise standard). But once this inconvenience is overcome, there is no reason for not to assess how they work. This chapter has provided a set of technicalities sufficient for making the Weak-Strong and the Strong-Weak interaction possible. Second, we have already accepted that the standards can shift from premises to conclusions, presumably because the context of these two kinds of occurrences of sentences, inferences, and metainferences (i.e., the context determined by premises and the one dictated by conclusions) might be different enough in order to demand less or more from a sentence, inference or metainference for them to be part of a sound argument (i.e. and inference or a metainference). This opens the possibility for a context to be also different in the way it interprets the logical constants that appear in it—i.e., either demanding a Weak Kleene way of understanding them or a Strong Kleene schema for interpreting them. And the context that the premise determines might be completely independent of the context governed by the conclusion, in the specific sense of having no influence whatsoever on it. Of course, this might be

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wrong. But in order to rule out the exploration of this possibility more than mere speculation is needed. Third, what Fjellstad called the “dual-valuation semantics” have been introduced by Humberstone in [1] and developed by Fjellstad himself in [3] and by Rosenblatt in [4]. The basic idea behind this kind of semantic apparatus is that it uses different but related valuations to evaluate the validity of a given inference. The semantic apparatus that we have introduced for evaluating the validity of metainferences with respect to Strong-Weak and Weak-Strong logics is a natural generalization of the dual-valuation semantics strategy. If dual-valuation semantics are regarded as a legitimate technique, it seems arbitrary to refrain from applying a similar strategy for the metainferential level. Fourth, part of what we have proved in this chapter is that some of the reasons for preferring logics that are purely defined in terms of Strong (Weak) Kleene logics can also be used to justify the interest in these new Strong-Weak or Weak-Strong logics. In particular, we have proved that just as it happens with their Strong or Weak relatives, some of these new logics also recover classical validities at the metainferential level. Finally, though we have not proved that there are some phenomena that are better modeled using these new logics, we claim that, on the one hand, they can also be applied for some of the purposes their Strong (Weak) Kleene relatives are used for, and, on the other hand, that there might be some areas of the inferential practice that might be better represented using these logics. If that application is discovered, we have already designed the technical apparatus for explaining how these logics work. So far, we have explored different hierarchies of metainferential logics based on a local way of understanding metainferential validity. But is local validity the only way of producing hierarchies of metainferential logics? In the next chapter, we will introduce metainferential logics based on different notions of metavalidity.

References 1. Humberstone, L. (1988). Heterogeneous logics. Erkenntnis 29(3), 395–435. 2. Fjellstad, A. (2015). How a semantics for tonk should be. The Review of Symbolic Logic 8(03), 488–505 (2015). https://doi.org/10.1017/S1755020314000513. 3. Fjellstad, A. (2017). Non-classical elegance for sequent calculus enthusiasts. Studia Logica 105(1), 93–119. https://doi.org/10.1007/s11225-016-9683-y. 4. Rosenblatt, L. (2019). Noncontractive classical logic. Notre Dame Journal of Formal Logic 60(4), 559–585. https://doi.org/10.1215/00294527-2019-0020.

Chapter 7

Hierarchies of Global and Absolutely Global Metainferential Logics

The aim of this chapter is to apply to some metainferential logics the notions of meta-validity presented in Chap. 2. First, we will explore the global2 notion (since it is a bit simpler to handle from a technical point of view), and we will show that the pure metainferential logic ST also recovers classical logic by substructural means, just as the ST-hierarchy does locally. We will then show that when the language is expanded with  and ⊥ constants—but not a λ constant—, and regarding schematic metainferences, we can define some mixed metainferential logics that capture the schematic metainferences of level 1 of classical logic using the global2 and the global notions. Finally, we will say a few words regarding the ST-hierarchy of metainferential logics specified in an absolutely global way. Moreover, we will mention that this hierarchy collapses with the local hierarchy (even without constants). This result, though, follows immediately from Fact 2.2.15 (Chap. 2), so we will not devote much time to it.1 We will begin by discussing the two global ways of understanding metainferential validity introduced in Chap. 2, given by Definitions 2.2.7 and 2.2.8. Recall that a metainference  ⇒n  is globally valid according to some logic L defined over a set of valuations V if and only if, if every valuation v ∈ V satisfies all of the premises in , then every valuation v ∈ V satisfies some conclusion in . On the other hand, it is globally2 valid if and only if, if either some premise is (globally2 ) invalid or some conclusion is (globally2 ) valid. These two notions of metainferential validity are quite different. The collapse results between the local and the global notions that we have introduced in Chap. 2 correspond to the notion of global validity, which has an obvious resemblance to the disjunctive way of reading multi-conclusion inferences. Nevertheless, the global2 notion seems to correspond better with the idea of preservation of validity from premises to conclusions. As we have previously shown, given the logics we are 1

Although in this chapter we will focus on Strong Kleene logics, all the results can be easily adapted to their Weak Kleene counterparts.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. Pailos and B. Da Ré, Metainferential Logics, Trends in Logic 61, https://doi.org/10.1007/978-3-031-44381-7_7

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working with, there are globally valid metainferences that do not have any valid conclusion. For instance, take ST. The following metainference is globally valid in it: ⇒p

p⇒

since any Strong Kleene valuation satisfies either ⇒ p or p ⇒ in ST. However, none of these inferences is valid in ST. Thus, the metainference is globally valid, but it is not globally2 valid. Although we think this is not the preferable generalization of the global notion to the S E T − S E T framework, we consider the global2 notion in this chapter because it still has a very natural interpretation, i.e., it is just preservation of validity. So, while the global notion in the S E T − S E T setting shares with the local notion the artificial flavor of involving a reference to valuations and satisfaction in its formulation, the global2 notion seems to model the somewhat more natural reasoning from valid premises to some valid conclusion.

7.1 Global Hierarchies When metainferential validity is understood locally, we have proved what we have called the Collapse Result:  ⇒  is locally valid in CL if and only if  ⇒  is locally valid in TS/ST. But we do not have the same Collapse result from a global perspective. Recall that global validities are included in local validity collapse according to Theorem 2.2.13. So, in this section, first we show that the logic ST in its global2 reading coincides with CL. Then, we show that when  and ⊥ are in the language and consider only schematic metainferences, some mixed logics such as TS/ST, LP/ST, K3/ST in their global2 and global readings also collapse with CL but only at the first level (schematic metainferences of level 1). Let’s start then with ST. Of course, as we mentioned, we assume that a logic determines its higher levels, so when we talk of ST we are implicitly committed with higher levels determined by ST, i.e., metainferences of level 1 determined by ST/ST, of level 2 by STST/STST, and so on. In this sense, ST can be seen as a mixed, but pure, metainferential logic: the standard for premises and conclusions is the same. For the S E T − M E T setting, it is clear that if two pure metainferential logics coincide, then their globally valid metainferences will be the same (e.g., Blake-Turner [1]). For the S E T − S E T setting, the generalization is pretty straightforward: Fact 7.1.1 Assuming some fixed propositional language, if two pure metainferential logics coincide in their inferential validities (i.e. if they share the set of valid inference tokens), then they also globally2 coincide in their metainferential validities. Proof Let L1 and L2 be two pure metainferential logics such that  ⇒  is valid in L1 if and only if it is valid in L2 . Now we prove that for every n ≥ 1,  ⇒n  is globally2 valid in L1 if and only if it is globally2 valid in L2 .

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81

For the base case, consider a metainference of level 1  ⇒1 . Since L1 is pure, the metainference is globally2 valid in L1 if and only if some premise is invalid in L1 or some conclusion is valid in L1 . But this is the case, if and only if some premise is invalid in L2 or some conclusion is valid in L2 . And, thus, this is the case if and only if it is globally2 valid in L2 . For the inductive step, assume  ⇒n  is globally2 valid in L1 if and only if it is globally2 valid in L2 . Let  ⇒n+1  be globally2 valid in L1 . This is the case if and only if some premise is globally2 invalid in L1 or some conclusion is globally2 valid in L1 . By the inductive hypothesis, this holds if and only if some premise is globally2 invalid in L2 or some conclusion is globally2 valid in L2 . This is equivalent  to claiming that  ⇒n+1  is globally2 valid in L2 . Therefore, given that ST and CL are two pure metainferential logics and validate the same inferences, the metainferences that both globally2 validate are also the same.2 However, things are not so straightforward when we move to mixed impure metainferential logics. For instance, let’s consider TS/ST.3 As TS has no validities— if the logic’s language lacks truth-value constants—, then every instance of every metainference schema of level 1 has invalid premises (since every inference in TS is invalid). Thus, every metainference schema with a non-empty set of premises becomes globally2 valid in TS/ST (and thus globally valid also). (As not every empty-premise metainference is globally2 valid in TS/ST, it is not strictly speaking a trivial logic in any of the global senses.) Nevertheless, as we will prove, with truth-value constants  and ⊥ (but without a 1/2-constant λ), and focusing solely on schemas, global2 (and also global) validity in LPST, K3/ST and TS/ST coincides in the metainferences of level 1 with CL.4 It is worth noticing that we need to consider schemas since, e.g., in TS the valid inferences are only those involving the truth-constants. So, for instance, the following metainference token: ⇒ p ∨ ¬p ⇒q

is clearly globally2 invalid in CL, since the premise is valid in CL but its conclusion does not hold. However, in TS/ST it is globally2 valid, since the premise is not valid in TS (the valuation v( p) = 21 is a counterexample of it). When we consider the schema: ⇒ φ ∨ ¬φ ⇒ψ 2 As we mentioned in Chap. 3, here we are assuming that the constant for the intermediate value is not expressible in the language. Otherwise, independently of the chosen notion for metainferential validity, ST and CL will not coincide at the inferential level. We are grateful to an anonymous reviewer for making us reflect on this point. 3 Here, as before, we are assuming that this logic determines the higher levels. So, for instance, the metainferences of level 2 are determined by TSST/TSST, those of level 3 by TSSTTSST/TSSTTSST, and so on. 4 We became aware of the result for TS/ST after an email exchange with Chris Scambler, who first discovered it.

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this is invalid in both logics, since every valuation is a counterexample in TS/ST of the following token: ⇒  ∨ ¬ ⇒⊥

So we are in a position to present the second collapse result regarding global2 validity: Fact 7.1.2 For every metainference schema  ⇒1 , provided the language has both constants  and ⊥, the following holds: 1.  ⇒1  is globally2 valid in CL if and only if it is globally2 valid in TS/ST. 2.  ⇒1  is globally2 valid in CL if and only if it is globally2 valid in LP/ST. 3.  ⇒1  is globally2 valid in CL if and only if it is globally2 valid in K3/ST. Proof We prove the result for the case of TS/ST, as the other two cases are similar. We need to prove that  ⇒1  is globally2 valid in CL if and only if it is globally2 in TS/ST. First, assume that  ⇒1  is globally2 valid in CL. This means that every token of it is globally2 valid in CL. Let   ⇒1  be one such token. Hence, as it is globally2 valid in CL either some premise is invalid in CL or some conclusion is valid in CL. On the one hand, if some conclusion is valid in CL, then it is valid in ST, since both logics are inferentially equivalent. On the other hand, suppose some premise, say  ⇒ , is invalid in CL. Then there is a Boolean bivaluation v such that for v(σ) = 1 for every σ ∈  and v(π) = 0 for every π ∈ . As we already mentioned, since SK-valuations are Boolean normal, the trivaluation v∗ such that v ∗ (ϕ) = v(ϕ) is an SK-valuation and is such that shows that  ⇒  is also invalid in TS. Therefore,   ⇒1  is globally2 valid in TSST. Since the token was arbitrary, the metainference schema  ⇒1  is globally2 valid in TSST. Second, assume on the contrary that  ⇒1  is globally2 invalid in CL. This means that there is a token of it that is globally2 invalid in CL. Let   ⇒1  be such a token. The idea now is to build another token   ⇒1  by replacing p by  for every p such that v( p) = 1 and by ⊥ for every p such that v( p) = 0, for every p propositional letter occurring in any formula in   ⇒1  . It is straightforward to check that the new token is clearly globally2 valid in TSST. Therefore, this is a witness of the global2 invalidity of the metainference schema in TSST.  Similar results hold for the global notion of validity: Fact 7.1.3 For every metainference schema  ⇒1  of level 1, provided the language has both constants  and ⊥, the following holds: 1.  ⇒1  is globally valid in CL if and only if it is globally valid in TS/ST. 2.  ⇒1  is globally valid in CL if and only if it is globally valid in LP/ST. 3.  ⇒1  is globally valid in CL if and only if it is globally valid in K3/ST.

7.1 Global Hierarchies

83

Proof Once again, we prove the result for the case of TS/ST, as the other two cases are similar. We need to prove that  ⇒1  is globally valid in CL if and only if it is globally valid in TS/ST. First, assume that  ⇒1  is globally valid in CL. This means that every token of it is globally valid in CL. Let   ⇒1  be one such token. Hence, as it is globally valid in CL, either some premise is invalid in CL or, for every valuation v, v satisfies some δ ∈  in CL. On the one hand, if for every valuation v, v satisfies some δ ∈  in CL, then, for every valuation v, v satisfies some δ ∈  in ST, since the only relevant additional valuations we need to check are the ones such that v(θ) = 21 , for some θ ∈ δ. But those valuations satisfy δ in ST. On the other hand, suppose some premise, say  ⇒ , is invalid in CL. Then there is a Boolean bivaluation v such that v(σ) = 1 for every σ ∈  and v(π) = 0 for every π ∈ . As we already mentioned, since SK-valuations are Boolean normal, the trivaluation v∗ such that v ∗ (ϕ) = v(ϕ) is an SK-valuation and is such that shows that  ⇒  is also invalid in TS. Therefore,   ⇒1  is globally valid in TSST. Since the token was arbitrary, the metainference schema  ⇒1  is globally valid in TSST. Second, assume on the contrary that  ⇒1  is globally2 invalid in CL. This means that there is a token of it that is globally2 invalid in CL. Let   ⇒1  be such a token. The idea now is to build another token   ⇒1  by replacing p by  for every p such that v( p) = 1 and by ⊥ for every p such that v( p) = 0, for every p propositional letter occurring in any formula in   ⇒1  . It is straightforward to check that the new token is clearly globally2 valid in TSST. Therefore, this is a  witness of the global2 invalidity of the metainference schema in TSST. Notice that the collapse for metainferences of level n > 1 does not hold for the globally2 notion of validity. a [[ϕ ⇒ ϕ] ⇒1 ∅] ⇒2 ∅ b [[ϕ, ϕ → ψ ⇒ ψ] ⇒1 ∅] ⇒2 ∅ c [[⇒ ϕ ∨ ¬ϕ] ⇒1 ∅] ⇒2 ∅ Metainference a is globally2 valid in CL since, for every instance of it, its premise is globally2 invalid in CL. However, in TS/ST it is globally2 invalid, given that in any instance that replaces ϕ by a propositional letter, the premise is globally2 valid. Regarding metainference b, it is globally2 valid in CL since, for every instance of it, its premise is globally2 invalid in CL. However, in LP/ST it is globally2 invalid, given that, in any instance that replaces ϕ and ψ by different propositional letters, the premise is globally2 valid. Finally, metainference c is globally2 valid in CL since, for every instance of it, its premise is globally2 invalid in CL. However, in K3/ST it is globally2 invalid, given that, in any instance that replaces ϕ by a propositional letter, the premise is globally2 valid.

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Therefore, the situation in the global field does not resemble the collapse when a local notion of metainferential validity is taken into account. But maybe the situation looks different when we deal with an absolutely global notion of metainferential validity. The next section explores that option.

7.2 Absolutely Global Hierarchies As it is well-known, under some standard assumptions, global metainferential validity is the counterpart of the proof-theoretic notion of admissibility. Nevertheless, some authors, e.g., Ripley in [2], argue that the really important metainferences are the derivable ones (given some calculus). Humberstone proves that derivability is equivalent to absolute global validity in two-valued frameworks. Moreover, Da Ré et al. in [3] extend this result for three-valued monotonic and reflexive (and monotonic and transitive) cases. In the sequel, we will explore a hierarchy of metainferential logics based on TS/ST in its absolutely global reading, and another one based on ST/TS, also in its absolutely global reading. At this point, we need to recall an important result presented in Chap. 2 (Fact 2.2.15): if a logic L is defined over a set of valuations V , the set of metainferences which are absolutely globally valid in ℘ (V ) is identical to the set of metainferences which are locally valid in V . This means that, if we define absolute global validity over the power set of a given set of valuations, absolute global validity and local validity will coincide. In other words, for the absolutely globally valid metainferences determined by the hierarchies based on TS/ST and ST/TS we obtain exactly the same results as in the local case. So let’s just define the logics using this notion of validity in order to exemplify how it works. Definition 7.2.1 A metainference  ⇒1  of level 1 is absolutely globally valid in TS/ST if and only if, for every set of valuations V ⊆ S K , if every γ ∈  is V -valid (i.e., satisfied by every valuation v ∈ V ) according to TS, then some δ ∈  is V -valid (i.e., satisfied by every valuation v ∈ V ) in ST. Therefore, those metainferences that are absolutely globally valid in TS/ST are exactly the classical metainferential validities of level 1, just as in the local reading based on SK-valuations, as a consequence of Fact 2.2.15. And, as before, we can define a hierarchy of logics recapturing more and more classical metainferences. And, just as we obtained a collapse result for the local notion, we obtain a collapse result for the absolutely global notion: Fact 7.2.2 For every metainference  ⇒n ,  ⇒n  is absolutely globally valid in CL if and only if  ⇒n  is absolutely globally valid in the hierarchy based on TS/ST.

References

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Also, as we did for the ST hierarchy, it is possible to define a logic STω as the union of all of the finite levels. This logic, of course, will recapture all the classical metainferences. On the other hand, we can also define a TS hierarchy which is metainferentially empty, and the union TSω which is metainferentially empty at every metainferential level. This hierarchy of logics also recovers all of the classical antivalidities, as shown for the local case. These comments intend to suggest that metainferential logics are not tied to the local notion of metainferential validity. Moreover, in the case of these two important hierarchies based on ST and TS, both pairs of hierarchies (i.e., the local and the absolutely global) exhibit the same behavior. It remains to be determined under what conditions the local and the absolutely global logics overlap—i.e., in particular, what happens if the absolutely global logics are not defined over the power set of the set of valuations that defines the corresponding local notion. As is well known, Humberstone, in [4], gives a negative answer to the question of whether the two notions are equivalent. This result is extended in Da Ré et al. [3] to reflexive and monotone three-valued schemas, on the one hand, and to transitive and monotone three-valued schemas, on the other. In that article, as well as in Teijeiro’s [5], some sufficient conditions for the overlap of the notions of local, global, and absolutely global validity are presented. A deeper exploration of these issues, unfortunately, is outside the scope of this book.

References 1. Blake-Turner, C. (2020). Deationism about logic. Journal of Philosophical Logic, 49(3), 551– 571. 2. Ripley, D. Uncut. Manuscript. 3. Da Ré, B., Szmuc, D., & Teijeiro, P. (2022). Derivability and metainferential validity. Journal of Philosophical Logic, 51(6). 4. Humberstone, L. (2000). Contra-classical logics. Australasian Journal of Philosophy, 78(4), 438–474. 5. Teijeiro, P. (2021). Strength and stability. Análisis Filosófico, 41(2), 337–349.

Chapter 8

Metainferential Sequent Calculi

As we have seen, there is a whole family of Strong Kleene mixed and impure metainferential logics. We have already presented the twelve metainferential mixed and impure consequence relations that can be characterized through four inferential logics with the previously mentioned—and well-known—three-valued semantics based on the Strong Kleene schema: ST, TS, K3, and LP. Nevertheless, and for a long time, these metainferential logics have lacked sequent calculi—or any proof-theoretical presentation—that can capture their validities or their invalidities. The main purpose of this chapter, then, is to introduce some of the sequent calculi for these logics that have recently been developed. One possible way of categorizing them is by checking whether they are unlabeled or labeled sequent calculi, on the other. We will present first one sequent calculus of the first kind—due to Rea Golan—, and one labeled sequent system—due to Pablo Cobreros, Luca Tranchini and Elio La Rosa—that will prove particularly useful when discussing the possible shortcomings of these mixed metainferential logics, and the truth theories based on them. Both of these calculi are sound and complete regarding (some version of) the local metainferential validities of these hierarchies of logics. Afterward, we will present a series of calculi we have developed that are sound and complete regarding the global metainferential validities (restricted to a S E T − M E T framework) and, moreover, invalidities of these logics. The main advantage of this last proposal is that it is completely general and independent of the schema used for semantically defining the logics. Before presenting these calculi, let’s mention that Andreas Fjellstad in [1] has also recently provided labeled calculi for the hierarchies of logics. We omit them in this presentation, given that, although developed independently of Cobreros et al. systems, they are also labeled sequent calculi.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. Pailos and B. Da Ré, Metainferential Logics, Trends in Logic 61, https://doi.org/10.1007/978-3-031-44381-7_8

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8.1 Sequent Calculi for Local Metainferential Validities Here we will describe two different sequent calculi for these logics. The first is due to Rea Golan [2]. The second one is due to Cobreros, Tranchini and La Rosa.1 Recently, Golan, in [2], has presented a series of sequent calculi, one for each logic in the hierarchy of ST’ metainferential logics that have been previously introduced. These calculi have a limited scope, at least in the following two senses. (1) He chooses a single-conclusion framework for metainferences. (2) He does not present sequent calculi for every Strong Kleene mixed metainferential logic. Instead, he focuses on the logics in the ST hierarchy.2 In these two features, Golan differentiates from both Cobreros, Tranchini and La Rosa, on the one hand, and our own systems, on the other. We will mention Golan’s main results, without digging into the details of his system. For each metainferential logic STn , Golan provides axioms in the form of metainferential schemas of level n, and rules in the form of metainferential schemas of level n + 1. One of Golan’s most ingenious tricks to define these logics is that rules of level n become axioms of level n + 1. Golan provides sequent rules and axioms for metainferences of all levels, including transitivity and monotonicity rules, and reflexivity. Additionally, Golan introduces a set Auxn of auxiliary axioms. Each of them has the form: (∗)

∅n−1  ⇒n−1 

In each Auxn ,  ⇒n−1  is derivable through a proof tree of metainferences of level n − 1. Moreover, ∅n−1 is the metainference of level n − 1 whose premises and conclusions are empty, i.e., ∅0 = ∅, ∅1 = (∅ ⇒1 ∅), ∅2 = ((∅ ⇒1 ∅) ⇒2 (∅ ⇒1 ∅)), etc. Golan introduces a proof system for every metainferential validity of every level for classical logic, CL. He calls it CL∞ , and it is defined through a hierarchy of proof systems for classical logic for each metainferential level. Golan says that CLn consists of CL0 —i.e., the proof system for classical inferential validities—along with all the structural rules up to level n. Nevertheless, this is questionable, as all Auxn (but the ones based on Identity) include occurrences of logical constants, for they are based on rules for those constants at the previous level. Golan then proves by induction CL∞ ’s soundness and completeness (once again, we omit the proof, which the reader can find in detail in [3]) with respect to the logic that starts with CL at the inferential level and has, for each metainferential level n + 1, the lifting of the previous level—i.e., if the standard for level n is C L n , 1

In what follows, we will keep the vocabulary choices made by the authors in each case. This does not imply that similar hierarchies of proof systems cannot be defined for other hierarchies of mixed metainferential logics. In fact, Golan himself defines one such hierarchy for LP and the hierarchy that results from it by lifting the previous level. (For more about the lifting operation, see Ripley’s [4] and [5].)

2

8.1 Sequent Calculi for Local Metainferential Validities

89

the standard for the next level is C L n /C L n . In a similar vein, Golan presents proof systems for LP and ST (which we will omit, too), and also for every logic in the ST hierarchy. To accomplish this goal, he needs to introduce two additional structural p (schemas of) rules: the substitution rules Subnc and Subn . Each one of them has a second premise introduced with a doubleline. We will present Subnc , so the reader can have a better idea of them.  ⇒n−1 ψ 1 ⇒n−1 φ1 . . . k ⇒n−1 φk  ⇒n−1 ψ  ⇒n−1 χ Subnc 1 ⇒n−1 φ1 . . . k ⇒n−1 φk  ⇒n−1 χ

The double-line inference states that the metainferences above and below the line are interderivable—and, therefore, equivalent. They are derivable from the Transitivity rules, but they are also weaker than them. The Transitivity rules work like the Cut rule, but instead of “cutting” a formula, they “cut” an inference or a metainferp ence (in the case of T ran n rules), or a set of inferences or a set of metainferences c (in the case of T ran n rules). And, more importantly, there is no way to derive the Meta-Cutn rules just with the Substitutivity rules. The absence of Transitivity rules makes it impossible, for any n, in the system Ln for the STn logic, to derive MetaCutn —although Meta-Cutn−1 will be derivable. Golan describes what each Ln looks ∞ , the proof system for STω . like, and then defines Lm ∞ captures exactly every classical metainferential (local) Golan proves that Lm ∞ if and only if it is validity—in the sense that a metainference is a theorem in Lm n (locally) valid in STω —, based on the fact that each Lm proves every classical validity of level n. In [3], Golan also shows how to extend this system to a language with a transparent truth predicate. He calls CMω the system that captures STω . CMω , extended with a transparent truth predicate, blocks the paradox because no transitivity principle is available as a rule—in the specific sense that they are no longer admissible—, ∞ . For Golan, this is a first major flaw of the although all of them are axioms of Lm system. Golan then introduces what he takes to be a second objection, but which should probably be framed as a specification of the alleged costs of the first one. The following quotes display Golan’s criticism. But there is a second price to pay here, which, I believe, is unbearable. For in its refusal to grant any transitivity principle the status of a rule, CMω goes substructural at all inferential levels, and so there is no room for real inferential action in this system. ([3, p.28]) In essence, then, all this system has is a bunch of axioms in the form of metainferences of various levels. I would like to argue that it is this second price that makes CMω a very poor candidate for a theory of truth. And poor candidate it is. After all, paradoxes like the liar can be said to pose a threat to the notion of truth only in a context where inferences can in principle be carried out. If there are no inferences, there is no way to derive the paradox and hence, one could argue, no paradox at all... Indeed, such a degenerate system can hardly be called a logic. ([3, p.29])

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And, finally, CMω has a unique device for expressing the validity of all transitivity principles: axioms in the form of metainferences of any arbitrary level. But all these axioms are no less “inert,” as they all fall short of launching any real inferential action. It is for this reason that CMω can be hardly called a logic, and that its solution to the paradox is dubious. (p. 30)

Regarding the first criticism, it is just not true that there are no rules in CMω . Weakening, Contraction, and every operational rule of every level are still rules— despite also being axioms. And, though it is true that everything that can be proved using these rules can also be proved as an axiom, this does not mean that there is no epistemic gain in the process of making the inference. In fact, there is epistemic gain in knowing the different ways in which something can be proved. Proofs are an important research item, and different proofs may not be equivalent. Thus, it seems wrong to conclude, from the fact that everything that can be proved can be proved as an axiom, that the system lacks genuine rules. There is another objection against Golan’s second criticism. He claims that, as his two-sided unlabeled sequent calculus for the truth theory based on CMω lacks rules, then STω —i.e., the theory for which the system is sound and complete—is not a genuine logic. This has a reductio flavor. Nevertheless, to conclude from these premises that STω itself is not a genuine logic, more is required. In particular, we need to prove that every sequent calculus for STω lacks rules. Below, we will talk about a labeled sequent calculus for STω expanded with a transparent truth predicate that has genuine rules. In particular, there are different versions of Cut that are valid in it, making the system wholly transitive, and, therefore, unsuspected from Golan’s own point of view. As we have mentioned, unlabeled sequent calculi are not the only available option for building proof theories for metainferential logics. Cobreros, Tranchini and La Rosa, in [6], propose a sequent calculus that captures every metainferential logic of every level based on the Strong Kleene logics ST, TS, LP and K3. For that purpose, they use both a labeled and a hybrid sequent calculus—in the sense that metainferences belonging to different levels can be part of the same higher-level metainference. They use a propositional language L with V as the set of propositional letters and the following grammar: A := V ||⊥|¬A|(A ∧ A)|(A ∨ A)|(A ⊃ A). A labeled formula is an expression of the form x : A, for each formula A ∈ L and x ∈ {s, t}—i.e., the symbols for being strictly or tolerantly satisfied.3 3

These labels purport to represent two different ways a valuation might satisfy a formula. In particular, a valuation v strictly satisfies a formula A (v s A) if and only if v(A) = 1, and v tolerantly satisfies a formula A (v t A) if and only if v(A) ∈ {1, 21 }. This helps understand the particular behavior of the formulas and inferences in the system. Nevertheless, there is no need to be aware of this link between labels and satisfaction relations in order to understand how the proof system works.

8.1 Sequent Calculi for Local Metainferential Validities

91

The authors’ approach is explicitly based on the one introduced by Girard, in [7], called G3SK, to capture the SK semantic setting. After introducing Girard’s approach, they extend it to the metainferential framework. The general form of an SK sequent is the following: s : 1 , t : 2 ⇒ s : 1 , t : 2 Each 1 , 2 , 1 and 2 are possibly empty. And a valuation v satisfies an SK sequent if and only if, if v s A for all A ∈ 1 and v t B for all B ∈ 2 , then either v s C for some C ∈ 1 or v s D for some D ∈ 2 . An SK sequent is valid if and only if it is satisfied in every valuation. A G3SK derivation, then, is a tree of SK sequents constructed in correspondence with the following propositional rules, and whose leaves are the following initial sequents: Definition 8.1.1 The calculus SG3SK is defined by the following rules: Initial sequents x : P, φ ⇒ ψ, x : P s : P, φ ⇒ ψ, t : P x : ⊥, φ ⇒ ψ

φ ⇒ ψ, x : 

Operational rules L¬

 ⇒ , x : A x : ¬A,  ⇒ 

L∧

x : A, x : B,  ⇒  x : (A ∧ B),  ⇒ 

L∨

x : A, , ⇒  x : B, , ⇒  x : (A ∨ B),  ⇒ 

R∧

 ⇒ x : A,   ⇒ x : B,   ⇒ x : (A ∧ B), 

R∨

 ⇒ x : A, x : B,   ⇒ x : (A ∨ B), 

L⊃

x : B, , ⇒   ⇒ , x : A, x : (A ⊃ B),  ⇒ 

R⊃

x : A,  ⇒ , x : B  ⇒ , x : (A ⊃ B)

R¬

x : A,  ⇒   ⇒ , x : ¬A

The operation x switches s from t and the other way around. The authors proved that an SK sequent is G3SK-derivable if and only if it is valid—i.e., that the calculus is sound and complete with respect to the semantic interpretation of SK sequents. One interesting fact about the interpretation of SK sequents is that the following form of Cut is sound:

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8 Metainferential Sequent Calculi Cut

 ⇒, , x : A x : A,  , φ ⇒  ,  ⇒ , 

However, of course, the strict-tolerant form of Cut is neither sound nor admissible with respect to the interpretation they provide for SK sequents. Cutst

 ⇒, , t : A s : A,  , φ ⇒  ,  ⇒ , 

And this is allegedly how the unsoundness of Cut in ST is reflected in G3SK. The next step is to generalize the notion of inference to higher (but finite) levels. Definition 8.1.2 N-inferences: • For every A ∈ L, A is a 0 inference. • If  and  are finite multisets of n inferences, then  ⇒  is an n + 1 inference. The authors finally introduce an extension of G3SK called G3SKω . The inferences that occur in the G3SKω sequents are also accompanied by an n level. An SKω sequent, for the authors, is an expression of the form  ⇒ , where  ⇒  are finite multisets containing labeled inferences of any level. The authors focus first on the general form of an SK1 sequent—which helps understand what the form of any SKn sequent is. s : 1 , t : 2 , tt : 1 , ts : 2 , st : 3 , ss : 4 ⇒ s : 1 , t : 2 , tt : 1 , ts : 2 , st : 3 , ss : 4 1 , 2 , 1 and 2 are possibly empty finite multisets of formulas. 1 , 2 , 3 , 4 , and also 1 , 2 and 1 , 2 , 3 , 4 , are possibly empty finite multisets of inferences of level 1. A valuation v satisfies an SK1 sequent if and only if, if v s A for all A ∈ 1 . ... and v ss B for all inferences B ∈ 4 , then either v s C for some C ∈ 1 , ... or v ss D for some inference D ∈ 4 . An SK1 sequent is valid if and only if it is satisfied in every valuation. From the notions of satisfaction and valuation of an SK1 sequent it is possible to recover all SK notions of satisfaction and validity of 1-inferences and 2-inferences, as expected, in a pretty straightforward way. And the notion of an SKω sequent is obtained by scaling the notion of an SK1 sequent. G3SKω is the result of adding to G3SK rules for introducing labeled inferences of level n both to the right and to the left of the (main) sequent symbol. R⇒

L⇒

l : ,  ⇒, , l : 

 ⇒ , ( ⇒l/l )

 ⇒ , l : A1 . . .  ⇒ , l : A p . . . l : B1 ,  ⇒ , . . . l : Bq ,  ⇒ 

(A1 , . . . , A p ⇒l/l B1 , . . . , Bq ),  ⇒ 

8.2 Sequent Calculi for Global Metainferential Validities (and Invalidities)

93

Regarding L⇒, the rule has p+q premises, and each Ai , B j are inferences of level n − 1. The authors prove that G3SKω is sound and complete with respect to SKω valid sequents. Thus, an SKω sequent is derivable in G3SKω if and only if it is valid in SKω . And G3SKω also admits higher-level forms of Cut—in fact, one for every label l of level n. Cut

 ⇒, , l : A l : A,  , φ ⇒  ,  ⇒ , 

Within this labeled sequent calculus, they are able to prove the following two facts, which show in what sense G3SKω can capture both the ST hierarchy and the TS hierarchy of metainferential logics. Fact 8.1.3 For all n let ln be the n-th element of the ST hierarchy. Thus, for all n-inferences  ⇒ , ln  ⇒  if and only if  ⇒  is valid in ln . Fact 8.1.4 For all n now let ln be the n-th element of the TS hierarchy. Thus, there is no structural (i.e., without truth-value constants) n-inference  ⇒  such that ln  ⇒ . Remember that each member ln of the TS hierarchy is empty with respect to the n-level of metainferences. So far, so good. We have presented two different sequent calculi that capture the local validities of the ST hierarchy. Moreover, the labeled sequent calculus described in Cobreros, Tranchini and La Rosa [8] allows us also to capture every validity of every metainferential logic based on ST, TS, K3, and LP. Now we will address the question of whether there is a calculus that can capture the global metainferential validities of these logics.

8.2 Sequent Calculi for Global Metainferential Validities (and Invalidities) In this subsection, we will introduce a procedure to build sequent calculi that are sound and complete regarding the global validities (and invalidities) of any metainferential logic of any level. Thus, throughout this section, unless otherwise stated, any occurrence of the term validity/invalidity must be understood as global validity/invalidity. It is worth mentioning that we will restrict ourselves in this section to a S E T − M E T framework, leaving for future work the calculi for multiple conclusions. This procedure is based on a recent article by Da Ré and Pailos [9], where the authors introduce such a mechanism for metainferential logics from the Strong Kleene family. However, here we will extend these results to any metainferential logic and we will use some specific logics to illustrate the method.4 4

Of course, there are some restrictions the inferential logics must satisfy for this method to work. Basically, since we need to formulate sound and complete sequent calculi for the invalidities, we

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Before we start, let’s briefly sketch the general strategy of the algorithm. Our goal is to build sound and complete sequent calculi for any metainferential logic X/Y of any level n ≥ 1, regarding the global validities. In the simplest case, where n = 1, we deal with a metainferential logic X/Y where X, Y are two inferential logics. Hence, for assessing the global validity of a metainference, we need to show that either its conclusion is valid in Y or some of its premises is invalid in X. In other words, if we could design sequent calculi capturing the validities and the invalidities of the inferential logics, we could use them (i.e., the inferential validity calculus and the inferential invalidity calculus) to develop a sequent calculus for the metainferential logic of level 1, X/Y. The main idea, then, is to translate the valid inferences of Y as a conclusion of a metainference of level 1 (globally valid in X/Y) and the invalid inferences of X as premises of a metainference of level 1 (globally valid in X/Y). But also, since we have a proof system for the validities of Y and the invalidities of X, we can construct a sequent calculus for the global invalidities of X/Y by translating the validities of X into premises of a metainference of level 1, and the invalidities of Y into a conclusion of a metainference of level 1. The idea is just to apply the algorithm, and using the calculi for the validities and the invalidities of the previous level we can build calculi for the validities and invalidities of the next level.5 Let’s see the details of all of this. Let’s start with some terminology. Sequents of level 0 (or simply sequents) are ordered pairs of (finite) sets of formulas, denoted by  ⇒ , where  and  are called, respectively, the antecedent and the succedent of the sequent. Although we have mentioned that in this section we do not use multiple conclusions for metainferences, we stick to the standard presentation of inferential logics with multiple conclusions. Antisequents of level 0 (or simply antisequents) are also ordered pairs of (finite) sets of formulas, denoted by   , where  and  are called, respectively, the antecedent and the succedent of the antisequent. In an abstract way, a sequent calculus consists of a set of Axioms (Ax) and Rules (R). In a sequent calculus for inferential logics, the Axioms are metainferences with empty premises and a sequent as a conclusion, and the rules are metainferences of level 1 with non-empty premises. For example, the following are the usual rules for conjunction, e.g., in the sequent calculus for valid inferences of the inferential logic ST: L∧

, φ, ψ ⇒  , φ ∧ ψ ⇒ 

R∧

 ⇒ φ,   ⇒ ψ,   ⇒ φ ∧ ψ, 

And the following are some key structural rules in the sequent calculus for invalid inferences of the inferential logic LP: need the inferential logic to be decidable. This requirement, though strong in general, is satisfied for all the logics that are used in the context of propositional metainferential logics. 5 This algorithm we are presenting is devised for the propositional metainferential logics that we have introduced. They are not meant to be applied for first level versions of them, as there cannot be sound and complete calculi for the inferential invalidities of these first-order logics.

8.2 Sequent Calculi for Global Metainferential Validities (and Invalidities) ESeq ,    

LK−

95

∅∅   ,  

RK−

Actually, all of these rules (those corresponding to the validities of ST and to the invalidities of LP) are sound for any of the inferential logics based on the Strong Kleene schema. Let’s call SX+ the sound and complete sequent calculus for the validities of X, and − SX the one corresponding to the invalidities. As an example, let’s introduce both calculi when X = K3.6 + Definition 8.2.1 The calculus SK3 is defined by:

Axioms I dK3

, φ ⇒ φ, 

EFSQ

φ, ¬φ,  ⇒ 

Rules L∧

, φ, ψ ⇒  , φ ∧ ψ ⇒ 

R∧

 ⇒ φ,   ⇒ ψ,   ⇒ φ ∧ ψ, 

L∨

, φ ⇒  , ψ ⇒  , φ ∨ ψ ⇒ 

R∨

 ⇒ φ, ψ,   ⇒ φ ∨ ψ, 

L¬¬

, φ ⇒  , ¬¬φ ⇒ 

L¬∧

, ¬φ ∨ ¬ψ ⇒  , ¬(φ ∧ ψ) ⇒ 

R¬∧

 ⇒ ¬φ ∨ ¬ψ,   ⇒ ¬(φ ∧ ψ), 

L¬∨

, ¬φ ∧ ¬ψ ⇒  , ¬(φ ∨ ψ) ⇒ 

R¬∨

 ⇒ ¬φ ∧ ¬ψ,   ⇒ ¬(φ ∨ ψ), 

R¬¬

 ⇒ φ,   ⇒ ¬¬φ, 

− is defined in the following way: Definition 8.2.2 The calculus SK3

Axiom − I dK3

6

φ1 , ...φn  ψ1 , ..., ψm

The soundness and completeness proof of each calculus can be found in [9].

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8 Metainferential Sequent Calculi

where φi ∈ {literals} and ψ j ∈ {literals}, and such that for every φi , ψ j , φi = ψ j , and for no φl , φl it is the case that φl = ¬φl Rules ESeq

LK−

,    

∅∅

RK−

  ,  

L∧−

, φ, ψ   , φ ∧ ψ  

R∧−

  φ,    φ ∧ ψ, 

L∨−

, φ   , φ ∨ ψ  

R∧−

  ψ,    φ ∧ ψ, 

L∨−

, ψ   , φ ∨ ψ  

R∨−

  φ, ψ,    φ ∨ ψ, 

L-I ¬¬−

, φ   , ¬¬φ  

R-I ¬¬−

  φ,    ¬¬φ, 

L¬∧−

, ¬φ ∨ ¬ψ   , ¬(φ ∧ ψ)  

R¬∧−

  ¬φ ∨ ¬ψ,    ¬(φ ∧ ψ), 

L¬∨−

, ¬φ ∧ ¬ψ   , ¬(φ ∨ ψ)  

R¬∨−

  ¬φ ∧ ¬ψ,    ¬(φ ∨ ψ), 

Before moving on, it is worth mentioning that the development of sequent calculi for invalidities is by far more uncommon than the study of sequent calculi for validities. However, we can mention some exceptions, like Tiomkin [10], Bonatti and Olivetti [11], Goranko [12], and, more recently, Carnielli and Pulcini [13] and Rosenblatt [14]. Thus, once we have sequent calculi for the validities and invalidities of the inferential logics, we can use them for building sequent calculi for the global validities and invalidities of the metainferential logics of level 1. Let’s recall that for single-conclusion metainferences of level n + 1: Definition 8.2.3 A metainference (of level n + 1)  ⇒n+1 φ is globally valid in X/Y if and only if either some γ ∈  is globally invalid in X, or the conclusion φ is globally valid in Y. And a metainference (of level n + 1) is globally invalid if and only if it is not globally valid. With the sequent calculi for the logics of level 0 X and Y, we can construct a sequent calculus for any metainferential logic based on any combination of X and Y.

8.2 Sequent Calculi for Global Metainferential Validities (and Invalidities)

97

So let’s start with the metainferential logics of level 1. In what follows, we will use  ⇒n δ (and  n δ) to denote (meta)sequents (antisequents) of level n > 0, where  is a set of sequents of level n − 1, and δ is a sequent of level n − 1. As before, we usually omit the subscript for sequents and antisequents of level 0. Finally, we use the symbol ; to separate the premises of a sequent or antisequent, and, in order to avoid any confusion, in some situations we will use brackets [] to separate the premises and the conclusion of a sequent (or an antisequent). Since we need to use some particular sequents in the general definitions, for the purposes of notation, we define the following abbreviation for the following sequents: • ∅0 = ∅ ⇒ ∅ • ∅i+1 = ∅ ⇒i+1 ∅i Note that the symbol ∅ denotes, as usual, the empty set, but ∅ j denotes a particular sequent of level j. Let’s begin with a logic X/Y, where X is the standard for the premises of the metainferences of X/Y and Y is the standard for the conclusion of X/Y. We need to define some functions to translate the axioms and the rules of SY+ and SX− , which are sequents of level 1—inferences between sequents (antisequents) of level 0—, into axioms and rules over sequents of level 2—inferences between sequents (antisequents) of level 1. The following functions are what we are looking for: Definition 8.2.4 Let τ1+ and τ1− be two functions from sequents of level 1 to sequents of level 2 defined in the following way:  τ1+ (

⇒1 δ) =

[∅ ⇒1 γ1 ; ...; ∅ ⇒1 γn ] ⇒2 [∅ ⇒1 δ] if  = ∅ if  = ∅ ∅ ⇒2 [∅ ⇒1 δ]

where in the first case  = {γ1 , .., γn } is a set of sequents (of level 0), and δ is a sequent (of level 0). And  τ1− (

⇒1 π ) =

[σ1 ⇒1 ∅0 ; . . . ; σn ⇒1 ∅0 ] ⇒2 [π ⇒1 ∅0 ] if  = ∅ if  = ∅ ∅ ⇒2 [π ⇒1 ∅0 ]

where in the first case  is a set of antisequents (of level 0) and {σ1 , .., σn } is the set resulting from replacing each antisequent symbol (of level 0) in  by the sequent symbol (of level 0). π is an antisequent (of level 0) and π is the result of replacing the antisequent symbol (of level 0) by the sequent symbol (of level 0). Finally, recall that ∅0 stands for the empty sequent ∅ ⇒ ∅. In a nutshell, τ1+ and τ1− translate sequents or antisequents, respectively, of level 1 into premises and conclusions of sequents of level 2, respectively, though they work very differently from each other. While the goal of τ1+ is to define higher-level rules that preserve validity, the aim of τ1− is to do a similar job with invalidities (though none of this means that every value of each of these two functions is a rule of these

98

8 Metainferential Sequent Calculi

two kinds in every case, but just when suitably applied). For example, the translation + is of the rule L∨ of the sequent calculus SK3 τ1+ ([, φ ⇒ ; , ψ ⇒ ] ⇒1 [, φ ∨ ψ ⇒ ]) = [∅ ⇒1 , φ ⇒ ; ∅ ⇒1 , ψ ⇒ ] ⇒2 [∅ ⇒1 , φ ∨ ψ ⇒ ]

With these functions in place, we can define the following calculus: + Definition 8.2.5 Let τ1+ , τ1− , as in Definition 8.2.4. The calculus SX/Y is defined by the following rules:

• Axioms: τ1+ (AY+ ) and τ1− (AX− ) with AX− , AY+ respectively being the axioms of SX− and SY+ , and with their respective provisos. + -rules: τ1+ (R), for every rule R of SY+ . • SX/Y − • SX/Y -rules: τ1− (R), for every rule R of SX− . • (Weakening Up): [ ⇒1 δ] ⇒2 [,  ⇒1 δ] • (Weakening Down): [ ⇒1 ∅0 ] ⇒2 [ ⇒1 π] where ,  are sets of sequents of level 0, and δ is a sequent of level 0, and π is a sequent of level 0 different from the empty sequent. Before showing that the calculus we have presented is sound and complete, let’s see some examples to illustrate how it works. To simplify the examples, let’s use the logic K3/K3, since we have already introduced the sequent calculi for the validities and the invalidities of K3. We know that the metainference [ p ∧ ¬ p ⇒ q; q ⇒ q ∨ s] ⇒1 [r ∧ q ⇒ r ∧ q] is globally valid in K3/K3, since the conclusion is valid in K3. Now we can prove + : that the corresponding sequent is provable in SK3/K3 τ1+ (AK3+ )

τ + (A

+

)

K3 1 ∅ ⇒1 [q, r ⇒ r ] ∅ ⇒1 [q, r ⇒ q] τ1+ (R∧K3+ ) ∅ ⇒ [r, q ⇒ r ∧ q] 1 τ1+ (L∧K3+ ) ∅ ⇒1 [r ∧ q ⇒ r ∧ q] W eakening U p [ p ∧ ¬ p ⇒ q; q ⇒ q ∨ s] ⇒1 [r ∧ q ⇒ r ∧ q]

Let’s see another example. We know that [ p ⇒ q ∨ ¬q; s ⇒ q ∨ s] ⇒1 [r ⇒ r ∧ p] is also globally valid in K3/K3 (since the first premise is invalid in K3). Now + : we can prove the corresponding sequent in SK3/K3 τ1− (AK3− ) − τ1 (R∨K3− )

[ p ⇒ q, ¬q] ⇒1 ∅0 [ p ⇒ q ∨ ¬q] ⇒1 ∅0 W eakening U p [ p ⇒ q ∨ ¬q; s ⇒ q ∨ s] ⇒1 ∅0 W eakening Down [ p ⇒ q ∧ r ; q ⇒ q ∨ s] ⇒1 [r ⇒ r ∧ p] Now, let’s present the first result that shows that the method works (this is just a generalization of the result reached by Da Ré and Pailos in [9]).

8.2 Sequent Calculi for Global Metainferential Validities (and Invalidities)

99

+ Theorem 8.2.6 SX/Y is sound and complete with respect to X/Y validities.

Proof The proof can be easily adapted from Da Ré and Pailos [9], so we leave it to the reader. Now that we have already presented the method for obtaining sequent calculi for the validities of metainferential logics of level 1, we can introduce the method for generating the sequent calculi for the invalidities. In order to do so, we need to use another translation function. Definition 8.2.7 Let τ1+− be the function from sequents of level 1 to sequents of level 2 defined in the following way: τ1+− ( ⇒1 δ ) = ∅ ⇒2 [ 1 δ] where  is a set of sequents (of level 0), and δ is a sequent or antisequent (of level 0), and δ is the sequent (of level 0) resulting from replacing the antisequent symbol (of level 0) by the sequent symbol (of level 0) in δ . This function allows us to translate the Axioms of the sequent calculi for validities and invalidities of level 0 into Axioms of the sequent calculi for invalidities of level 1. Also, in order to obtain the rules for the calculus, we need the following functions: +

Definition 8.2.8 Let τ1R be the function from sequents of level 1 to sequents of level 2 defined in the following way: +

τ1R ( ⇒1 δ) = [;  1 χ] ⇒2 [δ;  1 χ] where  and  are sets of sequents (of level 0), δ and χ are two sequents (of level 0). −

Definition 8.2.9 Let τ1R be the function from sequents of level 1 to sequents of level 2 defined in the following way: −

τ1R (γ ⇒1 δ) = [ 1 γ ] ⇒2 [ 1 δ ] where  is a set of sequents (of level 0), γ, δ are two antisequents and γ , δ are the two sequents resulting from replacing each antisequent symbol (of level 0) by a sequent symbol (of level 0) in γ, δ. Notice that, in these definitions  and χ are the context of the rules. For example, + + the translation by τ1R of the rule L∨ of SK3 is the following: +

τ1R ([, φ ⇒ ; , ψ ⇒ ] ⇒1 [, φ ∨ ψ ⇒ ]) = [, φ ⇒ ; , ψ ⇒ ;  1 χ] ⇒2 [, φ ∨ ψ ⇒ ;  1 χ] As in Definition 8.2.5, the idea is to use these functions to obtain the calculus.

100

8 Metainferential Sequent Calculi +

−

Definition 8.2.10 Let τ1+− , τ1R and τ1R as in the previous definitions. The calculus − is defined by the following rules: SX/Y • Axiom: τ1+− ({AiX+ } ⇒1 AY− )7 with {AiX+ } being any set of axioms from SX+ and AY− any axiom from SY− , with their respective provisos. + − − -rules: τ1R (R1 ), and τ1R (R2 ) for every rule R1 of SX+ and every rule R2 of • SX/Y SY− . • (Weakening In): [ 1 δ] ⇒2 [; {AiX+ } 1 δ], where {AiX+ } is any set of axioms from SX+ . • (Anti-Weakening Up): [;  1 δ] ⇒2 [ 1 δ] • (Anti-Weakening Down): [ 1 π] ⇒2 [ 1 ∅0 ] where ,  are sets of sequents (of level 0) and δ is a sequent (of level 0) and π is a sequent (of level 0) different from the empty sequent. Let’s see some examples of how this calculus works. As before, let’s take the logic K3/K3 to illustrate the proof system. We know that [ p ∧ q ⇒ p; q ⇒ q ∨ s] ⇒1 [r ⇒ r ∧ p] is globally invalid in K3/K3, since the premises are valid in K3 but the conclusion is not. − : Now we can prove its corresponding antisequent in SK3/K3 τ1+− (AK3+ K3− )

[ p, q ⇒ p; q ⇒ q, s] 1 [r ⇒ p] [ p ∧ q ⇒ p; q ⇒ q, s] 1 [r ⇒ p] τ1 (R∨K3+ ) [ p ∧ q ⇒ p; q ⇒ q ∨ s] 1 [r ⇒ p] − τ1R (R∧K3− ) [ p ∧ q ⇒ p, q ⇒ q ∨ s] 1 [r ⇒ r ∧ p] +

τ1R (L∧K3+ ) R+

Another example is the following. We know that [ p ⇒ p ∨ r ; q, p ⇒ p; p, q ⇒ p ∧ q] ⇒1 [r ⇒ s ∨ p] is invalid in K3/K3 (since the premises are valid in K3 but the conclusion is not). − : Now we can prove it in SK3/K3 τ1+− (AK3+ K3− )

[ p ⇒ p, r ; q, p ⇒ p; p, q ⇒ q] 1 [r ⇒ s, p] [ p ⇒ p, r ; p, q ⇒ p ∧ q] 1 [r ⇒ s, p] + τ1R (R∨K3+ ) [ p ⇒ p ∨ r ; p, q ⇒ p ∧ q] 1 [r ⇒ s, p] W eakening I n [ p ⇒ p ∨ r ; p, q ⇒ p ∧ q; q, p ⇒ p] 1 [r ⇒ s, p] − τ1R (R∨K3+ ) [ p ⇒ p ∨ r ; p, q ⇒ p ∧ q; q, p ⇒ p] 1 [r ⇒ s ∨ p] +

τ1R (R∧K3+ )

− Finally, the calculus SX/Y is sound and complete, but now for the set of invalidities.

7

Here we are taking AX+ and AY− as sequents (of level 0), which is an abuse of notation. Strictly speaking, they are sequents of level 1 with empty premises. So we could define a function that translates each sequent of level n with empty premises into a sequent of level n − 1, and apply the translation function τ1+− to these translated sequents. However, for the sake of simplicity we omit all of these details, since they do not make any substantial difference. What is important is that the instances of τ1+− ({AiX+ } ⇒1 AY− ) are sequents of level 1 with premises that are axioms of SX+ and conclusions that are axioms of SY− .

8.2 Sequent Calculi for Global Metainferential Validities (and Invalidities)

101

− Theorem 8.2.11 SX/Y is sound and complete with respect to X/Y-invalidities.

Proof It is easily adaptable from [9]. Once we have introduced the method for metainferential logics of level 1, we just need to generalize it to metainferential logics of level n. The only difference from the procedure for metainferential logics of level 1 is that now the translation functions are more general, but once the idea is understood for the first level it can be easily generalized. Let’s start then with the generalization of the translation functions. Let X/Y be a metainferential logic of level n, such that the logics X, Y are of level n − 1. In Definitions 8.2.13 and 8.2.17 we assume that the calculi SY+ , SY− , SX− , SX+ for the metainferential logics of level n − 1 are built using these definitions. Definition 8.2.12 Let τn+ and τn− be defined in the following way:  τn+ (

⇒n δ) =

[∅ ⇒n γ1 ; ...; ∅ ⇒n γm ] ⇒n+1 [∅ ⇒n δ] if  = ∅ if  = ∅ ∅ ⇒n+1 [∅ ⇒n δ]

where in the first case  = {γ1 , ..., γm } is a set of sequents (of level n − 1), and δ is a sequent (of level n − 1). And  τn− ( ⇒n π ) =

[σ1 ⇒n ∅n−1 ; . . . ; σm ⇒n ∅n−1 ] ⇒n+1 [π ⇒n ∅n−1 ] if   = ∅ if  = ∅ ∅ ⇒n+1 [π ⇒n ∅n−1 ]

where in the first case  is a set of antisequents (of level n − 1) and {σ1 , ..., σm } is the set resulting from replacing each antisequent symbol (of level n − 1) in  by the sequent symbol (of level n − 1). π is an antisequent (of level n − 1) and π is the result of replacing the antisequent symbol (of level n − 1) by the sequent symbol (of level n − 1). Finally, ∅n−1 is the symbol defined at the beginning of this section. Now we can define the following calculus for the validities of X/Y: + Definition 8.2.13 Let τn+ , τn− , as in Definition 8.2.12. The calculus SX/Y is defined by the following rules:

• Axioms: τn+ (AY+ ) and τn− (AX− ) with AX− , AY+ respectively being the axioms of SX+ and SY− , with their respective provisos. + -rules: τn+ (R), for every rule R of SY+ . • SX/Y − • SX/Y -rules: τn− (R), for every rule R of SX− . • (Weakening Up): [ ⇒n δ] ⇒n+1 [,  ⇒n δ] • (Weakening Down): [ ⇒n ∅n−1 ] ⇒n+1 [ ⇒n π] where ,  are sets of sequents of level n − 1, δ is a sequent (of level n − 1), and π is a sequent (of level n − 1) different from the sequent ∅n−1 .

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8 Metainferential Sequent Calculi

Just as we have done for metainferential logics of level 1, now we will introduce a calculus for the invalidities of a metainferential logic of level n. We need to start with a generalization of Definitions 8.2.7, 8.2.8 and 8.2.9. Definition 8.2.14 Let τn+− be a function from sequents and/or antisequents of level n to sequents of level n + 1 defined in the following way: τn+− ( ⇒n δ ) = ∅ ⇒n+1 [ n δ] where  is a set of sequents (of level n − 1), and δ is a sequent or antisequent (of level n − 1), and δ is the sequent (of level n − 1) resulting from replacing the antisequent symbol (of level n − 1) by the sequent symbol (of level n − 1) in δ . +

Definition 8.2.15 Let τnR be the function from sequents of level n to sequents of level n + 1 defined in the following way: +

τnR ( ⇒n δ) = [,  n χ] ⇒n+1 [δ,  n χ] where  and  are sets of sequents (of level n − 1), and δ and χ are sequents (of level n − 1). −

Definition 8.2.16 Let τnR be the function from sequents of level n to sequents of level n + 1 defined in the following way: −

τnR (γ ⇒n δ) = [ n γ ] ⇒n+1 [ n δ ] where  is a set of sequents (of level n − 1), γ, δ are two antisequents, and γ , δ are the two sequents resulting from replacing each antisequent symbol (of level n − 1) by a sequent symbol (of level n − 1) in γ, δ. With these new functions, we can provide a calculus for X/Y invalidities. +

−

Definition 8.2.17 Let τn+− , τnR and τnR as in the previous definitions. The calculus − is defined by the following rules: SX/Y • Axiom: τn+− ({AiX+ } ⇒n AY− ) with {AiX+ } being any set of axioms from SX+ and AY− any axiom from SY− , with their respecting provisos + − − -rules: τnR (R1 ), and τnR (R2 ) for every rule R1 of SX+ and every rule R2 of • SX/Y SY− . • (Weakening In): [ n δ] ⇒n+1 [, {AiX+ } n δ], where {AiX+ } is any set of axioms from SX+ . • (Anti-Weakening Up): [,  n δ] ⇒n+1 [ n δ] • (Anti-Weakening Down): [ n π] ⇒n+1 [ n ∅n−1 ] where ,  are sets of sequents (of level n − 1), δ is a sequent (of level n − 1), and π is a sequent (of level n − 1) different from the sequent ∅n−1 . − + and SX/Y . The following result What we have done is to build two calculi SX/Y shows that both are sound and complete for the respective validities and invalidities of the metainferential logic of level n X/Y.

References

103

+ Theorem 8.2.18 SX/Y is sound and complete with respect to X/Y validities and − SX/Y is sound and complete with respect to X/Y invalidities.

Proof The proof is the same as in [9]. Before concluding this chapter, let’s say a few words about how to extend these calculi to the SET—SET framework. Firstly, since global validity in the SET—MET framework is a degenerate case of global2 validity in the SET—SET framework, it is more or less straightforward to extend these calculi to the global2 notion. The idea would be just to allow weakening in the conclusions for the calculi for validities, and weakening restricted to the invalid axioms of the previous level in the calculi for the invalidities. These calculi would help reflect on the preservation of validity in the context of multiple conclusions. We leave the details to the reader. Regarding how to extend the calculi to the local or the (other) global notion, as far as we can see, it is not obvious at all how to do it in general, and so we leave it for future exploration.

References 1. Fjellstad, A. (2022). Metainferential reasoning on strong kleene models. Journal of Philosophical Logic, 51(6), 1327–1344. 2. Golan, R. (2022). Metainferences from a proof-theoretic perspective, and a hierarchy of validity predicates. Journal of Philosophical Logic, 51, 1295–1325. 3. Golan, R. (2023). On the metainferential solution to the semantic paradoxes. Journal of Philosophical Logic, 52(3), 797–820. 4. Ripley, D. (2022). One step is enough. Journal of Philosophical Logic, 51(6). 5. Ripley, D. A toolkit for metainferential logics. Manuscript. 6. Cobreros, P., Tranchini, L., & La Rosa, E. (2020). (I can’t get no) antisatisfaction. In Synthese (pp. 1–15). 7. Girard, J.-Y. (1987). Proof theory and logical complexity. Bibliopolis. 8. Cobreros, P., La Rosa, E., & Tranchini, L. (2021). Higher-level inferences in the strong-kleene setting: a proof-theoretic approach. Journal of Philosophical Logic, 1–36. 9. Da Ré, B., & Pailos, F. (2022). Sequent-calculi for metainferential logics. Studia Logica, 110(2), 319–353. 10. Tiomkin, M. (1987). Proving unprovability (No. CS Technion report CS0478). Technical report, Computer Science Department, Technion. 11. Bonatti, P. A., & Olivetti, N. (2002). Sequent calculi for propositional nonmonotonic logics. ACM Transactions on Computational Logic (TOCL), 3(2), 226–278. 12. Goranko, V. (1994). Refutation systems in modal logic. Studia Logica, 53(2), 299–324. 13. Carnielli, W. A., & Pulcini, G. (2017). Cut-elimination and deductive polarization in complementary classical logic. Logic Journal of the IGPL, 25(3), 273–282. 14. Rosenblatt, L. (2021). Towards a non-classical meta-theory for substructural approaches to paradox. Journal of Philosophical Logic, 1–49.

Chapter 9

Metainferential Theories of Truth

A central feature of metainferential logics is that they can be expanded with a transparent truth predicate.1 Some of them even retain every classical validity, usually up to a certain finite metainferential level. The remarkable exception is the logic STω , which can recapture every classical metainferential validity despite having a transparent truth predicate and a suitable self-referential procedure allowing us to express self-referential (sets of) sentences. Therefore, the truth theory based on STω seems the most attractive truth theory, at least if, in Hjortland’s words [1], the best truth theory is not only the one that both allows self-referential (sets of) sentences to occur in the language while avoiding triviality but also the one that keeps as much classical logic as possible. So far, though, we have not proved any of these claims. This chapter fulfills this promise and explains why metainferential logics are so important for truth theories in particular, and as the basis for solutions to semantic paradoxes, in general. Tarski’s theorem proves that a theory that includes classical logic, a transparent truth-predicate, and a self-referential procedure is trivial. Of course, this is a wellknown theorem, and we will not argue that it is false (how could we?). In addition, we will show in what sense classical logic should be understood, in order to accommodate a transparent truth predicate, avoiding Tarski’s result. For this purpose, we need first to understand when a truth predicate is transparent. A truth predicate T r is transparent if it satisfies the transparency condition, which states that, for every valuation v, every model M and every sentence A, v M (φ) = v M (T r (φ). As we will show, every logic in the ST hierarchy can be safely expanded with a transparent truth predicate. We will start by proving this for TS/ST, the first logic in the hierarchy, and we will call the resulting theory TS/ST+ .

1

In recent years there has been a lot of development on substructural theories of truth, i.e., theories of truth rejecting some of the structural metainferences: rejecting Cut Ripley [2, 3]; rejecting Contraction Zardini and Rosenblatt [4, 5], rejecting Reflexivity French [6] and rejecting Weakening Da Ré [7], among many others.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. Pailos and B. Da Ré, Metainferential Logics, Trends in Logic 61, https://doi.org/10.1007/978-3-031-44381-7_9

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As Kripke proves in [8], Tr can be understood as a fixed-point of a jump operator over the SK’s schema. But the SK schema is the one used to specify how ST, TS, and, moreover, TS/ST valuations work. Thus, TS/ST+ ’s truth predicate Tr can also be understood as a fixed-point, proving that the theory itself is non-trivial. Of course, the details should be handled with care. As T r is a predicate, valuations are no longer enough for interpreting TS/ST+ language. Models, then, should be included in the picture. Let L+ be the result of expanding TS/ST language L with a predicate T r and an infinite number of individual constants.2 L+ ’s interpretation will be partially constrained to ensure that paradoxical sentences will be around.3 In particular, some individual constants will be treated as distinguished names. Thus, we will fix a 1 − 1 function τ from names to formulas of L+ , and require, for every distinguished name n, that n should denote τ (n) in every model. This means that a set F O R(L+ ) of formulas of L+ must be a subset of the domain of every model, and that only infinite models will be admitted.4 Definition 9.0.1 A TS/ST-model for the language L+ is a structure < D, I > such that: • D is a domain such that F O R(L+ ) ⊆ D, and • I is an interpretation function such that: – – – – – – – – –

For an ordinary name a, I (a) ∈ D For a distinguished name n, I (n) = τ (n) n For an n-ary predicate P, I (P) ∈ {0, 21 , 1} D For propositional letters p, I ( p) ∈ {0, 21 , 1} For atomic formulas (that are not propositional letters) φ = P(t1 , t2 , ..., tn ), I (φ) = I (P)(I (t1 ), I (t2 ), ..., I (tn )) I (¬φ) = 1 − I (φ) I (φ ∧ ψ) = min(I (φ), I (ψ)) I (φ ∨ ψ) = max(I (φ), I (ψ)) I (φ → ψ) = max(1 − I (φ), I (ψ))

A TS/ST+ -model is a TS/ST-model that obeys the following restriction: for every pair of formulas φ, T r (φ), I (φ) = I (T r (φ)). This last condition ensures the transparency of T r . With these notions at hand, it is possible to prove that TS/ST+ is non-trivial—or, more accurately, satisfiable. Theorem 9.0.2 TS/ST+ is satisfiable. To keep things as simple as possible, TS/ST+ language will include neither quantifiers nor variables for names. 3 Here we follow the strategy used by Ripley [3]. This way of handling self-reference is also similar to the one used by Barwise and Etchemendy [9]. 4 This assumption is standard in works on theories of truth. See, for example, Kremer’s [10] and Ripley’s [3]. 2

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Proof TS/ST+ ’s satisfiability follows from the fact that the truth predicate can be interpreted as a fixed-point of a jump operator over the Strong Kleene schema, that is, the ones used by TS/ST and TS/ST+ models. In fact, TS/ST models are just simplifications of the ST models Ripley describes in [3].5 And ST models are just the Strong Kleene logic K3 (or LP) models expanded with distinguished names, following the lines of Kripke [8]. TS/ST+ models are the subset of TS/ST models (for L+ ) that satisfy transparency. Moreover, TS/ST+ models are only (simplified versions of) ST models for a first-order language with a transparent truth predicate. And, as Ripley shows in [3], these models are transparent not only because the truth predicate is transparent, but also because “no amount of adding T s [i.e., truth predicates like T r ] or removing them can make a valid argument invalid, or vice versa” (Ripley [3, p. 6]).  Recall that we also have the following Collapse Result. We know by Theorem 4.1.2—presented in the previous chapter–that, for every meta-metainference  ⇒2 ,  ⇒2  is valid in CL if and only if it is valid in ST2 .6 Thus, an obvious thing to do is expand ST2 with a transparent truth predicate. The resulting theory is called ST+ 2 , and it is also satisfiable. Theorem 9.0.3 ST+ 2 is satisfiable. + + Proof ST+ 2 language is L , i.e., TS/ST language. Moreover, ST2 models are the + same as TS/ST models, and the same relation exist between ST+ 2 and TS/ST + + models. Thus, the proof that ST2 is satisfiable is the same as the one for TS/ST . 

And we have also proved the more general Theorem 4.1.7, showing that for every level n (1 ≤ n < ω), a metainference of level n,  ⇒n  is valid in CL if and only if it is valid in STn . So we can take any of the logics of the hierarchy, say STn , expand it with a truth predicate, and adapt the proof of Theorem 9.0.3 in order to show that the resulting theory ST+ n is satisfiable. But we can move one step even further. We have also defined STω , a logic that recovers every classical metainference. One more attractive feature of STω is that not only is it a fully classical logic, but it also supports a transparent truth predicate. We will call the theory obtained from STω by expanding the language with a transparent truth predicate T r , STω+ —i.e., the theory that Pailos [11] calls CMω +. The following is an important result of this theory: Theorem 9.0.4 STω+ is satisfiable. Proof STω+ ’s satisfiability proof runs just as the corresponding proof for TS/ST+ . Notice that the language and the models of both theories are exactly the same.  This is because L+ includes neither quantifiers nor an identity predicate. In the previous chapter we were working with a propositional language. However, all of the results can be easily adapted to the language with predicates and constants, and so we omit repeating the proofs and assume that the theories here are formulated in this new enriched language without any loss.

5 6

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Therefore, STω+ is the truth theory we were looking for: not only does it validate every classically valid metainference of any level, but it also supports a transparent truth predicate. So far, we have shown how to build a transparent theory of truth based on the logic STω . However, one could wonder how this logic solves the paradoxes in relation to the other non-classical solutions, i.e., those based on LP, K3, or ST. In order to answer this question, let’s illustrate the point with a derivation of the Liar paradox (adapted7 from [12, p. 306]):

Id T r (λ) ⇒ T r (λ) R¬ ⇒ ¬T r (λ), T r (λ) R Tr ⇒ T r (λ) Cut

Id T r (λ) ⇒ T r (λ) L¬ ¬T r (λ), T r (λ) ⇒ L Tr T r (λ) ⇒

where λ is the Liar sentence, i.e., it is ¬T r λ. The way to block the paradox based on LP (e.g. [13]) consists in rejecting L¬. Let’s recall that LP is a paraconsistent logic where explosion fails. Thus, the negation of a formula receiving a designated value can also receive a designated value. Thus, the paradox is blocked. Of course, this is just an intuitive way of explaining how LP deals with paradoxes. A fixed-point construction can also be formally made for LP showing non-triviality. On the other hand, the way to deal with the paradox based on K3 is somewhat dual to the one of LP. Instead of rejecting L¬, K3 theorists will reject R¬. And this is perfectly coherent with K3 being a paracomplete logic, i.e., rejecting the law of excluded middle. From a semantic point of view, R¬ must be unsound since the negation of an undesignated formula can also be undesignated. Finally, the case of ST is more interesting, since it accepts all the classical rules for the connectives but blocks the paradox by rejecting the metarule of Cut. Thus, although the logic ST is transitive, once it is extended by a truth predicate it becomes nontransitive. One of the main advantages of this solution versus the previous two is that it can also deal with paradoxes seemingly not involving logical vocabulary, such as the V-Curry paradox (see, e.g., [14]). Therefore, if we were to try to maintain as many classical principles as we could, ST is better than its previous rivals since it validates all the classical inferences. However, it does not validate the metainference of Cut. Following this argument STω is preferable to ST since it validates all the classical metainferences, up to ω. We will now explain, as informally as possible, how STω blocks the paradox. First, notice that, as Identity is valid, the axioms of the derivation are also STω valid. Moreover, each inferential step corresponds to a STω -valid metainference. So, the following two inferences and metainferences are valid: 7

For the purposes of this chapter, we are not considering multisets, but sets.

9 Metainferential Theories of Truth

T r (λ) ⇒ T r (λ)

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R¬

T r (λ) ⇒ T r (λ) ⇒ T r (λ), ¬T r (λ)

But every instance of Cut is also valid, including: Cut

⇒ T r (λ)

⇒

T r (λ) ⇒

Moreover, each premise of that instance of Cut is also valid. T r (λ) ⇒

⇒ T r (λ)

Nevertheless, the empty inference should not be inferred. The proof-theoretic answer to this involves rephrasing the derivation in a calculus suitable for representing STω . There are some options here, as we mentioned in the previous chapter. Ultimately, what allows STω to block the paradox involves the special way it deals with this particular instance of Cut. Here is the thing. This instance of Cut is STω -valid (or, better, valid in the truth theory based on it), as it is valid in (the truth theory based on) TS/ST. And the premises of this instance of Cut are STω -valid (or, better, valid in the truth theory based on it) because they are valid in (the truth theory based on) ST. But in order for us to legitimately infer the empty inference from both of these facts, we will need not that these premises are ST-valid, but that they should be TS-valid. But ⇒ T r (λ) is not TS-valid (in fact, it is TS-antivalid). Thus, though the premises of this instance of Cut are valid qua inferences, they are not valid qua premises of an instance of Cut, which is what we need in order to legitimately infer the conclusion, i.e., the empty inference. In fact, no valuation v satisfies one of the premises of this instance of Cut (i.e.,⇒ T r (λ)). Thus, we lack what is necessary for “triggering” the inferential move represented by that instance of Cut, because not only is one premise not valid (in the relevant standard, the standard for premises of metainferences of level 1, TS): indeed, no valuation satisfies it (i.e., it is antivalid in TS). This is the informal explanation of why the paradox is blocked in STω : the derivation is blocked in the very last step, despite Cut (in general, and specifically this instance of it) being valid and having valid premises. Before ending, we would like to address two possible objections. They were originally raised by an anonymous referee that evaluated the article that is the basis of this chapter, Pailos’ [11]. The first one is the following. She has suggested that TS/ST does not actually recovers Cut, because what we have called Cut does not give the correct metainferential rendering of the unrestricted Cut rule. As far as we know, the discussion about Cut has always focused on rule versions (additive or multiplicative, or even mixed forms) of the metainferential schema that we have called Cut. This is exactly how Cut is presented, for example, in Ripley’s [3, 15]. Surely, the different metainferential schemas that are instances of what we have called Meta-Cut n are related to Cut in some way or another. Nevertheless, we are not inclined to claim that they are, as she has suggested, the same rule—or instances of a possible unrestricted Cut rule. Each instance of Meta-Cut n has its own unique level. We cannot tell how we are supposed to identify rules (i.e., metainferences)

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across levels. In any case, this is not an easy task. But until that issue is settled, we cannot justifiably claim that there is such an “unrestricted Cut rule”. In any case, if, as she has claimed, Meta-Cut (i.e., Meta-Cut 1 ) is the rule that truly expresses the unrestricted Cut rule, then, as ST2 validates it, ST2 truly recovers Cut. However, she might probably claim that the metainference that really expresses Cut (or, as she claims, gives the correct metainferential rendering of the unrestricted Cut rule) in ST2 is Meta-Cut 2 . And, more generally, given that no logic STn validates Meta-Cut n , then no logic in the hierarchy really recovers Cut. But it is worth noticing that none of these objections can be raised against STω . As STω validates any metainference that is valid in at least one of the logics STn in the ST-hierarchy, then, as every metainferential version of Cut is recovered at some point in the hierarchy, STω itself validates every single one of them. Moreover, as it has been proved, STω can be safely expanded with a transparent truth predicate. The resulting theory, STω+ , is satisfiable. But this result has nothing to do with the non-transitive nature of STω+ . In fact, STω+ is as transitive as STω , because both of them validate every metainferential version of Cut, including (what we have called) Cut itself. The main reason that explains STω+ ’s satisfiability is that it is defined through SK-models and its logical constants are the usual ones. In particular, no constant like a strong negation can be defined in these languages—i.e., a unary constant ∼ such that for every formula A and valuation v, v(∼ A) = 0 if and only if v(A) = 1, and in any other case, v(∼ A) = 1.8 Every paradoxical sentence that can be expressed with the vocabulary of the theory, like the Liar or any Curry-sentence, might (or will) receive the value 1/2 in every valuation. The second objection points directly to STω . It claims that it is not correct to say that STω is fully classical just because it not only recovers every classical inference but also every metainference of any level. If STω were a real fully classical logic, then it would behave just like any presentation of classical logic. For example, it would not be satisfiable when a transparent truth predicate is added to it. But, as any standard presentation of classical logic, such as a Hilbert-style axiomatic calculus for classical logic, cannot be expanded with a transparent truth predicate, this shows that classical logic and STω are two different things. This point is contentious, for at least three reasons. The first one is the following. Some authors, such as Ripley and his co-authors in Cobreros, Ripley, Egré and van Rooij [3], defend that different presentations of the same logics may behave differently with respect to the same language. Then, it is perfectly possible that, for example, a two-valued presentation of classical logic—e.g., CL—cannot be safely expanded with a transparent truth predicate, while that kind of predicate can be added to a three-valued presentation of it—e.g., ST—without the resulting theory becoming trivial. We do not endorse this position. Nevertheless, we do think that it is an open debate whether a logic is sensitive to the different ways it might be presented— e.g., in a sequent calculus, in a natural deduction system, in an axiomatic one, in a two-valued semantics, in a three-valued semantics, etc.

8

For more about strong or classical negations, see Omori [16].

References

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The second reason is that in order to establish whether or not two logics are in fact the same, a criterion for distinguishing between logics is required. Here, we have implicitly assumed an extensional criterion. This norm proposes the following: for any two logics L 1 and L 2 , L 1 is the same logic as L 2 if and only if they validate exactly the same sentences, inferences and metainferences of level n, for any level n. If this is the right way to distinguish between logics, then it is correct to claim that CL and STω are the same logic. But there are other criteria that have been proposed. For example, Dicher and Paoli [17], claim that a logic is a similarity class of abstract consequence relations. And according to that criterion, for example, ST and CL are two different logics—in fact, ST is just LP. If Dicher and Paoli’s criterion is the right one, then every logic in the hierarchy is also equivalent to LP. We are not sure whether or not STω and CL belong to the same similarity class of abstract consequence relation. In any case, whether or not Dicher and Paoli’s criterion is the right way to distinguish between logics, or whether the extensional criterion is the right one, or whether none of them gets things right, is a complex issue. Thus, we want to defend the following, weaker version of our original claim. Suppose the extensional criterion is the right way to distinguish logics. In that case, it is certainly true that CL and STω are the same logic—and it would be right to claim that STω is fully classical. But if the antecedent of this conditional is false, then CL and STω may be different logics. We think that Dicher and Paoli’s criterion is not a good way to identify logics for many reasons. Just to mention one: according to them, there cannot be substructural logics, because every abstract consequence relation needs to be reflexive, monotone, and transitive. Thus, if one thinks that substructural logics are logics, then one needs to reject Dicher and Paoli’s criterion.9 Finally, even if it is true that STω is not equivalent to CL, it still can be true that STω is fully classical. The point, here, is what characterizes a logic as classical. If this means being equivalent to CL, then STω might not be classical—at least, for example, if two logics cannot be equivalent if they behave differently when they are expanded with the same vocabulary. But if what defines a logic as classical is that it should validate the same set of metainferences (of every level) as CL, then STω is certainly classical—even if it turns out not equivalent to CL.

References 1. Hjortland, O. T. (2021). Theories of truth and the maxim of minimal mutilation. Synthese, 199(Suppl 3), 787–818. 2. Cobreros, P., et al. (2014). Reaching transparent truth. Mind, 122(488), 841–866. 3. Ripley, D. (2012). Conservatively extending classical logic with transparent truth. Review of Symbolic Logic, 5(2), 354–378. 4. Zardini, E. (2011). Truth without contra(di)ction. The Review of Symbolic Logic, 4(04), 498– 535. 5. Rosenblatt, L. (2019). Noncontractive classical logic. Notre Dame Journal of Formal Logic, 60(4), 559–585. https://doi.org/10.1215/00294527-2019-0020 9

See [18] for more reasons against Dicher and Paoli’s criterion.

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6. French, R. (2016). Structural reexivity and the paradoxes of self-reference. Ergo, an Open Access Journal of Philosophy, 3. 7. Da Ré, B. (2021). Structural weakening and paradoxes. Notre Dame Journal of Formal Logic, 62(2), 369–398. 8. Kripke, S. (1975). Outline of a theory of truth. Journal of Philosophy, 72(19), 690–716. 9. Barwise, J., & Etchemendy, J. (1987). The liar: An essay on truth and circularity. Oxford, UK: Oxford University Press. 10. Kremer, M. (1988). Kripke and the logic of truth. Journal of Philosophical Logic, 17, 225–278. 11. Pailos, F. (2020). A fully classical truth theory characterized by substructural means. The Review of Symbolic Logic, 13(2), 249–268. 12. Ripley, D. (2015). Comparing substructural theories of truth. Ergo, 2(13), 299–328. 13. Priest, G. (1979). The logic of paradox. Journal of Philosophical logic, 8(1), 219–241. 14. Beall, J. C., & Murzi, J. (2013). Two flavors of curry’s paradox. Journal of Philosophy, 110, 143–65. 15. Ripley, D. (2013). Paradoxes and failures of cut. Australasian Journal of Philosophy, 91(1), 139–164. 16. De, M., & Omori, H. (2015). Classical negation and expansions of Belnap-Dunn logic. Studia Logica, 103(4), 825–851. 17. Dicher, B., & Paoli, F. (2019). ST, LP and tolerant metainferences. In Graham priest on dialetheism and paraconsistency (pp. 383–407). Springer. 18. Barrio, E., Pailos, F., & Szmuc, D. (2019). (Meta)inferential levels of entailment beyond the Tarskian paradigm. Synthese. https://doi.org/10.1007/s11229-019-02411-6

Chapter 10

Philosophical Reflections: Applications and Discussions

All the facts and results that we have presented and displayed reveal a partial answer to a general concern about this project: why should we care about metainferential logics? Why are they interesting, and worth our time? In the Introduction, we presented different applications of metainferential logics, which jointly provide an answer to these questions. We have shown, among other things, how to successfully apply the ST-hierarchy of metainferential logics to deal with semantic paradoxes, by building a theory of transparent truth, how the TS-hierarchy gives a new and deeper sense of how a logic can be empty, how metainferential logics based on structural logics fail to validate some structural metainferences, and even how some metainferential logics built using substructural logics can validate all the classical metainferences. We have also mentioned other important applications of metainferential logics to some philosophical and technical problems, such as how a logic should be characterized, how to analyze the debate between global and local validity, or how to understand abstract features of consequence relations. In the sequel, we will present another three main applications of metainferential logics: (i) the consequences of metainferential logics for the debate around logical pluralism, (ii) what it means for two metainferential logics to be dual,1 and (iii) the consequences of metainferential logics regarding how to understand what a logic is, and, specifically, what a paraconsistent logic and what a paracomplete logic are. Finally, we will discuss some more important philosophical issues revolving around metainferential logics that have still not been addressed.

1

As a reviewer rightly points out, this is not really a philosophical application. We do think, though, that this shows one fruitful technical application of metainferential logics, one that sheds light on the relationship between some logics that were traditionally thought of as dual, despite the lack of a precise way of specifying in which sense these logics were dual. We have considered, in particular, the case of ST and TS.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. Pailos and B. Da Ré, Metainferential Logics, Trends in Logic 61, https://doi.org/10.1007/978-3-031-44381-7_10

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10.1 Metainferential Logics and the Monism/Pluralism Debate. New (Metainferential) Collapses Logical pluralism is the thesis that there is more than one correct logic.2 One of the most important lines against logical pluralism is the one represented by the Collapse Argument. This argument has been widely discussed, for example, in the works by Williamson [1], Priest [2], Read [3], and Keefe [4]. This is a usual version of the Collapse Argument—see, for example, Caret [5, p. 4]. Suppose a pluralist who accepts that two or more logics are correct with respect to the same context C, believes the premises of an argument judged as valid-in-C by one of the logics she regards as correct—and she knows this—, although it is not valid-in-C according to some other logic she regards as correct. What should she do with the conclusion of this argument? Should she accept it or not? Either way, the pluralist position collapses into the system that results from collecting all the arguments that she accepts—which will be equal to at most one of the logics she accepts. Barrio, Pailos and Szmuc [6] claim that this kind of reasoning needs to be substantially refined. This account assumes that for two logics to be different, they need to have different sets of valid inferences. But as we have shown, there are different logics with the same set of valid inferences. These logics disagree regarding the set of valid metainferences. A pluralist embracing a pair of logics that are different in this sense will not be subject to the present form of the Collapse Argument. Thus, the Collapse Argument (or this version of it) seems not to cover all possible cases of pluralism. Nevertheless, the authors show that the phenomenon can be generalized, given that different logics share their valid inferences and metainferences up to some finite level, but not their valid metainferences. And they provide a recipe for building metainferential versions of the original Collapse Argument. The Collapse Argument seems to rest on some strong presuppositions. In this regard, Stei [7, p. 3] noticed that among these we have that logical consequence is global in scope, that logical consequence is normative, and that there is rivalry between different correct consequence relations. Moreover, rivalry also seems to presuppose that the logics need to share the same vocabulary and have the same domain of application. But, once these issues are dealt with, what remains is a highly intuitive idea: that there is rivalry between two logical systems if there is at least one inference such that it is valid in one logic, but not in the other. Thus, one of the most important presuppositions of the Collapse Argument is identifying a logical system with its set of valid inferences. Admitting that two logics may have the same set of valid inferences while nevertheless being different logics seems to be an option that is not touched by the Collapse Argument, in its present form. This opens the door for a pluralist to endorse different 2

This thesis has been defended in many places. But the particular version of it that started the debate around the Collapse Argument is the one introduced in Beall and Restall [8].

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logics with the same valid inferences. Thus, there will be no inference whatsoever regarding which these logics disagree. One example of this is provided by a potential pluralist who embraces both CL and ST. As we already know by now, the set of valid metainferences is different in both cases. In particular, Cut is valid in the former, but not in the latter. This pluralist seems to be untouched by the Collapse Argument. Such a pluralist will never find herself faced with an inference that is considered valid by one logic but not by the other. One available option for the anti-pluralist could be to insist on claiming that the above fact shows nothing more than the identity of CL and ST. Nevertheless, as CL is prone to trivialization when faced with paradoxes, while ST is not, this claim is hard to defend. The best response for the anti-pluralist is to develop a new anti-pluralist argument that helps her cope with this case. In the sequel, we will introduce such criticism as characterized by Barrio, Pailos and Szmuc [6]. It can be interpreted as extending the Collapse Argument to the metainferential realm. It is based on a way of identifying logics that is capable of distinguishing between systems that coincide in their valid inferences, and that will tell CL and ST apart. The key idea is that, if two logics agree with respect to their valid inferences but disagree regarding their valid metainferences, then they are in fact different logics. This criterion has been both introduced and defended in Barrio, Pailos, and Szmuc [9] and Pailos [10]. And this will help the anti-pluralist to rehash the Collapse Argument in such a way that a pluralist embracing CL and ST cannot escape from it. This makes it possible to devise a new identity criterion for logics by stating that two logics are identical only if they have the same set of valid inferences, and the same set of valid metainferences. And similarly, that rivalry between two logics happens if and only if they are not identical concerning this new identity criterion. It is worth noticing that this new version of the Collapse Argument extends the limits of rivalry between logics well beyond the traditional version of the Argument, which focuses only on inferences. As powerful as the original Collapse is, it cannot be used to dismiss every form of pluralism (about the same context), as there are logics that are typically regarded as different—and that are applied mainly to different purposes— while sharing their validities. This is the case, in particular, of the rivalry between CL and ST. With the metainferential way of characterizing logics, these two are different, specifically, as regards the things they make valid. Those things are now not limited to inferences, but they also include metainferences. Thus, rivalry can emerge also in the metainferential realm. What makes this new version interesting, then, is that it generalizes the Collapse, covering now cases that seem to be, in the traditional version of the Collapse, a way out for the pluralist. This seems enough to justify its interest for anyone who finds the monist/pluralist debate philosophically relevant.3 3

As will soon be apparent, the answer to the new version of the Collapse Argument is true to the premises of the argument (as opposed, for instance, to a contextualist way of rejecting the Collapse Argument, which only admits one true logic for each context), in the sense that it does not reject any one of them.

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So now CL and ST become rival logics and, hence, the anti-pluralist can propose what the authors called a refined version of the Collapse Argument. The key idea of that argument is the following. Suppose that the pluralist takes as correct two logics that have the same set of valid inferences but disagree concerning their metainferences. Consider, now, one metainference that is valid according to one logic but not the other, and a scenario when the pluralist believes all the premises of that metainference. Should she accept the conclusion of the metainference? It is not possible to give a pluralist answer to this question. Thus, the pluralist position collapses into a monism. Nevertheless, this new refinement might not be enough, as is probably expected by now. For, as we know, TS/ST is a logic that has the same inferential and metainferential validities as CL, but which is, nevertheless, not identical to CL, because it differs in its metainferential validities from level 2 and above. Thus, a pluralist that accepts both CL and TS/ST is not touched by either the Collapse Argument or the previously sketched refinement, as there is no inference or metainference valid in one logic but not in the other. But, of course, a new refinement of the Collapse Argument is possible. This will have to consider not only valid inferences and valid metainferences, but also valid metametainferences—or metainferences of level 2—when providing identity and rivalry criteria for logical systems. The new identity criterion specifies that two logics are identical only if they have the same set of valid inferences, the same set of valid metainferences, and the same set of valid metametainferences. Rivalry between two logics arises considering any difference in any of these three levels. As we know, there is an actual difference between CL and TS/ST regarding metametainferences. Looking through the lens of this new criterion, CL and TS/ST are indeed rival logics, so the anti-pluralist can easily re-instantiate the Collapse Argument in an even more refined form. For let’s suppose there is a pluralist that takes these two logics as correct, and believes all the premises of a given metametainference that is valid according to one logic but not the other. Should she accept the conclusion of that metametainference? There cannot be a pluralist response to this question (or so the argument goes). Thus, once again, the pluralist position collapses into a monist stance. For sure, the argument can be mimicked at higher and higher levels. As we know, for any finite level n, it is possible to find a logic that matches classical logic up to that level, but whose set of validities of higher levels is properly included in CL. And a pluralist that takes as correct both CL and such a metainferential logic might avoid a new metainferential version of the Collapse Argument that only considers inferences and metainferences up to level n. According to the authors, this means that the anti-pluralist should adopt a logical identity criterion that covers every metainferential level. Based on this new criterion, two logics are identical only if they have the same set of valid inferences and the same set of valid generalized metainferences of all levels—i.e. if they have the same set of valid inferences and the same set of valid metainferences of level n, for all n ∈ ω. Accordingly, it is possible to defend that there is rivalry between two logics if and only if they are not identical concerning the fully generalized criterion of identity

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detailed above. So not only will CL and ST be judged as rival logics, but also CL and TS/ST, as well as CL and any logic that does not coincide with it regarding all the inferences and metainferences of every level. As Barrio, Pailos and Szmuc [6] stated, this is the basis of the Collapse Argument in its strongest and most general form, which is the following. Suppose the pluralist embraces two logics that are not identical, such that there is at least a certain metainferential level n such that its sets of valid metainferences are not the same. Suppose, additionally, that the pluralist believes all the premises of a given metainference of level n that is valid according to one logic but not according to the other. Is she or is she not entitled to accept the conclusion of the given metainference? There cannot be pluralism about this, as it seems that the pluralist cannot get away with it by using any of the previously discussed strategies. But maybe this is just because we are not picking the right kind of pluralist. And this is what Barrio, Pailos and Szmuc [6] got wrong. The pluralist that might cause trouble to all of these kinds of Collapse Arguments should take as correct two different logics that have the same metainferential validities of any finite level, such as CL and STω . There is no chance of disagreement here because they do not disagree on any (meta)inferential validity at all.4 There are some ways for the anti-pluralist to reject this conclusion. First, she might strengthen the way of characterizing a logic. Thus, she might claim that two logics are identical if and only if they share the same set of metainferential validities. But this might be too extreme, for it implies that CL and STω are in fact the same logic (while still defending that the non-trivial truth-theory based on STω recovers everything important about CL, thus being a better fit from an anti-exceptionalist point of view than, at least, any other inferential solution). And there are important reasons for rejecting this view. Some of the differences are quite obvious. For instance, while the latter is characterized by three-valued Strong Kleene matrices, the former is characterized by two-valued Boolean matrices. But as these logics, not only have the same inferential validities, but also the same set of metainferential validities at any level, if a logic is characterized solely regarding its validities, then, they should be considered just as two different presentations of the same logic, though with important differences, as the former, but not the latter, can be expanded non-trivially with a transparent truth-predicate. This, indeed, is the position defended by Ripley in [11] (regarding CL and ST). To maintain that there are different logics means either that a logic should be characterized intensionally—and not only based on what it validates—, or extensionally, but taking into account also other things besides its validities. In fact, as we have already mentioned, Scambler [12] defends that a logic should also be characterized through what he calls its antivalidities, while Barrio and Pailos [13] also argue for including contingencies as part of the picture. These authors associate validities with what a logic (or a logician who supports that logic) 4

Of course, this path could continue to grow if we chose to extend the hierarchy of metainferential levels beyond the finite ones. For the reasons already mentioned in Chap. 4, we chose to work only with finite levels, and thus CL and STω are the last relevant pair of logics (of this type) we will assess here.

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accepts (just by supporting that logic), antivalidities with what a logic rejects, and contingencies with the kind of things a logic neither accepts nor rejects. Finally, Cobreros, Tranchini, and La Rosa [14] propose three other conditional properties closely related to both validities and antivalidities. If any of these new properties should be taken into account when characterizing a logic, then there seems to be no room for stating that STω and CL are in fact just different presentations of the same logic. And a pluralist who takes both of them as correct is free from the hierarchy of Collapse Arguments presented in Barrio, Pailos and Szmuc [6].

10.2 Duality Between Metainferential Logics There are many ways of defining duality for logics. In [15], Da Ré, Pailos, Szmuc and Teijeiro have explored one of these different possibilities, namely negation duality, and applied it to metainferential logics. According to this view, inferential duality is defined as follows: Definition 10.2.1 Two inferential logics L1 and L2 are dual if and only if for every pair of set of sentences , : •  ⇒  is valid in L1 if and only if ¬() ⇒ ¬() is valid in L2 , and •  ⇒  is valid in L2 if and only if ¬() ⇒ ¬() is valid in L1 where ¬() = {¬γ : γ ∈ } and similarly ¬() = {¬δ : δ ∈ }. Given this definition, as noted by [16], it is easy to check that LP and K3 are dual, while ST and TS are self-dual. However, while, like inferential logics, ST coincides with classical logic and TS is empty, their main interests lie in their metainferential properties as logics of level 1. So, in [15], Da Ré, Pailos, Szmuc and Teijeiro wonder how to extend the notion of duality to metainferential logics. The first problem is that the very notion of negation duality for metainferences depends for its formulation on the presence of some kind of negation for (meta)inferences, and it is not obvious at all how to formulate such a negation. The way they do it is by defining positive and negative (meta)inferences. Positive (meta)inferences, denoted by  ⇒+ n , are interpreted just as regular (meta)inferences, i.e. what we have been denoting by  ⇒n , and so their satisfaction conditions in each logic are defined as above. Regarding the negative (meta)inferences, denoted by  ⇒− n , their satisfaction conditions are the following5 : Definition 10.2.2 Given a logic L1 , a valuation v satisfies the negative + (meta)inference  ⇒− n  if and only if v satisfies ⇒n γ, for all γ ∈ , and v satisfies 5

We adapt the following definition from Da Ré, Pailos, Szmuc and Teijeiro [15], in the way Da Ré, Rubin and Teijeiro do it in [17].

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δ ⇒+ n for all δ ∈ . And of course a negative (meta)inference is valid in a logic if every valuation satisfies it. At first sight this definition could seem a bit artificial, in the sense that it is not clear in what sense this defines a negation. However, a closer inspection of the definition via some examples helps enlighten the situation. For instance, let’s consider CL. In CL we have the following: Exclusion: it is not the case that there is a Boolean valuation v such that at the − same time it satisfies  ⇒+ n  and  ⇒n . − Exhaustion: every Boolean valuation v satisfies either  ⇒+ n  or  ⇒n ,

Both facts can be easily checked. When considering ST and TS as logics of level 1 we obtain the following. In ST exclusion does not hold (but exhaustion does), whereas in TS exhaustion does not hold (but exclusion does), and this is the same that happens in LP and K3 regarding the negation of the Strong Kleene schema! So, the next step is to define negation duality for metainferential logics. Definition 10.2.3 Two metainferential logics L1 and L2 of level n + 1 are dual if and only if for every pair of positive or negative (meta)inferences of level n , : + •  ⇒+ n+1  is locally valid in L1 if and only if ∗ ⇒n+1 ∗ is locally valid in L2 , and − •  ⇒− n+1  is locally valid in L2 if and only if ∗ ⇒n+1 ∗ is locally valid in L1

where ∗ is just the result of changing the sign of  (i.e. if  is a positive (negative) meta(inference), ∗ is its corresponding negative (positive) (meta)inference), and similarly for ∗. Notice that we do not need to take into account negative metainferences of level n + 1: given the satisfaction conditions in Definition 10.2.2 for duality, it is sufficient to consider positive metainferences. Now we are in a position to introduce the main result: Fact 10.2.4 ([15]) ST and TS as metainferential logics of level 1 are dual. Proof The proof is very simple and can be found in Da Ré, Pailos, Szmuc, Teijeiro [15].  Before closing this brief section, notice that once we are taking into account metainferential logics, two logics may be dual at one level but differ at another level. For instance, as we saw, ST and TS as inferential logics (or metainferential logics of level 0 if you prefer) are not dual (actually each one is self-dual) but, as Fact 10.2.4 shows, as metainferential logics of level 1 they are dual. It is fairly easy to check that this also does not hold for ST and TS as metainferential logics of level 2. So, we leave it open for future work to define hierarchies of dual logics, i.e. two hierarchies such that each logic in one hierarchy is dual to the corresponding logic in the other hierarchy (hint: as metainferential logics of level 2, TS/ST and ST/TS are dual).

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10.3 What Is a (Paraconsistent or Paracomplete) Logic? Paraconsistent logics are usually characterized as the ones that reject the classically valid principle of Explosion, which states that from a contradiction, everything follows. Barrio, Pailos and Szmuc [18], however, offer a new characterization of what paraconsistent logics are, which we will call BPS-paraconsistency.6 In a nutshell, they claim that a logic L is BPS-paraconsistent if either the inferential or the metainferential formulation of Explosion is invalid in it. While the one on the left below is a traditional formulation of Explosion, the one on the right is the metainferential form of Explosion for metainferences of level 1, according to the authors. φ, ¬φ ⇒ ψ

⇒ φ ⇒ ¬φ ⇒ψ

Explosion, in its traditional, inferential form, can be understood as saying that contradiction implies triviality—i.e., that an inference with an inconsistent premise set implies any conclusion. More formally, they present the following, extended way of understanding what a paraconsistent logic is: Definition 10.3.1 A logic L is BPS-paraconsistent if and only if (at least) one of the following holds: • φ, ¬φ ⇒ ψ is invalid in L • [⇒ φ, ⇒ ¬φ] ⇒1 ⇒ ψ is invalid in L. Therefore, according to this new criterion, LP remains paraconsistent, as it is both inferentially and metainferentially BPS-paraconsistent. TS is also paraconsistent, but just because it is inferentially, but not metainferentially, BPS-paraconsistent. Finally, ST, despite being inferentially classical, is also BPS-paraconsistent, but not because it is inferentially explosive—which is not—, but metainferentially explosive. The authors also suggested that the definition of BPS-paraconsistency can be extended to higher levels. Nevertheless, they do not develop it. We will present a simple way to do it, which adapts a definition introduced by Scambler [12]. Definition 10.3.2 For any formula φ, φ0 = φ and φn+1 = ∅ ⇒n φn . Thus, the following extended definition of BPS-paraconsistency can be displayed. Definition 10.3.3 A logic L is BPS-paraconsistent if and only if (at least) one of the following holds:

6

Of course the authors just talk about paraconsistency simpliciter.

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• φ, ¬φ ⇒ ψ is invalid in L • [φn , ¬φn ] ⇒n+1 ψn , for any 1 < n < ω, is invalid in L. With this new definition at hand, we might try to give different measures of paraconsistency relative to this new definition. Nevertheless, we will not pursue this research line here. Interestingly, we have a ready-made definition of paracomplete logic. Definition 10.3.4 A logic L is BPS-paracomplete if and only if (at least) one of the following holds • ∅ ⇒ φ ∨ ¬φ is invalid in L • ∅ ⇒n+1 (φ ∨ ¬φ)n , for any 0 < n < ω, is invalid in L. Adapting the previous notions, we can conclude that K3 and TS are BPSparacomplete at every level.7 Notice that, from Definitions 10.3.3 and 10.3.4, ST is only metainferentially BPSparaconsistent, while TS is BPS-paracomplete at every level. Some people could take this to be somewhat unreasonable given the duality between these two logics. However, this tension is only apparent. ST and TS are inferentially self-dual. Actually, while ST is neither inferentially paraconsistent nor inferentially paracomplete (since it is classical logic!), TS is both inferentially paraconsistent, and paracomplete (since it is an empty logic!). Thus, the duality between these two systems can hold only at the metainferential level.8 So far, we have been using the prefix BPS for the concepts to indicate that they are inspired by the way contradictions are conceived in Barrio, Pailos and Szmuc [18], i.e. using the negation connective. However, as we mentioned in the previous section, there are other ways for expressing contradictions for metainferential levels. In [17], Da Ré, Rubin and Teijeiro have offered a different characterization of what a metainferentially paraconsistent logic is, which we will call DRT-paraconsistency, in order to distinguish it from the previous characterization. As in the BPS definition, the authors define DRT-paraconsistency in terms of the failure of a metainferential version of Explosion. However, they disagree on the way meta Explosion is defined. As introduced in Sect. 10.2, using the terminology from Da Ré, Pailos, 7

An anonymous reviewer suggests to us that Ripley [19], relates the failure of Cut to a failure of some (metainferential) form of Excluded Middle. In that article, Ripley conceptualizes Cut as an extensibility constraint, meaning that if the conclusion of Cut fails, then it is either in bounds to accept the Cut formula, or it is in bounds to deny it. This can be understood as a form of excluded middle because it is not always the case that for every formula, it is in bounds either to accept it or to deny it. So when evaluated from this other perspective, ST has a paracomplete flavor, which might feel as a tension with its (BPS-)paraconsistent behavior. Notice, though, that this paracomplete feeling emerges from the particular philosophical interpretation of inferences supported by Ripley, while its BPS-paraconsistency derives only from what the theory validates. Thus, the tension between these two features of ST does not result in any kind of formal flow of the theory. 8 We would like to thank an anonymous reviewer for pointing this out to us.

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Szmuc and Teijeiro [15], the authors extend the language with negative and positive (meta)inferences, and define DRT-paraconsistency as follows: Definition 10.3.5 A logic L is DRT-paraconsistent if and only if Explosion is inferentially invalid or the following metainference: Metan E x plosion

 ⇒+ n−1 

 ⇒− n−1  +/−

 ⇒n−1 

is locally invalid, for some n with n ≥ 1. Notice that Metan Explosion does not depend on any particular vocabulary (in particular, even if a logic lacks a negation connective it can be metainferentially paraconsistent). Also, applying this definition, LP is DRT-paraconsistent because the inference of Explosion fails, although Meta1 Explosion is locally valid in LP/LP. Moreover, ST is DRT-paraconsistent because Meta1 Explosion is locally invalid in ST/ST. For more details on the comparison between DRT and BPS paraconsistency, see Da Ré, Rubin and Teijeiro [17].9 Also, there is another related concept, pure paraconsistency, which was originally introduced in Da Ré [20], and applied and extended in Da Ré, Rubin and Teijeiro [17]: Definition 10.3.6 A logic is purely paraconsistent if and only if not only is explosion invalid at the inferential level, but also Metan Explosion fails at every level n. Notice that neither LP nor ST are purely paraconsistent according to these definitions. In [17], Da Ré, Rubin and Teijeiro introduce a logic which is paraconsistent at every level. On the other hand, the idea of defining a DRT-paracomplete logic is straightforward. Since Da Ré, Rubin and Teijeiro [17] adopt the S E T − S E T framework, we can define paracompleteness as follows: Definition 10.3.7 A logic L is DRT-paracomplete if and only if LEM is inferentially invalid (i.e. the inference ⇒ ϕ ∨ ¬φ) or the following metainference: Metan L E M

 ⇒+ n−1 

∅

 ⇒− n−1 

is locally invalid, for some n with n ≥ 1. Of course, as before K3 and TS are DRT-paracomplete. We could also easily define a notion of pure paracompleteness. It is more or less obvious that under this 9

There is yet another way of defining metainferential paraconsistency using the inferential negation introduced by Fiore, Pailos, and Rubin [21]. We leave this exploration for future work.

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definition TS would be purely paracomplete. And this seems also an interesting (technical) application. As a final comment, we remain neutral regarding the dispute between the definitions developed by BPS and DRT as one of the authors of this book is also one of the authors of BPS, while the other is one of the authors of DRT. Leaving this aside, we think each position has its own merits, and we leave it to the reader to decide in favor of one of these two concepts, or to develop a new one.

10.4 Further Issues In this section, we will focus on two possible philosophical topics revolving around metainferential logics, and we will finally mention further explorations. The first topic we want to address is whether a logic can be understood as an inferential (satisfaction or validity) standard—i.e., whether we might remain agnostic about which metainferences are or are not valid. The following quote by Ripley gives a positive answer to this issue. ... For example, [12]... says “Absent any other reasons for suspicion one should probably take [X] [i.e., a list of satisfaction or counterexample relations, one for each metainferential level] to be what someone has in mind if they only specify X [i.e., a satisfaction or counterexample relation for inferences].” I don’t think this tendency is warranted. Most of the time, when someone has specified a meta0 counterexample relation (which is to say an ordinary counterexample relation), they do not have the world of all higher minferences [i.e. higher-level metainferences], full counterexample relations [i.e., counterexample relations that include a counterexample relation for every metainferential level], etc, in mind at all. They are often focused on validity for meta0 inferences (which is to say inferences). (Ripley [22, p.12]).)

Though at least one of us thinks that, in a sense, people do have in mind [X] when they say X, we will not argue for that here. We just want to defend that they had better make some commitments regarding higher metainferential levels if they want to avoid unpleasant consequences. More specifically, we think the following position, captured by these two quotes, should be revised: As I’ve pointed out, an advocate of ST as a useful meta0 counterexample relation has thereby taken on no commitments at all regarding metan counterexample relations for 1 ≤ n. (Ripley [22], p. 16).) Lifting depends on information carried by a counterexample relation that is not there in the consequence relation it determines. Or: if someone specifies just a metan consequence relation, they have not thereby settled on any particular metan+1 consequence relation. (Ripley [23])

This discussion might be less abstract if we focus on a particular example. Thus, we will consider the case of LP. If LP is just its valid inferences, or even its valid inferences plus every (globally valid) rule it is closed under, then it will not matter how it is locally and metainferentially described. Thus, the logics mentioned in the next paragraph will be different, but acceptable, ways to present LP. They just differ

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on their commitments regarding metainferences (of level 1). Some of them are very awkward. Are we still willing to categorize them as just versions of LP? Consider for instance four of the logics introduced in Chap. 3: LP/LP, K3/LP, ST/LP and TS/LP. Assuming a local notion of inferential validity, and focusing on metainferences, these four logics behave quite differently, as the following results show. Fact 10.4.1 Cut is invalid in K3/LP. Fact 10.4.2 Every structural metainferential schema—but Identity—is invalid in ST/LP. Fact 10.4.3 All the structural metainferences are valid in TS/LP, which is, then, a fully structural logic. But there are even more bizarre logics (or maybe different metainferential presentations of the same logic) that yield even stranger results, but nevertheless share LP’s inferential validities. Specifically, we are talking about the Weak-Strong family of logics that we have introduced in Chap. 5. One of those logics is LPw /LP. This is one notable fact about it. Fact 10.4.4 The following metainference: (Meta − Simpli f ication)

⇒ϕ∧ψ ⇒ϕ

is locally invalid in LPw /LP. Therefore, while some of these metainferential logics fail to validate some structural properties, we can also have a divergence in the validity of metainferences involving logical vocabulary. This is, at first, an unpleasant result. Of course, what we have said relies on a local way of understanding metainferential validity. As we have seen in Chap. 7, this is not mandatory. But even if we switch to a global or an absolutely global notion of metainferential validities, these problems will remain. Let’s illustrate our point with the logic TS/LP in its global2 reading. In this logic, every metainference schema will be globally2 valid, since no inference is valid in TS. So, while LP is not commonly thought of as a trivial metainferential logic TS/LP, globally2 validates every metainferential schema with a non-empty set of premises. Nothing of what we have said, though, explains why we should care about metainferences of higher levels. Nevertheless, this kind of awkward examples can be emulated at higher metainferential levels, by higher metainferential logics. And none of these rely on the particular behavior of LP, as we can repeat these results for ST, K3 or TS, and extend them to higher metainferential levels.

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10.5 Future Work Metainferential logics are just starting to develop. As such, we see two main new results that can be achieved in this field. First, the construction of new metainferential logics. Second, new applications to areas and problems that have yet not been addressed with these tools. As an example of the first, we will mention the following. Both ST and TSbased hierarchies are uniform and recursive. They are uniform in the sense that each step in the hierarchy extends what the previous step has accomplished. And they are recursive in the sense that there is a well-defined recipe to build each logic in the hierarchy in terms of the previous ones. But there is no reason to think that every hierarchy of metainferential logics behaves like this. Hierarchies might not be uniform, and might not be recursive either. One path worth exploring is to find out what conditions need to be fulfilled in order to have a uniform and recursive hierarchy. One of the two uniform and recursive hierarchies that have been mentioned—i.e., the ST hierarchy—, presents new and more fulfilling solutions to traditional problems in the field of philosophical logic, like semantic paradoxes (because, in the terms used by Hjortland [24], it mutilates less classical logic than its inferential rivals). The other hierarchy, the one based on TS, gives new insights into the nature of logical theories (because it shows what a truly empty logic might look like). A second unexplored aspect refers to metainferential logics built by means of inferential logics with models with fewer or more than three values.10 Logics can be characterized by different many-valued semantics. In particular, Cobreros, Ripley, Egré and van Rooij [25], and Ripley [11, 19], have supported the idea that ST is a three-valued presentation of classical logic. Thus, it is possible to claim that a metainferential logic characterized by, for example, K3 and a bivalent presentation of classical logic, is the same as the one characterized by K3 and ST. But, as Ripley and French [26], and Chemlá, Egré, and Spector [27] have argued, there are logics that cannot be characterized with a semantic apparatus with fewer than three values. Also, there are others that cannot be characterized by a semantics with fewer than four values. It is worth exploring possible techniques to characterize metainferential many-valued logics that are not three-valued. Finally, there are some novel applications that have not been explored yet. For example, whether or not a difference in metainferential validities (or even in metainferential antivalidities) affects the meaning of the logical constants, if it does it at all. If it does, then this means that the meaning of logical constants cannot be captured in 10

This does not mean at all that we think that more than three values are needed. In fact, Cobreros, Ripley, Egré and van Rooij [28] show that the theory of vagueness based on the three-valued ST that they defend in [29] and the theory of vagueness with uncountable truth values defended by Smith [30] are in fact two different logics with what they call a parameterized consequence. They prove that, against Smith’s claim, not every theory of vagueness based on a three-valued setting faces what he calls the jolt problem. In fact, the theory of vagueness based on ST is free of that problem. Moreover, it also validates the principle of tolerance used in the sorites paradox and recovers the valid inferences of the continuum-valued framework that Smith uses. Therefore, they conclude, three values are enough, at least for the kind of theory of vagueness that Smith had in mind.

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a set of introduction and/or elimination rules, and that, probably, structural rules do also play a part in this story. Moreover, modal metainferential logics or non-mixed metainferential logics (for example, of an intuitionistic kind) are still unexplored possibilities.11 Another sort of immediate application of metainferential logics is to the vagueness phenomena. It might be the case that a more adequate explanation of the sorites paradoxes does not lie in the failure of Cut, as defended in Ripley [29], but, maybe, in the failure of Meta-Cut.12 These are just a few possibilities worth mentioning, but most probably there are many more that we are unaware of.

References 1. Williamson, T. (1987). Equivocation and existence. Proceedings of the Aristotelian Society, 88, 109–127. 2. Priest, G. (2006). Logic: One or many? In: J. Woods & B. Brown (Eds.) Logical consequence: Rival approaches. Proceedings of the 1999 Conference of the Society of Exact Philosophy. Stanmore: Hermes. 3. Read, S. (2006). Monism: The one true logic. In D. de Vidi & T. Kenyon (Eds.), A logical approach to philosophy: Essays in memory of graham solomon. Dordrecht: Springer. 4. Keefe, R. (2014). What logical pluralism cannot be. Synthese, 191(7), 1375–1390. 5. Caret, C. R. (2017). The collapse of logical pluralism has been greatly exaggerated. Erkenntnis, 82(4), 739–760. 6. Barrio, E., Pailos, F., & Szmuc, D. (2018). Substructural logics, pluralism and collapse. Synthese, 1–17. 7. Stei, E. (2020). Rivalry, normativity, and the collapse of logical pluralism. Inquiry, 63(3–4), 411–432. https://doi.org/10.1080/0020174X.2017.1327370 8. Beall, J., & Restall, G. (2006). Logical pluralism. Oxford: Oxford University Press. 9. Barrio, E., Pailos, F., & Szmuc, D. (2020). A hierarchy of classical and paraconsistent logics. Journal of Philosophical Logic, 49(1), 93–120. https://doi.org/10.1007/s10992-019-09513-z 10. Pailos, F. (2020). A fully classical truth theory characterized by substructural means. The Review of Symbolic Logic, 13(2), 249–268. 11. Ripley, D. (2012). Conservatively extending classical logic with transparent truth. Review of Symbolic Logic, 5(2), 354–378. 12. Scambler, C. (2020). Classical logic and the strict tolerant hierarchy. Journal of Philosophical Logic, 49(2), 351–370. 13. Barrio, E., & Pailos, F. (2021). Validities, antivalidities and contingencies: A multi-standard approach. Journal of Philosophical Logic. https://doi.org/10.1007/s10992-021-09610-y 14. Cobreros, P., Tranchini, L., & La Rosa, E. (2020) (I Can’t Get No) Antisatisfaction. Synthese, 1–15. 15. Da Ré, B., et al. (2020). Metainferential duality. Journal of Applied Non-classical Logics, 30(4), 312–334. 16. Cobreros, P., et al. (2012). Tolerant, classical, strict. Journal of Philosophical Logic, 41(2), 347–385. 17. Da Ré, B., Rubin, M., & Teijeiro, P. (2022). Metainferential paraconsistency. Logic and Logical Philosophy, 31(2), 235–260. 18. Barrio, E., Pailos, F., & Szmuc, D. (2018). What is a paraconsistent logic? Contradictions, from consistency to inconsistency (pp. 89–108). Springer. 11

However, we are aware that Isabella McAllister is working on both modal and intuitionistic metainferential logics. 12 This path has been partially explored by Porter [31].

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19. Ripley, D. (2013). Paradoxes and failures of cut. Australasian Journal of Philosophy, 91(1), 139–164. 20. Da Ré, B. (2019). Paraconsistencia total. Revista de Humanidades de Valparaíso, 13. 21. Fiore, C., Pailos, F., & Rubin, M. (2023). Inferential constants. Journal of Philosophical Logic, 52(3), 767–796. 22. Ripley, D. (2022). One step is enough. Journal of Philosophical Logic, 51(6). 23. Ripley, D. A toolkit for metainferential logics. Manuscript. 24. Hjortland, O. T. (2021). Theories of truth and the maxim of minimal mutilation. Synthese, 199(Suppl 3), 787–818. 25. Cobreros, P., et al. (2014). Reaching transparent truth. Mind, 122(488), 841–866. 26. French, R., & Ripley, D. (2019). Valuations: Bi, tri, and tetra. Studia Logica, 107(6), 1313– 1346. 27. Chemla, E., Égré, P., & Spector, B. (2017). Characterizing logical consequence in many-valued logic. Journal of Logic and Computation, 27(7), 2193–2226. 28. Cobreros, P., et al. (2022). Tolerance and degrees of truth. arXiv: 2207.12786 [math.LO]. 29. Cobreros, P., et al. (2015). Vagueness, truth and permissive consequence. In Unifying the philosophy of truth (pp. 409–430). Springer. 30. Smith, N. J. J. (2008). Vagueness and degrees of truth. Oxford University Press. 31. Porter, B. (2022). Supervaluations and the strict-tolerant hierarchy. Journal of Philosophical Logic, 51(6), 1367–1386.

Chapter 11

Concluding Remarks

The validity of an inference has traditionally been associated with the preservation of truth from premises to conclusions. Mixed logics call against this way of understanding this relation: impure mixed logics cannot be understood as preserving some set of designated values from premises to conclusions. Also, some of them have proved to be extremely interesting, and have multiple applications, despite lacking this feature of designated-values preservation. ST, in particular, shares all the inferential validities of classical logic, but, unlike it, can be expanded in a non-trivial way with a transparent truth predicate. Thus, if a logic is just a set of inferential validities, then it is no longer true that classical logic cannot be non-trivially extended with a transparent truth predicate. The debate about what a logic is, and, in particular, whether the inferential level is enough for defining a logic, or whether we should also take into account at least the (first) metainferential level, had, with the hierarchy of metainferential logics based on TS/ST—a metainferential mixed and impure logic—, an unexpected turn. This hierarchy seems to show that it is not enough to take into consideration the first metainferential level, but that all of them, infinite in number, must be taken into account. The focus, then, shifts to metainferential logics, the main subject of this book. Just as impure mixed logics do not cease to be logics because their consequence relation cannot be understood as preserving one and the same set of designated values—i.e., because the standard for conclusions does not determine the standard for premises—, also impure metainferential logics do not abandon their status as logics just because the validity of metainferences is not determined by the standard of validity for inferences—i.e., because the standard for conclusions of a metainference does not determine the standard for premises of a metainference.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. Pailos and B. Da Ré, Metainferential Logics, Trends in Logic 61, https://doi.org/10.1007/978-3-031-44381-7_11

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The preceding paragraph shed light on a most overlooked aspect of the metainferential logics discussed in this book: that they can all be understood as mixed logics, only not as mixed logics of the inferential type. Many of the most interesting of them are, moreover, impure. Metainferential logics–at least the ones discussed in this book–represent, thus, an extension of the set of mixed logics. In fact, they can be seen as one more link in a chain of ways of liberalizing how consequence relation might be understood, which has the multi-standard logics introduced by Barrio and Pailos in [1] and by Pailos in [2]–many of which are composed of metainferential logics–as its most recent form. Multi-standard logics have been applied to the problem of defining logics that recover not only all classical metainferential validities but also their antivalidities and contingencies. Barrio and Pailos [1] developed a family of logics that can accomplish this goal. All of them employ, as standards, transfinite metainferential logics. And Pailos [2] presents another family of multi-standard logics. Its members lack not only metainferential validities but also antivalidities and contingencies, proving a deeper sense in which a logic can be empty. Again, all these logics use transfinite metainferential logics as standards of validity, antivalidity, and contingency (and more). Multi-standard logics, therefore, represent another way of applying metainferential logics. These are not the only applications for these logics. As we have already mentioned, metainferential logics have been developed as a new way of characterizing a logic (e.g., Barrio, Pailos, and Szmuc in [3]), as a way of analyzing the debate between global and local validity Barrio et al. [4], as a key for a new version of the Collapse Argument against logical pluralism (as shown by Barrio, Pailos, and Szmuc in [5]), and as a useful way of discriminating between various substructural solutions to semantic paradoxes (as shown in Pailos [6]). Moreover, it is not implausible to think of metainferential logics as a central feature of new solutions to vagueness-related phenomena, validity-related paradoxes and even to apply them to an intuitionistic and modal framework. We would like to mention some new applications of metainferential logics that we are aware people are working on. The first one concerns the validity paradox, and how these hierarchies of metainferential logics work when a suitable validity predicate is added to the language. Can they avoid the validity paradox? The problem, as is framed in Beall and Murzi [7], is that no operational—i.e., non-substructural— solution to semantic paradoxes can avoid it, because it does not rely on properties of the logical constants but on structural features of the Tarskian consequence relation. Possible solutions to this paradox imply giving up one of those features, which means going substructural. In this way, we also get a uniform solution to every semantic paradox, in the sense that each one of them is treated the same way: by giving up one (and the same) structural feature: Identity, Weakening, Contraction, Exchange, or Cut.1 Nevertheless, Porter, in [8], argues against these solutions by developing new higher-level validity paradoxes that cannot be answered just by giving up lower-level versions of structural rules—and, in particular, they cannot be solved by first-level 1

For more about this diagnosis, see Cobreros et al. in [9] or in [10].

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substructural logics, like ST or TS. This means that, in order to answer Porter’s challenge, a metainferential validity-theory that is substructural at every level needs to be developed. We think that both a non-transitive (all the way up) and a nonreflexive (all the way up) validity theory might be built based on the metainferential logics presented in this book. Nevertheless, this solution is not fully developed yet. The second new type of metainferential logics that might be worth exploring are modal metainferential logics. A major problem with a possible metainferential modal logic, regarding the logics mostly explored in the literature, is that these use a local way of understanding metainferential validity, i.e., as preservation of satisfaction over a valuation, in each valuation. But if we use a local definition of metainferential validity in a modal framework, then metainferences traditionally regarded as valid, such as Necessitation, turn out to be invalid. Two possible ways to go for metainferential modal logics are either explaining why things like Necessitation should fail or changing a local setting for either a global or an absolutely global one. We think both options are worth exploring. As we have already mentioned, we are aware that Isabella McAllister is exploring metainferential modal logics (in a spirit closely related to the multi-standard logics mentioned above). But there is more. In Roffé and Pailos [11], the authors present a way of translating the metainferences of a mixed metainferential system into formulas of an extendedlanguage system that contains new operators (one for each satisfaction standard). This allows us to represent metainferential logics in sentences, which is the first step toward building axiomatic systems for these logics. We believe that all of the things we have mentioned here—and all the applications of metainferential logics displayed throughout the book—show the fruitfulness of these new systems of logics. As we have also pointed out, the options open to metainferential logics are wide and varied. We hope to be able to address some of these projects in the near future.

References 1. Barrio, E., & Pailos, F. (2021). Validities, antivalidities and contingencies: a multi-standard approach. Journal of Philosophical Logic. https://doi.org/10.1007/s10992-021-09610-y 2. Pailos, F. (2022). Empty logics. Journal of Philosophical Logic, 51(6). 3. Barrio, E., Pailos, F., Szmuc, D. (2020). A hierarchy of classical and paraconsistent logics. Journal of Philosophical Logic, 49(1). https://doi.org/10.1007/s10992-019-09513-z, pp. 93120. 4. Barrio, E., Pailos, F., & Szmuc, D. (2019). (Meta)inferential levels of entailment beyond the Tarskian paradigm. In Synthese. https://doi.org/10.1007/s11229-019-02411-6 5. Barrio, E., Pailos, F., Szmuc, D. (2018). Substructural logics, pluralism and collapse. In Synthese (pp. 1–17). 6. Pailos, F. (2020). A fully classical truth theory characterized by substructural means. The Review of Symbolic Logic, 13(2), 249–268. 7. Beall, J. C., & Murzi, J. (2013). Two avors of curry paradox. Journal of Philosophy, 110, 143–65. 8. Porter, B. (2020). A metainferential hierarchy of validity curry paradoxes, The logic and metaphysics workshop. CUNY: New York, United States.

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9. Ripley, D. (2012). Conservatively extending classical logic with transparent truth. Review of Symbolic Logic, 5(2), 354–378. 10. Cobreros, P., et al. (2014). Reaching transparent truth. Mind, 122(488), 841–866. 11. Roffé, A. J., & Pailos, F. (2021). Translating metainferences into formulae: satisfaction operators and sequent calculi. Australasian Journal of Logic, 7(3), 18.

Author Index

B Barrio, Eduardo, 5, 6, 10–12, 32, 35, 49, 54, 55, 57, 58, 114, 115, 117, 118, 120, 121, 130 Barwise, Jon, 106 Beall, J. C., 108, 114, 130 Blake-Turner, Christopher, 80 Bochvar, Dimitri, 64 Boghossian, Paul, 10 Bonatti, Piero, 96 Broom, John, 10

Fjellstad, Andreas, 31, 72, 78, 87 Frankowski, Szymon, 14, 31 French, Rohan, 32, 105, 125

C Caret, Colin, 114 Carnielli, Walter, 96 Chemlá, Emmanuel, 2, 14, 32, 125 Cobreros, Pablo, 4–6, 9, 31, 32, 34, 38, 43, 45, 87, 88, 90, 93, 110, 118, 125, 130

H Hallden, Sören, 64 Hjortland, Ole, 4, 5, 105, 125 Hlobil, Ulf, 10, 11, 31, 60 Humberstone, Lloyd, 14, 72, 78, 84, 85

D Da Ré, Bruno, 12, 14, 19, 64, 84, 85, 93, 98, 99, 105, 118, 119, 121, 122 Dicher, Bogdan, 1, 2, 10, 11, 13, 14, 32, 54, 111

K Keefe, Rosanna, 22, 114 Kortenbach, Bas, 20, 22 Kremer, Michael, 106 Kripke, Saul, 106, 107

E Egré, Paul, 2, 5, 6, 9, 14, 31, 32, 110, 125, 130 Etchemendy, Jon, 106

F Ferguson, Thomas, 59, 64 Fine, Kit, 22 Fiore, Camillo, 122

G Garson, James, 14, 16 Girard, Jean-Yves, 91 Girard, Patrick, 32 Golan, Rea, 4, 25, 26, 87–89 Goranko, Valentin, 96

L La Rosa, Elio, 4, 6, 34, 38, 43, 45, 87, 88, 90, 93, 118

M Malinowski, Grzegorz, 14, 32 McAllister, Isabella, 58–60, 126 Moore, Eoin, 51 Murzi, Julien, 108, 130

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. Pailos and B. Da Ré, Metainferential Logics, Trends in Logic 61, https://doi.org/10.1007/978-3-031-44381-7

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134 N Neta, Ram, 10 O Olivetti, Nicola, 96 Omori, Hitoshi, 110 P Pailos, Federico, 3, 5, 6, 10–12, 14, 15, 33–35, 48, 49, 55, 57, 58, 63, 64, 93, 98, 99, 107, 109, 114, 115, 117–122, 130, 131 Paoli, Francesco, 1, 2, 10, 11, 13, 14, 32, 54, 64, 111 Porter, Brian, 58, 59, 126, 130 Prawitz, Dag, 58 Priest, Graham, 6, 14, 114 Pulcini, Gabriele, 96 Pynko, Alexej, 32

Author Index Schroeder-Heister, Peter, 14 Shramko, Yaroslav, 32 Smiley, Timothy, 58 Smith, Nicholas, 125 Spector, Benjamin, 2, 14, 32, 125 Stei, Eric, 114 Szmuc, Damián, 5, 6, 10–12, 14, 19, 49, 64, 84, 85, 114, 115, 117–122, 130

T Tajer, Diego, 32, 54 Tarski, Alfred, 105 Teijeiro, Paula, 2, 12–14, 19, 23–25, 84, 85, 118, 119, 121, 122 Tiomkin, Michael, 96 Tranchini, Luca, 4, 6, 34, 38, 43, 45, 87, 88, 90, 93, 118

U Urbas, Igor, 24

R Ramírez-Cámara, Elisángela, 59 Read, Stephen, 114 Restall, Greg, 58, 114 Ripley, Dave, 1, 5, 6, 9, 11, 15, 17, 31–33, 37, 43, 48, 49, 54, 58, 67, 68, 84, 88, 105–107, 109, 110, 117, 123, 125, 126, 130 Roffé, Ariel, 131 Rosenblatt, Lucas, 32, 54, 72, 78, 96, 105 Rubin, Mariela, 118, 121, 122 Rumfitt, Ian, 58

W Wansing, Heinrich, 14, 32 Williamson, Timothy, 16, 114 Wright, Crispin, 10

S Scambler, Chris, 11, 12, 15, 16, 35, 49, 55, 57–59, 117, 120, 123

Z Zardini, Elia, 11, 105

V van Rooij, Robert, 5, 6, 9, 31, 32, 110, 125, 130